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abstract: 'There is increasing evidence for specific cortical and subcortical large-scale human epileptic networks to be involved in the generation, spread, and termination of not only primary generalized but also focal onset seizures. The complex dynamics of such networks has been studied with methods of analysis from graph theory. In addition to investigating network-specific characteristics, recent studies aim to determine the functional role of single nodes—such as the epileptic focus—in epileptic brain networks and their relationship to ictogenesis. Utilizing the concept of betweenness centrality to assess the importance of network nodes, previous studies reported the epileptic focus to be of highest importance prior to seizures, which would support the notion of a network hub that facilitates seizure activity. We performed a time-resolved analysis of various aspects of node importance in epileptic brain networks derived from long-term, multi-channel, intracranial electroencephalographic recordings from an epilepsy patient. Our preliminary findings indicate that the epileptic focus is not consistently the most important network node, but node importance may drastically vary over time.'
address: |
^1^Department of Epileptology, University of Bonn,\
Sigmund-Freud-Stra[ß]{}e 25, 53105 Bonn, Germany\
^2^Helmholtz-Institute for Radiation and Nuclear Physics, University of Bonn,\
Nussallee 14–16, 53115 Bonn, Germany\
^3^Interdisciplinary Center for Complex Systems, University of Bonn,\
Br[ü]{}hler Stra[ß]{}e 7, 53175 Bonn, Germany\
^\*^E-mail: `[email protected]`
author:
- 'G. GEIER^1,2,\*^, M.-T. KUHNERT^1,2,3^, C. E. ELGER^1^, K. LEHNERTZ^1,2,3^'
title: ON THE CENTRALITY OF THE FOCUS IN HUMAN EPILEPTIC BRAIN NETWORKS
---
[14cm]{}(3cm,27cm) R. Tetzlaff and C. E. Elger and K. Lehnertz (2013), *Recent Advances in Predicting and Preventing Epileptic Seizures*, page 175–185, Singapore, World Scientific.\
Copyright 2013 by World Scientific.
Introduction
============
Over the last decade network analysis has proven to be an invaluable tool to advance our understanding of complex dynamical systems in diverse scientific fields[@Strogatz2001; @Albert2002; @Newman2003; @Boccaletti2006a; @Arenas2008; @Fortunato2010; @Newman2012] including the neuroscienes[@Reijneveld2007; @Bullmore2009; @Sporns2011a; @Stam2012]. Specific aspects of functional brain networks—with nodes that are usually associated with sensors capturing the dynamics of different brain regions and with links representing interactions[@Pikovsky2001; @Kantz2003; @Pereda2005; @Hlavackova2007; @Marwan2007; @Lehnertz2009b; @Friedrich2011; @Lehnertz2011b] between pairs of brain regions—were reported to differ between epilepsy patients and healthy controls[@Chavez2010; @Horstmann2010; @Ansmann2012] which supports the concept of an epileptic network[@Bertram1998; @Bragin2000; @Spencer2002; @Lemieux2011; @Berg2011]. Moreover, epileptic networks during generalized and focal seizures (including status epilepticus) were shown to possess topologies that differ from those during the seizure-free interval[@Ponten2007; @Schindler2008a; @Ponten2009; @vanDellen2009; @Kramer2010; @Kuhnert2010; @Bialonski2011b; @Kramer2011; @Gupta2011]. Most of the aforementioned studies investigated network-specific characteristics such as the average shortest path length or the clustering coefficient. Network theory, however, also provides concepts and tools to assess various aspects of importance (e.g. centralities) of a node in a network[@Freeman1979; @Bonacich1987; @Koschutzki2005; @Estrada2010; @Kuhnert2012], but by now, there are only a few studies that investigated node-specific characteristics of epileptic networks[@Kramer2008; @Wilke2011; @Varotto2012], and these studies investigated the dynamics of functional brain networks during seizures only. Refs. and reported on highest centrality values for the (clinically defined) epileptic focus which would support the notion of a crucial network node that facilitates seizure activity.
We here report preliminary findings obtained from a time-resolved analysis of node importance in functional brain networks derived from long-term, multi-channel, intracranial electroencephalographic (iEEG) recordings from an epilepsy patient. Investigating various centrality aspects, we provide first evidence that the epileptic focus is not consistently the most important node (i.e., with highest centrality), but node importance may drastically vary over time.
Methods
=======
Inferring Weighted Functional Networks
--------------------------------------
We analyzed iEEG data from a patient who underwent presurgical evaluation of drug-resistant epilepsy of left mesial-temporal origin and who is completely seizure free after selective amygdalohippocampectomy. The patient had signed informed consent that the clinical data might be used and published for research purposes. The study protocol had previously been approved by the local ethics committee. iEEG was recorded from $N=60$ channels (chronically implanted intrahippocampal depth and subdural grid and strip electrodes) and the total recording time amounted to about 1.7 days, during which three seizures were observed. iEEG data were sampled at using a analog-to-digital converter, filtered within a frequency band of , and referenced against the average of two recording contacts outside the focal region.
Following previous studies[@Horstmann2010; @Kuhnert2010; @Kuhnert2012; @Ansmann2012] we associated each recording site with a network node and defined functional network links between any pair of nodes $j$ and $k$—regardless of their anatomical connectivity—using the mean phase coherence $R_{j,k}$ as a measure for signal interdependencies[@Mormann2000]. We used a sliding window approach with non-overlapping windows of $M=4096$ data points (duration: ) each to estimate $R_{j,k}$ in a time-resolved fashion, employing the Hilbert transform to extract the phases $\Phi$ from the windowed iEEG. The elements of the interdependence matrix [[**I**]{}]{}then read:
$$\label{eq:R}
R_{jk}=
\left|\left(\frac{1}{M}\sum_{m=0}^{M-1}{\exp i\left(\Phi_j(m)-\Phi_k(m)\right)}\right)\right|.$$
In order to derive an adjacency matrix [[**A**]{}]{}from [[**I**]{}]{}(i.e, an undirected, weighted functional network) and to account for the case that the centrality metrics could reflect trivial properties of the weight collection[@Ansmann2011] we sort $\left\{R_{jk} \;\middle|\; j < k \right\}$ in ascending order and denote with $\upsilon_{jk}$ the position of $R_{jk}$ in this order (rank). We then consider $A_{jk}=2\upsilon_{jk}/(N(N-1))$, $j \neq k$, and $A_{jj}=0$. This approach leads to a weight collection with entries being uniformly distributed in the interval $[0,1]$.
Estimating Centrality
---------------------
The importance of a network node may be assessed via centrality metrics[@Freeman1979; @Bonacich1987; @Koschutzki2005; @Estrada2010]. Degree, closeness, and betweenness centrality are frequently used for network analyses, and for these metrics generalizations to weighted networks have been proposed (see Ref. for an overview).
If a node is adjacent to many other nodes, it possesses a high degree centrality. When investigating weighted networks, however, the number of neighboring nodes is not a sensible measure and one may consider [*strength centrality*]{} of node $j$ instead[@Barrat2004b]
$$\mathcal{C}^S(j) = \frac{\sum_{k}{a_{jk}}}{N-1}.
\label{eq:cs}$$
Assessing node importance in weighted networks via closeness and betweenness centrality requires the definition of shortest paths. This can be achieved by assuming the “length” of a link to vary inversely with its weight[@Newman2004]. The [*closeness centrality*]{} of node $j$ is defined as
$$\mathcal{C}^C(j) = \frac{N-1}{\sum_k{d_{jk}}},
\label{eq:cc}$$
where $d_{jk}$ denotes the length of the shortest path from node $j$ to node $k$.
The [*betweenness centrality*]{} of node $j$ is the fraction of shortest paths running through that node. $$\mathcal{C}^B(j) = \frac{2}{(N-1)(N-2)} \sum_{h=0}^{N}\sum_{\substack{
k=0\\ k\neq j}}^{N}\frac{\eta_{hk}(j)}{\eta_{hk}}.
\label{eq:bc}$$
Here, $\eta_{hk}(j)$ denotes the number of shortest paths between nodes $h$ and $k$ running through node $j$, and $\eta_{hk}$ is the total number of shortest paths between nodes $h$ and $k$. We used the algorithm proposed by Brandes[@Brandes2001] to estimate the aforementioned centralities. Fig. \[img:centr\] illustrates the centrality metrics [$\mathcal{C}^S$]{}, [$\mathcal{C}^C$]{}, and [$\mathcal{C}^B$]{}for the nodes of an exemplary network.
![ Values of degree centrality (top), closeness centrality (middle) and betweenness centrality (bottom) for nodes of an exemplary binary network. The most important node (highest centrality) is indicated by an arrow.[]{data-label="img:centr"}](Figure_01_a.eps){width="0.7\columnwidth"}
Results
=======
In Figs. \[img:dc\_example\], \[img:cc\_example\], and \[img:bc\_example\] we show the temporal evolutions of [$\mathcal{C}^S$]{}, [$\mathcal{C}^C$]{}and [$\mathcal{C}^B$]{}over for three selected nodes from the exemplary epileptic brain networks investigated here. We chose one node from within the epileptic focus (upper plots of figures), another node from the immediate surrounding of the epileptic focus (middle plots of figures), and a third one which was associated with a recording site far off the epileptic focus (lower plots in figures). All centrality metrics exhibited large fluctuations over time, both on shorter and longer time scales. The temporal evolutions of [$\mathcal{C}^S$]{}and [$\mathcal{C}^C$]{}were quite similar, while [$\mathcal{C}^B$]{}behaved differently from the two other metrics. The similarity between [$\mathcal{C}^S$]{}and [$\mathcal{C}^C$]{}was to be expected, at least to some degree (see the discussion in Ref. ), since they characterize the role of a node as a starting or end point of a path. On the other hand, [$\mathcal{C}^B$]{}characterizes a node’s share of all paths between pairs of nodes that utilize that node.
For this patient, we could not observe any clear cut changes of the centrality metrics prior to seizures that would indicate a preictal state. Moreover, none of the metrics exhibited features in their temporal evolutions that would constantly indicate the network nodes associated with the epileptic focus (or its immediate neighborhood) as important nodes. Rather, their importance may drastically vary over time.
To demonstrate that our exemplary results hold for all nodes of the epileptic brain networks investigated here, we show, in Fig. \[img:boxplots\], findings obtained from an exploratory data analysis. The main statistical characteristics of centralities of each node (maximum and minimum value, the median, and the quartiles estimated from the respective temporal evolutions) indicated that neither the epileptic focus nor its immediate surrounding can be considered as important, and that the different centrality metrics ranked different nodes as most important.
Conclusion
==========
We have investigated various aspects of centrality of individual nodes in epileptic brain networks derived from long-term, multi-channel iEEG recordings from an epilepsy patient. Utilizing different centrality metrics, we observed nodes far from the clinically defined epileptic focus and its immediate surrounding to be the most important ones. Although our findings must, at present, be regarded as preliminary, they are nevertheless in stark contrast to previous studies[@Wilke2011; @Varotto2012] that reported highest node centralities for the epileptic focus only. It remains to be investigated whether the different findings can be attributed to the dynamics of different epileptic brains or to, e.g., differences in network inference. One also needs to take into account that there are a number of potentially confounding variables whose impact on estimates of different centrality metrics is still poorly understood.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Deutsche Forschungsgemeinschaft (Grant No. LE660/4-2).
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---
abstract: 'Properties of hybrid stars with a mixed phase composed of asymmetric nuclear matter and strange quark matter are studied. The quark phase is investigated by the quark quasiparticle model with a self-consistent thermodynamic and statistical treatment. We present the stability windows of the strange quark matter with respect to the interaction coupling constant versus the bag constant. We find that the appearance of the quark-hadron mixed phases is associated with the meta-stable or unstable regions of the pure quark matter parameters. The mass-radius relation of the hybrid star is dominated by the equation of state of quark matter rather than nuclear matter. Due to the appearance of mixed phase, the mass of hybrid star is reduced to 1.64 M$_{\odot}$ with radius $10.6$ km by comparison with neutron star.'
address: 'Department of Physics and Institute of Theoretical Physics, Shanxi University, Taiyuan 030006, China'
author:
- 'Xin-Jian Wen'
title: Equation of state in hybrid stars and the stability window of quark matter
---
Thermodynamic and statistical physics of quasiparticle model; Equation of state of strange quark matter; Phase diagram of hybrid stars
INTRODUCTION {#sec:intro}
============
The appearance of quark matter or hadron-quark mixed phase in the massive neutron stars is a hot topic in the study of compact objects. The baryon densities of the stars cover a larger range from very low densities in the outer part to the order of about ten times the saturation density in the inner core. To study the structure of compact stars, the key point is to find a reliable form of the equation of state (EOS) that determines the characteristic of the constituent matter [@Read09]. Unfortunately, however, there is no single theory to cover the large density range with respect to quark degrees of freedom. In literature, the low-density phase can be described by quantum hadrodynamics (QHD). At high densities, a new form of matter, called strange quark matter (SQM), might exist and be more stable than nuclear matter ($^{56}$Fe) [@bodmer; @witten1984; @Liu1984; @Crawford1992; @Lugones02]. For more theoretical and experimental results, see [@Berger1987; @Madsen1993; @Schaffner-Bielich1997; @Madsen2001]. A family of compact stars consisting completely of the deconfined mixture of $u$-, $d$-, $s$- quarks has been called “strange stars" [@Alcock1986; @Haensel1986; @RxXu09]. If the hypothesis of stable strange quark matter is correct [@witten1984], the possibility of phase transition exists in principle [@Bombaci2008]. The compact star with a “quark matter core", either as a hadron-quark mixed phase or as a pure quark phase, are called “hybrid stars" (HyS) [@Glendenning1996; @Goyal2004]. Recently, it was argued that the interior core of a low-mass compact star could be dominated by the Color-Flavor-Locked (CFL) quark matter [@Stejner2005; @Alford2003]. In contrast, it was remarked that the CFL phase would make the HyS unstable [@klahn2007] and the possible configuration of compact stars, such as the strange hadrons, hyperonic matter [@Sahu2001] and quark matter core, can soften the equation of states of neutron stars [@Sahu2001; @Shen2002; @Burgio2002]. However, Alford et al pointed out the “masquerade effect" [@alford2005] that the hybrid star has a mass-radius relation similar to that of the pure neutron star. Also, the isolated neutron star RX J1856-375 with a large radius and/or mass [@trumper2004] is a possible candidate of HyS, which implies a constraint on testing a rather stiff equation of state at high density [@klahn2006]. Regardless of whether the quark matter is ruled out, it seemed that the soft equation of state were ruled out in the center of compact stars [@ozel2006].
To get a reliable equation of state in the microscopic calculation of interacting dense hadronic matter, Wiringa et.al added the three-body potential to the nucleon Hamiltonian and gave the light nuclei binding energy and nuclear matter saturation properties [@Wiringa1988]. The Brueckner theory with three-body forces has been used recently to study the mixed phase of hadrons and quarks in compact stars [@peng2008]. In literature, there is another successful method called relativistic mean-field (RMF) theory. It is a powerful tool in describing various aspects of nuclear physics, such as the properties of nuclear matter, finite nuclei, and neutron stars, as well as the dynamics of heavy-ion collisions [@Sahu2000; @glendenning1987]. Recently, the model has been extended to include the density-dependent meson-nucleon coupling constant in finite nuclei [@Ring2002] and nuclear matter [@klahn2006; @Meng2004].
In the theoretical description of the deconfined quark matter, we can resort to the phenomenological models constrained by experimental information. There are many successful works considering the medium-effect of quark masses. One of them obtains the confinement by the density dependence of quark masses [@QMDD]. Another one is the quark quasiparticle model, where the vacuum energy density is not constant but density-dependent. For the medium dependence of the quasiparticle masses, it was derived at the zero-momentum limit of the dispersion relations from an effective quark propagator by resuming one-loop self-energy diagrams in the hard-dense-loop approximation (HDL) [@Schertler1997]. We have recently extended the model to include the important finite-size effect [@wen2009] and magnetized strange quark matter (MSQM) [@wen2012].
In this paper, we study the effect of the interaction coupling constants on the equation of mixed phase of nuclear and quark matter and the properties of hybrid stars. For the nuclear EOS, we adopt the relativistic nuclear field theory solved at the mean-field level, and especially the Baym-Pethick-Sutherland (BPS) model for densities below $5\times 10^{14}$g cm$^{-3}$ [@Baym1971; @ruster2006]. In describing quark matter, we employ the quark quasiparticle model instead of the conventional bag model [@Burgio2002]. The density-dependent bag function is obtained self-consistently rather than artificially.
This paper is organized as follow. In Section \[Sec:thermo\], we briefly give a short introduction to the relativistic nonlinear mean-field model describing the nuclear matter. In Section \[QM\], we introduce the treatment of SQM in the framework of the quark quasiparticle model and present the stability window of quark matter. In Section \[comstar\], we display the phase diagram of the mixed phase and discuss the chemical potential behavior. With the equations of state, we investigate the influence of coupling constant and bag constant on the mass-radius relation. The last section is a short summary.
The nuclear EOS in the relativistic mean-field model {#Sec:thermo}
====================================================
The relativistic nuclear field theory is solved at the mean-field level. The in-medium interaction of nucleons can be realized through the exchanges of the $\sigma$, $\omega$, and $\rho$ mesons. The bulk matter is assumed to be electrically neutralized and in the lowest energy state, i.e. in general $\beta$-equilibrium. The influence of the temperature can be neglected [@Manka2000]. The Lagrange density for this model is written as [@Manka2001; @Bednarek2001] $$\begin{aligned}
{\cal{L}}_{RMF} &=&\frac{1}{2} \left(
\partial_{\mu}\sigma\partial^{\mu}\sigma +
m_\rho^2\rho_\mu\rho^\mu\right) -U(\sigma)+V(\omega)
-\frac{1}{4} \left[\Omega_{\mu\nu}\Omega^{\mu\nu}+ R^a_{\mu\nu}R^{a\mu\nu}+F_{\mu\nu}F^{\mu\nu} \right] \nonumber\\
& &+\bar{\psi} \left(i\gamma^{\mu}\partial_{\mu}-m_{N}+g_{\sigma
N}\sigma -g_{\omega N}\gamma^\mu\omega_\mu -\frac{1}{2} g_{\rho
N}\gamma^\mu \rho_\mu^a\cdot \tau^a -e\gamma^\mu Q_e
A_\mu\right)\psi \nonumber\\
&& +\bar{\psi}_e(i\gamma^\mu \partial_\mu-m_e)\psi_e.\end{aligned}$$ where the nucleon field $\psi$ has a form of column vector for protons and neutrons, and the field tensors are given by $$\begin{aligned}
\Omega_{\mu\nu} &=&
\partial_{\nu}\omega_{\mu}-\partial_{\mu}\omega_{\nu},\\
R^a_{\mu\nu} &=&
\partial_{\mu}\rho^a_{\nu}-\partial_{\nu}\rho^a_{\mu}, \\
F_{\mu\nu} &=& \partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu}.\end{aligned}$$ The non-linear potential functions are contained for the meson $\sigma$ [@Bodmer1991] and $\omega$ as $$\begin{aligned}
U(\sigma)&=&\frac{1}{2} m_\sigma^2\sigma^2
+\frac{1}{3} g_3\sigma^3 +\frac{1}{4} g_4\sigma^4,\\
V(\omega)&=&\frac{1}{2} m^2_\omega \omega_\mu\omega^\mu+\frac{1}{4}
c_4 (\omega_\mu\omega^\mu)^2.\end{aligned}$$ The parameters $m_N$, $m_\sigma$, $m_\rho$, $m_\omega$ are the masses of nucleon, scalar meson $\sigma$, isovector-vector meson $\rho$, and isoscalar-vector meson $\omega$ respectively. In principle, they can be fixed algebraically by the properties of bulk nuclear matter, such as, the binding per nucleon, the saturation baryon density, the effective mass of the nucleon at saturation, and the compression modulus. The isovector $\rho$ field vanishes for symmetry nuclear matter.
The scalar density and conserved baryon number density are expressed with the use of Fermi integrals $$\begin{aligned}
\rho_s&=&\frac{4}{(2\pi)^3}\int_0^{p_f}
\frac{M^*}{\sqrt{p^2+M^{*2}}}dp^3,\\
\rho_N&=&\rho_p+\rho_n=\frac{\nu^3_p}{3\pi^2}+\frac{\nu^3_n}{3\pi^2},\end{aligned}$$ where the effective nucleon mass is defined by $M^*\equiv
M-g_{\sigma N}\sigma_0$, with $\nu_p$ and $\nu_n$ being the Fermi momenta of protons and neutrons respectively.
The energy density and pressure of nuclear matter can be obtained from the energy-momentum tensor. Including the contribution of nucleons and mesons, the total energy density $\epsilon^\mathrm{HP}$ and the pressure $P^\mathrm{HP}$ are [@Glendenning1996; @alford2001] $$\begin{aligned}
\epsilon^\mathrm{HP}&=&\frac{2}{(2\pi)^3}\left[\int_0^{\nu_p}\sqrt{p^2+{M^*}^2}dp^3
+ \int_0^{\nu_n}\sqrt{p^2+{M^*}^2}dp^3 \right]
\nonumber \\
&& +\frac{1}{2} m_\sigma^2\sigma^2 +\frac{1}{3} g_3\sigma^3
+\frac{1}{4} g_4\sigma^4 +\frac{1}{2} m^2_\omega
\omega^2_0+\frac{3}{4} c_4 \omega_0^4 +\frac{1}{2}m^2_\rho
\rho_0^2,\\
P^\mathrm{HP}&=&\frac{1}{3}\frac{2}{(2\pi)^3}\left[\int_0^{\nu_p}\frac{p^2}{\sqrt{p^2+{M^*}^2}}dp^3
+ \int_0^{\nu_n}\frac{p^2}{\sqrt{p^2+{M^*}^2}}dp^3 \right]
\nonumber \\
&& -\frac{1}{2} m_\sigma^2\sigma^2 -\frac{1}{3} g_3\sigma^3
-\frac{1}{4} g_4\sigma^4 +\frac{1}{2} m^2_\omega
\omega^2_0+\frac{1}{4} c_4 \omega_0^4+\frac{1}{2}m^2_\rho
\rho_0^2.\label{pressN}\end{aligned}$$ If the electron contribution is included, $P_\mathrm{e}=\mu_e^4/(12\pi^2)$. For symmetric nuclear matter, the number density of protons and neutrons are equal, and correspondingly the Fermi momenta are also equal, i.e., $\nu_p=\nu_n$. In this case, the isovector meson has no contribution.
From the equation of motion for nucleons, the Fermi energy of nucleons, or equivalently, the chemical potential $\mu_N$ can be expressed as $$\begin{aligned}
\mu_N=g_{\omega N}\omega_0+\frac{1}{2}g_{\rho
N}\rho_{03}\tau_{3N}+\sqrt{\nu^2+{M^*}^2}.\end{aligned}$$
In this paper, we choose four typical sets of parameters (TM1, NL3, BKA20, and TW-99) [@Meng2004; @BKA]. They stand for different stiffness equations of state. The messes for $\sigma$, $\omega$, and $\rho$ mesons are $m_\sigma=509$MeV, $m_\omega=782$MeV, and $m_\rho=770$MeV with nucleon mass $M=939$MeV. For getting the EOS in the low-density, i.e., the crust equation of state of compact stars, we can resort to the BPS model [@Baym1971] in calculations.
Quark quasiparticle model for quark matter EOS {#QM}
==============================================
At high density, nuclear matter is expected to undergo a phase transition to a deconfined phase. If the quark chemical potential exceeds the strange quark mass, the system can lower its Fermi energy by converting the down quark into strange quarks. Recently, we have developed the quark quasiparticle model in studying strangelets [@wen2009] and MSQM [@wen2012]. In the quasiparticle model, the effective quark mass following from the hard-dense-loop (HDL) approximation of quark self-energy at zero temperature is expressed as [@Schertler1997; @Pisarski1989], $$\begin{aligned}
m_i^*=\frac{m_i}{2}+ \sqrt{\frac{m^2_i}{4}+
\frac{g^2\mu_i^2}{6\pi^2}}\, ,\label{mass1}\end{aligned}$$ where $m_i$ is the corresponding current mass of quarks, $g=\sqrt{4\pi\alpha_s}$ denotes the strong interaction coupling constant. The effective quark mass $m_i^*$ increases with $g$, $m_i$ and the quark chemical potential $\mu_i$. In this paper, we treat $g$ as a free parameter in the range of $(0,5)$. For light quarks ($u$ and $d$ quarks), we take the current mass to be zero, and Eq. (\[mass1\]) is reduced to the simple form $$\label{miud}m_i^*=\frac{g\mu_i}{\sqrt{6}\pi}\equiv\alpha\mu_i, \ \
(i=u, d)$$
The quasiparticle contribution to the thermodynamic potential density is given as [@wen2009]. $$\begin{aligned}
\Omega_i &=&
-\frac{d_i T}{2\pi^2}
\int_0^{\infty}
\left\{
\ln\left[1+e^{-(\epsilon_{i,p}-\mu_i)/T}\right]
+\ln\left[1+e^{-(\epsilon_{i,p}+\mu_i)/T}\right]
\right\}
p^2 \mbox{d}p,\end{aligned}$$where $\epsilon_{i,p}=\sqrt{p^2+{m_i^*}^2}$ and $T$ is the temperature. At zero temperature, the integration can be calculated out to give $$\begin{aligned}
\Omega_i &=&-\frac{d_i}{48\pi^2}
\Bigg[
|\mu_i|\sqrt{\mu_i^2-{m_i^*}^2}\left(2\mu_i^2-5{m_i^*}^2\right)
+3{m_i^*}^4\ln\frac{|\mu_i|+\sqrt{\mu_i^2-{m_i^*}^2}}{m_i^*}
\Bigg],\end{aligned}$$ where $m_i$ and $\mu_i$ are, respectively, the particle masses and chemical potentials. $d_i$ is the degeneracy factor with $d_i=2(\rm{spin})\times 3(\rm{color})=6$ for quarks and $d_i=2$ for electrons. When the variable set (T, V, {$\mu_i$}) is chosen as the independent state variables, the thermodynamic potential is the characteristic function. In the quasiparticle model, the total thermodynamic potential density can be written as $$\label{Omegtot}
\Omega=\sum_i \left [\Omega_i(\mu_i,m_i^*)+B_i(\mu_i)\right]+B_0,$$ Where $B_i(\mu_i)$ is the medium-dependent quantity determined by thermodynamic consistency. A little later, we will see that $B_i(\mu_i)$ is given by an indefinite integration. When it is expressed by a definite integration, an integration constant is needed. Therefore, $B_0$ is from the sum of relevant integration constants, and we treat it as a free input parameter. In the quasiparticle model, the particle number density should be of the same form as that of a Fermi gas with the normal particle mass replaced by the effective quasiparticle mass,i.e., $$\begin{aligned}
\label{nderi}
n_i=-\frac{\partial \Omega_i}{\partial \mu_i}
=\frac{d_i}{6\pi^2}\left(\mu_i^2-{m_i^*}^2\right)^{3/2}.\end{aligned}$$ On the other hand, we know from the fundamental thermodynamics that $$\label{ni2}
n_i
=-\left.\frac{\mathrm{d}\Omega}{\mathrm{d}\mu_i}\right|_{\mu_{k\neq
i}} =-\frac{\partial\Omega_i}{\partial \mu_i}
-\frac{\partial \Omega_i}{\partial m_i^*}\frac{\partial m_i^*}{\partial \mu_i}
-\frac{\mathrm{d} B_i}{\mathrm{d} \mu_i}.$$
Equating the last equality in Eq. (\[ni2\]) with the first equality in Eq. (\[nderi\]), we immediately have $$\label{Biexp}
\frac{\mathrm{d} B_i}{\mathrm{d} \mu_i} =-\frac{\partial
\Omega_i}{\partial m_i^*}\frac{\partial m_i^*}{\partial \mu_i}
\ \ \mbox{i.e.} \ \
B_i=-\int_{m_i^*}^{\mu_i}
\frac{\partial \Omega_i}{\partial m_i^*}
\frac{\partial m_i^*}{\partial \mu_i}
\mbox{d}\mu_i,$$ where the derivative of the thermodynamic potential density with respect to the quark effective mass can be analytically expressed by $$\begin{aligned}
\frac{\partial \Omega_i}{\partial m_i^*}&=&\frac{d_i
m_i^*}{4\pi^2} \left[\mu_i \sqrt{\mu_i^2-{m_i^*}^2} -{m_i^*}^2 \ln
\frac{\mu_i+\sqrt{\mu_i^2-{m_i^*}^2}}{m_i^*}
\right]. \label{dodm}\end{aligned}$$
Accordingly, the pressure $P^\mathrm{QP}$, energy density $\epsilon^\mathrm{QP}$, and baryon density $\rho^{QP}$ for SQM at zero temperature are written as, $$\begin{aligned}
P^\mathrm{QP}&=&-\sum_i \left [\Omega_i(\mu_i,m_i^*)+B_i(\mu_i)\right]-B_0,\label{pressQ}\\
\epsilon^\mathrm{QP} &=& \sum_i \left [\Omega_i(\mu_i,m_i^*)+B_i(\mu_i)\right]+\sum_i \mu_i n_i+B_0, \label{energy}\\
\rho^\mathrm{QP}&=&\frac{1}{3}\sum_i n_i.\end{aligned}$$
Because the current mass of light quarks is nearly zero, one can have an analytical expression for light quarks by combining Eqs.(\[miud\]) and (\[Biexp\]) [@wen2009; @Schertler1997], giving $$\begin{aligned}
\label{Bexp}
B_i^*(\mu_i)&=&-\int^{\mu_i}_0 \left.\frac{\partial \Omega}{\partial
m^*_i}\right|_{T=0,\mu_i}\frac{\mathrm{d} m^*_i}{\mathrm{d}\mu_i} \mathrm{d}\mu_i \nonumber \\
&=&-\frac{d_i}{16\pi^2} \left[\alpha^2\beta-\alpha^4
\ln(\frac{\beta+1}{\alpha}) \right]\mu_i^4, \ \ \ (i=u,d),\end{aligned}$$ where $\alpha=g/(\sqrt{6}\pi)$ and $\beta=\sqrt{1-\alpha(g)^2}$ are $g$-dependent functions.
For massive quarks, $B^*(\mu)$ has been observed as the result [@Schertler1997], $$\begin{aligned}
B_s^*(\mu_s)&=&-\frac{d_s}{16\pi^2}\Big[
\frac{\sqrt{(m_{s}^*-m_s)(\beta^2 m_s^*-m_s)}}{24 \alpha^2\beta^4}
\Big] \times \sum_{n=0}^3 a_n m_s^{3-n}{m_s^*}^n -{m_s^*}^4\ln\big( \frac{k_F+\mu}{m_s^*} \big) \nonumber\\
&& +\frac{5 \alpha^4-12 \alpha^2+8}{16 \beta^5}m_s^4 \times\ln\Big(
\frac{\beta\sqrt{m_s^*-m_s}+\sqrt{\beta^2m_s^*-m_s} }{\alpha^2m_s}
\Big),\end{aligned}$$ where the coefficient $a_n$ can be related to the coupling constant through four polynomial function as in [@Schertler1997], which is very different from the gaussian parametrization of the density-dependent bag constant [@maieron2004].
With the above quark mass formulas and thermodynamic treatment, one can get the properties of bulk quark matter. According to the Witten-Bodmer hypothesis, it is required that the energy per baryon of two-flavor quark matter is bigger than 930 MeV in order not to contradict with standard nuclear physics, but that of symmetric three-flavor quark matter is less than 930 MeV. For the co-existence of nuclear matter and quark matter in the process of phase transition, the allowed values of model parameters should be chosen in the metastable or unstable regions. In Fig. \[fig B0window\], we present the stability windows of SQM in the $B_0^{1/4}$-$g$ plane. The area below the dotted line is forbidden because the energy per baryon $\epsilon^{QP} /\rho^{QP}$ of 2-flavor quark matter is less than $930$ MeV. The $\epsilon^{QP} /\rho^{QP}$ of 3-flavor quark matter is smaller than $930$ MeV below the solid line and is smaller than 939 MeV in the narrow area below the dashed line. In the top area the strange quark matter is unstable.
![Stability windows in the parameter space (B$_0^{1/4}$, $g$). Below the dotted line is the forbidden area where two-flavor quark matter is absolutely stable. Above the solid or dashed line, SQM is meta-stable or unstable, and SQM is absolutely stable between the the dotted and solid curves. []{data-label="fig B0window"}](B0window.eps){width="7.cm" height="7cm"}
Hadron-quark mixed phase and hybrid stars {#comstar}
=========================================
Transition from nuclear matter to quark matter
----------------------------------------------
We suppose that the compact star composed of quark matter core and nuclear matter surface. The nuclear matter includes protons, neutrons and electrons. The quark matter consists of a mixture of quarks ($u$, $d$, and $s$) and electrons. In the interface of the two phases, there may be a mixed phase of nuclear and quark matter, which is important in understanding the properties of hybrid stars. To describe the structure of the mixed phase, we define the fraction $\chi$ occupied by quark matter by $\chi\equiv
V^\mathrm{QP}/(V^\mathrm{QP}+V^\mathrm{HP}).$ Then we have the total baryon number density $\rho$, energy density $\epsilon$, and electrical charge $Q$ as, $$\begin{aligned}
\rho_B&=&(1-\chi)\rho^\mathrm{HP}+\chi \rho^\mathrm{QP},\label{baryoneq}\\
\epsilon&=&(1-\chi)\epsilon^\mathrm{HP}+\chi\epsilon^\mathrm{QP},\\
Q&=&(1-\chi)Q^\mathrm{HP}+\chi Q^\mathrm{QP},\label{chargeq}\end{aligned}$$where $Q=0$ according to the bulk charge neutrality requirement. The critical density $\rho_c^\mathrm{HP}$ for pure nuclear matter is determined by $\chi=0$. With increasing density, the deconfinement phase transition takes place. Consequently, quark matter appears in the mixed phase. When $\chi$ increases up to $1$, the quark phase dominates the system completely at the critical density $\rho_c^\mathrm{QP}$.
From the quark constituent of nucleons, we have the chemical potential relations through the linear combinations, $$\label{mupnexp}
\mu_p= 2\mu_u +\mu_d, \ \
\mu_n= \mu_u+2\mu_d.$$ Incorporated with the $\beta$-equilibrium condition $\mu_d=\mu_s=\mu_u+\mu_e$, there are only two independent chemical potentials, e.g. $\mu_u$ and $\mu_e$. The Gibbs condition for the phase equilibrium between nuclear matter and quark matter is that the system should be in thermal, chemical, and mechanical equilibrium. Therefore, in addition to the common zero temperature, we also have the two conditions $$\begin{aligned}
\mu^\mathrm{HP}&=&\mu^\mathrm{QP}, \label{baronpot}
\\
P^\mathrm{HP}(\mu_p,\mu_n,\mu_e)&=&P^\mathrm{QP}(\mu_u,\mu_d,\mu_s,\mu_e),
\label{presseq}\end{aligned}$$ where $P^\mathrm{HP}$ and $P^\mathrm{QP}$ denotes the pressure in the nuclear phase and in the quark phase, respectively. They are given by Eqs. (\[pressN\]) and (\[pressQ\]) separately. The electron is uniformly distributed in the system and the pressure is common for both phases. Eq. (\[baronpot\]) means that the baryon chemical potential in both phases are the same. It can be shown that Eq. (\[baronpot\]) is equivalnet to Eq. (\[mupnexp\]) [@peng2008]. For a given total baryon number density, one can obtain the two independent chemical potentials and $\chi$ by solving the equations (\[baryoneq\]), (\[chargeq\]) and (\[presseq\]).
![ The electron chemical potential $\mu_e$ (left axis) and the fraction of SQM $\chi$ (right axis). Parameters are the same as for Fig. \[fig\_energ\].[]{data-label="fig muechi"}](energy.eps){width="7.cm" height="7cm"}
![ The electron chemical potential $\mu_e$ (left axis) and the fraction of SQM $\chi$ (right axis). Parameters are the same as for Fig. \[fig\_energ\].[]{data-label="fig muechi"}](muechi.eps){width="7.cm" height="7cm"}
In Fig. \[fig\_energ\], the system energy per baryon is displayed as a function of the density for $g=3$, $B_0^{1/4}=150$ MeV, and TM1 parameter set. The mixed phase (shaded area) exists in the range of about $1\sim 3$ times the nuclear saturation density. In this density range, the energy per baryon of mixed phase (solid line) is lower than that of both the pure nuclear (dashed line) and the pure quark phase (dotted line). So the appearance of the mixed phase in neutron star matter is energetically favored for a proper parameter in meta-stable or unstable regions. This observation is consistent with that in Ref. [@peng2008]. If one chooses the absolutely stable parameter-sets ($g$, $B_0^{1/4}$), no mixed phase will exist and the hybrid star is unstable and will collapse to a strange star. In the middle density range, the mixed phase starts at the nuclear critical density $\rho_c^\mathrm{HP}$ where $\chi=0$ and ends at the quark critical density $\rho_c^\mathrm{QP}$ where $\chi=1$. The critical points are marked with full dots in Fig. \[fig\_energ\]. By comparing the different equation of state for nuclear matter, we find that the solid lines between the $\rho_c^\mathrm{HP}$ and $\rho_c^\mathrm{QP}$ is shorter for harder EOS of nuclear matter. It is also observed that the quark critical density $\rho_c^\mathrm{QP}$ varies only slightly with the RMF parameters.
To provide a better understanding of the Glendenning hypothesis of global charge neutrality, we give the electron chemical potential and the quark fraction versus the total density in Fig. \[fig muechi\]. In the shaded area of mixed phase, the increasing contribution of quark phase is a complementarity to the decrease of electrons. The electron chemical potential $\mu_e$ (dashed line) decreases rapidly from the maximum value $85$ MeV to $20$ MeV. And finally it becomes very small in the pure quark phase where the quark fraction $\chi$ is unity, where the three-flavor quark matter occupies the whole space.
The structure of hybrid stars {#Sec:numerical}
-----------------------------
With the above equation of state, we now study the structure of compact stars. As usually done, we assume that the hybrid star is a spherically symmetric distribution of mass in hydrostatic equilibrium. The equilibrium configurations are obtained by solving the Tolman-Oppenheimer-Volkoff (TOV) equation for the pressure $P(r)$, the energy density $\epsilon(r)$ and the enclosed mass $m(r)$: $$\frac{dP(r)}{dr} =-\frac{Gm(r)\epsilon(r)}{r^2}
\frac{[1+P(r)/\epsilon(r)][1+4\pi r^3 P(r)/m(r)]}{1-2Gm(r)/r}$$ where $G=6.707\times 10^{-45}$MeV$^{-2}$ is the gravitational constant, $r$ is the distance from the center of the star. The subsidiary condition is $$\frac{dm(r)}{dr}=4\pi r^2 \epsilon(r).$$ Giving the stellar radius $R$, which is defined by zero pressure at the stellar surface, the gravitational mass is given by $$M(R)\equiv 4\pi\int^R _0r^2 \epsilon(r)\mbox{d}r.$$
![ The mass-radius relation of hybrid stars with different parameter sets of the nuclear matter. The maximum mass, marked with a full dot on each cure, do not change significantly with the nuclear parameters.[]{data-label="fig HyS2"}](NSmassR.eps){width="7.cm" height="7cm"}
![ The mass-radius relation of hybrid stars with different parameter sets of the nuclear matter. The maximum mass, marked with a full dot on each cure, do not change significantly with the nuclear parameters.[]{data-label="fig HyS2"}](mass1p64.eps){width="7.cm" height="7cm"}
Before investigating the properties of the hybrid star, let’s give the mass-radius relations for compact stars composed of pure nuclear matter. The TOV equation is solved with the nuclear matter EOS introduced in Sec. \[Sec:thermo\]. We take the four parameter sets in the calculation: TM1, NL3, BKA20, TW-99. The maximum masses of neutron stars are located in $2\sim 3$ times the solar mass M$_\odot$ with the corresponding radii in the range (12.3 $\sim$ 14.1 km) in Fig. \[fig NSmass\]. For harder EOS of the parameter set TW-99, the maximum star mass approaches the value $2.86$ M$_\odot$, which is bigger than the mass $2.06$ M$_\odot$ with the parameter set TM1. Different with the self-bound stars, the equation of state of nuclear matter at lower density is calculated in the BPS model.
To investigate the influence of model parameter on the mass-radius relation of hybrid stars, we should consider different EOS of the nuclear matter and strange quark matter. Firstly, the parameter set ($g=3$, $B_0^{1/4}=150$ MeV) is employed on the quark side, while different parameter sets on the nuclear side are adopted in the calculation. The corresponding mass-radius relations are plotted in Fig. \[fig HyS2\] where the maximum star mass are marked with full dots. Even though the nuclear parameter is adopted for different sets, the maximum mass is very close to the same value $1.64$ M$_\odot$. Doing calculations with other choices of the parameter $B_0$ and $g$, we find the result that the maximum mass does not be significantly influenced by the parameter uncertainty of the nuclear equation of state.
Now we fixed the TM1 parameter set for the nuclear EOS while changing the values of $g$ and $B_0$ on the quark side. In this case the maximum mass of hybrid stars is calculated in Fig. \[fig HyS1\]. The distinction between Fig. \[fig HyS2\] and \[fig HyS1\] can be understood from the comparison of EOS. In the upper panel of Fig. \[fig eos\], we can know that the narrow range on the energy density axis is affected by nuclear parameter sets, while a larger range is greatly dominated by quark matter parameter sets. For a bigger value of the maximum of hybrid stars, the corresponding density range of the mixed phase is wider. In the parameter range, the maximum mass is in the range $(1.5 \sim 2.04)
M_\odot$ and radius in the range ($9.64 \sim 12.4)$ km, which comprises the pulsar PSR J1614-2230 with $1.97\pm 0.04 M_\odot$ [@1614]. For a completely relation between the maximum mass of hybrid stars and the parameter set of SQM, we show the contour plots of the maximum mass in the panel with $B_0^{1/4}$ and $g$ on the horizontal and vertical axis in Fig. \[fig contour\]. According to the stability window shown in Fig .\[fig B0window\], the maximum mass of hybrid stars has been marked on each line in the allowed area.
![The pressure versus the energy density for the equations of state of hybrid stars. Figure (a) and (b) are, respectively, for Fig. \[fig HyS2\] and \[fig HyS1\]. []{data-label="fig eos"}](massTM1.eps){width="7.cm" height="7cm"}
![The pressure versus the energy density for the equations of state of hybrid stars. Figure (a) and (b) are, respectively, for Fig. \[fig HyS2\] and \[fig HyS1\]. []{data-label="fig eos"}](pressenergy.eps){width="7.cm" height="7cm"}
![ The contour plots of the maximum mass of hybrid stars in the panel (B$_0^{1/4}$, $g$) . Based on the stability window in Fig. \[fig B0window\], the maximums of hybrid stars are shown on each line in the allowed area.[]{data-label="fig contour"}](fig8.eps){width="7.cm" height="7cm"}
Summary {#Sec:conls}
=======
We have studied the hybrid stars with mixed phase of nuclear matter and SQM. In the outer layer, the nuclear matter is described by the relativistic mean-field theory and crust EOS by the BPS model. In the inner core, the quark matter is investigated by the quasiparticle model. According to the Witten-Bodmer hypothesis, we present the stability window of quark matter in the coupling constant $g$ versus bag constant $B_0^{1/4}$ panel. The mixed phase exists in the range $1 \sim 3$ times the nuclear saturation density, which is energetically favorable by comparison with pure nuclear matter or SQM. We point that the mixed phase of hybrid stars is located in the meta-stable or unstable regions of pure quark matter. Otherwise the hybrid star will collapse to a strange star and no mixed phase could exist.
We show that the maximum mass and radius of hybrid stars are controlled by the equation of state of quark matter [@Narain2006] rather than that of the nuclear model. Or more exactly in the present framework, they are mainly controlled by the coupling constant $g$ and bag constant $B_0$ in the quasiparticle model. For a smaller coupling constant, the maximum mass is 1.54 M$_\odot$ with radius about 9.64 km. The larger coupling constant will broaden the range of the mixed phase and increase the maximum mass of a hybrid star up to $2.0$ M$_\odot$. So the quark matter with strong coupling can meet the constraint at high density in compact stars [@klahn2006] and corroborate the “masquerade effect" in the previous studies [@alford2005]. For even high density, the possible CFL matter, especially the magnetized CFL matter [@Ferrer2011], should be considered, which will be investigated in our future work.
The authors would like to thank support from the National Natural Science Foundation of China (11005071 and 11135011) and the Shanxi Provincial Natural Science Foundation (2011011001).
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|
---
abstract: 'Grain boundaries in YBa$_2$Cu$_3$O$_{7-x}$ superconducting films are considered as Josephson junctions with a critical current density $j_c(x)$ alternating along the junction. A self-generated magnetic flux is treated both analytically and numerically for an almost periodic distribution of $j_c(x)$. We obtained a magnetic flux-pattern similar to the one which was recently observed experimentally.'
address: |
School of Physics and Astronomy,\
Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
author:
- 'R. G. Mints and Ilya Papiashvili'
title: 'Self-generated magnetic flux in YBa$_2$Cu$_3$O$_{7-x}$ grain boundaries'
---
[2]{}
Introduction
============
Grain boundaries in high-$T_c$ cuprates are interesting and important for both fundamental physics and applications of high-temperature superconductivity.[@Lin; @Tsu; @Mil] Conventional models of strongly coupled Josephson junctions are applicable to describe electromagnetic properties of grain boundaries in thin films of high-$T_c$ superconductors.[@Cha; @Gro] A remarkable exception of this rule is the \[001\]-tilt boundary in YBa$_2$Cu$_3$O$_{7-x}$ films with a misorientation angle close to 45$^{\circ}$.[@Iva; @Che; @Hum; @Cop; @Man] Indeed, these grain boundaries have an anomalous dependence of the critical current $I_c$ on an applied magnetic field $H_a$. Contrary to a usual Fraunhofer-type dependence $I_c(H_a)$ with a major central peak at $H_a=0$ and minor symmetric side-peaks the asymmetric 45$^{\circ}$ \[001\]-tilt grain boundaries demonstrate a pattern without a central dominant peak. Instead two symmetric major side-peaks appear at certain applied magnetic fields $H_a=\pm\, H_{\rm sp}\ne
0$.[@Iva; @Che; @Hum; @Cop; @Man; @Min]
Several mechanisms have been suggested to explain this phenomena. [@Hum; @Cop; @Min] The anomalous dependence $I_c(H_a)$ with symmetric major side-peaks is obviously a result of a specific heterogeneity of electrical properties of the asymmetric 45$^{\circ}$ \[001\]-tilt grain boundaries. Two fundamental experimental observations in conjunction explain this heterogeneity in a natural way.[@Cop; @Man] [*First*]{}, a fine scale faceting of grain boundaries was discovered in experiments using the transmission electron microscopy (TEM) technique. [@Mil; @Jia; @Ros; @Tra; @Seo] These facets have a typical length-scale $l$ of the order of 10–100 nm and a wide variety of orientations relative to the axis of symmetry of the superconductor. [*Second*]{}, quite a few of recent experiments provide an evidence of a predominant $d_{x^2-y^2}$ wave symmetry of the order parameter in many of the high-$T_c$ cuprates. In some experimental studies the symmetry of the order parameter is more complicated and is shown to be a certain mixture of the $d_{x^2-y^2}$ and $s$ wave components.[@Tsu; @DJM; @Wol; @Bra; @Igu; @Van; @Ish; @Mul]
These two fundamental experimental observations indicate the existence of two contributions to the phase difference of the order parameter across the grain boundary. Indeed, consider a meandering grain boundary in a film of a superconductor with the $d_{x^2-y^2}$ wave symmetry of the order parameter and assume that there is a certain magnetic flux inside the grain boundary. In this case there is a phase difference $\varphi$ caused by the magnetic flux and at the same time there is an additional phase difference $\alpha$ caused by the misalignment of the anisotropic $d_{x^2-y^2}$ wave superconductors. Therefore, the tunneling current density $j_c$ is defined by the total phase difference $\Delta =\varphi -\alpha$. A model describing this Josephson current density $j(x)$ results from an assumption that $j(x)\propto\sin\,[\varphi (x)-\alpha (x)]$, where $x$ is along the grain boundary line.[@Man] The local values of the phase difference $\alpha (x)$ depend on the relative orientation of neighboring facets. In the case of an asymmetric 45$^{\circ}$ grain boundary we have $\alpha (x)=0$ or $\pi$, and therefore $j(x)\propto\sin\varphi
(x)\cos\alpha (x)$.[@Man] In other words in the framework of a model relating $j(x)$ to the orientation of the facets we arrive to $j(x)=j_c(x)\sin\varphi (x)$ with an alternating critical current density $j_c(x)\propto\cos\alpha(x)$. The dependence $j_c(x)$ is imposed by the sequence of facets along a grain boundary line. If this sequence is periodic or almost periodic then the function $j_c(x)$ is a periodic or almost periodic alternating function. The typical length-scale for $j_c(x)$ is of the same order as the length of the facets $l$, [*i.e.*]{}, this length-scale is of 10–100 nm.
Variation of orientation of facets along a meandering grain boundary leads to formation of local superconducting current loops even in the absence of an applied magnetic field if the total phase difference $\Delta\ne 0$. It was predicted, in particular, that these current loops can generate a certain magnetic flux at a contact of two facets with $\alpha =0$ and $\alpha =\pi$.[@Cop]
Self-generated randomly distributed magnetic flux was discovered in asymmetric 45$^{\circ}$ \[001\]-tilt grain boundaries in YBa$_2$Cu$_3$O$_{7-x}$ superconducting films in the absence of an applied external magnetic field.[@JMa] This flux $\phi_s(x)$ changes its sign randomly and has an amplitude of variations less than the flux quantum $\phi_0$. The average value of $\phi_s(x)$ along the grain boundary is nearly zero. Noticeable, that this disordered non-quantized magnetic flux was observed only for the samples exhibiting the anomalous dependence of the critical current $I_c(H_a)$ on magnetic field with the two symmetric major side-peaks.
It was shown analytically that under certain conditions a stationary state with a self-generated flux exists for a Josephson junction with a periodically alternating critical current density $j_c(x)$.[@RGM] The same spatial distributions of $j_c(x)$ result in an anomalous dependence of the critical current $I_c(H_a)$ on the applied magnetic field.[@Min] Numerical calculations show that two symmetric major side-peaks appear for a periodically alternating $j_c(x)$. The randomness of the critical current density $j_c(x)$ smears these peaks but leaves their position in place at weak randomness. We therefore conclude that the experimental observation of the well pronounced major side-peaks on the curve $I_c(H_a)$ [@Iva; @Che; @Hum; @Cop; @Man] means that the alternating critical current density is a periodic or almost periodic function of $x$. A noticeable randomness of $j_c(x)$ would smear out the dominant side-peaks.[@Min]
In this paper we calculate both analytically and numerically the self-generated flux $\phi_s$ in a Josephson junction with an almost periodically alternating critical current density $j_c(x)$. The paper is organized as follows. First, we review briefly the case when $j_c(x)$ is a periodic alternating function.[@RGM] We derive the main equations of the two-scale perturbation theory and apply these equations to analyze the non-quantized self-generated flux. This approach forms a basis for the following analytical calculations. Next, we treat the self-generated flux for the case of an almost periodic alternating critical density $j_c(x)$ and start with a qualitative approach to the problem. We review then the results of our numerical simulations which verify the qualitative consideration and exhibit a magnetic flux-pattern which is similar to the one that was recently observed experimentally.[@JMa]
Main equations
==============
It is convenient for the following analyses to write the function $j_c(x)$ in the form $$j_c=\langle j_c\rangle\,[1+g(x)],
\label{eq1}$$ where $\langle j_c\rangle$ is the average value of the critical current density $j_c(x)$ over an interval with a length $L\gg l$ $$\langle j_c\rangle ={1\over L}\,\int_0^Lj_c(x)\,dx.
\label{eq2}$$
The function $g(x)$ introduced in Eq. (\[eq1\]) alternates on a typical length scale of $l$. Note that by definition the average value of $g(x)$ is zero, [*i.e.*]{}, $\langle g(x)\rangle =0$. The maximum value of $|g(x)|$ varies from $|g(x)|_{\rm max}\gtrsim 1$ to $|g(x)|_{\rm max}\gg 1$. We assume also that $\lambda\ll
l\ll\Lambda_J$, where $\lambda$ is the London penetration depth and $$\Lambda_J^2={c\,\phi_o\over 16\pi^2\lambda\langle j_c\rangle}
\label{eq3}$$ is an effective Josephson penetration depth. It is worth to mention that in the case of an alternating current density this effective penetration depth is not a local characteristics of a tunnel junction. It is rather an effective quantity defined on the same typical length scale as $\langle j_c\rangle$.
The phase difference $\varphi(x)$ satisfies the equation $$\Lambda_J^2\,\varphi''-[1+g(x)]\sin\varphi=0.
\label{eq4}$$ In the limiting case $l\ll\Lambda_J$ it is convenient to write a solution of this equation as a sum of a certain smooth function $\psi(x)$ with a length scale of order $\Lambda_J$ and a rapidly oscillating function $\xi(x)$ with a length scale of order $\l$[@RGM] $$\varphi(x)=\psi(x)+\xi(x).
\label{eq5}$$ We assume also that $|\xi(x)|\ll|\psi(x)|$. After substituting Eq. (\[eq5\]) into Eq. (\[eq4\]) and keeping the terms up to the first order in $\xi (x)$ we obtain $$\Lambda_J^2\psi ''+\Lambda_J^2\xi '' -
[1+g(x)][\sin\psi+\xi\cos\psi]=0.
\label{eq6}$$
Note, that experimentally[@JMa] the self-generated flux was observed by a SQUID pickup loop with a size of several $\Lambda_J\gg~l$. It means that this method is averaging out the fast alternating flux defined by the phase $\xi (x)$ and measures the spatially smooth flux defined by $\psi (x)$.
Next we consider briefly the case of a [*periodically*]{} alternating critical current density $j_c(x)$ which forms the basis of the following analysis of a general case with $j_c(x)$ being a randomly alternating function.
Periodically alternating critical current density
=================================================
Two-scale perturbation theory
-----------------------------
If the critical current density $j_c(x)$ is a periodic function, then $g(x)$ also is a periodic function. In this case an approximate solution of Eq. (\[eq6\]) can be obtained based on a two-scale perturbation theory.[@Lan] As a first step in order to apply this approach to Eq. (\[eq6\]) we separate the fast alternating terms with a typical length scale $l$ and the smooth terms varying with a typical length scale $\Lambda_J$ $$(\Lambda_J^2\psi''-\sin\psi -g\xi\cos\psi )+
(\Lambda_J^2\xi'' -g\sin\psi)=0.
\label{eq7}$$
In Eq. (\[eq7\]) we omitted two out of three fast alternating terms of Eq. (\[eq6\]) since they are proportional to $\xi (x)$ and therefore are smaller than the term proportional to $g (x)$. Next, we note that the terms included into the first pair of brackets in Eq. (\[eq7\]) cancel each other independently on the terms included into the second pair of brackets in Eq. (\[eq7\]) as these two type of terms have very different length scales $l$ and $\Lambda_J$ and $l\ll\Lambda_J$.[@Lan] The same reasoning is applicable to the terms included into the second pair of brackets in Eq. (\[eq7\]). As a result we obtain the following two equations for $\xi (x)$ and $\psi (x)$[@RGM] $$\begin{aligned}
& \Lambda_J^2\xi ''=g(x)\sin\psi,\label{eq8}\\
& \Lambda_J^2\psi''-\sin\psi-
\langle g(x)\xi(x)\rangle\cos\psi=0.\label{eq9}\end{aligned}$$ It is worth to note that we obtain the [*two*]{} functions $\psi (x)$ and $\xi (x)$ from [*one*]{} equation (\[eq7\]) as only two type of terms with different typical length scales $l$ and $\Lambda_J$ appear in Eq. (\[eq6\]). If $g(x)$ would have a wide range of typical length scales the above separation would not be possible.
Introducing the Fourier transform of $g(x)$ as $$g(x)=\sum_{-\infty}^\infty g_ke^{ikx}
\label{eq10}$$ we find the solution of Eq. (\[eq8\]) in the form $$\xi (x)=-{\sin\psi\over\Lambda_J^2}\,
\sum_{-\infty}^\infty {g_ke^{ikx}\over k^2}=-\xi_g(x)\,\sin\psi,
\label{eq11}$$ where the sums in Eqs. (\[eq10\]) and (\[eq11\]) are over the wave vectors $k=2\pi n/{\cal L}$, ${\cal L}$ is the length of the junction, and $n$ is an integer. It is worth mentioning that the function $\xi_g (x)$ is defined only by the alternating components of the critical current density $j_c(x)$. Also while deriving Eq. (\[eq11\]) we ignored the spatial dependence of $\sin\psi$. This can be done since on the length-scale $l$ the variation of the smooth function $\sin\psi (x)$ is of order $l/\Lambda_J\ll 1$. The alternating part of the critical current density has the typical wave numbers $k\sim 1/l$. Therefore, using Eq. (\[eq11\]) we estimate $\xi (x)$ as $$\xi (x)\sim -\sin\psi{l^2\over\Lambda_J^2}\,g(x).
\label{eq12}$$ It follows from this estimate that the typical values of the phase difference $\xi (x)$ are small $(\langle|\xi (x)|\rangle\ll 1)$ if $$\langle |g(x)|\rangle\ll {\Lambda_J^2\over l^2}.
\label{eq13}$$
Next, using Eq. (\[eq11\]) we rewrite Eq. (\[eq9\]) for the smooth phase shift $\psi (x)$ in the form $$\Lambda_J^2\psi''-\sin\psi+\gamma\sin\psi\cos\psi=0,
\label{eq14}$$ where $$\gamma=\langle g(x)\xi_g(x)\rangle=
-{1\over \sin\psi}\,\langle g(x)\xi(x)\rangle
\label{eq15}$$ is a constant. A similar derivation for the current density across the tunnel junction results in $$j(x)=\langle j_c\rangle\sin\psi (1 -\gamma\cos\psi).
\label{eq16}$$
The magnetic field $B_s(x)$ generated by the alternating component of the critical current $\langle j_c\rangle\,g(x)$ is given by $$B_s={4\pi\over c}\,\langle j_c\rangle\int g(x)\,dx=
-{\phi_0\over 4\pi\lambda}\,{d\xi_g\over dx}
\label{eq17}$$ and the averaged field $\langle B_s(x)\rangle =0$. The alternating magnetic flux $\phi_s$ produced by the field $B_s$ is equal to $$\phi_s=-{\phi_0\over 2\pi}\,\xi_g.
\label{eq18}$$ Combining Eqs. (\[eq15\]) and (\[eq17\]) we find for $\gamma$ the formula $$\gamma={c\lambda\over\phi_0\langle j_c\rangle }\,\langle B_s^2\rangle=
{\langle B_s^2\rangle\over\langle B_J^2\rangle},
\label{eq19}$$ where we introduce a characteristic magnetic field $$B_J={4\pi\over c}\,\langle j_c\rangle\Lambda_J.
\label{eq20}$$ It follows from equation (\[eq19\]) that $\gamma$ is a positive constant which can be estimated as $$\gamma\sim {l^2\over\Lambda_J^2}\,\langle g^2\rangle.
\label{eq21}$$
The energy of a Josephson junction $\cal E$ takes the form[@Bar] ${\cal E}={\cal E}_0+{\cal E}_{\varphi}$, where ${\cal E}_0$ is independent on $\varphi (x)$ and $${\cal E}_{\varphi}={\langle j_c\rangle\over 2e}\,\int dx
\Bigl\{{1\over 2}\,\Lambda_J^2\varphi'^2-[1+g(x)]\cos\varphi\Bigr\}.
\label{eq22}$$ Using Eqs. (\[eq8\]), (\[eq11\]), and the definition of $\gamma$, we obtain the energy ${\cal E}_{\varphi}$ in terms of the smooth phase shift $\psi (x)$ $${\cal E}_{\varphi}={\hbar\langle j_c\rangle\over 4e}\int dx
\Bigl\{\Lambda_J^2\psi'^2-2\cos\psi
-\gamma\sin^2\psi\Bigr\}.
\label{eq23}$$ Note, that solutions $\psi (x)$ of Eq. (\[eq14\]) correspond to the minima and to the maxima of the energy functional (\[eq23\]).
Non-quantized self-generated flux
---------------------------------
Let us apply Eqs. (\[eq8\]), (\[eq9\]), and (\[eq15\]) to a consider the stationary states of a Josephson junction with a certain length ${\cal L}\gg \Lambda_J$ in an absence of applied magnetic field. In this case the average flux inside the junction is zero and thus an alternating self-generated flux $\phi_s (x)$ appears simultaneously with a certain phase $\psi ={\rm const}$ as it follows from Eqs. (\[eq11\]) and (\[eq18\]).
In the stationary state with $\psi ={\rm const}$ the values of $\psi$ are determined by Eq. (\[eq14\]) which takes the form $$\sin\psi\,(1-\gamma\cos\psi)=0.
\label{eq24}$$ Note, that this equation means also that the current density $j(x)$ across the tunnel junction is equal to zero.
In the case $\gamma <1$ equation (\[eq24\]) has two solutions, namely, $\psi =0$ and $\psi=\pi$ and thus, as follows from Eq. (\[eq11\]), there is no self-generated flux. It is also worth mentioning that the energy of a Josephson junction ${\cal E}$ has a minimum for $\psi =0$ and maximum for $\psi =\pi$.
In the case $\gamma\ge 1$ there are four solutions of equation (\[eq24\]), namely, $\psi =-\psi_{\gamma},\,0,\,\psi_{\gamma},\,\pi$, where $$\psi_{\gamma}=\arccos\,(1/\gamma).
\label{eq25}$$ The energy ${\cal E}[\psi(x)]$ has a minimum for $\psi=\pm\,\psi_{\gamma}$ and a maximum for $\psi =0,\pi$. The self-generated flux $$\phi_s=-{\phi_0 \xi\over 2\pi}=-{\phi_0\xi_g\over 2\pi}\sin\psi_\gamma=
\mp\,\phi_0{\xi_g\over 2\pi}\,{\sqrt{\gamma^2-1}\over\gamma}
\label{eq26}$$ arises in the two stationary states with $\psi=\pm\,\psi_{\gamma}$, each of these states corresponds to a minimum energy $\cal E$. The assumption $\langle\xi (x)\rangle\ll 1$ restricts the value of $\gamma$. However, it follows from Eqs. (\[eq13\]) and (\[eq21\]) that $\langle\xi (x)\rangle\ll 1$ and $\gamma >1$ hold simultaneously only if $${\Lambda_J\over l}<\langle |g(x)|\rangle\ll{\Lambda_J^2\over l^2}.
\label{eq27}$$ Using equations (\[eq12\]) and (\[eq26\]) we estimate $|\phi_s(x)|$ as $$|\phi_s(x)|\sim\phi_0\,{\sqrt{\gamma^2-1}\over\gamma}\,
{l^2\over\Lambda_J^2}\,|g(x)|\ll\phi_0.
\label{eq28}$$
The above results hold only for a periodic critical current density $j_c(x)$ and the predicted self-generated flux $\phi_s(x)$ has a typical amplitude of variations which is much less than the one observed experimentally.[@JMa]
Non-periodic alternating critical current density
=================================================
The above analytical approach to the problem of a self-generated flux in a non-uniform Josephson junction is based on an assumption that the critical current density $j_c(x)$ is an alternating periodic function. This model allows for analytical calculation and provides a reasonable preliminary insight into the properties of an idealized Josephson junction with an alternating $j_c(x)$. At the same time this simple model fails for a quantitative description of any real system with a certain randomness of the spatial distribution of Josephson critical current density $j_c(x)$. In this section we generalize the above approach assuming that the alternating critical current density $j_c(x)$ is almost periodic, [*i.e.*]{}, we assume that there is a typical length of interchange of sign of $j_c(x)$ which is distributed randomly near some mean value $l$.
In the case of an almost periodic $j_c(x)$ we can not apply the two-scale perturbation theory in the same way as we did it in the previous section. Indeed, an arbitrary solution of Eq. (\[eq8\]) takes the form $$\xi(x)={\sin\psi\over\Lambda_J^2}\,G(x),
\label{eq29}$$ where $$G(x)=\int_{a}^x\!\! dx'\!\! \int_{a'}^{x'} \!\! dx''\,g(x''),
\label{eq30}$$ and the integration constants $a$ and $a'$ are defined by the boundary conditions. The random function $G(x)$ increases with an increase of the integration interval. In general, the value of $|G(x)|$ can become arbitrarily large if the length of the tunnel junction $\cal L$ becomes large enough. This is in a contradiction with our main assumption that the phase $\xi (x)$ is a small and fast varying component of the total phase difference $\varphi (x)$. To solve this contradiction we write $\xi (x)$ as $$\xi(x)={\sin\psi\over \Lambda^2_J}\left[ G(x)-G_a(x)\right].
\label{eq31}$$ The function $G_a(x)$ is a smoothing average of $G(x)$ over an interval with a certain length ${\cal L}_a$, where $l<{\cal L}_a\ll {\cal L}$.
The procedure of filtering out the smooth part of $G(x)$ is especially evident if we use Fourier series for $g(x)$ and $G(x)$. Introducing Fourier transform for $g(x)$ as $$g(x)=\sum_{-\infty}^\infty g_k\,e^{ikx}
\label{eq32}$$ we find Fourier series for the function $G(x)$ in the form $$G(x)=-\sum_{-\infty}^\infty {g_k\over k^2}\,e^{ikx},
\label{eq33}$$ where the sums in Eqs. (\[eq32\]) and (\[eq33\]) are over the wave vectors $k=2\pi n/{\cal L}$ and $n$ is an integer. The smooth part of the function $G(x)$ can be obtained by extracting the fast Fourier harmonics, [*i.e.*]{}, by extracting from the sum (\[eq33\]) terms with wave vectors $|k|>k_a\sim 2\pi/{\cal L}_a$. As a result we find for $G_a(x)$ and $\xi(x)$ the series $$\begin{aligned}
G_a(x)&=&-\sum_{-k_a}^{k_a} {g_k\over k^2}\,e^{ikx},
\label{eq34}\\
\xi(x)&=&{\sin\psi\over\Lambda_J^2}\,
\Biggl (\sum_{-\infty}^{-k_a} + \sum_{k_a}^{\infty} \Biggr )
\Biggl [{g_k\over k^2}\,e^{ikx}\Biggr ].
\label{eq35}\end{aligned}$$
The small and fast alternating part $\xi (x)$ of the phase difference $\varphi (x)$ is thus defined by Eq. (\[eq31\]). This equation is a straightforward generalization of the two-scale perturbation theory approach to a real case of an almost periodic $g(x)$.
Next, we use Eq. (\[eq31\]) to derive an equation for the smooth part $\psi (x)$ of the phase $\varphi (x)$. [*First*]{}, combining Eqs. (\[eq31\]), (\[eq7\]), and (\[eq30\]) we arrive to the relation $$\Lambda^2_J\psi''-[1+G_a''(x)]\sin\psi-g(x)\xi(x)\sin\psi=0.
\label{eq36}$$ [*Second*]{}, we average Eq. (\[eq36\]) over an interval with a certain length $L$ assuming that $l\ll L\ll {\cal L}_a$. This averaging results in an equation describing the phase $\psi(x)$ $$\Lambda^2_J\psi''-[1+G_a''(x)]\sin\psi +
\gamma (x)\sin\psi\cos\psi=0,
\label{eq37}$$ where $\gamma (x)$ is defined by Eq. (\[eq15\]). It is worth mentioning that equation (\[eq37\]) differs from the analogues equation (\[eq9\]) by an additional term $G_a''(x)$ in the coefficient before $\sin\psi$ and by the fact that the parameter $\gamma =\gamma (x)$ is a function of the coordinate $x$ along the junction.
The coefficient $1+G_a''(x)$ is defined by magnetic field $B_c(x)$ which would be generated by a current with the density $j_c(x)$. Indeed, using Eqs. (\[eq1\]), (\[eq30\]), and Maxwell’s equation $dB_c/dx=4\pi j_c/c$ we obtain for $G(x)$ $$\begin{aligned}
G(x)&=&{c\over 4\pi \langle j_c\rangle}\,\int dx'\,[B_c(x')-B_a(x')]=
\nonumber\\
&=&{2\pi\Lambda_J^2\over\phi_0}\,[\phi_c(x)-\phi_a(x)],
\label{eq38}\end{aligned}$$ where the magnetic field $B_a(x)$ would be generated by a constant current density $\langle j_c\rangle$, [*i.e.*]{}, $dB_a/dx=4\pi
\langle j_c\rangle/c$, and $\phi_c(x)$ and $\phi_a(x)$ are the fluxes of the fields $B_c(x)$ and $B_a(x)$. It follows now from Eq. (\[eq38\]) that $$1+G_a''(x)={2\pi\Lambda^2_J\over \phi_0}\,\langle\phi_c(x)\rangle''.
\label{eq39}$$
As shown above the value of the parameter $\gamma$ is determining the existence or absence of the self-generated flux. Using Eq. (\[eq8\]) and the definition of $\gamma (x)$ given by Eq. (\[eq15\]) we obtain for $\gamma (x)$ an expression $$\gamma (x)=-{\Lambda^2_J\over\sin^2\psi}\,\langle \xi''\xi\rangle =
{\Lambda^2_J\over\sin^2\psi}\,\langle {\xi'}^2\rangle >0
\label{eq40}$$ demonstrating that the condition $\gamma (x)>0$ holds also in the case of an almost periodic critical current density.
Self-generated flux in a stationary state
-----------------------------------------
Let us now consider stationary solutions for the smooth part $\psi (x)$ of the phase difference $\varphi (x)$ qualitatively. Assume [*first*]{} that there are sufficiently large intervals with lengths $L_i\gg\Lambda_J$, where the function $\psi(x)$ is constant or varies with a typical space-scale of order ${\cal L}_i\gg\Lambda_J$. In this case equation (\[eq37\]) reduces to $$\big[1+G_a''(x)-\gamma(x)\cos\psi\big]\sin\psi=0.
\label{eq41}$$
-.75
=
This equation is similar to Eq. (\[eq24\]) which we derived for the case of a periodic critical current density $j_c(x)$ and has different solutions depending on the value of the parameter $$\gamma_r (x)={\gamma(x)\over 1+G_a''(x)}. \label{eq42}$$
In the regions with $\gamma_r (x)<1$ equation (\[eq41\]) has two solutions $\psi =0$ and $\psi =\pi$ and therefore as it follows from Eq. (\[eq29\]) there is no self-generated flux in these regions.
The energy of a Josephson junction ${\cal E}_\varphi$ given by Eq. (\[eq22\]) can be written in terms of the smooth part of the phase difference $\psi (x)$. In the case of an almost periodic critical current density $j_c(x)$ this equation reads $$\begin{aligned}
& &{\cal E}_{\varphi}={\hbar\langle j_c\rangle\over 4e}\times
\nonumber \\
& &\int dx\Bigl\{\Lambda_J^2\psi'^2-2\cos\psi [1+G_a''(x)] -
\gamma (x)\sin^2\psi\Bigr\}
\label{eq43}\end{aligned}$$ and if $\gamma_r (x)<1$, then the energy ${\cal E}_{\varphi}$ has a minimum for $\psi =0$ and a maximum for $\psi =\pi$.
In the regions where the function $\gamma_r(x)>1$ equation (\[eq41\]) has four solutions $\psi =-\psi_\gamma (x),\,0,\,\psi_\gamma (x),\,
\pi$, with $$\psi_{\gamma} (x)=\arccos\biggl[{1\over\gamma_r(x)}\biggr]=
\arccos\biggl[{1+G_a''(x)\over\gamma (x)}\biggr].
\label{eq44}$$ The energy ${\cal E}_{\varphi}$ has a minimum for $\psi
=\pm\,\psi_\gamma (x)$ and a maximum for $\psi =0,\,\pi$. The self-generated flux is thus non-zero in the regions with $\gamma_r>1$. This flux has a fast and a smoothly varying parts defined by $\xi (x)$ and $\psi_\gamma (x)$.
0.75
The randomness of the function $g(x)$ causes variation of $\gamma_r (x)$ along the junction. As a result of this variation intervals with $\gamma_r (x)>1$ are interlaced with intervals with $\gamma_r (x)\leq 1$. As it was mentioned above, in the case of $\gamma_r >1$ the energy of the Josephson junction ${\cal E}_\varphi$ has a minimum for $\psi =\pm\,\psi_{\gamma}(x)$ and a maximum for $\psi =0,\,\pi$. When the value of $\gamma_r (x)$ changes from $\gamma_r (x)>1$ to $\gamma_r (x)\leq 1$ the energy ${\cal E}_\varphi$ still has a maximum if $\psi=\pi$, but a state with $\psi=0$ becomes a state with a minimum energy.
The above results provide a qualitative description of experimentally observable flux distribution along a Josephson junction with an almost periodic alternating critical current density. This flux distribution spatially averaged by the measurement tools is defined by the function $\psi(x)$ (see Fig. \[fig\_1\]). Inside the intervals with $\gamma_r(x)>1$ the phase $\psi(x)$ tends to one of the solutions $\pm\,\psi_{\gamma}(x)$. The profile of the function $\psi(x)$ correlates with the profile of $\psi_{\gamma}(x)$, though does not coincide with it exactly because the solution $\psi(x)=\psi_{\gamma}(x)$ was obtained under assumption $\psi''=0$, which does not hold exactly for the intervals $\gamma_r(x)>1$. The smooth part of the phase difference inside the intervals with $\gamma_r(x)\leq 1$ is $\psi = 0$ which is consistent with the assumption $\psi''=0$.
The value of $\psi_\gamma$ increases quite fast with an increase of the parameter $\gamma$ (see Fig. \[fig\_2\]). In particular, for $\gamma =2$ the value of $\psi_\gamma$ is already about 0.75 of its maximum value $\pi /2$. This means that for most of the experimentally observable peaks of the self-generated flux the values of $\psi$ will be close to $\pi /2$ which corresponds to a magnetic flux $\phi_0 /4$. In some places of the junction the phase $\psi$ changes from $-\psi_\gamma$ to $\psi_\gamma$. The flux localized in this area of the junction will be close to $\phi_0/2$.
Numerical simulations
=====================
The finite difference scheme
----------------------------
To study the self-generation of magnetic flux in a tunnel junction with an alternating critical current density numerically we introduced time dependence into the main equation (\[eq4\]) $$\ddot{\varphi} +\alpha\dot{\varphi}-\Lambda_J^2\psi''
+[1+g(x)]\sin\varphi=0,
\label{eq45}$$ where $\alpha\sim 1$ is a decay constant. This approach allows to study both dynamics and statics of the system.
The term $\alpha\dot{\varphi}$ introduces dissipation. As a result of this dissipation the relaxation of the system ends up in one of the stable stationary states described by a certain solution of the static equation (\[eq4\]). Moreover, for a given distribution of the critical current density $j_c(x)$ we obtained different static solutions when we start the numeric simulation from different initial states. We compare and classify these solutions based on the features of the function $j_c(x)$. Indeed, this function essentially describes the pinning properties of the junction. Therefore a variety of initial conditions can converge to a similar flux pinning pattern.
To solve Eq. (\[eq45\]) numerically we use the finite difference scheme.[@Abl] We adopted this method to our case and checked stability and convergency of the obtained solutions. As a result we arrived to the following scheme $$\begin{aligned}
&\varphi&\rightarrow{\varphi_{m+1}^n+\varphi_{m-1}^n \over 2}
\equiv\tilde{\varphi}_m^n\\
&\dot{\varphi}&\rightarrow
{\tilde{\varphi}_m^n-\varphi_m^{n-1} \over\tau}\\
&\ddot{\varphi}&\rightarrow{\varphi_m^{n+1}+\varphi_m^{n-1}-2
\tilde\varphi_m^n \over\tau^2}\\
&\varphi''&\rightarrow{\varphi_{m+1}^n+\varphi_{m-1}^n-2
\varphi_m^n \over h^2},\end{aligned}$$ where $f^n_m=f(x_m,t_n)$, $\tau$ and $h$ are steps along $t$ and $x$ correspondingly. Next, we choose units providing $\Lambda_J=1$ and set $h$=$\tau$. As a result we arrive to the following finite difference scheme $$\begin{aligned}
\varphi_m^{n+1}=&-&(1- \alpha\tau)\,\varphi_m^{n-1}+
(2-\alpha\tau)\,\tilde{\varphi}_m^n
\nonumber\\
&-&\tau^2(1-g_m)\sin\tilde{\varphi}_m^n.
\label{eq50}\end{aligned}$$
Stationary solutions
--------------------
Initially a certain random function $g(x)$ is generated for an interval with a length $L$ with a given values of $l$ and $\delta l$ (a typical length-scale of the function $g(x)$ and its dispersion), $g$ and $\delta g$ (amplitude of the function $g(x)$ and its dispersion). This allows to calculate the function $\gamma_r (x)$ for the whole interval. An initial state $\varphi_0(x)$ is prepared as a random or some specific function. Finally the dynamical rules (\[eq50\]) are applied to the initial state iteratively until a stationary state is established.
In Fig. \[fig\_3\] we show one of the stationary solutions obtained by a numerical simulation and the function $\gamma_r (x)$ calculated for this solution. It is clearly seen from Fig. \[fig\_3\] that $\varphi(x)$ arises at the places where $\gamma_r (x)$ exceeds $1$. Heights of the peaks are less than $\pi/2$, and thus the corresponding magnetic flux amplitudes are less than $\phi_0/4$.
In general a different initial state of the same sample, [*i.e.*]{}, for the same function $\gamma_r (x)$, generates a different stationary state. Our numerical simulations show that these different states differ only by sign of some peaks of $\varphi(x)$, but the shapes and locations stay unchanged.
We have compared our results with the experimental data.[@JMa] The typical amplitude of the flux variations measured by a SQUID pickup loop with a size of several $\Lambda_J$ is about $0.25$ of $\phi_0$ with rare narrow picks with an amplitude about $0.5\,\phi_0$ which is in a good agreement with our calculations.
SUMMARY
=======
We treated a Josephson junction with an alternating critical current density $j_c(x)$ as a model for considering electromagnetic properties of grain boundaries in YBa$_2$Cu$_3$O$_{7-x}$ superconducting films. The study is mainly focused on a specific case of an almost periodically alternating function $j_c(x)$. We demonstrated both analytically and numerically that under certain conditions a self-generated flux pattern arise for this type of spatial distribution of the critical current density $j_c(x)$. The obtained flux pattern with two types of interlacing flux domains is similar to the one which was recently observed experimentally in YBa$_2$Cu$_3$O$_{7-x}$ superconducting films in the absence of an external magnetic field. The typical amplitude for the magnetic flux peaks is of order $\phi_0/4$ and $\phi_0/2$ and the typical distance between the peaks depends on the spatial distribution of $j_c(x)$.
One of us (RGM) is grateful to E.H. Brandt, J.R. Clem, H. Hilgenkamp, V.G. Kogan, and J. Mannhart for useful and stimulating discussions. This research was supported in part by grant No. 96-00048 from the United States – Israel Binational Science Foundation (BSF), Jerusalem, Israel.
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|
---
abstract: 'The topological phase transitions in static and periodically driven Kitaev chains are investigated by means of a renormalization group (RG) approach. These transitions, across which the numbers of static or Floquet Majorana edge modes change, are accompanied by divergences of the relevant Berry connections. These divergences at certain high symmetry points in momentum space form the basis of the RG approach, through which topological phase boundaries are identified as a function of system parameters. We also introduce several aspects to characterize the quantum criticality of the topological phase transitions in both static and Floquet systems: a correlation function that measures the overlap of Majorana-Wannier functions, the decay length of the Majorana edge mode and a scaling law relating the critical exponents. These indicate a common universal critical behavior for topological phase transitions, in both static and periodically driven chains. For the latter, the RG flows additionally display intriguing features related to gap closures at non-high symmetry points due to momentarily frozen dynamics.'
author:
- Paolo Molignini
- Wei Chen
- 'R. Chitra'
bibliography:
- 'many-body-biblio.bib'
title: ' Universal quantum criticality in static and Floquet-Majorana chains'
---
Introduction
============
The discovery of topological order has enriched the theory of phase transitions with a new fundamental paradigm [@Wen:1990]. Contrary to the traditional Landau formalism based on spontaneous symmetry breaking [@Landau; @Miransky-book], topologically ordered systems are not described by local order parameters but by robust ground state degeneracy, quantized geometric phases [@Thouless:1982; @Wen:1989; @Qi:2008] and often long-range quantum entanglement [@Wen:1995; @XieChen:2010; @Fidkowski:2010]. Moreover, bulk-edge correspondences lead to a wide variety of edge states in topological systems [@Wen:1991; @Kane-Mele2005]. Topological systems can exhibit charge fractionalization, as well as excitations with exotic abelian and non-abelian statistics [@Goldman:1995; @dePicciotto:1997; @Martin:2004; @Moore; @Kitaev2001; @Mong:2014]. These can be harnessed for revolutionary applications, such as spintronics with edge currents [@Sato-Spintronics-book], topological quantum memory devices with highly-entangled matter [@Dennis:2002], and, most notably, fault-tolerant topological quantum computation [@Kitaev:2003; @Nayak:2008; @Mong:2014].
Recently, the exploration of topological order has been extended to Floquet systems, where it can be generated through driving in otherwise topologically trivial systems. Prominent examples are Floquet topological insulators, and Floquet topological superconductors that host Floquet-Majorana modes [@Lindner; @Kitagawa; @Liu; @Cayssol:2013; @Harper:2017; @Roy:2017; @Yao:2017]. Floquet systems in one dimension often exhibit a tunable topology wherein, the number of edge modes can be systematically increased by manipulating the intensity or the frequency of the drive [@Liu; @Thakurathi; @Molignini:2017]. This results in a series of gap closures in the quasienergy spectrum which signal out-of-equilibrium Floquet topological phase transitions (TPT) between topologically inequivalent phases [@Graf:2013; @Molignini:2017].
The primary focus of literature has been on topological classifications of these phases based on symmetries rather than the nature of the transitions themselves [@ChiuReview:2016]. Recently, a renormalization group approach was proposed to study the nature of TPTs in static systems [@Chen:2016; @Chen-Sigrist:2016]. It essentially exploits the idea that, since the topological invariant generally takes the form of an integration over a certain curvature function, TPTs can be identified through appropriate deformations of the curvature function, analogous to stretching a messy string to reveal the number of knots it contains. This method, termed the curvature renormalization group (CRG) approach, has been successful in describing TPTs in a variety of interacting and noninteracting static models [@Chen:2016; @Chen-Sigrist:2016; @Kourtis:2017; @Chen:2018].
In this article, we extend the CRG scheme to Floquet systems. To benchmark our method, we study TPTs in both the static and periodically driven Kitaev chains. The scheme is based on deformations of the Berry connection of the appropriate Bloch or Floquet-Bloch eigenstate of the static or effective Floquet hamiltonian, which plays the role of the curvature function in this problem. Though winding numbers based on this curvature function incompletely reproduce the TPTs in Floquet systems [@Thakurathi; @Rudner:2013], the curvature function always diverges at certain high symmetry points (HSPs) in momentum space as one approaches the TPTs. We will show that the latter feature determines the critical points of our CRG and suffices to obtain the full topological phase diagram in an extremely simplified manner. We find that TPTs across which the number of edge Majorana modes change by one are characterized by certain universal features: divergence of the Majorana-Wannier state correlation length and a scaling law that constrains the critical exponents. Additionally, the fixed points of the CRG flow reveal another, more subtle type of instability, where the driving-induced dynamics is frozen revealing minimal correlations. Intriguingly, some of these fixed lines are also associated with gap closings in the quasi-energy spectrum opening up the potential for new kinds of TPTs in driven systems.
The article is structured as follows. In Sec. \[sec:generic\_critical\_behavior\], we present an overview of the CRG method based on the Berry connection function. In Sec. \[sec:static\_kitaev\_chain\], we present an illustration of the method by applying it to the static Kitaev chain for fermions or, equivalently, the XY spin-$\frac12$ chain in a transverse magnetic field. In Sec. \[sec:periodically\_driven\_Kitaev\], we apply the methodology to the periodically driven Kitaev chain and present a general analysis of the Floquet TPTs which exist in these systems. Finally, Sec. \[sec:conclusions\] summarizes the main results of the article and offers a glimpse in open questions and possible future directions.
Topological phase transitions and a renormalization group approach\[sec:generic\_critical\_behavior\]
=====================================================================================================
Here, we briefly review the CRG approach expostulated in [@Chen:2016; @Chen-Sigrist:2016; @Chen:2017], which is designed to capture the critical behaviour of a static system close to a TPT. We consider a topological system, whose critical behavior at the TPTs is driven by a set of tuning parameters ${\bf M}=\left(M_{1},M_{2},...\right)$ in the Hamiltonian. For example, if we consider the XY spin chain in a transverse field, the tuning parameters are the magnetic field and the anisotropy, or in the equivalent Kitaev chain model, the parameters are the chemical potential and the pairing gap.
We denote by $F(k, {\bf M})$, the generic curvature function in one dimension that is synonymous with the notion of the curvature of a closed string whose integral counts the number of knots it contains. We will elaborate on the relation between $F$ and the Berry connection in $k$-space in Sec. \[sec:static\_Kitaev\_chain\]. For static systems, this curvature function determines the topological properties of the system via the winding number defined by $$W = \int_{-\pi}^{\pi} \frac{\mathrm{d}k}{2\pi}F(k, {\bf M}).
\label{winding_number_definition}$$ Phases with different $W$ are separated by TPTs. As discussed extensively in Refs. [@Chen:2016; @Chen-Sigrist:2016; @Chen:2017], near HSPs $k_0$ of the underlying lattice, $F(k, {\bf M})$ typically displays the Ornstein-Zernike form $$\begin{aligned}
F(k_{0}+\delta k,{\bf M})=\frac{F(k_{0},{\bf M})}{1+\xi_{k_0}^{2}\delta k^{2}}\;.
\label{Ornstein_Zernike_fit}\end{aligned}$$ Let ${\bf M}_c$ denote the critical point where the system undergoes a TPT associated with a gap closure at a certain $k_0$.
When ${\bf M}\rightarrow{\bf M}_{c}$, the length scale $\xi_{k_0} \to \infty$ resulting in a narrowing of the Lorentzian of Eq. (\[Ornstein\_Zernike\_fit\]) and a divergence of the curvature function as ${\bf M}_c$ is approached from below or above: $$\begin{aligned}
&&\lim_{{\bf M}\rightarrow{\bf M}_{c}^{+}}F(k_{0},{\bf M}) = -\lim_{{\bf M}\rightarrow{\bf M}_{c}^{-}}F(k_{0},{\bf M}) = \pm\infty \, .
\label{Fk0_xi_divergence}\end{aligned}$$
Close to the TPT, we expect the following divergent behavior: $$\begin{aligned}
F(k_{0},{\bf M})\propto |{\bf M}-{\bf M}_{c}|^{-\gamma}\;,\;\;\;\xi_{k_0}\propto |{\bf M}-{\bf M}_{c}|^{-\nu}\;.
\label{Fk0_xi_critical_exponent}\end{aligned}$$ with exponents $\gamma$ and $\nu$ characterizing the underlying TPT. The conservation of the topological invariant as ${\bf M}\rightarrow{\bf M}_{c}$, however, imposes a scaling law $\gamma=\nu$ [@Chen:2017].
The exponents $\nu$ and $\gamma$ are synonymous to those assigned to correlation length and susceptibility exponents within the Landau paradigm. To see this, we consider the Fourier transform of the curvature function: $$\begin{aligned}
\lambda_{R}=\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}e^{ikR}F(k,{\bf M})\, .
\label{lambdaR_Fourier_trans}\end{aligned}$$ The quantity $\lambda_R$ yields a [ *[ Majorana-Wannier state correlation function]{}*]{} that exemplifies the proximity of the system to a TPT. Inserting Eq. (\[Ornstein\_Zernike\_fit\]) into Eq. ([\[lambdaR\_Fourier\_trans\]]{}), we see that the correlation function decays exponentially $\lambda_R \propto \exp(-R/\xi_{k_0})$. This justifies the notion of $\xi_{k_0}$ as the correlation length of the TPT with the associated critical exponents $\nu$. On the other hand, the curvature function at HSP is the integration of the correlation function $\int \lambda_{R}\mathrm{d}R=F(k_{0}=0,{\bf M})$, which plays the role of the susceptibility in the Landau order parameter paradigm, and hence the exponent $\gamma$ is assigned.
Based on the divergence described by (\[Fk0\_xi\_divergence\]), the CRG scheme has been proposed to identify the TPTs [@Chen:2016; @Chen-Sigrist:2016; @Chen:2017]. The method is based on the iterative search for the trajectory in parameter space (RG flow) along which the divergence of the curvature function is reduced but the topology remains unchanged. Under this invariant procedure the system will gradually move away from the critical point and the TPTs can be identified. The RG flow is obtained by demanding that at a given parameter set ${\bf M}$, the next parameter set ${\bf M}^{\prime}$ in the iteration satisfies $$F(k_0, {\bf M}^{\prime}) = F(k_0 + \delta k, {\bf M}),
\label{cond-RG-eqn}$$ where $k_{0}$ is a HSP and $\delta k$ is a small deviation from it. It can be rigorously shown that $F(k_{0},{\bf M})$ gradually broadens under this procedure [@Chen:2016], as schematically depicted in Fig. \[plots-Berry-conn-static\].
![The curvature function $F(k,{\bf M})$ near the HSP $k_{0}=0$ for the static Kitaev chain plotted for several values of $\mu_0$ at $\Delta=0.7$. The critical point is located at $\mu_{0}=1$. As approaching the critical point, the curvature function develops a divergence at the HSP (compare orange line and red line), and the divergence flips sign as the system crosses the critical point (compare red line and blue line). The CRG procedure demands the $F(k_{0}+\delta k,{\bf M})$ (red dot) to be equal to $F(k_{0},{\bf M}^{\prime})$ (orange dot), as indicated by the dashed line, through one obtains the CRG flow ${\bf M}\rightarrow{\bf M}^{\prime}$ along which the divergence is reduced. []{data-label="plots-Berry-conn-static"}](BerryConnectionStatic2.pdf){width="0.8\columnwidth"}
Writing $\mathrm{d}M_{i} = M_{i}^{\prime} - M_{i}$ and $\delta k^2 \equiv \mathrm{d}l$, and expanding Eq. (\[cond-RG-eqn\]) to leading order yields the generic RG equation for the parameters ${\bf M}$: $$\frac{\mathrm{d}M_{i}}{\mathrm{d} l} = \frac{1}{2} \frac{\partial^2_k F(k, {\bf M}) \big|_{k=k_0}}{\partial_{M_{i}} F(k_{0}, {\bf M})},
\label{generic_RG_equation}$$ The critical and fixed points of the flow are defined by the following conditions [@Kourtis:2017] $$\begin{aligned}
{\rm critical\;point}:&&\left|\frac{d{\bf M}}{dl}\right|\rightarrow\infty,\;{\rm flow\;directs\;away},
\nonumber \\
{\rm\;fixed\;point}:&&\left|\frac{d{\bf M}}{dl}\right|\rightarrow 0,\;{\rm flow\;directs\;into}.
\label{identifying_Mc_Mf}\end{aligned}$$
The TPTs are signalled by the critical points of the flow that form a $(d_{M}-1)$-dimensional surface in the $d_{M}$-dimensional parameter space. The fixed points of the flow are instead related to regions of low-correlation where $\xi_{k_0}$ vanishes. In the following sections, the CRG method will be applied to investigate the TPTs in both the static and the periodically driven Kitaev chain.
Static Kitaev chain {#sec:static_kitaev_chain}
===================
Majorana edge modes
-------------------
We consider the static Kitaev chain described by the following 1D spinless $p$-wave superconducting Hamiltonian, $$\begin{aligned}
\mathcal{H}_{0} &= \sum_{n=1}^{N-1}\left[ t \left( f_n^{\dagger} f_{n+1} + f_{n+1}^{\dagger} f_n \right) + \Delta \left( f_n f_{n+1} + f_{n+1}^{\dagger} f_n^{\dagger} \right)\right] \nonumber \\
& \qquad - \mu_0 \sum_{n=1}^{N} (2 f_n^{\dagger} f_n - 1),
\label{ham-fermionic-static}\end{aligned}$$ where $f_n, f_{n}^{(\dagger)}$ are spinless fermionic creation and annihilation operators, $t$ is the hopping, $\Delta$ is the $p$-wave pairing between spinless fermions, and $\mu_0$ is the static chemical potential [@Kitaev2001]. Note that the fermionic chain can be exactly mapped to a spin-$\frac{1}{2}$ XY chain in a transverse field via a Jordan-Wigner transformation. Throughout this work we will set $t=1$.
The Kitaev chain undergoes a TPT to a topologically nontrivial phase with edge localized Majorana fermions. This can be seen by splitting each fermion into two real Majorana fermions $$w_{2n-1} = f_n + f_n^{\dagger}, \qquad w_{2n} = i(f_n - f_n^{\dagger}),
\label{Majorana_real_space}$$ that satisfy the anti-commutation relations $\left\{w_n, w_m \right\} = 2 \delta_{nm}$ and $w_n = w_n^{\dagger}$, *i.e.* they are their own antiparticle. In this representation, the Hamiltonian reads $$\begin{aligned}
\label{Hamiltonian_Majorana_static}
\mathcal{H}^{\text{M}}_{0} &= i \sum_{n=1}^{N-1} \bigg[ \frac{t - \Delta}{2} w_{2n} w_{2n+1} - \frac{t + \Delta}{2} w_{2n-1} w_{2n+2} \bigg] \nonumber \\
& \quad + i \mu_0 \sum_{n=1}^{N} w_{2n-1} w_{2n} = i \sum_{m,n}^{2N} w_m A_{mn}(t) w_n.\end{aligned}$$ The topologically nontrivial phase with zero-energy edge localized Majoranas appear for $|\frac{\mu_0}{t}| < 1$[@Kitaev2001].
To investigate the topology, it is useful to rewrite the Hamiltonian in in Fourier space $$\begin{aligned}
\mathcal{H}_o &= 2(t - \mu_0) f_0^{\dagger} f_0 + 2(-t - \mu_0) f_{\pi}^{\dagger} f_{\pi} + \nonumber \\
& \quad + \sum_{0<k<\pi} \left( f_{k}^{\dagger} f_{-k} \right) h_k \left( \begin{array}{c} f_k \\ f_{-k}^{\dagger} \end{array} \right)
\label{ham-fermionic-k}\end{aligned}$$ where, $f_{k} = \frac{1}{\sqrt{N}} \sum_{n=1}^N f_n e^{ikn}$ and $$h_k = a_{2,k} \tau^y + a_{3,k} \tau^z = 2\Delta \sin k \tau^y + 2(t \cos k - \mu_0) \tau^z
\label{Dirac_model_static_Kitaev}$$ and $\tau^a$ are the standard Pauli matrices. The BdG-Hamiltonian $h_k$ can be interpreted as a vector in this Pauli space [@Thakurathi] which subtends an angle $\phi_k$ in the $yz$-plane. Integrating this angle variable over the Brillouin zone yields the desired winding number $$W = \frac{1}{\mathrm{Vol}(BZ)} \int_{BZ} \mathrm{d} \phi_k.
\label{winding_number}$$ By mapping out $W$ across the parameter space spanned by ${\bf M}=(\mu_0,\Delta)$, one obtains a topologically nontrivial phase with $W=1$ at $|\mu_0|<|t|$, and a trivial phase with $W=0$ at $|\mu_0|>|t|$. The winding number also equals the number of Majorana edge modes $W=\mathcal{M}$ and hence correctly represents the topological invariant in the static case. Thus the Majorana number jumps by $|\Delta{\cal M}| = 1$ across the TPT.
CRG analysis
------------
To extract the critical behavior around the TPT, we first obtain the curvature function from Eq. (\[winding\_number\]), defined as $$\begin{aligned}
F(k, {\bf M}) &\equiv \frac{\mathrm{d} \phi_k}{\mathrm{d}k}= \frac{a_{2,k} a_{3,k}' - a_{2,k}' a_{3,k}}{a_{3,k}^2 + a_{2,k}^2}
\nonumber \\
&= \frac{\mathrm{d}}{\mathrm{d} k} \arctan \left( \frac{t \cos k - \mu_0}{\Delta \sin k} \right)
\nonumber \\
&= \frac{\Delta ( \mu_0 \cos k - t)}{(t \cos k - \mu_0)^2 + \Delta^2 \sin^2 k}.
\label{Scaling_function_static_Kitaev}\end{aligned}$$ The winding number in Eq. (\[winding\_number\]) is given by the momentum-space integration of the curvature function in Eq. (\[winding\_number\_definition\]). Expanding around the high symmetry points, $k_{0}=0$ or $\pi$, one can verify that the curvature function indeed manifests the Ornstein-Zernike form of Eq. (\[Ornstein\_Zernike\_fit\]). Some plots of $F(k, {\bf M})$ for different values of $\mu_0$ and $t=1$, $\Delta=0.7$ are shown in Fig. \[plots-Berry-conn-static\].
To implement the CRG procedure, we insert Eq. (\[Scaling\_function\_static\_Kitaev\]) into Eq. (\[cond-RG-eqn\]), and expand around the two HSPs $k_{0}=0$ and $k_{0}=\pi$ separately. Choosing the CRG parameter $M=\mu_{0}$ and using Eq. (\[generic\_RG\_equation\]), we obtain the following RG equation: $$\frac{\mathrm{d} \mu_{0}}{\mathrm{d} l} = \pm t + \frac{\mu_{0}}{2} \mp \frac{\Delta^{2}}{t \mp \mu_{0}},
\label{RG_eq_static_mu}$$ where the upper sign is for $k_{0}=0$ and the lower sign is for $k_{0}=\pi$. This RG flow identifies the critical points $\mu_{0} =\pm t$ according to the rules in Eq. . The fixed points of Eq. define ellipses centered at $(\mu_{0}, \Delta) =t (\mp\frac{1}{2}, 0)$: $$\left( \frac{\mu_{0} \pm \frac{1}{2}}{\frac{3}{2}} \right)^2 + \left( \frac{\Delta}{\frac{3}{2\sqrt{2}}} \right)^2 = t^2.
\label{RG_eq_static_ellipses}$$ with upper sign for $k_{0}=0$ and lower sign for $k_{0}=\pi$.
The CRG procedure applied to the pairing gap $\Delta$ leads to $$\frac{\mathrm{d} \Delta}{\mathrm{d} l} = \mp \Delta \frac{\pm \Delta^2 + (t \pm \mu_{0}/2)(\mu_{0} \mp t)}{(t \mp \mu_{0})^2}.
\label{RG_eq_static_Delta}$$ Eq. (\[RG\_eq\_static\_Delta\]) represents the same set of critical points (vertical lines at $\mu_{0}=\pm t$) and fixed points (the two ellipses), as shown in Fig. \[plots-RG-sol-static\]. Note that the critical lines are independent of $\Delta$.
![RG flow of the static Kitaev chain described by Eqs. (\[RG\_eq\_static\_mu\]) and (\[RG\_eq\_static\_Delta\]), using $k_{0}=0$ and $k_{0}=\pi$. The flow direction is indicated by the arrows, and the color scale indicates the flow rate in log scale. The yellow lines are the critical points $\mu_{0}=\pm t$ where the flow rate diverges, with $t=1$ set to be the energy unit. The blue ellipses are the fixed points described by Eq. (\[RG\_eq\_static\_ellipses\]) where the flow rate vanishes, which are stable in some regions and unstable in the other. []{data-label="plots-RG-sol-static"}](RG_flow_static_Kitaev.pdf){width="\columnwidth"}
Criticality - Majorana-Wannier state correlation function \[sec:Majorana\_Wannier\_correlation\]
------------------------------------------------------------------------------------------------
Next, we introduce a correlation function to quantify the proximity to a TPT, according to Eq. (\[lambdaR\_Fourier\_trans\]). Our intuition is based on previous investigations of other 1D non-superconducting systems, such as the Su-Schrieffer-Heeger (SSH) model, in which a Wannier state correlation function based on the Fourier transform of Berry connection is proposed [@Chen:2017]. For an analogous construction for the Kitaev chain, we need to find an appropriate gauge for the gauge-dependent Berry connection, such that we recover the curvature function in Eq. (\[Scaling\_function\_static\_Kitaev\]) and the Ornstein-Zernike form cf. Eq. (\[Ornstein\_Zernike\_fit\]).
Firstly, we observe that the Dirac Hamiltonian in Eq. has a filled-band eigenstate $$\begin{aligned}
|u_{k-}\rangle=\frac{1}{\sqrt{2a_{k}(a_{k}+a_{3,k})}}
\left(
\begin{array}{c}
ia_{2,k} \\
a_{3,k}+a_{k}
\end{array}
\right)\;,
\label{wave_fn_uk}\end{aligned}$$ where $a_{k}=\sqrt{a_{2,k}^{2}+a_{3,k}^{2}}$. However, the corresponding Berry connection is trivial since $A_{k}=\langle u_{k-}|i\partial_{k}|u_{k-}\rangle=0$. To obtain a Berry connection of the form of Eq. (\[Scaling\_function\_static\_Kitaev\]), we transform to the “correct" choice of gauge: $$\begin{aligned}
|\tilde{u}_{k-}\rangle=\frac{1}{\sqrt{2}a_{k}}
\left(
\begin{array}{c}
-a_{k} \\
a_{3,k}+ia_{2,k}
\end{array}
\right)=V_{k}|u_{k-}\rangle\;,
\label{Kitaev_chain_gauge_transformation}\end{aligned}$$ such that the Berry connection of this state is equal to (half of) the curvature function in Eq. (\[Scaling\_function\_static\_Kitaev\]) $$\begin{aligned}
&&\tilde{A}_{k}=\langle \tilde{u}_{k-}|i\partial_{k}|\tilde{u}_{k-}\rangle
\nonumber \\
&&=\frac{a_{2,k}\partial_{k}a_{3,k}-a_{3,k}\partial_{k}a_{2,k}}{2a_{k}^{2}}
=\frac{F(k,{\bf M})}{2}
\nonumber \\
&&=\langle u_{k-}|i\partial_{k}|u_{k-}\rangle+\langle u_{k-}|\left(iV_{k}^{\dag}\partial_{k}V_{k}\right)|u_{k-}\rangle
\nonumber \\
&&=\langle u_{k-}|\left(iV_{k}^{\dag}\partial_{k}V_{k}\right)|u_{k-}\rangle\;.
\label{Berry_connection_gauge_choice}\end{aligned}$$ Interestingly, $|\tilde{u}_{k-}\rangle$ is not an eigenstate of our Hamiltonian in Eq. (\[Dirac\_model\_static\_Kitaev\]), but an eigenstate of $$\begin{aligned}
&&\tilde{h}(k)=Rh(k)R^{-1}=a_{3,k}\tau^{x}+a_{2,k}\tau^{y}\;,
\nonumber \\
&&R=e^{-i\tau_{y}\pi/4}=\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
1 & -1 \\
1 & 1
\end{array}
\right)\;,\end{aligned}$$ *i.e.*, rotating the particle-hole basis such that $a_{3,k}$ becomes the $a_{1,k}$ component. The eigenstate basis of $|\tilde{u}_{k-}\rangle$ is no longer the Nambu spinor $(f_{k},f_{-k}^{\dag})^{T}$, but the rotated spinor: $$\begin{aligned}
R\left(
\begin{array}{c}
f_{k} \\
f_{-k}^{\dag}
\end{array}
\right)=\frac{1}{\sqrt{2}}\left(
\begin{array}{c}
f_{k}-f_{-k}^{\dag} \\
f_{k}+f_{-k}^{\dag}
\end{array}
\right)\;.
\label{Nambu_to_Majorana_rotation}\end{aligned}$$ This new basis has a nice physical interpretation as it is related to the momentum space operator of the real space Majorana fermions in Eq. (\[Majorana\_real\_space\]), $$\begin{aligned}
f_{k}\pm f_{-k}^{\dag}=\sum_{n}e^{-ikr_{n}}\left(f_{n}\pm f_{n}^{\dag}\right)\;.\end{aligned}$$ In summary, the gauge choice that leads to $|\tilde{u}_{k-}\rangle$ consists of (i) a rotation from the Nambu spinor to the Majorana basis, and (ii) a gauge choice imposing an eigenstate specified by Eq. (\[Kitaev\_chain\_gauge\_transformation\]). This ensures that the curvature function in Eq. (\[Scaling\_function\_static\_Kitaev\]) and the Berry connection shown in Eq. (\[Berry\_connection\_gauge\_choice\]) are exactly equivalent.
The Majorana-Wannier states can now be defined in the $|\tilde{u}_{k-}\rangle$ basis $$\begin{aligned}
&&|\tilde{u}_{k-}\rangle=\sum_{R}e^{-ik({\hat r}-R)}|R\rangle\;,
\nonumber \\
&&|R\rangle=\int dk e^{ik({\hat r}-R)}|\tilde{u}_{k-}\rangle\;.
\label{Wannier_state_definition}\end{aligned}$$ This permits a direct transposition of the statement in the theory of charge polarization [@King-Smith:1993; @Resta:1994] to this Majorana problem: in this choice of gauge described in the previous paragraph, the winding number $W$ in Eq. (\[winding\_number\_definition\]) is equal to the charge polarization of the Majorana-Wannier state $$\begin{aligned}
W=\int_{-\pi}^{\pi}\frac{{\rm d}k}{2\pi}\langle\tilde{u}_{k-}|i\partial_{k}|\tilde{u}_{k-}\rangle=\langle 0|{\hat r}|0\rangle\;,
\label{theory_of_charge_polarization}\end{aligned}$$ where $|0\rangle$ denotes the Majorana-Wannier function centered at the home cell $r=0$, and ${\hat r}$ is the position operator.
Correspondingly, the Fourier transform of the curvature function yields a Majorana-Wannier state correlation function [@Chen:2017] which in conjunction with the Ornstein-Zernike form of the curvature function in Eq. (\[Ornstein\_Zernike\_fit\]) yields $$\begin{aligned}
\lambda_{R}&=&\int dk \langle \tilde{u}_{k-}|i\partial_{k}|\tilde{u}_{k-}\rangle e^{ikR}=\langle R|{\hat r}|0\rangle
\nonumber \\
&=&\int dr W^{\ast}(r-R)\; r\;W(r)\propto e^{-R/\xi_{k_0}}\;,
\label{Majorana_Wannier_correlation_static}\end{aligned}$$ where $\xi_{k_0}$ is the correlation length at the relevant HSP. Note that the Majorana-Wannier state correlation function is nonzero in both topologically trivial and nontrivial phases, unlike the Majorana edge states that only appears in the topologically nontrivial phase. Near the critical point $\mu_{0}\rightarrow\pm t$, a straightforward expansion of Eq. (\[Scaling\_function\_static\_Kitaev\]) around the HSP into the Ornstein-Zernike form of Eq. yields $$\begin{aligned}
&&\lim_{\mu_{0}\rightarrow\pm t}\xi_{k_0}=\left|\frac{\Delta}{\mu_{0}\mp t}\right|\;,
\nonumber \\
&&\lim_{\mu_{0}\rightarrow\pm t}F(k_{0},{\bf M})=\pm\frac{\Delta}{\mu_{0}\mp t}\;.
\label{static_Kitaev_xi_Fk0}\end{aligned}$$ In the topologically nontrivial phase, the correlation length $\xi_{k_0}$ coincides with the localization length of the Majorana edge states, as proved explicitly in Appendix \[appendix:decay\_length\_Majorana\_static\]. Comparing the above with Eq. (\[Fk0\_xi\_critical\_exponent\]), we immediately see that the critical exponents defining the Kitaev chain TPT ($|\Delta{\cal M}|=1$) are $\gamma=\nu=1$, compatible with the scaling law imposed by the conservation of the topological invariant within a phase. These critical exponents are the same as those obtained for other TPTs in noninteracting static 1D Dirac models such as the SSH model [@Chen:2017], indicating that all these models belong to the same universality class.
Periodically driven Kitaev chain {#sec:periodically_driven_Kitaev}
================================
Floquet-Majorana fermions
-------------------------
In this section, we extend the CRG approach to periodically driven Kitaev chain [@Thakurathi; @Molignini:2017], which is known to host Floquet-Majorana modes. The Hamiltonian describing the driven Kitaev chain is the same as in Eq. (\[ham-fermionic-static\]) plus a time modulation of the chemical potential: $$\begin{aligned}
\mathcal{H}(t) \equiv \mathcal{H}_{0}[ \mu_0 \to \mu(t)].
\label{ham-fermionic}\end{aligned}$$ For concreteness, we choose the driving to be a sequence of Dirac pulses with period $T$, $\mu(t) = \mu_0 + \mu_1 T \sum_{m \in \mathbb{Z}} \delta(t - m T)$. The momentum space equivalent of Eq. (\[Dirac\_model\_static\_Kitaev\]) is obtained using $h_k(t) = 2\Delta \sin k \tau^y + 2(t \cos k - \mu(t)) \tau^z$.
To study the generation of Majorana modes, we first obtain an effective Floquet Hamiltonian [@Molignini:2017] describing the stroboscopic physics $$h_{\text{eff},k} = i \log U_k(T,0),$$ from the time-ordered evolution operator $$U_k(T,0) \equiv \mathcal{T} \left[ \exp \left( -i \int_0^T h_k(t) \mathrm{d}t \right) \right]$$ since $U_k(NT, (N-1)T) = [U_k(T,0)]^N$ by virtue of the Floquet theorem. The eigenvalues of $h_{\text{eff}}$ define the quasienergies $\epsilon_{\alpha, k}$ of the Floquet-state solutions $\Psi_{\alpha}(k,t) = \exp(-i\epsilon_{\alpha,k} t) \Phi_{\alpha}(k,t)$, where $\Phi_{\alpha}(k,t) = \Phi_{\alpha}(k, t+ T)$ [@Haenggi]. The quasienergies are defined up to a multiple of $\frac{2\pi}{T}=\omega$ because of the $T$-periodicity of the Floquet modes $\Phi_{\alpha}(k,t)$. It is customary to restrict them to take values in a first “[[Floquet-]{}]{}Brillouin zone” of quasienergies $\epsilon_{\alpha} \in (-\omega/2, \omega/2)$.
Akin to the static case, it is possible to identify two kinds of zero-quasienergy Majorana modes in the Floquet spectrum, which are then termed Floquet-Majorana fermions (FMF’s). If $\gamma_{\epsilon}^{\dagger}(t)$ denotes the creation operator of a Floquet mode $\Phi_{\epsilon}(t)$ associated to the quasienergy $\epsilon$ (we drop the index $\alpha$ for simplicity), then particle-hole symmetry implies $\gamma_{\epsilon}(t) = \gamma^{\dagger}_{-\epsilon}(t)$ [@Liu]. Hence, for $\epsilon=0$, we recover a zero-quasienergy Majorana mode as $\gamma_0(t) = \gamma_0^{\dagger}(t)$. However, since quasienergies are defined within $(-\omega/2, \omega/2)$, the same situation can occur for $\epsilon = \omega/2$ with $e^{-i \omega t /2} \gamma_{\omega/2} = \left(e^{-i \omega t /2} \gamma_{\omega/2} \right)^{\dagger}$ [@Liu]. Accordingly, the eigenvalues of $U(T,0) =\prod_k U_k(T,0)$ will then be either $e^{i 0 T} = 1$ or $e^{i \omega/2 T} = e^{i\pi} = -1$ and the corresponding FMF’s are labeled as $0$-FMF’s or $\pi$-FMF’s.
FMF’s have characteristics similar to their static counterparts: they have zero quasienergy (modulo $\omega/2$), have real wavefunctions (stemming from $\gamma = \gamma^{\dagger}$) and are localized near the ends of the chain [@Thakurathi]. However, in contrast to the static case, it is possible to generate a hierarchy of FMF’s by simply tuning the system’s parameters over a wide range. The system can hence be made topological even when the undriven phase has trivial topology and belongs to the $\mathbb{Z}$ class, in contrast to the simpler $\mathbb{Z}_2$ categorization of the static system [@Thakurathi; @Molignini:2017].
For the $\delta$-driving considered here, the Floquet operator in $k$-space takes the form $$U_k(T,0) = \left( \begin{array}{cc} A & B \\ -B & A^{*} \end{array} \right)$$ where $A=e^{2i \mu_1 T} \left( \cosh \omega(k) - i\frac{\beta(k)}{\omega(k)} \sinh \omega(k) \right)$, $B=-\frac{\alpha(k)}{\omega(k)} \sinh \omega(k)$, $\alpha(k) = 2T \Delta \sin k$, $\beta(k) = 2T(t \cos k - \mu_0)$ and $\omega^2(k) = \beta^2(k) - \alpha^2(k)$. Diagonalizing $U_k(T,0)$, we obtain the explicit form of the effective Hamiltonian as $$h_{\mathrm{eff},k} = \frac{2\log \lambda^{-}}{\lambda^{+} - \lambda^{-}} \left[ \tilde{a}_{2,k} \tau^y + \tilde{a}_{3,k} \tau^z \right]
\label{heff_definition}$$ where $\tilde{a}_{2,k} = B$ and $\tilde{a}_{3,k} = \Im[A]$ and $\lambda^{\pm}= \Re[A] \pm \sqrt{ \Re[A]^2 - (|A|^2 + |B|^2)}$ are the eigenvalues of $U_k(T,0)$.
Based on the similarity of $h_{\text{eff},k}$ to the static $h_k$, one could generalize the definition of the winding number introduced in Eq. to the driven case. Nevertheless, it has been recently shown that this construction fails to correctly count the number of FMF’s in certain driving regimes [@Thakurathi; @Rudner:2013]. An alternative method, obviating the calculation of micromotion, was proposed in Ref. [@Thakurathi] for the one dimensional case. It defines a finite line segment between the two points (modified according to our definition of the Hamiltonian) $$\begin{aligned}
b_0(T, \mu_1)=\frac{2T}{\pi}\left(\mu_{0}-t+\mu_{1}\right)\;,
\nonumber \\
b_\pi(T, \mu_1)=\frac{2T}{\pi}\left(\mu_{0}+t+\mu_{1}\right)\;,\end{aligned}$$ corresponding to the cases when the Floquet evolution operator $U_k(T,0) = \pm \mathds{1}$, which are realized at the HSPs $k=0$ and $k=\pi$ (hence the labelling). The topological invariant is then constructed from a non-trivial counting of the odd and even integers smaller/bigger than a certain threshold and encompassed by the segment, and can be written as $ \mathcal{M} = N_{0} + N_{\pi}$, where $N_{0(\pi)}$ counts the number of FMF’s with Floquet eigenvalue $\pm 1$ [@Thakurathi].
By computing both $N_0$ and $N_{\pi}$, we map out the $(T,\mu_1)$-phase diagram of the inequivalent topological regions of the effective Floquet Hamiltonian. Fig. \[top-invariants1\] depicts such a phase diagram and clearly illustrates a mismatch with the one obtained from the conventional winding number calculations. Generally, the static parameters $\Delta$ and $\mu_0$ strongly influence the phase diagrams. The total number of FMF’s per edge $\mathcal{M} \in \mathbb{Z}$ as opposed to the static case where $\mathcal{M} \in \mathbb{Z}_2$ and can change by an integer across the boundaries (see Fig. \[top-invariants1\]). Typically, from the phase diagram Fig. \[top-invariants1\](c), we note that across the TPT $|\Delta{\cal M}| =1$. Sometimes anomalous transition regions exist at higher periods where $\Delta\mathcal{M}=2$. In what follows, we primarily focus on TPTs with $|\Delta{\cal M}| =1$ and discuss the anomalous lines with $|\Delta{\cal M}| = 2$ in section \[sec:frozen-dynamics\].
![Illustration of the number of FMF’s of the driven Kitaev chain for $\mu_0=0.1$, plotted as a function of the driving parameters $T$ and $\mu_1$. The driving was performed starting from a static topological region. Note that the topological phase diagrams are independent of $\Delta$. (a) The FMF’s with Floquet eigenvalues $+1$. (b) The FMF’s with Floquet eigenvalues $-1$. (c) The phase diagram of the system according to the total number of FMF’s for each phase. (d) The winding number $W$ stemming from the Berry connection. One sees that the winding number $W$ does not fully coincide with the correct number of FMF’s.[]{data-label="top-invariants1"}](top-indices2.pdf){width="0.99\columnwidth"}
To understand the TPTs, it is instructive to analyze the Floquet quasienergy dispersion, or equivalently the eigenvalues $e^{i \theta_k}$ of the Floquet operator $U_k$. They can be compactly written as [@Molignini:2017] $$\begin{aligned}
\cos \theta_k &= \cos(2\mu_1 T) \cos(T E_{k}) + \nonumber \\
& \quad + \sin(2\mu_1 T) \frac{2(t \cos k - \mu_0)}{E_{k}} \sin(T E_{k})
\label{quasienergy_delta_func_driving}\end{aligned}$$ with the static energy dispersion $$E_{k} = 2 \sqrt{(t \cos(k) - \mu_0)^2 + \Delta^2 \sin^2(k)}.$$ In accordance with the bulk-edge correspondence [@Kitaev-table], TPT’s should be signalled by a closing of the gap in the quasienergy spectrum. We note that instances of TPTs not associated with gap closing have been discovered in systems where the symmetry of the Hamiltonian changes across the topological phase boundary [@Ezawa:2013]. Our results for the quasienergy spectra for long but finite Kitaev chains are shown in Fig. \[quasienergy-spectra\]. Note that there are gap closings at $0$ or $\pi$ quasienergies, reflecting the creation or annihilation of $0,\pi$-Majorana modes.
Additionally, a systematic analysis of the bulk quasienergy dispersion $\theta_k$ for different values of the static parameters $\Delta$ and $\mu_0$, reveals gap closures with linear dispersions around $\theta_{k}=0$ and the zone edge ($\theta_{k}=\pi$) whenever a new $0$-FMF ($\pi$-FMF) is generated or destroyed. These gap closures related to TPTs, specifically appear at HSP $k=0$ and $k=\pi$ in analogy with the static case. This behaviour, shown in Fig. \[gap-closings-quasienergy\](a), is in agreement with expectations for the universality class $|\Delta{\cal M}|=1$.
![Depiction of quasienergy spectra — *i.e.* eigenvalues of the Floquet operator — as function of the driving intensity $\mu_1$ for an open chain of $N=100$ fermions. The other system’s parameters are chosen as follows: a) $\mu_0=0.1$, $\Delta=0.1$, $T=1.0$, b) $\mu_0=0.5$, $\Delta=0.9$, $T=1.9$. The TPT’s generating or removing $0$-FMF’s ($\pi$-FMF’s) are marked by gap closings at $0$ ($\pi$) with the corresponding appearance or disappearance of eigenvalues at $0$ ($\pi$). Note that there are instances of gap closings, marked by red circles, not associated with a change in the topological invariants. Those phase transitions appear in observables such as correlators and are detected by the CRG scheme.[]{data-label="quasienergy-spectra"}](quasienergy-spectra2.pdf){width="\columnwidth"}
![ (Top) Gap-closing manifests in the quasienergy dispersion $\theta_k$, and (bottom) the corresponding divergence of the curvature function $F(k,{\bf M})$ in the periodically driven Kitaev chain: (a) At $\Delta=0.5$, $\mu_0=0.1$, $\mu_1=0.9$, $T=2.0$, which is the critical point of gap-closing at $k_{0}=0$ and creating a $0$-FMF. The inset of the quasienergy plot shows that the gap-closing at $k_{0}=0$ is in fact linear at low energy, although it looks quadratic at larger scale. (b) At $\Delta=0.1$, $\mu_0=0.1$, $\mu_1=0.78$, $T=2.0$, where the gap closes at non-HSPs (see also Fig. \[figure:backfolding\]). []{data-label="gap-closings-quasienergy"}](gap-closings-quasienergy_3.pdf){width="0.99\columnwidth"}
Note that gap closures can also occur in $\theta_k$ at *non*-HSP (see Fig. \[gap-closings-quasienergy\](b)). These features at non-HSP are visible in the quasienergy spectra but are *not* systematically associated with a change in the $N_0 + N_{\pi}$ (see Fig. \[top-invariants1\]d)). We will show in the next section that the CRG scheme is capable of capturing the physics of both topological and nontopological gap closures at HSP and non HSP.
RG flow for the Floquet effective Hamiltonian of the driven Kitaev chain
------------------------------------------------------------------------
In the previous sections, we saw that driving can indeed generate a hierarchy of Floquet-Majorana modes $\mathcal M$ with TPTs between zones with differing $\mathcal M$. We now demonstrate that this complex phase diagram — cf. Fig. \[top-invariants1\] — can be obtained by the CRG procedure outlined in Sec. \[sec:static\_kitaev\_chain\]. As in the static case, the effective Floquet Hamiltonian $h_{\text{eff}}$ in Eq. defines an angle function $\phi_k$ that represents the angle that $h_{\text{eff}}$ spans in the $yz$-plane. The curvature function $F(k,{\bf M}) \equiv \frac{\mathrm{d}\phi_k}{\mathrm{d} k}$ is calculated from the angle function as
$$\begin{aligned}
F(k,{\bf M}) &= \frac{\mathrm{d}}{\mathrm{d}k} \arctan \left( \frac{\tilde{a}_{3,k}}{\tilde{a}_{2,k}} \right) = \frac{\mathrm{d}}{\mathrm{d}k} \arctan \left[ \frac{ \cos(2\mu_1T) (t \cos (k) - \mu_0) \sin(T E_{k}) - \frac{\sin(2\mu_1 T)}{2} \cos(T E_{k}) E_{k}}{\Delta \sin (k) \sin(T E_{k})} \right].
\label{Scaling_function_deriven_Kitaev}\end{aligned}$$
Sample curvature functions are shown in the [ the bottom panels of]{} Fig. \[gap-closings-quasienergy\].
![ (a) The topological phase boundaries in the ${\bf M}=(T,\mu_1)$ parameter space for the periodically driven Kitaev chain at $\mu_{0}=0.1$. White lines signal the creation of a $0$-FMF and black lines the creation of a $\pi$-FMF. (b) The magnitude of the residual correlator $C_{\text{res}}$ in the same parameter space, taken from Ref. [@Molignini:2017]. (c) The RG flow obtained from $k_{0}=0$. The color codes are log of the numerator $\log \left[\partial_{k}^{2}F(k,{\bf M})|_{k=0}\right]$ in Eq. (\[generic\_RG\_equation\]), with orange the high value and blue the low value. The bright lines (critical points of $0$-FMF) coincide with the white lines in (a), and blue lines (stable or unstable fixed points) are close to the white lines in (b) (minimum of $C_{\text{res}}$). (d) The RG flow obtained from $k_{0}=\pi$, whose bright lines (critical points of $\pi$-FMF) correspond to the black lines in (a)[]{data-label="figure:RG-comparison"}](RG-comparison6.pdf){width="0.99\columnwidth"}
In the periodically driven case, the parameters that define the CRG flow are ${\bf M} = (T, \mu_1)$, *i.e.* the period and intensity of the driving. As for the static case, we use Eq. to obtain the flow equations for the system in the ${\bf M}=(T,\mu_{1})$ parameter space. However, since the analytical expressions of the derivatives of the curvature function are cumbersome, we numerically evaluate them on a discrete mesh of points: $$\begin{aligned}
\frac{dM_{i}}{dl}=\frac{\Delta M_{i}}{\Delta k^{2}}\frac{F(k_{0}+\Delta k,{\bf M})-F(k_{0},{\bf M})}{F(k_{0},{\bf M}+\Delta {\bf M}_{i})-F(k_{0},{\bf M})}\;,
\label{RG_eq_numerical}\end{aligned}$$ where $\Delta k$, $\Delta{\bf M}=(\Delta T,\Delta\mu_{1})$ are grid spacings. The advantage of Eq. (\[RG\_eq\_numerical\]) is that at each mesh point of the ${\bf M}=(T,\mu_{1})$ parameter space, one only requires to calculate two points in the momentum space $k_{0}$ and $k_{0}+\Delta k$ without explicitly performing the integration of winding number in Eq. (\[winding\_number\_definition\]), rendering a very convenient numerical tool to identify TPT. The resulting RG flows for both HSP $k=0$, $k=\pi$ are shown in Figs. \[figure:RG-comparison\](c) and (d) for $\mu_0=0.1$. Using the criteria outlined in Eq. , we distinguish the set of critical and fixed points as the bright lines of maximal flow and the dark lines of zero flow, respectively.
### Critical Lines
Comparing with the phase diagram obtained Fig. \[top-invariants1\](c), we see that the critical lines of the CRG method correctly capture the phase boundaries where $\Delta\mathcal{M}=1$. The CRG scheme is also able to track which type of FMFs are created or annihilated at the critical boundaries. A direct comparison of the flow diagrams with Fig. \[figure:RG-comparison\](a) and (b) reveals that the phase boundaries where the number of 0($\pi)$-FMF’s $N_{0}$($N_\pi$) changes are delineated by the critical lines of the CRG flow around the HSP $k_0=0(\pi)$, respectively. This correlation holds true also for all ranges of the parameters $\mu_0,\mu_1$ and $T$.
Analytical expressions for the critical flow lines where $N_0$ or $N_{\pi}$ changes by one can also be obtained by analyzing the divergences of $\lim_{k \to 0, \pi} F(k, T, \mu_1)$ as a function of $T, \mu_1$. The TPTs occur at the boundaries defined by the simple equations: $$\begin{aligned}
\label{anal}
0-\text{FMF}: \quad \mu_1(T) &= \frac{m_0 \pi}{2T} + (t - \mu_0), \quad m_0 \in \mathbb{Z} \nonumber \\
\pi-\text{FMF}: \quad \mu_1(T) &= \frac{m_{\pi} \pi}{2T} - (t + \mu_0), \quad m_{\pi} \in \mathbb{Z},\end{aligned}$$ which agrees with the boundaries delineated by $\mathcal M$ in Fig. \[figure:RG-comparison\](a) and the residual correlator in Fig. \[figure:RG-comparison\](a)-(b). We note that there are fixed lines ${\bf M}_{f}$ in close proximity to some dominant critical lines ${\bf M}_{c}$( not visible in the flow) and hence subsumed by the criticality. To distinguish ${\bf M}_{c}$ from ${\bf M}_{f}$, very fine spacings of $\Delta k$, $\Delta T$ and $\Delta\mu_{1}$ must be used, as demonstrated in Fig. \[fig:driven\_RGflow\_detail\] (a). A striking feature of is that as in the static case, the critical lines and hence the topological phase diagram in the Floquet case are independent of the anisotropy/p-wave parameter $\Delta$. This is a non-trivial prediction which can easily be verified using the quasienergy spectra.
Signatures of these transitions in a driven-dissipative setup were recently studied in [@Molignini:2017; @Prosen2011]. It was shown that the weighted sum of the Majorana correlator $C_{ij}(t) =\left<\omega_i \omega_j\right> - \delta_{ij}$ was capable of delineating the boundaries between different topological phases. This sum, termed the *residual correlator*, is defined as $C_{\text{res}} \propto \sum_{|j-k| \ge N/2} |C_{jk}|$ [@Prosen2011; @Molignini:2017] and effectively filters out short-range correlations and offers a measure of long-range correlations. A typical plot of $C_{\text{res}}$ from Ref. [@Molignini:2017] is shown in Fig. \[figure:RG-comparison\]. Clearly, the residual correlator is able to precisely track the critical topological phase boundaries. Note that lines of pronounced low correlation are seen in $C_{\text{res}}$ which are unrelated to the critical lines. We will show in Sec. \[sec:frozen-dynamics\] that these low correlation lines are related to fixed lines of the CRG and frozen dynamics.
![ (a) The CRG flow along $T$-direction at fixed $\mu_{1}=0.1$ close to the critical point ${\bf M}_{c}=(T_{c},\mu_{1c})\approx(1.9635,0.1)$. The flow has been obtained from $k_0 =0$ by numerically evaluating the derivatives with a very find grid $\Delta k=0.0001$ and $\Delta T=0.00001$. One sees that the critical point $\mu_{1c}\approx 1.9635$ at which $dT/dl$ diverges and the fixed point $\mu_{1f}\approx 1.9637$ are extremely close. (b) The inverse of the Wannier state correlation length $\xi_{k_0}^{-1}$ and that of the curvature function at HSP $F(k_{0},T,\mu_{1})^{-1}$, both vanish linearly as $T\rightarrow T_{c}$, indicating their critical exponents $\gamma=\nu=1$.[]{data-label="fig:driven_RGflow_detail"}](driven_RGflow_detail.pdf){width="\columnwidth"}
### Wannier state correlation functions and critical exponents
To characterize the TPTs for the Floquet chain, we derive the equivalent Majorana-Wannier state correlation function. We follow the same procedure outlined for the static case in Sec. \[sec:Majorana\_Wannier\_correlation\]. Let $|u_{k-}\rangle$ denote the lowest eigenstate of $h_{\text{eff},k}$. It has the form of Eq. (\[wave\_fn\_uk\]) and a vanishing Berry connection $A_{k}=\langle u_{k-}|i\partial_{k}|u_{k-}\rangle=0$. We use the gauge transformation Eq. (\[Kitaev\_chain\_gauge\_transformation\]) to obtain $\tilde{h}_{\text{eff},k}=Rh_{\text{eff},k}R^{-1}=\tilde{a}_{3,k}\tau^{x}+\tilde{a}_{2,k}\tau^{y}$. The eigenstate of $\tilde{h}_{\text{eff},k}$, $|\tilde{u}_{k-}\rangle$ is again expressible in the basis of Floquet-Majorana fermions as in Eq. (\[Nambu\_to\_Majorana\_rotation\]). The nonvanishing Berry connection of Eq. (\[Scaling\_function\_deriven\_Kitaev\]) for the driven case is now given by $\tilde{A}_{k}=\langle \tilde{u}_{k-}|i\partial_{k}|\tilde{u}_{k-}\rangle=F(k,{\bf M})/2$. The Majorana-Wannier state $|R\rangle$ and the corresponding Majorana-Wannier state correlation function can then be defined using Eqs. (\[Wannier\_state\_definition\]) and (\[Majorana\_Wannier\_correlation\_static\]). $|R\rangle$ is now a time-independent *stroboscopic* function encoding the physics in $h_{\text{eff}}$.
To obtain the critical exponents of the Floquet TPTs, we compute the Majorana-Wannier state correlation length. We find that $$\begin{aligned}
\xi_{k_0} \approx\left|\frac{(\partial_{k}\tilde{a}_{2,k})_{k_{0}}}{\tilde{a}_{3,k_{0}}}\right|
=\left|F(k_{0},{\bf M})\right|\;,
\label{driven_xi_Fk0_bk}\end{aligned}$$ implying that the critical exponents of the TPT $\gamma=\nu$. The detailed calculations are performed in Appendix \[appendix:decay\_length\_Majorana\_driven\]. Similar to the static case, there are two correlation lengths $\xi_0$ and $\xi_{\pi}$ depending on which HSP is considered. At the TPTs, only the $\xi_{k_0}$ associated with the HSP $k_0$ at which the gap closes in the quasienergy spectrum will diverge. Close to the TPT driven by a control parameter $M_i$ as calculated in Appendix \[appendix:decay\_length\_Majorana\_driven\], $$\begin{aligned}
\xi_{k_0} \propto \frac1 {\vert M_i -M_{ic}\vert}
\label{driven_xi_Mi}\end{aligned}$$ where $M_{ic}$ is the critical value of the parameter. This demonstrates that the critical exponents of the Floquet TPT are $\nu=\gamma=1$. This result is further bolstered by the numerical extraction of the exponents from the Ornstein-Zernike fit to the curvature function $F(k,{\bf M})$ in Eq. (\[Ornstein\_Zernike\_fit\]). From Fig. \[fig:driven\_RGflow\_detail\] (b), we see that $\xi_{k_0}^{-1}$ is linear in $M_i -M_{ic}$ only in a very narrow range near ${\bf M}_{c}$, indicating that the critical region is in general very small.
In Appendix \[appendix:decay\_length\_Majorana\_driven\], we show that the Floquet Majorana edge state [*being created at the TPTs*]{} has a decay length that coincides with the Majorana-Wannier state correlation length. Across each critical boundary in Fig. \[figure:RG-comparison\] [*only one Majorana edge state is created*]{}, *i.e.* $|\Delta{\cal M}|=1$, and this state determines the critical behavior. The change in Majorana number $|\Delta{\cal M}|=1$ in conjunction with the critical exponents $\nu=\gamma=1$ means that the driven system still belongs to the same universality class as the static Majorana chain, despite the intrinsically more complex phase diagram of the Floquet Majorana chain cf. Fig. \[figure:RG-comparison\].
### Fixed lines and frozen dynamics {#sec:frozen-dynamics}
The fixed lines of the CRG resolve the puzzle of low correlations in $C_{\text{res}}$ discussed earlier. Specifically, comparing Figs. \[figure:RG-comparison\](b), (c) and (d) we see that the combined fixed lines of the flow ${\bf M}_{f}$ for $k_0=0$ and $k_0=\pi$ precisely encompass the regions of reduced correlations in $C_{\text{res}}$. This correspondence is sound, because the fixed lines represent points where the Majorana-Wannier state correlation length vanishes (see next section).
Insight into the nature of the fixed lines (FL) can be obtained by examining the behavior of the Berry connection as one traverses these lines. We find that the Berry connection shows divergences at non HSPs which then flip sign across the FL if $\mathcal{M} > 1$. When $\cal{M} \le 1$ no divergence occurs like along the fixed lines in the static case.
![Comparison between the topological invariant $\mathcal{M}=N_{0} + N_{\pi}$ and the residual correlator $C_{res}$ for the $(T,\mu_1)$-phase diagram of the driven Kitaev chain with $\Delta=0.1$, $\mu_0=0.5$. The topological invariant seems to indicate additional phase boundaries at higher periods, where the number of FMF’s jumps by two. These additional phase boundaries coincide with the fixed lines of the CRG flow and the lines of low correlation in $C_{res}$.[]{data-label="top-invariants2"}](top-indices-mu0-0-5.pdf){width="0.99\columnwidth"}
![Plot of the Berry connection $F(k, {\bf M})$ across the FL for $\Delta=0.1$, $\mu_0=0.1$, and at a) $\mu_1=1.5$ and b) $\mu_1=1.0$. Panel a) shows the Berry connection for a region with one FMF (see also Fig. \[top-invariants1\]). The same number of positive and negative peaks flips sign. Hence, the topological index $W=-1$ is unchanged during this transition. Panel b) shows instead a region with two FMF’s, while the topological index $W$ changes from $0$ to $+2$. Note that the smaller peaks centered around $k \approx 1.4$ have the same weight (area under the peak) as the diverging peaks that flip sign. []{data-label="figure:peak-flip-DeltaW"}](BC-flips-wider2.pdf){width="\columnwidth"}
Remarkably, the CRG approach based on the expansion around HSPs captures very well the divergences at non-HSPs. This can be attributed to the conservation of the winding number $W$ within a phase, such that when the divergence at non-HSP occurs, the curvature function at HSP will be affected. We now discuss the possible physical mechanism leading to the FL and how to obtain an analytical forms for their equations. A closer inspection of the quasienergy dispersion reveals that, for $\mu_1(T) = \frac{\pi m}{2T}$ with $m \in \mathbb{Z}$, the second summand vanishes because of the sine being zero. We are thus left with [@Prosen2011] $$\label{theta}
\theta_k= \begin{cases}
T E_{k}, \quad &m \in 2\mathbb{Z} \\
\pi - T E_{k}, \quad &m \in 2\mathbb{Z}+1.
\end{cases}$$ The lines described by the form of $\mu_1(T)$ above correspond to the location of the FL. Consequently, along FLs the driving is momentarily frozen and the quasienergy dispersion is the static energy dispersion $E_{k}$, where the relevant topology can host at most one Majorana mode per edge. This also explains why gap closings appear at non-HSPs: along the FL, for $k$ values where $\vert E_{k} \vert > \pi$, the constraint $-\pi\le \theta_k \le \pi$ requires that $|E_{k}|$ to be folded back to the first Floquet-Brillouin zone, as illustrated in Fig. \[figure:backfolding\]. The nodes in the folding, which occur at non-HSPs, reminisce gapless points. It remains to be seen if the peculiar behavior of the FL associated with frozen dynamics is a feature of the type of driving applied, or if it can be extended to other driving protocols such a two- or multistep-driving.
A natural question is whether the FLs signal additional TPTs. Comparing Figs. \[top-invariants2\](a) and \[top-invariants2\](b), we clearly see that the FLs are not systematically associated with a change in the number of FMF’s given by $\mathcal{M} = N_{0} + N_{\pi}$. However, there exist regions, where the FLs signal a change in FMFs with $\Delta\mathcal{M}=2$. An example is the yellow regions in Fig. \[top-invariants2\](a) where $\mathcal{M}=3$ jumps to $\mathcal{M}=1$ . Such $|\Delta{\cal M}|=2$ jumps are also seen for other parameters. We have verified that these jumps are indeed systematically associated with gap closings at $0$ or $\pm \pi$ in the Floquet quasienergy spectrum for both open and closed systems with $N=2000$. The corresponding wave functions are real and show edge localization for open chains (for details see Appendix \[appendix:figures\_2Maj\_jumps\]). These gap closures indicate that sometimes FLs signal transitions where $\Delta\mathcal{M} \neq \pm 1$ (see Fig. \[gap-closings-quasienergy\] and \[quasienergy-spectra\]). It follows that these do not belong to the universality class of the TPTs where $\Delta\mathcal{M} =\pm 1$.
These intriguing features merit further study to verify if indeed a driven system can host multiple universality classes or if they are in anyway related to the anomalous topological phases discussed in driven two dimensional systems[@Rudner:2013], where new FMF’s modes are indeed created but the *difference* of the FMF’s $|N_0 - N_{\pi}|$ remains unchanged (*e.g.* $N_0=1, N_{\pi}=0 \to N_0 =1, N_{\pi}=2$).
![Backfolding of the upper band of (a) the static energy dispersion into (b) the first Floquet-Brillouin zone corresponding to the LCFLs where the dynamics is frozen. This backfolding procedure induces apparent gap closings at non-HSPs. The parameters are $\Delta=0.1$, $\mu_0=0.1$, $\mu_1=1.26$ and $T=2.5$.[]{data-label="figure:backfolding"}](backfolding-quasienergy.pdf){width="\columnwidth"}
Conclusions {#sec:conclusions}
===========
In summary, we have applied the CRG approach to study TPTs in static and periodically driven Kitaev chains. The [method]{}, though insensitive to micromotion, provides a simple and efficient way to obtain the full topological phase diagram of the driven system. Comparing our results with quasienergy spectra calculations, as well as exact Majorana correlation functions, we find that the CRG scheme correctly captures topological phase boundaries. The critical points of the CRG flow correspond to TPTs where the number of localized edge Majorana modes changes by one. Extending the notions of charge polarization and Majorana-Wannier states to the effective Floquet-Bloch eigenstates, we show that the TPTs in the driven case are signalled by a divergence of the correlation length of the Majorana-Wannier state correlation function. A calculation of the critical exponents reveals that TPTs in both the static and periodically driven chain belong to the same universality class.
The fixed lines of the CRG flow, on the other hand, reflect the frozen dynamics of the system, where the quasienergy dispersion maps back to the static dispersion. The fixed lines provide an explanation for previously unexplained features seen in Majorana correlations in open systems. Surprisingly, some of the FLs indicate new topological instabilities where the total number of $0-$ and $\pi-$FMF’s changes by $2$. Because of the simultaneous appearance of pairs of $0$ or $\pi$-FMF’s, these transitions are speculated to be of the anomalous kind, across which the difference $|N_0 - N_{\pi}|$ stays constant.
An interesting question awaiting exploration is whether, for such intriguing systems, the CRG methodology captures the complexity of the topology stemming from the underlying micromotion. In particular, given the preliminary results indicating additional transitions where $|\Delta{\cal M}| = 2$, it would be intriguing to apply the CRG method to 2D models that are known to host anomalous topological phases [@Rudner:2013; @Tauber:2018].
We anticipate that our CRG method may be broadly applied to investigate TPTs and universality classes in Floquet systems subject to other types of periodic driving, such as square waves or multistep driving, or models defined in higher spatial dimensions. Furthermore, the discretized RG equation, Eq. (\[RG\_eq\_numerical\]), offers a very efficient numerical tool to identify TPTs especially in driven higher dimensional systems.
Acknowledgments
===============
We kindly acknowledge financial support by Giulio Anderheggen and the ETH Zürich Foundation. The authors would like to thank Manisha Thakurathi, Aline Ramires, Luca Papariello, Ivo Maceira, and Clément Tauber for fruitful discussions.
Decay length of the Majorana edge state close to the static TPTs {#appendix:decay_length_Majorana_static}
================================================================
To show that the correlation length of the Wannier state correlation function coincides with the decay length of the Majorana edge state in the static Kitaev chain, we address the edge state explicitly. We first consider the edge state when the system is about to have a gap-closing at an HSP $k_{0}=\left\{0,\pi\right\}$. We aim at solving for the zero energy edge state satisfying $({\bf a}_{k}\cdot{\boldsymbol\tau})\psi(x)=0$ for a Hamiltonian defined in the positive half-space $x\geq 0$. Expanding the Hamiltonian around $k_{0}$ to leading order and project the Hamiltonian into real space by $k=-i\partial_{x}$ gives $$\begin{aligned}
&&a_{2,k_{0}+k}=k\left(\partial_{k}a_{2,k}\right)_{k_{0}}=-i\left(\partial_{k}a_{2,k}\right)_{k_{0}}\partial_{x}\;,
\nonumber \\
&&a_{3,k_{0}+k}=a_{3,k_{0}}+\frac{k^{2}}{2}\left(\partial_{k}^{2}a_{3,k}\right)_{k_{0}}
\nonumber \\
&&=a_{3,k_{0}}-\frac{1}{2}\left(\partial_{k}^{2}a_{3,k}\right)_{k_{0}}\partial_{x}^{2}
\;,
\nonumber \\
&&\left\{-i\left(\partial_{k}a_{2,k}\right)_{k_{0}}\tau^{y}\partial_{x}\right.
\nonumber \\
&&\left.+\left[a_{3,k_{0}}-\frac{1}{2}\left(\partial_{k}^{2}a_{3,k}\right)_{k_{0}}\partial_{x}^{2}\right]\tau^{z}\right\}\psi(x)=0\;,
\nonumber \\
\label{edge_state_h_expansion}\end{aligned}$$ where $\left(\partial_{k}a_{2,k}\right)_{k_{0}}$ denotes $\partial_{k}a_{2,k}$ evaluated at $k=k_{0}$. Multiplying the equation by $\tau^{y}$, we see that the edge state is an eigenstate of $\tau^{x}$, with ansatz $\psi=\chi_{\eta}\phi(x)\propto\chi_{\eta}e^{-x/\xi_{k_0}}$, where $\tau^{x}\chi_{\eta}=\eta\chi_{\eta}=\pm\chi_{\eta}$.
The solution for the decay length is $$\begin{aligned}
\xi_{k_0, \pm}^{-1}&=&\frac{(\partial_{k}a_{2,k})_{k_{0}}}{\eta (\partial_{k}^{2}a_{3,k})_{k_{0}}}
\nonumber \\
&&\pm \frac{|(\partial_{k}a_{2,k})_{k_{0}}|}{|\eta| (\partial_{k}^{2}a_{3,k})_{k_{0}}}
\sqrt{1+2\eta^2 \frac{a_{3,k}(\partial_{k}^{2}a_{3,k})_{k_{0}}}{(\partial_{k}a_{2,k})_{k_{0}}^{2}}}.\;\;\;\end{aligned}$$ Demanding the sum of the two $\xi_{k_0,+}^{-1}+\xi_{k_0, -}^{-1}=2\eta(\partial_{k}a_{2,k})_{k_{0}}/(\partial_{k}^{2}a_{3,k})_{k_{0}}>0$ yields $\eta={\rm Sgn}\left[(\partial_{k}a_{2,k})_{k_{0}}/(\partial_{k}^{2}a_{3,k})_{k_{0}}\right]$. There can be two cases: (a) If $(\partial_{k}^{2}a_{3,k})_{k_{0}}>0$, then $\eta={\rm Sgn}\left[(\partial_{k}a_{2,k})_{k_{0}}\right]$ and hence the longer one $$\begin{aligned}
\xi_{k_0,-}=-\frac{|(\partial_{k}a_{2,k})_{k_{0}}|}{a_{3,k_{0}}}\end{aligned}$$ is identified as the decay length, where we have expanded the square root to obtain this solution. The decay length must be positive, and $a_{3,k_{0}}<0$ must be satisfied in order for the edge state to exist. (b) If $(\partial_{k}^{2}a_{3,k})_{k_{0}}<0$, then $\eta=-{\rm Sgn}\left[(\partial_{k}a_{2,k})_{k_{0}}\right]$ and hence the longer one $$\begin{aligned}
\xi_{k_0,+}=\frac{|(\partial_{k}a_{2,k})_{k_{0}}|}{a_{3,k_{0}}}\end{aligned}$$ is the decay length. Demanding it to be positive, one sees that only when $a_{3,k_{0}}>0$ does the edge state exist. In summary, across the TPT $(\partial_{k}^{2}a_{3,k})_{k_{0}}a_{3,k_{0}}$ changes sign, and the edge state exists in the phase that has $$\begin{aligned}
(\partial_{k}^{2}a_{3,k})_{k_{0}}a_{3,k_{0}}<0\;,\end{aligned}$$ with a decay length $$\begin{aligned}
\xi_{k_0}=\left|\frac{(\partial_{k}a_{2,k})_{k_{0}}}{a_{3,k_{0}}}\right|\;.
\label{static_decay_length_general}\end{aligned}$$ In the static Kitaev chain, using Eq. (\[Dirac\_model\_static\_Kitaev\]) and $k_{0}=\left\{0,\pi\right\}$ yields $$\begin{aligned}
&&k_{0}=0:\;{\rm exits\;when}\;\mu_{0}<t\;{\rm and}\;\xi_{0}=\left|\frac{\Delta}{t-\mu_0}\right|\;,
\nonumber \\
&&k_{0}=\pi:\;{\rm exits\;when}\;\mu_{0}>-t\;{\rm and}\;\xi_{\pi}=\left|\frac{\Delta}{t+\mu_0}\right|\;.
\nonumber \\\end{aligned}$$ If $\eta=1$, then one may choose the spinor of the edge state to be $\chi_{\eta}=(1,1)^{T}/\sqrt{2}$, meaning that the edge state annihilation operator $\psi=\left(f+f^{\dag}\right)/\sqrt{2}=\psi^{\dag}$ is its own creation operator. Likewisely, if $\eta=-1$, then one may choose $\chi_{\eta}=(i,-i)^{T}/\sqrt{2}$ and hence the edge state annihilation operator $\psi=\left(if-if^{\dag}\right)/\sqrt{2}=\psi^{\dag}$ is again its own creation operator. Thus the annihilation operator of the edge state is a Majorana fermion. Comparing the decay length $\xi_{k_0}$ with Eq. (\[static\_Kitaev\_xi\_Fk0\]), it is evident that the decay length of the Majorana edge state that appears in the open boundary condition coincides with the correlation length $\xi$ of the Wannier state correlation function defined in the closed boundary condition.
Decay length of the Floquet-Majorana edge state close to the driven TPTs {#appendix:decay_length_Majorana_driven}
========================================================================
The same analysis is also applicable to the periodically driven case, in which case we look for the localized zero Floquet energy edge state satisfying $h_{\text{eff},k}\psi=(\tilde{\bf a}_{k}\cdot{\boldsymbol\tau})\psi=0$, or equivalently from Eq. (\[heff\_definition\]), $$\begin{aligned}
\left[\tilde{a}_{2,k}\tau^{y}+\tilde{a}_{3,k}\tau^{z}\right]\psi(x)=0\;.
\label{driven_Dirac_real_space}\end{aligned}$$ Following the same calculation from Eq. (\[edge\_state\_h\_expansion\]) to (\[static\_decay\_length\_general\]), we see that across the TPTs $(\partial_{k}^{2}\tilde{a}_{3,k})_{k_{0}}\tilde{a}_{3,k_{0}}$ changes sign, and the edge state exists in the phase that has $$\begin{aligned}
(\partial_{k}^{2}\tilde{a}_{3,k})_{k_{0}}\tilde{a}_{3,k_{0}}<0\;,
\label{driven_Majorana_existence_criterion}\end{aligned}$$ as we have verified numerically, and the decay length is $$\begin{aligned}
\xi_{k_0}=\left|\frac{(\partial_{k}\tilde{a}_{2,k})_{k_{0}}}{\tilde{a}_{3,k_{0}}}\right|\;.
\label{driven_decay_length_general}\end{aligned}$$ Comparing with Eq. (\[driven\_xi\_Fk0\_bk\]), the correspondence between the decay length and the Majorana-Wannier state correlation length $\xi$ is evident.
We proceed to discuss the critical exponent of $\xi_{k_0}=\xi$ near the TPTs in the driven case. From Eq. (\[heff\_definition\]), we see that at the HSP $k=\left\{0,\pi\right\}$, the $\tilde{a}_{2,k}$ vanishes at any $\left\{T,\mu_{1}\right\}$, so the gap-closing at $k_{0}=\left\{0,\pi\right\}$ is entirely determined by when the $\tilde{a}_{3,k}$ term vanishes. First let us consider the TPT caused by tuning $T$ but holding $\mu_{1}$ fixed. The critical point $T_{c}$ thus satisfies $\tilde{a}_{3,k_{0},T_{c}}=0$. Expand $\tilde{a}_{3,k}$ near the critical point $T_{c}$ yields $$\begin{aligned}
\tilde{a}_{3,k_{0},T_{c}+\delta T} &&= \tilde{a}_{3,k_{0},T_{c}}+\delta T\;(\partial_{T}\tilde{a}_{3,k_{0},T})_{T_{c}}
\nonumber \\
&&=\delta T\;(\partial_{T}\tilde{a}_{3,k_{0},T})_{T_{c}}\;,\end{aligned}$$ provided the leading order expansion does not vanish, which is true for this Floquet Majorana problem. Therefore, the decay length in Eq. (\[driven\_decay\_length\_general\]) near the critical point scales like $$\begin{aligned}
\xi_{k_0}|_{T_{c}+\delta T}\propto\frac{1}{\delta T\;(\partial_{T}\tilde{a}_{3,k_{0},T})_{T_{c}}}\propto\frac{1}{T-T_{c}}\;,\end{aligned}$$ indicating its critical exponent is $\nu=1$ when $T$ approaches $T_{c}$. The same argument also holds when one varies $\mu_{1}$ across the critical point $\mu_{1c}$ holding $T$ fixed, in which case we expand $$\begin{aligned}
\tilde{a}_{3,k_{0},\mu_{1c}+\delta\mu_{1}} &&=\tilde{a}_{3,k_{0},\mu_{1c}}+\delta \mu_{1}\;(\partial_{\mu_{1}}\tilde{a}_{3,k_{0},\mu_{1}})_{\mu_{1c}}
\nonumber \\
&&=\delta \mu_{1}\;(\partial_{\mu_{1}}\tilde{a}_{3,k_{0},\mu_{1}})_{\mu_{1c}}\;.\end{aligned}$$ The decay length near the critical point scales like $$\begin{aligned}
\xi_{k_0}|_{\mu_{1c}+\delta\mu_{1}}\propto\frac{1}{\delta \mu_{1}\;(\partial_{\mu_{1}}\tilde{a}_{3,k_{0},\mu_{1}})_{\mu_{1c}}}\propto\frac{1}{\mu_{1}-\mu_{1c}}.\end{aligned}$$ In short, whether approaching the phase boundary $\left\{T_{c},\mu_{1c}\right\}$ by varying $T$ or $\mu_{1}$, the critical exponent of the edge state decay length is $\nu=1$.
Behavior of quasienergy spectrum and eigenfunctions across $|\Delta \mathcal{M}|=2$ transitions {#appendix:figures_2Maj_jumps}
===============================================================================================
As explained in the main text, simultaneous divergences at non-HSP in the Berry connection can lead to the appearance of regions where the number of Floquet-Majorana modes jumps by two, *i.e.* $|\Delta{\cal M}| = 2$. To understand the character of these transitions we illustrate here some graphical results pertaining to these regions. To verify whether the number of FMF’s indeed changes across the FLs, we have calculated the quasienergy spectrum for chains up to $N=2000$ (see Fig. \[figure:quasienergy-ribs\]). We find that in the 2-FMF region for $\mu_0=0.1$, two pairs of eigenvalues $\pm \epsilon_i$ approach $\pm \pi$ within $10^{-5}$, while a finite gap ($10^{-2}$) remains at 0. The eigenvalues close to $\pi$ are separated from the next eigenvalues by a finite gap of the same order. Similarly, in the 3-FMF region for $\mu_0=0.5$, a pair of eigenvalues approaches 0, while two pairs approach $\pm \pi$ within $10^{-5}$. They are again separated from the next eigenvalues by a gap of at least $10^{-2}$. We have confirmed that these eigenvalues converge to $0$ or $\pi$ as $N \to \infty$, while the gaps stay finite. However, the relative magnitude of the gaps is too small to convincingly substantiate the appearance of isolated quasienergies at 0 and $\pm \pi$. In order to understand the character of those additional asymptotic zero-energy Floquet modes, we have furthermore plotted their eigenvectors and discovered that they can always be chosen to be purely real. Additionally, they appear to be localized at the edges, although their localization length, contrary to the FMF’s obtained deep into the topological phases, can stretch over hundreds of sites and hence tend to hybridize them out of the 0- or $\pi$-energy (see Fig. \[figure:eigenfunctions-ribs\].) The asymptotic eigenmodes seem to increase their localization in the limit $N \to \infty$. In summary, current results reach the limits of our numerical accuracy and we can therefore not conclusively confirm that the asymptotic zero-energy Floquet modes can be classified as true FMF’s. Typically, the fixed lines do not manifest in the topological phase diagram derived from $\mathcal{M} = N_{0} + N_{\pi}$. However, for certain values of the static parameter $\mu_0$, parts of the lines are detected at higher periods $T \gtrsim 2.0$ as additional transitions where the value of $\mathcal{M}$ jumps by two (see *e.g.* figure \[top-invariants2\]). It is however unclear to us whether this principle remains in other Floquet systems or other types of driving potential.
![Normalized quasienergy spectrum $\epsilon_i/\pi$ across the LCFL at $\mu_0=0.5$, $T=2.4$, $\mu_1=0.6-0.7$ for open boundary conditions (panels a) and b)) and periodic boundary conditions (panels c) and d)). The spectrum is shown in proximity of quasienergy 0 (panels b) and d)) and $\pi$ (panels a) and c)). A small but sizeable gap closes at $\mu_1 \approx 0.65$ in correspondence to the fixed point of the CRG flow, also for periodic boundary conditions.[]{data-label="figure:quasienergy-ribs"}](zoom-gap-mu0-0-5.pdf){width="\columnwidth"}
![Illustration of the eigenfunctions corresponding to the quasienergies $\pi$ ($\psi_1$ and $\psi_2$) and 0 ($\psi_3$) below the FL at $T=2.4$, $\mu_1=0.6$ for a chain of $N=2000$ fermions. The other parameters are $\Delta=0.1$ and $\mu_0=0.5$. Note that the additional $\pi$-modes tend to localize at the edges, but their localization length is much larger than the one of the $0$-mode (bottom panel). These modes are expected to localize asymptotically as $N \to \infty$.[]{data-label="figure:eigenfunctions-ribs"}](eigenvectors-N-2000.pdf){width="\columnwidth"}
|
---
abstract: 'The first discovered accreting millisecond pulsar, , went into X-ray outburst in April 2015. We triggered a 100 ks [[*XMM-Newton*]{}]{} ToO, taken at the peak of the outburst, and a 55 ks [[*NuSTAR*]{}]{} ToO, performed four days apart. We report here the results of a detailed spectral analysis of both the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra. While the [[*XMM-Newton*]{}]{} spectrum appears much softer than in previous observations, the [[*NuSTAR*]{}]{} spectrum confirms the results obtained with [[*XMM-Newton*]{}]{} during the 2008 outburst. We find clear evidence of a broad iron line that we interpret as produced by reflection from the inner accretion disk. For the first time, we use a self-consistent reflection model to fit the reflection features in the [[*NuSTAR*]{}]{} spectrum; in this case we find a statistically significant improvement of the fit with respect to a simple Gaussian or diskline model to fit the iron line, implying that the reflection continuum is also significantly detected. Despite the differences evident between the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra, the smearing best-fit parameters found for these spectra are consistent with each other and are compatible with previous results. In particular, we find an upper limit to the inner disk radius of $\sim 12~R_g$. In all the cases, a high inclination angle ($>50^\circ$) of the system is required. This inclination angle, combined with measurements of the radial velocity of the optical companion, results in a low value for the neutron star mass ($<0.8\,M_\odot$), a result that deserves further investigation.'
author:
- |
T. Di Salvo$^{1}$[^1], A. Sanna$^{2}$, L. Burderi$^{2}$, A. Papitto$^{3}$, R. Iaria$^{1}$,\
[ ]{}\
$^1$Università degli Studi di Palermo, Dipartimento di Fisica e Chimica, via Archirafi 36 - 90123 Palermo, Italy\
$^2$Università degli Studi di Cagliari, Dipartimento di Fisica, SP Monserrato-Sestu, KM 0.7, 09042 Monserrato, Italy\
$^{3}$INAF – INAF - Osservatorio Astronomico di Roma, via di Frascati 33, I-00044 Monteporzio Catone, Roma, Italy
bibliography:
- 'ms\_saxj\_NuSTAR.bib'
title: '[[*NuSTAR*]{}]{} and [[*XMM-Newton*]{}]{} broad–band spectrum of during its latest outburst in 2015'
---
line: formation — line: identification — stars: neutron — stars: individual: — stars: magnetic fields — X-ray: general — X-ray: binaries
Introduction
============
An accreting millisecond pulsar (hereafter AMSP) is a neutron star (NS) accreting mass from a low mass companion star ($\le 1 M_\odot$) and rotating at millisecond periods, as it is witnessed by the coherent pulsations in its X-ray light curve. This phenomenon is relatively rare among Low Mass X-ray Binaries (LMXBs) and is caused by the NS magnetic field, which is strong enough (given the accretion rate) to effectively funnel the accreting matter onto the magnetic poles. Most of the AMSPs are transient X-ray sources with recurrence times between two and more than ten years, and their outbursts usually last from a week to two months at most. (J1808 hereafter) is the first discovered AMSP [@Wijnands.etal:98]. The observed X-ray coherent pulsations are a fundamental probe of its dynamical and orbital state. A 60 ks long [[*XMM-Newton*]{}]{} observation of this source revealed a broad ($FWHM \sim 2$ keV) emission line at the energy of iron fluorescence emission [@Papitto.etal:09 P09 hereafter]. This broad line has been confirmed by Suzaku, which observed the source the day after with compatible spectral parameters within the errors [@Cackett.etal:09].
Such broadened features are ubiquitous among accreting compact objects. First discovered in Active Galactic Nuclei , these were subsequently observed in Galactic X-ray binaries containing black holes [e.g. @Miller.etal:2004] or NS . To explain the broadness of these features it is usually assumed that they originate from reprocessed emission of the accretion disc, illuminated by the primary Comptonized spectrum. Experimental results support the view that, in several cases, the accretion disk is truncated at few gravitational radii from the central mass. Here Doppler shifts and relativistic boosting in a fast rotating plasma, along with the gravitational red-shift caused by the strong gravitational field of the compact object, distort the line profile asymmetrically, broadening its shape up to 1 keV. Their shape thus depends on the geometry of the reflecting region of the disc, on the Keplerian velocity in the disc, and on its ionization state. In this case the line parameters allow to derive several important physical features with unprecedented accuracy. We just mention here the inclination of the disk with respect to the line of sight, which, in some cases , is derived with an accuracy of just few degrees, and, most important, the inner disk radius, which, in some sources, is determined within 1-2 Rg (where $Rg = G M / c^2$ is the gravitational radius). For these systems the compelling discovery of a fast spinning (extreme Kerr) black hole has been claimed based on the fact that the inner disc radius derived from the fit of the iron line lies below 6 Rg, which is the last stable orbit for a non-rotating (Schwarzschild) black hole . Moreover, iron lines play a particularly important role if observed in pulsars, as they carry information about the magnetospheric radius. This radius indicates where in the disk stresses exerted by the magnetic field start to remove angular momentum from the matter. According to accretion theory, it depends on the magnetic field strength, on the accretion rate, and on the details of the interaction between the magnetic field and the accreting matter.
Studies of the iron line profile could be hence a very powerful diagnostic tool to investigate the behaviour of matter in extreme gravitational fields and to effectively constrain the compactness (mass to radius ratio) of NSs or the spin parameter of black holes in X-ray binaries. However, some authors argued against the disk origin of broad iron lines, addressing the asymmetric broadening of these lines as caused by Compton down–scattering in a outflowing wind (e.g. @Titarchuk.etal:2009; see, however, @Cackett.Miller:13) or to pile-up distortions in CCD spectra (@Ng.etal:10; see, however, @Miller.etal:10 for extensive simulations of pile-up effects). Strong pile-up effects on the line profile can be excluded; in fact, it has been shown that there is a good agreement between spectral parameters derived from CCD-based spectra with those derived from gas-based spectrometers [see e.g. @Cackett.etal:12; @Egron.etal:13]. More recently, observations with [[*NuSTAR*]{}]{}, which does not suffer from pile-up, have confirmed that iron line profiles in most LMXBs appear broad and asymmetric [see e.g. @Miller.etal:13; @Degenaar.etal:15; @King.etal:16; @Sleator.etal:16; @Ludlam.etal:17 and references therein].
Nevertheless an asymmetric broad iron line may be the effect of either reflection from a Keplerian disc or Compton broadening and/or down–scattering. However, if the origin of this line is from disc reprocessing, one would also expect the presence of a spectral hump between 20 and 40 keV due to Compton scattering of the primary spectrum by the disc. Indeed this reflection hump has been observed in the spectrum of some NS LMXBs [e.g. @Barret.etal:00; @Piraino.etal:99; @Yoshida.etal:93; @Fiocchi.etal:07; @DiSalvo.etal:2015], usually with reflection amplitudes (defined in terms of the solid angle $\Omega/2\pi$ subtended by the reflector as seen from the corona) lower than 0.3, indicating a spherical geometry of the illuminating corona. Therefore, the use of broad-band, moderately high energy resolution spectra, together with the use of self-consistent reflection models able to simultaneously fit the iron line profile and the related Compton hump, are fundamental in order to probe the consistency of the parameters of the whole reflection component and the reliability of the disk parameters derived from the so-called Fe-line method [see also @Matranga.etal:2017].
To date 22 AMSPs have been discovered since 1998 , the last ones discovered in 2018 (IGR J17379$-$3747, @Sanna.etal:2018, and IGR J17591$-$2342, @Sanna.etal:2018b), and three transitional pulsars, including IGR J18245$-$2452, the only transitional pulsars that went into an X-ray outburst [@Papitto.etal:2013a]. Spectral studies at high resolution are fundamental in order to characterise their emission during outbursts [see e.g. @Poutanen:2006 for a review]. Besides an energetically dominating Comptonized component, one or two soft components are often detected if enough statistics and spectral resolution is guaranteed. These soft components are interpreted as the emission arising from the accretion disc and from the NS hot spots. One of the most frequently recurring AMSP is which goes into outburst more or less regularly every 2–3 years. The light curve shape is also very regular with outburst peak fluxes between 60 and 80 mCrab (2–10 keV), and a subsequent slow decay on a timescale of $10-15$ days until the source decays below 16 mCrab and enters a low luminosity flaring state. Its spin frequency is constantly decreasing at a rate ($\sim 5 \times 10^{-16}$ Hz/s) compatible with the one expected from dipole emission of a $\sim 10^8$ G rotating pulsar [@Hartman.etal:2009; @Sanna.etal:2017b]. While this is the most probable explanation for such a deceleration, it was also proposed that the NS spin-down may be due to the emission of continuous Gravitational Waves [@Bildsten:1998]. This source is also charactised by a puzzling fast orbital period evolution [@DiSalvo.etal:2008; @Hartman.etal:2009; @Burderi.etal:2009; @Patruno.etal:2012b; @Sanna.etal:2017b]. The time scale of this evolution is so short (few $\times 10^6$ yr) that a non conservative evolution or large short-term angular momentum exchange between the mass donor and the orbit, caused by gravitational quadrupole coupling due to variations in the oblateness of the companion, are indicated as possible explanations [see @Sanna.etal:2017b for a discussion]. Such a conclusion might establish an evolutionary link between (at least some) AMSPs and the so-called Black Widow Pulsars. To be confirmed, this scenario needs more measures, as quasi–cyclic period variations are expected in binaries [@Arzoumanian.etal:1994].
was observed with XMM-Newton during its 2008 outburst (see P09 for details). During this 60 ks observation a broad iron line ($\sigma = 1.1 \pm 0.1$ keV) was detected at an energy of $\sim 6.4$ keV. Modelling this line according to the disc reflection hypothesis (diskline model) allowed P09 to place the inner radius of the reflecting region between 6 and 12 Rg and the outer radius at about 200 Rg. As is a pulsar, the inner radius can be interpreted as the magnetospheric radius, which is predicted by accretion theories to lie exactly between the coronation radius, which in the case of is at about 30 km, and the NS surface in order to allow the observation of X-ray coherent pulsations from the source [@Ghosh.Lamb:1979]. The statistics of the 2008 observation was nevertheless too low to discriminate between a symmetric and an asymmetric profile, although a disc interpretation is strongly favoured, as Compton broadening is not a viable explanation in a source whose Comptonized component originates at a large temperature [$kT_e \ge 30$ keV, see e.g. @Gierlinski.etal:2002].
The main goal of this paper is to characterize the broad–band X-ray spectrum of the transient AMSP , and in particular the iron line and other reflection features, with a larger statistics than in previous observations, taking advantage from the large exposure of the 80 ks–[[*XMM-Newton*]{}]{} observation and the broad–band coverage provided by the [[*NuSTAR*]{}]{} observation performed during the latest outburst from the source. This allowes us to acquire the source broadband spectrum and to constrain the reflection component properties such as the broad Fe emission line together with the expected Compton hump, therefore allowing to infer the properties of the accretion flow close to the NS. We report here on a detailed study of the reflection features and the fit, with a self-consistent reflection model, of both the iron line profile and the associated Compton reflection hump at energies above 10 keV. In this spectrum, which includes hard–band data (up to $50-70$ keV), the overall fractional amount of reflection is well determined by fitting the Compton hump. We can therefore test whether the observed iron line is consistent with this fractional amount of reflection. In this way we can confirm independently (fitting a different outburst state and using different instruments) the inner disk parameters already obtained with [[*XMM-Newton*]{}]{} and Suzaku for the 2008 outburst.
Observations
============
, went into X-ray outburst in April 2015, after more than three years from its previous outburst in 2008. was observed by [[*XMM-Newton*]{}]{} on 2015 April 11 (ObsID: 0724490201) for a total observing time of about 110 ks, as a result of an anticipated target of opportunity (ToO) observation approved to observe the source during an outburst. During the observation an abrupt drop-off of the count rate was visible in the EPIC/pn light curve caused by a problem with the Star Tracker, which led the satellite to be off-target for about seven hours, resulting in 80 ks effective on-source exposure. During the observation, the EPIC/pn camera was operated in timing mode to prevent photon pile-up and to allow the analysis of the coherent and aperiodic timing behaviour of the source [see @Sanna.etal:2017b for the timing analysis of these data]. The EPIC/MOS cameras were switched off during the observation in order to allocate as much telemetry as possible to the pn in the case of high count rate, and the Reflection Grating Spectrometer (RGS) was operated in the standard spectroscopy mode.
We have extracted source, background spectra and response matrices using the Science Analysis Software (SAS) v.16.1.0, setting the parameters of the tools accordingly. We produced a calibrated photon event file using reprocessing tools [*epproc*]{} and [*rgsproc*]{} for the pn and RGS data, respectively. Before extracting the spectra, we searched for contaminations due to background solar flares detected in the 10-12 keV Epic-pn light-curve, but we did not find periods with high background. We also looked for the presence of pile-up in the pn spectrum; we have run the task [*epatplot*]{} and we did not find any significant contamination. The count-rate registered in the pn observation was around 450 c/s, which is below the limit for avoiding contamination by pile-up. Therefore, the source spectra were extracted from a rectangular region between $RAWX \geq 23$ and $RAWX \leq 49$. We selected only events with PATTERN $\leq 4$ and FLAG $= 0$ as a standard procedure to eliminate spurious events. We extracted the background spectrum from a region included between $RAWX \geq 5$ and $RAWX \leq 10$. Finally, using the task [*rgscombine*]{} we have obtained the added source spectrum for RGS1+RGS2, the relative added background spectrum along with the relative response matrices. We have fitted RGS spectrum in the 0.5-1.8 keV energy range, whereas the pn spectrum in the 2.4-10 keV energy range. The spectral analysis of the [[*XMM-Newton*]{}]{}/EPIC-pn spectrum was restricted to 2.4-10 keV to exclude the region around the detector Si K-edge (1.8 keV) and the mirror Au M-edge (2.3 keV) that could affect our analysis, as well as to exclude possible residuals of instrumental origin below 2 keV that usually appear in case of bright sources observed in timing mode [see e.g. @D_Ai.etal:10; @Egron.etal:13].
In this paper we also analyze data collected by the [[*NuSTAR*]{}]{} satellite. A ToO was requested to observe the source during the 2015 outburst in order to complement the [[*XMM-Newton*]{}]{} spectrum with high energy coverage. The [[*NuSTAR*]{}]{} observation, obtained as Director Discretionary Time (DDT), was performed four days after the [[*XMM-Newton*]{}]{} observation, on 2015 April 15 (ObsID: 90102003002), for a total observing time of 55 ks, resulting in roughly 49 ks of exposure per telescope. Science data were extracted using NuSTARDAS (NuSTAR Data Analysis Software) v1.7.1. Source data have been extracted from a circular region with 120“ radius whereas the background has been extracted from a circular region with 60” radius in a position far from the source. With the aim to get “STAGE 2” events clean, we run the [*nupipeline*]{} with default values of the parameters and with the parameter [*SAAMODE*]{} set to [*optimized*]{} in order to eliminate high background events caused by the SAA passage. The average count rate during the [[*NuSTAR*]{}]{} observation was $\sim 35-40$ c/s.
A type-I burst is present during the [[*NuSTAR*]{}]{} observation, at about 14 ks from the beginning of the observation. The burst profile is not complete since the rise phase was in coincidence with a gap in the light curve. The peak of the burst seems to reach approximately 200 c/s, about a factor 4 the level of the persistent emission, and it lasted about 200 s. We eliminated a time interval of 250 s starting from 5 s before the rise of the burst, and checked that the spectra did not change significantly. Spectra for both detectors, FPMA and FPMB, were extracted using the [*nuproducts*]{} command. Corresponding response files were also created as output of [*nuproducts*]{}. A comparison of the FPMA and FPMB spectra, indicated a good agreement between them. We have therefore created a single added spectrum, with its corresponding background spectrum, ancillary response file and matrix response, using the [*addascaspec*]{} command. In this way, we obtain a summed spectrum for the two [[*NuSTAR*]{}]{} modules [see e.g. @Miller.etal:13]. We fitted this spectrum in the 3-70 keV energy range, where the emission from the source dominates over the background.
In Figure \[lcurve\] we show the light curve during the 2015 outburst of obtained with the instruments, XRT and BAT, on board the Swift satellite. In this light curve the dates of the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} observations are indicated with stars.
![Swift/BAT and XRT light curve during the 2015 outburst of . The dates of the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} observations are indicated with stars.[]{data-label="lcurve"}](lc_bat_xrt2.eps){width="10cm"}
Spectral Analysis and Results
=============================
For spectral analysis, the EPIC/pn energy channels were grouped in order to have at least 20 counts per energy channel and to oversample the energy resolution element by no more than three channels. RGS and [[*NuSTAR*]{}]{} spectra were grouped in order to have at least 20 counts per energy channel. The X-ray spectral package we use to model the observed emission is XSPEC v.12.9.1. For each fit we have used [*phabs*]{} in XSPEC to model the photoelectric absorption due to neutral matter, with photoelectric cross sections from @Balucinska:1992 and element abundances from @Anders:1989.
The [[*NuSTAR*]{}]{} spectrum
-----------------------------
We have started analyzing the broad-band spectrum acquired with [[*NuSTAR*]{}]{} in the energy range $3-70$ keV. We fit the continuum emission with a soft blackbody and the Comptonization model [*nthComp*]{} in XSPEC [@Zycki.etal:99], modified at low energy by photoelectric absorption caused by neutral matter. This is a standard model to fit the broad-band continuum emission in NS LMXBs of the atoll class both in the soft and in the hard states [see e.g. @Piraino.etal:07; @Di_Salvo.etal:09; @Egron.etal:13; @Sanna.etal:2013; @DiSalvo.etal:2015 and references therein] and provides also a good fit to the broad-band continuum emission in AMSPs [see e.g. @Papitto.etal:10; @Papitto.etal:13b; @Papitto.etal:2016; @Sanna.etal:2016]. Because of the lack of sensitivity of [[*NuSTAR*]{}]{} at low energies, we had to fix the photoelectric equivalent hydrogen column density at $0.21 \times 10^{22}$ cm$^{-2}$, which is the best fit value for this parameter obtained for the X-ray spectrum of observed by [[*XMM-Newton*]{}]{} during the 2008 outburst (see P09). The soft blackbody component has a best fit temperature of $\sim 0.7$ keV, the seed photons for Comptonization, assumed to have a blackbody spectrum, have a temperature lower than $\sim 0.3$ keV; these photons are comptonized in a hot corona with an electron temperature of $\sim 30$ keV and a moderate optical depth, corresponding to a photon index of the Comptonization spectrum of $\Gamma \sim 1.8$. This continuum model gave, however, an unacceptable fit, corresponding to a $\chi^2 / dof = 1679 / 1325$, because of the presence of evident localized residuals in the $5-9$ keV range, clearly caused by the presence of a broad iron line, whose profile is evident in Figure \[nuspec\] (left panel).
![[**Left panel:**]{}[[*NuSTAR*]{}]{} spectrum in the energy range 3 - 70 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the continuum model of when the Fe line normalization is set to 0. [**Right panel:**]{} [[*XMM-Newton*]{}]{} spectrum in the energy range 0.6 - 10 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the continuum model of . In both cases the model consists of a blackbody (dotted line) and the Comptonization component [*nthComp*]{} (solid line), both multiplied by the photoelectric absorption component modeled with [*phabs*]{}. Evident residuals are present at the expected iron line energy of $6.4-6.7$ keV. Residuals between 2.4 and 4.5 keV are also present in the pn spectrum.[]{data-label="nuspec"}](NuSTAR_data_res_line.ps "fig:"){width="6cm"} ![[**Left panel:**]{}[[*NuSTAR*]{}]{} spectrum in the energy range 3 - 70 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the continuum model of when the Fe line normalization is set to 0. [**Right panel:**]{} [[*XMM-Newton*]{}]{} spectrum in the energy range 0.6 - 10 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the continuum model of . In both cases the model consists of a blackbody (dotted line) and the Comptonization component [*nthComp*]{} (solid line), both multiplied by the photoelectric absorption component modeled with [*phabs*]{}. Evident residuals are present at the expected iron line energy of $6.4-6.7$ keV. Residuals between 2.4 and 4.5 keV are also present in the pn spectrum.[]{data-label="nuspec"}](XMM_res_lines.ps "fig:"){width="6cm"}
To fit these residuals we first added to our continuum model a [*Gaussian*]{} component to model the broad iron line profile visible in the residuals. The addition of this component resulted in a significant improvement of the fit and to an acceptable $\chi^2 / dof = 1316 / 1322$. The Gaussian centroid was at $6.51 \pm
0.08$ keV, its width $\sigma = 0.65 \pm 0.10$ keV, and its equivalent width was $101 \pm 13$ eV. We also tried to fit the residuals at the iron line energy with a [*diskline*]{} profile [@Fabian.etal:89] instead of a Gaussian. The addition of this component resulted in a further improvement of the fit, $\chi^2 / dof = 1306 / 1319$. The [*diskline*]{} component is characterized by a centroid energy of $6.3-6.5$ keV, indicating neutral and/or weakly ionized iron (Fe I-XV), an emissivity index, describing the emissivity of the disk as a function of the emission radius $\propto r^{Betor}$, of $Betor \sim -(1.8 - 2.2)$, an inner and outer radius of the emitting region of $R_{in} \le 7$ R$_g$ and $R_{out} \sim 290 - 1100$ R$_g$, respectively, where R$_g = G M_{NS}/c^2$ is the gravitational radius, and a large inclination angle of the system with respect to the line of sight, $i \ge 70^\circ$. The best fit parameters are reported in Table \[Tab1\]. These parameters are very similar (compatible well within the errors) to the best fit parameters of the iron line component obtained from the [[*XMM-Newton*]{}]{} spectrum of during the 2008 outburst (cf. P09); in particular the previous [[*XMM-Newton*]{}]{} spectrum also gave a quite small outer disk radius of $140 - 360$ R$_g$ and a high inclination angle of $i \ge 60^\circ$.
The broad iron line profile seems to be compatible with reflection of the main Comptonization spectrum off the accretion disk, where the broadness of the profile is induced by the fast motion of the matter in the inner disk, and related (mildly) relativistic effects. However, if the iron line is produced by reflection then a Compton hump should be visible in the high energy part of the spectrum given that the source is in a hard state. To test this hypothesis we tried to substitute the [*diskline*]{} component with a self-consistent reflection model to fit both the iron line and the Compton hump. In particular, we used the [*relxillCp*]{} model [@Garcia.etal:2014], which models the irradiation of the accretion using an [*nthComp*]{} Comptonization continuum (Cp). Note that at the moment it is not possible to fit the temperature of the seed photons for the Comptonization component, that is fixed at 50 eV (Thomas Dauser, private communication). This should be anyway a good approximation for the case of given the low value we find for this temperature (see Table \[Tab1\] and \[Tabtot\]).
The [*relxillCp*]{} model allows to determine the reflection fraction, defined as ratio of intrinsic intensity emitted towards the disk compared to that escaping to infinity [see @Dauser.etal:2016 for more details], the inclination angle of the system and the ionization parameter of the disk, given by $\log \xi$, where $\xi = L_X /(n_e r^2)$, where $L_X$ is the luminosity of the incident X-ray spectrum, $n_e$ is the electron number density in the emitting region, and $r$ is the distance between the illuminating source and the emitting region. This reflection model includes the smearing component [*relconv*]{}[^2] in order to take into account Doppler and relativistic effects caused by the fast motion of the matter in the disk. Unless specified otherwise, the two emissivity indices (defined in order to be negative as in the case of the [*diskline*]{} parameter [*Betor*]{}) have been constrained in order to assume the same value and the dimension-less spin parameter a has been fixed to 0.
This model gives a good fit to the data, corresponding to a $\chi^2 /
dof = 1300/1320$. We find that fixing the iron abundance, $A_{Fe}$, at 2 times the solar value the fit slightly improved returning a $\chi^2 / dof = 1297 / 1320$, lower with respect to the previous fit with a [*diskline*]{}, corresponding to a $\Delta \chi^2 = 9$ with one extra degree of freedom. Letting the iron abundance free to vary, we find that its best fit value was $\sim 3$ corresponding to a $\chi^2$ very similar to the previous one, but the uncertainty on this parameter is quite large ($A_{Fe} \simeq 1 - 4$), so we preferred to keep fixed this parameter at 2 times the solar value. The outer disk radius has been left free to vary, but its uncertainty could not be determined because the $\chi^2$ was quite insensitive to its value; probably the energy resolution of [[*NuSTAR*]{}]{} is not enough to constrain this parameter. The spectral results are reported in Table \[Tab1\], and in Figure \[fig3\] (right panel) we show the [[*NuSTAR*]{}]{} spectrum, the best-fit model, and the residuals in units of $\sigma$ with respect to the best-fit model. We extrapolated the total $0.5-200$ keV observed luminosity of the source during the [[*NuSTAR*]{}]{} observation, corresponding to $(3.78 \pm 0.15) \times 10^{36}$ ergs/s assuming a distance to the source of 3.5 kpc [@Galloway.etal:2006].
----------- ---------------------------------------------- ------------------------- ------------------------
DISKLINE RELXILLCP
Component Parameter [[*NuSTAR*]{}]{} [[*NuSTAR*]{}]{}
($3-70$ keV) ($3-70$ keV)
phabs $N_H$ ($\times 10^{22}$ cm$^{-2}$) $0.21$ (fixed) $0.21$ (fixed)
bbody $kT_{BB}$ (keV) $0.697 \pm 0.015$ $0.672 \pm 0.007$
bbody L$_{BB}$ ($L_{36}/D_{10}^2$) $2.14^{+0.09}_{-0.04}$ $1.79 \pm 0.04$
bbody R$_{BB}$ (km) $2.93 \pm 0.14$ $2.88 \pm 0.07$
nthComp $kT_{seed}$ (keV) $< 0.29$ $-$
nthComp $\Gamma$ $1.819 \pm 0.006$ $1.868 \pm 0.015$
nthComp $kT_e$ (keV) $26.1^{+2.3}_{-1.8}$ $39^{+13}_{-3}$
nthComp Flux ($10^{-10}$ ergs cm$^{-2}$ s$^{-1}$) $9.5 \pm 1.4$ $-$
diskline $E_{line}$ (keV) $6.38 \pm 0.10$ $-$
diskline $I_{line}$ ($10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $8.6 \pm 1.2$ $-$
diskline $EqW$ (eV) $124 \pm 19$ $-$
Smearing $Betor$ $-2.04^{+0.19}_{-0.15}$ $-1.95 \pm 0.12$
Smearing $R_{in}$ ($G M / c^2$) $< 7$ $14.9 \pm 2.5$
Smearing $R_{out}$ ($G M / c^2$) $520^{+550}_{-240}$ $1000$ (INDEF)
Smearing Incl (deg) $> 70$ $50^{+22}_{-5}$
RelxillCp Refl Frac $-$ $0.62 \pm 0.04$
RelxillCp Fe abund $-$ $2$ (fixed)
RelxillCp $\log \xi$ $-$ $2.76^{+0.10}_{-0.07}$
RelxillCp Norm ($\times 10^{-3}$) $-$ $3.73^{+0.07}_{-0.14}$
total Flux ($10^{-9}$ ergs cm$^{-2}$ s$^{-1}$) $2.01 \pm 0.12$ $2.02 \pm 0.09$
total $\chi^2$ (dof) $1305.68~(1319)$ $1296.66~(1320)$
----------- ---------------------------------------------- ------------------------- ------------------------
: The best fit parameters of the spectral fitting of the [[*NuSTAR*]{}]{} ($3-70$ keV energy band) spectrum of . In all the cases the continuum emission is described by a combination of a blackbody and the Comptonization component [*nthComp*]{}, modified at lower energy by photoelectric absorption from neutral matter modeled with [*phabs*]{}. The reflection component is fitted with a [*diskline*]{} component, or with the self-consistent reflection model [*relxillCp*]{}. The blackbody luminosity is given in units of $L_{36}/D_{10}^2$, where $L_{36}$ is the bolometric luminosity in units of $10^{36}$ ergs/s and $D_{10}$ the distance to the source in units of 10 kpc. The blackbody radius is calculated in the hypothesis of spherical emission and for a distance of 3.5 kpc. Smearing indicate the smearing component of the [*diskline*]{} and [*relxillCp*]{} models, respectively. Flux in the nthComp component is calculated in the $1-10$ keV range, while total flux is calculated in the $1.6-70$ keV band. Uncertainties are given at $90\%$ confidence level. INDEF means that the error on the parameter could not be calculated being the $\chi^2$ quite insensitive to its value.
\[Tab1\]
The [[*XMM-Newton*]{}]{} spectrum
---------------------------------
In order to check the results obtained from the [[*NuSTAR*]{}]{} spectrum, we have fitted separately the [[*XMM-Newton*]{}]{} spectra (RGS and pn, energy range $0.6-10$ keV), using the same continuum model. Again, fitting only the continuum results in an unacceptable fit, with $\chi^2/dof = 3489/1458$, and clear residuals are present at the iron line energy and lower energies, at $\sim 2.6$ keV, $\sim 3.3$ keV, and $\sim 4$ keV (see Fig. \[nuspec\], right panel). These low-energy residuals are similar to those observed in other bright LMXBs of the atoll class in the soft state, as 4U 1705-44 [see e.g. @Di_Salvo.etal:09; @Egron.etal:13] or GX 3+1 [e.g. @Piraino.etal:12; @Pintore.etal:2015]. We therefore added to the continuum model four [*diskline*]{} components to fit the iron line and the other low energy features. The smearing parameters of the [*diskline*]{} fitting the low-energy lines were fixed to be the same of those of the [*diskline*]{} fitting the iron line. In this way we get a significative improvement of the fit, corresponding to $\chi^2/dof = 2094/1446$. The best fit parameters of this fit are reported in Table \[Tabtot\], and in Figure \[fig3\] (left panel) we show the [[*XMM-Newton*]{}]{} spectrum, the best-fit model, and the residuals in units of $\sigma$ with respect to the best-fit model.
Note that the electron temperature of the Comptonization component is much lower with respect to that measured with [[*NuSTAR*]{}]{} four days later, indicating that could have been in a soft state at the time of the [[*XMM-Newton*]{}]{} observation, in agreement with the presence of ionized discrete features in the spectrum. Note also that the [[*XMM-Newton*]{}]{} observation was acquired at the peak of the outburst and this is the first time that the spectrum has been observed with good energy resolution at the peak of an outburst. Substituting the [*diskline*]{} used for the iron line with the reflection model [*relxillCp*]{}, keeping the smearing parameters fixed to the corresponding smearing parameters of the other, low-energy [*disklines*]{}, returns a $\chi^2/dof = 2100/1447$, that is slightly worse than before ($\Delta \chi \simeq 6$), but with one extra degree of freedom. All the best-fit parameters are consistent within the errors with the best-fit values of the previous fit with disklines. As regards the reflection parameters, we fixed the iron abundance to 2, as in the case of the [[*NuSTAR*]{}]{} spectrum, the reflection fraction is compatible to that obtained with [[*NuSTAR*]{}]{}, although with a larger uncertainty, while the ionization parameter results to be much higher, $\log \xi = 3.6 - 3.8$, in agreement with the presence of features from highly ionized elements in the [[*XMM-Newton*]{}]{} spectrum. The best fit parameters of this fit are reported in Table \[Tabtot\]. Note that some residuals are still present at $\sim 7$ keV, this is also visible in Figure \[fig3\]. To fit these residuals we tried to add a Gaussian emission line at that energy, obtaining a $\chi^2/dof = 2072/1444$; the improvement of the fit is barely significant, corresponding to an F-test probability of chance improvement of $\sim 1.2 \times 10^{-4}$. We also tried to fit the iron abundance obtaining a preference for an overabundance, $A_{Fe} = 3.4 \pm 0.7$, but without a statistically significant improvement of the fit (F-test probability of chance improvement $\sim 1.3 \times 10^{-3}$ for the addition of one parameter).
Combined analysis of the [[*NuSTAR*]{}]{} and [[*XMM-Newton*]{}]{} spectra
--------------------------------------------------------------------------
In order to increase the statistics at the iron line energy and for the whole reflection component, we tried to fit together the [[*NuSTAR*]{}]{} and [[*XMM-Newton*]{}]{} spectra, using the best-fit models obtained above. The two observations are not perfectly simultaneous, the [[*NuSTAR*]{}]{} observation being performed four days after the [[*XMM-Newton*]{}]{} observation taken at the peak of the outburst. Both the best fit continuum emission and the emission lines are very different between the two spectra. We therefore left most of the parameters free to vary between the two spectra and only few parameters were constrained to assume the same value for the two spectra; these are the equivalent hydrogen column density, $N_H$ of the interstellar absorption, all the parameters of the smearing component and the iron abundance. Both the ionization parameter of the reflection component and the reflection fraction were quite different between the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra (cf.Table \[Tab1\] and Table \[Tabtot\]) and therefore we let these parameters free to vary between the two spectra. The low-energy disklines are not required by the [[*NuSTAR*]{}]{} spectrum and hence are not included in the fit of the [[*NuSTAR*]{}]{} spectrum. Again, letting the iron abundance free to vary (but forced to assume the same value for the two spectra) we find an improvement of the fit, corresponding to a $\chi^2/dof = 3394.30/2770$ (corresponding to an F-test probability of chance improvement of $4 \times 10^{-5}$ for the addition of one parameter). The best-fit value was $A_{Fe} \simeq 3.4$, while all the other parameters did not change significantly. However, we could not determine the error on this parameter, and therefore we preferred to keep the iron abundance fixed to two times the Solar value. In Table \[Tabtot\] (last column) we report the best-fit parameters obtained for this fit. In Figure \[fig4\] we show the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra together with the best fit model and residuals in units of sigma with respect to this model.
----------- ----------------------------------------------------- ------------------------ ------------------------ --------------------------- ---------------------------
DISKLINE RELXILLCP
Component Parameter [[*XMM-Newton*]{}]{} [[*XMM-Newton*]{}]{} [[*XMM-Newton*]{}]{} [[*NuSTAR*]{}]{}
($0.6-10$ keV) ($0.6-10$ keV)
phabs $N_H$ ($\times 10^{22}$ cm$^{-2}$) $0.164 \pm 0.006$ $0.142 \pm 0.011$
bbody $kT_{BB}$ (keV) $0.103 \pm 0.002$ $0.109 \pm 0.003$ $0.112^{+0.003}_{-0.002}$ $0.680^{+0.002}_{-0.003}$
bbody L$_{BB}$ ($L_{36}/D_{10}^2$) $10.3^{+0.9}_{-0.6}$ $4.4^{+1.2}_{-0.9}$ $5.4 \pm 1.1$ $1.64 \pm 0.11$
bbody R$_{BB}$ (km) $295 \pm 17$ $172 \pm 25$ $180 \pm 20$ $2.69 \pm 0.09$
nthComp $kT_{seed}$ (keV) $0.15^{+0.02}_{-0.04}$ $-$
nthComp $\Gamma$ $1.904 \pm 0.008$ $1.78 \pm 0.04$ $1.73 \pm 0.03$ $1.88 \pm 0.02$
nthComp $kT_e$ (keV) $4.7 \pm 0.3$ $7.2 \pm 0.8$ $6.2 \pm 0.5$ $36.9^{+6.4}_{-4.6}$
nthComp Flux ($10^{-10}$ ergs cm$^{-2}$ s$^{-1}$) $14.7 \pm 0.9$ $-$ $-$ $-$
diskline $E_{line}$ (keV) $2.665 \pm 0.012$ $2.666 \pm 0.015$ $2.664 \pm 0.016$ $-$
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $21.5 \pm 1.9$ $21 \pm 3$ $18 \pm 3$ $-$
diskline $EqW$ (eV) $41 \pm 6$ $42 \pm 6$ $35 \pm 11$ $-$
diskline $E_{line}$ (keV) $3.294 \pm 0.019$ $3.31 \pm 0.02$ $3.28 \pm 0.02$ $-$
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $15.0 \pm 1.5$ $16 \pm 3$ $11.6 \pm 1.6$ $-$
diskline $EqW$ (eV) $38 \pm 6$ $50 \pm 8$ $33 \pm 11$ $-$
diskline $E_{line}$ (keV) $3.99 \pm 0.04$ $4.06 \pm 0.04$ $3.99 \pm 0.05$ $-$
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $8.0^{+0.7}_{-0.5}$ $9.5 \pm 1.3$ $6.3^{+1.1}_{-0.9}$ $-$
diskline $EqW$ (eV) $30 \pm 4$ $42 \pm 7$ $31 \pm 9$ $-$
diskline $E_{line}$ (keV) $6.70 \pm 0.05$ $-$ $-$ $-$
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $5.4 \pm 0.6$ $-$ $-$ $-$
diskline $EqW$ (eV) $58 \pm 8$ $-$ $-$ $-$
Smearing $Betor$ $-2.16 \pm 0.10$ $-2.20 \pm 0.04$
Smearing $R_{in}$ ($G M / c^2$) $10^{+8}_{-3} $ $<7.2$
Smearing $R_{out}$ ($G M / c^2$) $1032^{+750}_{-270}$ $>900$
Smearing Incl (deg) $> 58$ $59.8^{+4.2}_{-1.6}$
RelxillCp Refl Frac $-$ $0.69^{+0.27}_{-0.19}$ $0.9^{+0.4}_{-0.3}$ $0.22^{+0.11}_{-0.04}$
RelxillCp Fe abund $-$ $2$ (fixed)
RelxillCp $\log \xi$ $-$ $3.72 \pm 0.09$ $3.78 \pm 0.08$ $2.4 \pm 0.3$
RelxillCp Norm ($\times 10^{-3}$) $-$ $2.64^{+0.25}_{-0.36}$ $2.4 \pm 0.2$ $3.79^{+0.10}_{-0.07}$
total Flux ($10^{-9}$ ergs cm$^{-2}$ s$^{-1}$) $1.59 \pm 0.02$ $1.60 \pm 0.07$ $1.59 \pm 0.22$ $2.03 \pm 0.04$
total $\chi^2$ (dof) $2094.07~(1446)$ $2100.49~(1447)$
----------- ----------------------------------------------------- ------------------------ ------------------------ --------------------------- ---------------------------
: The best fit parameters of the spectral fitting of the [[*XMM-Newton*]{}]{} ($0.6-10$ keV energy band) and [[*XMM-Newton*]{}]{} + [[*NuSTAR*]{}]{} ($0.6-70$ keV energy band) spectra of . In all the cases the continuum emission is described by a combination of a blackbody and the Comptonization component [*nthComp*]{}, modified at lower energy by photoelectric absorption from neutral matter modeled with [*phabs*]{}. The reflection component is fitted with [*diskline*]{} components or with the self-consistent reflection model [*relxillCp*]{}. The blackbody luminosity is given in units of $L_{36}/D_{10}^2$, where $L_{36}$ is the bolometric luminosity in units of $10^{36}$ ergs/s and $D_{10}$ the distance to the source in units of 10 kpc. The blackbody radius is calculated in the hypothesis of spherical emission and for a distance of 3.5 kpc. Fluxes in the nthComp component are calculated in the $1-10$ keV range, while total flux is calculated in the $0.6-10$ keV band for the [[*XMM-Newton*]{}]{} spectrum and in the $1.6-70$ keV band for the [[*NuSTAR*]{}]{} spectrum. Uncertainties are given at $90\%$ confidence level.
\[Tabtot\]
![[**Left:**]{} [[*XMM-Newton*]{}]{} RGS (black points) and pn (red points) spectra of in the energy range 0.6 - 10 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the best-fit model (see Table \[Tabtot\], third column). The model consists of a blackbody (dotted line), the Comptonization component [*nthComp*]{} (solid line) and four disklines (dashed lines) describing the reflection component, all multiplied by photoelectric absorption. [**Right:**]{} [[*NuSTAR*]{}]{} spectrum in the energy range 3 - 70 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the best-fit model shown in the last column of Table \[Tab1\]. The model components are also shown. From the left to the right we see the blackbody component (dotted line), the Comptonization component plus the smeared reflection component modeled by [*relxillCp*]{} (solid line). []{data-label="fig3"}](XMM_res_disklines.ps "fig:"){width="6cm"} ![[**Left:**]{} [[*XMM-Newton*]{}]{} RGS (black points) and pn (red points) spectra of in the energy range 0.6 - 10 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the best-fit model (see Table \[Tabtot\], third column). The model consists of a blackbody (dotted line), the Comptonization component [*nthComp*]{} (solid line) and four disklines (dashed lines) describing the reflection component, all multiplied by photoelectric absorption. [**Right:**]{} [[*NuSTAR*]{}]{} spectrum in the energy range 3 - 70 keV (top) and residuals in units of $\sigma$ (bottom) with respect to the best-fit model shown in the last column of Table \[Tab1\]. The model components are also shown. From the left to the right we see the blackbody component (dotted line), the Comptonization component plus the smeared reflection component modeled by [*relxillCp*]{} (solid line). []{data-label="fig3"}](NuSTAR_data_res_relxill_Fe2.ps "fig:"){width="6cm"}
![[[*XMM-Newton*]{}]{} (black and red points) and [[*NuSTAR*]{}]{} (green points) spectra of (top) and residuals in units of $\sigma$ (middle) with respect to the best-fit model (see Table \[Tabtot\], last column). The model consists of a blackbody (with different temperatures for the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra, dotted lines), the Comptonization component plus the smeared reflection component modeled by [*relxillCp*]{} (solid line), all multiplied by photoelectric absorption. Three disklines (indicated with dashed lines) are used to fit the [[*XMM-Newton*]{}]{} spectra but are not required for the [[*NuSTAR*]{}]{} spectrum. The total model is plotted on top of the data. Note that each spectrum is convolved with its response matrix and effective area, as well as the corresponding model and model components. In the bottom panel we show the residuals in units of $\sigma$ with respect to the best-fit model including the absorption components (see Tab. \[Tabfin\]).[]{data-label="fig4"}](nus_xmm_relxill_diskl_new.ps "fig:"){width="12cm"} ![[[*XMM-Newton*]{}]{} (black and red points) and [[*NuSTAR*]{}]{} (green points) spectra of (top) and residuals in units of $\sigma$ (middle) with respect to the best-fit model (see Table \[Tabtot\], last column). The model consists of a blackbody (with different temperatures for the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra, dotted lines), the Comptonization component plus the smeared reflection component modeled by [*relxillCp*]{} (solid line), all multiplied by photoelectric absorption. Three disklines (indicated with dashed lines) are used to fit the [[*XMM-Newton*]{}]{} spectra but are not required for the [[*NuSTAR*]{}]{} spectrum. The total model is plotted on top of the data. Note that each spectrum is convolved with its response matrix and effective area, as well as the corresponding model and model components. In the bottom panel we show the residuals in units of $\sigma$ with respect to the best-fit model including the absorption components (see Tab. \[Tabfin\]).[]{data-label="fig4"}](nus_xmm_abs_res.ps "fig:"){width="4.8cm"}
Form visual inspection of Fig. \[fig3\] and \[fig4\] some residuals are still evident in the [[*XMM-Newton*]{}]{} RGS (at $\sim 0.9$, $\sim 1.3$ and $\sim 1.6$ keV) and pn spectra (at $\sim 7$ keV). In order to fit these residuals we tentatively add to the previous model three (Gaussian) absorption lines and an edge. The edge has a best-fit energy of $\sim 7.4$ keV and may be associated to mildly ionized iron (Fe V-XV). The centroid energies of the three absorption lines are 0.947 keV (possibly from Ne IX, resonance transition rest-frame energy 0.922 keV), 1.372 keV (possibly from Mg XI, resonance transition rest-frame energy 1.352 keV) and 1.596 keV, respectively. The latter is remarkably close to the resonant line of Al XII (rest-frame 1.598 keV). Note, however, that this element has a low cosmic abundance ($\sim 3 \times 10^{-6}$ in number of atoms with respect to H) and that this line has a significance of about $3.5 \sigma$; there is the possibility that this line is of instrumental origin, and we prefer not to discuss it further.
The first two absorption lines appear, instead, relatively broad ($\sigma_{Ne} \simeq 0.035$ keV and $\sigma_{Mg} \simeq 0.015$ keV, respectively), corresponding to velocity dispersion of $\sim 4\%$ and $\sim 1\%$ of the velocity of light, respectively. If our identification of the first two absorption lines is correct, than their energies appear to be blue-shifted with respect the corresponding rest-frame energies of $\sim 0.025$ keV and $\sim 0.02$ keV, respectively, corresponding to a velocity of $\sim 2.7 \%$ and $\sim 1.5 \%$ of the speed of light, possibly indicating the presence of an outflowing, weakly relativistic wind. Note that, if the iron edge, that we detect at $7.39$ keV, is indeed a blue-shifted neutral iron edge (rest-frame energy 7.112 keV), than it would correspond to a velocity of $\sim 3.9 \%\, c$.
The addition of these components improved the quality of the fit, returning a $\chi^2/dof = 3186/2759$, implying a decrease by $\Delta \chi^2 = 229$ for the addition of 12 parameters with respect to the previous best-fit. In this last fit, we decided to fix the outer disk radius at 1000 R$_g$ and the iron abundance at 2 times the Solar value, and to include the low-energy disklines also in the fitting of the [[*NuSTAR*]{}]{} spectrum. The line at $2.6$ keV is indeed below the energy band used for the [[*NuSTAR*]{}]{} spectrum and therefore it was not included. The other two disklines were included with all the parameters fixed to those of the [[*XMM-Newton*]{}]{} spectrum, except for the normalization that was left free to vary. We find that the line at 3.3 keV is indeed not necessary in the [[*NuSTAR*]{}]{} spectrum, with an upper limits on its equivalent width of $30$ keV, while the addition of the line at $4.1$ keV in the [[*NuSTAR*]{}]{} spectrum is significant at $\sim 3 \sigma$ confidence level. All the other parameters are very similar (compatible within the associated errors) to those of the previous best-fit; the most significant difference is in the ionization parameter of the [[*NuSTAR*]{}]{} spectrum for which we only get an upper limit of $\log \xi < 2$, in agreement with the centroid energy of the iron line at $\sim 6.4$ keV. The results of this fit are shown in Table \[Tabfin\], and the residuals with respect to the best fit model are shown in the bottom panel of Fig. \[fig4\].
----------- ----------------------------------------------------- --------------------------------- ---------------------------
Component Parameter [[*XMM-Newton*]{}]{} [[*NuSTAR*]{}]{}
phabs $N_H$ ($\times 10^{22}$ cm$^{-2}$)
edge $E_{Fe}$ (keV)
edge $\tau$ ($\times 10^{-2}$)
gauss $E_{line}$ (keV)
gauss $\sigma$ (keV)
gauss $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$)
gauss $EqW$ (eV)
gauss $E_{line}$ (keV)
gauss $\sigma$ (keV)
gauss $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$)
gauss $EqW$ (eV)
gauss $E_{line}$ (keV)
gauss $\sigma$ (keV)
gauss $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$)
gauss $EqW$ (eV)
diskline $E_{line}$ (keV)
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $32 \pm 6$ $-$
diskline $EqW$ (eV) $72 \pm 12$ $-$
diskline $E_{line}$ (keV)
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $21.9 \pm 0.3$ $<22$
diskline $EqW$ (eV) $71 \pm 14$ $<29$
diskline $E_{line}$ (keV)
diskline $I_{line}$ ($\times 10^{-4}$ ph cm$^{-2}$ s$^{-1}$) $10.8 \pm 1.8$ $5.8^{+5.2}_{-2.8}$
diskline $EqW$ (eV) $48 \pm 10$ $37 \pm 19$
Smearing $Betor$
Smearing $R_{in}$ ($G M / c^2$)
Smearing $R_{out}$ ($G M / c^2$)
Smearing Incl (deg)
bbody $kT_{BB}$ (keV) $0.111 \pm 0.004$ $0.50^{+0.07}_{-0.33}$
bbody L$_{BB}$ ($L_{36}/D_{10}^2$) $2.7 \pm 0.8$ $1.6^{+10}_{-0.3}$
bbody R$_{BB}$ (km) $130 \pm 21$ $4.9^{+110}_{-1.5}$
nthComp $\Gamma$ $1.77^{+0.02}_{-0.04} \pm 0.04$ $1.939^{+0.008}_{-0.005}$
nthComp $kT_e$ (keV) $7.3 \pm 0.8$ $69^{+25}_{-15}$
RelxillCp Refl Frac $0.78^{+0.7}_{-0.2}$ $0.35 \pm 0.06$
RelxillCp Fe abund
RelxillCp $\log \xi$ $3.64 \pm 0.12$ $<2$
RelxillCp Norm ($\times 10^{-3}$) $2.7 \pm 0.2$ $4.14^{+0.10}_{-0.09}$
total $\chi^2$ (dof)
----------- ----------------------------------------------------- --------------------------------- ---------------------------
: The best fit parameters of the spectral fitting of the [[*XMM-Newton*]{}]{} ($0.6-10$ keV energy band) and [[*XMM-Newton*]{}]{} + [[*NuSTAR*]{}]{} ($0.6-70$ keV energy band) spectra of . The continuum emission is described by a combination of a blackbody and the Comptonization component [*nthComp*]{} included in the self-consistent reflection model [*relxillCp*]{}, modified at lower energy by photoelectric absorption from neutral matter modeled with [*phabs*]{}. The blackbody luminosity is given in units of $L_{36}/D_{10}^2$, where $L_{36}$ is the bolometric luminosity in units of $10^{36}$ ergs/s and $D_{10}$ the distance to the source in units of 10 kpc. The blackbody radius is calculated in the hypothesis of spherical emission and for a distance of 3.5 kpc. Uncertainties are given at $90\%$ confidence level.
\[Tabfin\]
Discussion
==========
In this paper we have analyzed broad-band X-ray spectra acquired during the 2015 outburst of the AMSP observed by [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{}. The [[*XMM-Newton*]{}]{} ToO was performed at the peak of the outburst on 2015 April 11 for a total observing time of 110 ks, which resulted in an effective on-source exposure of $\sim 80$ ks. The [[*NuSTAR*]{}]{} observation was performed approximately four days later, on 2015 April 15, and resulted in 49 ks of exposure per each of the [[*NuSTAR*]{}]{} modules. In this way, we have obtained a broad-band (from 0.6 to 70 keV), moderately high-resolution spectrum of the source.
Comparison with previous spectral results
-----------------------------------------
has been previously observed with good energy resolution by [[*XMM-Newton*]{}]{} and [*Suzaku*]{}, approximately 1 day apart, during the 2008 outburst. In that occasion, a broad iron line was detected in both the [[*XMM-Newton*]{}]{} and [*Suzaku*]{} spectra, with very similar profiles, and was fitted by a [*diskline*]{} [see P09, @Cackett.etal:09]. The inner disk radius derived in this way was R$_{in} = 8.7 \pm 0.4$ R$_g$ using [[*XMM-Newton*]{}]{} (P09) and R$_{in} = 13.2 \pm 2.5$ R$_g$ from a joint fit of the [[*XMM-Newton*]{}]{} and [*Suzaku*]{} spectra [@Cackett.etal:09], respectively. In both cases, the inner disk radius was consistent with being inside the co-rotation radius, which is the radius at which the magnetosphere rotation velocity equals that of an assumed Keplerian disc, R$_{co} = (G M_{NS}
/ \Omega^2)^{1/3}$, where $M_{NS}$ is the NS mass and $\Omega$ its spin angular velocity. For the case of , the co-rotation radius is R$_{co} = 31\,
m_{1.4}^{1/3}$ km, where $m_{1.4}$ is the NS mass in units of $1.4\, M_\odot$. This has to be compared with the inner disk radius inferred from the reflection component, that is R$_{in} < 25.6\, m_{1.4}$ km (at $90\%$ confidence level, P09). This is thought to be a necessary condition in order to observe coherent pulsations in accreting pulsars and to avoid efficient propeller ejection of matter due to the centrifugal barrier [according to the standard theory of accretion onto fast rotators, see e.g. @Ghosh.Lamb:1979]. Assuming that the inner disk (as measured by the Fe line) is truncated at the magnetospheric radius this implies a magnetic field strength of $\sim 3 \times 10^8$ Gauss at the magnetic poles [@Cackett.etal:09]. Interestingly, this estimate is consistent with other independent estimates of the magnetic field strength in this source based on completely different arguments . Also, both the [[*XMM-Newton*]{}]{} and [*Suzaku*]{} spectra gave a low ionization state of iron, inferred from the line centroid always consistent with 6.4 keV (corresponding to the rest-frame energy of the K$\alpha$ transition of neutral or moderately ionized iron), and a high inclination angle of the system with respect to the line of sight, always above $50^\circ$.
Our analysis of the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra of during the 2015 outburst gave remarkably similar results as regards the smearing parameters, both when fitting the iron line profile with a [*diskline*]{} and when using a self-consistent reflection model. These results are also very similar to those obtained for the 2008 [[*XMM-Newton*]{}]{} observation (P09). To show the agreement between these results we report in Table \[tab3\] the results obtained by P09 from the [[*XMM-Newton*]{}]{} observation performed in 2008 and our results from the [[*NuSTAR*]{}]{} spectrum obtained in 2015 when fitting the line profile with a [*diskline*]{}. Again we find a low ionization parameter, a large inclination angle ($i > 70^\circ$), and a small inner disk radius (less than 7 R$_g$, corresponding to R$_{in} < 15\, m_{1.4}$ km). Even when we fit together the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra with a self-consistent reflection model, we get disk parameters very similar to those obtained with the simple [*diskline*]{}. In particular for the [[*NuSTAR*]{}]{} spectrum, the ionization parameter, $\log \xi$, is less than 2.7, the emissivity index of the disk is around 2 (compatible with the presence of a central illuminating source), the inclination angle is quite high, $> 50$ deg, and the inner disk radius is constrained between (best estimate) 7.5 and 11.5 R$_g$, corresponding to $16-24$ km for a 1.4 M$_\odot$ NS, well within the co-rotation radius. Note that, despite the fact that the spectrum is typical of a hard state, the inner disk radius is quite close to the NS surface, implying that the disk is truncated not too far from the compact object in the hard state. This is also observed in other NS/LMXBs in the hard state and, as noted above, is a necessary condition to avoid a strong propeller effect in the case of a pulsar.
0.5cm
Parameter [[*XMM-Newton*]{}]{} (2008) [[*NuSTAR*]{}]{} (2015) [[*XMM-Newton*]{}]{} (2015)
------------------------- ----------------------------- ------------------------- -----------------------------
$E_{Fe}$ (keV) $6.43 \pm 0.08$ $6.38 \pm 0.10$ $6.70 \pm 0.05$
$Betor$ $-2.3 \pm 0.3$ $-2.0 \pm 0.2$ $-2.16 \pm 0.10$
$R_{in}$ ($G M / c^2$) $8.7 \pm 0.4$ $< 7$ $10^{+8}_{-3}$
$R_{out}$ ($G M / c^2$) $127 \div 318$ $280 \div 1070$ $760 \div 1780$
Incl (deg) $> 58$ $> 70$ $>58$
EqW (eV) $120 \pm 20$ $120 \pm 20$ $58 \pm 8$
Bol $L_X$ (erg/s) $6.6 \times 10^{36}$ $3.6 \times 10^{36}$ $3.1 \times 10^{36}$
: Comparison of the best-fit [*diskline*]{} parameters obtained for the 2008 outburst as observed in the 60-ks [[*XMM-Newton*]{}]{} observation [@Papitto.etal:09] and for the 2015 outburst as observed by [[*NuSTAR*]{}]{} (this paper). For comparison we also show the best-fit [*diskline*]{} parameters obtained for the 2015 80-ks [[*XMM-Newton*]{}]{} observation, when the source appears to be in a soft state. Bol $L_X$ is the bolometric luminosity extrapolated in the $0.05 - 150$ keV energy range during the observation and assuming a distance to the source of $3.5$ kpc.
\[tab3\]
A small inner disk radius is also implied for most of the other AMSPs for which a spectral analysis has been performed and a broad iron line has been detected in moderately high resolution spectra. The AMSP IGR J17511-3057, observed by [[*XMM-Newton*]{}]{} for 70 ks and [*RXTE*]{} [@Papitto.etal:10], shows both a broad iron line and the Compton hump at $\sim 30$ keV. In this case, the inner disc radius was at $\ge 40$ km for a $1.4\, M_\odot$ NS, with an inclination angle between $38^\circ$ and $68^\circ$ [see also @Papitto.etal:2016]. The AMSP and transitional pulsar IGR J18245-2452 observed by [[*XMM-Newton*]{}]{} [@Papitto.etal:2013a], showed a broad iron line at 6.7 keV (identified as K$\alpha$ emission from Fe XXV) with a width of $\sim 1.6$ keV, corresponding to R$_{in} \simeq 17.5$ R$_g$ or $\sim 36.7$ km for a $1.4\, M_\odot$ NS. For comparison, the inner disk radius derived from the blackbody component was $28 \pm 5$ km. The (intermittent) AMSP HETE J1900-2455, observed by [[*XMM-Newton*]{}]{} for $\sim 65$ ks [@Papitto.etal:13b], showed a broad iron line at 6.6 keV (Fe XXIII-XXV) and an intense and broad line at $\sim 0.98$ keV, visible both in the pn and in the RGS spectrum, compatible with being produced in the same disk region. In this case, the inner disc radius was $25 \pm 15\, R_g$, with an inclination angle between $27^\circ$ and $34^\circ$. The (intermittent) AMSP SAX J1748.9-2021, observed by [[*XMM-Newton*]{}]{} for $\sim 115$ ks and [*INTEGRAL*]{} [@Pintore.etal:2016], was caught at a relatively high luminosity of $\sim 5 \times 10^{37}$ erg/s corresponding to $\sim 25\%$ of the Eddington limit for a $1.4\, M_\odot$ NS, and, exceptionally for an AMSP, showed a spectrum compatible with a soft state. The broad-band spectrum is in fact dominated by a cold thermal Comptonization component ($\sim 2$ keV) and shows an additional hard X-ray emission described by a power-law (photon index $\Gamma \sim 2.3$), typically detected in LMXBs in the soft state [see e.g. @DiSalvo.etal:2000]. In addition, a number of broad (Gaussian $\sigma = 0.1 - 0.4$ keV) emission features, likely associated to reflection processes, have been observed in the XMM-Newton spectrum. A broad iron line was observed at an energy of $\sim 6.7-6.8$ keV, consistent with a Fe XXV K$\alpha$ transition produced in the disc at a distance of $\sim 20-43\, R_g$ ($\sim 42 - 90$ km) with an inclination angle of $\sim 38-45^\circ$. The other broad emission lines may be associated to K-shell emission of highly ionized elements, and are compatible with coming from the same emission region as the iron line. A moderately broad, neutral Fe emission line has been observed during the 2015 outburst of IGR J00291+5934 observed by [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} [@Sanna.etal:2016]. Fitted with a Gaussian profile the line centroid was at an energy of $6.37 \pm 0.04$ keV with a $\sigma = 80 \pm 70$ eV, while using a [*diskline*]{} profile, the line parameters were poorly constrained. Finally, the newly discovered AMSP MAXI J0911-655, observed by [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} [@Sanna.etal:2016b], shows the presence of a weak, marginally significant and relatively narrow emission line in the range $6.5-6.6$ keV, modelled with a gaussian profile with $\sigma$ ranging between 0.02 and 0.2 keV, which was identified with K$\alpha$ emission from moderate to highly ionized iron.
Detailed discussion of the [[*NuSTAR*]{}]{} and [[*XMM-Newton*]{}]{} spectra: similarities and differences
----------------------------------------------------------------------------------------------------------
The use of a self-consistent reflection model instead of a [*diskline*]{} for the [[*NuSTAR*]{}]{} spectrum of gives an improvement of the fit (corresponding to a $\Delta \chi^2 \simeq 9$ with one parameter less). This demonstrate that the Compton hump is significantly detected in the spectrum and that the line profile parameters are in good agreement with those needed to fit the whole reflection component. The best-fit results strongly suggest a moderate overabundance of iron by approximately a factor between 2 and 3. However, this overabundance may be indicative of a disk density higher than the value of $10^{15}$ cm$^{-3}$ assumed in the [*relxill*]{} model. Indeed, @Garcia.etal:2016 have shown that the iron abundance is sensitive to the density used in calculating the reflection model, and that when assuming a higher-density disk a lower iron abundance is obtained [see also @Garcia.etal:2018 and references therein]. Hence, this overabundance should be confirmed using appropriate reflection models in which the density in the disk can be varied. There are versions of the [*relxill*]{} model which allow to vary the disk density, although at the moment these models have a high energy cutoff of the illuminating continuum fixed to 300 keV, that is much higher than the temperature of the Comptonization continuum we find in the spectrum of .
We have also fitted the value of the outer radius of the emitting region in the disk; this parameter should be always let free to vary in high statistics spectra, since it strongly correlates with the inclination angle of the system and fixing this parameter may result in an artificially narrow uncertainty for the inclination angle. In all our fitting, the value of the outer radius of the emitting region in the disk, was quite undetermined, although compatible (within the large uncertainties) with the value derived by P09 during the 2008 outburst ($130 \div 320\, R_g$). Usually in other bright LMXBs, when it is possible to let this parameter free, the best fit value is most of the times very high, above $2500\, R_g$ [see e.g. @Di_Salvo.etal:09; @Iaria.etal:2009], as it should be expected if the entire disk is illuminated and emits a reprocessed spectrum. However, is a transient with short outbursts, and therefore during the outburst at least the innermost part of the disk is emptying in few days. This means that the disc, over which reflection takes place, may have a ring-like shape with an outer disk radius which is relatively close to the inner disc radius. Therefore, it is possible that the small outer disk radius inferred from the reflection component in during the 2008 outburst may be an indication that the accretion disk was already emptying or that some disk parameters may change abruptly towards the outer disk in short-duration outbursts. Unfortunately, we were not able confirm this result with these observations. Future observations at high energy resolution and high statistics (perhaps taken during the decay phase of the outburst) might be able to confirm this finding which may give important information on the evolution of the accretion disk in this kind of transients.
The best-fit continuum model that we find for the [[*NuSTAR*]{}]{} spectrum of is also very similar to that already used by P09 to fit the [[*XMM-Newton*]{}]{} spectrum during the 2008 outburst, with the difference that P09 used two soft components, a blackbody (at a temperature kT $\sim 0.4$ keV) and a multicolor disk blackbody (at kT$_{in} \sim 0.2$ keV), and a power-law to fit the Comptonization component, while we use a blackbody and a Comptonization model that include a soft (Wien spectrum) component as a seed photon. In our fit, the seed photon temperature for Comptonization (kT$_{seed} < 0.29$ keV) is comparable to the disk blackbody temperature in the P09 deconvolution, possibly indicating that both the disk and the NS contribute to the seed photons for Comptonization. The 2015 [[*NuSTAR*]{}]{} spectrum also requires a blackbody component with a temperature of $\sim 0.7$ keV, slightly higher than the blackbody temperature reported by P09, and a spherical radius of the emitting region of $\sim 3$ km, a factor $\sim 2$ lower than the blackbody radius reported by P09. The value of the equivalent hydrogen column to the source is very precisely determined by the [[*XMM-Newton*]{}]{} spectrum, $N_H \simeq 0.15 \times 10^{22}$ cm$^{-2}$, and is slightly lower than that derived by P09 during the 2008 outburst.
On the other hand the [[*XMM-Newton*]{}]{} spectrum of taken in 2015 looks quite different from the 2015 [[*NuSTAR*]{}]{} spectrum and the 2008 [[*XMM-Newton*]{}]{} spectrum of the source. The blackbody component is found at a low temperature, $\sim 0.1$ keV, very close to the seed photon temperature ($kT_{seed} \simeq 0.15$ keV), and the corresponding radius of the blackbody emitting region, assuming a spherical geometry and not considering color corrections, results to be $150 - 200$ km, much larger than the inner radius inferred from reflection features (cf. Tab. \[Tabtot\]). The Comptonization spectrum appears much softer, with an electron temperature around $5-8$ keV. Moreover, several emission lines are observed in the [[*XMM-Newton*]{}]{} spectrum, identified with $K\alpha$ transitions of highly ionized (He-like or H-like) elements (S XVI $-$ rest-frame energy $2.623$ keV, Ar XVIII $-$ rest-frame energy $3.323$ keV, Ca XIX$-$XX $-$ rest-frame energy $3.902$ and $4.108$ keV, respectively, and Fe XXV $-$ rest-frame energy $6.70$ keV). The smearing parameters of these lines are compatible to be the same, and appear very similar to what we find for the smearing parameters of the [[*NuSTAR*]{}]{} spectrum (emissivity index -$2.1 - 2.3$, inner radius $6 - 18$ R$_g$, outer radius $\sim 1000$ R$_g$, inclination $50 - 65$ deg). When we try to fit the reflection component (Fe line and Compton hump) with the self-consistent reflection model [*relxill*]{} we find parameters very similar to those obtained for the [[*NuSTAR*]{}]{} spectrum, except for a high value of the ionization parameter ($\log \xi \sim 3.7$), in agreement with the high energies of the emission lines, which would require $\log \xi > 2$. Most probably the [[*XMM-Newton*]{}]{} spectrum of taken in 2015 corresponds to a transition spectrum, in line with the fact that the [[*XMM-Newton*]{}]{} observation was taken at the very beginning of the outburst, that evolved to the more standard [[*NuSTAR*]{}]{} spectrum a few days after. If this is the case, experienced a soft to hard transition at the beginning of the outburst, that has never been observed before for an AMSP. Note that a spectral transition has been observed for the 11 Hz X-ray pulsar IGR J17480-2446 in the globular cluster Terzan 5 during its X-ray outburst in 2010 [@Papitto.etal:2012]. However, in that case, a hard to soft state transition was observed during the outburst rise. Unfortunately, the lack of high-energy coverage strictly simultaneous to the [[*XMM-Newton*]{}]{} observation of does not allow us to put further constraints on the high-energy spectrum, or to look for the presence of hard continuum components or a complex reflection component.
Binary inclination and mass of the neutron star
-----------------------------------------------
As stated above, despite the differences between the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra we find quite similar values for the parameters of the reflection component and relativistic smearing. We therefore tried to fit simultaneously these spectra in order to increase the statistics of the reflection features and improve the constraints on the corresponding best-fit parameters. We therefore let free to vary all the parameters of the continuum emission, except for the $N_H$, that was not constrained in the [[*NuSTAR*]{}]{} spectrum alone, and tied together the parameters of the relativistic smearing, with the iron abundance fixed at 2 times the Solar value. Fitting the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra together, we could get a precise estimate of the inner disk radius, which is constrained between 7.5 and 11.5 $R_g$, while the best estimate of the system inclination, constrained between $58^\circ$ and $64^\circ$, comes from the fitting with [*relxillCp*]{} of the [[*XMM-Newton*]{}]{} spectrum (see Tab. \[Tabtot\]). We also find evidence in the [[*XMM-Newton*]{}]{} spectrum of the presence of some absorption discrete features (see Tab. \[Tabfin\]), namely an absorption edge at $\sim 7.4$ keV from neutral or mildly ionized iron and at least two absorption lines, possibly from $K\alpha$ transitions of highly ionized (He-like) Ne IX (at 0.947 keV) and Mg XI (at 1.372 keV). These lines appear relatively broad (implying a velocity dispersion of $\sigma_v \sim 1 \% \,c$) and blue-shifted at a velocity a few percent the speed of light. If confirmed, these lines may suggest the presence of a weakly relativistic outflowing wind towards the observer. Absorption lines from ionized elements are usually observed in high-inclination ($60-70^\circ$) sources [see e.g. @Pintore.etal:2014 and references therein] and therefore their presence in the [[*XMM-Newton*]{}]{} spectrum may support the possibility of a high inclination, $i \sim 60^\circ$, in .
This estimate is compatible with previous estimates based on the fitting of the iron line profile during the 2008 outburst observed by [[*XMM-Newton*]{}]{} , and is also consistent with the inclination of $60^\circ
\pm 5^\circ$ given by @Ibragimov.Poutanen:2009 from a detailed analysis of the 2002 outburst from the source, as well as with the inclination range from $36^\circ$ to $67^\circ$ given by @Deloye.etal:2008 studying the optical modulation along the orbital period of the system observed during a quiescence period in 2007; these authors also suggest a pulsar mass $> 2.2\,
M_\odot$. A high inclination is qualitatively in agreement with the claim of a massive NS ($> 1.8\, M_\odot$) and a low mass companion star, a brown dwarf with $< 0.1\, M_\odot$, as suggested by @Bildsten.etal:2001 [see also @DiSalvo.etal:2008; @Burderi.etal:2009]. Finally, a recent estimate of the inclination angle to the system comes from a time-resolved optical imaging of SAX J1808.4-3658 during its quiescent state and 2008 outburst. A Markov chain Monte Carlo technique has been used to fit the multi-band light curve of the source in quiescence with an irradiated star model, and a tight constraint of $50^{+6}_{-5}$ deg has been derived on the inclination angle [@Wang.etal:2013]. This implies a constraint on the mass of the pulsar and its companion star, which are inferred to be $0.97^{+0.31}_{-0.22}\, M_\odot$ and $0.04^{+0.02}_{-0.01}\, M_\odot$ (both at $1\sigma$ confidence level), respectively.
However, high values for the inclination angle of the system look at odd when considered together with optical estimates of the radial velocity of the companion star. From phase resolved optical spectroscopy and photometry of the optical counterpart to , obtained during the 2008 outburst, @Elbert.etal:2009 reveals a focused spot of emission at a location consistent with the secondary star. The velocity of this emission is estimated at $324 \pm 15$ km/s; applying a “K-correction”, the authors estimate the velocity of the secondary star projected on to the line of sight to be $370 \pm 40$ km/s [see also @Cornelisse.etal:2009]. This estimate, coupled with a high inclination angle of the system, gives very low values for the NS mass, and has been used to argue against the presence of a heavy NS in this system. In fact, the pulsar mass can be estimated using the following relation: $M_1 \sin^3 i / (1+q)^2 = K_2^3 P_{orb}/(2\pi G)$, where $M_1$ is the pulsar mass, $q = M_2/M_1$ is the mass ratio of the system, $P_{orb}$ is the orbital period of the system, and $K_2$ is the radial velocity of the companion star of mass $M_2$. Using the estimated radial velocity of the companion star together with our best-fit value for the inclination angle, we find a pulsar mass in the range: $M_1 = 0.5 \div 0.8 \, M_\odot$. This range of masses for a NS is unacceptable and casts serious doubts on the estimates of the radial velocity of the companion and/or on a high inclination angle for the system.
A possibility we can imagine is that the reflection is measuring the inclination with respect to the sight of the inner part of the accretion disk, that may be different from the binary inclination. If the inner accretion disk is tilted with respect to the orbital plane, for instance because of the action of the NS magnetic field, such that the inner disk is observed at high inclination, than this could explain why measured inclination of the inner disk can be different from the binary inclination. However, this would not explain the high inclination angle measured by @Wang.etal:2013 during X-ray quiescence.
The other possibility is that the problem comes from measurements of the companion radial velocity. Note that the reported measurements of the radial velocity $K_2$ are still affected by large uncertainties. This is because these measurements are taken during X-ray outburst and are affected by the presence of the accretion disk and the strong irradiation of the companion star. These estimates should therefore be confirmed in order to obtain a reliable estimate of the NS mass.
Conclusions
===========
In summary, we have reported a detailed spectral analysis of the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra of during the latest outburst in 2015. The main results of this study are described in the following. The [[*XMM-Newton*]{}]{} spectrum, taken for the first time at the beginning of the outburst, appears to be much softer than what is usually found for this source and quite puzzling, while the broad-band [[*NuSTAR*]{}]{} spectrum, acquired a few days after, gives results perfectly compatible with those found from the [[*XMM-Newton*]{}]{} observation performed in 2008. Despite the differences present between the [[*XMM-Newton*]{}]{} and [[*NuSTAR*]{}]{} spectra taken in 2015, we could fit simultaneously the smeared reflection component in these spectra. In particular, we find that the reflection component requires a ionization parameter of $\log \xi \sim 2.4$ for the [[*NuSTAR*]{}]{} spectrum and a higher value, $\log \xi \sim 3.8$ for the [[*XMM-Newton*]{}]{} spectrum, and strong evidence of an overabundance of iron by a factor two with respect to the solar abundance, although this may be due to a relatively high density in the disk. Also, the smearing parameters are very similar to those found with [[*XMM-Newton*]{}]{} during the 2008 outburst. The emissivity index of the disc is $\sim -2$, consistent with a dominating illuminating central source, and we find that the upper limit to the inner disk radius is $\sim 12~R_g$, compatible with an inner disk radius smaller than the corotation radius. We also give a precise measure of the inclination angle of the system, which results around $60^\circ$, in agreement with previous spectral results, as well as with the results of fitting the reflection component in each spectrum with empirical models (disklines). A high-inclination angle for this system is also supported by the presence of absorption discrete features in the [[*XMM-Newton*]{}]{} spectrum, although these detections should be confirmed by further spectroscopic studies. The high inclination of the system with respect to our line of sight, when combined with available measurements of the radial velocity of the optical companion, poses, however, a problem as regards the correct determination of the mass of the NS in this systems, and therefore deserves further investigation.
[^3]
[^1]: E-mail:[email protected]
[^2]: More details can be found in the following webpage: http://www.sternwarte.uni-erlangen.de/ dauser/research/relxill/
[^3]: We thank the unknown referee for her/his suggestions that certainly improved the quality of the manuscript. We also thank Dr. Matranga for collaborating to a first draft of this paper. We acknowledge financial contribution from the agreement ASI-INAF I/037/12/0. A.P. acknowledges funding from the European Union’s Horizon 2020 Framework Programme for Research and Innovation under the Marie Sklodowska-Curie Individual Fellowship grant agreement 660657-TMSP-H2020-MSCA-IF-2014. We acknowledges support from the HERMES Project, financed by the Italian Space Agency (ASI) Agreement n. 2016/13 U.O, as well as fruitful discussion with the international team on “The disk-magnetosphere interaction around transitional millisecond pulsars” at the International Space Science Institute, Bern.
|
---
abstract: 'Reconstruction of directional fields is a need in many geometry processing tasks, such as image tracing, extraction of 3D geometric features, and finding principal surface directions. A common approach to the construction of directional fields from data relies on complex optimization procedures, which are usually poorly formalizable, require a considerable computational effort, and do not transfer across applications. In this work, we propose a deep learning-based approach and study the expressive power and generalization ability.'
author:
- 'Maria Taktasheva[^1]'
- 'Albert Matveev[ ^fnsymbol[1]{}^]{}'
- Alexey Artemov
- Evgeny Burnaev
title: Learning to Approximate Directional Fields Defined over 2D Planes
---
[^1]: These two authors contribute equally to the work.
|
GUTPA/01/04/02
.1in
[**J.L. Chkareuli**]{}
[*Institute of Physics, Georgian Academy of Sciences, 380077 Tbilisi, Georgia\
*]{}
[**C.D. Froggatt**]{}
[*Department of Physics and Astronomy\
Glasgow University, Glasgow G12 8QQ, Scotland\
*]{}
[**H.B. Nielsen**]{}
[*Niels Bohr Institute,\
Blegdamsvej 17-21, DK 2100 Copenhagen, Denmark*]{}
In this letter we reconsider the role of Lorentz invariance in the dynamical generation of the observed internal symmetries. We argue that, generally, Lorentz invariance can only be imposed in the sense that all Lorentz non-invariant effects caused by the spontaneous breakdown of Lorentz symmetry are physically unobservable. Remarkably, the application of this principle to the most general relativistically invariant Lagrangian, with arbitrary couplings for all the fields involved, leads by itself to the appearance of a symmetry and, what is more, to the massless vector fields gauging this symmetry in both Abelian and non-Abelian cases. In contrast, purely global symmetries are only generated as accidental consequences of the gauge symmetry.
It is still a very attractive idea that a local symmetry for all the fundamental interactions of matter and the corresponding massless gauge fields could be dynamically generated (see [@book] and extended references therein). In particular there has been considerable interest [@bj] in the interpretation of gauge fields as composite Nambu-Jona-Lasinio (NJL) bosons [@njl], possibly associated with the spontaneous breakdown of Lorentz symmetry (SBLS). However, in contrast to the belief advocated in the pioneering works [@bj], there is a generic problem in turning the composite vector particles into genuine massless gauge bosons [@suzuki].
In this note we would like to return to the role of Lorentz symmetry in a dynamical generation of gauge invariance. We argue that, generally, Lorentz invariance can only be imposed in the sense that all Lorentz non-invariant effects caused by its spontaneous breakdown are physically unobservable. We show here that the physical non-observability of the SBLS, taken as a basic principle, leads to genuine gauge invariant theories, both Abelian and non-Abelian, even though one starts from an arbitrary relativistically invariant Lagrangian. In the original Lagrangian, the vector fields are taken as massive and all possible kinetic and interaction terms are included. However, when SBLS occurs and its non-observability is imposed, the vector bosons become massless and the only surviving interaction terms are those allowed by the corresponding gauge symmetry. Thus, the Lorentz symmetry breaking does not manifest itself in any physical way, due to the generated gauge symmetry converting the SBLS into gauge degrees of freedom of the massless vector bosons. Remarkably, even global symmetries are not required in the original Lagrangian—the SBLS induces them automatically as accidental symmetries accompanying the generated gauge theory.
In order to consider general interactions between a vector field and fermionic matter, it is convenient to use 2-component left-handed Weyl fields $\psi _{Li}$ to represent the fermions. For simplicity we shall consider the case of two Weyl fields ($i=1,2$), which will finally be combined to form a Dirac-like field $\psi $ = ${{\psi _{L1} \choose \psi _{L2}^{\dagger }}}$ in Weyl representation.
The most general Lagrangian density, only having terms of mass dimension 4 or less, for a theory containing a pure spin-1 vector field and two Weyl fermions is: $$\begin{aligned}
L(A,\psi ) &=&-\frac{1}{4}F_{\mu \nu }F_{\mu \nu }+\frac{1}{2}M^{2}A_{\mu
}^{2}+i\sum_{j=1}^{2}\psi _{Lj}^{\dagger }\sigma _{\mu }\partial _{\mu }\psi
_{Lj} \nonumber \\
&&-\sum_{j,k=1}^{2}\epsilon ^{\alpha \beta }
(m_{jk}\psi _{Lj\alpha }\psi _{Lk\beta } + m_{jk}^{\star }
\psi _{Lj\alpha }^{\dagger }\psi_{Lk\beta }^{\dagger }) \nonumber \\
&&+\sum_{j,k=1}^{2}e_{jk}A_{\mu }\psi _{Lj}^{\dagger }\sigma _{\mu }\psi
_{Lk}+\frac{f}{4}A_{\mu }^{2}\cdot A_{\mu }^{2} \label{L}\end{aligned}$$ with the Lorentz condition ($\partial _{\mu }A_{\mu }=0$) imposed as an off-shell constraint, singling out a genuine spin-1 component in the four-vector $A_{\mu }$. This constraint also ensures that, after an appropriate scaling, the kinetic term for $A_{\mu}$ can be written in the usual $-\frac{1}{4}F_{\mu \nu }F_{\mu \nu}$ form. Note that $m_{jk}=m_{kj}$, as a consequence of Fermi statistics. It is always possible to simplify this Lagrangian density, by defining two new left-handed Weyl spinor fields which transform the “charge term” $\sum_{j,k=1}^{2}e_{jk}A_{\mu }\psi
_{Lj}^{\dagger }\sigma _{\mu }\psi _{Lk}$ into the diagonal form $%
\sum_{k=1}^{2}e_{k}A_{\mu }\psi _{Lk}^{\dagger }\sigma _{\mu }\psi _{Lk}$.
Let us consider now the SBLS in some detail. We propose that the vector field $A_{\mu }$ takes the form $$A_{\mu }=a_{\mu }(x)+n_{\mu } \label{f}$$ when the SBLS occurs. Here the constant Lorentz four-vector $n_{\mu }$ is a classical background field appearing when the vector field $A_{\mu }$ develops a vacuum expectation value (VEV). Substitution of the form (\[f\]) into the Lagrangian (\[L\]) immediately shows that the kinetic term for the vector field $A_{\mu }$ translates into a kinetic term for $a_{\mu }$ ($F_{\mu \nu }^{(A)}=F_{\mu \nu }^{(a)}$), while its mass and interaction terms are correspondingly changed. As to the interaction term, one can always make a unitary transformation to two new Weyl fermion fields ${\bf \Psi }_{Lk}$ $$\psi _{Lk}=\exp [ie_{k}\omega (x)]\textrm{ }{\bf \Psi }_{Lk}\
\textrm{, \qquad}
\omega (x)=n\cdot x \label{psi}$$ so that the Lorentz symmetry-breaking term $n_{\mu }\cdot
\sum_{k=1}^{2}e_{k}\psi _{Lk}^{\dagger }\sigma _{\mu }\psi _{Lk}$ is exactly cancelled in the Lagrangian density $L(a_{\mu }+n_{\mu },\psi ).$ This cancellation occurs due to the appearance of a compensating term from the fermion kinetic term, provided that the phase function $\omega (x)$ is chosen to be linear[@ferrari] in the coordinate four-vector $x_{\mu }$ (as indicated in Eq. \[psi\]). However, in general, the mass terms will also be changed under the transformation (\[psi\]): $$m_{jk}\psi _{Lj\alpha }\psi _{Lk\beta }\rightarrow m_{jk}\exp \left[
i(e_{j}+e_{k})n\cdot x\right] {\bf \Psi }_{Lj\alpha }{\bf \Psi }_{Lk\beta }
\label{mass}$$ If $e_{j}+e_{k}\neq 0$ for some non-zero mass matrix element $m_{jk}$, the transformed mass term will manifestly depend on $n_{\mu }$ through the translational non-invariant factor $\exp \left[ i(e_{j}+e_{k})n\cdot
x\right] $, which in turn will visibly violate Lorentz symmetry. So our main assumption of the unobservability of SBLS implies that we can only have a non-zero value for $m_{jk}$ when $e_{j}+e_{k}=0$.
After imposing these conditions on the charges, the remaining traces of SBLS are contained in the vector field mass term and the $A_{\mu }^{2}\cdot
A_{\mu }^{2}$ term. Thus the remaining condition for the non-observability of SBLS becomes: $$\lbrack M^{2}\textrm{ }+f(a^{2}+(n\cdot a)+n^{2})](n\cdot a)=0 \label{cons**}$$ An extra gauge condition $n\cdot a$ $\equiv n_{\mu }\cdot a_{\mu }=0$ would be incompatible with the Lorentz gauge ($\partial _{\mu }A_{\mu }=0$) already imposed on the vector field $a_{\mu }$. Therefore, the only way to satisfy Eq. (\[cons\*\*\]) is to take $M^{2}=0$ and $f=0$. Otherwise it would either represent an extra gauge condition on $a_{\mu }$, or it would impose another dynamical equation in addition to the usual Euler equation for $a_{\mu }$.
Thus imposing the non-observability of SBLS, the Lorentz gauge restriction and the presence of terms of only dimension 4 or less has led us to the Lagrangian density for chiral electrodynamics, having the form: $$\begin{aligned}
L &=&-\frac{1}{4}F_{\mu \nu }F_{\mu \nu }+i\sum_{k=1}^{2}{\bf \Psi ^{\dagger
}}_{Lk}\sigma _{\mu }(\partial _{\mu }-ie_{k}a_{\mu }){\bf \Psi }_{Lk}
\nonumber \\
&&-\sum_{j,k=1}^{2}(m_{jk}{\bf \Psi }_{Lj\alpha }{\bf \Psi }_{Lk\beta
}\epsilon ^{\alpha \beta }+h.c.) \label{Lcqed}\end{aligned}$$ with the restriction that $m_{jk}=0$ unless $e_{j}+e_{k}=0$. In general, i.e. when $\sum_{k}e_{k}^{3}\neq 0$, even this Lagrangian density will lead to the observability of the SBLS, because of the presence of Adler-Bell-Jackiw anomalies [@ABJ] in the conservation equation for the current $j_{\mu }^{A}=\sum_{k}e_{k}{\bf \Psi ^{\dagger }}_{Lk}\sigma _{\mu}
{\bf \Psi }_{Lk}$ coupled to $A_{\mu }$. We are now interpreting the ${\bf
\Psi }_{Lk}$ as the physical fermion fields. However, in momentum representation, the transformation (\[psi\]) corresponds to displacing the momentum of each fermion by an amount $e_{k}n_{\mu }$ This induces a breakdown of momentum conservation, which can only be kept unobservable as long as the charge associated with the current $j_{\mu }^{A}$ is conserved. This means that an anomaly in the current conservation will also violate momentum conservation by terms proportional to $n_{\mu }$. Such a breaking of momentum conservation would also give observable Lorentz symmetry violation. So the only way to satisfy our non-observabilty of SBLS principle is to require that the no gauge anomaly condition $$\sum_{k}e_{k}^{3}=0 \label{anomaly}$$ be fulfilled. For the simple case of just two Weyl fields, this means that the two charges must be of equal magnitude and opposite sign, $e_{1}+e_{2}=0$. This is also precisely the condition that must be satisfied for a non-zero mass matrix element $m_{12}=m_{21}\neq 0$. If the charges are non-zero, the diagonal (Majorana) mass matrix elements vanish, $m_{11}=m_{22}=0$, and the two Weyl fields correspond to a massive particle described by the Dirac field ${\bf \Psi }$ = ${{\bf \Psi }_{L1} \choose {\bf \Psi }_{L2}^{\dagger }}$. Thus we finally arrive at gauge invariant QED as the only version of the theory which is compatible with physical Lorentz invariance when SBLS occurs.
Let us now consider the many-vector field case which can result in a non-Abelian gauge symmetry. We suppose there are a set of pure spin-1 vector fields, $A_{\mu }^{i}(x)$ with $i=1,...N$, satisfying the Lorentz gauge condition, but not even proposing a global symmetry at the start. The matter fields are collected in another set of Dirac fields $\psi
=(\psi ^{(1)},...,\psi ^{(r)})$. Here, for simplicity, we shall neglect terms violating fermion number and parity conservation. The general Lagrangian density $L(A_{\mu }^{i},\psi)$ describing all their interactions is given by: $$\begin{aligned}
L = -\frac{1}{4}F_{\mu \nu }^{i}F_{\mu \nu}^{i}
+\frac{1}{2}(M^{2})_{ij}A_{\mu }^{i}A_{\mu }^{j}
+\alpha ^{ijk}\partial _{\nu }A_{\mu
}^{i}\cdot A_{\mu }^{j}A_{\nu }^{k} && \nonumber \\
+\beta ^{ijkl}A_{\mu }^{i}A_{\nu }^{j}A_{\mu }^{k}A_{\nu }^{l}+i
\overline{\psi }\gamma \partial \psi -\overline{\psi }m\psi +A_{\mu }^{i}
\overline{\psi }\gamma _{\mu }T^{i}\psi && \label{LN}\end{aligned}$$ Here $F_{\mu \nu }^{i}=\partial _{\mu }A_{\nu }^{i}-\partial _{\nu }A_{\mu
}^{i}$ , while $(M^{2})_{ij}$ is a general $N\times N$ mass-matrix for the vector fields and $\alpha ^{ijk}$ and $\beta ^{ijkl}$ are dimensionless coupling constants. The $r\times r$ matrices $m$ and $T^{i}$ contain the still arbitrary fermion masses and coupling constants describing the interaction between the fermions and the vector fields (all the numbers mentioned are real and the matrices Hermitian, as follows in this case from the Hermiticity of the Lagrangian density).
We assume that the vector fields $A_{\mu }^{i}$ each take the form $$A_{\mu }^{i}(x)=a_{\mu }^{i}(x)+n_{\mu }^{i} \label{ab}$$ when SBLS occurs; here the constant Lorentz four-vectors $n_{\mu }^{i}$ ($i=1,...N$) are the VEVs of the vector fields. Substitution of the form (\[ab\]) into the Lagrangian density (\[LN\]) shows that the kinetic term for the vector fields $A_{\mu }^{i}$ translates into a kinetic term for the vector fields $a_{\mu }^{i}$ ($F_{\mu \nu }^{(A)}=F_{\mu \nu }^{(a)}$), while their mass and interaction terms are correspondingly changed. Now we consider at first just infinitesimally small $n_{\mu }^{i}$ four vectors. Furthermore we introduce a stronger form of the non-observability of SBLS principle, requiring exact cancellations between non-Lorentz invariant terms of the same structure in the Lagrangian density $L(a_{\mu}^{i}+n_{\mu }^{i},\psi )$ for [*any*]{} set of infinitesimal vectors $n_{\mu }^{i}$. Then we define a new set of vector fields ${\bf a}_{\mu }^{i}$ by the infinitesimal transformation
$$a_{\mu }^{i}={\bf a}_{\mu }^{i}-\alpha ^{ijk}\omega ^{j}(x){\bf a}_{\mu}^{k}
\textrm{ , \quad }\omega ^{i}(x)=n_{\mu }^{i}\cdot x_{\mu } \label{rot}$$
which includes the above coupling constants $\alpha ^{ijk}$ and the linear “gauge” functions $\omega ^{i}(x)$. We require that the Lorentz symmetry-breaking terms in the cubic and quartic self-interactions of the vector fields ${\bf a}_{\mu }^{i}$, including those arising from their kinetic terms, should cancel for [*any*]{} infinitesimal vector $n_{\mu}^{i}$. This condition is satisfied if and only if the coupling constants $\alpha^{ijk}$[** **]{} and $\beta ^{ijkl}$ satisfy the following conditions ([**a**]{}) and ([**b**]{}):
([**a**]{}) $\alpha ^{ijk}$ is totally antisymmetric (in the indices $i$, $j$ and $k)$ and obeys the structure relations:
$$\alpha ^{ijk}{\bf \equiv }\alpha ^{[ijk]}\equiv \alpha _{[jk]}^{i}
\textrm{ ,\quad }\left[ \alpha ^{i},\alpha ^{j} \right] =
-\alpha ^{ijk}\alpha ^{k}
\label{alg}$$
where the $\alpha ^{i}$ are defined as matrices with elements $(\alpha^{i})^{jk}=\alpha ^{ijk}$.
([**b**]{}) $\beta ^{ijkl}$ takes the factorised form:
$$\beta ^{ijkl}=-\frac{1}{4}\alpha ^{ijm}\cdot \alpha ^{klm}. \label{fac}$$
It follows from ([**a**]{}) that the matrices $\alpha ^{k}$ form the adjoint representation of a Lie algebra, under which the vector fields transform infinitesimally as given in Eq. (\[rot\]). In the case when the matrices $\alpha ^{i}$ can be decomposed into a block diagonal form, there appears a product of symmetry groups rather than a single simple group.
Let us turn now to the mass term for the vector fields in the Lagrangian $L(a_{\mu }^{i}+n_{\mu }^{i},\psi )$. When expressed in terms of the transformed vector fields ${\bf a}_{\mu }^{i}$ (\[rot\]) it contains SBLS remnants, which should vanish, of the type:
$$(M^{2})_{ij}(\alpha ^{ikl}\omega ^{k}{\bf a}_{\mu }^{l}{\bf a}_{\mu }^{j}
\textrm{ }{\bf +a}_{\mu }^{i}n_{\mu }^{j}){\bf =}0 \label{mmm}$$
Here we have used the symmetry feature $(M^{2})_{ij}=(M^{2})_{ji}$ for a real Hermitian matrix $M^{2}$ and have retained only the first-order terms in $n_{\mu}^{i} $. These two types of remnant have different structures and hence must vanish independently. One can readily see that, in view of the antisymmetry of the structure constants, the first term in Eq. (\[mmm\]) may be written in the following form containing the commutator of the matrices $M^{2}$ and $\alpha ^{k}$:
$$\left[ M^{2},\alpha ^{k}\right]_{jl}\omega ^{k}{\bf a}_{\mu }^{l}
{\bf a}_{\mu }^{j}=0 \label{com}$$
It follows that the mass matrix $M^{2}$ should commute with all the matrices $\alpha ^{k}$, in order to satisfy Eq. (\[com\]) for all sets of “gauge” functions $\omega ^{i}=n_{\mu }^{i}\cdot x_{\mu }$. Since the matrices $\alpha ^{k}$ have been shown to form an irreducible representation of a (simple) Lie algebra, Schur’s lemma implies that the matrix $M^{2}$ is a multiple of the identity matrix, ($M^{2})_{ij}={\bf M}^{2}\delta_{ij}$, thus giving the same mass for all the vector fields. It then follows that the vanishing of the second term in Eq. (\[mmm\]) leads to the simple condition: $${\bf M}^{2}(n^{i}\cdot {\bf a}^{i})=0 \label{m}$$ for any infinitesimal $n_{\mu }^{i}$. Since the Lorentz gauge condition ($\partial _{\mu }{\bf a}_{\mu}^{i}=0$) has already been imposed, we cannot impose extra gauge conditions of the type $n^{i}\cdot{\bf a}^{i}=n_{\mu }^{i}\cdot {\bf a}_{\mu }^{i}=0$. Thus, we are necessarily led to:
([**c**]{}) massless vector fields, $(M^{2})_{ij}={\bf M}^{2}\delta_{ij\textrm{ }}=0$.
Finally we consider the interaction term between the vector and fermion fields in the ”shifted” Lagrangian density $L(a_{\mu }^{i}+n_{\mu
}^{i},\psi )$. In terms of the transformed vector fields ${\bf a}_{\mu}^{i}$ (\[rot\]), it takes the form
$$({\bf a}_{\mu }^{i}-\alpha ^{ijk}\omega ^{j}{\bf a}_{\mu }^{k}\textrm{ }
+n_{\mu }^{i}{\bf )\cdot }\overline{\psi }\gamma _{\mu }T^{i}\psi
\label{fer}$$
It is readily confirmed that the Lorentz symmetry-breaking terms (the second and third ones) can be eliminated, when one introduces a new set of fermion fields ${\bf \Psi }$ using a unitary transformation of the type:
$$\psi =\exp \left[ iT^{i}\omega ^{i}(x)\right]
{\bf \Psi }\textrm{ , \qquad }\omega^{i}(x)=n^{i}\cdot x \label{ff}$$
One of the compensating terms appears from the fermion kinetic term and the compensation occurs for any set of “gauge” functions $\omega^i(x)$ if and only if:
([**d**]{}) the matrices $T^{i}$ form a representation of the Lie algebra with structure constants $\alpha ^{ijk}$:
$$\left[ T^{i},T^{j}\right] = i\alpha ^{ijk}T^{k}. \label{TTT}$$
In general this will be a reducible representation but, for simplicity, we shall take it to be irreducible here. This means that the matter fermions ${\bf \Psi }$ are all assigned to an irreducible multiplet determined by the matrices $T^{i}$. At the same time, the unitary transformation (\[ff\]) changes the mass term for the fermions to
$$\overline{{\bf \Psi }}\left( m+i\omega ^{k}\left[ m,T^{k}\right]
\right){\bf \Psi }
\label{k}$$
The vanishing of the Lorentz non-invariant term (the second one) in Eq. (\[k\]) for any set of “gauge” functions $\omega^i(x)$ requires that the matrix $m$ should commute with all the matrices $T^{k}$. According to Schur’s lemma, this means that the matrix $m$ is proportional to the identity, thus giving:
([**e**]{}) the same mass for all the fermion fields within the irreducible multiplet determined by the matrices $T^{i}$: $$m_{rs}={\bf m}\delta _{rs}.$$ In the case when the fermions are decomposed into several irreducible multiplets, their masses are equal within each multiplet.
Now, collecting together the conditions ([**a**]{})-([**e**]{}) derived from the non-observability of the SBLS for [*any*]{} set of infinitesimal vectors $n_{\mu }^{i}$ applied to the general Lagrangian density (\[LN\]), we arrive at a truly gauge invariant Yang-Mills theory for the new fields ${\bf a}_{\mu }^{i}$ and ${\bf \Psi }$:
$$L_{YM}={\bf -}\frac{1}{4}{\bf F}_{\mu \nu }^{i}{\bf F}_{\mu \nu }^{i}+
i\overline{{\bf \Psi }}\gamma \partial {\bf \Psi -m}\overline{{\bf \Psi }}
{\bf \Psi +ga}_{\mu }^{i}\overline{{\bf \Psi }}{\bf \gamma }_{\mu }
{\bf T}^{i}{\bf \Psi } \label{fin}$$
Here ${\bf F}_{\mu \nu }^{i}=\partial_\mu {\bf a}_{\nu }^{i} -
\partial_{\nu }{\bf a}_{\mu }^{i} {\bf +ga }^{ijk}{\bf a}_{\mu }^{j}
{\bf a}_{\nu}^{k}$ and ${\bf g}$ is a universal gauge coupling constant extracted from the corresponding matrices $\alpha ^{ijk}={\bf ga }^{ijk}$ and $T^{i}={\bf gT}^{i}$.
Let us now consider the generalisation of the vector field VEVs from infinitesimal to finite background classical fields $n_{\mu }^{i}$. Unfortunately one cannot directly generalise the SBLS form (\[ab\]) to all finite $n_{\mu }^{i}$ vectors. Otherwise, the $n_{\mu }^{i}$ for the different vector fields might not commute under the Yang-Mills symmetry and might point in different directions in Lorentz space, giving rise to a non-vanishing field strength $F_{\mu \nu }^{k}$ in the corresponding vacuum. Such a vacuum would not be Lorentz invariant, implying a real physical breakdown of Lorentz symmetry. This problem can be automatically avoided if the finite SBLS shift vector $n_{\mu }^{i}$ in the basic equation (\[ab\]) takes the factorised form $n_{\mu }^{i}=n_{\mu }\cdot f^{i}$ where $n_{\mu}$ is a constant Lorentz vector as in the Abelian case, while $f^{i}$ ($%
i=1,2,...N$) is a vector in the internal charge space. Using the Lagrangian density (\[fin\]) derived for infinitesimal VEVs, it is now straightforward to show that there will be no observable effects of SBLS for [*any*]{} set of finite factorised VEVs $n_{\mu }^{i}=n_{\mu }\cdot f^{i}$. For this purpose, we generalise Eq. (\[rot\]) to the finite transformation:
$$a_{\mu }\cdot \alpha =\exp [(\omega \cdot \alpha )]{\bf a}_{\mu }\cdot
\alpha \exp [-(\omega \cdot \alpha )] \label{finiterot}$$
In conclusion, we have shown that gauge invariant Abelian and non-Abelian theories can be obtained from the requirement of the physical non-observability of the SBLS rather than by using the Yang-Mills gauge principle. Thus the vector fields become a source of the symmetries, rather than local symmetries being a source of the vector fields as in the usual formulation. Imposing the condition that the Lorentz symmetry breaking be unobservable of course restricts the values of the coupling constants and mass parameters in the Lagrangian density. These restrictions may naturally also depend on the direction and strength of the Lorentz symmetry breaking vector field VEVs, whose effects are to be hidden. This allows us a choice as to how strong an assumption we make about the non-observability requirement. Actually, in the Abelian case, we just assumed this non-observability for the physical vacuum (\[f\]) that really appears. However we needed a stronger assumption in the non-Abelian case: the SBLS is unobservable in any vacuum for which the vector fields have VEVs of the factorised form $n_{\mu }^{i}=n_{\mu }\cdot f^{i}$. This factorised form is a special case in which the $n_{\mu }^{i}$ commute with each other.
We did not specify here mechanisms which could induce the SBLS—rather we studied general consequences for the possible dynamics of the matter and vector fields, requiring it to be physically unobservable. We address this and other related questions elsewhere[@long].
We are indebted to Z. Berezhiani, H. Durr, O. Kancheli, K. Maki, P. Minkowski and D. Sutherland for stimulating discussions and useful remarks.
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M. Suzuki, see ref. [@bj].
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|
---
abstract: 'We investigate two simple prescriptions to account for the Pauli principle in a three-body cluster model employing a new method based on an adiabatic hyperspherical expansion to solve the Faddeev equations in coordinate space. The resulting wave functions are computed and compared. They are furthermore tested on halo nuclei by calculations of momentum distributions and invariant mass spectra arising after fragmentation of fast $^6$He and $^{11}$Li in collisions with light targets. The prescriptions are very accurate and the available measured quantities are remarkably well reproduced when final state interactions are included.'
author:
- |
E. Garrido, D.V. Fedorov and A.S. Jensen\
Institute of Physics and Astronomy, Aarhus University,\
DK-8000 Aarhus C, Denmark
title: 'The Pauli principle in a three-body cluster model and the momentum distributions after fragmentation of $^6$He and $^{11}$Li '
---
-1.2cm
25.60.-t, 21.45.+v, 21.60.Gx
Introduction
============
Three-body models are useful to describe the relative wave function of three particles when their intrinsic structure remains unchanged [@ric92; @zhu93; @fed94a]. The intrinsic and the relative degrees of freedom are then assumed to be completely decoupled. However, when the particles themselves are composite structures (clusters) and contain identical fermions this assumption violates the Pauli principle related to fermions distributed in different clusters. This violation typically leads to unphysical deeply bound two-cluster states. In the two-body problem this violation can be easily cured by removing this unphysical state from the active space available for the system.
In the three-cluster problem most of the usual approaches to include the Pauli principle basically fall into two categories: i) to project out the undesired overlap of the three-body wave function with the Pauli forbidden two-body state[@kuk78; @mar82; @ban83] and ii) to modify the two-body interaction in such a way that the unphysical bound cluster-cluster state is avoided, provided the low energy scattering properties [@joh90] or the phase shifts [@bay87; @fie90] remain unchanged. The prescriptions basically amount to either excluding occupation of Pauli forbidden two-body states in the three-body wave function or excluding the apriori appearance of such undesired states. The two prescriptions give somewhat different results and it is not yet clear which one is preferable. The projection technique, although seems to be more straightforward, is in practice technically more difficult.
The problem of treating the intercluster Pauli principle again attracted attention in investigations of the structure of the three-body halo nuclei [@zhu93] where most of the information is obtained by studying momentum distributions of fragments of these nuclei in fast collisions with light targets. Originally these momentum distributions were expected to reveal the structure of the initial three-body wave function [@kob88]. This is still true although more elaborate analyses are required. The correct account for the two-body interaction between fragments in the final state turned out to be essential when low lying resonances are present [@gar96; @kor94].
The purpose of the present paper is twofold. First we describe how the projecting out technique can be easily incorporated in the adiabatic hyperspherical expansion of the Faddeev equations in coordinate space [@fed93b]. This method to solve the Faddeev equations has already been used in several investigations of halo nuclei [@fed94a; @fed94c; @fed95].
Secondly we investigate how the differences in the prescriptions are reflected in the momentum distributions of fragments from the break-up reactions of three-body halos. This question is important for an unambiguous interpretation of the wealth of experimental data on three-body halos like $^6$He ($^4$He+n+n) and $^{11}$Li ($^9$Li+n+n).
After the introduction we shall in section 2 describe the method and compare potentials and wave functions for two different prescriptions. In section 3 we use these wave functions in computations of momentum distributions in nuclear break-up reactions of three-body halo nuclei. Finally in section 4 we give a brief summary and the conclusions.
Three-cluster problem with Pauli forbidden two-cluster states
=============================================================
#### Adiabatic hyperspherical expansion of the Faddeev equations. {#adiabatic-hyperspherical-expansion-of-the-faddeev-equations. .unnumbered}
We shall use the hyperspherical coordinates ($\rho$, $\alpha$, $\Omega_x$, $\Omega_y$) defined in refs. [@zhu93; @fed94a]. The volume element is given by $\rho^5 d\Omega d\rho$, where $d\Omega=\sin^2\alpha \cos^2\alpha d\alpha d\Omega_x d\Omega_y$. The total wave function $\Psi_{J M}$ of the three-body system (with total spin $J$ and projection $M$) is written as a sum of three components, which in turn for each hyperradius $\rho$ are expanded in a complete set of generalized angular functions $\Phi^{(i)}_{n}(\rho,\Omega_i)$ $$\Psi_{J M}= \frac {1}{\rho^{5/2}}
\sum_n f_n(\rho)
\sum_{i=1}^3 \Phi^{(i)}_{n}(\rho ,\Omega_i) \; ,
\label{tot}$$ where $\rho^{-5/2}$ is related to the volume element.
These wave functions satisfy the angular part of the three Faddeev equations: $${\hbar^2 \over 2m}\frac{1}{\rho^2}\hat\Lambda^2 \Phi^{(i)}_{n}
+V_{jk} (\Phi^{(i)}_{n}+\Phi^{(j)}_{n J M}+
\Phi^{(k)}_{n})\equiv {\hbar^2 \over
2m}\frac{1}{\rho^2} \lambda_n(\rho) \Phi^{(i)}_{n} \; ,
\label{ang}$$ where $\{i,j,k\}$ is a cyclic permutation of $\{1,2,3\}$, $m$ is an arbitrary normalization mass, and $\hat\Lambda^2$ is the $\rho$-independent part of the kinetic energy operator. The analytic expressions for $\hat{\Lambda}^2$ and the kinetic energy operator can for instance be found in [@fed94a].
The radial expansion coefficients $f_n(\rho)$ are obtained from a coupled set of “radial” differential equations [@fed94a], i.e. $$\begin{aligned}
\label{rad}
\left(-\frac{\rm d ^2}{\rm d \rho^2}
-{2mE\over\hbar^2}+ \frac{1}{\rho^2}\left( \lambda_n(\rho) - Q_{n n} +
\frac{15}{4}\right) \right)f_n(\rho) =
\nonumber \\
\sum_{n' \neq n} \left(
2P_{n n'}{\rm d \over\rm d \rho}
+Q_{n n'}
\right)f_{n'}(\rho) \; ,\end{aligned}$$ where the functions $P$ and $Q$ are defined as angular integrals: $$P_{n n'}(\rho)\equiv \sum_{i,j=1}^{3}
\int d\Omega \Phi_n^{(i)\ast}(\rho,\Omega)
{\partial\over\partial\rho}\Phi_{n'}^{(j)}(\rho,\Omega) \; ,$$ $$Q_{n n'}(\rho)\equiv \sum_{i,j=1}^{3}
\int d\Omega \Phi_n^{(i)\ast}(\rho,\Omega)
{\partial^2\over\partial\rho^2}\Phi_{n'}^{(j)}(\rho,\Omega) \; .$$
The radial equations in eq.(\[rad\]) reveal that $\hbar^2(\lambda_n(\rho) - Q_{n n} + \frac{15}{4})/(2m\rho^2)$ is the diagonal part of the effective radial potential. Its behavior is decisive for the properties of the three-body system. The $\lambda$-spectrum at both $\rho = 0$ and $\rho = \infty$ is identical to the hyperspherical spectrum. In addition, for every bound two-body state, there exists one $\lambda$-value which bends over and diverges parabolically as function of $\rho$ for $\rho \rightarrow \infty$, see [@fed94c]. Such a level corresponds at large distances to the three-body structure where the two-body subsystem is in the corresponding bound state, whereas the third particle is far away.
#### Treatment of the Pauli principle. {#treatment-of-the-pauli-principle. .unnumbered}
The first prescription is a two-body potential without Pauli forbidden states but with the same low energy properties as the original deep two-body potential with the forbidden state. This corresponds to the use of an additional Pauli repulsion. We have discarded the apparently rigorous procedure of constructing the phase equivalent potential from the original two-body potential. The reasons are that the original interaction in any case appears in the form of an effective (mean) potential adjusted to reproduce specific measured quantities. Furthermore, the binding energy of the three-body system must be accurately reproduced to provide the correct structure. An additional fine tuning is therefore almost inevitable. The strictly phase equivalent potential is substantially more difficult to obtain and use. Under these circumstances we prefer to take the pragmatic approach and construct phenomenological potentials adjusted to have specifically chosen crucial properties.
The second prescription is a unique feature of the adiabatical expansion method. Since in this method each of the two-body bound states gives rise to a separate diverging $\lambda$-value, the prescription is simply to omit the $\lambda$-values corresponding to the Pauli forbidden two-body bound states from the expansion (\[tot\]). Then all the Pauli forbidden states are automatically excluded in the three-body wave functions.
Two-body potentials
===================
We shall consider the two halo nuclei $^6$He and $^{11}$Li, both approximated as three-body systems consisting of a core ($^4$He and $^9$Li) surrounded by two valence neutrons. This approximation works remarkably well for these systems and furthermore due to relatively weak binding of the systems only the low energy parameters of the two-body potentials are essential for the structure of the systems [@zhu93]. As the core is inert in our approximation it is sufficient for our purpose to assume zero spin of the core for $^{11}$Li.
In both cases the lowest neutron s-states in the core are filled. Therefore to conform with the Pauli principle the halo neutron should be prevented from occupying the lowest neutron-core s-state. We achieve this goal by use of two different neutron-core potentials with the same low energy properties. One of the potentials will have a Pauli forbidden bound state the other will not. This Pauli forbidden state will be consequently removed from the active space.
#### Neutron-neutron potential. {#neutron-neutron-potential. .unnumbered}
As indicated in [@zhu93] and proved by our test runs the particular radial shape of the $n-n$ interaction is not important for $^6$He and $^{11}$Li ground state properties as long as the low energy $n-n$ scattering parameters are correct. We therefore use a simple potential similar to the one in [@fed95] which reproduces the experimental s- and p-wave scattering lengths and effective ranges. It contains central, spin-orbit (${\bf L \cdot S}$), tensor ($S_{12}$) and spin-spin (${\bf
s}_1\cdot{\bf s}_2$) interactions and is explicitly given as $$\begin{aligned}
V_{nn}(r)=37.05 \exp(-(r/1.31)^2)
-7.38\exp(-(r/1.84)^2) \nonumber \\
-23.77\exp(-(r/1.45)^2){\bf L \cdot S}
+7.16\exp(-(r/2.43)^2) S_{12}\nonumber \\+
\left(49.40\exp(-(r/1.31)^2)+
29.53\exp(-(r/1.84)^2) \right) {\bf s}_1 \cdot {\bf s}_2,\end{aligned}$$ where the strengths are in MeV and ranges in fm. Its scattering lengths[^1] $a$ and effective ranges $r_e$ are (in fm) $a(^1S_0)$= 18.45, $r_e(^1S_0)$= 2.83, $a(^3P_0)$= 3.38, $r_e(^3P_0)$= 1.10, $a(^3P_1)$= $-2.02$, $r_e(^3P_1)$= $-2.94$, $a(^3P_2)$= 0.31, $r_e(^3P_2)$= 18.73.
#### Neutron-$^4$He potential. {#neutron-4he-potential. .unnumbered}
For $^6$He the low energy properties of the neutron-core subsystem in the s- and p-waves are rather well known and reflected in an s-wave scattering length of $a_s=-3.07 \pm 0.02$ fm and two resonance energies $E$ and widths $\Gamma$ of $E(p_{3/2})=0.77$ MeV, $\Gamma(p_{3/2})=0.64$ MeV and $E(p_{1/2}) \approx 1.97$ MeV, $\Gamma(p_{1/2}) \approx 5.22$ MeV, see [@ajz88]. The $p_{1/2}$-$p_{3/2}$ splitting demands a spin-orbit force or equivalently (with the same range of the interactions) different strengths in the $p_{1/2}$ and $p_{3/2}$ channels. The negative s-wave scattering length can be reproduced by a repulsive potential or by an attractive potential with a bound state which in this case is the Pauli forbidden state occupied by the core neutrons.
We introduce central l-dependent and spin-orbit components in the potential with a gaussian shape $S$exp$(-r^2/b^2)$. The range was chosen to be $b=2.33$ fm for all components (similar to [@zhu93]) except for the repulsive s-wave where the range $b$ was changed to 3.34 fm in order to reproduce the same effective range as for attractive potential. The strength parameters are then defined by the fit to the specified scattering length and positions of the resonances.
The p-wave strengths are $S(p_{1/2})=-48.675$ MeV and $S(p_{3/2})=-53.175$ MeV (or, equivalently, the central p-wave strength of -51.675 Mev and the spin-orbit strength of -3.00 MeV) which provide the p-wave resonances $E(p_{1/2})=1.94$ MeV, $\Gamma(p_{1/2})=4.0$ MeV and $E(p_{3/2})=0.77$ MeV, $\Gamma(p_{3/2})=0.73$ MeV. These parameters are the same for both repulsive and attractive s-wave potential.
The s-wave strength for the attractive potential (further referred to as “attractive") is $S(s_{1/2})=-39.2$ MeV ($r_e$=1.41 fm and $a=-3.07$ fm, one Pauli forbidden bound state). Without additional fit these s- and p-wave potentials provide the binding energy B($^6$He)=1.0 MeV and r.m.s. radius R($^6$He)=2.45 fm, which is close to experimental data B($^6$He)=$0.97\pm$ 0.04 MeV and R($^6$He)$=2.57\pm~0.10$ fm.
The repulsive s-wave potential which reproduces the experimental scattering length and the same effective range as the attractive potential slightly underbindes $^6$He (by approximately 200 keV). Such underbinding for repulsive potentials is not unusual for $^6$He. To alleviate this problem people normally increase the range of all potentials by a few per cent [@zhu93]. However this procedure shifts the positions of the p-resonances from their experimental values. As we shall see below these positions are extremely important for the momentum distributions. We therefore leave the p-waves unchanged and instead reduce slightly the repulsion in the s-wave.
The resulting potential is further referred to as “repulsive" and has the s-wave range and strength $b=3.34$ fm, $S(s_{1/2})=9.70$ MeV, ($a=-2.58$ fm, $r_e=$0.67 fm) which leads to the binding energy B($^6$He)=1.0 MeV and r.m.s. radius R($^6$He)=2.50 fm.
#### Neutron-$^9$Li potential. {#neutron-9li-potential. .unnumbered}
For $^{11}$Li the low energy neutron-core data are less known, although evidence is accumulating for a low lying virtual s-state at $E(s)
\approx 0.15 \pm 0.15$ MeV and the lowest p-resonance at $E(p) \approx
0.6 \pm 0.2$ MeV, see [@boh93; @you94; @zin95; @abr95]. The Pauli forbidden states are in this case both the lowest s-state and the $p_{3/2}$-state. For simplicity the latter is in the calculations placed at high positive energy and thereby removed from the active space by a large inverse spin-orbit potential [@fed95]. For the s-waves we use a shallow potential without bound state and a deep potential with a Pauli forbidden state.
The radial shapes of the neutron-core interactions are also assumed to be gaussians, i.e. $S$exp$(-r^2/b^2)$ with $b=2.55$ fm [@joh90] except for the deep s-wave potential with a bound state where we use $b=1.49$ fm to maintain the same effective range as for the shallow potential.
We adopt further the usual assumption that the neutron-core interactions do not depend on the spin of the $^9$Li-core. A more realistic study of $^{11}$Li properties taking into account the spin of $^{9}$Li has been made in [@gar96; @fed95].
Again the spin-orbit neutron-core potential effectively only gives different strengths for the two different p-waves. With the choice of range for the radial potentials there are only three strength parameters left each related to a resonance, a virtual state or a scattering length. The spin-orbit force is used to remove the Pauli forbidden $p_{3/2}$-state from the active space. The two remaining strength parameters then determine the s- and p$_{1/2}$-state as well as the binding energy of $^{11}$Li. One of them must be used to fine tune the binding energy B of the total three-body system B($^{11}$Li)$=0.295\pm~0.035$ MeV. With correct binding energy the root mean square radius is then always reproduced within the experimental uncertainty, which is R($^{11}$Li)$=3.1\pm~0.3$ fm.
The potential with a bound Pauli forbidden s-state (further referred to as “deep") has the parameters $b=1.49$ fm for the s-wave and $b=2.55$ fm for p-waves, $S(s_{1/2})=-176.608$ MeV (one bound state, $a$=8.738 fm, virtual level at 0.20 MeV), $S(p_{3/2})=9.55$ MeV ($E(p_{3/2}$) is high and uninteresting) and $S(p_{1/2})=-38.34$ MeV ($E(p_{1/2})=0.77$ MeV, $\Gamma(p_{1/2})=0.89$ MeV). The binding energy and root mean square radius is then computed to be B($^{11}$Li)=0.30 MeV and R($^{11}$Li)=3.34 fm.
The shallow potential without bound states but with the same low energy properties as the deep potential overbinds $^{11}$Li by some 180 keV. In contrast to $^6$He the dominating component is now the s-wave in the neutron-core subsystem. Therefore to fine tune the binding energy we modify now the p-wave potential namely the position of the $p_{1/2}$ resonance while keeping the other properties unmodified.
The fine tuned potentials without a bound state (further referred to as “shallow") has the parameters $b=2.55$ fm, $S(s_{1/2})=-7.14$ MeV (no bound state, $a$=8.738 fm, virtual level at 0.20 MeV), $S(p_{3/2})=9.55$ MeV ($E(p_{3/2})$ is high) and $S(p_{1/2})=-35.45$ MeV ($E(p_{1/2})=1.7$ MeV, $\Gamma(p_{1/2})=4.2$ MeV). The binding energy and root mean square radius is now computed to be B($^{11}$Li)= 0.30 MeV and R($^{11}$Li)= 3.35 fm.
Effective three-body adiabatic potentials
=========================================
#### $^6$He. {#he. .unnumbered}
The effective radial potential in eq.(\[rad\]) is the crucial quantity which in turn mainly is determined by the $\lambda$-spectrum. The two prescriptions for dealing with the Pauli principle lead to different spectra as seen in fig. \[1a\] for $^6$He. The purely repulsive neutron-core s-wave potential (solid curves) results in the steep increase of the lowest $\lambda$-value at small $\rho$. The increase is quickly interrupted by two (avoided) crossings involving the next two $\lambda$-values, which in contrast are moving steeply down due to the strong p-wave attraction leading to a low lying p-resonance in $^5$He (see the parameters given in the previous section). They therefore must contain a substantial amount of p-wave and the related components must be dominating in the wave function which accordingly also has about 87% p-wave and 13% s-wave in the relative neutron-core system. The curves eventually return back recovering the hyperspherical spectrum for $\rho = \infty$. This occurs without (avoided) level crossings. The interaction between crossing levels is vanishing or small indicating different symmetries of these levels. The related almost preserved quantum numbers are the neutron-core relative angular momenta $l_x=0$ and 1 corresponding to states originating from zero and from 12. The two levels originating from 32 are apparently at small $\rho$ dominated respectively by s- and p-wave components. At larger $\rho$ the highest $\lambda$ bends over due to p-wave admixture and the related strong attraction.
The $\lambda$-spectrum for the other potential with sufficient attraction in the neutron-core s-wave to support one bound s-state is also shown as the dashed curves in fig. \[1a\]. The lowest $\lambda$-value is not shown in the figure. It is mainly s-waves in the neutron-core subsystem and decreases at small $\rho$ and diverges parabolically for $\rho \rightarrow \infty$ as clear signals of strong attraction and a resulting bound state. All states built on this (not plotted) level are Pauli forbidden and the level is therefore excluded in the following calculations. The two next $\lambda$-values on the other hand remain almost unchanged (apart from the avoided crossings) indicating a dominating p-wave content (about 93%) in the relative neutron-core subsystem. This behavior is consistent with an almost unchanged energy of the lowest lying p-resonance in the two-body subsystem. As the lowest lying levels dominate the wave function the behavior of these effective potentials also guarantee an almost identical wavefunction in the two approaches. Also the two highest $\lambda$-values remain essentially unchanged. The third $\lambda$-value start out as an s-wave as seen from the strong decrease. At larger distances it is then replacing the s-wave originating from zero (crossing two times) from the purely repulsive potential.
#### $^{11}$Li. {#li. .unnumbered}
For $^{11}$Li the low lying s-state of $^{10}$Li results in a structure where the dominating neutron-core relative configuration is an s-state. This is rather different from $^6$He where the predominant neutron-core relative configurations were p-states. The $\lambda$-spectra for the two potentials corresponding to the different prescriptions are shown in fig. \[1b\]. The shallow attractive potential without a bound s-state (solid curves) results in a rather slowly changing lowest $\lambda$-value which contains essentially only neutron-core s-waves. This would by far carry the largest probability in the resulting wave function. The two highest $\lambda$-values are also smooth functions of $\rho$ while the second level steeply decreases at small $\rho$ indicating attraction in the corresponding partial wave.
The other potential, which has a Pauli forbidden bound neutron-core s-state, has much stronger s-wave attraction and a $\lambda$-spectrum, see dashed curves fig. \[1b\], with a fast moving level originating from 32. This level responds to the large s-wave attraction and crosses quickly the two levels originating from 12 before replacing the lowest level from the shallow potential. This level will dominate the configuration of the radial wave function. The lowest $\lambda$-value originating from zero is not plotted and also omitted in the following calculations. It decreases at small $\rho$ and diverges parabolically at large $\rho$. The four remaining levels all have similarly behaving counter parts in the other spectrum. Again we see the (avoided) crossings indicating almost conserved quantum numbers on the levels. As before they can be traced back to the neutron-core relative angular momentum $l_x=0$ and 1.
#### Radial functions. {#radial-functions. .unnumbered}
The radial wave function corresponding to the lowest $\lambda$ is shown in fig. \[2\] for the two nuclei. This component is responsible for more than 90% of the wave function. Only the result of one of the prescriptions is shown for each case, since the curves would be difficult to distinguish especially at larger distances where they are very close. This is closely related to the fact that the binding energies and radii are the same in both prescriptions. The wave function is spatially more extended for $^{11}$Li than for $^6$He, again reflecting the difference in binding energy.
Momentum distributions
======================
The momentum distributions after fragmentation of halo nuclei is a very direct way of gaining detailed information about the relative wave function of these nuclei. It was originally expected simply to provide the Fourier transform, but more complicated and detailed analyses are needed. Such computations are fortunately available at this moment and we shall exploit this avenue to test our prescriptions.
#### The sudden approximation. {#the-sudden-approximation. .unnumbered}
We consider a process where a high-energy three-body halo projectile instantaneously looses one of the particles without disturbing the remaining two. We also assume a light target and we shall therefore neglect the Coulomb dissociation process, which then only contributes marginally.
We work in the center of mass system of the three-body projectile and denote by $\mbox{\bf k}_y$ and $\mbox{\bf k}_x$ the total and relative momentum of the two remaining particles in the final state. The transition matrix of the reaction in the sudden approximation is then given by $$M^{JM}_{s_x \sigma_x s_y \sigma_y}(\mbox{\bf k}_y, \mbox{\bf k}_x)
\propto \langle
e^{i \mbox{\scriptsize \bf k}_y \cdot \mbox{\scriptsize \bf y} }
\chi_{s_y \sigma_y} w_{s_x \sigma_x}(\mbox{\bf k}_x, \mbox{\bf x})
| \Psi_{JM} \rangle \; ,
\label{tran}$$ where $\Psi_{JM}$ is the three-body wave function and $w_{s_x
\sigma_x}$ is the final state [*distorted*]{} two-body wave function [@gar96] corresponding to the two remaining particles with the distance [**x**]{}, total spin and projection equal to $s_x$ and $\sigma_x$. The distance between the center of mass of the two-body system and the removed particle is [**y**]{} and $\chi_{s_y \sigma_y}$ is the spin wave function of the third particle where $s_y$ and $\sigma_y$ are the related total spin and projection.
The cross section or momentum distribution is now obtained by squaring the transition matrix and subsequently averaging over initial states and summing over final states: $$\frac{d^6\sigma}{d\mbox{\bf k}_x d\mbox{\bf k}_y} \propto
\sum_M \sum_{s_x \sigma_x \sigma_y}
|M^{J M}_{s_x \sigma_x s_y \sigma_y}(\mbox{\bf k}_x, \mbox{\bf k}_y)|^2 \; .
\label{mom}$$
Using the momentum $\mbox{\bf p} (=a \mbox{\bf k}_x + b \mbox{\bf
k}_y)$ of one of the particles relative to the center of mass of the projectile as the variable instead of $\mbox{\bf k}_x$, we obtain the relation $$\frac{d^6\sigma}{d\mbox{\bf p} d\mbox{\bf k}_y} =
\frac{1}{a^3} \frac{d^6\sigma}{d\mbox{\bf k}_x d\mbox{\bf k}_y}
\propto \frac{1}{a^3}
\sum_M \sum_{s_x \sigma_x \sigma_y}
|M^{J M}_{s_x \sigma_x s_y \sigma_y}(\mbox{\bf k}_x,
\mbox{\bf k}_y)|^2 \; ,
\label{rot}$$ where $a^3$ arises from the Jacobi determinant for the transformation. The differential cross section in eq.(\[rot\]) should be integrated over all unobserved variables, i.e. $\mbox{\bf k}_y$ and some of the components of [**p**]{}. Note that we have not specified any coordinate system, and the axis [**x**]{}, [**y**]{}, and [**z**]{} are therefore completely arbitrary. Thus, in our approximation the longitudinal and transverse momentum distributions are identical.
After neutron removal fragmentation reactions is usual to define the invariant mass $E_{\mbox{\scriptsize core+n}}$ as $$E_{\mbox{\scriptsize core+n}}=\left( (E_{\mbox{\scriptsize core}}+
E_{\mbox{\scriptsize n}})^2 + c^2 (\mbox{\bf p}_{\mbox{\scriptsize
core}}+\mbox{\bf p}_{\mbox{\scriptsize n}})^2 \right)^{1/2} -
(M_{\mbox{\scriptsize core}} + M_{\mbox{\scriptsize n}})c^2,
\label{invdef}$$ where $E_{\mbox{\scriptsize core,n}}$, $\mbox{\bf p}_{\mbox{\scriptsize
core,n}}$, and $M_{\mbox{\scriptsize core,n}}$ denote the energy, three-momentum, and rest mass of the core and the neutron, respectively.
Computing the invariant mass in the frame of the two-body system after the fragmentation ($\mbox{\bf p}_{\mbox{\scriptsize core}}+\mbox{\bf
p}_{\mbox{\scriptsize n}}=0$) we have $$p_{\mbox{\scriptsize core}}^2 c^2 = p_{\mbox{\scriptsize n}}^2 c^2 =
\frac{1}{m} \frac{M_{\mbox{\scriptsize core}} M_{\mbox{\scriptsize n}}}
{M_{\mbox{\scriptsize core}} + M_{\mbox{\scriptsize n}}} k_x^2 c^2 =
E_{\mbox{\scriptsize core}}^2 - M_{\mbox{\scriptsize core}}^2 c^4 =
E_{\mbox{\scriptsize n}}^2 - M_{\mbox{\scriptsize n}}^2 c^4,$$ and $$dE_{\mbox{\scriptsize core+n}} = dE_{\mbox{\scriptsize core}} +
dE_{\mbox{\scriptsize n}} = \frac{E_{\mbox{\scriptsize core}} +
E_{\mbox{\scriptsize n}} }{E_{\mbox{\scriptsize core}}
E_{\mbox{\scriptsize n}} }
\frac{M_{\mbox{\scriptsize core}} M_{\mbox{\scriptsize n}}}
{m (M_{\mbox{\scriptsize core}}+M_{\mbox{\scriptsize n}})}
k_x dk_x$$
The invariant mass spectrum is then defined $$\frac{d\sigma}{dE_{\mbox{\scriptsize core+n}}}=
\frac{E_{\mbox{\scriptsize core}} E_{\mbox{\scriptsize n}} }
{E_{\mbox{\scriptsize core}} + E_{\mbox{\scriptsize n}} }
\frac{m (M_{\mbox{\scriptsize core}} + M_{\mbox{\scriptsize n}})}
{M_{\mbox{\scriptsize core}} M_{\mbox{\scriptsize n}}}
\frac{1}{k_x} \frac{d\sigma}{dk_x}
\label{invsp}$$ where $d\sigma/dk_x$ is obtained from eq.(\[mom\]) after integrating over the unobserved quantities.
#### $^6$He fragmentation. {#he-fragmentation. .unnumbered}
In fig. \[3\] the computed longitudinal core momentum distribution after neutron removal in a fast nuclear break-up reaction of $^6$He on a light target is shown. The results of the two prescriptions are exhibited both with and without inclusion of the final state interaction. As expected the effect of the final state interaction is visible but not extremely important even with this fairly light core. The results of the two prescriptions are very close when the final state interaction is included. When it is neglected the intrinsic differences in the wave functions obtained with the two different prescriptions to account for the Pauli principle produce a clear distinction. The difference is in the additional node of the forbidden state prescription compared to the reduced attraction prescription. The experimental data in fig. \[3\] are taken from [@koba92], and correspond to the transverse core momentum distribution after fast $^6$He fragmentation on a carbon target. The present model computations do not distinguish between directions, but the results are expected to be more appropriate for the longitudinal directions, which unfortunately are unavailable at this moment. However, the agreement with the measured results is still encouraging for several reasons. First, we have basically no free parameters. Second, the experimental transverse momentum distributions are expected to be broader than the longitudinal ones [@hum95]. Third, no broadening due to neglected effects and experimental resolution is included in the computation.
In figs. \[4a\] and \[4b\] we show corresponding computed longitudinal and radial ($p^\bot = (p_x^2 + p_y^2)^{1/2}$) neutron momentum distributions for a neutron removal $^6$He fragmentation reaction. Again the two prescriptions give very similar results when the final state interaction is included. In this case we obtain a very large influence of the final state interaction which reduce the full width at half maximum by a factor 2-3. Inclusion of the final state interaction is therefore necessary to obtain the observed agreement with the measured values. We want here to emphasize that the computed results are found in a consistent model where the same two-body potential is responsible for both the three-body structure of the initial halo state and the final state interaction after break-up. The experimental data in fig. \[4a\] are obtained from [@koba93], and correspond to the transverse neutron momentum distribution after $^6$He fragmentation on a carbon target at 800 MeV/u. Again the computed curve is expected to compare more favourably with the experimental longitudinal neutron momentum distribution. However, these data are not presently available.
Although measured values are not available we show in figs. \[5a\] and \[5b\] the computed longitudinal and radial neutron momentum distributions for core break-up reactions of $^6$He. In this case the core is violently removed, and the final neutron-neutron interaction is relevant. The two prescriptions again lead to similar results. The final state interaction can not be neglected due to the low lying virtual s-state in the neutron-neutron system.
Key quantities for a number of different cases are given in table I, where the $^6$He binding energy and radius are kept at the measured values. For the case of the repulsive s-wave potential we have first considered two cases (first two rows of table I) where both the $p_{1/2}$ and the $p_{3/2}$-resonance energies lie within the experimental values. In these two cases the computed scattering length for the s-wave is smaller than the experimental value of $-3.07$. Increasing the absolute size of the computed s-wave scattering length (keeping fixed the energy of the lowest $p_{3/2}$-resonance) reduces the energy of the $p_{1/2}$-resonance (case 3 in the table). When the scattering length is equal to the experimental value the energy of the second p-resonance is below the experimental energy. Cases 4 to 6 in table I are analogous to cases 1 to 3, but with an s-wave repulsive potential giving a clearly worse agreement with the experimental s-wave scattering length. However the changes in the full width at half maximum of the momentum distributions, the p-wave content and root mean square radius, are very small, reflecting that the properties the s-wave interaction do not play an essential role in the structure of $^6$He.
For an attractive s-potential with one bound state (two last rows of table I) it is possible to reproduce both the experimental s-wave scattering length and the two p-wave resonance energies. For the two cases 7 and 8 shown in table I the variation of the energy of the $p_{1/2}$-resonance only produces a change in the $^6$He binding energy of about 70 keV, but in both cases consistent with a computed value of 1.0 MeV. The cases 1 and 7 correspond to the parameters given in section 3, and they are the two cases considered for $^6$He in figs. \[3\], \[4a\], \[4b\], \[5a\], and \[5b\].
#### $^{11}$Li fragmentation. {#li-fragmentation. .unnumbered}
The interaction parameters for the schematic $^{11}$Li computation, where the core spin is assumed to be zero, only provide the s- and $p_{1/2}$-strengths for fine tuning. We already placed the $p_{3/2}$-state at a very high energy and thereby removed it from the active space. In contrast to $^6$He the dominating component is now an s-wave in the neutron-core subsystem. In table II we show the relevant quantities for three different computations. The first row corresponds to a computation with a shallow s-wave potential without bound states while the second and third rows correspond to computations with deep s-wave potentials with one bound state. The effective range and scattering length are the same in the three cases and the s-wave interaction places the lowest virtual s-state at 200 keV. Once the s-wave potential has been fixed the energy of the $p_{1/2}$-resonance must then be around 1.7 MeV for the shallow potential case in order to fit the $^{11}$Li binding energy. When the deep potential is used we consider two situations. First we keep the energy of the $p_{3/2}$-resonance unchanged. Then the $p_{1/2}$-energy is reduced to 0.8 MeV in order to recover the $^{11}$Li binding energy. In the second case we force the $p_{1/2}$-resonance energy to take the same value as for the shallow potential. The correct $^{11}$Li binding energy is in this case obtained by reducing the energy of the $p_{3/2}$-resonance. The cases 1 and 2 correspond to the parameters given in section 3 and they will be the two cases considered for $^{11}$Li in the next figures (more elaborated and detailed computations for $^{11}$Li fragmentation reactions may be found elsewhere, see [@gar96]).
------ -------------- --------- -------------- ------------------- ------------ ------------ ------------- ------
case $E(s_{1/2})$ $a$ $E(p_{1/2})$ $\Gamma(p_{1/2})$ $\Gamma_c$ $\Gamma_n$ $p$-content R
(MeV) (fm) (MeV) (MeV) (MeV/c) (MeV/c) (%) (fm)
1 5.27 $-2.58$ 1.94 4.02 101.6 72.6 87.7 2.50
2 5.13 $-2.21$ 2.55 7.38 100.5 74.8 86.2 2.50
3 5.09 $-3.07$ 1.28 1.77 103.0 68.1 89.4 2.50
4 10.6 $-2.04$ 1.94 4.02 98.8 72.9 86.7 2.53
5 10.8 $-1.80$ 2.55 7.38 98.3 74.5 85.5 2.53
6 9.3 $-2.51$ 1.10 1.34 99.2 67.1 88.5 2.53
7 5.08 $-3.07$ 1.94 4.02 107.6 67.4 92.8 2.45
8 5.08 $-3.07$ 2.55 7.38 106.2 68.1 92.7 2.49
------ -------------- --------- -------------- ------------------- ------------ ------------ ------------- ------
: Key quantities for $^{6}$He corresponding to various neutron-core interactions which are chosen as gaussians in each partial wave. The neutron-neutron interaction is given in Section 3. For the neutron-$^4$He subsystem we give energies of the virtual $s_{1/2}$-state, s-wave scattering lengths, energies and widths of the $p_{1/2}$-resonance. The $p_{3/2}$-resonance energy and width are in all cases 0.77 MeV and 0.73 MeV, respectively. The full width at half maximum for core and neutron momentum distributions is denoted $\Gamma_c$ and $\Gamma _n$. We also give the probability for finding the neutron-$^{4}$He subsystem in a $p$-wave in the three-body wave function of $^{6}$He. The remaining probablity is found in s-waves. The total binding energy of the three-body system is in all cases equal to about 1.0 MeV and the root mean square radius is given in the last column. Cases 1-6 and 7-8 correspond to potentials without bound and with one bound s-state, respectively.
Looking at the root mean square radius we observe that it only varies between about 3.30 fm and 3.40 fm independently of the prescription and mainly sensitive to the amount of s-state in the neutron-core subsystem, see also [@fed95]. The deep potential gives almost the same radius as the shallow potential. This is in contradiction with the large radii ($\geq$ 3.50 fm) obtained in similar Faddeev calculations, where the lowest s-state is projected out [@tho94]. These surprisingly large radii are hard to reconcile with the general asymptotic relation between binding energy and radius for weakly bound halo nuclei [@fed94a; @fed94c]. Incidentally a similarly large radius ($3.55 \pm 0.10$ fm) is also obtained in recent Glauber analyses of the measured reaction cross section [@alk96]. The reliability of such analyses still remains to be investigated.
------ -------------- ------- -------------- ------------------- ------------ ------------ ------------- ------
case $E(s_{1/2})$ $a$ $E(p_{1/2})$ $\Gamma(p_{1/2})$ $\Gamma_c$ $\Gamma_n$ $p$-content R
(MeV) (fm) (MeV) (MeV) (MeV/c) (MeV/c) (%) (fm)
1 0.200 8.738 1.68 4.16 57.6 34.8 17.3 3.34
2 0.200 8.738 0.77 0.89 56.1 38.2 21.0 3.35
3 0.200 8.738 1.68 4.16 55.6 35.6 15.6 3.37
------ -------------- ------- -------------- ------------------- ------------ ------------ ------------- ------
: Key quantities for $^{11}$Li corresponding to various neutron-core interactions which are chosen as gaussians in each partial wave. The neutron-neutron interaction is given in section 3. For the neutron-$^{9}$Li subsystem we give energies of the virtual $s_{1/2}$-state, s-wave scattering lengths, energies and widths of the $p_{1/2}$-resonance. The width and energy of the $p_{3/2}$-resonance is unphysical and not included. The full width at half maximum for core and neutron momentum distributions is denoted $\Gamma_c$ and $\Gamma
_n$. We also give the probability for finding the neutron-$^{10}$Li subsystem in a $p$-wave in the three-body wave function of $^{11}$Li. The remaining probability is found in s-waves. The total binding energy of the three-body system is in all cases equal to about 0.30 MeV and the root mean square radius is given in the last column. Cases 2 and 3 correspond to potentials with one bound s-state.
In fig. \[6\] we show the computed longitudinal core momentum distributions after neutron removal in a fast nuclear break-up reaction of $^{11}$Li on a light target. We assume that the $^{9}$Li-core has spin zero. The model is then without the proper spin couplings and the spin splitting of the neutron-core relative states [@fed95]. Although simplified the model is still not far from being realistic. The two prescriptions again give very similar results and the effect of the final state interaction is now a little smaller than for $^{6}$He due to the larger mass of the core nucleus. The experimental distributions are essentially reproduced, and correspond to a $^{11}$Li fragmentation reaction on Al at 468 MeV/u and 648 MeV/u [@geiss96].
In figs. \[7a\] and \[7b\] we show computed longitudinal and radial ($p^\bot
=(p^2_x+p^2_y)^{1/2}$) neutron momentum distributions for a neutron removal reaction of $^{11}$Li. The cases shown in the figure correspond to rows 1 and 2 of table II. The difference in energy of the $p_{1/2}$ resonance in the neutron-core subsystem produces the difference in the momentum distribution when the final state interaction is included. In fact the momentum distributions obtained in the third case of table II are almost indistinguishable from those obtained in the first case (the energy of the $p_{1/2}$ resonance is the same in these two cases). The final state interaction is then important due to the presence of low lying virtual s-states and p-resonances, and very sensitive to the energies of these low lying states.
For completeness we show in figs. \[8a\] and \[8b\] the longitudinal and radial neutron momentum distribution from a $^{11}$Li core break-up reaction. We again observe an important effect produced by the inclusion of the final neutron-neutron interaction (the neutron-neutron interaction has a low lying virtual s-state), and an almost identical result for both the shallow and deep neutron-$^9$Li s-wave potential. In figs. \[7b\] and \[8b\] the experimental data are fairly well reproduced, and correspond to a $^{11}$Li fragmentation reaction at 280 MeV/u on a carbon target [@zin95; @nil95]
#### Invariant mass spectrum. {#invariant-mass-spectrum. .unnumbered}
Computing the invariant mass $E_{\mbox{\scriptsize core+n}}$ (eq.(\[invdef\])) in the frame of the two-body system in the final state ($\mbox{\bf p}_{\mbox{\scriptsize core}}+\mbox{\bf
p}_{\mbox{\scriptsize n}}=0$) we can interpret $E_{\mbox{\scriptsize
core+n}}$ as the kinetic energy of the neutron-core system in the final state: $$E_{\mbox{\scriptsize core+n}} \approx \frac{p_{\mbox{\scriptsize
core}}^2}{2 \mu} = \frac{p_{\mbox{\scriptsize n}}^2}{2 \mu}
= \frac{k_x^2}{2 m}$$ where $\mu$ and $m$ are the reduced mass of the two-body system and the arbitrary normalization mass, respectively. The energy of a resonance is defined as the $k_x^2/2 m$ value for which the cross section has a maximum. As a consequence the invariant mass spectrum (\[invsp\]) will show a peak at the $E_{\mbox{\scriptsize core+n}}$ energy equal to a resonance energy.
In fig. \[9\] we show the invariant mass spectrum (\[invsp\]) from a $^6$He neutron removal reaction. The cases with repulsive s-potential (solid line) and attractive s-potential with a Pauli forbidden state (short-dashed line) are shown. The parameters correspond to cases 1 and 7 in table I, respectively. As seen from the figure, both curves are very similar, and show a peak around $E_{\mbox{\scriptsize core+n}}=0.8$ MeV, that corresponds to the energy of the $p_{3/2}$ resonance. Another p-resonance is present at 2 MeV in $^5$He. However its width is much larger than before, and therefore its effect on the spectrum is much more spreaded out. To illustrate how the invariant mass spectrum is sensitive to the resonance energies we also show the case where the neutron-$^4$He potential introduced in ref. [@zhu93] is used (long-dashed line). The suggested increase by 3% of the range of the neutron-$^4$He potential makes the energy of the $p_{3/2}$ resonance too small (around 0.3 MeV), giving rise to the pronounced peak in the invariant mass spectrum at that energy. The different height of the curves in fig. \[9\] comes from the fact that all the three curves are normalized to 1.
In fig. \[10\] we show the same spectrum as in fig. \[9\] from a neutron removal $^{11}$Li reaction. The cases of the shallow and deep potentials are shown (cases 1 and 2 in table II). The difference between these two cases comes from the different energy of the lowest p-resonance.
For the deep potential (dashed line) the $p_{1/2}$ resonance is at 0.77 MeV, creating a clear shoulder in the curve at that energy. Note that the contribution to the wave function from the p-wave (21% probability) is given directly by the schematic model. It is determined by the energies of the virtual $s_{1/2}$ state and the $p_{1/2}$ resonance together with the requirement of fitting the binding energy and radius of $^{11}$Li. Therefore the shoulder in the distribution is less pronounced than in calculations where the p-wave content is chosen to be higher, for instance in [@han95], where a rather arbitrary 50% contribution for the p-waves is chosen. For the shallow potential (solid curve) the large width of the resonance at 1.7 MeV broadens the corresponding peak at that energy and it disappears completely. For both the deep and the shallow potential the lowest peak comes from the low lying virtual s-state in $^{10}$Li. However in this case the position of the peak is not directly related to the energy of the virtual state. A virtual state produces an increase
of the momentum distributions at zero momentum. In fact, the value of the invariant mass spectrum divided by $\sqrt{E_{\mbox{\scriptsize core+n}}}$ is not zero at zero energy, and the lower the virtual s-state the larger the value at the origin. In the inset of fig. \[10\] we show the deep potential case where we have separated the contributions from the s and the p-waves. It is then clear how the s-wave is responsible for the first peak, while the shoulder is produced by the $p_{1/2}$ resonance at 0.77 MeV. The experimental points [@eml96] correspond to a $^{11}$Li fragmentation process on carbon. A better agreement between the computed curves and the experimental data is possible when the spin of the core is taken into account in the description of $^{11}$Li and $^{10}$Li.
Summary and conclusions
=======================
We study two prescriptions to take the Pauli principle into account in the three-body cluster model where more than one cluster contain nucleons. One is to exclude the Pauli forbidden two-body state from the active space available for the three-body system, the other is to construct a two-body potential without the Pauli forbidden state but with the same low energy properties.
We solve the Faddeev equations in coordinate space by means of the adiabatic hyperspherical expansion. We calculate the angular eigenvalues of the Faddeev equations which are closely related to the effective radial potentials. This spectrum is computed for the two different potentials with identical low energy properties and either without or with one Pauli forbidden bound s-state. Each bound state gives rise to a specifically behaving angular eigenvalue. The technically inexpensive prescription to account for the Pauli principle is then simply to omit the angular eigenvalue which corresponds to the Pauli forbidden bound state in the calculations of the three-body radial wave functions.
We apply these prescriptions to analyses of the high energy fragmentation reactions of two halo nuclei $^6$He and $^{11}$Li within a three-body neutron+neutron+core model. The potentials of the two prescriptions are adjusted to reproduce the same s- and p-wave low energy data.
For $^6$He these data are known experimentally and the interactions are therefore fixed. The potential with forbidden state then gives correct binding energy and root mean sqare radius for $^6$He. The equivalent potential without a bound state slightly underbinds the system. A small reduction of the repulsive core is needed to obtain the correct binding. This automatically provides almost the same reasonable root mean square radius. After this small adjustment the wave functions for the two nuclei for the two types of potential are compared and found remarkably similar.
For $^{11}$Li we use the currently accepted experimentla data for the lowest s- and p- levels in the neutron core subsystem to adjust the neutron core potentials. With these levels being close to experimental data the deep potentials provide reasonable binding energy and root mean square radius. The equivalent potential without a bound state slightly overbinds the system. After a small attenuation of the position of the neutron core $p_{3/2}$ level this prescription also provides the correct binding and size of the system.
In general the two prescriptions without any fine tuning provide very close but still distinguishable ground state properties of the three-body system. After small adjustments the properties become remarkably similar.
We then compute neutron and core momentum distributions in nuclear break-up reactions of these halo nuclei. We use the sudden approximation and include final state interactions, which are crucial for the neutron distributions as the full width at half maximum for the neutrons from $^6$He is reduced by a factor of more than 2 from the value corresponding to omitting the final state interaction.
The two prescriptions give almost identical results and the measured momentum distributions are rather accurately reproduced. These computations are carried out in a consistent model where the same two-body potential is used both for the initial three-body halo structure and for the final state interaction after the break-up process.
The final state invariant mass spectra for $^5$He and $^{10}$Li are computed and their features discussed. A virtual s-state shows up as a pronounced peak close to zero energy. This peak arises from the phase espace factor which vanishes at zero energy, and its position is unrelated with the energy of the virtual state. Higher angular momenta show up as distinct peaks at the resonance energy. These spectra are then sensitive to the continuum structure of the two-body system.
In conclusion the two prescriptions to account for the Pauli principle work remarkably well for the two test examples $^6$He and $^{11}$Li. In both cases the lowest Pauli forbidden s-state is occupied by nucleons in the core nucleus. However, the valence neutrons occupy predominantly neutron-core relative p-states and s-states, respectively for $^6$He and $^{11}$Li. The wave functions are further tested in connection with our fragmentation model and the measured momentum distributions are nicely reproduced. For $^6$He these quantities are computed for the first time whereas $^{11}$Li was investigated earlier in a more sophisticated model. Although fairly realistic, we used here only a schematic model, because the main purpose was to demonstrate the reliability to account for the Pauli pinciple. In any case the results provide additional support both for the prescriptions and for the fragmentation model.
#### Acknowledgments. {#acknowledgments. .unnumbered}
We want to thank B. Jonson, K. Riisager, and H. Emling for useful discussions and for making the latest unpublished experimental data available. One of us (E.G.) acknowledges support from the European Union through the Human Capital and Mobility program contract nr. ERBCHBGCT930320.
[99]{} Jean-Marc Richard, Phys. Rep. 212 (1992) 1 M.V. Zhukov, B.V. Danilin, D.V. Fedorov, J.M. Bang, I.J. Thompson and J.S. Vaagen, Phys. Rep. 231 (1993) 151 D.V. Fedorov, A.S. Jensen and K. Riisager, Phys. Rev. C49 (1994) 201 V.I. Kukulin and V.N. Pomerantsev, Ann of Phys. 111 (1978) 330 S. Marsh, Nucl. Phys. A389 (1982) 45 J. Bang, J.J Benayoun, C. Gignoux and I.J. Thompson, Nucl. Phys. A405 (1983) 126 L. Johannsen, A.S. Jensen and P.G. Hansen, Phys. Lett. B244 (1990) 357 D. Baye, Phys. Rev. Lett. 58 (1987) 2738 H. Fiedeldey, S.A. Sofianos, A. Papastylianos, K. Amos and L.J. Allen, Phys. Rev. C42 (1990) 411 T. Kobayashi, O. Yamakawa, K. Omata, K. Sugimoto, T. Shimoda, N. Takahashi and Tanihata, Phys. Rev. Lett. 60 (1988) 2599 E. Garrido, D.V. Fedorov and A.S. Jensen, Phys. Rev. C53 (1996) 3159 A.A. Korsheninnikov and T. Kobayashi, Nucl. Phys. A567 (1994) 97 D.V. Fedorov and A.S. Jensen, Phys. Rev. Lett. 71 (1993) 4103 D.V. Fedorov, A.S. Jensen and K. Riisager, Phys. Rev. C50 (1994) 2372 D.V. Fedorov, E. Garrido, and A.S. Jensen, Phys. Rev. C51 (1995) 3052 F. Ajzenberg-Selove, Nucl. Phys. A490 (1988) 1 Bohlen et al. Z.Phys. A344 (1993) 381 B.M. Young et al. Phys. Rev. C49 (1994) 279 M. Zinser et al., Phys. Rev. Lett. 75 (1995) 1719 S.N. Abramovich, B.Ya. Guzhovskij and L.M. Lazarev, Phys. Part. Nucl. 26 (1995) 423 T. Kobayashi, Nucl. Phys. A538 (1992) 343c F. Humbert et al., Phys. Lett. B347 (1995) 198 T. Kobayashi, Nucl. Phys. A553 (1993) 465c I.J. Thompson and M.V. Zhukov, Phys. Rev. C49 (1994) 1904 J.S. Al-khalili and J.A. Tostevin, Phys. Rev. Lett. 76 (1996) 3903 H. Geissel and W. Schwab, private communication T. Nilsson et al., Europhys. Lett. 30 (1995) 19 P.G. Hansen, Proceedings International Conference on Exotic Nuclei and Atomic Masses (M. de Saint Simon and O. Sorlin, Arles, France, 1995), 175 H. Emling, private communication, and the S034 collaboration to be published
[^1]: our sign convention is $k^{2l+1}\cot(\delta)\stackrel{k\rightarrow 0}{\rightarrow}
1/a+r_ek^2/2$
|
---
abstract: 'Numerical mode matching (NMM) methods are widely used for analyzing wave propagation and scattering in structures that are piecewise uniform along one spatial direction. For open structures that are unbounded in transverse directions (perpendicular to the uniform direction), the NMM methods use the perfectly matched layer (PML) technique to truncate the transverse variables. When incident waves are specified in homogeneous media surrounding the main structure, the total field is not always outgoing, and the NMM methods rely on reference solutions for each uniform segment. Existing NMM methods have difficulty handing gracing incident waves and special incident waves related to the onset of total internal reflection, and are not very efficient at computing reference solutions for non-plane incident waves. In this paper, a new NMM method is developed to overcome these limitations. A Robin-type boundary condition is proposed to ensure that non-propagating and non-decaying wave field components are not reflected by truncated PMLs. Exponential convergence of the PML solutions based on the hybrid Dirichlet-Robin boundary condition is established theoretically. A fast method is developed for computing reference solutions for cylindrical incident waves. The new NMM is implemented for two-dimensional structures and polarized electromagnetic waves. Numerical experiments are carried out to validate the new NMM method and to demonstrate its performance.'
author:
- 'Wangtao Lu [^1]'
- 'Ya Yan Lu[^2]'
- '[Dawei Song]{} [^3]'
bibliography:
- 'wt.bib'
title: A numerical mode matching method for wave scattering in a layered medium with a stratified inhomogeneity
---
Introduction
============
Wave scattering problems in a layered medium with a penetrable or impenetrable inhomogeneity appear in numerous scientific and engineering applications [@chew95]. Classical numerical methods such as the finite difference method, the finite element method (FEM) [@mon03], and the spectral method are very versatile, but are not always the most efficient, since they need to discretize the whole computational domain. For piecewise homogeneous structures, the boundary integral equation (BIE) methods [@cai02; @brulyoperaratur16; @laigreone16; @luluqia18] are highly competitive since they discretize only the interfaces and the boundary of the inhomogeneity. If the structure can be divided into a number of segments or regions where the governing equation becomes separable, the mode matching method, a.k.a mode expansion method or modal method [@botcramcp81; @li93a; @shestesan82], and its many numerical variants [@chiyehshi09; @gra99; @gragui96; @kno78; @lalmor96; @li96; @lushilu14; @mor95; @sonyualu11; @derdezoly98; @biebae01; @biederbaeolydez01] may be used. Typically, these methods are applicable if the structure is piecewise uniform along one spatial direction. In each uniform segment, the wave field is expanded in eigenmodes of a related transverse differential operator, and the expansion coefficients are solved from a linear system obtained by matching the wave field at the interfaces between neighboring segments. The classical mode matching method solves the eigenmodes analytically. The numerical mode matching (NMM) methods solve the eigenmodes by numerical methods, and they are easier to implement and applicable to more general structures. The mode matching method and its variants have the advantage of avoiding discretizing one spatial variable. They are widely used in engineering applications, since many designed structures are indeed piecewise uniform.
For numerical simulations of waves, the perfectly matched layer (PML) [@ber94] is an important technique for truncating unbounded domains. It is widely used with standard numerical methods, such as FEM, that discretize the whole computational domain. The BIE methods usually automatically take care of the radiation conditions at infinity, but for scattering problems in layerd media, PML can also be used to efficiently truncate interfaces that extend to infinity [@luluqia18]. For NMM methods, PML was first applied to study piecewise uniform waveguides [@derdezoly98; @biebae01; @biederbaeolydez01]. An optical waveguide is an open structure, i.e., the transverse domain perpendicular to the waveguide axis is unbounded. Analytic mode matching method is difficult to use, since the transverse operator has a continuous spectrum and field expansions contain integrals related to the radiation modes. When a PML is used to truncate the transverse domain, typically with a zero Dirichlet boundary condition at the external boundary of the PML, the continuous spectrum is discretized, and the field expansions are approximated by sums of discrete eigenmodes.
For many applications, an incident wave is specified in the homogeneous media surrounding the scatterer, then the total wave field in each uniform segment does not satisfy outgoing radiation conditions in the transverse directions, and is incompatible with the eigenmodes computed using a PML. To overcome this difficulty, we can find a reference solution for the given incident wave in each uniform segment, and then expand the difference between the total field and the reference solution in the PML-based eigenmodes [@lushilu14]. Typically, the field difference in each segment is indeed outgoing in the transverse directions, and a NMM method based on this approach works reasonably well. But unfortunately, the method breaks down in special circumstances where the field difference in a segment contains a component that is exactly or nearly invariant in the transverse direction, i.e., a component with a zero transverse wavenumber. This happens if the incident wave has the critical incident angle for the onset of total internal reflection in the exterior segments. In that case, the field difference in any interior segment contains a non-propagating and non-decaying component with a zero or near zero transverse wavenumber. This difficulty also arises when the incident wave is nearly parallel to the uniform direction, i.e., a gracing incidence. In that case, the field difference in an interior segment also contains a plane wave component with a near zero transverse wavenumber.
In this paper, we develop a new NMM method to overcome the above difficulty. Our approach is to use a Robin boundary condition for the PML in the interior segments. The boundary condition is designed to ensure that the field component with a zero or near zero transverse wavenumber is not reflected by the PML. A similar Robin-type condition for PML was previously used by one of the authors to preserve a weakly confined guided mode propagating in optical waveguides [@holu03]. For the exterior segments, we keep the simple zero Dirichlet boundary condition. To give the method a theoretical foundation, we analyze the effectiveness of the PML using hybrid Dirichlet-Robin boundary conditions. It is shown that the error induced by the PML decays exponentially with the thickness or the absorbing coefficient of the PML. For scattering problems with incident waves from a point or line source, the NMM method faces an additional difficulty, namely, the computation of the reference solutions, especially for the segment involving the inhomogeneity. The traditional approach that turns a point or line source to plane waves by Fourier transform is not very efficient. We develop an efficient method for computing the reference solutions based the PML technique and the method of separation of variables.
The rest of this paper is organized as follows. In Section 2, we formulate the scattering problem, review the PML theory. In Section 3, we describe an NMM method. In Section 4, we derive the new Robin-type boundary condition and show that the solution based on a PML and a hybrid Dirichlet-Robin condition converges to the true scattering solution exponentially. In Section 5, we develop an efficient method for computing reference solutions when the incident wave is a line source. In Section 6, we present a few numerical examples to validate our NMM method and to illustrate its performance. The paper is concluded by some remarks and discussions in Section 7.
Problem formulation
===================
To simplify the presentation, we begin with a scattering problem in a two-layer medium. The physical structure is characterized by a $z$-invariant dielectric function $$\begin{aligned}
\label{eq:eps:func}
\varepsilon(x,y) = \left\{
\begin{array}{lc}
\varepsilon_{+}=n_+^2, & (x,y)\in \mathbb{R}_+^2 \backslash \bar{D},\\
\tilde{\varepsilon}(y), & (x,y)\in D,\\
\varepsilon_{-}=n_-^2, & (x,y)\in \mathbb{R}_-^2\backslash \bar{D},
\end{array}
\right.\end{aligned}$$ where $\mathbb{R}^2_{\pm}=\{(x,y)\in\mathbb{R}^2:\pm y>0\}$, $\tilde{ \varepsilon }(y)\geq 1$ is piecewise smooth on $(y_0,y_1)$, $D$ is a rectangle $(-x_0,x_0) \times (y_0,y_1)$ with $x_0>0$, $y_1\geq 0$ and $y_0\leq 0$, and it corresponds to a stratified inhomogeneity. In $\mathbb{R}_+^2/\bar{D}$, we specify a plane incident wave $u^{\rm
inc}=e^{i(\alpha x -\beta_+ y)}$, where $\alpha=k_0n_+\sin\theta$, $\beta_+=k_0n_+\cos\theta$, and $\theta\in(-\pi/2,\pi/2)$ is the incident angle. The total wave field $u^{\rm tot}$ satisfies the Helmholtz equation $$\begin{aligned}
\label{eq:gov:problem}
\Delta u^{\rm tot} + k_0^2 \varepsilon(x,y) u^{\rm tot} = 0,\end{aligned}$$ where $\Delta=\partial_x^2 + \partial_y^2$, and $k_0$ is the free-space wavenumber. Across an interface or discontinuity, we have the following transmission condition $$\begin{aligned}
\label{eq:trans:cond}
[u^{\rm tot}] = 0, \quad \left[\frac{\partial u^{\rm tot}}{\partial{\bm \nu}} \right]= 0,\end{aligned}$$ where ${\bm \nu}$ is the unit normal vector on the interface pointing towards $\mathbb{R}_-^2$, and $[\cdot]$ denotes the jump of the quantity across the interface. For electromagnetic waves in the $E$ polarization, $u^{\rm
tot}$ is the $z$ component (the only nonzero component) of the electric field.
At infinity, the scattered wave field defined as $$u^s=\left\{ \begin{array}{lc}
u^{\rm tot} - u^{\rm tot}_0, & {\rm in}\ \mathbb{R}^2/\bar{D},\\
u^{\rm tot}, &{\rm in}\ D,
\end{array}
\right.$$ satisfies the half-plane Sommerfeld radiation condition in both $\mathbb{R}^2_+$ and $\mathbb{R}^2_-$, i.e., $$\begin{aligned}
\label{eq:half:Som:cond}
\lim_{r\rightarrow \infty}\sqrt{r}(\partial_r u^s - ik_0n_{\pm}u^s) = 0,\quad r=\sqrt{x^2+y^2},\quad (x,y)\in\mathbb{R}_{\pm}^2/\bar{D}.\end{aligned}$$ Here, $u^{\rm tot}_0$ is the solution for the same incident wave in the background two-layer medium without the inhomogeneity. More precisely, we have $$\begin{aligned}
\label{eq:sol:cond:c}
u^{\rm tot}_0 &= \left\{\begin{array}{lc}
e^{i(\alpha x - \beta_+ y)} + Re^{i(\alpha x + \beta_+ y)} &(x,y)\in\mathbb{R}_+^2,\\
(R+1)e^{i(\alpha x - \beta_-y)} &(x,y)\in\mathbb{R}_-^2,
\end{array}\right.\\
\label{eq:sol:para}
\beta_- &= \sqrt{k_0^2\varepsilon_--\alpha^2},\quad R =
\frac{\beta_+-\beta_-}{\beta_++\beta_-}.\end{aligned}$$ According to [@roazha92; @chezhe10; @baohuyin18], we have the following existence and uniqueness results:
For any incident plane wave with $k_0>0$, the scattering problem (\[eq:gov:problem\]), (\[eq:trans:cond\]), (\[eq:half:Som:cond\]) has a unique solution $u^{\rm tot}$ in $H_{\rm loc}^{1}(\mathbb{R}^2)$.
Since $u^s$ is outgoing, the PML technique [@ber94] can be used to truncate $\mathbb{R}^2$. Let us define the following complex coordinate stretching functions $$\begin{aligned}
\label{eq:pml:xy}
\tilde{x}(x) = x + i\int_{0}^{x}\sigma_1 (t) dt,\quad
\tilde{y}(y) = y + i\int_{0}^{y}\sigma_2 (t) dt,\end{aligned}$$ where $\sigma_l(t) = \sigma_l(-t)$ for all $t$, $\sigma_l(t)=0$ for $|t|\leq
L_l/2$, and $\sigma_l(t)>0$ for $|t|>L_l/2$, and $L_l>0$ for $l=1$, $2$. Notice that the rectangle $B_1= (-L_1/2,L_1/2) \times (-L_2/2,L_2/2)$ encloses the inhomogeneity $D$, and the rectangle $B_2=(-L_1/2-d_1,L_1/2+d_1)
\times (-L_2/2-d_2,L_2/2+d_2)$ is used to truncate $\mathbb{R}^2$. Based on Green’s representation formula, the extension of $u^s$ in $B_2$ can be defined, then $$\tilde{u}^s(x,y):=u^s(\tilde{x}(x),\tilde{y}(y))$$ satisfies the following PML-Helmholtz equation $$\begin{aligned}
\label{eq:upml:1}
&\nabla\cdot(A\nabla\tilde{u}^s) + \alpha_1(x)\alpha_2(y)k_0^2\varepsilon(x,y)\tilde{u}^s = 0,\\
\label{eq:upml:2}
&\tilde{u}^{s}(x,0+) = \tilde{u}^{s}(x,0-), \quad \partial_{\tilde{y}} \tilde{u}^s(x,0+) = \partial_{ \tilde{y} } \tilde{u}^s(x,0-),\ {\rm in}\ \mathbb{R}/[-x_0,x_0].\end{aligned}$$ where $A={\rm diag}(\alpha_2(y)/\alpha_1(x),\alpha_1(x)/\alpha_2(y))$, and $\alpha_l=1+i\sigma_l$. Typically, a zero Dirichlet boundary condition is enforced on $\Gamma_2=\partial B_2$, i.e., $$\begin{aligned}
\label{eq:upml:3}
\tilde{u}^s(x,y) = 0,\quad{\rm on}\quad \Gamma_2.\end{aligned}$$ The following theorem characterizes the exponential convergence of the PML solution.
Let $\sigma_1=\sigma_2\equiv\sigma$ and $d_1=d_2 =d$ in the PML for some positive constants $\sigma$ and $d$, and let $\bar{\sigma}=\sigma d$ such that $\gamma_0\bar{\sigma}\geq \max(k_{\rm min}^{-1}, d)$, where $\gamma_0 = d /
\sqrt{(L_1+d)^2+(L_2+d)^2}$. Then for sufficiently large $\bar{\sigma}$, the PML problem (\[eq:upml:1\]-\[eq:upml:3\]) has a unique solution $\tilde{u}^s$ in $H^1(B_2)$. Moreover, there exists a constant $C$, which depends only on $\gamma_0$, $k_{\rm max}/k_{\rm min}$, and $L_2/L_1$, but independent of $n_-$, $n_+$, $L_1$, $L_2$, and $d$, such that $$\begin{aligned}
\label{eq:est:ut:u}
||u^s-\tilde{u}^s||_{H^1(B_1)}\leq C(1+\hat{C}^{-1})\gamma_1(1+k_{\rm min}L_1)^3\alpha_m^3\nonumber\\
(1+\bar{\sigma}/L_1)^2e^{-k_{\rm min}\gamma_0\bar{ \sigma }}||\tilde{u}^s||_{H^{1/2}(\Gamma_1)},
\end{aligned}$$ where $k_{\rm min}=k_0\min(n_-,n_+)$, $k_{\rm max}=k_0\max(n_-,n_+)$, $\gamma_1=e^{L_2\sqrt{|k_{\rm max}^2-k_{\rm min}^2|}/2}$, $\alpha_m=\sqrt{1+\sigma^2}$, and $\hat{C} =
\frac{\min(1,\sigma^3)}{2(1+\sigma^2)^2\max(1,k_{\rm max}^2d^2)}$.
The PML problem can be considered in $B_2/\bar{D}$ by regarding $D$ as an obstacle and enforcing the Dirichlet boundary condition $\tilde{u}^s|_{\partial D}=u^s|_{\partial
D}$. Evidently, this theorem follows directly from Theorem 7.2 in [@chezhe10].
Thanks to Theorem 2.2, $\tilde{u}^s$ converges to $u^s$ exponentially in $B_1$. Therefore, we only need to deal with $\tilde{u}^s$ in the bounded domain $B_2$ instead of $u^s$ in $\mathbb{R}^2$.
Numerical mode matching method
==============================
For the scattering problem formulated above, the NMM methods are applicable, since the structure is uniform in $x$ in three different segments corresponding to $x<-x_0$, $-x_0 < x < x_0$ and $x> x_0$, respectively. Since a PML is used in the NMM method, we define the three segments by $S_1=\{(x,y)|-d_1-L_1/2<x< -x_0\}\cap
B_2$, $S_2=\{(x,y)|-x_0< x< x_0\}\cap B_2$ and $S_3=\{(x,y)|x_0<x<d_1+L_1/2\}\cap B_2$. It is clear that $\varepsilon(x,y)=\varepsilon_i(y)$ in $S_i$ is independent of $x$, for $i=1$, 2, 3. Accordingly, $\Gamma_2$ (the boundary of $B_2$) is decomposed into three parts $\Gamma_{2}^1$, $\Gamma_{2}^2$ and $\Gamma_{2}^3$. In particular, $$\Gamma_2^2=\{(x,d_2+L/2)||x|<x_0\}\cup\{(x,-d_2-L/2)||x|<x_0\}.$$
In the last several decades, many different NMM methods have been developed. These methods use different numerical methods to solve the eigenmodes in the uniform segments, and also use different techniques to impose the continuity conditions at the interfaces between the neighboring segments. Our NMM method is similar to the one presented in [@lushilu14], and its basic steps are summarized below.
We consider segments $S_1$ and $S_3$ first. According to Eqs. (\[eq:upml:1\]-\[eq:upml:3\]), $\tilde{u}^s$ in $S_i$ ($i=1$, 3) solves $$\begin{aligned}
\label{eq:us:i1}
&\nabla\cdot(A\nabla\tilde{u}^s) + k_0^2\alpha_1(x)\alpha_2(y)\varepsilon_i(y) \tilde{u}^s = 0,\\
\label{eq:us:i2}
&\tilde{u}^{s}(x,0+) = \tilde{u}^{s}(x,0-), \quad \partial_{\tilde{y}} \tilde{u}^s(x,0+) = \partial_{ \tilde{y} } \tilde{u}^s(x,0-),\\
\label{eq:us:i3}
&\tilde{u}^{s}(x, d_2+L_2/2) = \tilde{u}^{s}(x,-d_2-L_2/2)=0.\end{aligned}$$ By the method of separation of variables, inserting $\tilde{u}^s(x,y)=\phi(y)\psi(x)$ into (\[eq:us:i1\]-\[eq:us:i3\]), we obtain the following eigenvalue problem for $\phi(y)$ $$\begin{aligned}
\label{eq:phi:te}
&\frac{1}{\alpha_2}\frac{d}{dy}\left( \frac{1}{\alpha_2}\frac{d\phi}{dy} \right) + k_0^2\varepsilon_i(y)\phi(y)=\delta\phi,\\
\label{eq:phi:te2}
&\phi(0+) = \phi(0-), \phi'(0+) = \phi'(0-),\\
\label{eq:phi:te3}
&\phi(d_2+L_2/2) = \phi(-d_2-L_2/2) = 0,\end{aligned}$$ and the associated equation for $\psi(x)$ $$\begin{aligned}
\label{eq:psi:te}
\frac{1}{\alpha_1}\frac{d}{dx}\left( \frac{1}{\alpha_1}\frac{d\psi}{dx} \right) + \delta\psi = 0.\end{aligned}$$
The above Sturm-Liouville eigenvalue problem (\[eq:phi:te\]-\[eq:phi:te3\]) for $\phi$ is not self-adjoint, thus $\delta$ is in general complex. Nevertheless, $\delta$ can be forced to the upper half-plane based on the following proposition.
Under the same assumptions as Theorem 2.2, we have that for sufficiently large $\bar{\sigma}$, ${\rm Im}(\delta)\geq 0$.
See Proposition A.1 in Appendix A.
As in [@sonyualu11], we employ a pseudospectral method to find the numerical eigenmodes. Assuming $N$ eigenpairs $\{\delta_j, \phi_j(y)\}$ for $j=1$, ..., $N$, are obtained based on the $N$ collocation points $\{y^j\}_{j=1}^{N}\subset [-d_2-L_2/2, d_2+L_2/2]$, we approximate $\tilde{u}^s$ by $$\begin{aligned}
\label{eq:region1}
\tilde{u}^s \approx \sum_{j=1}^{N}\left[ c_j^{(1)}
e^{-i\sqrt{\delta_j}(\tilde{x}(x)-\tilde{x}(-x_0))} + d_j^{(1)}
e^{i\sqrt{\delta_j}(\tilde{x}(x)-\tilde{x}(-d_1-L_1/2))} \right] \phi_j(y)\end{aligned}$$ in $S_1$, and by $$\begin{aligned}
\label{eq:region3}
\tilde{u}^s \approx \sum_{j=1}^{N} \left[ c_j^{(3)}
e^{-i\sqrt{\delta_j}(\tilde{x}(x)-\tilde{x}(d_1+L_1/2))} + d_j^{(3)}
e^{i\sqrt{\delta_j}(\tilde{x}(x)-\tilde{x}(x_0))} \right] \phi_j(y)\end{aligned}$$ in $S_3$, where $\sqrt{\delta_j}$ is defined to be in the branch with ${\rm
Im}(\sqrt{\delta_j})\geq 0$ and hence with ${\rm Re}(\sqrt{\delta_j})\geq 0$ according to Proposition 1. Based on the zero Dirichlet boundary condition at $x=\pm(d_1+L_1/2)$, we get $$\begin{aligned}
\label{eq:dj1}
d_j^{(1)} &= -c_j^{(1)} e^{-i\sqrt{\delta_j}(\tilde{x}(-d_1-L_1/2)-\tilde{x}(-x_0))},\\
\label{eq:cj3}
c_j^{(3)} &= -d_j^{(3)}e^{i\sqrt{\delta_j}(\tilde{x}(d_1+L_1/2)-\tilde{x}(-x_0))}.\end{aligned}$$ Therefore, $$\begin{aligned}
|d_j^{(1)}|=|c_j^{(1)}| e^{-{\rm Im}(\sqrt{ \delta_j })(L_1/2-x_0+d_1) - {\rm Re}(\sqrt{ \delta_j })\int_{L_1/2}^{L_1/2+d_1}\sigma(t)dt}\approx 0,\\
|c_j^{(3)}|=|d_j^{(3)}| e^{-{\rm Im}(\sqrt{ \delta_j })(L_1/2-x_0+d_1) - {\rm Re}(\sqrt{ \delta_j })\int_{L_1/2}^{L_1/2+d_1}\sigma(t)dt}\approx 0,\end{aligned}$$ for sufficiently large $\sigma$ and $d_1$. Consequently, we can assume that there are no terms with coefficients $d_j^{(1)}$ and $c_j^{(3)}$ in Eqs. (\[eq:region1\]) and (\[eq:region3\]), respectively. Physically, this corresponds to the fact that $u^s$ should not contain incoming waves in the two exterior segments.
In segment $S_2$, we have $$\begin{aligned}
\label{eq:eps2}
\varepsilon_2(y) = \left\{
\begin{array}{lc}
n_+^2, & y>y_1,\\
\tilde{\varepsilon}(y), & y_0<y<y_1,\\
n_-^2, & y<y_0.
\end{array}
\right.\end{aligned}$$ The method of separation of variables is not applicable to $\tilde{u}^s$, since it does not satisfy the homogeneous transmission conditions (\[eq:trans:cond\]) at $y=y_0$ and $y=y_1$. Instead, we need to subtract from $u^{\rm tot}$ a wave field $u^{\rm tot}_2$ which solves the scattering problem for the same incident wave and a layered profile $\varepsilon(x,y)=\varepsilon_2(y)$ in $\mathbb{R}^2$. We let $u^{\rm tot}_2$ be the solution with the same $x$-dependence as the incident wave. More details are given Proposition A.2.
For $u_2^s = u^{\rm tot} - u_2^{\rm tot}$, we enforce the same zero Dirichlet boundary condition $$\label{eq:dir:cond}
\tilde{u}_2^s=0,\quad{\rm on}\ \Gamma_2^2,$$ where $\tilde{u}_2^s(x,y)=u_2^s(\tilde{x}(x),\tilde{y}(y))$. The method of separation of variables can be applied to $\tilde{u}_2^s$. Based on the same discretization points $\{y^j\}_{j=1}^N$, we obtain $N$ eigenpairs $\{\delta_j^{(2)},\phi_j^{(2)}(y)\}_{j=1}^{N}$ in $S_2$, then $$\begin{aligned}
\label{eq:region2}
\tilde{u}_2^s \approx \sum_{j=1}^{N}\left[ c_j^{(2)} e^{-i\sqrt{\delta_j^{(2)}}(x-x_0)} + d_j^{(2)}
e^{i\sqrt{\delta_j^{(2)}}(x+x_0)} \right] \phi_j^{(2)}(y).\end{aligned}$$
On the two interfaces between $S_2$ and the other two segments $S_1$ and $S_3$, i.e. at $x=\pm x_0$, we have the transmission conditions $$\begin{aligned}
\label{eq:gov:tran1}
[\tilde{u}_s(\pm x_0,y) - \tilde{u}_s^2(\pm x_0,y)] &= [\tilde{f}(\pm x_0,y)],\\
\label{eq:gov:tran2}
[\partial_x \tilde{u}_s(\pm x_0,y) - \partial_x \tilde{u}_s^2(\pm
x_0,y)] &= [\tilde{g}(\pm x_0,y)], \end{aligned}$$ where $\tilde{f}(x,y)=f(x,\tilde{y}(y))$, $\tilde{g}(x,y)=g(x,\tilde{y}(y))$, and $$f(x,y)=u^{\rm tot}_2(x,y) - u^{\rm tot}_0(x,y),\quad g(x, y)=\partial_x u^{\rm tot}_2(x,y) - \partial_x u^{\rm tot}_0(x,y).$$ Collocating (\[eq:gov:tran1\]) and (\[eq:gov:tran2\]) at $y=y^j$ for $j=1,\ldots, N$, and using Eqs. (\[eq:region1\]), (\[eq:region3\]) and (\[eq:region2\]), we obtain a linear system $$\begin{aligned}
{\bm A}\left[\begin{array}{c}
{\bm c}^{(1)}\\
{\bm c}^{(2)}\\
{\bm d}^{(2)}\\
{\bm d}^{(3)}\\
\end{array}\right] = {\bm b},\end{aligned}$$ where ${\bm A}$ is a $4N\times 4N$ matrix, ${\bm b}$ is a $4N\times 1$ matrix, ${\bm c}^{(i)}=[c_1^{(i)}, \ldots,
c_N^{(i)}]^{T}$, etc. Solving the above system, we get $\tilde{u}^s$ in $S_1$ and $S_3$, and $\tilde{u}^s_2$ in $S_2$, thus $u^{\rm tot}$ can be found in the physical domian $B_1$.
In the above, the NMM method is only presented for the case of a single inhomogeneous segment in a two-media layered background. It is straightforward to extend the NMM method to structures with multiple inhomogeneous segments that are uniform along the same direction. The NMM method can also be used to study scattering problems in the $H$ polarization (the only nonzero component of the magnetic field is its $z$ component) and problems involving perfect electrical conductor (PEC) or perfect magnetic conductor (PMC) scatterers.
A Robin-type boundary condition
===============================
As we mentioned in the introduction, the NMM method based on the zero Dirichlet condition (\[eq:dir:cond\]) usualy works, but in some special circumstances, it exhibits a slow convergence and even a divergence, since $u_2^s$ may not be strictly outgoing. It should be pointed out that there is no contradiction with Theorem 2.2, since that theorem is about applying the PML to $u^s$, but the NMM method applies the PML to $u_2^s$ for the interior segment $S_2$.
In fact, it is easy to deduce that $$\begin{aligned}
\label{eq:us2:asym:p}
u_2^s&= (R-R_2e^{-2i\beta_+ y_1})e^{i(\alpha x+\beta_+ y)} + u^s,\quad{\rm for}\
y>y_1,\\
u_2^s&=(T - T_2e^{i\beta_-y_0-i\beta_+ y_1})e^{i(\alpha x - \beta_-
y)}+u^s,\quad{\rm for}\ y<y_0, \end{aligned}$$ where $R$ and $R_2$ ($T$ and $T_2$) are the reflection (transmission) coefficients in the reference solutions $u_0^{\rm tot}$ and $u_2^{\rm
tot}$, respectively. Therefore, $u_2^s$ can be decomposed as a scattered wave field and an up-going plane wave with $y$-wavenumber $\beta_+$ for $y>y_1$ or a down-going plane wave with $y$-wavenumber $\beta_-$ for $y<y_0$. Consequently, only when $\beta_+$ and $\beta_-$ are sufficiently far away from zero, does $u_2^s$ attenuate in the PML. However, this is not ture for the following two cases:
- For gracing incidences with $\theta$ close to $\pm\pi/2$, $\beta_+$ is close to $0$;
- For $n_-< n_+$ and at the critical angles $\theta = \pm
\arcsin (n_-/n_+)$ for the onset of total internal reflection, $\beta_-=0$.
Notice that $\tilde{u}^s\approx 0$ at the exterior boundary of the PML, therefore $$\begin{aligned}
\tilde{u}_2^s&\approx (R-R_2e^{-2i\beta_+ y_1})e^{i(\alpha x+\beta_+
\tilde{y})},\quad{\rm on}\ y=d_2+L_2/2,\\
\tilde{u}_2^s&\approx(T - T_2e^{i\beta_-y_0-i\beta_+ y_1})e^{i(\alpha x -
\beta_- \tilde{y})},\quad{\rm on}\ y=-d_2-L_2/2.\end{aligned}$$ However, the NMM method is not compatible with the above inhomogeneous boundary conditions. To overcome this difficulty, the following result is needed.
The scattered fields $u^s$ and $u_2^s$ satisfy $$\begin{aligned}
\label{eq:us:u}
\partial_{y} u_2^s - i\beta_+ u_2^s = \partial_{y} u^s - i\beta_+ u^s,\quad{\rm on}\ y=d_2+L_2/2,\\
\label{eq:us:b}
\partial_{y} u_2^s + i\beta_- u_2^s = \partial_{y} u^s + i\beta_- u^s,\quad{\rm on}\ y=-d_2-L_2/2.\end{aligned}$$ Correspondingly, $$\begin{aligned}
\label{eq:tus:u}
(A\nabla \tilde{u}_2^s)\cdot{\bf \nu} - i\beta_+ \tilde{u}_2^s =
(A\nabla\tilde{u}^s)\cdot{\bf \nu} - i\beta_+ \tilde{u}^s,\quad{\rm on}\
y=d_2+L_2/2,\\
\label{eq:tus:b}
(A\nabla \tilde{u}_2^s)\cdot{\bf \nu} - i\beta_- \tilde{u}_2^s =
(A\nabla\tilde{u}^s)\cdot{\bf \nu} - i\beta_- \tilde{u}^s,\quad{\rm on}\
y=-d_2-L_2/2.
\end{aligned}$$
The proof is straightforward.
The above proposition suggests the following homogeneous Robin boundary conditions, $$\begin{aligned}
\label{eq:cond:tus:1}
\frac{1}{\alpha_2}\partial_y\tilde{u}_2^s-i\beta_+\tilde{u}_2^s&=(A\nabla \tilde{u}_2^s)\cdot{\bf \nu} - i\beta_+ \tilde{u}_2^s = 0,\quad{\rm on}\quad y=d_2+L_2/2,\\
\label{eq:cond:tus:2}
\frac{1}{\alpha_2}\partial_y\tilde{u}_2^s+i\beta_+\tilde{u}_2^s&=(A\nabla \tilde{u}_2^s)\cdot{\bf \nu} - i\beta_- \tilde{u}_2^s = 0,\quad{\rm on}\quad y=-d_2-L_2/2.\end{aligned}$$ Based on the pseudospectral method [@sonyualu11] and the above boundary conditions, we can find the eigenmodes $\phi_j^{(2)}$ ($1 \le j \le N$), and expand $\tilde{u}_2^s$ in $S_2$ in these eigenmodes.
Although different boundary conditions are used on $\Gamma_2$, the following theorem ensures that $\tilde{u}^s$ still converges to $u^s$ exponentially.
Under the same assumptions as Theorem 2.2, we have that for sufficiently large $\bar{\sigma}$, the PML problem (\[eq:upml:1\]), (\[eq:upml:2\]) equipped with the following hybrid Dirichlet-Robin boundary condition $$\begin{aligned}
\label{eq:general:cond}
\left\{\begin{array}{lc}
\tilde{u}^s = 0,&{\rm on}\quad \Gamma_2/\overline{{\Gamma}}\\
( A\nabla\tilde{u}^s )\cdot {\bm \nu} = iW^2 \tilde{u}^s,&{\rm on}\quad \Gamma,
\end{array}
\right.
\end{aligned}$$ where $W\in L^{\infty}(\Gamma)$ is real-valued and $\Gamma\subset\Gamma_2$ is an open bounded set, has a unique solution $\tilde{u}^s$ in $H^1(B_2)$. Moreover, there exists a constant $C$, which depends only on $||W^2||_{L^\infty(\Gamma)}$, $\gamma_0$, $k_{\rm
max}/k_{\rm min}$, and $L_2/L_1$, but independent of $n_-$, $n_+$, $L_1$, $L_2$, and $d$, such that $$\begin{aligned}
||u^s-\tilde{u}^s||_{H^1(B_1)}\leq C(1+\hat{C}^{-1})\gamma_1(1+k_{\rm min}L_1)^3\alpha_m^3\nonumber\\
(1+\bar{\sigma}/L_1)^2e^{-k_{\rm min}\gamma_0\bar{ \sigma }}||\tilde{u}^s||_{H^{1/2}(\Gamma_1)}.
\end{aligned}$$
This can be proved by the similar argument as the proof of Theorem 7.2 in [@chezhe10]. We here only mention significant modifications. For consistency and simplicity, we will load the whole notations from [@chezhe10] and will use them only in this proof so that $x=(x_1,x_2)$ now denotes a point but not a scalar, etc..
The PML equation in the PML layer (see Eqs. (5.1-5.3) in [@chezhe10]) for the generalized boundary condition (\[eq:general:cond\]) should be revised to $$\begin{aligned}
\label{eq:proof:pmleq}
&\nabla\cdot(A\nabla w) + \alpha_1\alpha_2k^2w = 0,\quad {\rm in}\quad \Omega_{\rm PML} = B_2\backslash\bar{B}_1,\\
\label{eq:proof:pmleq2}
&[w]_\Sigma=\left[ \frac{\partial w}{\partial x_2} \right]_\Sigma=0\quad {\rm on}\quad \Sigma\cap\Omega_{\rm PML},\\
\label{eq:proof:pmleq3}
&w=0\ {\rm on}\ \Gamma_1,\quad w=q\ {\rm on}\ \Gamma_2\backslash\bar{\Gamma},\quad (A\nabla w)\cdot{\bf \nu}-iW^2 w=\tilde{q},\ {\rm on}\quad\Gamma,
\end{aligned}$$ where $q\in H^{1/2}(\Gamma_2/\bar{\Gamma})$ and $\tilde{q}\in
H^{-1/2}(\Gamma)$. Then, the related sesquilinear form $c: H^{1}(\Omega_{\rm
PML})\times H^{1}(\Omega_{\rm PML})\rightarrow \mathbb{C}$ becomes $$c(\varphi,\psi) =
\int_{\Omega_{PML}}(A\nabla\varphi\cdot\nabla\bar{\psi}-\alpha_1\alpha_2k^2\varphi\bar{\psi})dx
- i\int_{\Gamma}W^2\varphi\bar{\psi}ds,\forall\varphi,\psi\in
H_{0/\Gamma}^1(\Omega_{\rm PML}),$$ where $H_{0/\Gamma}^1(\Omega_{\rm PML}):=\{v\in H^{1}(\Omega_{\rm PML}):
v=0\ {\rm on}\ \Gamma_1\cup\Gamma_2/\bar{\Gamma}\}$. The weak formulation of (\[eq:proof:pmleq\]-\[eq:proof:pmleq3\]) is: Find $w\in
H^{1}(\Omega_{\rm PML})$ such that Eq.(\[eq:proof:pmleq3\]) is satisfied and that $$\begin{aligned}
\label{eq:proof:weakform}
c(w,v) = \langle \tilde{q}, v\rangle|_{\Gamma},\quad\forall v\in H_{0/\Gamma}^1(\Omega_{\rm PML}).
\end{aligned}$$ Correspondingly, the weighted $H^1$-norm is revised to $$|||\varphi|||_{H^1(\Omega)}=\left( ||\nabla\varphi||^2_{L^2(\Omega)} +
||k\varphi||^2_{L^2(\Omega)} +
||W\varphi||_{\Gamma}^2/(1+\sigma^2)^2 \right)^{1/2},$$ and the equivalent norm on $H^{1}(\Omega_{\rm PML})$ becomes $$||\varphi||_{*,\Omega_{\rm PML}} = \left(
||A\nabla\varphi||^2_{L^2(\Omega_{\rm PML})} +
||k\alpha_1\alpha_2\varphi||^2_{L^2(\Omega_{\rm PML})} +
||W\varphi||_{\Gamma}^2 \right)^{1/2}.$$
Next, one sees that Lemma 5.1 in [@chezhe10] still holds with the space $H_0^1$ replaced by $H_{0/\Gamma}^1$. The proof relies on the following estimates $$||\varphi||_{L^2(\Omega_1)}^2\leq d_1^2\left| \left|
\frac{\partial\varphi}{\partial x_1} \right|
\right|^2_{L^2(\Omega_1)},\quad ||\varphi||_{L^2(\Omega_2)}\leq
d_2^2\left| \left| \frac{\partial\varphi}{\partial x_2}\right| \right|^2,$$ which were proved by using $\varphi=0$ on $\Gamma_2$ in [@chezhe10]. However, we remark that these two estimates still hold even when $\varphi\neq0$ on $\Gamma_2$ since we always have $\varphi=0$ on $\Gamma_1$ for $\varphi\in
H_{0/\Gamma}^1(\Omega_{\rm PML})$.
Thus, it is clear that Lemma 5.2 in [@chezhe10] holds with the space of $\zeta$ replaced by “for any $\zeta\in H^1(\Omega_{\rm PML})$ such that $\zeta=0$ on $\Gamma_1$, $\zeta=q$ on $\Gamma_2/\bar{\Gamma}$, and $A(\nabla\zeta)\cdot{\bf \nu}-iW^2\zeta=\tilde{q}$ on $\Gamma$.”
Next, Lemma 7.1 in [@chezhe10] holds after one replaces $X(f)$ with $$\begin{aligned}
\tilde{X}(f):=\{&\zeta\in H^1(\Omega_{\rm PML}): \zeta=0\ {\rm on}\
\Gamma_1, \zeta
= \mathbb{E}(f)\ {\rm on}\ \Gamma_2/\bar{\Gamma}, \\
&(A\nabla\zeta)\cdot {\bf \nu} - iW^2\zeta =
(A\nabla_x\mathbb{E}(f))\cdot {\bf \nu} -
iW^2\mathbb{E}(f)\ {\rm on}\ \Gamma\}.
\end{aligned}$$ Here, we will have $$\begin{aligned}
\inf_{\zeta\in \tilde{X}}||\zeta||_{*,\Omega_{\rm PML}}\leq
C(1+k_1L_1)\alpha_m^2(& ||\mathbb{E}(f)||_{H^{1/2}(\Gamma_2/\bar{\Gamma})} \nonumber\\
&+ ||(A\nabla_x\mathbb{E}(f))\cdot{\bf \nu}-iW^2\mathbb{E}(f)||_{H^{-1/2}}(\Gamma)),
\end{aligned}$$ where $C$ now depends on the norm $||W^2||_{L^2(\Gamma_2)}$ considering the modified norm $||\cdot||_{*,\Omega_{\rm PML}}$. Since $\mathbb{E}(f)$ is smooth on $\Gamma$, $$(A\nabla_x\mathbb{E}(f))\cdot{\bf \nu}-iW^2\mathbb{E}(f)\in
L^{2}(\Gamma)\cap L^{\infty}(\Gamma),$$ so that $$||(A\nabla\mathbb{E}(f))\cdot{\bf
\nu}-iW^2\mathbb{E}(f)||_{H^{-1/2}(\Gamma)}\leq
C||\mathbb{E}(f)||_{W^{1,\infty}(\Gamma)},$$ and hence Lemma 7.1 in [@chezhe10] follows which proves the theorem.
If we define $W$ in $\Gamma_2^2$ by $$\begin{aligned}
W(x,y) = \left\{
\begin{array}{lc}
\sqrt{\beta_+}, & {\rm on}\ y=d_2 + L_2/2,\\
\sqrt{\beta_-}, & {\rm on}\ y=-d_2 - L_2/2,
\end{array}
\right.\end{aligned}$$ then Theorem 3 is applicable to our scattering problem. Consequently, with the hybrid Dirichlet-Robin condition (\[eq:general:cond\]), $\tilde{u}^s$ still exponentially converges to $u^s$ in the physical domain $B_1$.
Theorems 2.2 and 3.1 are established for PMLs with constant and equal $\sigma_1$ and $\sigma_2$. In practice, we may set $\sigma_1(x)$ and $\sigma_2(y)$ as continuous functions to increase flexibility. For example, we may choose $$\begin{aligned}
\label{eq:pml:setup}
\sigma_l(t) = \sigma\left( \frac{t - L_l/2}{d_l} \right)^m,\quad{\rm in}\ L_l/2<|t|<L_l/2+d_l,\end{aligned}$$ for a positive constant $\sigma$ and an integer $m\geq 0$, where $m=0$ corresponds to the constant case.
Cylindrical incident waves
==========================
The NMM methods are typically implemented for plane incident waves. For other incident waves, such as point or line sources and Gaussian beams, the NMM methods may be used with a Fourier transform that rewrites the incident wave as a superposition of plane waves. This approach is not very efficient, since it is necessary to solve the problem for many different incident plane waves. Alternatively, we can try to find a reference solution for the given non-plane incident wave in each uniform segment. This task is nontrivial for the interior segment corresponding to the inhomogeneity. In the following, we present an efficient method for computing the reference solutions when the incident wave is a cylindrical wave generated by a line source.
The incident cylindrical wave is $u^{\rm
inc}=\frac{i}{4}H_0^{(1)}(k_0n_+\rho(x,y))$ corresponding to a line source at $(x^*,y^*)\in\mathbb{R}_+^2/\bar{D}$, where $\rho(x,y)=\sqrt{(x-x^*)^2 +
(y-y^*)^2}$. The governing Helmholtz equation becomes $$\begin{aligned}
\label{eq:gov:ps}
\Delta u^{\rm tot} + k_0^2\varepsilon(x,y)u^{\rm tot} = -\delta(x-x^*)\delta(y-y^*).\end{aligned}$$ Considering the location of the source, we have the following three cases:
- If $|x^*|<x_0$, we set $u_0^{\rm tot}\equiv 0$ and find a nonzero $u_2^{\rm tot}$;
- If $|x^*|>x_0$ and $y^*>y_1$, we set $u_2^{\rm tot}\equiv
0$ and find a nonzero $u_0^{\rm tot}$;
- If $|x^*|=x_0$, then we have to find nonzero $u_0^{\rm tot}$ and $u_2^{\rm tot}$.
We consider the typical case (a), where $u^{\rm tot}_2$ must be computed in segment $S_2$. The NMM method requires $u_2^{\rm tot}$ and its $x$-derivative at $x=\pm x_0$ to evaluate $\tilde{f}$ and $\tilde{g}$ in Eqs. (\[eq:gov:tran1\]) and (\[eq:gov:tran2\]).
Following the one-dimensional profile $\varepsilon_2(y)$ given in (\[eq:eps2\]), $\mathbb{R}^2$ can be split into three layers $y<y_0$, $y_0<y<y_1$, and $y>y_1$. The wave field $$u^{s} = \left\{
\begin{array}{lc}
u_2^{\rm tot} - u^{\rm inc},&{\rm in}\ y>y_1,\\
u_2^{\rm tot},&{\rm otherwise},
\end{array}
\right.$$ is outgoing as $y \to \pm \infty$. Using the same PML as before and applying the technique of separation of variables to $\tilde{u}^s$, we obtain the following eigenvalue problem $$\begin{aligned}
\label{eq:eigprob:x1}
&\frac{1}{\alpha_1}\frac{d}{dx}\left( \frac{1}{\alpha_1} \frac{d\psi}{dx}\right) = \delta \psi,\\
\label{eq:eigprob:x2}
&\psi(-L_1/2-d_1) = \psi(L_1/2+d_1) = 0,\end{aligned}$$ and its associated equation $$\begin{aligned}
\label{eq:asso:phi}
\frac{1}{\alpha_2}\frac{d}{dy}\left( \frac{1}{\alpha_2} \frac{d\phi}{dy}\right) + (k_0^2\varepsilon_2(y)+\delta) \phi = 0,\\
\label{eq:asso:phi2}
\phi(-L_2/2-d_2) = \phi(L_2/2+d_2) = 0.\end{aligned}$$ Different from the main step of the NMM method, the separation of variables here leads to an eigenvalue problem for $\psi$ (a function of $x$), instead of $\phi$ which is not continuous at $y=y_1$.
The eigenvalue problem for $\psi$ can be solved by a pesudospectral method as in [@sonyualu11]. If $M$ numerical eigenpairs $\{\delta_j,\psi_j(x)\}_{j=1}^M$ are obtained corresponding to the collocation points $\{x_j\}_{j=1}^M\subset[-L_1/2-d_1,L_1/2+d_1]$, we approximate $\tilde{u}^s$ by $$\begin{aligned}
\label{eq:us:top}
\tilde{u}^s \approx \left\{\begin{array}{lc}
\sum_{j=1}^M c_j^{t}e^{i\sqrt{k_0^2\varepsilon_+ + \delta_j}(y-y_1)}\psi_j(x), & y>y_1,\\
\sum_{j=1}^M \phi_j(y)\psi_j(x), & y_0\leq y\leq y_1,\\
\sum_{j=1}^M
c_j^{b}e^{-i\sqrt{k_0^2\varepsilon_- +
\delta_j}(y-y_0)}\psi_j(x), & y<y_0,
\end{array}\right.\end{aligned}$$ where the square roots have nonnegative imaginary parts. Notice that only outgoing waves are retained in the top and bottom layers.
The functions $\phi_j(y)$ satisfy $$\label{eq:phixj}
\frac{d^2\phi_j}{dy^2} + [ k_0^2\tilde{\varepsilon}(y) + \delta_j] \phi_j = 0.$$ Since $\tilde{u}^s$ satisfies the transmission condition at $y=y_0$, we have $$\phi_j(y_0) = c_j^b,\quad \phi_j'(y_0+) =
-c_j^bi\sqrt{k_0^2\varepsilon_-+\delta_j}.$$ Therefore, we enforce the following Robin boundary condition $$\label{eq:RBC:bot}
\phi_j'(y_0+) = -i\sqrt{k_0^2\varepsilon_-+\delta_j}\phi_j(y_0).$$ At $y=y_1$, we can find the coefficients $\{c_j^{\rm ps},
d_j^{\rm ps}\}_{j=1}^N$ such that $$\begin{aligned}
\sum_{j=1}^{N}c_j^{\rm ps}\psi_j(x) &= \frac{i}{4}H_0^{(1)}(k_0n_+\rho(\tilde{x}(x),y_1)),\\
\sum_{j=1}^{N}d_j^{\rm ps}\psi_j(x) &=
\frac{d}{dy}\frac{i}{4}H_0^{(1)}(k_0n_+\rho(\tilde{x}(x),y))|_{y=y_1}\end{aligned}$$ are exactly satisfied at the collocation points $\{ x_j \}_{j=1}^N$. Since $u^{\rm tot}$ satisfies the transmission condition on $y=y_1$, we have $$c_j^t + c_j^{\rm ps} = \phi_j(y_1),\quad
ic_j^t\sqrt{k_0^2\varepsilon_++\delta_j} + d_j^{\rm ps} = \phi_j'(y_1-).$$ Eliminating $c_j^t$, the above yields the following Robin boundary condition, $$\begin{aligned}
\label{eq:RBC:top}
\phi_j'(y_1-) - i\sqrt{k_0^2\varepsilon_++\delta_j}\phi_j(y_1) = d_j^{\rm ps} - i\sqrt{k_0^2\varepsilon_++\delta_j}c_j^{\rm ps}.\end{aligned}$$
As shown in Proposition A.2, the boundary value problem (\[eq:phixj\]), (\[eq:RBC:bot\]) and (\[eq:RBC:top\]) has a unique solution. Using a pesudospectral method, we solve this boundary value problem and obtain $\phi_j(y)$ at the collocation points $\{y^j\}_{j=1}^N\cap[y_0,y_1]$. Finally, since $c_j^b=\phi_j(y_0)$ and $c_j^t = \phi_j(y_1)-c_j^{\rm ps}$, we have $\tilde{u}^s$ and $u_2^{\rm tot}$ in $B_2$. The other two cases (b) and (c) are similar; we omit the details here.
Numerical examples
==================
In this section, we carry out several numerical experiments to exhibit the performance of our NMM method. In all examples, the physical domain is chosen to be $(-2.5,2.5)\times(-2.5,2.5)$, and the free-space wavenumber $k_0=2\pi/\lambda$ with wavelength $\lambda=1.13$.
[**Example 1.**]{} In the first example, the background two-layer medium is separated by interface $y=0$, with $\varepsilon(x,y)=4$ in the top, and $\varepsilon(x,y)=1$ in the bottom. The inhomgeneity filled in domain $D=(-0.5, 0.5)\times (-1, 1)$ is the same as the medium in the top, so that it functions as a local perturbation to the interface $y=0$; see the dashed lines in Fig. \[fig:ex1:0\]. The PML-BIE method recently developed in [@luluqia18] is applicable to this problem and is used to validate our NMM method. Using $1000$ points to discretize the interface in the PML-BIE method, we obtain a reference solution $u^{\rm
tot}_{\rm ref}$. To quantify the accuracy of the NMM method, we define the following relative error, $$\begin{aligned}
\label{eq:rel:err}
{ e_{\rm rel}} = \frac{\max_{(x,y)\in S}|u^{\rm tot}_{\rm ref}(x,y) - u^{\rm
tot}_{\rm NMM}(x,y)|}{\max_{(x,y)\in S}|u^{\rm tot}_{\rm ref}(x,y)|}.\end{aligned}$$ Notice that $e_{\rm rel}$ compares the numerical solution by the NMM method with the reference solution on the set $S=\{(x, y)|x=\pm 0.5, y = \pm 2.5, -1,
0\}$. The choice of $S$ is typical, since it contains all corners on the interfaces and some points at the interior boundary of the PML.
First, we validate the Robin-type boundary condition. We choose $\sigma=70$, $d=0.05$, and $m=0$ to set up the PML. For both $E$ and $H$ polarizations, we choose $N=950$ eigenmodes in each segment, and compute $e_{\rm rel}$ for incident angle $\theta$ varying in $[0,\pi/2)$. The results are shown in Figs. \[fig:ex1:2\](a) and \[fig:ex1:2\](b).
(a)![Example 1: (a) and (b): Relative error versus incident angle $\theta$ using Robin and Dirichlet boundary conditions on $\Gamma_2^2$, for $N=950$, $\sigma=70$, $d=0.05$, and $m=0$. (a) $E$ polarization; (b) $H$ polarization. (c) and (d): Relative error versus PML thickness $d$ at $\theta=0$ and $\theta = \pi/6$, for $N=950$, $m=0$ and $\sigma=70$: (c) $E$ polarization; (d) $H$ polarization.[]{data-label="fig:ex1:2"}](fig/ex1/plane_te_0_comp.pdf "fig:"){width="40.00000%"} (b)![Example 1: (a) and (b): Relative error versus incident angle $\theta$ using Robin and Dirichlet boundary conditions on $\Gamma_2^2$, for $N=950$, $\sigma=70$, $d=0.05$, and $m=0$. (a) $E$ polarization; (b) $H$ polarization. (c) and (d): Relative error versus PML thickness $d$ at $\theta=0$ and $\theta = \pi/6$, for $N=950$, $m=0$ and $\sigma=70$: (c) $E$ polarization; (d) $H$ polarization.[]{data-label="fig:ex1:2"}](fig/ex1/plane_tm_0_comp.pdf "fig:"){width="40.00000%"}\
(c)![Example 1: (a) and (b): Relative error versus incident angle $\theta$ using Robin and Dirichlet boundary conditions on $\Gamma_2^2$, for $N=950$, $\sigma=70$, $d=0.05$, and $m=0$. (a) $E$ polarization; (b) $H$ polarization. (c) and (d): Relative error versus PML thickness $d$ at $\theta=0$ and $\theta = \pi/6$, for $N=950$, $m=0$ and $\sigma=70$: (c) $E$ polarization; (d) $H$ polarization.[]{data-label="fig:ex1:2"}](fig/ex1/plane_conv_te.pdf "fig:"){width="40.00000%"} (d)![Example 1: (a) and (b): Relative error versus incident angle $\theta$ using Robin and Dirichlet boundary conditions on $\Gamma_2^2$, for $N=950$, $\sigma=70$, $d=0.05$, and $m=0$. (a) $E$ polarization; (b) $H$ polarization. (c) and (d): Relative error versus PML thickness $d$ at $\theta=0$ and $\theta = \pi/6$, for $N=950$, $m=0$ and $\sigma=70$: (c) $E$ polarization; (d) $H$ polarization.[]{data-label="fig:ex1:2"}](fig/ex1/plane_conv_tm.pdf "fig:"){width="40.00000%"}\
It is clear that in the vicinity of the critical angle $\theta=\pi/6$, where total internal reflection first occurs, or $\theta=\pi/2$, which gives horizontally propagating incident plane waves, the Robin boundary condition produces a much smaller $e_{\rm rel}$ and significantly outperforms the Dirichlet boundary condition.
At both the critical incident angle $\theta=\pi/6$ and the normal incidence with $\theta=0$, we study the relation between $e_{\rm rel}$ and the PML thickness $d$ for a fixed $\sigma=70$. The numerical results are shown in Figs. \[fig:ex1:2\](c) and \[fig:ex1:2\](d), where both axes are scaled logarithmically. When $d$ is small, we expect that the error is dominated by the truncation of the PML. The results in Figs. \[fig:ex1:2\](c) and \[fig:ex1:2\](d) indicate that $e_{\rm rel}$ initially decays exponentially as $d$ is increased. This behavior is expected from Theorem 3.1.
Finally, we compare the numerical solutions by the NMM and PML-BIE methods for two types of incident waves: a plane wave with the critical incident angle $\theta=\pi/6$, and a cylindrical wave excited by a source at $(0.2,1)$. The results are shown in Fig. \[fig:ex1:0\].
(a)![Example 1: Numerical solutions of a scattering problem in $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(0.2, 1)$. For both (a) and (b), the reference solution by the PML-BIE method is shown on the left and the numerical solution by the NMM method is shown on the right.[]{data-label="fig:ex1:0"}](fig/ex1/Ref_numel.pdf "fig:"){width="40.00000%"} ![Example 1: Numerical solutions of a scattering problem in $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(0.2, 1)$. For both (a) and (b), the reference solution by the PML-BIE method is shown on the left and the numerical solution by the NMM method is shown on the right.[]{data-label="fig:ex1:0"}](fig/ex1/numel_crt.pdf "fig:"){width="40.00000%"} (b)![Example 1: Numerical solutions of a scattering problem in $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(0.2, 1)$. For both (a) and (b), the reference solution by the PML-BIE method is shown on the left and the numerical solution by the NMM method is shown on the right.[]{data-label="fig:ex1:0"}](fig/ex1/Ref_numel_ps.pdf "fig:"){width="40.00000%"} ![Example 1: Numerical solutions of a scattering problem in $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(0.2, 1)$. For both (a) and (b), the reference solution by the PML-BIE method is shown on the left and the numerical solution by the NMM method is shown on the right.[]{data-label="fig:ex1:0"}](fig/ex1/numel_crt_ps.pdf "fig:"){width="40.00000%"}
For each case, the PML-BIE solution is shown on the left and the NMM solution is shown on the right. Clearly, the solutions obtained by the two numerical methods are nearly indistinguishable from each other.
[**Example 2.**]{} The dielectric function $\varepsilon(x,y)$ is profiled by Fig. \[fig:ex2:0\](c), where $\varepsilon(x,y)$ is $4$ in the top layer, $1$ in the bottom layer, and $(1.5+y)^2$ in the inhomogeneity $D=(-0.5,0.5)\times(-0.5,0.5)$.
Since $\varepsilon(x,y)$ is variable in $D$, the PML-BIE method is no longer applicable. We use the NMM method to find the total field $u^{\rm tot}$ for the $E$ polarization and for two different incident waves: a plane wave at the critical incident angle $\theta=\pi/6$, and a cylindrical wave excited by a source at $(0.2,1)$.
For these two incident waves, using $N=534$ eigenmodes in each segment, and using $m=0$, $\sigma=70$ and $d=1$ to set up the PML, we obtain two numerical solutions, relatively, as shown in Figs. \[fig:ex2:0\](a) and \[fig:ex2:0\](b).
(a)![Example 2: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; b) cylindrical wave for a source at $(0.2, 1)$; (c) profile of the dielectric function $\varepsilon(x,y)$; (d) relative error $e_{rel}$ versus PML thickness $d$. In (a) and (b), we take $N=534$, $m=0$, $\sigma=70$ and $d=1$.[]{data-label="fig:ex2:0"}](fig/ex2/numel_crt.pdf "fig:"){width="38.00000%"} (b)![Example 2: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; b) cylindrical wave for a source at $(0.2, 1)$; (c) profile of the dielectric function $\varepsilon(x,y)$; (d) relative error $e_{rel}$ versus PML thickness $d$. In (a) and (b), we take $N=534$, $m=0$, $\sigma=70$ and $d=1$.[]{data-label="fig:ex2:0"}](fig/ex2/numel_crt_ps.pdf "fig:"){width="38.00000%"} (c)![Example 2: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; b) cylindrical wave for a source at $(0.2, 1)$; (c) profile of the dielectric function $\varepsilon(x,y)$; (d) relative error $e_{rel}$ versus PML thickness $d$. In (a) and (b), we take $N=534$, $m=0$, $\sigma=70$ and $d=1$.[]{data-label="fig:ex2:0"}](fig/ex2/medium.pdf "fig:"){width="42.00000%"} (d)![Example 2: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; b) cylindrical wave for a source at $(0.2, 1)$; (c) profile of the dielectric function $\varepsilon(x,y)$; (d) relative error $e_{rel}$ versus PML thickness $d$. In (a) and (b), we take $N=534$, $m=0$, $\sigma=70$ and $d=1$.[]{data-label="fig:ex2:0"}](fig/ex2/plane_cylindrical_conv_te.pdf "fig:"){width="42.00000%"}
Using the above two numerical solutions as reference solutions, relatively, we compute $e_{\rm rel}$ defined in Eq. (\[eq:rel:err\]), but for $S=\{(x,y)|x=\pm 0.5, y=\pm 0.5, \pm 2.5\}$ for numerical solutions with values of $d$ less than $1$, for the two incident waves, relatively; see Fig. \[fig:ex2:0\](d). As before, when $d$ is small, the relative error decays exponentially, since it is dominated by the truncation of the PML.
[**Example 3.**]{} The dielectric function $\varepsilon(x,y)$ is profiled by Fig. \[fig:ex3:0\] (c), where two $y$-dependent inhomogeneities are embedded in the background medium with three layers. Here, $\varepsilon(x,y)$ is $4$, $2.25$ and $1$ in the top, inner, and bottom layers, respectively, and in the two inhomogeneities, $$\varepsilon(x,y)=\left\{\begin{array}{ll}
(1+\sin^2(\pi y/2))^2, & (x,y)\in D_1=(-1.5,-0.5)\times(-1,1);\\
(1+\cos^2(\pi y/2))^2, & (x,y)\in D_2=(0.5,1.5)\times(-1,1).\\
\end{array}\right.$$
Using $N=633$ eigenmodes in each segment, and using $m=0$, $\sigma=70$, and $d=1$ to set up the PML, we calculate the total field $u^{\rm tot}$ for the $E$ polarization and for two different incident waves: a plane wave with the critical incident angle $\theta= \pi/6$, and a cylindrical wave excited by a source at $(-0.7, 1.2)$. The results are shown in Figs. \[fig:ex3:0\](a) and \[fig:ex3:0\](b).
(a)![Example 3: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(-0.7, 1.2)$; (c) profile of the dielectric function $\varepsilon(x,y)$. Other parameters: $N=633$, $m=0$, $\sigma=70$, and $d=1$.[]{data-label="fig:ex3:0"}](fig/ex3/numel_crt.pdf "fig:"){width="38.00000%"} (b)![Example 3: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(-0.7, 1.2)$; (c) profile of the dielectric function $\varepsilon(x,y)$. Other parameters: $N=633$, $m=0$, $\sigma=70$, and $d=1$.[]{data-label="fig:ex3:0"}](fig/ex3/numel_crt_ps.pdf "fig:"){width="38.00000%"} (c)![Example 3: Numerical solutions of a scattering problem in the $E$ polarization: (a) plane wave at critical incident angle $\theta=\pi/6$; (b) cylindrical wave for a source at $(-0.7, 1.2)$; (c) profile of the dielectric function $\varepsilon(x,y)$. Other parameters: $N=633$, $m=0$, $\sigma=70$, and $d=1$.[]{data-label="fig:ex3:0"}](fig/ex3/medium.pdf "fig:"){width="42.00000%"}
Conclusion
==========
The NMM methods are widely used in engineering applications for simulating propagation and scattering of linear electromagnetic, acoustic and elastic waves. These methods are restricted to special structures, but are more efficient than the standard numerical methods when they are applicable, since no discretization is needed for one spatial variable. In this paper, a new NMM method is developed to overcome a limitation of existing NMM methods due to the existence of a non-propagating and non-decaying wave field component. A Robin-type boundary condition is used to ensure that the wave field component with zero or near-zero transverse wavenumber is not reflected by a truncated PML. A theoretical foundation of the new NMM is established by a theorem which reveals the exponential convergence of the PML solution with the hybrid Dirichlet-Robin boundary conditions. In addition, for scattering problems with cylindrical incident waves, we developed a fast method to compute reference solutions needed in the NMM methods. Numerical examples are presented to validate the NMM method and illustrate its performance.
We have implemented the new NMM method for two-dimensional structures with one or more inhomogeneities, for electromagnetic waves in $E$ and $H$ polarizations, and for both plane and cylindrical incident waves. The NMM methods are also applicable to three-dimensional (3D) rotationally symmetric structures that are piecewise uniform in the radial variable [@hulinluoh18; @lushilu14]. There is also a related method for more general 3D structures without the rotational symmetry [@shilu15; @shilulu16]. The techniques developed in this paper, namely, the Robin-type boundary condition for terminating the PML and the fast method for computing reference solutions for cylindrical incident waves, should also be useful in these NMM and related methods for 3D structures.
Proof of Propositions
=====================
Under the same assumptions as Theorem 2.2, we consider the eigenvalue problem (\[eq:phi:te\]-\[eq:phi:te3\]) with $\varepsilon_i$ replaced by $$\varepsilon_{\rm gen}(y) = \left\{
\begin{array}{lc}
\varepsilon_+,& y>y_1,\\
\varepsilon_-,& y<y_0,\\
\varepsilon_{\rm PHY}(y), & y_0\leq y\leq y_1,
\end{array}
\right.$$ where $\varepsilon_{\rm PHY}(y)$ can be any piecewise smooth and positive function. Then, one and only one of the following two cases occurs:
- There exist $\sigma^0>0$ and $d^0>0$ such that if $\sigma>\sigma^0$ or if $d>d^0$, then ${\rm Im}(\delta)\geq 0$ for any eigenpair $\{\phi,\delta\}$ that solves (\[eq:phi:te\]-\[eq:phi:te3\]);
- For a fixed $d>0$ ($\sigma>0$), there exist a sequence of $\{\sigma^n\}_{n=1}^{\infty}$($\{d^n\}_{n=1}^{\infty}$, respectively) that approaches infinity as $n\rightarrow\infty$ such that there exists a sequence of associated eigenpairs $\{\phi^n,\delta^n\}$ satisfying ${\rm
Im}(\delta^n)<0$.
If case (b) holds, then ${\rm Im}(\delta^n)\rightarrow 0$ as $n\rightarrow\infty$ and for sufficiently large $n$, $${\rm Re}(\delta^n)\in[\max(k_0^2\varepsilon_+,k_0^2\varepsilon_-),
\max(k_0^2\varepsilon_{\rm PHY})).$$
It is clear that if case (a) does not hold, then case (b) must hold. We now prove ${\rm Im (\delta^n)}\rightarrow 0$ as $n\rightarrow\infty$. Integrating (\[eq:phi:te\]) with $\bar{\phi}$ on $[-L_2/2,L_2/2]$ and using integration by parts yield $$\begin{aligned}
\label{eq:det:delta}
\delta\int_{-L_2/2}^{L_2/2} |\phi|^2dy =& -\int_{-L_2/2}^{L_2/2}|\phi'(y)|^2dy +
k_0^2\int_{-L_2/2}^{L_2/2}\varepsilon_{\rm PHY}|\phi|^2dy \nonumber\\
&+ \left.\left(
\frac{d\phi}{dy}\bar{\phi} \right)\right|_{-L_2/2}^{L_2/2}.
\end{aligned}$$ In ${\rm PML}_y=(-L_2/2-d,-L_2/2)\cup(L_2/2,L_2/2+d)$, $\phi(y)$ has the following general solution form, $$\phi(y) = c^1_{\pm}e^{\pm ik^*_{\pm} ( \tilde{y}\mp L_2/2)} +
c^2_{\pm}e^{\mp ik^*_{\pm} ( \tilde{y}\mp L_2/2)},\quad{\rm in}\ {\rm PML}_{y}\cap \mathbb{R}_{\pm},$$ where $k^*_{\pm}=\sqrt{k_0^2\varepsilon_{\pm}-\delta}$ with ${\rm
Re}(k^*_{\pm})\geq 0$. The homogeneous Dirichlet boundary condition at $y=\pm(L_2/2+d)$ implies that $$c^2_{\pm} = -c^1_{\pm} e^{\pm i 2k_{\pm}^* d(1+i\sigma)},$$ so that, by a straightforward calculation, one obtains $$\begin{aligned}
\label{eq:est:imag}
\left.\left(
\frac{d\phi}{dy}\bar{\phi} \right)\right|_{-L_2/2}^{L_2/2} = \sum_{l=\pm}|c_l^1|^2\Big[(1 - e^{-4d({\rm Im}(k^*_{l}) + {\rm Re}(k^*_l)\sigma)})ik^*_l \nonumber\\
- 2k^*_l e^{-2d({\rm Im}(k^*_l) + {\rm Re}(k^*_l)\sigma)}\sin(2d({\rm Re}(k^*_l)-{\rm Im}(k^*_l)\sigma))\Big].
\end{aligned}$$ If ${\rm Im}(\delta)<0$, then ${\rm Im}(k_{\pm}^{*})>0$. Next, we show that ${\rm Re}(k_\pm^{*,n})\rightarrow 0$ as $n\rightarrow\infty$, where $k_{\pm}^{*,n}=\sqrt{k_0^2\varepsilon_{\pm}-\delta^n}$ with nonnegative real part. Otherwise, suppose we have a subsequence $\{n_k\}_{k=1}^{\infty}$ such that ${\rm Re}(k_{\pm}^{*,n_k})\rightarrow c_{\pm}^{0}> 0$ as $n_k\rightarrow\infty$. Considering the imaginary part of (\[eq:est:imag\]), $${\rm Im}\left.\left(
\frac{d\phi^{n_k}}{dy}\bar{\phi}^{n_k} \right)\right|_{-L_2/2}^{L_2/2}\rightarrow
|c_+^1|^2c_+^0 + |c_-^1|^2c_-^0> 0,$$ which implies that ${\rm Im}(\delta^{n_k})>0$. Considering the real part of (\[eq:est:imag\]), $${\rm Re}\left.\left(\frac{d\phi}{dy}\bar{\phi}
\right)\right|_{-L_2/2}^{L_2/2}\leq\sum_{l=\pm}|c_l^1|^2\Big[-(1 -
e^{-4d({\rm Im}(k^*_{l}) + {\rm Re}(k^*_l)\sigma)}){\rm Im}(k_l^*) \nonumber\\
+ 2{\rm Re}(k^*_l)\Big].$$ For sufficiently large $n$, ${\rm Re}(k^{*,n}_l)$ can be arbitrarily small such that $${\rm Re}\left.\left(\frac{d\phi^{n}}{dy}\bar{\phi}^n
\right)\right|_{-L_2/2}^{L_2/2}\leq 0.$$ Therefore, considering the real part of (\[eq:det:delta\]), $${\rm Re}(\delta^n)< k_0^2\max(\varepsilon_{\rm PHY}),$$ since $\frac{d}{dy}\phi^{n}(y)\neq 0$ in $[-L_2/2,L_2/2]$. On the other hand, $${\rm Re}(\delta) = k_0^2\varepsilon_{\pm} - {\rm Re}(k^*_{\pm})^2 + {\rm
Im}(k^*_{\pm})^2\geq k_0^2\varepsilon_{\pm} - {\rm Re}(k^*_{\pm})^2.$$ so that ${\rm Re}(\delta^n)\geq k_0^2\varepsilon_{\pm}$ and that $${\rm Im}(k_{\pm}^{*,n})<k_0^2\max(\varepsilon_{PHY})-k_0^2\varepsilon_{\pm} +
{\rm Re}(k_{\pm}^{*,n})^2,$$ Consequently, as $n\rightarrow \infty$, we have ${\rm Im}(\delta^n)=-2{\rm Im}(k_{\pm}^{*,n}){\rm
Re}(k_{\pm}^{*,n})\rightarrow 0$.
If $\tilde{ \varepsilon }(y)$ is smooth on $[y_0,y_1]$, the following field $$\begin{aligned}
\label{eq:u2:closed:form}
u^{\rm tot}_2 = e^{-i\beta_+ y_1}\left\{
\begin{array}{ll}
e^{i\alpha x}( e^{-i\beta_+ (y-y_1))} + R_2e^{i\beta_+ (y-y_1)})& {\rm if}\ y\geq y_1,\\
e^{i\alpha x} f(y),&{\rm if}\ y_0<y<y_1,\\
T_2e^{i(\alpha x - \beta_-(y-y_0))},&{\rm if}\ y\leq y_0.
\end{array}
\right.
\end{aligned}$$ solves the scattering problem (\[eq:gov:problem\]) and (\[eq:trans:cond\]) with $\varepsilon(x,y)=\varepsilon_2(y)$ in $\mathbb{R}^2$, where $\beta_-$ was defined in (\[eq:sol:para\]), the unknown function $f\in C^{2}[y_0,y_1]$ is uniquely determined by the following boundary value problem $$\begin{aligned}
%&1+R_2 = \bar{u}(y_1), \quad ik_0n_+\cos\theta(R_2-1) = \bar{u}'(y_1-),\\
\label{eq:bvp:1}
&f'' + (k_0^2\varepsilon_2 - \alpha^2) f = 0,\\
%&T_2 = \bar{u}(y_0), \quad -k_*T_2 = \bar{u}'(y_0+).\\
\label{eq:bvp:2}
&f'(y_0+) = -i\beta_-f(y_0), \\
\label{eq:bvp:3}
&f'(y_1-) = i\beta_+(f(y_1)-2),
\end{aligned}$$ and $$R_2 = f(y_1) - 1,\quad T_2 = f(y_0).$$
The verification that $u_2^{\rm tot}$ defined in (\[eq:u2:closed:form\]) is indeed a solution is straightforward. One only needs to prove that the boundary value problem (\[eq:bvp:1\]-\[eq:bvp:3\]) has a unique solution. By the standard ODE theory, one needs to show that equation (\[eq:bvp:1\]) with the following homogeneous Robin boundary conditions $$f'(y_0+) = -i\beta_-f(y_0),\quad f'(y_1-) =
i\beta_+f(y_1),$$ has only the trivial solution $f=0$. To show this, integrating (\[eq:bvp:1\]) with $\bar{f}$ on $[y_0,y_1]$ yields, by integration by parts, $$\begin{aligned}
\int_{y_0}^{y_1} f'\bar{f}'dy -
(k_0^2\tilde{\varepsilon}-\alpha^2)f\bar{f}dy -
(i\beta_+f(y_1)\bar{f}(y_1) + i\beta_-f(y_0)\bar{f}(y_0)) = 0.
\end{aligned}$$ Considering the imaginary part of the left-hand side, we have $$\beta_+|f(y_1)|^2 + \beta_-|f(y_0)|^2 = 0,$$ so that $f(y_1)=0$ since $\beta_+>0$. Notice that $\beta_-$ could be zero when total internal reflection occurs. Then, $f'(y_1-)=i\beta_+f(y_1)=0$ indicates that $f=0$ on $[y_0,y_1]$ which completes the proof.
$u_2^{\rm tot}$ in (\[eq:u2:closed:form\]) plus any guided mode or any surface mode, if there exists, still solves the scattering problem.
[^1]: School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Email: [email protected] (corresponding author).
[^2]: Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China. Email: [email protected].
[^3]: Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China. Email: [email protected].
|
---
abstract: 'The main result of this article provides a characterization of reductive homogeneous spaces equipped with some geometric structure (non necessarily pseudo-Riemannian) in terms of the existence of certain connection. The result generalizes the well-known result of Ambrose and Singer for Riemannian homogeneous spaces, as well as its extensions for other geometries found in the literature. The manifold must be connected and simply connected, the connection has to be complete and has to satisfy a set of geometric partial differential equations. If the connection is not complete or the manifold is not simply-connected, the result provides a characterization of reductive locally homogeneous manifolds. Finally, we use these results in the local framework to classify with explicit expressions reductive locally homogeneous almost symplectic, symplectic and Fedosov manifolds.'
author:
- 'J.L. Carmona Jiménez and M. Castrillón López'
nocite: '[@*]'
title: 'The Ambrose-Singer Theorem for general homogeneous manifolds with applications to symplectic geometry'
---
**Key words.** Ambrose-Singer theorem, Fedosov manifolds, homogeneous manifolds, homogeneous structures, locally homogeneous manifolds, symplectic manifolds.
Introduction {#Section I}
============
Locally symmetric spaces are characterized, as it is well known since Élie Cartan [@C1929], either by the existence of local geodesic involutions or by having parallel Riemann curvature tensor. The global version of this classical result requires some conditions on the topology of the manifold: connectedness, simply connectedness and completeness. Recall that, in this global version, symmetric spaces become a special type of homogeneous Riemannian manifolds. In 1958, Ambrose-Singer [@AS1958] generalized the result for arbitrary homogeneous manifold, still assuming the same topological conditions:
\[Theorem I1\] Let $(M,g)$ be a connected and simply-connected complete Riemannian manifold. Then, the following statements are equivalent:
1. The manifold $M$ is homogeneous.
2. The manifold $M$ admits a linear connection ${\tilde{{\nabla}}}$ satisfying: $$\label{Equation 1}
{\tilde{{\nabla}}}R = 0,\quad {\tilde{{\nabla}}}S = 0,\quad {\tilde{{\nabla}}}g = 0,$$
where $R$ is the curvature tensor of the Levi-Civita connection ${\nabla}^{LC}$ and $S = {\nabla}^{LC} - {\tilde{{\nabla}}}$.
In fact, the classification of the tensor $S$ (also known as *homogeneous structure tensor*) into $O(n)$-irreducible classes explicitly, $n=\mathrm{dim}M$, provides interesting and powerful geometric results taking benefit from the interplay between PDEs, Algebra and Geometry, an ambitious program that formally started with [@TV1983] (for a recent reference giving a panoramic view of most of these geometric results, the reader can go to [@CC2019]). It is important to note that if $M$ is not simply-connected or complete, the existence of ${\tilde{{\nabla}}}$ is still extremely useful, as it characterizes locally homogeneous manifolds, a kind of spaces that are more than a mere local version global spaces (there are locally homogeneous spaces that are not locally isometric to global homogeneous spaces).
Important extensions of the Theorem of Ambrose-Singer have been carried out in the literature. On one hand, the characterization of (local) homogeneity on pseudo-Riemannian manifolds was developed in [@GO1997]. This situation shows a relevant difference with the original Riemannian work since the existence of the metric connection with parallel torsion and curvature characterizes (locally) homogeneous spaces of reductive type only. As we know, the Lie algebra of the group acting transitively on reductive spaces can be decomposed into two factors, invariant under the adjoint action of the isotropy subgroup. Since every Riemannian homogeneous manifold is automatically reductive, this particularity only shows up when dealing with metrics with signature. The second main extension of the homogeneous structure tensors was given when additional geometric structures are considered together with the pseudo-Riemannian metric, see [@K1980] and [@L2014]. With geometric structure they mean a reduction of the orthogonal frame bundle, that is, a $G$-structure, for a subgroup $G$ of the orthogonal group of the corresponding signature. This reduction is understood to be determined by the existence of a tensor or set of tensors on the manifold characterizing the frames of the corresponding reduction. From that point of view, the group $G$ is the stabilizer of a canonical tensor (or set of canonical tensors) on $\mathbb{R}^n$ by the natural action of $O(p,q),\, p+q=n$. When this geometry structure is included in the picture, the notion of homogeneous spaces requires the transitive action of an isometry group that also conserves the geometric tensors on $M$. Important instances of this situation include Kähler, quaternion-Kähler, Sasaki or $G_2$ spaces among others. The generalization now requires the existence of a metric connection making parallel curvature and the torsion and the tensors defining the geometry (see [@K1980]).
The main goal of this article is the presentation of a complete generalization of all these result in the case of homogeneous spaces in broad sense, that is, independently of the presence of a pseudo-Riemannian metric on the manifold (see Theorem \[Theorem AG2\] below). More specifically, we here give a characterization of reductive and homogeneous spaces equipped with a structure defined by a tensor (or a set of tensors) non-necessarily associated to a $G$-structure, through the existence of a complete connection satisfying certain conditions of the Ambrose-Singer type. With homogeneity we understand that a Lie group acts transitively and leaves the tensors invariant. For the local version of the results, we can drop again the topological conditions on the manifold as well as the completeness of the connection to have only the so-called notion of AS-manifold. In that case, reductivity must be defined carefully (in particular, we follow some of the ideas in [@L2015]) and we show that every reductive locally homogeneous manifold in the broad sense can be equipped with as Ambrose Singer connection.
As a particular instance of our results, if one of the tensors is a pseudo-Riemannian metric, we recover all the traditional theorems in the literature. With respect to that, we also explore the case where the manifold is also equipped with an additional connection for which its (local) transformations are affine for the Ambrose Singer connection. Then, the difference tensor of the connection and the Ambrose-Singer connection is a generalization of the homogeneous structure tensor.
One of the main goals of these results aim at following the fruitfully philosophy exploited in the pseudo-Riemannian case, that provides geometric and explicit results through the classifications of homogeneous structure tensors (in particular, the reader can go to [@BGO2011] for survey of some results). Here, we show a similar idea with the classification of the torsion of the Ambrose-Singer connection or, whenever these ir another fixed connection, the corresponding classification of homogeneous structure tensors. This is an ambitious project to develop in future works. Nevertheless, we tackle in this article a purely non-metric example: almost symplectic, symplectic and Fedosov manifolds, where both the classes of torsion and homogeneous structures are given together with some first geometric result.
A generalization of Ambrose-Singer Theorem {#Section AG}
==========================================
The main result
---------------
Let $G$ be a Lie group acting transitively on a smooth manifold $M$. Choosing a point $p_0\in M$, we can identify $M$ with $G/H$ where $H\subset G$ is the isotropy subgroup of $p_0$. Note that $M$ need not be pseudo-Riemannian and $G$ is not necessarily a group of isometries. The manifold is said to be reductive homogeneous if there is a Lie algebra decomposition ${\mathfrak{g}}= {\mathfrak{h}}\oplus \mathfrak{m}$ for certain vector subspace $\mathfrak{m}\subset {\mathfrak{g}}$ such that $\mathrm{Ad}_h ({\mathfrak{m}}) = {\mathfrak{m}}$, $\forall h \in H$. In this case, the subspace $\mathfrak{m}$ can be identified with $\mathfrak{m}$ through the map $\mathfrak{m}{\longrightarrow}T_{p_0}M$, $X\mapsto {\frac{d}{dt} \Big |_{t=0}}\mathrm{exp}(tX) p_0$.
The action of $G$ on $M$ naturally lifts to the frame bundle ${\mathcal{L}(M)}$. It is well known that there is a unique connection in ${\mathcal{L}(M)}$, that is, a unique linear connection ${\tilde{{\nabla}}}$, such that for every reference $u$ at $p\in M$ and for each $X \in {\mathfrak{m}}$, the orbit $\exp(tX)\cdot u$ is horizontal. This is called the *canonical connection* of the reductive decomposition ${\mathfrak{g}}= {\mathfrak{h}}\oplus \mathfrak{m}$. This connection satisfies the following important result.
\[Proposition AG1\] Let $M=G/H$ be a reductive homogeneous manifold equipped with the canonical connection ${\tilde{{\nabla}}}$ and let $K$ be an invariant tensor field on $M$ with respect to the action of $G$. Then ${\tilde{{\nabla}}}K = 0$.
In this work, a $n$-dimensional manifold $M$ with a *geometric structure* is understood as a manifold equipped with a tensor or a set of tensors $P_1,\ldots,P_r$, $r\in\mathbb{N}$. This definition is initially more relaxed than the classical notion of geometric structure in the literature (see for example de [@ML2004]). More precisely, a traditional approach defines a geometric structure as a reduction of the frame bundle through a canonical model linear tensor $P_0\in (\otimes ^s (\mathbb{R}^n)^*)\otimes (\otimes ^l \mathbb{R}^n)$ in $\mathbb{R}^n$. If $H$ is its stabilizer by the natural action of $ Gl(n,\mathbb{R})$ on $(\otimes ^r (\mathbb{R}^n)^*)\otimes (\otimes ^l \mathbb{R}^n)$, a $(r,l)$-tensor $P$ on $M$ defines a traditional geometric structure with model $P_0$ if the map $$k:\mathcal{L}(M){\longrightarrow}(\otimes ^s (\mathbb{R}^n)^*)\otimes (\otimes ^l \mathbb{R}^n)$$ is defined as $$k(u)(v_1,...,v_s,\, \alpha _1,...,\alpha _l)=P(u(v_1),...,u(v_s),\, (u^*)^{-1}(\alpha_1),...,(u^*)^{-1}(\alpha_l)),$$ takes values in the $Gl(n,\mathbb{R})$-orbit of $P_0$. In particular, the subset $Q=k^{-1}(P_0)\subset \mathcal{L}(M)$ is a $H$-reduction of the frame bundle.
Essential examples of this situation cover the (pseudo-)Riemannian, Kähler, complex, symplectic or Poisson manifolds, among others. Note that some of these examples are metric, in the sense that one of the tensors $P_i$ is a (pseudo)-Riemannian metric, but some other instances are not metric. The Poisson case shows a geometric structure that is not necessarily traditional since the bivector tensor associated to a Poisson structure may have singularities that are incompatible with the existence of a model linear tensor $P_0$.
\[Theorem AG2\] Let $M$ be a connected and simply-connected manifold and let $P_1, ..., P_r$ tensor fields defining a geometric structure on $M$. Then, the following statements are equivalent:
1. The manifold $M=G/H$ is reductive homogeneous with $G$-invariant tensor fields $ P_1, ..., P_r $.
2. The manifold $M$ admits a complete linear connection ${\tilde{{\nabla}}}$ satisfying: $$\label{Equation 2}
{\tilde{{\nabla}}}{\tilde{R}}= 0,\quad {\tilde{{\nabla}}}{\tilde{T}}= 0,\quad {\tilde{{\nabla}}}P_i = 0 \quad i = 1, ... r,$$
where ${\tilde{R}}$ and ${\tilde{T}}$ are the curvature and torsion tensors of ${\tilde{{\nabla}}}$.
Proof of the main result
------------------------
Suppose $M=G/H$ is a reductive homogeneous manifold with $G$-invariant tensors fields $ P_1, ..., P_r $. If $ {\tilde{{\nabla}}}$ is the canonical connection associated to the reductive decomposition, it is well know that the canonical connection leaves invariant ${\tilde{R}}$ and ${\tilde{T}}$, that is $ {\tilde{{\nabla}}}{\tilde{R}}= 0,\, {\tilde{{\nabla}}}{\tilde{T}}= 0$. We also have ${\tilde{{\nabla}}}P_i = 0,\, i=1,...,r$, from Proposition \[Proposition AG1\]. The completeness of this connection comes from [@KN1963 Ch. X, Cor. 2.5].
Conversely, let $ {\tilde{{\nabla}}}$ be a complete connection on $M$ satisfying $ {\tilde{{\nabla}}}{\tilde{R}}= 0,\, {\tilde{{\nabla}}}{\tilde{T}}= 0, \, {\tilde{{\nabla}}}P_i = 0,\, i=1,...,r$. We fix a frame $u_0 \in \mathcal{L}(M)$. Let $ (\tilde{P}(u_0) {\longrightarrow}M, \widetilde{Hol} (u_0)) $, $ \tilde{P}(u_0)\subset \mathcal{L}(M)$, be the holonomy bundle of the connection $ {\tilde{{\nabla}}}$. To simplify the notation, we denote $ \tilde{P}(u_0) $ by $ \tilde{P} $ and the subgroup $ \widetilde{Hol}(u_0) $ by $ \tilde{H} $. We will denote by $ \tilde{{\mathfrak{h}}} $ and $ {\mathfrak{h}}$ the Lie algebras of $\tilde{H}$ and $H$, respectively.
We now proceed by parts.
**A construction of a complete distribution in $ \tilde{P} $:**
On one hand, if we choose $ \{A_1, ..., A_m \} $ a basis of $ \tilde{{\mathfrak{h}}} $, the associated fundamental vector fields $ \{A_1^*, ... , A_m^* \} $ in $ \tilde{P} $ are complete. On the other hand, for the canonical basis $ \{e_1, ..., e_n \} $ of ${{\mathbb{R}}}^n $, the standard vector fields on $ {\mathcal{L}(M)}$, $$B (e_1) = B_1, \quad \ldots \quad B (e_n) = B_n.$$ are complete on ${\mathcal{L}(M)}$ since $ {\tilde{{\nabla}}}$ is a complete connection (see [@KN1963 Vol. I, Prop. 6.5, p. 140]). Note that, since ${\tilde{{\nabla}}}$ restricts to $\tilde{P}$ and each $B_i$ is horizontal with respect to it, these standard vector fields are tangent to $\tilde{P} \subset {\mathcal{L}(M)}$. Hence $ \{A^*_1, ..., A^*_m, B_1, ..., B_n \}$ is span complete distribution on $\tilde{P}$.
**The structure coefficients of the generating vectors are constant:**
We have $$[A_k^*, A_l^*] = [A_k, A_l]^*,\quad [A_k^*, B_i] = B(A_k(e_i)).$$ We now check that $[B_i,B_j]$ has constant coefficients. We denote by $\theta$ the contact form on ${\mathcal{L}(M)}$ and by $\omega$ the connection form associated to ${\tilde{{\nabla}}}$. The curvature and torsion of $\omega$ are denoted by $ \Omega $ and $ \Theta $, respectively. Then, $$\begin{aligned}
\Theta(B_i, B_j) & = - \theta([B_i, B_j]) \in {{\mathbb{R}}}^n, \\
\Omega(B_i, B_j) & = - \omega([B_i, B_j]) \in \tilde{{\mathfrak{h}}}.\end{aligned}$$ Hence, the splitting $[B_i, B_j] = [B_i, B_j]^h+[B_i, B_j]^v$ with respect to $\omega$ can be written as $$[B_i, B_j] = B(\theta([B_i,B_j]))+\omega ([B_i, B_j])^*
=-B(\Theta(B_i, B_j)) - (\Omega(B_i, B_j))^*.$$ For every horizontal vector $ \overline{X} \in T_u \tilde{P}$ $$\begin{aligned}
\overline{X} (\Theta_u(B_i, B_j)) &= u^{- 1} (({\tilde{{\nabla}}}_X {\tilde{T}}) (X_i, X_j)) = 0, \\
\overline{X} (\Omega_u(B_i, B_j)e_k) &= u^{- 1} (({\tilde{{\nabla}}}_X {\tilde{R}}) (X_i, X_j, X_k)) = 0,\end{aligned}$$ where $ X,\, X_i,\, X_j,\, X_k \in T_{\pi(u)} M $ are the projections of $ \overline{X},\, \,B_i,\, B_j,\, B_k $, respectively. Then $ \Theta (B_i, B_j) $ and $ \Omega (B_i, B_j) e_k $ are constants and hence $[B_i, B_j]$ is a combination of $\{A^*_1, ..., A^*_m$, $B_1, ..., B_n \}$ with constant coefficients.
**$\mathbf{M}$ is a homogeneous space.**
Let $ G $ be the universal covering of $ \tilde{P} $ and let $ \rho: G {\longrightarrow}\tilde{P} $ be the covering map. The vector fields $ \overline{A_k^*}$ and $\overline{B_i}$ on $G$ projecting to $\overline{A_k^*}$ and $\overline{B_i}$ are complete and the coefficients of the brackets are constant. Hence ([@T1992 p. 10, Prop. 1.9]), given a chosen point $e \in \rho ^{-1}(u_0)$, we can endow $G$ with a structure of Lie group with neutral element $ e$ and such that the Lie algebra $ {\mathfrak{g}}$ of $G$ is generated by $ \{(\overline{A_k^*})_e, (\overline{B_i})_e \} $. As $ [A_k^*, A_l^*] = [A_k, A_l]^* $, we can consider the Lie subalgebra $ {\mathfrak{g}}_0 \subset {\mathfrak{g}}$ generated by $ \{(\overline {A_k^*})_e \} $ and let $ G_0 \subset G $ be the associated Lie subgroup to ${\mathfrak{g}}_0$.
\[Lemma AG3\] The manifold $M$ is diffeomorphic to $ G / G_0 $ and hence it is homogeneous.
The map $\pi_1 = \pi \circ \rho : G {\longrightarrow}M$ is a fibration of $M$. We take its exact homotopy sequence: $$\xymatrix{... \ar[r]
& \Pi_1(M,y) \ar[r] \ar@{=}[d]
& \Pi_0(\pi_1^{-1}(y),b) \ar[r]
& \Pi_0(G,b) \ar@{=}[d] \\
& 0
&
& 0}$$ where $b\in G$ and $\rho(b) = y$. We infer that $\Pi_0 (\pi_1^{-1}(y), b) = 0$, that is, $\pi_1^{-1}(y)$ is connected. Since $\pi_1$ is continuous, we obtain that it is closed as well. Finally, by the equality $\pi_{1*}(\overline{A_k^*}) = 0$, we can deduce that the fibres are isomorphic to $G_0$ and closed.
We define $$\begin{aligned}
p : G/G_0 &\rightarrow M \\
[b] &\longmapsto \pi_1(b).\end{aligned}$$ This map $p$ is well defined. Indeed, if we take a fixed point $b_0 \in G_0$ and we express it as $b_0 = \exp(Y_1) ... \exp(Y_s)$, with $Y_1,...,Y_s \in \{\pi_{1*}(\overline{A_k^*}) = 0\}$, then we have $\pi(b\cdot b_0) = \pi (b)$, for all $b\in G$. Furthermore, $p$ is a diffeomorphism since it is bijective and its differential is an linear isomorphism at each point. The injectivity of the differential can be obtained from the fact that $\pi_1^{-1}(y)$ is isomorphic to $G_0$, where surjectivity is straightforward since $\rho $ and $\pi$ are both surjective.
**The structure tensors are invariant and $\mathbf{M}$ is reductive.**
\[Lemma AG4\] For any $ a \in G $, the lift $ \tilde{L_a}: \mathcal{L}(M){\longrightarrow}\mathcal{L}(M)$ of the map $$\begin{aligned}
L_a: M & {\longrightarrow}M \\
[b] & \longmapsto [a \cdot b]\end{aligned}$$ restricts to the reduction bundle $\tilde{L}_a: \tilde{P} {\longrightarrow}\tilde{P} $.
Let $ \mathbf{L}_a $ be the left multiplication on $G$ by $ a \in G $. Note that $ L_a \circ \pi_1 = \pi_1 \circ \mathbf{L}_a $. Then $$L_{a *} \circ \pi_{1 *} (\overline{B_i})_b = \pi_{1 *} \circ \mathbf{L}_{a *} (\overline{B_i})_b = \pi_{1 *} (\overline{B_i})_{ab},$$ and $$\begin{aligned}
L_{a *} \rho(b) & = (L_a([b]); L_{a *} \circ \pi_{1 *} (\overline{B_1})_b, ..., L_{a *} \circ \pi_{1*} (\overline{B_n})_b) = \\
& = ([ab]; \pi_{1*} (\overline{B_1})_{ab}, ..., \pi_{1 *} (\overline{B_n})_{ab}) = \\
& = \rho(ab) \in \tilde{P}.\end{aligned}$$ Hence if $ y = \rho(b) $, $$\tilde{L_a} (\rho (b)) = \rho (ab).$$
Since $ \tilde{P}$ is included in the reduction of $\mathcal{L}(M)$ defined by the tensors $P_1,...,P_r$, we have that $ \tilde{L_a} $ preserves them.
On the other hand, $ \tilde {P} $ is a Lie group. The action of $G$ on $ \tilde{P} $ introduced in the previous lemma \[Lemma AG4\] is transitive, since $ \tilde{L_a} (\rho (b)) = \rho (ab) $, and also effective because it is constructed by linear transformation. In particular, the Lie algebra of $ \tilde {P} $ is isomorphic to the Lie algebra of its universal covering $ G $, through the isomorphism $ \rho_{e *} $ in the neutral element.
Finally, we have $ {\mathfrak{g}}= {\mathfrak{g}}_0 \oplus {\mathfrak{m}}$, where $ {\mathfrak{m}}$ is the subspace generated by $ \{B_i \} $, which clearly satisfies $ [{\mathfrak{g}}_0, {\mathfrak{m}}] \subset {\mathfrak{m}}$. Since $ G_0 $ is connected, we have the ${\mathrm{Ad}}$ invariance of $ {\mathfrak{m}}$, and the proof of Theorem \[Theorem AG2\] is completed.
\[Remark AG5\] The case with no geometric structure on $M$ ($r=0$) was treated in [@KN1963 Vol II, Ch. X, Th. 2.8] or [@KK1980]. There, the authors characterize connected and simply connected reductive homogeneous manifolds $M=G/H$ by the existence of a complete connection ${\tilde{{\nabla}}}$ such that $\, {\tilde{{\nabla}}}{\tilde{R}}= 0,\, {\tilde{{\nabla}}}{\tilde{T}}= 0$. Theorem \[Theorem AG2\] thus provides the generalization of this result to manifolds endowed with additional structures ($r\geq 0$).
\[Definition AG6\] Let $(M,P_1,...,P_r)$ be a manifold equipped with a geometric structure defined by a set of tensors $P_1,...,P_r$. A connection ${\tilde{{\nabla}}}$ is called *generalized Ambrose-Singer connection* if it satisfies that: $${\tilde{{\nabla}}}{\tilde{R}}= 0,\quad {\tilde{{\nabla}}}{\tilde{T}}= 0,\quad {\tilde{{\nabla}}}P_i = 0,\quad i= 1,...,r$$ where ${\tilde{R}}$ and ${\tilde{T}}$ are the curvature and torsion of ${\tilde{{\nabla}}}$.
For short, generalized Ambrose-Singer connection are called AS-connection, and the manifold $(M,P_1,...,P_r)$ where it is defined, an AS-manifold.
We note that Theorem \[Theorem AG2\] generalizes Ambrose-Singer Theorem \[Theorem I1\] on Riemannian manifolds $(M,g)$ by setting $r=1$ and $P_1=g$. In this case, the AS conditions , that is, ${\tilde{{\nabla}}}{\tilde{R}}= 0,\, {\tilde{{\nabla}}}{\tilde{T}}= 0,\, {\tilde{{\nabla}}}g= 0$, are known to be equivalent to the more classical conditions ${\tilde{{\nabla}}}R = 0,\, {\tilde{{\nabla}}}S = 0,\, {\tilde{{\nabla}}}g= 0$, where $S={\tilde{{\nabla}}}-\nabla ^{LC}$, and $R$ is the curvature of the Levi-Civita connection $\nabla ^{LC}$. We now show that this equivalence can be analysed from a broader perspective for manifolds equipped with a fixed connection of which the group acting transitively must be of affine transformations. More precisely, we have the following result.
\[Theorem AG7\] Let $M$ be a connected and simply-connected manifold with an affine connection ${\nabla}$ and let $P_1, ..., P_r$ tensor fields defining a geometric structure on $M$. Then, the following statements are equivalent:
1. The manifold $M=G/H$ is reductive homogeneous with $G$-invariant tensor fields $ P_1, ..., P_r $. Being $G$ a subgroup of $\mathrm{Aff}(M,{\nabla})$.
2. The manifold $M$ admits a complete linear connection ${\tilde{{\nabla}}}$ satisfying: $$\label{Equation 3}
{\tilde{{\nabla}}}R = 0, \quad {\tilde{{\nabla}}}T = 0,\quad {\tilde{{\nabla}}}S = 0,\quad {\tilde{{\nabla}}}P_i = 0 \quad i = 1, ... r,$$
where $R$ and $T$ are the curvature and torsion of ${\nabla}$ and $S= {\nabla}- {\tilde{{\nabla}}}$ is the tensor.
Let $G\subset \mathrm{Aff}(M, {\nabla})$ be a group acting transitively on $M$ and preserving $P_1, ... , P_r$. Additionally, $G$ preserves the tensor $S = {\nabla}- {\tilde{{\nabla}}}$. Hence, by Theorem \[Theorem AG2\] we have that $${\tilde{{\nabla}}}{\tilde{R}}= 0,\quad {\tilde{{\nabla}}}{\tilde{T}}= 0,\quad {\tilde{{\nabla}}}S = 0,\quad {\tilde{{\nabla}}}P_i = 0,\quad i= 1,...,r$$ which are equivalent to $${\tilde{{\nabla}}}R = 0, \quad {\tilde{{\nabla}}}T = 0,\quad {\tilde{{\nabla}}}S = 0,\quad {\tilde{{\nabla}}}P_i = 0 \quad i = 1, ... , r$$ by the following observation, $$T_{X}Y -\tilde{T}_{X}Y = S_{X}Y -S_{Y}X, \quad \tilde{R}_{XY} = R_{XY} + [S_X, S_Y] + S_{S_{X}Y-S_{Y}X}.$$ Conversely, by Theorem \[Theorem AG2\], we have that there exist a Lie group $G$ preserving $S$, $P_1, ..., P_r$. Every transformation of $G$ on $M$ preserves $S$ and is an affine transformation of ${\tilde{{\nabla}}}$. Hence, $G$ preserves $S + {\tilde{{\nabla}}}= {\nabla}$ which means they are affine transformations of ${\nabla}$.
\[Remark AG8\] In particular, Theorem \[Theorem AG7\] covers the case of homogeneous Riemannian manifold when $r=1$, $P_1 = g$ the metric tensor and ${\nabla}= {\nabla}^{LC}$ is the Levi-Civita connection.
\[Definition AG9\] Let $(M,P_1,...,P_r)$ be a manifold equipped with a geometric structure defined by a set of tensors $P_1,...,P_r$ and an affine connection ${\nabla}$ and an AS-connection ${\tilde{{\nabla}}}$. Then, we call *Homogeneous Structure of ${\nabla}$* to the tensor $S = {\nabla}- {\tilde{{\nabla}}}$ and it is satisfied: $${\tilde{{\nabla}}}R = 0, \quad {\tilde{{\nabla}}}T = 0,\quad {\tilde{{\nabla}}}S = 0,\quad {\tilde{{\nabla}}}P_i = 0 \quad i = 1, ... r.$$ where $R$ and $T$ are the curvature and torsion of ${\tilde{{\nabla}}}$.
In particular, homogeneous structure of ${\nabla}^{LC}$ are the classical homogeneous structures of Riemannian manifolds.
In the following sections, for the sake of brevity and simplicity, we consider $M$ with a geometrical structure defined by one tensor $K=(P_1, ..., P_r)$, the following results being analogous for a finite set of tensors $(P_1, ..., P_r)$.
Reductive locally homogeneous manifolds {#Section RL}
=======================================
The conditions involved in Theorem \[Theorem AG2\] are of three different types. First, there is a group of partial differential equations expressed as the vanishing of some covariant derivatives. Second, the completeness of the AS-connection. And finally, a couple of topological conditions (connectedness and simply-connectedness) of the manifold $M$. Connectedness is not an issue, since one usually works with connected components. With respect to simply connectedness, even though essential, it is a condition that can be implemented by working with the universal cover of the manifold, and then project the structures back to the original space. The projection will probably imply that the space is locally homogeneous only, but locally isomorphic to the global homogeneous cover. The completeness, however entails more delicate information since non-complete AS connections may induce locally homogeneous manifolds that are not locally isomorphic to homogeneous spaces. In the Riemannian case we have the following classical result (see [@T1992]).
\[Theorem RL1\] Let $(M,g)$ be a Riemannian manifold. Then, $(M,g)$ is a locally homogeneous manifold if and only if there exists a linear connection ${\tilde{{\nabla}}}$ satisfying, ${\tilde{{\nabla}}}{\tilde{R}}= 0$, ${\tilde{{\nabla}}}{\tilde{T}}= 0 $ and ${\tilde{{\nabla}}}g = 0$, where ${\tilde{R}}$ and ${\tilde{T}}$ are the curvature and torsion of ${\tilde{{\nabla}}}$.
However, if one wants to move forward, the generalization to pseudo-Riemannian manifolds with signature implies the understanding of the notion of reductibity in the local case. That construction was recently achieved in [@L2015]. We generalize below the definition of reductive locally homogeneous manifolds with non-necessarily metric structure, and we also characterize these manifolds through a transitive Lie pseudo-group and an AS-connection.
First of all, we are going to determine the notation related to this section. Let $M$ be a manifold with a geometric structure defined by a tensor or a set of tensors $K=(P_1,...,P_r)$.
\[Definition RL2\] Let $(M,K)$ a manifold with a geometric structure defined by $K$. A *pseudo-group* ${\mathcal{G}}$ is a collection of locally diffeomorphism, $\varphi: U_{\varphi} {\longrightarrow}M$, such that:
- Identity: $Id_M \in {\mathcal{G}}$.
- Inverse: If $\varphi \in {\mathcal{G}}$, then $\varphi ^{-1} \in {\mathcal{G}}$.
- Restriction: If $\varphi \in {\mathcal{G}}, \varphi: U {\longrightarrow}M$ and $V \subset U$, then $\varphi|_V \in {\mathcal{G}}$.
- Continuation: If $\mathrm{dom}(\varphi) = \bigcup U_k$ and $\varphi|_{U_k} \in {\mathcal{G}}$, then $\varphi \in {\mathcal{G}}$.
- Composition: If $\mathrm{im}(\varphi) \subset \mathrm{dom}(\psi)$, then $\psi \circ \varphi \in {\mathcal{G}}$.
In addition, we will require that ${\mathcal{G}}$ leave $K$ invariant, that is, for every $\varphi \in {\mathcal{G}}$, we have that $\varphi ^* K = K$.
A space endow with geometric structure $(M,K)$ is a *locally homogeneous manifold* if there exist a Lie pseudo-group ${\mathcal{G}}$ acting transitively on $M$. In order to define a reductive locally homogeneous manifold, we have to know:
- The meaning of isotropy representation related to pseudo-groups.
- The meaning of adjoin function.
We again fix a frame $u_0\in \mathcal{L}(M)$ over $p_0\in M$. We define ${\mathcal{G}}(p_0)$ as the set of transformations which that $p_0$ belongs to the domain of $\varphi$ and ${\mathcal{G}}(p_0,p_0) \subset {\mathcal{G}}(p_0)$ the set of transformations such that $\varphi (p_0)=p_0$. The quotient $H(p_0) = {\mathcal{G}}(p_0, p_0) / \sim$ with respect to the relation $\varphi \sim \psi \iff \varphi |_U = \psi |_U$ for certain neighbourhood $U$ of $p_0$, is a topological group. We say that the action of ${\mathcal{G}}$ on $M$ is *effective* and *closed* if the map $$\begin{aligned}
H(p_0) &{\longrightarrow}{\mathrm{GL}}(n,{{\mathbb{R}}}) \label{Equation 4}\\
\varphi &\longmapsto u_0^{-1} \circ \varphi_* \circ u_0 \nonumber\end{aligned}$$ is a monomorphism and the image $\mathbf{H}(u_0)$ is closed, respectively, in particular, $\mathbf{H}(u_0)$ is a Lie subgroup of ${\mathrm{GL}}(n,{{\mathbb{R}}})$. The morphism will be called the isotropy representation of ${\mathcal{G}}$ on $(M,K)$.
\[Proposition RL3\] The action of ${\mathcal{G}}$ on $M$ is effective if and only if for every $\varphi, \psi \in {\mathcal{G}}$ such that $\varphi(p_0) = \psi(p_0)$ and $\varphi_{*,p_0} = \psi_{*,p_0}$, then $\varphi = \psi$ in a open neighbourhood of $p_0$.
It is obvious that if we have the second condition then we have an effective action.
Conversely. If $\varphi, \psi \in {\mathcal{G}}$ are such that $\varphi(p_0) = \psi(p_0)$ and $\varphi_{*,p_0} = \psi_{*,p_0}$, we have $\psi \circ \varphi ^{-1}\in H(p_0), \, \psi \circ \varphi ^{-1}(p_0) = p_0$ and $(\psi \circ \varphi ^{-1})_{*,p_0} = Id_{T_{p_0}M}$. Then, $\psi \circ \varphi ^{-1}= Id_M$ in a open neighbourhood of $p_0$.
We now consider $$\label{Equation RL2}
P(u_0): = \{ \varphi_* \circ u_0 : {{\mathbb{R}}}^n {\longrightarrow}T_{f(p_0)}M : \varphi\in {\mathcal{G}}(p_0) \}.$$ This bundle is a reduction of $({\mathcal{L}(M)}{\longrightarrow}M, {\mathrm{GL}}(n,{{\mathbb{R}}}))$ to the group $\mathbf{H}(u_0)$.
\[Proposition RL4\] If $u_0,u_1\in \mathcal{L}(M)$ are two frames on $p_0$ and $p_1\in M$ respectively, then $$P(u_1)=P(u_0)g,$$ where $g$ is the element in ${\mathrm{GL}}(n,{{\mathbb{R}}})$ such that $\psi_* u_0 = u_1 g^{-1} $, with $\psi \in \mathcal{G}$, $\varphi (p_0)=p_1$.
We define the homomorphism $\sigma: H(p_0){\longrightarrow}H(p_1)$, $\varphi \longmapsto \psi_* \circ \varphi \circ \psi ^{-1}_*$. For the sake of simplicity, we also denote by $\sigma: \mathbf{H}(u_0){\longrightarrow}\mathbf{H}(u_1)$ the induced homomorphism by the identification . It is a matter of checking that $R_g:\mathcal{L}(M){\longrightarrow}\mathcal{L}(M)$ induces a principal bundle isomorphism between $P(u_0)$ and $P(u_1)$ with associated Lie group homomorphism $\sigma$.
In particular, the groups $\mathbf{H}(u_0)$ and $\mathbf{H}(u_1)$ are always isomorphic. Because of this, we may simply write $\mathbf{H}$ for any $\mathbf{H}(u_0)$.
Given an element $\varphi \in H(p_0)$, we define $$\begin{aligned}
{\mathrm{Ad}}_{\varphi}: T_{u_0}P(u_0) &{\longrightarrow}T_{u_0}P(u_0) \label{Equation RL3} \\
{\frac{d}{dt} \Big |_{t=0}}(\varphi_t)_*(u_0) &\longmapsto {\frac{d}{dt} \Big |_{t=0}}(\varphi \circ\varphi_t \circ \varphi^{-1})_* (u_0)\end{aligned}$$ where $\varphi _t \in \mathcal{G}$, $t$ belonging to certain interval $(-\epsilon , \epsilon)$.
\[Definition RL5\] Let $(M,K)$ a manifold with a geometric structure, we will say that is *reductive locally homogeneous manifold* if there exist a Lie pseudo-group ${\mathcal{G}}$ acting transitively, effectively and closed on $M$, and we can decompose $T_{u_0}P = {\mathfrak{h}}+ {\mathfrak{m}}$, where ${\mathfrak{h}}$ is the lie algebra associated to $H(p_0)$ and ${\mathfrak{m}}$ is a ${\mathrm{Ad}}(H(p_0))$-invariant subspace.
The definition depends at first sight on the chosen frame $u_0$. However, this dependence is not real as the following result proves.
\[Proposition RL6\] Let $u_0$, $u_1 \in \mathcal{L}(M)$ two linear frames. Then $T_{u_1}P(u_1)$ decomposes as ${\mathfrak{h}}+ {\mathfrak{m}}_1$ for an ${\mathrm{Ad}}(H(p_1))$-invariant subspace ${\mathfrak{m}}_1$ if and only $T_{u_0}P(u_0)$ decomposes as ${\mathfrak{h}}+ {\mathfrak{m}}_0$ for an ${\mathrm{Ad}}(H(p_0))$-invariant subspace ${\mathfrak{m}}_0$.
Given the decomposition $T_{u_0} P(u_0) = {\mathfrak{h}}+ {\mathfrak{m}}_0$ such that ${\mathrm{Ad}}(H(p_0))_{\varphi} ({\mathfrak{m}}_0) \subset {\mathfrak{m}}_0$, we write $T_{u_1} P(u_1) = {\mathfrak{h}}+ {\mathfrak{m}}_1$ with ${\mathfrak{m}}_1 = \Psi_* {\mathfrak{m}}_0$, where $\Psi = R_g \circ \psi_*$ and $\psi\in\mathcal{G}$ is that $\psi_* u_0= u_1 g^{-1}$ and $g\in GL(n,\mathbb{R})$. The subspace ${\mathfrak{m}}_1$ is ${\mathrm{Ad}}(H(p_1))$-invariant. Indeed, for any element $X = {\frac{d}{dt} \Big |_{t=0}}\varphi_t (u_0)\in {\mathfrak{m}}_0$ and $\varphi \in H(p_0)$, we have that $$\begin{aligned}
&\Psi_* ({\mathrm{Ad}}(H(p_0))_{\varphi}(X)) =\\
&= {\frac{d}{dt} \Big |_{t=0}}R_g \circ \psi_* \circ \varphi_* \circ (\varphi_t)_* \circ \varphi^{-1}_* (u_0) = \\
&= {\frac{d}{dt} \Big |_{t=0}}\psi_* \circ \varphi_* \circ \psi^{-1}_* \circ \psi_* \circ (\varphi_t)_* \circ \psi^{-1}_* \circ \psi_* \circ \varphi^{-1}_* \circ \psi^{-1}_* \circ u_1 = \\
&= {\mathrm{Ad}}(H(p_1))_{(\psi_* \circ \varphi \circ \psi^{-1}_* )}({\frac{d}{dt} \Big |_{t=0}}\psi_* \circ \varphi_t \circ \psi^{-1}_* ( \psi_* u_0g)) = \\
&= {\mathrm{Ad}}(H(p_1))_{(\psi_* \circ \varphi \circ \psi^{-1}_* )}({\frac{d}{dt} \Big |_{t=0}}R_g \circ \psi_* \circ \varphi_t ( u_0)) = \\
&= {\mathrm{Ad}}(H(p_1))_{\sigma(\varphi)}(\Psi_*(X)).
\end{aligned}$$
Now we give a local version of Theorem \[Theorem AG2\] above. Furthermore, it provides a generalization of the Ticerri’s result Theorem \[Theorem RL1\].
\[Theorem RL7\] Let $(M,K)$ be a differentiable manifold with a geometric structure $K$. Then the following assertions are equivalents:
1. The manifold $(M,K)$ is a reductive locally homogeneous space, associated to the Lie pseudo-group ${\mathcal{G}}$.
2. There exists a connection ${\tilde{{\nabla}}}$ such that: $${\tilde{{\nabla}}}{\tilde{R}}= 0, \quad {\tilde{{\nabla}}}{\tilde{T}}= 0, \quad {\tilde{{\nabla}}}K = 0,$$ where ${\tilde{R}}$ and ${\tilde{T}}$ are the curvature and torsion of ${\tilde{{\nabla}}}$ respectively.
Given a Lie pseudo-group $\mathcal{G}$ acting transitively on $(M,K)$ in a reductive locally fashion, let $(P {\longrightarrow}M, \mathbf{H})$ the principal bundle associate to the structure of reductive locally homogeneous space as in , for a fixed frame $u_0 \in {\mathcal{L}(M)}$. We define a horizontal distribution $D$ in $P$ by $D_u = \Psi _* ({\mathfrak{m}})$, $\Psi = \psi _*$, for the unique $\psi \in \mathcal{G}$ such that $\psi _* (u_0) =u$, where $T_{u_0}P={\mathfrak{h}}+ {\mathfrak{m}}$ is the reductive decomposition. The distribution $D$ is also $\mathbf{H}$-invariant, that is, given $Y=\Psi_* (X)\in D_u$, $X \in {\mathfrak{m}}$, we have that $(R_h)_* (Y) \in D_{u\cdot h} $, for $h\in \mathbf{H}$. Indeed, we write $X = {\frac{d}{dt} \Big |_{t=0}}(\varphi_t)_* (u_0)$ and, by , $h = u_0^{-1} \circ \varphi_* \circ u_0$ for certain $\varphi \in H(p_0)$. Then $$\begin{aligned}
(R_h)_* (Y) &= (R_h\circ \Psi)_* (X) = {\frac{d}{dt} \Big |_{t=0}}R_h \circ \psi_* \circ (\varphi_t)_* (u_0) =\\
&= {\frac{d}{dt} \Big |_{t=0}}\psi_* \circ (\varphi_t)_* (u_0 \circ u_0^{-1} \circ \varphi_* \circ u_0) =\\
&= {\frac{d}{dt} \Big |_{t=0}}\psi_* \circ (\varphi_t)_* \circ \varphi_* \circ u_0 =\\
&= {\frac{d}{dt} \Big |_{t=0}}\psi_* \circ \varphi_* \circ \varphi^{-1}_* (\varphi_t)_* \circ \varphi_* \circ u_0 =\\
&= (\psi_* \circ \varphi_*)_* {\mathrm{Ad}}(H(p_0))_{\varphi^{-1}}(X).
\end{aligned}$$ As ${\mathrm{Ad}}(H(p_0))_{\varphi^{-1}}(X) \in {\mathfrak{m}}$ by reductive condition, and $\psi \circ \varphi \in {\mathcal{G}}$ we get the invariance. This means that $D$ can be understood as a linear connection $\tilde{\nabla}$.
We now show that $${\tilde{{\nabla}}}{\tilde{R}}= 0, \quad {\tilde{{\nabla}}}{\tilde{T}}= 0, \quad {\tilde{{\nabla}}}K = 0.$$ For $p,q\in M$, let $\gamma$ be a path connecting them. The horizontal lift $\tilde{\gamma}$ with respect to $\tilde{\nabla}$ from $u\in P_p$ to $v\in P_q$, can be regarded as the the parallel transportation $T_pM {\longrightarrow}T_qM$. But since $v = \psi _* u$, for an element $\psi \in \mathcal{G}$, we have that the parallel transportation is exactly $\psi _*$. We have that $\psi_*$ preserves $K$ and the connection ${\tilde{{\nabla}}}$ (and hence, its curvature and torsion) by construction. Therefore $K$, $\tilde{R}$ and $\tilde{T}$ are invariant under parallel transportation and their covariant derivatives vanish.
Conversely, given a linear connection ${\tilde{{\nabla}}}$ such that ${\tilde{{\nabla}}}{\tilde{R}}= 0, \, {\tilde{{\nabla}}}{\tilde{T}}= 0, \, {\tilde{{\nabla}}}K = 0$, let $\mathcal{G}$ the its Lie pseudo-group of local transvections. Since ${\tilde{{\nabla}}}K=0$, the elements of $\mathcal{G}$ preserve $K$. Furthermore, see [@KN1963 V. I, p. 262, Cor. 7.5], $\mathcal{G}$ acts transitively.
To finish the proof we only have to show the reductive condition. Let $(\tilde{P}(u_0) {\longrightarrow}M, {\widetilde{Hol}}(u_0))$ be the holonomy reduction of the frame bundle associated to ${\tilde{{\nabla}}}$ and an element $u_0 \in {\mathcal{L}(M)}$. We first prove that $\mathcal{G}(p_0)$ acts transitively on $\tilde{P}(u_0)$, being $p_0 = \pi(u_0)$. Given $v\in \tilde{P}(u_0) $, there exists a horizontal curve connecting $u_0$ with $v$. The projection to $M$ of that curve can be regarded as a parallel transportation from $p_0$ to $q = \pi(v)$ that, in addition, preserves curvature and torsion. Hence by [@KN1963 V. I, p. 261, Thm. 7.4] there exist a local transvection $\psi \in \mathcal{G}(p_0)$ from $p_0$ to $q$ such that $\psi _*$ is that parallel transportation. Therefore, $\varphi _* (u_0) = v$ and $\mathcal{G}(p_0)$ acts transitively on $\tilde{P}(u_0)$. By construction, $P(u_0)$ (see ) coincides with $\tilde{P}(u_0)$. In particular, ${\widetilde{Hol}}(u_0) = \mathbf{H}(u_0)$ which is closed and the effective condition it is satisfied because Proposition \[Proposition RL3\].
Finally, if we consider $T_{u_0} \tilde{P}(u_0) = {\mathfrak{h}}+ {\mathfrak{m}}$, where ${\mathfrak{m}}$ is the horizontal distribution of ${\tilde{{\nabla}}}$. To prove that ${\mathfrak{m}}$ is $\mathrm{Ad}(H(p_0))$-invariant, let $X = {\frac{d}{dt} \Big |_{t=0}}(\varphi_t)_* (u_0) \in {\mathfrak{m}}$, $\varphi \mathcal{G}(p_0)$ such that $\varphi(p_0)=p_0 $ and $h = u_0^{-1} \circ (\varphi^{-1})_* \circ u_0 \in \mathbf{H}(u_0)$. We consider, $$\begin{aligned}
{\mathrm{Ad}}(H(p_0))_{\varphi}(X) &= {\frac{d}{dt} \Big |_{t=0}}\varphi_* \circ (\varphi_t)_* \circ \varphi^{-1}_* (u_0)= \\
&= {\frac{d}{dt} \Big |_{t=0}}\varphi_* \circ (\varphi_t)_* \circ u_0 \circ u_0^{-1} \circ \varphi^{-1}_* u_0= \\
&= {\frac{d}{dt} \Big |_{t=0}}R_h \circ \varphi_* \circ (\varphi _t)_*(u_0) = (R_h \circ \varphi_*)_* (X).
\end{aligned}$$ Hence, $(R_h \circ \varphi_*)_* (X)$ belongs to the horizontal distribution (${\mathfrak{m}}$), because affine transvections preserve the horizontal distribution.
If we apply this last Theorem in the framework of Theorem \[Theorem AG7\] above, we get the following result.
\[Theorem RL8\] Let $(M, K)$ be a differentiable manifold with an affine connection ${\nabla}$. Then the following assertions are equivalents:
1. The manifold $(M,K)$ is a reductive locally homogeneous space, associated to a Lie pseudo-group contained in $\mathrm{Aff}_{loc}(M,{\nabla})$.
2. There exists a connection ${\tilde{{\nabla}}}$ such that: $${\tilde{{\nabla}}}{\tilde{R}}= 0, \quad {\tilde{{\nabla}}}{\tilde{T}}= 0,\quad {\tilde{{\nabla}}}S = 0, \quad {\tilde{{\nabla}}}K = 0,$$ or $${\tilde{{\nabla}}}R = 0, \quad {\tilde{{\nabla}}}T = 0,\quad {\tilde{{\nabla}}}S = 0, \quad {\tilde{{\nabla}}}K = 0,$$ where $R,T$ and ${\tilde{R}}, {\tilde{T}}$ are the curvature and torsion tensor of ${\nabla}$ and ${\tilde{{\nabla}}}$, respectively, and $S= {\nabla}- {\tilde{{\nabla}}}$ is the tensor.
\[Definition RL9\] Let $(M,K, {\nabla})$ a manifold endowed with a geometrical structure and an affine connection ${\nabla}$. We will say that $(M,K, {\nabla})$ is a *reductive locally homogeneous manifold with ${\nabla}$* if it is reductive locally homogeneous associated to a Lie pseudo-group contained in $\mathrm{Aff}_{loc}(M,{\nabla})$.
AS-manifolds and Homogeneous Structures {#Section AS}
=======================================
Theorem \[Theorem AG2\] provides a characterization of reductive homogeneity in terms of a set of tensor differential equations for complete connections on connected and simply-connected manifolds. The goal now is the results obtained by AS-tensor equations when we drop the mentioned three condition. In other words, we are going to work from an infinitesimal, or even pointwise, point of view.
Let $V$ be a vector space of dimension $n$. Let $${\tilde{R}}: V \wedge V {\longrightarrow}\mathrm{End}(V), \quad {\tilde{T}}: V {\longrightarrow}\mathrm{End}(V),$$ be linear homomorphism and let $K$ be a set of linear tensors on $V$. We will say that $({\tilde{R}},{\tilde{T}})$ is an *infinitesimal model associated to $K$* if it satisfies $$\begin{aligned}
&{\tilde{T}}_XY +{\tilde{T}}_YX = 0, \label{Equation 6}\\
&{\tilde{R}}_{XY}Z + {\tilde{R}}_{YX}Z = 0, \label{Equation 7}\\
&{\tilde{R}}_{XY} \cdot {\tilde{T}}= {\tilde{R}}_{XY} \cdot {\tilde{R}}= 0,\label{Equation 8}\\
&{\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} {\tilde{R}}_{XY}Z + {\tilde{T}}_{{\tilde{T}}_XY}Z = 0, \label{Equation 9}\\
&{\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} {\tilde{R}}_{{\tilde{T}}_XY Z} = 0, \label{Equation 10}\\
&{\tilde{R}}_{XY}\cdot K = 0, \label{Equation 11}\end{aligned}$$ where ${\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}}$ is the cyclic sum, and ${\tilde{R}}_{XY}$ acts in a natural way in the tensor algebra of $V$ as a derivation. In addition, we say that two infinitesimal model $(V, {\tilde{R}},{\tilde{T}})$ and $(V', {\tilde{R}}',{\tilde{T}}')$ are *isomorphic* if there exist a linear isomorphism $f: V {\longrightarrow}V'$ such that $$\label{Equation 12}
f\, {\tilde{R}}= {\tilde{R}}', \quad f\, {\tilde{T}}= {\tilde{T}}', \quad f\, K = K'.$$ This notion of infinitesimal model is a generalization of the one given, for example, in [@N1954].
\[Theorem AS1\] Given a point $p_0\in M$ of an AS-manifold $(M,K, {\tilde{{\nabla}}})$, then $(V = T_{p_0}M, {\tilde{T}}_{p_0}, {\tilde{R}}_{p_0})$ is an infinitesimal model associated to $K_{p_0}$, where ${\tilde{R}}$ and ${\tilde{T}}$ are the curvature and torsion of ${\tilde{{\nabla}}}$.
Let $(M,K, {\tilde{{\nabla}}})$ be an AS-manifold. It satisfies $${\tilde{{\nabla}}}{\tilde{R}}= 0, \quad {\tilde{{\nabla}}}{\tilde{T}}= 0, \quad {\tilde{{\nabla}}}K = 0,$$ given a point $p_0\in M$ and we recall $V=T_{p_0}M$, ${\tilde{R}}_0 = {\tilde{R}}_{p_0}$, $ {\tilde{T}}_0 = {\tilde{T}}_{p_0}$ and $K_0=K_{p_0}$, hence, $({\tilde{R}}_0,{\tilde{T}}_0)$ is an infinitesimal model. Indeed, we deduce and from the skew-symmetric definition of torsion and curvature. Equations and come from ${\tilde{{\nabla}}}{\tilde{R}}= 0, \, {\tilde{{\nabla}}}{\tilde{T}}= 0,\, {\tilde{{\nabla}}}K = 0$. Finally, equations and are the Bianchi identities.
Note that, Theorem \[Theorem AS1\] provides an infinitesimal model for every point in an AS-manifolds. Now, we show it does not matter the chosen point $p_0\in M$. Indeed,
\[Theorem AS2\] Let $(M , K, {\tilde{{\nabla}}})$ be an AS-manifold. Given two different points $p_0$, $p_1 \in M$ their associated infinitesimal model are isomorphic.
By [@KN1963 Vol. 1, p. 262, Cor. 7.5], there exists a locally affine transformation $\varphi$ sending $p_0$ to $p_1$. Because of being affine, we have that $\varphi_*$ is a linear isomorphism between $T_{p_0}M$ and $T_{p_1}M$ satisfying $\varphi_* {\tilde{T}}_{p_0} = {\tilde{T}}_{p_1}$ and $\varphi_* {\tilde{R}}_{p_0} = {\tilde{R}}_{p_1}$. By ${\tilde{{\nabla}}}K = 0$, we conclude that $\varphi_* K_{p_0} = K_{p_1}$.
Hence, associated to any AS-manifold there exist unique infinitesimal model up to isomorphism. Furthermore, when different manifolds have isomorphic associated infinitesimal models, from [@KN1963 Vol. 1, p.261, Thm. 7.4] we get the following result.
\[Theorem AS3\] Let $(M , K, {\tilde{{\nabla}}})$ and $(M', K', {\tilde{{\nabla}}}')$ be two AS-manifold and let $p_0 \in M$ and $p_0' \in M'$ be two points, such that their associated infinitesimal models are isomorphic. Hence, there exists an local affine diffeomorphism between $p_0$ and $p'_0$ sending to $K_{p_0}$ to $K'_{p'_0}$.
So, we define the notion of AS-isomorphism between AS-manifolds.
\[Definition AS4\] Let $(M , K, {\tilde{{\nabla}}})$ and $(M', K', {\tilde{{\nabla}}}')$ be two AS-manifold and let $p_0 \in M$ and $p_0' \in M'$ be two point. We say $(M , K, {\tilde{{\nabla}}})$ and $(M', K', {\tilde{{\nabla}}}')$ are AS-isomorphic if there exists an local affine diffeomorphism between $p_0$ and $p_0'$ sending $K$ to $K'$.
Conversely, the following Theorem we show that every infinitesimal model is isomorphic to the infinitesimal model of one and only one simply-connected, connected and reductive homogeneous space. Hence, in particular, every AS-manifold is AS-isomorphic to one simply-connected, connected and reductive homogeneous space.
\[Theorem AS5\] Let $V$ be a vector space and $({\tilde{R}}_0,{\tilde{T}}_0)$ an infinitesimal model associated to tensors $K_0$. Then, there is a unique (up to affine-isomorphism) connected, simply connected and reductive homogeneous manifold $M$ with a geometrical structure defined by the tensors field $K$ and a point $p_0 \in M$ such that $$K_{p_0}= K_0,$$ as well as a complete AS-connection ${\tilde{{\nabla}}}$ for which the curvature ${\tilde{R}}$ and torsion ${\tilde{T}}$ verify that ${\tilde{R}}_{p_0} = {\tilde{R}}_0$ and ${\tilde{T}}_{p_0} = {\tilde{T}}_0$.
We fix a point $p_0\in M$, from [@KK1980 p. 34, Thm. I.17], there exists a simply-connected and connected reductive homogeneous manifold $(M,{\tilde{{\nabla}}})$ with ${\tilde{{\nabla}}}{\tilde{R}}= 0, \, {\tilde{{\nabla}}}{\tilde{T}}= 0$ and ${\tilde{T}}_{p_0} = {\tilde{T}}_0, \, {\tilde{R}}_{p_0} = {\tilde{R}}_0$. In addition , from [@KK1980 p. 43, Prop. I.39], the manifold is equipped with a set of tensors $K$ such that $K_{p_0}= K_0, \, {\tilde{{\nabla}}}K =0$.
\[Corollary AS6\] Let $(M,K)$ and $(M',K')$ two simply-connected and connected reductive homogeneous manifolds which are AS-isomorphic. Hence, there is an affine diffeomorphism between $M$ and $M'$ sending $K$ to $K'$.
Summarizing, there exists a surjective morphism between the class of AS-manifolds and the class of infinitesimal models such that for the class of AS-manifolds with a fixed infinitesimal model there is one and only one connected, simply-connected and reductive homogeneous manifold.
Finally, from every infinitesimal model $({\tilde{R}}, {\tilde{T}})$ on $V$ associated to $K$, we can construct a transitive Lie algebra using the so-called *Nomizu construction*, see [@N1954]. Let $${\mathfrak{g}}_0 = V \oplus {\mathfrak{h}}_0,$$ where ${\mathfrak{h}}_0 = \{ A \in \mathfrak{end}(V): A \cdot {\tilde{R}}= 0,\, A \cdot {\tilde{T}}= 0,\, A \cdot K = 0 \}$, equipped with the Lie bracket $$\begin{aligned}
&[A,B] = AB -BA, &A,B \in {\mathfrak{h}}_0, \\
&[A,X] = AX, &A \in {\mathfrak{h}}_0, \, X\in V, \\
&[X,Y] = -{\tilde{T}}_XY +{\tilde{R}}_{XY}, &X,Y \in V.\end{aligned}$$ Alternatively, we can also consider the so-called *transvection algebra* ${\mathfrak{g}}_0 '= V \oplus {\mathfrak{h}}_0' $, where ${\mathfrak{h}}_0'$ is the lie algebra of endomorphism generated by ${\tilde{R}}_{XY}$ with $X,Y \in V$, equipped with brackets as above. In particular, this Lie algebra coincides with the holonomy algebra of the connection ${\tilde{{\nabla}}}$. Then, we have shown that every infinitesimal model has Nomizu and transvection constructions.
Two Nomizu constructions $({\mathfrak{g}}_0, {\mathfrak{h}}_0, {\tilde{T}}, {\tilde{R}}, K)$ and $({\mathfrak{g}}_0',{\mathfrak{h}}_0', {\tilde{T}}', {\tilde{R}}', K')$ are isomorphic if there exist a Lie algebra isomorphism $F: {\mathfrak{g}}_0 {\longrightarrow}{\mathfrak{g}}_0'$ such that $F (V)= V'$, $F$ sends $K$ to $K'$ and $F({\mathfrak{h}}_0) = {\mathfrak{h}}_0'$.
\[Proposition AS7\] Two infinitesimal model are isomorphic if and only if their Nomizu constructions are isomorphic.
Suppose that $V$ and $V'$ are two vector space with two infinitesimal models $({\tilde{R}},{\tilde{T}})$ and $({\tilde{R}}',{\tilde{T}}')$. Then there is a isomorphism $f: V {\longrightarrow}V'$ such that satisfies . We thus consider $\tilde{f}: {\mathfrak{g}}_0 {\longrightarrow}{\mathfrak{g}}_0'$ such that $\tilde{f}|_V = f$ and $\tilde{f}|_{{\mathfrak{h}}_0}(A)= f \circ A \circ f^{-1}$.
Conversely, given a Lie algebra homomorphism $F: {\mathfrak{g}}_0 {\longrightarrow}{\mathfrak{g}}_0'$ such that $F(V)= V'$ and $F({\mathfrak{h}}_0) = {\mathfrak{h}}_0'$, then $f = F|_V$ is the isomorphism. Indeed, by definition $f$ sends $K$ to $K'$ and, taking into account that $F$ is a Lie algebra morphism, we obtain that $f$ sends ${\tilde{R}}$ to ${\tilde{R}}'$ and ${\tilde{T}}$ to ${\tilde{T}}'$.
Surprisingly, the converse is no true: two different Nomizu constructions could arise the same Lie algebra, see [@L2014 p. 36].
Obviously, by Proposition \[Proposition AS7\] and Corollary \[Corollary AS6\] we have that connected, simply-connected and reductive homogeneous manifolds with isomorphic Nomizu constructions are isomorphic. The same applies to the transvection algebras
\[Remark AS8\] To study the different representations of a reductive homogeneous space $M$ as a coset $G/H$ we can study the different Nomizu constructions. In particular, if $M$ is connected and simply-connected, there is a bijection between different representations of homogeneous spaces as $G/H$ and different Nomizu constructions on $M$.
We finally the particular case where $M$ is a manifold with a geometric structure $K$ equipped with a connection ${\nabla}$ and an AS-connection ${\tilde{{\nabla}}}$ such that, $${\tilde{{\nabla}}}{\tilde{R}}= 0, \quad {\tilde{{\nabla}}}{\tilde{T}}= 0,\quad {\tilde{{\nabla}}}S = 0,\quad {\tilde{{\nabla}}}K = 0,$$ where $S= {\nabla}- {\tilde{{\nabla}}}$. So we can consider, $$T_{X}Y = \tilde{T}_{X}Y + S_{X}Y -S_{Y}X, \quad R_{XY} = \tilde{R}_{XY} + [S_Y, S_X] + S_{S_{Y}X-S_{X}Y}.$$ where $R$ and $T$ are the curvature and torsion of ${\nabla}$.
\[Corollary AS9\] Let $(M, S, K, {\tilde{{\nabla}}})$ and $(M', S', K', {\tilde{{\nabla}}}')$ be two AS-manifolds with homogeneous structures $S$ and $S'$. Then, there exist an AS-isomorphism between $M$ and $M'$ if and only if there exists an affine local diffeomorphism between $(M,{\nabla})$ and $(M', {\nabla}')$ sending $S$ to $S'$ and $K$ to $K'$.
Given a fixed point $p_0\in M$, by Theorem \[Theorem AS1\], we can consider an infinitesimal model $(T_{p_0}M,{\tilde{R}}_{p_0}, {\tilde{T}}_{p_0})$ associated to $S_{p_0}$ and $K_{p_0}$ with $$\begin{aligned}
(T_{p_0})_{X}Y &= (\tilde{T}_{p_0})_{X}Y + (S_{p_0})_{X}Y -(S_{p_0})_{Y}X,\\
(R_{p_0})_{XY} &= (\tilde{R}_{p_0})_{XY} + [(S_{p_0})_Y, (S_{p_0})_X] + (S_{p_0})_{(S_{p_0})_{Y}X-(S_{p_0})_{X}Y},\end{aligned}$$ where $R_{p_0}$ and $T_{p_0}$ are the curvature and torsion of ${\nabla}$ in $p_0$.
\[Corollary AS10\] Let $(V, {\tilde{R}}, {\tilde{T}})$ and $(V', {\tilde{R}}', {\tilde{T}}')$ be two infinitesimal models associated to $S$, $K$ and $S'$, $K'$, respectively, with $$\begin{aligned}
T_{X}Y &= \tilde{T}_{X}Y + S_{X}Y -S_{Y}X, \quad R_{XY} = \tilde{R}_{XY} + [S_Y, S_X] + S_{S_{Y}X-S_{X}Y},\\
T'_{X}Y &= {\tilde{T}}'_{X}Y + S'_{X}Y -S'_{Y}X, \quad R'_{XY} = \tilde{R}'_{XY} + [S'_Y, S'_X] + S'_{S'_{Y}X-S'_{X}Y}.
\end{aligned}$$ Hence, there exist a isomorphism of infinitesimal models if and only if there exist an linear isomorphism $f: V {\longrightarrow}V'$ such that, $$\label{Equation 13}
f\, R = R', \quad f\, T = T' \quad f\, S = S', \quad f\, K=K'.$$
{#Section IS}
Let $(V, \omega)$ be a symplectic vector space. Based on the classifications given in [@AP2011], we give bellow explicit expressions of the invariant $\mathrm{Sp}(V)$-submodules of $S^2 V^* \otimes V^*$ and $\wedge^2 V^* \otimes V^*$. For that, we identify a symplectic vector space $V$ and its dual $V^*$ as $$\begin{aligned}
(\cdot)^*: V &{\longrightarrow}V^* \\
X &\longmapsto X^*(Y) = \omega(X,Y).\end{aligned}$$ Furthermore, we can transfer the symplectic form to $V^*$ as $\omega^*(X^*,Y^*) = \omega(X,Y)$, that is, we regard $(V, \omega)$ and $(V^*, \omega^*)$ as symplectomorphic.
For the sake of symplcity, from now on, we denote $\omega_{XY} = \omega(X,Y)$.
\[Theorem IS1\] The space of contorsion-like tensors has the decomposition in irreducible $\mathrm{Sp}(V)$-submodules as $$S^2 V^* \otimes V^* = \mathcal{S}_1 (V) + \mathcal{S}_2 (V) + \mathcal{S}_3 (V)$$ where, $$\begin{aligned}
\mathcal{S}_1(V) &= \{ S \in S^2 V^* \otimes V^*:
\: S_{XYZ} = \omega_{ZY}\omega_{XU}+ \omega_{ZX}\omega_{YU}, U \in V \},
\\
\mathcal{S}_2(V) &= \{ S \in S^2V^* \otimes V^*:\:
{\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} S_{XYZ} = 0,\, \mathrm{s}_{13}(S)=0\},
\\
\mathcal{S}_3(V) &= \{ S \in S^2V^* \otimes V^*:\: S_{XYZ}= S_{XZY} \} = S^3 V^*,
\end{aligned}$$ and $$\mathrm{s}_{13}(S)(Z)=\sum _{i=1}^n \left(S_{e_i Z e_{i+n}}-S_{e_{i+n} Z e_{i}}\right),$$ for a symplectic base $\{ e_1,\ldots,e_n,e_{n+1},\ldots, e_{2n}\}$.
The dimensions of the subspaces are $$\dim (\mathcal{S}_1(V)) =2n, \quad
\dim (\mathcal{S}_2(V)) =\frac{8}{3}(n^3-n), \quad
\dim (\mathcal{S}_3(V)) =\small{\begin{pmatrix}
2n+2\\
3
\end{pmatrix}}.$$
Given a symplectic basis $\{ e_1, \ldots ,e_n, e_{n+1}, \ldots, e_{2n}\}$ of $V$, we define the morphisms $$\begin{aligned}
\varphi : S^2 V^* \otimes V^* &{\longrightarrow}V^* \\
(u_1^* u_2 ^* \otimes v^*) & \longmapsto \omega_{u_1,v} u_2^* + \omega_{u_2, v} u_1^*,\\
\pi : S^2 V^* \otimes V^* &{\longrightarrow}S^3 V^* \\
(u_1^* u_2 ^* \otimes u^*) & \longmapsto \frac{1}{3} u^* u_1^* u_2^*,
\end{aligned}$$ and $$\begin{aligned}
\xi : V^* &{\longrightarrow}S^2 V^* \otimes V^* \\
u^* & \longmapsto \frac{1}{2n+1} \sum_{i=1}^{n} e_i^* u^* \otimes e_{i+n}^* - e_{i+n}^* u^* \otimes e_{i}^*.
\end{aligned}$$ By Theorem 1.1 in [@AP2011], applied to $(V^*,\omega^*)$, we decompose $$S^2 V^* \otimes V^* = S^3 V^* + \mathcal{A}' + V^*$$ where $\mathcal{A}' = \ker(\varphi)\cap \ker(\pi)$ and $V^*$ is isomorphic to $\mathrm{im}(\xi)$. We define $\mathcal{S}_1(V):= V^*$, $\mathcal{S}_2(V):= \mathcal{A}'$ and $\mathcal{S}_3 (V) := S^3 V^*$.
For the explicit expression of $\mathcal{S}_1(V)$, given $W^* \in V^*$ we have $$\begin{aligned}
\xi(W^*)_{XYZ} &= \frac{1}{2n+1}( \sum_{i=1}^{n}\, x_{i+n} \omega_{WY} z_i + y_{i+n}\omega_{WX} z_i \\
& \quad - x_i\omega_{WY} z_{i+n} - y_i \omega_{WX} z_{i+n})\\
&= \frac{1}{2n +1} ( \omega_{ZX} \omega_{WY} + \omega_{ZY} \omega_{WX}).
\end{aligned}$$ Hence, taking $U = \frac{1}{2n +1} W $, we get the required result for $\mathcal{S}_1(V)$.
With respect to the explicit expressions of $\mathcal{S}_2(V)$, for $$S =\frac{1}{2} \sum_{i,j,k=1}^{2n} S_{e_i e_j e_k} e_i^*e_j^*\otimes e_k^* \in S^2 V^* \otimes V^*,$$ we have $$\begin{aligned}
\varphi(S)
&= \frac{1}{2} \sum_{i,j,k=1}^{2n} S_{e_i e_j e_k} \mathrm{s}_{13}(e_i^*e_j^*\otimes e_k^*) = \\
&= \frac{1}{2} \sum_{i,j,k=1}^{2n} S_{e_i e_j e_k} ( \omega_{e_ie_k} e_j^* + \omega_{e_je_k} e_i^*) = \\
&= \sum_{i,j,k=1}^{2n} \frac{1}{2} (S_{e_i e_j e_k}+ S_{e_j e_i e_k})\omega_{e_ie_k}e_j^* =\\
&= \sum_{j=1}^{2n} \sum_{i=1}^{n} \frac{1}{2} ( S_{e_i e_j e_{i+n}} + S_{e_j e_i e_{i+n}} - S_{e_{i+n} e_j e_i} - S_{e_j e_{i+n} e_i})e_j^* =\\
&= \sum_{j=1}^{2n} \sum_{i=1}^{n} (S_{e_i e_j e_{i+n}}- S_{e_{i+n}e_j e_i})e_j^*.
\end{aligned}$$ Hence, $S\in \mathrm{ker}\varphi$ if and only if $\mathrm{s}_{13}(S)=0$ as in the statement. Moreover, $ \frac{1}{3} e_i^*e_j^*e_k^* = e_i^*e_j^*\otimes e_k^* + e_k^*e_i^*\otimes e_j^* +e_j^*e_k^*\otimes e_i^*$ and therefore $$\pi(S)_{XYZ} = {\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} S_{XYZ},$$ so that we have the expression for the tensors in $\mathcal{S}_2(V)$. The dimensions come from Theorem 1.1 of [@AP2011].
Now, using these expressions we are going to give the explicit classes of torsion-like tensors.
\[Theorem IS2\] The space of torsion-like tensors has the decomposition in irreducible $\mathrm{Sp}(V)$-submodules as $$\wedge^2 V^* \otimes V^* =
\tilde{\mathcal{T}}_1(V)+\tilde{\mathcal{T}}_2(V)+\tilde{\mathcal{T}}_3(V)+\tilde{\mathcal{T}}_4(V)$$ where $$\begin{aligned}
\tilde{\mathcal{T}}_1(V) = \{{\tilde{T}}\in \wedge^2 V^* \otimes V^*&:\: {\tilde{T}}_{XYZ}= 2 \omega_{XY}\omega_{ZU} + \omega_{XZ}\omega_{YU} - \omega_{YZ} \omega_{XU}, \, U \in V\}, \\
\tilde{\mathcal{T}}_2(V) = \{ {\tilde{T}}\in \wedge^2 V^* \otimes V^* &:\: {\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} {\tilde{T}}_{XYZ} = 0, \, \mathrm{t}_{12}({\tilde{T}})= 0\},\\
\tilde{\mathcal{T}}_3(V) = \{{\tilde{T}}\in \wedge^2 V^* \otimes V^* &:\: {\tilde{T}}_{XYZ} =\omega_{XY}\omega_{UZ} + \omega_{YZ} \omega_{UX} + \omega_{ZX}\omega_{UY}, \, U \in V\},\\
\tilde{\mathcal{T}}_4(V) = \{{\tilde{T}}\in \wedge^2 V^* \otimes V^*&:\: {\tilde{T}}_{XYZ} = - {\tilde{T}}_{XZY}, \, \mathrm{t}_{12} ({\tilde{T}}) = 0\},
\end{aligned}$$ and $$\begin{aligned}
\mathrm{t}_{12}({\tilde{T}})(Z)&=\sum _{i=1}^n \left({\tilde{T}}_{e_i e_{i+n} Z}\right),
\end{aligned}$$ for a symplectic basis $\{ e_1,\ldots,e_n,e_{n+1},\ldots, e_{2n}\}$. In addition,
For a symplectic basis $\{ e_1, \ldots ,e_n, e_{n+1}, \ldots, e_{2n}\}$ of $V$, we define the morphisms $$\begin{aligned}
A_2: S^2 V^* \otimes V^*& {\longrightarrow}\: \wedge ^2 V^* \otimes V^* \\
(u_1^*u_2^* \otimes v^*)& \longmapsto \: v^* \wedge u_1^* \otimes u_2^* + v^* \wedge u_2^*\otimes u_1 ^*, \\
C: \wedge^2 V^* \otimes V^* & {\longrightarrow}V^* \\
(u_1^*\wedge u_2^* \otimes v^*) & \longmapsto \omega_{u_1 u_2}v^* + \omega_{vu_1}u_2^* + \omega_{u_2 v}u_1^*,
\end{aligned}$$ and $$\begin{aligned}
\eta: V^* &{\longrightarrow}\wedge^3 V^* \\
u^* &\longmapsto \sum_{i=1}^{n} e_i^* \wedge e_{i+n}^* \wedge u^*.
\end{aligned}$$ By Theorem 1.2 of [@AP2011], applied to $(V^*,\omega^*)$, we decompose $$\wedge^2 V^* \otimes V^* = V_1^*+\mathcal{A}'+V_2^*+\mathcal{T}'$$ where, $V_1^* = A_2(\mathcal{S}_1(V))$, $\mathcal{A}' = A_2(\mathcal{S}_2(V))$, $\mathcal{T}' = \ker C \cap \wedge^3 V^*$ and $V_2^*= \mathrm{Im}(\eta)$ is the vector space such that $V_2^* \subset \wedge^3 V^*$ and $V_2^* + \mathcal{T}' = \wedge^3 V^*$. We define $\tilde{\mathcal{T}}_1(V):= V_1^*$, $\tilde{\mathcal{T}}_2(V):= \mathcal{A}'$, $\tilde{\mathcal{T}}_3 (V) := V_2^*$ and $\tilde{\mathcal{T}}_4(V) : = \mathcal{T}'$.
First, as $$\label{Equation A2}
A_2(S)_{XYZ} = S_{YZX}- S_{XZY},$$ we get the expression for the tensors in $\tilde{\mathcal{T}}_1(V)$ in view of the expression of $\mathcal{S}_1(V)$ in Theorem \[Theorem IS1\].
Indeed, by equation , we infer the explicit expression of $\tilde{\mathcal{T}}_1(V)$.
To study the explicit expression of $\mathcal{T}_2(V)$, we have to consider the following exact sequence, [@AP2011 Eq. (1.3)], $$0 {\longrightarrow}S^3 V^* \xrightarrow{A_1} S^2V^* \otimes V^* \xrightarrow{A_2} \wedge^2 V^* \otimes V^* \xrightarrow{A_3} \wedge^3 V^* {\longrightarrow}0$$ where $A_1 = \pi$ and $A_3(u_1^* \wedge u_2^* \otimes v^*) = u_1^* \wedge u_2 ^* \wedge v^*$. Note that, $ u_1^* \wedge u_2 ^* \wedge v^* = u_1^* \wedge u_2 ^* \otimes v^* + v^* \wedge u_1^* \otimes u_2^* + u_2^* \wedge v^* \otimes u_1^*$, hence, $$A_3({\tilde{T}})_{XYZ} = {\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} {\tilde{T}}_{XYZ}.$$
Therefore, $\tilde{\mathcal{T}}_2(V)$ is generated by ${\tilde{T}}_{XYZ} = S_{YZX}- S_{XZY}$ with $S \in S^2 V^* \otimes V^*$ and $\mathrm{s}_{13}(S)= 0$. The first condition is equivalent to ${\tilde{T}}\in \mathrm{ker}(A_3)$, or equivalently, $ {\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} {\tilde{T}}_{XYZ} = 0$. The second condition is equivalent $\mathrm{t}_{12} (T) = 0$ straightforwardly. For the explicit expressions of $ \tilde{\mathcal{T}}_4(V)$, given $${\tilde{T}}= \frac{1}{2} \sum_{i,j,k=1}^{2n} \tilde{T}_{e_i e_j e_k} e_i^*\wedge e_j^* \otimes e_k^* \in \wedge^2 V^* \otimes V^*,$$ we have $$\begin{aligned}
C ({\tilde{T}})
&= \frac{1}{2} \sum_{i,j,k=1}^{2n} {\tilde{T}}_{e_i e_j e_k} C(e_i^*\wedge e_j^* \otimes e_k^*) = \\
&= \frac{1}{2} \sum_{i,j,k=1}^{2n} {\tilde{T}}_{e_i e_j e_k} \left(\omega_{e_i e_j} e_k^* + \omega_{e_k e_i} e_j^* + \omega_{e_j e_k} e_i^*\right) =\\
&= \frac{1}{2}\left( \sum_{i,j,k=1}^{2n} {\tilde{T}}_{e_i e_j e_k} \omega_{e_i e_j} e_k^* + \sum_{i,j,k=1}^{2n} {\tilde{T}}_{e_i e_j e_k} \omega_{e_k e_i} e_j^* + \sum_{i,j,k=1}^{2n} {\tilde{T}}_{e_i e_j e_k} \omega_{e_j e_k} e_i^*\right) =\\
&= \frac{1}{2} \sum_{i,j,k=1}^{2n} \left({\tilde{T}}_{e_i e_j e_k} + {\tilde{T}}_{e_k e_i e_j}+{\tilde{T}}_{e_j e_k e_i}\right) \omega_{e_i e_j} e_k^* = \\
&= \sum_{k=1}^{2n} \sum_{i=1}^{n} \frac{1}{2}\left({\tilde{T}}_{e_i e_{i+n} e_k} + {\tilde{T}}_{e_k e_i e_{i+n}}+ {\tilde{T}}_{e_{i+n} e_k e_i}- {\tilde{T}}_{e_{i+n} e_i e_k} - \tilde{T}_{e_k e_{i+n} e_i} - \tilde{T}_{e_i e_k e_{i+n}} \right) e_k^* =\\
&= \sum_{k=1}^{2n} \sum_{i=1}^{n} \left({\tilde{T}}_{e_i e_{i+n} e_k} + {\tilde{T}}_{e_k e_i e_{i+n}}+ {\tilde{T}}_{e_{i+n} e_k e_i} \right) e_k^*.
\end{aligned}$$ Therefore, for $\tilde{T}\in \wedge ^3 V^*$, $C({\tilde{T}})=0$ is equivalent to $\mathrm{t}_{12} ({\tilde{T}}) = 0$.
Finally, with respect to the explicit expressions of $\tilde{\mathcal{T}}_3(V)$, given $U^* \in V^*$ with dual element $U \in V$, $$\begin{aligned}
\eta (U^*)
&= \sum_{i=1}^{n} e_i^* \wedge e_{i+n}^* \wedge U^* =\\
&= \sum_{i=1}^{n} \left(e_i^* \wedge e_{i+n}^* \otimes U^* + U^* \wedge e_i^* \otimes e_{i+n}^* + e_{i+n}^* \wedge U^* \otimes e_i^*\right),
\end{aligned}$$ evaluating in $X,Y,Z$, we infer, $$\begin{aligned}
\eta(U^*)_{XYZ} &= \sum_{i=1}^{n}\hspace{0.4cm} \left((x_i y_{i+n}- x_{i+n} y_i) \omega_{UZ} + \right.\\
&\hspace{1cm}+ (\omega_{UX} y_{i+n} - \omega_{UY} x_{i+n}) (-z_i) + \\
&\hspace{1cm}\left. + (- x_i \omega_{UY} - (-y_i) \omega_{UX}) z_{i+n}\right)= \\
&= \omega_{XY}\omega_{UZ} + \omega_{YZ} \omega_{UX} + \omega_{ZX}\omega_{UY}
\end{aligned}$$ Therefore, $\tilde{\mathcal{T}}_3(V)$ has the claimed form.
\[Remark IS3\] We have the following sums
- $\tilde{\mathcal{T}}_1(V) + \tilde{\mathcal{T}}_2(V) = \{{\tilde{T}}\in \wedge^2 V^* \otimes V^* :\: {\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} T_{XYZ} = 0\}$,
- $\tilde{\mathcal{T}}_2(V)+ \tilde{\mathcal{T}}_4(V) = \{{\tilde{T}}\in \wedge^2 V^* \otimes V^* :\: \mathrm{t}_{12} ({\tilde{T}}) = 0\}, $
- $\tilde{\mathcal{T}}_3(V) + \tilde{\mathcal{T}}_4(V) = \wedge^3 V^*$.
The first two come directly from the expressions of the classes in the previous Theorem. With respect to the last identity, we note that $\mathrm{Id}_{V^*} = \frac{1}{3(n-1)} C \circ \eta$, so that we can decompose $\wedge^3 V^* = \ker C + \mathrm{Im} \,\eta = \tilde{\mathcal{T}}_3(V) + \tilde{\mathcal{T}}_4(V)$.
Classifications for almost symplectic and Fedosov AS-manifolds {#Section C}
==============================================================
Almost symplectic AS-manifolds
------------------------------
We now want to classify the infinitesimal models in the case of vector spaces $V$ endowed with a linear symplectic tensor $K=\omega$. If $(V, {\tilde{R}}, {\tilde{T}})$ and $(V', {\tilde{R}}', {\tilde{T}}')$ are two infinitesimal models associated to symplectic linear tensors $\omega$ and $\omega'$, respectively, with $\mathrm{dim}V=\mathrm{dim}V'$, since there are symplectomorphisms between $V$ and $V'$, we can identify $V'$ with $V$ and $\omega'$ with $\omega$. From , isomorphisms $f:V{\longrightarrow}V$ of almost symplectic infinitesimal models satisfy $$f\, {\tilde{R}}= {\tilde{R}}', \quad f\, {\tilde{T}}= {\tilde{T}}', \quad f\, \omega = \omega,$$ and in particular $f \in \mathrm{Sp}(V,\omega)=\mathrm{Sp}(V)$. If we decompose curvature-like or torsion-like tensor spaces in $\mathrm{Sp}(V)$-irreducible submodules, then we get a necessary condition to be isomorphic as models, by virtue of Theorem \[Theorem AS3\], also as AS-manifolds.
For the classification of the torsion ${\tilde{T}}$ into $\mathrm{Sp}(V)$-classes, we will work both with $(1,2)$-tensors and $(0,3)$-tensors given by the isomorphism $${\tilde{T}}_{XYZ}= \omega({\tilde{T}}_XY, Z), \quad X,Y,Z \in V.$$
Let $(M, \omega)$ an almost symplectic AS-manifold. We denote by $\tilde{\mathcal{T}}$ the set of *homogeneous almost symplectic torsions*, that is, the torsions of an AS-connection on $(M, \omega)$. Given any $p_0 \in M$, from Theorem \[Theorem AS1\], $(V=T_{p_0}M,\tilde{R}_{p_0},{\tilde{T}}_{p_0})$ is an infinitesimal model associated to $\omega _{p_0}$. Thus $T_{p_0} \in \wedge^2 V \otimes V$, and the classification given in Theorem \[Theorem IS2\] gives us the following decomposition $$\tilde{\mathcal{T}} =
\tilde{\mathcal{T}}_1+\tilde{\mathcal{T}}_2+\tilde{\mathcal{T}}_3+\tilde{\mathcal{T}}_4$$ where $$\begin{aligned}
\tilde{\mathcal{T}}_1 &= \{{\tilde{T}}\in \tilde{\mathcal{T}} :\: {\tilde{T}}_{XYZ}= 2 \omega_{XY}\omega_{ZU} + \omega_{XZ}\omega_{YU} - \omega_{YZ} \omega_{XU}, \, U \in \mathfrak{X}(M) \}, \\
\tilde{\mathcal{T}}_2 &= \{ {\tilde{T}}\in \tilde{\mathcal{T}}:\: {\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} {\tilde{T}}_{XYZ} = 0, \, \mathrm{t}_{12}({\tilde{T}})= 0\}\\
\tilde{\mathcal{T}}_3 &= \{{\tilde{T}}\in \tilde{\mathcal{T}}:\: {\tilde{T}}_{XYZ} =\omega_{XY}\omega_{UZ} + \omega_{YZ} \omega_{UX} + \omega_{ZX}\omega_{UY}, \, U \in \mathfrak{X}(M)\},\\
\tilde{\mathcal{T}}_4 &= \{{\tilde{T}}\in \tilde{\mathcal{T}} :\: {\tilde{T}}_{XYZ}=- {\tilde{T}}_{XZY},\, \mathrm{t}_{12} ({\tilde{T}}) = 0\}.\end{aligned}$$
\[Definition C1\] Let ${\tilde{T}}\in \tilde{\mathcal{T}}$ be an homogeneous almost symplectic torsion. It is of *type i* if ${\tilde{T}}$ lies in $\tilde{\mathcal{T}}_i$ and correspondingly is of *type i+j* if lies in $\tilde{\mathcal{T}}_i + \tilde{\mathcal{T}}_j$ with $i$, $j\in \{1,2,3,4\}$ and $i\neq j$.
Summarizing, we described almost symplectic AS-manifolds in sixteen classes defined by its torsion tensor.
\[Theorem C2\] Let $(M, \omega)$ be an almost symplectic AS-manifold. Then, $(M, \omega)$ is a symplectic manifold if and only if the torsion of ${\tilde{{\nabla}}}$ lies in $\tilde{\mathcal{T}_1} + \tilde{\mathcal{T}_2}$.
If $(M, \omega)$ is a symplectic manifold, there is a torsion-free symplectic connection ${\nabla}$ (see [@AP2011 Theorem 2.1]). The difference $S = {\nabla}- {\tilde{{\nabla}}}$ is a $(1,2)$-tensor such that ${\tilde{T}}_X Y = S_Y X - S_XY$. Then ${\tilde{T}}_{XYZ} = A_2 (-S) = S_{XZY}- S_{YZX}$, where $S_{XYZ} = \omega(S_ZX, Y)$ and ${\tilde{T}}_{XYZ} = \omega({\tilde{T}}_XY,Z)$. In particular, ${\tilde{T}}$ lies in $\mathcal{T}_1 + \mathcal{T}_2$.
Conversely, if ${\tilde{T}}$ lies in $\mathcal{T}_1 + \mathcal{T}_2$, then there exists at least one tensor $S \in S^2 T^*M \otimes T^*M$, such that, ${\tilde{T}}_{XYZ} = S_{YZX} -S_{XZY}$. We can consider the tensor $S_XY$ with $ \omega(S_Z X, Y) = S_{XYZ}$. It satisfies that ${\tilde{T}}_X Y = S_XY - S_Y X$ with $\omega({\tilde{T}}_XY,Z) = {\tilde{T}}_{XYZ}$ and preserves the symplectic form. The connection ${\nabla}= {\tilde{{\nabla}}}- S$ is symplectic.
Fedosov AS-manifolds
--------------------
We now want to study infinitesimal models associated to a linear symplectic tensor $\omega$ and a homogeneous structure $S$ as in Corollary \[Corollary AS10\]. Let $(V, {\tilde{R}}, {\tilde{T}})$ and $(V', {\tilde{R}}', {\tilde{T}}')$ be two infinitesimal models associated to (1,2) linear tensors $S$ and $S'$, respectively, with $$\begin{aligned}
T_{X}Y &= \tilde{T}_{X}Y + S_{X}Y -S_{Y}X, \quad R_{XY} = \tilde{R}_{XY} + [S_Y, S_X] + S_{S_{Y}X-S_{X}Y},\\
T'_{X}Y &= {\tilde{T}}'_{X}Y + S'_{X}Y -S'_{Y}X, \quad R'_{XY} = \tilde{R}'_{XY} + [S'_Y, S'_X] + S'_{S'_{Y}X-S'_{X}Y},\end{aligned}$$ and also associated to symplectic linear tensors $\omega$ and $\omega'$, respectively, with $\mathrm{dim}V=\mathrm{dim}V'$. Since there are symplectomorphisms between $V$ and $V'$, we can identify $V$ with $V'$ and $\omega$ with $\omega'$. Therefore, by , there is a linear isomorphism $f: V {\longrightarrow}V$ such that, $$\label{Equation 15}
f\, R= R', \quad f\, T = T', \quad f\, S = S', \quad f\, \omega = \omega,$$ and in particular $f \in \mathrm{Sp}(V,\omega)=\mathrm{Sp}(V)$. If we decompose contorsion-like, curvature-like or torsion-like tensor spaces in $\mathrm{Sp}(V)$-irreducible submodules, then we get a necessary condition to be isomorphic as models, by virtue of Theorem \[Theorem AS3\], also as AS-manifolds.
Let $(M,\omega, {\nabla})$ be a Fedosov manifold [@GRS1998] of dimension $2n$ with homogeneous structure $S = {\nabla}-{\tilde{{\nabla}}}$ satisfying $${\tilde{{\nabla}}}R = 0, \quad {\tilde{{\nabla}}}S = 0, \quad {\tilde{{\nabla}}}\omega = 0.$$ Note that we drop condition ${\tilde{{\nabla}}}T=0$ as $\nabla$ is torsionless. Since $\nabla \omega =0$, the second condition is equivalent to $S \cdot \omega = 0$. We will work with $S$ both as a $(1,2)$-tensor and a $(0,3)$-tensor by the isomorphism $$S_{XYZ} = \omega(S_Z X, Y).$$ The condition $ S \cdot \omega = 0$ is equivalent to $$S_{XYZ} = S_{YXZ}$$ that is, $S\in S^2 T^*M \otimes T^*M$.
From Theorem \[Theorem AS1\], given $p_0\in M$, we can consider the infinitesimal model $(V = T_{p_0}M,{\tilde{R}}_{p_0}, {\tilde{T}}_{p_0})$ associated to $S_{p_0}$ and $\omega_{p_0}$ with $$\begin{aligned}
(T_{p_0})_{X}Y &= (\tilde{T}_{p_0})_{X}Y + (S_{p_0})_{X}Y -(S_{p_0})_{Y}X,\\
(R_{p_0})_{XY} &= (\tilde{R}_{p_0})_{XY} + [(S_{p_0})_Y, (S_{p_0})_X] + (S_{p_0})_{(S_{p_0})_{Y}X-(S_{p_0})_{X}Y},\end{aligned}$$ where $R_{p_0}$ and $T_{p_0}$ are the curvature and torsion of ${\nabla}$ in $p_0$ and $S_{p_0} \in S^2 V^* \otimes V^*$. We denote by $\mathcal{S}$ the set of homogeneous structures on a Fedosov manifold $(M, \omega, {\nabla})$. Hence, by Theorem \[Theorem IS1\], we have the following classification of homogeneous structure tensors in $\mathrm{Sp}(V)$-invariant subspaces: $$\mathcal{S} = \mathcal{S}_1 + \mathcal{S}_2 + \mathcal{S}_3,$$ where $$\begin{aligned}
\mathcal{S}_1&= \{ S \in \mathcal{S}:
\: S_{XYZ} = \omega_{ZY}\omega_{XU}+ \omega_{ZX}\omega_{YU}, U \in \mathfrak{X}(M) \},
\\
\mathcal{S}_2 &= \{ S \in \mathcal{S}:\:
{\underset{\text{\tiny{{$XYZ$}}}}{\text{\Large{$\mathfrak{S}$}}}} S_{XYZ} = 0,\, \mathrm{s}_{13}(S)=0\},
\\
\mathcal{S}_3 &= \{ S \in \mathcal{S} :\: S_{XYZ}= S_{XZY} \},\end{aligned}$$ and $$\mathrm{s}_{13}(S)(Z)=\sum _{i=1}^n \left(S_{e_i Z e_{i+n}}-S_{e_{i+n} Z e_{i}}\right),$$ for a symplectic basis $\{ e_1,\ldots,e_n,e_{n+1},\ldots, e_{2n}\}$ of $T_{p_0}M$.
\[Definition C3\] Let $S \in \mathcal{S}$ be an homogeneous Fedosov structure. It is of *type i* if $S$ lies in $\mathcal{S}_i$ and correspondingly is of *type i+j* if lies in $\mathcal{S}_i + \mathcal{S}_j$ with $i$, $j \in \{1,2,3\}$ and $i \neq j$.
Hence, Fedosov homogeneous structure are classified into eight different classes.
\[Remark C4\] In [@V1985] the author gives a decomposition of the curvature tensor of a symplectic connection in two $\mathrm{Sp}(V)$-irreducible submodules: Ricci type and Ricci flat. Hence, by virtue of and Theorem \[Theorem AS3\], there is as much as four different classes of symplectic curvature tensor of Fedosov AS-manifolds. We can combine this idea to refine the classification in Definition \[Definition C3\] to get as much as thirty two different classes of Fedosov AS-manifolds.
With respect to the classification of homogeneous structures in Definition \[Definition C3\] and the classification of torsions ${\tilde{T}}$ o AS-manifolds in Definition \[Definition C1\], we have the following result which is a consequence of the expression of $A_2$ in .
\[Proposition C5\] Let $(M,\omega ,\nabla )$ a Fedosov manifold equipped with homogenous structure $S$.
- If $S\in \mathcal{S}_1$, then the torsion ${\tilde{T}}$ of ${\tilde{{\nabla}}}=\nabla -S$ belongs to $\tilde{\mathcal{T}}_1$.
- If $S\in \mathcal{S}_2$, then the torsion ${\tilde{T}}$ of ${\tilde{{\nabla}}}=\nabla -S$ belongs to $\tilde{\mathcal{T}}_2$.
- If $S\in \mathcal{S}_3$, then the torsion ${\tilde{T}}$ of ${\tilde{{\nabla}}}=\nabla -S$ vanishes. The manifold $(M, \omega, {\tilde{{\nabla}}})$ is a Fedosov manifold with parallel curvature.
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V. F. Kiričenko. *On homogeneous Riemannian spaces with an invariant structure tensor*, Sov. Math., Dokl. **21** (1980), 734-737.
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abstract: |
We discuss the hydrodynamic radius $R_H$ of polymer chains in good solvent, and show that the leading order correction to the asymptotic law $R_H
\propto N^\nu$ ($N$ degree of polymerization, $\nu \approx 0.59$) is an “analytic” term of order $N^{-(1 - \nu)}$, which is directly related to the discretization of the chain into a finite number of beads. This result is further corroborated by exact calculations for Gaussian chains, and extensive numerical simulations of different models of good–solvent chains, where we find a value of $1.591 \pm 0.007$ for the asymptotic universal ratio $R_G / R_H$, $R_G$ being the chain’s gyration radius. For $\Theta$ chains the data apparently extrapolate to $R_G / R_H \approx 1.44$, which is different from the Gaussian value $1.5045$, but in accordance with previous simulations. We also show that the experimentally observed deviations of the initial decay rate in dynamic light scattering from the asymptotic Benmouna–Akcasu value can partly be understood by similar arguments.
author:
- |
Burkhard Dünweg[^1], Dirk Reith, Martin Steinhauser, and Kurt Kremer\
Max–Planck–Institut für Polymerforschung\
Ackermannweg 10, D–55128 Mainz, Germany
title: Corrections to Scaling in the Hydrodynamic Properties of Dilute Polymer Solutions
---
[2]{}
Introduction and Summary {#sec:intro}
========================
It is well–known that the average size $R$ of an isolated flexible uncharged polymer chain in good solvent is asymptotically proportional to $N^\nu$, where $N$ is the degree of polymerization, and $\nu \approx 0.5877$ [@limadrassokal]. This law holds for any measure of the chain size, the most popular of which are the mean square end–to–end distance, $$\label{eq:endenddistance}
\left< R_E^2 \right> = \left< r_{1 N}^2 \right> ,$$ the mean square radius of gyration, $$\label{eq:gyrationradius}
\left< R_G^2 \right> = \frac{1}{2 N^2} \sum_{ij} \left< r_{i j}^2 \right> ,$$ and the hydrodynamic radius $$\label{eq:hydrodynradius}
\left< \frac{1}{R_H} \right> = \frac{1}{N^2} \sum_{i \ne j}
\left< \frac{1}{r_{i j}} \right> .$$ In these equations, we have assumed that the chain is composed of $N$ monomers (i. e. $N - 1$ bonds) at positions $\vec r_i$, $i = 1, \ldots, N$, and $r_{i
j} = \left\vert \vec r_i - \vec r_j \right\vert$. Experimentally, the gyration radius is determined from small–angle scattering experiments; for small wave numbers $q$ the single–chain static structure factor behaves like [@doiedw] $$\begin{aligned}
\label{eq:structurefactor}
S(q) & = & \frac{1}{N} \sum_{i j}
\left< \exp \left[ i \vec q \cdot \vec r_{i j} \right] \right>
\nonumber \\
& = & N \left[ 1 - \frac{q^2}{3} \left< R_G^2 \right> + O(q^4) \right] . \end{aligned}$$ Conversely, the hydrodynamic radius is determined via small–angle dynamic light scattering experiments, where the dynamic structure factor $$\label{eq:dynstrucfac}
S(q,t) = \frac{1}{N}
\sum_{i j} \left< \exp\left[ i \vec q \cdot
\left( \vec r_i(t) - \vec r_j (0) \right) \right] \right>$$ is measured. In the small $q$ limit, it decays like $$\label{eq:decaydynstruc1}
\frac{S(q,t)}{S(q,0)} = \exp \left( - D q^2 t \right),$$ where $D$ is the chain diffusion constant, which, within the framework of Kirkwood–Zimm theory [@doiedw] is related to $R_H$ via $$\label{eq:kirkwoodformula}
D = \frac{D_0}{N} + \frac{k_B T}{6 \pi \eta} \left< \frac{1}{R_H} \right> ,$$ where $D_0$ is the monomer diffusion constant (usually this contribution is neglected), $k_B$ is Boltzmann’s constant, $T$ the temperature and $\eta$ the solvent viscosity. In principle, it is possible to obtain $R_H$, as a static quantity, also from purely static scattering, as is seen from the relation $$\label{eq:rhfromsofq}
\left< \frac{1}{R_H} \right> = \frac{2}{\pi N} \int_0^\infty dq
\left( S(q) - 1 \right),$$ but for technical reasons, this has so far not been applied in experiments.
When analyzing data for the chain size, one has to take into account that the law $R \propto N^\nu$ holds only in the asymptotic limit $N \to \infty$, while for finite chain lengths deviations occur. This is particularly important for computer simulations, where data with high statistical accuracy can be obtained. For this reason, corrections to scaling have been worked out in great detail, and exploited in high–resolution numerical studies, for the end–to–end distance and the gyration radius, where the relation $$\label{eq:knowncorrtoscaling}
\left< R_E^2 \right> = A N^{2 \nu}
\left( 1 + \frac{B}{N^\Delta} + \ldots \right)$$ (and analogously for $R_G$) holds [@sokal]. Here $A$ and $B$ are non–universal amplitudes, while $\Delta$ is a universal correction–to–scaling exponent, whose value is difficult to determine beyond the accuracy $\Delta \approx 0.5$ ($\Delta \approx 0.56$ according to Ref. , $\Delta \approx 0.43$ according to Ref. ). The omitted terms are further powers $N^{-\Delta - 1}$, $N^{-\Delta - 2}$, …, as well as $N^{-\Delta_2}$, $N^{-\Delta_2 - 1}$, …(i. e. there are further larger correction–to–scaling exponents), plus so–called “analytic” terms $N^{-1}$, $N^{-2}$, …[@sokal]. The important point to notice is that the “analytic” corrections will arise even for a Gaussian chain, and are due to the fact that the chain consists of a finite number of beads. This will be demonstrated explicitly in Sec. \[sec:hydradgaussian\]. Conversely, the “non–analytic” corrections are due to the fact that, in the language of renormalization group (RG) theory, the chain’s Hamiltonian is not identical to the fixed point Hamiltonian. The exponent $\Delta$ is related to the largest sub–leading eigenvalue of the RG transformation at the fixed point. In first order $\epsilon$ expansion its value is [@schaefer] $\Delta = \nu \omega$ with $1 / \nu = 2 - \epsilon /
4 + O(\epsilon^2)$ and $\omega = \epsilon + O (\epsilon^2)$, where $\epsilon =
4 - d$ and $d$ is the spatial dimension. Higher–order calculations [@zinnjustin] have resulted in $\Delta = \nu \omega = 0.588 \times 0.82 =
0.482$. We adopt here the convention (which we view as quite natural) to distinguish the terms by their different origins, and call correction terms “analytic” corrections if they are present even in the Gaussian limit, while we call “non–analytic” corrections those terms which occur exclusively for excluded–volume chains. As we will see in Sec. \[sec:hydradgoodsolv\], “analytic” corrections defined in this way do not necessarily imply integer powers of $N$.
As for the hydrodynamic radius of self–avoiding walks (SAWs), there is no high–resolution numerical study available, and corrections to scaling have not yet been dealt with systematically. This is somewhat unfortunate, since the corrections are unusually large for $R_H$, and of experimental relevance. For a good solvent chain, one expects again $N^{-\Delta}$ etc. terms, plus “analytic” corrections. It is the main purpose of the present paper to show that the leading–order term of these latter corrections is now given by $$\label{eq:analcorrhydrad}
\left< \frac{1}{R_H} \right> = \frac{A}{N^\nu}
\left( 1 - \frac{B}{N^{1 - \nu}} + \ldots \right) ,$$ where $B$ is usually positive. We will show in Secs. \[sec:hydradgoodsolv\] and \[sec:altderiv\] that this form is a straightforward consequence of discretizing the chain into beads. As $1 - \nu \approx 0.41$, this will, for long chains, ultimately dominate over the $N^{-\Delta}$ term, where the exponent is (according to Refs. , and ) slightly larger. Nevertheless, the exponents are so close that in most cases one will observe contributions from both terms. On the other hand, it is a well–known empirical fact that for many experimental systems, as well as for most computer models, the corrections to scaling of the gyration radius are quite weak, such that the $N^{-\Delta}$ term should have a rather small amplitude. One could therefore expect that the corresponding amplitude of the $N^{-\Delta}$ contribution in $R_H$ is also quite small. Then the most likely candidate for explaining the experimental and numerical observation that $R_H$ is usually subject to very large corrections to scaling [@schaefer] would actually be the “analytic” $N^{- (1 - \nu)}$ term.
For a [*Gaussian*]{} chain we are able to solve the problem exactly, see Sec. \[sec:hydradgaussian\]: $$\label{eq:gaussmainz}
\left< \frac{b}{R_H} \right> =
\frac{8}{3} \left( \frac{6}{\pi} \right)^{1/2} N^{-1/2}
\left( 1 - B N^{-1/2} + \ldots \right) ,$$ with $B = - (3/4) \zeta(1/2) \approx 1.095266$ (here $\zeta$ is Riemann’s zeta function), and $b$ denoting the root mean square bond length. As $\nu = 1/2$ for a Gaussian chain, this form is consistent with Eq. \[eq:analcorrhydrad\].
The difficulties in observing the asymptotic $N^\nu$ scaling of $R_H$ have a long history. Adam and Delsanti [@adamdelsanti1; @adamdelsanti2] performed dynamic light scattering experiments and found an effective power law $R_H
\propto N^{0.55}$. This is quite typical, and has been found in many other experiments, too [@nemoto; @venkataswamy; @simone], although an exponent of $0.61$ [@tsunashima1] has been reported as well. A reduction of the effective exponent is indeed expected, as seen from Eq. \[eq:analcorrhydrad\], and is also observed in Brownian Dynamics simulations [@depablo]. As a caveat, note that a scattering experiment does not measure $R_H$, but rather the diffusion constant $D$. This quantity has an additional $D_0 / N$ contribution (Eq. \[eq:kirkwoodformula\]), which is of the same order as the leading correction of Eq. \[eq:analcorrhydrad\]. Therefore, the corrections in $D$ are weaker than those in $R_H$. Nevertheless, the $D_0 / N$ term is typically not large enough to fully compensate the corrections in $R_H$. This is easily seen for the Gaussian case from Eqs. \[eq:kirkwoodformula\] and \[eq:gaussmainz\]: The monomer diffusion constant $D_0$ can be written as $D_0 = k_B T / (6 \pi \eta a)$, which defines a monomer Stokes radius $a$. Thus $$\begin{aligned}
\frac{D}{D_0} & = & N^{-1} \\
& + & \frac{a}{b} \left( 3.6853 N^{-1/2}
- 4.0364 N^{-1} + \ldots \right) . \nonumber\end{aligned}$$ Since $a$ should be of the order of the bond length $b$, one sees that a large $N^{-1}$ contribution remains.
A first attempt to explain the experimental observation is due to Weill and des Cloizeaux [@weilldescloizeaux]. They conjectured that $\nu_{eff} =
0.55$ is due to non–perfect solvent quality, and a crossover between good solvent behavior ($\nu \approx 0.6$, large length scales) and $\Theta$ solvent ($\nu = 0.5$, small length scales). In particular, they pointed out that the averaging over $1 / r$ assigns a very large statistical weight to the small distances. Although this latter argument is true, and generally accepted as the basic origin for the slow convergence of $R_H$, the explanation in terms of solvent quality has turned out to be incorrect. In Ref. it was demonstrated that $R_H$ should [ *not*]{} be much more susceptible to solvent quality effects than $R_E$ or $R_G$ — the enhanced sensitivity of $R_H$ to the small distances is balanced by the fact that $R_E$ and $R_G$ are more sensitive to the decreased swelling of the chain near its ends: A SAW is [*inhomogeneous*]{}, i. e. $\left< r_{ij}^2
\right>$ depends on the position of the $ij$ bond on the chain, and is systematically larger in the interior, as has been shown both numerically [@schaeferbaumgaertner; @kkdipl] and analytically [@schaefer; @schaeferbaumgaertner].
Furthermore, Schäfer and Baumgärtner [@schaeferbaumgaertner] performed a detailed RG calculation and predicted in one–loop order for the universal amplitude ratio $$\begin{aligned}
\label{eq:rgrhoneloop}
\rho_\infty & = & \lim_{N \to \infty} \rho(N)
= \lim_{N \to \infty} \frac{R_G(N)}{R_H(N)} \nonumber \\
& \approx & 1.06 \times \frac{8}{3 \sqrt{\pi}}
\approx 1.06 \times 1.5045 \approx 1.595 ;\end{aligned}$$ here $8 / (3 \sqrt{\pi})$ is the exact random walk (RW) value. Other RG studies resulted in $\rho_\infty = 1.562$ [@oonokohmoto] (this value was later revised to 1.51, see Ref. ) and $\rho_\infty =
1.62$ [@douglasfreed], while a semi–empirical relation based on fitting the distribution function of internal distances to light scattering data yields $\rho_\infty = 1.5955$ [@tsunashima1; @tsunanew]. A value of $\rho_\infty \approx 1.6$ was also found in Brownian Dynamics simulations [@depablo]. [From]{} Eq. \[eq:analcorrhydrad\] it is clear that $\rho$ should be subject to an $N^{-(1 - \nu)}$ correction for finite chain length; nevertheless, experiments have so far not reported a systematic dependence on molecular weight. Typically, values around $\rho \approx 1.5$ [@simone; @tsunashima1], or $\rho \approx 1.6$ / $\rho \approx 1.3$ for different solvents [@venkataswamy] are found in the good–solvent regime. In view of the inaccuracies of the experiments ($R_G$ typically has an error of $5 \%$ [@venkataswamy]) the inability to observe a systematic behavior in $N$ is not very surprising.
In order to contribute to the resolution of these questions, we have performed computer simulations of very different models of polymer chains, both for SAWs and for $\Theta$ chains, and calculated $R_G$ and $R_H$, as outlined in Sec. \[sec:numres\]. To provide a complete and well–converged data set represents the second main goal of our paper. To our knowledge, our results are the most accurate data obtained on $R_H$ so far. Concerning $R_G$, however, our data are less accurate than those of Li [*et al.*]{} [@limadrassokal], which we still view as the most precise numerical study on the SAW problem so far. Therefore we have taken their values for the exponents $\nu$ and $\Delta$ for our fits. We find $\rho_\infty = 1.591
\pm 0.007$ for good–solvent chains, in very good agreement with Ref. . (Note that our error estimate is probably overly optimistic, since it only includes statistical errors and completely neglects systematic errors.)
Theoretical and numerical investigations on corrections to scaling in $R_H$ have first focused on the RW case. The work by Guttman [*et al.*]{} [@guttman1; @guttman2; @akcasnew] showed by analytical calculation that a Gaussian chain should obey Eq. \[eq:gaussmainz\]. The prefactor of the correction was first [@guttman1] determined only approximately, $B \approx
1.125$, while later [@akcasnew] it was given exactly in terms of an integral. Furthermore, Monte Carlo (MC) simulations of lattice chains at the $\Theta$ point revealed that in this case the ratio $R_G / R_H$ apparently does [*not*]{} converge to its Gaussian value $1.5045$, but rather to roughly $1.4$. Our simulations (see Sec. \[sec:numres\]) find a similar behavior ($\rho_\infty \approx 1.44$). We believe that this can be explained qualitatively from RG arguments [@schaefer] as follows: The asymptotic behavior is expected to be governed by the Gaussian fixed point, and thus $\rho_\infty$, as a universal amplitude ratio, is expected to assume the Gaussian value. However, the numerical extrapolation will only produce this value if all relevant correction terms, i. e. the analytic $N^{-1/2}$ term, plus the non–analytic corrections, are consistently taken into account. Neither the data analysis by Guttman [*et al.*]{} [@guttman1; @guttman2], nor ours, fulfill this requirement, as both just fit to Eq. \[eq:analcorrhydrad\] with $\nu = 1/2$, and thus are expected to produce substantial systematic errors in $\rho_\infty$. To do this in a better way is practically impossible, since (i) our $\Theta$ data have insufficient statistical accuracy to allow for additional fit parameters, (ii) the precise form of the correction terms is unknown for $R_H$ (in contrast to $R_E$ and $R_G$, for which the leading–order terms have been calculated by tricritical field theory [@hager], with the interesting feature that they are universal), and (iii) the non–analytic corrections vary extremely slowly (logarithmically) with $N$, such that either one would need unrealistically long chains to ensure dominance of the leading orders, or an expansion up to unrealistically high order. These problems have been elucidated in quite some detail for $R_E$ and $R_G$ [@hager], explaining previous difficulties in the interpretation of highly accurate MC data on $\Theta$ chains [@grassberger]. In this context, it should be mentioned that experiments [@schmidtburchard; @tsunashima2] typically find a value of $\rho = 1.3$, i. e. a similar reduction as in the good solvent case.
Later, MC data were taken of excluded–volume (EV) chains with SAW statistics. Schäfer and Baumgärtner [@schaeferbaumgaertner] used chains of up to $161$ monomers, with an EV strength particularly close to the SAW fixed point, such that poor–solvent effects can be ruled out. The inhomogeneous swelling was demonstrated, and the $R_H$ data were fitted with Eq. \[eq:analcorrhydrad\]. This was done with an empirical correction–to–scaling exponent of $1/2$ instead of $1 - \nu$. The same evidence was shown in the simulation data by Batoulis and Kremer [@batouliskremer] of chains of length of up to $N \approx 400$. Ladd and Frenkel [@laddfrenkel] simulated chains of length of up to $N = 1025$ and were able to describe their $R_H$ data via Eq. \[eq:analcorrhydrad\], with $A = 3.84$ and $B = 1.06$, but without detailed justification of their use of the correct $1 - \nu$ exponent. Schäfer and Baumgärtner [@schaeferbaumgaertner] concluded from both their analytical studies and their simulation data that not the solvent quality, but rather the chain’s microstructure is responsible for the slow convergence. Our reasoning (Secs. \[sec:hydradgaussian\]–\[sec:altderiv\]), which is similar to the one by Guttman [*et al.*]{} [@guttman1; @guttman2; @akcasnew], exactly supports this picture: The corrections are due to the fact that the chain is discretized into beads, or, in other words, to the fact that there is a lower length scale cutoff for the frictional properties. However, the notion of “stiffness”, which is often used in this context [@schaeferbaumgaertner], is, in our view, somewhat misleading: As outlined in Sec. \[sec:stiffness\], we expect a large local chain stiffness to [*decrease*]{} the correction until it ultimately even changes its sign. The same conclusion has been found by Akcasu and Guttman [@akcasnew] for stiff chains without excluded volume.
In the context of dynamic light scattering of dilute polymer solutions there is yet another unresolved puzzle. As Akcasu [*et al.*]{} have shown [@akcasugurol], the initial decay rate of the dynamic structure factor, $$\label{eq:omegaofqdef}
\Omega (q) = \left. \frac{d}{dt} \frac{S(q,t)}{S(q,0)} \right\vert_{t = 0} ,$$ can be written as $$\label{eq:omegaofqakcasu}
\Omega (q) = \frac{\sum_{ij}
\left< \vec q \cdot \tensor D_{ij} \cdot \vec q
\exp( i \vec q \cdot \vec r_{ij} ) \right> }{
\sum_{ij} \left<
\exp( i \vec q \cdot \vec r_{ij} ) \right> } ,$$ where $\tensor D_{ij}$ is the diffusion tensor. Equation \[eq:omegaofqakcasu\] is a rigorous result, the only assumption being that the chain dynamics can be described by Kirkwood’s diffusion equation [@doiedw]. Usually, $\tensor D_{ij}$ is taken as the Oseen tensor, $$\label{eq:oseendef}
\tensor D_{ij} = D_0 \delta_{ij} \tensor 1 + (1 - \delta_{ij})
\frac{k_B T}{8 \pi \eta r_{ij}}
( \tensor 1 + \hat r_{ij} \otimes \hat r_{ij} ) ,$$ where $\hat r_{ij} \otimes \hat r_{ij}$ denotes the tensor product of the unit vector in $\vec r_{ij}$ direction with itself. In this case, Eq. \[eq:omegaofqakcasu\] is just the $q > 0$ generalization of Eq. \[eq:kirkwoodformula\]. It can then be shown [@doiedw; @benmounaakcasu] that for $q$ in the scaling regime $R_G^{-1} \ll q \ll b^{-1}$ ($b$ denoting the bond length), or, strictly spoken, in the limit $q b \to 0$, $q R_G \to \infty$, the relation $$\label{eq:plateau}
\Omega (q) = C \frac{k_B T}{\eta} q^3$$ holds, where the numerical constant $C$ only depends on chain statistics: $C =
0.0625$ for RW statistics ($\nu = 1/2$) and $C = 0.0788$ for SAWs ($\nu =
0.6$). This has been tested by light scattering experiments both for good solvents [@nemoto; @tsunashima1; @hanakcasu; @bhatt1; @bhatt2] and for $\Theta$ solvents [@tsunashima2; @hanakcasu]. In both cases the relation is verified with reasonable accuracy, but with a prefactor $C$ which is systematically smaller than the theoretical prediction. The reasons for this shift are not clear; an attempt by a generalized theory which introduces draining [@shiwa] so far had only limited success [@tsunashima3]. In Sec. \[sec:initdecay\] we show that the deviation can partly be explained by the fact that in reality neither $q b = 0$ nor $q R_G = \infty$ holds. Taking these nonidealities crudely into account, we find a shift in the same direction, which is however smaller than the experimental one. Nevertheless, we believe that this third main result is of direct relevance for the analysis of experimental data. There are also some indications from Molecular Dynamics simulations [@bdphd] that the description in terms of the Kirkwood theory is insufficient on these length and time scales.
Analytical Theory {#sec:theory}
=================
Hydrodynamic Radius of a Gaussian Chain {#sec:hydradgaussian}
---------------------------------------
For a Gaussian chain with root mean square bond length $b$, we have $$\left< r_{ij}^2 \right> = b^2 \left\vert i - j \right\vert$$ and $$\left< r_{ij}^{-1} \right> = 6^{1/2} \pi^{-1/2}
b^{-1} \left\vert i - j \right\vert^{-1/2} ,$$ and hence $$\left< R_E^2 \right> = b^2 \left(N - 1 \right)
= b^2 N \left(1 - \frac{1}{N} \right)$$ and $$\begin{aligned}
\left< R_G^2 \right> & = & \frac{b^2}{N^2} \sum_{i < j} (j - i)
= \frac{b^2}{N^2} \sum_{n = 1}^{N - 1} n (N - n) \nonumber \\
& = & \frac{1}{6} b^2 N \left( 1 - \frac{1}{N^2} \right) ,\end{aligned}$$ where we have used elementary summation formulae. For the hydrodynamic radius, we find analogously $$\left< R_H^{-1} \right> = \sqrt{ \frac{6}{\pi} }
\frac{2}{b N^2} \sum_{n = 1}^{N - 1} n^{-1/2} (N - n) .$$ According to the Euler–Maclaurin formula (see Appendix \[sec:euler\]), Eq. \[eq:eulermaclaurinpower\], the sums can be expanded as $$\begin{aligned}
\sum_{n = 1}^{N - 1} n^{-1/2} & = &
2 N^{1/2} - \frac{1}{2} N^{-1/2}
+ \zeta\left(\frac{1}{2}\right)
+ O(N^{-3/2}) , \nonumber \\
\sum_{n = 1}^{N - 1} n^{+1/2} & = &
\frac{2}{3} N^{3/2} - \frac{1}{2} N^{1/2}
\nonumber \\
& + & \zeta\left(- \frac{1}{2}\right)
+ O(N^{-1/2}) .\end{aligned}$$ Hence, $$\sum_{n = 1}^{N - 1} n^{-1/2} (N - n) =
\frac{4}{3} N^{3/2} + N \zeta \left( \frac{1}{2} \right) + O(N^0)$$ and $$\begin{aligned}
\left< R_H^{-1} \right> & = & \sqrt{ \frac{6}{\pi} }
\frac{8}{3 b} N^{-1/2} \times \nonumber \\
& & \left( 1 + \frac{3}{4} \zeta \left( \frac{1}{2} \right) N^{-1/2}
+ O(N^{-3/2}) \right) ,\end{aligned}$$ which is the result anticipated in Eq. \[eq:gaussmainz\].
Hydrodynamic Radius of a Good Solvent Chain {#sec:hydradgoodsolv}
-------------------------------------------
For a linear SAW, the main difficulty is the fact that, unlike for a RW, $\left< r_{ij}^2 \right>$ and $\left< r_{ij}^{-1} \right>$ do [*not*]{} [@schaefer; @schaeferbaumgaertner; @kkdipl] just depend on $\left\vert i - j
\right\vert$, but rather on the positions relative to the ends as well. In order to obtain the leading–order analytic corrections due to discretization, we can restrict the discussion to the leading–order scale–invariant behavior, i. e. we can assume that the SAW is strictly scale invariant with the exponent $\nu$, with no non–analytic corrections. If we would include the latter, they would just generate further additive terms in our expressions. In what follows, we therefore omit them, for the sake of simplified notation, but keep in mind that they have to be added at the end in order to obtain the full expressions. We thus assume the relations $$\begin{aligned}
\phi_G \left(\lambda x, \lambda y \right) & = & \lambda^{2 \nu}
\phi_G \left( x, y \right) , \\
\phi_H \left(\lambda x, \lambda y \right) & = & \lambda^{- \nu}
\phi_H \left( x, y \right) ,\end{aligned}$$ where we have introduced the notation $$\begin{aligned}
\phi_G \left(i, j \right) & = & \left< r_{ij}^2 \right> , \\
\phi_H \left(i, j \right) & = & \left< r_{ij}^{-1} \right> .\end{aligned}$$ The definitions of $R_G$ and $R_H$ lead us to study the sum $$\sigma (N) = \sum_{n = 2}^N \sum_{m = 1}^{n - 1} \phi( m, n )$$ for $\phi = \phi_G$ and $\phi = \phi_H$, respectively, by means of the Euler–Maclaurin expansion of Appendix \[sec:euler\]. Treating the inner sum first, we find $$\label{eq:eulersaw1}
\sum_{m = 1}^{n - 1} \phi(m, n) = \mbox{\rm const.} + \varphi(n)$$ with the formal expansion $$\label{eq:eulersaw2}
\varphi(n) = \int_1^n dx \phi(x, n)
+ \frac{1}{12} \left. \frac{d}{dx} \phi (x, n) \right\vert_{x = n}
+ \ldots ,$$ since $\phi(n,n)$ vanishes. Note also that the constant in Eq. \[eq:eulersaw1\] does not depend on $n$, hence $$\begin{aligned}
\sigma (N) & = & (N - 1) \mbox{\rm const.} +
\sum_{n = 2}^N \varphi(n) \nonumber \\ & = &
(N - 1) \mbox{\rm const.} +
\int_2^{N + 1} dy \varphi (y) \nonumber \\
& - & \frac{1}{2} \varphi(N + 1)
+ \mbox{\rm const.} + \ldots .\end{aligned}$$ Inserting Eq. \[eq:eulersaw2\], we find
$$\begin{aligned}
\sigma (N)
& = & \int_2^{N + 1} dy \int_1^y dx \phi(x, y) \nonumber \\
& + & \int_2^{N + 1} dy
\frac{1}{12} \left. \frac{d}{dx} \phi (x, y) \right\vert_{x = y}
\nonumber \\
& - & \frac{1}{2} \int_1^{N + 1} dx \phi(x, N + 1) \nonumber \\
& + & N \mbox{\rm const.} \nonumber \\
& + & \mbox{\rm const.} + \ldots .\end{aligned}$$
After transformation to the reduced variables $u = x / N$ and $v = y / N$, and exploiting the scaling behavior of $\phi$, it is possible to determine the order of each term. For the gyration radius, we find $O(N^{2 + 2\nu})$, $O(N^{2 \nu})$, $O(N^{1 + 2\nu})$, $O(N^{1})$, $O(N^{0})$, respectively, for the five terms in order of their appearance. Conversely, for $\left< R_H^{-1} \right>$ the different scaling behavior of $\phi$ implies $O(N^{2 - \nu})$, $O(N^{- \nu})$, $O(N^{1 - \nu})$, $O(N^{1})$, $O(N^{0})$ for the corresponding orders. For $\left< R_G^2 \right>$ the leading order is $O(N^{2 + 2\nu})$, while the next sub–leading order is $O(N^{1 + 2\nu})$, resulting in a leading order correction of $O(N^{-1})$. For $\left< R_H^{-1}
\right>$, the leading order is $O(N^{2 - \nu})$, followed by the $O(N^{1})$ term. Thus the correction to scaling for $R_G$ has order $O(N^{-(1 - \nu)})$. This proves Eq. \[eq:analcorrhydrad\]. Of course, this consideration does not prove that the amplitude $B$ in Eq. \[eq:analcorrhydrad\] is positive; however, this is expected from the result for Gaussian chains.
Alternative Derivation {#sec:altderiv}
----------------------
Equation \[eq:analcorrhydrad\] can also be derived in a more heuristic way, starting from Eq. \[eq:rhfromsofq\]. Figure \[fig:sofq\] shows the typical behavior of $S(q)$: For wave numbers $q$ with $R_G^{-1} \ll q \ll b^{-1}$ the structure factor exhibits a power–law decay $q^{-1/\nu}$ which indicates the chain’s fractal geometry, while for larger $q$ it oscillates around unity. We therefore can introduce a cutoff wavenumber $q_0$ from which on there is no further contribution to the integral, i. e. $q_0$ is the smallest of all the $\hat q$’s with the property $\int_{\hat q}^\infty dq (S(q) - 1) = 0$. Hence, $$\frac{1}{R_H} =
\frac{2}{\pi N} \int_0^{q_0} dq S(q) - \frac{2 q_0}{\pi N} .$$ It is physically clear that for a flexible chain $q_0$ must be roughly $(2
\pi) / b$, apart from a numerical prefactor of order unity. Moreover, the fractal $q^{-1/\nu}$ decay of $S(q)$ roughly extends up to $q_0$, at which point $S(q) \approx 1$ is reached. We now introduce a modified structure factor $\tilde S (q)$, which is identical to $S(q)$ up to $q = q_0$, but extends the $q^{-1 / \nu}$ decay up to $q = \infty$. In this latter regime, we have $$\tilde S(q) = \alpha \left( \frac{q}{q_0} \right)^{-1/\nu} ,$$ where $\alpha$ is a numerical prefactor of order unity. Therefore we can write $$\frac{1}{R_H} =
\frac{2}{\pi N} \int_0^{\infty} dq \tilde S(q)
- \frac{2}{\pi N} \int_{q_0}^\infty dq \tilde S(q)
- \frac{2 q_0}{\pi N} .$$ Evaluating the second integral, and writing $\tilde S (q)$ in scaling form, $$\label{eq:strufacscalingform}
\tilde S(q) = N s(q R_G)$$ (here we have again assumed strict scale invariance, i. e. absence of non–analytic corrections to scaling, for the same reason as outlined at the beginning of the previous subsection), one finds $$\frac{R_G}{R_H} = \frac{2}{\pi} \int_0^{\infty} dx s(x)
- \frac{2}{\pi} \left( \alpha \frac{\nu}{1 - \nu} + 1 \right)
\frac{q_0 R_G}{N} ,$$ i. e. again a negative correction of order $O(N^{-(1 - \nu)})$.
Effect of Chain Stiffness {#sec:stiffness}
-------------------------
The advantage of the approach of the previous subsection is that it can be easily generalized to study the influence of local structure, since it is well–known how this is reflected in $S(q)$. For a locally stiff chain with a persistence length large compared to the bond length $b$, one expects that $q_0$ is roughly unchanged with respect to the flexible case. However, the $q^{-1 / \nu}$ decay does no longer extend down to $q \approx q_0$, but only to $q \approx q_1$, where $q_1$ is a crossover wave number, whose inverse is a typical length scale below which stiffness effects are important. With $\tilde
S (q)$ being again the continuation of the $q^{-1 / \nu}$ decay up to $q =
\infty$, we have $$\begin{aligned}
\label{eq:splittheintegral}
\frac{1}{R_H} & = & \frac{2}{\pi N} \int_0^\infty dq \tilde S (q)
- \frac{2}{\pi N} \int_{q_1}^\infty dq \tilde S (q)
\nonumber \\
& + & \frac{2}{\pi N} \int_{q_1}^{q_0} dq S (q)
- \frac{2 q_0}{\pi N} .\end{aligned}$$ We now assume $$\tilde S(q) = \alpha \frac{q_0}{q_1}
\left( \frac{q}{q_1} \right)^{-1 / \nu}$$ for $q > q_1$, and $$S(q) = \beta \left( \frac{q}{q_0} \right)^{-1}$$ for $q_1 < q < q_0$. Here, $\alpha$ and $\beta$ denote prefactors of order unity, and the $q^{-1}$ decay results from the local stretching. Evaluating the integrals, and using Eq. \[eq:strufacscalingform\], one thus finds $$\begin{aligned}
\frac{R_G}{R_H} & = & \frac{2}{\pi} \int_0^{\infty} dx s(x) \\
&& - \frac{2}{\pi} \left( \alpha \frac{\nu}{1 - \nu} + 1
- \beta \ln \frac{q_0}{q_1} \right) \frac{q_0 R_G}{N} . \nonumber\end{aligned}$$ In order to compare with the flexible case, we still have to take into account that stiffness tends to increase the gyration radius, by roughly a factor of $(q_0 / q_1)^{1 - \nu}$: $$\begin{aligned}
\frac{R_G}{R_H} & = & \frac{2}{\pi} \int_0^{\infty} dx s(x) \\
&& - \frac{2}{\pi} \left( \frac{q_0}{q_1} \right)^{1 - \nu}
\left( \alpha \frac{\nu}{1 - \nu} + 1
- \beta \ln \frac{q_0}{q_1} \right) \frac{q_0 R_G^{(0)}}{N} ,
\nonumber\end{aligned}$$ where $R_G^{(0)}$ denotes the gyration radius in the flexible case.
The prefactor of the correction term hence depends on the stiffness parameter $q_0 / q_1$ in a non–trivial way; for small $q_0 / q_1$ both an increase and a decrease are possible, depending on the parameters. For sufficiently large stiffness one always obtains a decrease of the correction, and ultimately even a change of its sign.
Initial Decay Rate {#sec:initdecay}
------------------
In this subsection, we are concerned with the initial decay rate $\Omega (q)$, see Eq. \[eq:omegaofqakcasu\]. Splitting the sum in the numerator into diagonal and off–diagonal terms, one finds $$\begin{aligned}
\Omega (q) & = & \frac{D_0 q^2}{S(q)} \\
& + & \frac{1}{N S(q)}
\sum_{i \ne j}
\left< \vec q \cdot \tensor D_{ij} \cdot \vec q
\exp( i \vec q \cdot \vec r_{ij} ) \right> . \nonumber\end{aligned}$$ Following Refs. , we use the Fourier representation of the Oseen tensor for the off–diagonal elements, $$\tensor D_{ij} = \frac{k_B T}{\eta}
\frac{1}{(2 \pi)^3} \int d^3 k
\frac{\tensor 1 - \hat k \otimes \hat k}{k^2}
\exp(i \vec k \cdot \vec r_{ij}) ,$$ to find $$\begin{aligned}
\Omega (q) & = & \frac{D_0 q^2}{S(q)} + \frac{1}{S(q)} \frac{k_B T}{\eta}
\frac{1}{(2 \pi)^3} \int d^3 k \nonumber \\
& & \times \frac{q^2 - (\hat k \cdot \vec q)^2}{k^2}
\left( S(\vec k + \vec q) - 1 \right) .\end{aligned}$$ We now focus attention on the dimensionless quantity $$\begin{aligned}
C(q) & = & \frac{\eta}{q^3 k_B T} \Omega(q) \nonumber \\
& = & \frac{1}{6 \pi q a S(q)} +
\frac{1}{S(q)} \frac{1}{(2 \pi)^3} \int d^3 k \nonumber \\
&& \times \frac{1 - (\hat k \cdot \hat q)^2}{q k^2}
\left( S(\vec k + \vec q) - 1 \right) ,\end{aligned}$$ where we again have expressed the monomer diffusion constant $D_0$ in terms of a Stokes radius $a$. After transforming to the dimensionless integration variable $$\vec x = \frac{ \vec k + \vec q }{q}$$ and performing the angular integration, one has [@doiedw; @hammouda] $$\begin{aligned}
C(q) & = & \frac{1}{6 \pi q a S(q)} \\
&& + \frac{1}{S(q)} \frac{1}{(2 \pi)^2}
\int_0^\infty dx f(x) \left( S(q x) - 1 \right)
\nonumber\end{aligned}$$ with $$f(x) = x^2 \left( \frac{1 + x^2}{2 x}
\ln \left\vert \frac{1 + x}{1 - x} \right\vert - 1 \right) .$$ This function can be expanded as $$f(x) = \sum_{n = 0}^\infty \left(
\frac{1}{2 n + 1} + \frac{1}{2 n + 3} \right) x^{2 n + 4}$$ for $x < 1$, and $$f(x) = \sum_{n = 0}^\infty \left(
\frac{1}{2 n + 1} + \frac{1}{2 n + 3} \right) x^{- 2 n}$$ for $x > 1$.
In order to make further progress, we have to specify the structure factor $S(q)$. This shall be done by the most simplistic model which takes into account both finite bead size and finite chain length (see also Fig. \[fig:sofq\]): $$\label{eq:simplemodelstructurefactor}
S(q) = \left\{
\begin{array}{l l}
N & \hspace{0.5cm}
q < \frac{2 \pi}{a} N^{- \nu} \\
\left( \frac{q a}{2 \pi} \right)^{-1 / \nu} & \hspace{0.5cm}
\frac{2 \pi}{a} N^{- \nu}
< q < \frac{2 \pi}{a} \\
1 & \hspace{0.5cm}
q > \frac{2 \pi}{a}
\end{array}
\right. .$$ We now calculate $C(q)$ in the scaling regime $R_G^{-1} \ll q \ll a^{-1}$. Defining the $x$ values where $S(q x)$ changes its behavior as $$\begin{aligned}
x_1 & = & \frac{2 \pi}{q a N^\nu} \ll 1 , \\
x_2 & = & \frac{2 \pi}{q a} \gg 1 ,\end{aligned}$$ we can write $(q a)^{-1} = x_2 / (2 \pi)$, $S(q)^{-1} = x_2^{-1 / \nu}$, $N / S(q) = x_1^{- 1 / \nu}$; hence $$\begin{aligned}
C(q) & = & \frac{1}{12 \pi^2} x_2^{1 - 1 / \nu} \nonumber \\
& + & \frac{1}{(2 \pi)^2} x_1^{- 1 / \nu} \int_0^{x_1} dx f(x)
\nonumber \\
& + & \frac{1}{(2 \pi)^2} \int_{x_1}^{x_2} dx f(x) x^{-1 / \nu}
\nonumber \\
& - & \frac{1}{(2 \pi)^2} x_2^{-1 / \nu} \int_0^{x_2} dx f(x) . \end{aligned}$$ Since $x_1 \ll 1$ and $x_2 \gg 1$, we can write $$\begin{aligned}
\int_0^{x_1} dx f(x) & \approx & \frac{4}{15} x_1^5, \\
\int_0^{x_2} dx f(x) & \approx & \frac{4}{3} x_2,\end{aligned}$$ where we have taken just the leading–order terms of the expansions of $f$; this results in $$\begin{aligned}
C(q) & \approx & \frac{1}{12 \pi^2} x_2^{1 - 1 / \nu} \nonumber \\
& + & \frac{1}{ 15 \pi^2} x_1^{5 - 1 / \nu}
\nonumber \\
& + & \frac{1}{(2 \pi)^2} \int_{x_1}^{x_2} dx f(x) x^{-1 / \nu}
\nonumber \\
& - & \frac{1}{3 \pi^2} x_2^{1 -1 / \nu} . \end{aligned}$$ In the asymptotic limit $q R_G \to \infty$, i. e. $x_1 \to 0$, and $q a \to 0$, i. e. $x_2 \to \infty$, this obviously converges to the asymptotic value $$\begin{aligned}
C_{as} & = & \frac{1}{(2 \pi)^2} \int_0^\infty dx f(x) x^{- 1 / \nu}
\nonumber \\
& = & \left\{
\begin{array}{l l l}
1 / 16 & = 0.0625 & \hspace{0.5cm} \nu = 1 / 2 \\
\sqrt{3} / (7 \pi) & \approx 0.0788 & \hspace{0.5cm} \nu = 3 / 5
\end{array}
\right. .\end{aligned}$$ Focusing now on the correction, i. e. $\Delta C(q) = C(q) - C_{as}$, we find $$\begin{aligned}
\Delta C(q) & \approx & \frac{1}{12 \pi^2} x_2^{1 - 1 / \nu} \nonumber \\
& + & \frac{1}{ 15 \pi^2} x_1^{5 - 1 / \nu}
\nonumber \\
& - & \frac{1}{(2 \pi)^2} \int_{0}^{x_1} dx f(x) x^{-1 / \nu}
\nonumber \\
& - & \frac{1}{(2 \pi)^2} \int_{x_2}^{\infty} dx f(x) x^{-1 / \nu}
\nonumber \\
& - & \frac{1}{3 \pi^2} x_2^{1 - 1 / \nu} ;\end{aligned}$$ taking again the leading–order terms for the remaining integrals results in $$\begin{aligned}
\label{eq:shiftresult}
\Delta C(q) & \approx & - \frac{1}{12 \pi^2} \frac{3 + \nu}{1 - \nu}
x_2^{1 - 1 / \nu} \nonumber \\
&& - \frac{1}{15 \pi^2} \frac{1}{5 \nu - 1}
x_1^{5 - 1/ \nu} .\end{aligned}$$ One thus sees that [*both*]{} finite chain length [*and*]{} finite bead size have the tendency to decrease $C$, as observed in the experiments. The latter effect is clearly more important, as $x_1$ enters only via a relatively high power. Further insight is gained by numerical evaluation of the shift for reasonable parameter values.
Tsunashima [*et al.*]{} [@tsunashima1] performed their experiments with polyisoprene chains of size $R_G = 210 nm$. Typical scattering wavenumbers in their plateau regime were given by $q R_G = 4 \ldots 8$; the experimental observation in this regime was $C \approx 0.06$, i. e. a shift of $\Delta C
\approx - 2 \times 10^{-2}$. In what follows, we consider the value $q R_G
\approx 6$. Thus $$x_2 = \frac{2 \pi}{q R_G} \frac{R_G}{a} \approx 500,$$ where we have estimated the monomer size $a$ as $0.45 nm$ [@tsunaprivate]. Inserting this into Eq. \[eq:shiftresult\], we find for the $x_2$ contribution a value of $\Delta C \approx - 1 \times 10^{- 3}$, i. e. one order of magnitude smaller than the experimental value.
It is not completely clear if a more thorough treatment of the integral would fully account for the deviation; our guess is that it would probably not. Molecular Dynamics data [@bdphd] seem to rather indicate that for typical systems (i. e. on not yet asymptotic length scales) the coupling between polymer and solvent is more complex than the simple Kirkwood description. Nevertheless, we consider our result as important for the interpretation of experimental data: There is obviously a substantial contribution to $\Delta C$ which stems from the finite bead size, and which is only weakly $q$–dependent. A plateau–like shape of $C(q)$ alone apparently does not guarantee asymptotic behavior. Clearly more work has to be done to fully resolve the puzzle, but we believe our considerations show that theories which neglect the influence of finite bead size (and, to a lesser degree, of finite chain length) are simply not accurate enough to describe experimental data even of rather long chains.
Numerical Results {#sec:numres}
=================
In our numerical studies, we have used three different polymer chain models, which we will denote as model A, B, and C.
[*Model A*]{} is a bead–spring model in the continuum. $N$ monomers are connected via an anharmonic (“finitely extensible nonlinear elastic”) spring potential, $$\label{eq:fene}
U_{FENE} = \left\{
\begin{array}{ll}
-\frac{1}{2} k R_{0}^2 \ln
\left[ 1-\left(\displaystyle\frac{r}{R_0}\right)^2 \right]
& \hspace{0.5cm} r < R_0\\
\infty & \hspace{0.5cm} r \geq R_0
\end{array}
\right. ,$$ where we use the standard parameters [@kremer] $k = 30$, $R_0 = 1.5$ in dimensionless units. Between all monomers there is an additional non–bonded potential $$\label{eq:ljcos}
U_{LJcos} = \left\{%
\begin{array}{l}
4 \left[ \left( \displaystyle\frac{1}{r} \right)^{12}
-\left( \displaystyle\frac{1}{r} \right)^6
+ \frac{1}{4} \right] - \lambda ,
\hspace{0.2cm} r \le 2^{1/6} \\
\frac{1}{2} \lambda \left[ \cos ( \alpha r^2 +\beta ) - 1 \right] ,
\hspace{0.2cm} 2^{1/6} \le r \le 1.5 \\
\\
0 , \hspace{0.2cm} r \ge 1.5 ,
\end{array}\right.$$ where $\alpha$ and $\beta$ are determined as the solutions of the linear set of equations $$\begin{aligned}
\label{eq:lineq}
2^{1/3}\alpha + \beta &=& \pi\\
2.25 \alpha + \beta &=& 2\pi ,\end{aligned}$$ i. e. $\alpha = 3.1730728678$ and $\beta = -0.85622864544$. This potential has originally been constructed to simulate amphiphilic systems [@thoso]. The parameter $\lambda$ serves to control the strength of the attractive interaction and is varied instead of the temperature, which is fixed at $k_B T
= 1$. For sufficiently strong $\lambda$, the chain assumes a collapsed state, while $\lambda = 0$ corresponds to good solvent. We used a combination of stochastic dynamics [@kremer] and the pivot algorithm [@sokal]. Applying standard methods [@kremer] on data of chains of length of up to $N = 2000$, we located the $\Theta$ point at $\lambda = 0.65 \pm 0.02$. In the good solvent limit, and at $\lambda = 0.65$, we also ran an $N = 5000$ chain.
[*Model B*]{} is a mesoscopic model for an aqueous solution of the sodium salt of poly (acrylic acid) (PAA), whose input parameters have been derived from an extensive atomistic simulation of an aqueous PAA solution ($T=333.15$ K and $p=1$ atm) in the highly diluted regime, such that the ion concentration (number of charges on the chain, plus counterions) is $0.4$ mol/l [@biermann01s]. From this simulation, structural averages like the distributions of bond angles or radial distribution functions between monomers were extracted. We mapped this system to the mesoscale by replacing one repeating unit (i. e. one monomer) by one bead. As center of the coarse–grained (CG) beads, the monomer center of mass (excluding the sodium ion) was chosen. Bonded as well as non–bonded terms were parameterized by systematically varying the interactions until the structure of the atomistic model was reproduced [@reith00s]. This also allowed us to neglect all explicit water molecules and sodium ions (necessarily present in the parent atomistic simulation) in subsequent CG simulations. Their effect on the PAA chain conformation is, however, implicitly present in the model. This means that a system of roughly $10^{4}$ atoms could be reduced to a system which consists of only $23$ “super atoms”. As in model A, we used both stochastic
dynamics and pivot Monte Carlo moves. The final force field was utilized to calculate $R_G$, $R_H$ and other static properties like the structure factor for PAA strands of length 8 to 3155 repeating units [@simone]. The numerical results agree well with light scattering data on dilute PAA solutions with corresponding mean molar weights. In particular, the hydrodynamic radii of six different PAA–salt samples with molecular weights in the range from $18100$ to $296600$ g/mol were measured. For four samples, the molar masses $M_{W}$ and the radii of gyration $R_G$ were measured as well. The PAA samples were of polydispersity $D_P$ between $1.5$ and $1.8$ and diluted in aqueous NaCl–containing solution ($0.1-1$ mol/l) [@simone].
Finally, [*model C*]{} is the SAW on the face–centered cubic lattice, which we prefer over the simple cubic for reasons of increased local flexibility, which in turn means proximity to the SAW fixed point. Units of length are defined in such a way that the bond length is $\sqrt{2}$. The chains of length $N = 64, 128, \ldots 32768$ were generated by using a dimerization procedure [@sokal]. Up to $N = 8192$ the statistical sample always consisted of $M =
1024$ chains, while
$M = 1085$ for $N = 16284$ and $M = 296$ for $N = 32768$.
In what follows, we outline our $R_G$ and $R_H$ data for these three models. Figures \[fig:martintheta\] and \[fig:martingood\] summarize our results for model A at $\Theta$ condition $\lambda = 0.65$ (Fig. \[fig:martintheta\]) and at good solvent condition $\lambda = 0$ (Fig. \[fig:martingood\]), respectively. For the $\Theta$ chains, we obtained very good fits with the functions $\left< R_G^2 \right> = 0.2834 N - 0.53$ and $\left< R_H^{-1} \right> = 2.710 N^{-1/2} - 3.74 N^{-1}$, while for the good solvent data the analogous fits are $\left< R_G^2 \right> = 0.2706 N^{1.1754}
- 0.32 N^{0.62}$ and $\left< R_H^{-1} \right> = 3.131 N^{-0.5877} - 3.04
N^{-1}$. These fit functions are also shown in Figs. \[fig:martintheta\] and \[fig:martingood\]. The ratio $\rho = R_G / R_H$, as it results from these data, is shown in the subsequent Figs. \[fig:ratiomartintheta\] and \[fig:ratiomartingood\] for $\Theta$ and good solvent conditions, respectively. It should be noted that the numerical resolution (for each of our models) is clearly by far not competitive with the study by Li [*et al.*]{} [@limadrassokal]. For this reason, we did not attempt to determine the exponents from our data, but rather used the values for $\nu$ and $\Delta$ from Ref. . We did not include an $N^{-\Delta}$ term in the fit for $R_H$
in the SAW case, although such a term is expected to be present. The reason is that our model A data are too inaccurate to allow for such a three–parameter fit in a stable way. Similarly, we ignored the non–analytic corrections to scaling in the $\Theta$ case, for essentially the same reason, as has been discussed in some more detail in Sec. \[sec:intro\]. Taking the statistical inaccuracies of the data, and of the resulting fit parameters into account, we obtain for the asymptotic amplitude ratio $\rho = R_G / R_H$ the values $\rho = 1.44 \pm 0.01$ at the $\Theta$ point, and $\rho = 1.63 \pm 0.01$ in the excluded–volume case. The actual error in $\rho$ is expected to be significantly larger, since neither the uncertainties in the exponents and in the location of the $\Theta$ point, nor systematic errors due to higher–order corrections to scaling have been taken into account. This is particularly apparent in the $\Theta$ case, where one expects in the asymptotic long–chain limit rather the Gaussian value $1.5045$, but also obvious in the SAW case, where the results on the longer chains of model C yield a considerably smaller value for $\rho$.
The most interesting aspect of model B is that it closely
resembles a real system, and a quantitative comparison with experiments is possible [@simone]. In Fig. \[fig:RGRH\_dirk\] we show simulation results for $R_G$, $R_H$, and their ratio. The data are taken as published in Ref. . For the ratio, experimental results are also included. The scaling $N^{-(1 - \nu)}$, and the extrapolation to $\rho = 1.61 \pm 0.02$ is nicely borne out by the simulation data. The experiments are too inaccurate to demonstrate a clear systematic trend. In spite of this, an extrapolation yields $\rho \approx 1.5-1.6$, which means that the theoretical calculations are supported by data of a real chemical system.
Our model C data (SAW) comprise the largest range of chain lengths of our three models, combined with precise estimates of statistical errors, which allows a more detailed data analysis. For our $R_G$ data, we obtained the fit $\left< R_G^2 \right> = A N^{1.1754} + B N^{0.62}$ with $A = 0.3341 \pm
0.0023$, $B = - 0.20 \pm 0.05$, where we again use the exponents from Ref. . The deviation $\chi^2$ (sum of the residuals squares, normalized by the variances) has the value $\chi^2 = 9.4$ (10 data points). The corresponding quality of fit $Q$, which is the probability to observe the measured $\chi^2$ value, or a larger one, is $Q = 0.31$. Our data, in a representation which emphasizes the corrections to scaling, are shown in Fig. \[fig:bdrg2corscal\]. It is seen that these are indeed weak, highlighting the difficulties in determining an accurate value for the correction–to–scaling exponent.
Turning to our $R_H$ data from model C, we first did a nonlinear two–parameter fit $\left< R_H^{-1} \right> = A N^{-\nu_{eff}}$, resulting in $\nu_{eff} = 0.55$. However, this fit is very poor, with a least–square sum $\chi^2 = 433$. Conversely, a linear two–parameter fit $\left< R_H^{-1}
\right> = A N^{-0.5877} + B N^{-1}$ yields a rather good value $\chi^2 = 11.8$ ($Q = 0.16$), with $A = 2.732 \pm 0.005$, $B = -3.10 \pm 0.06$, demonstrating also numerically that $R_H$ data should be interpreted in terms of corrections to scaling, instead of an effective exponent. Actually, one should expect the presence of an additional correction of order $N^{-\Delta}$, $\Delta \approx
0.56$. Since this correction tends to decrease $R_G$ (see Fig. \[fig:bdrg2corscal\]), it should also decrease $R_H$, i. e. increase $\left<
R_H^{-1} \right>$, or weaken the analytic $N^{-(1 - \nu)}$
term. Thus, in a regression $\left< R_H^{-1} \right> = A N^{-0.5877} + B N^{-\phi}$, where we keep $\phi$ fixed, one should obtain the best fit for a value of $\phi$ slightly smaller than unity. This is indeed what we observe, as seen from Fig. \[fig:bdqvalues\], where we plot the quality $Q$ of such a fit as a function of $\phi$. This figure also clearly rules out a single correction to scaling with an exponent of $1/2$ or even larger. We thus attempted a three–parameter fit $\left< R_H^{-1} \right> = A N^{-0.5877} + B N^{-1} + C N^{-1.15}$ to also take the $N^{-\Delta}$ term into account. The result of this fit, which seems to be reasonably stable, is $A = 2.753 \pm 0.008$, $B = - 4.3 \pm 0.4$, $C =
2.2 \pm 0.7$, with $\chi^2 = 5.0$, and a very good quality $Q = 0.66$. We thus use this fit to demonstrate the corrections to scaling of $\left< R_H^{-1}
\right>$ in Fig. \[fig:bdrhcorscal\], where the presence of the $N^{-\Delta}$ term shows up in a slight curvature. Finally, we also used this fit, combined with the corresponding one for $R_G$ (see Fig. \[fig:bdrg2corscal\]), to describe the data on the ratio $\rho = R_G / R_H$, as shown in Fig. \[fig:bdratio\], where the asymptotic value is $1.591 \pm
0.007$. Again we feel that the real uncertainty is larger, due to lack of control of the systematic errors. We also checked that both the quality of fit, and the value of $\rho$ did not change significantly when we reduced the exponent $\Delta$ to its theoretical value [@zinnjustin] $\Delta = 0.482$.
To summarize, we have collected our most important numerical results, the extrapolated $\rho$ values, in Table \[tab:rgrhvalues\].
Model $R_G / R_H$
-------------- -------------------
A (SAW) $1.63 \pm 0.01$
B (SAW) $1.61 \pm 0.02$
C (SAW) $1.591 \pm 0.007$
A ($\Theta$) $1.44 \pm 0.01$
: Asymptotic universal ratio $R_G / R_H$ as estimated by numerical simulations of various models (see text). Error bars take into account statistical uncertainties only, while systematic errors in the extrapolation procedure are neglected.[]{data-label="tab:rgrhvalues"}
Acknowledgments {#acknowledgments .unnumbered}
===============
Stimulating discussions with J. Horbach, A. J. C. Ladd, J. J. de Pablo, and S. Wiegand are gratefully acknowledged. We thank L. Schäfer, A. Z. Akcasu and Y. Tsunashima for useful remarks and hints to the literature, G. Besold for a critical reading of the manuscript, and DSM and the BMBF Competence Center in Materials Simulations for financial support.
Euler–Maclaurin Formula {#sec:euler}
=======================
Quite usually, sums are approximated via the corresponding integrals. The Euler–Maclaurin formula [@olver; @graham], which we outline here for the convenience of the reader, constructs a systematic asymptotic expansion around that approximation. Defining a difference operator $\Delta$ via $$\Delta f(x) = f(x + 1) - f(x),$$ one obviously has $$\Delta F(N) = f(N)$$ for $$F(N) = \sum_{n = n_0}^{N - 1} f(n) ,$$ and thus $$F(N) = \Delta^{-1} f(N) + \mbox{\rm const.} .$$ On the other hand, $$\Delta = \exp \left( \frac{d}{dx} \right) - 1$$ or $$\begin{aligned}
\Delta^{-1} & = & \left( \frac{d}{dx} \right)^{-1}
\left( \frac{d}{dx} \right)
\left[ \exp \left( \frac{d}{dx} \right) - 1 \right]^{-1}
\nonumber \\
& = & \int dx
\sum_{k = 0}^\infty \frac{B_k}{k !} \left( \frac{d}{dx} \right)^k ,\end{aligned}$$ where $B_k$ are the Bernoulli numbers defined via the Taylor expansion of $x /
(e^x - 1)$: $B_0 = 1$, $B_1 = -1/2$, $B_2 = 1/6$, $B_4 = -1/30$, …, $B_3
= B_5 = B_7 = \ldots = 0$. Hence, $$\Delta^{-1} = \int dx - \frac{1}{2} + \frac{1}{12} \frac{d}{dx}
- \frac{1}{720} \left( \frac{d}{dx} \right)^3 + \ldots$$ and thus $$\begin{aligned}
\label{eq:eulermaclaurin}
\sum_{n = n_0}^{N - 1} f(n) & = & \int_{n_0}^N dx f(x) - \frac{1}{2} f(N)
+ \mbox{\rm const.} \nonumber \\
& + & \frac{1}{12} \left. \frac{d}{dx} f(x) \right\vert_{x = N}
- \frac{1}{720} \left. \frac{d^3}{dx^3} f(x) \right\vert_{x = N}
\nonumber \\ & + & \ldots ,\end{aligned}$$ where the “integration” constant is determined via (perhaps numerical) comparison of both sides. For a power law with $q < -1$ one thus finds from the definition of the Riemann zeta function $$\begin{aligned}
\label{eq:eulermaclaurinpower}
\sum_{n = 1}^{N - 1} n^q & = & \frac{N^{q + 1}}{q + 1} - \frac{1}{2} N^q
+ \zeta(-q) + \frac{1}{12} q N^{q - 1} \nonumber \\
& - & \frac{1}{720} q (q - 1) (q - 2) N^{q - 3} + \ldots .\end{aligned}$$ By analytic continuation with respect to $q$, this result holds for general $q$ [@olver].
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[^1]: corresponding author, Electronic Mail: `[email protected]`
|
---
abstract: 'We adapt a method of Voisin to powers of abelian varieties in order to study orbits for rational equivalence of zero-cycles on very general abelian varieties. We deduce that a very general abelian variety of dimension at least $2k-2$ has gonality at least $k+1$. This settles a conjecture of Voisin. We also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality. In particular, we obtain a strengthening of the Sommese bound on the degree of irrationality of abelian varieties. In the appendix we present some new identities in the Chow group of zero-cycles of abelian varieties.'
address: 'Department of Mathematics, University of Chicago, IL, 60637'
author:
- Olivier Martin
title: On a Conjecture of Voisin on the Gonality of Very General Abelian Varieties
---
Introduction
============
In his seminal 1969 paper [@M] Mumford shows that the Chow group of zero-cycles of a smooth projective surface over $\mathbb{C}$ with $p_g>0$ is very large. Building on the work of Mumford, in [@R1] and [@R2] Roĭtman studies the map $$\text{Sym}^k (X)\to CH_0(X)$$ for $X$ a smooth complex projective variety. He shows that fibers of this map, which we call orbits of degree $k$ for rational equivalence[^1], are countable unions of Zariski closed subsets. Moreover, he defines birational invariants $d(X)$ and $j(X)\in \mathbb{Z}_{\geq 0}$ such that for $k\gg 0$ the minimal dimension of orbits of degree $k$ for rational equivalence is $k(\dim X-d(X))-j(X)$. Roĭtman’s generalization of Mumford’s theorem is the following statement: $$\text{If }H^0(X,\Omega^q)\neq 0 \text{ then }d(X)\geq q.$$ In particular, if $X$ has a global holomorphic top form then a very general $x_1+\ldots+x_k\in \text{Sym}^kX$ is contained in a zero-dimensional orbit.\
Abelian varieties are among the simplest examples of varieties admitting a global holomorphic top form. In this article we will focus our attention on this example and take a point of view different from the one mentioned above. Instead of fixing an abelian variety $A$ and considering the minimal dimension of orbits of degree $k$ for rational equivalence, we will be interested in the maximal dimension of such orbits for a very general abelian variety $A$ of dimension $g$ with a given polarization $\theta$.\
This perspective has already been studied by Pirola, Alzati-Pirola, and Voisin (see respectively [@P],[@AP],[@V]) with a view towards the gonality of very general abelian varieties. The story begins with [@P] in which Pirola shows that, given a very general abelian variety $A$, curves of geometric genus less than $\dim A-1$ are rigid in the Kummer variety $K(A)=A/\{\pm 1\}$. This allows him to show:
\[P\] A very general abelian variety of dimension $\geq 3$ does not have positive dimensional orbits of degree $2$. In particular it does not admit a non-constant morphism from a hyperelliptic curve.
There are several ways in which one might hope to generalize this result. For instance, one can ask for the gonality of a very general abelian variety of dimension $g$. We define the gonality of a smooth projective variety $X$ as the minimal gonality of the normalization of an irreducible curve $C\subset X$. Note that an irreducible curve $C\subset X$ whose normalization $\widetilde{C}$ has gonality $k$ provides us with a positive dimensional orbit in $\text{Sym}^k \widetilde{C}$, and thus with a positive dimensional orbit in $\text{Sym}^k X$. Hence, one can give a lower bound on the gonality of $X$ by giving a lower bound on the degree of positive dimensional orbits. This suggest to consider the function $$\mathscr{G}(k):=\text{min}\begin{cases}\begin{rcases}g\in \mathbb{Z}_{>0}: \text{a very general abelian variety of dimension } g\\ \text{ does not have a positive dimensional orbit of degree } k\end{rcases},\end{cases}$$ and attempt to find an upper bound[^2] on $\mathscr{G}(k)$. Indeed, a very general abelian variety of dimension at least $\mathscr{G}(k)$ must have gonality at least $k+1$.\
In this direction, a few years after the publication of [@P], Alzati and Pirola improved on Pirola’s results in [@AP], showing that a very general abelian variety $A$ of dimension $\geq 4$ does not have positive dimensional orbits of degree $3$, i.e.that $\mathscr{G}(3)\leq 4$. This suggest that for any $k\in \mathbb{N}$ a very general abelian variety of sufficiently large dimension should not admit a positive dimensional orbit of degree $k$, i.e.that $\mathscr{G}(k)$ is finite for any $k\in \mathbb{Z}_{>0}$. This problem was posed in [@BPEL] and answered positively by Voisin in [@V].
\[Vmainthm\] A very general abelian variety of dimension at least $2^{k}(2k-1)+(2^{k}-1)(k-2)$ does not have a positive dimensional orbit of degree $k$, i.e. $\mathscr{G}(k)\leq 2^{k}(2k-1)+(2^{k}-1)(k-2).$
Voisin provides some evidence suggesting that this bound can be improved significantly. Her main conjecture from [@V] is the following linear bound on the gonality of very general abelian varieties:
A very general abelian variety of dimension at least $2k-1$ has gonality at least $k+1$.\[Vweakconj\]
The central result of this paper is the proof of this conjecture. It is obtained by generalizing Voisin’s method to powers of abelian varieties. This allows us to rule out the existence of positive dimensional $\text{CCS}_k$ in very general abelian varieties of dimension $g$ for $g$ large compared to $k$.
For $k\geq 3$, a very general abelian variety of dimension at least $2k-2$ has no positive dimensional orbits of degree $k$, i.e. $\mathscr{G}(k)\leq 2k-2$.
This gives the following lower bound on the gonality of very general abelian varieties:
For $k\geq 3$, a very general abelian variety of dimension $\geq 2k-2$ has gonality at least $k+1$. In particular Conjecture [\[Vweakconj\]](Vweakconj) holds.
Another approach to generalizing Theorem [\[P\]](P) is the following: Observe that a nonconstant morphism from a hyperelliptic curve $C$ to an abelian surface $A$ gives rise to a positive dimensional orbit of the form $2\{0_A\}$. Indeed, translating $C$ if needed, we can assume that a Weierstrass point of $C$ maps to $0_A$. This suggests to consider the maximal $g$ for which a very general abelian variety $A$ of dimension $g$ contains an irreducible curve $C\subset A$ whose normalization $\pi: \widetilde{C}\to C$ admits a degree $k$ morphism to $\mathbb{P}^1$ with a point of ramification index at least $l$. Here we say that $p\in \widetilde{C}$ has ramification index at least $l$ if the sum of the ramification indices of $\widetilde{C}\to \mathbb{P}^1$ at all points in $\pi^{-1}(\pi(p))$ is at least $l$. This maximal $g$ is less than $$\mathscr{G}_l(k):=\text{min}\begin{cases}\begin{rcases}g\in \mathbb{Z}_{>0}: \text{a very general abelian variety of dimension } g \text{ does not }\\\text{ admit a positive dimensional orbit of the form } |\sum_{i=1}^{k-l} \{a_i\}+l\{a_0\}|\end{rcases},\end{cases}$$ where, given a smooth projective variety $X$ and $\{x_1\}+\ldots+\{x_k\}\in \text{Sym}^k X$, we denote by $|\{x_1\}+\ldots+\{x_k\}|$ the orbit of $\{x_1\}+\ldots+\{x_k\}$, namely the orbit containing this point. Clearly, we have $$\mathscr{G}_1(k)=\mathscr{G}_0(k)=\mathscr{G}(k),$$ and $$\mathscr{G}_l(k)\leq \mathscr{G}(k).$$ In [@V] the author shows the following:
The following inequality holds: $$\mathscr{G}_l(k)\leq 2^{k-l}(2k-1)+(2^{k-l}-1)(k-2).$$ In particular $$\mathscr{G}_k(k)\leq 2k-1.$$
We can improve on Voisin’s result to show the following:
A very general abelian variety $A$ of dimension at least $2k+2-l$ does not have a positive dimensional orbit of the form $|\sum_{i=1}^{k-l}\{a_i\}+l\{0_A\}|$, i.e. $\mathscr{G}_{l}(k)\leq 2k+2-l.$\
Moreover, if $A$ is a very general abelian variety of dimension at least $k+1$ the orbit $|k\{0_A\}|$ is countable, i.e. $\mathscr{G}_{k}(k)\leq k+1.$
Taking a slightly different perpective, one can consider the maximal dimension of orbits of degree $k$ of abelian varieties. One of the main contributions of [@V] is the following extension of results of Alzati-Pirola (the cases $k=2,3$):
\[k-1\] Orbits of degree $k$ on an abelian variety have dimension at most $k-1$.
As observed by Voisin, one sees easily[^3] that this bound cannot be improved if one considers all abelian varieties. Moreover, it cannot be improved for very general abelian surfaces as shown in Example . Nevertheless Theorem [\[Vmainthm\]](Vmainthm) shows that it can be improved for very general abelian varieties of large dimension. In this direction we have:
Orbits of degree $k$ on a very general abelian variety of dimension $\geq k-1$ have dimension at most $k-2$.
Recall that the degree of irrationality $\text{Irr}(X)$ of a variety $X$ is the minimal degree of a dominant morphism $X\to \mathbb{P}^{\dim X}$. The previous theorem provides the following improvement of the Sommese bound $\text{Irr}(A)\geq \dim A+1$ (see the appendix to [@BE]) on the degree of irrationality of abelian varieties.
If $A$ is a very general abelian variety of dimension $g\geq 3$, then $$\textup{Irr}(A)\geq g+2.$$
It is very likely that one can do better by studying the Gauss map of $\text{CCS}_k$[^4] of very general abelian varieties.\
Finally, in the appendix we present the following generalization of Proposition 0.9 of [@V] along with similar results:
Consider an abelian variety $A$ and effective zero-cycles $\sum_{i=1}^k\{x_i\}$, $\sum_{i=1}^k \{y_i\}$ on $A$ such that $$\sum_{i=1}^k \{x_i\}= \sum_{i=1}^k \{y_i\}\in CH_0(A).$$ Then for $i=1,\ldots, k$ $$\prod_{j=1}^k(\{x_i\}-\{y_j\})=0\in CH_0(A),$$ where the product is the Pontryagin product.
In the last part of this introduction we will sketch Voisin’s proof of Theorem [\[Vmainthm\]](Vmainthm) and give an idea of the methods we will use to show Theorem [\[2k-2\]](2k-2). Voisin considers what she calls *naturally defined subsets of abelian varieties*. Given a universal abelian variety $\mathcal{A}/S$ of dimension $g$ with a fixed polarization $\theta$ there are subvarieties $S_\lambda\subset S$ along which $\mathcal{A}_s\sim \mathcal{B}_s^{\lambda}\times \mathcal{E}_s^{\lambda}$, where $\mathcal{B}^{\lambda}/S_\lambda$ is a family of abelian varieties of dimension $(g-1)$ and $\mathcal{E}^\lambda/S_\lambda$ is a family of elliptic curves. Let $S_\lambda(B)=\{s\in S_\lambda: B^\lambda_s=B\}$.\
Voisin shows that, given a naturally defined subset $\Sigma_{\mathcal{A}}\subsetneq \mathcal{A}$, there is an $S_\lambda$ such that the image of the restriction of $p_\lambda: (\mathcal{B}^\lambda_s\times \mathcal{E}_{s}^{\lambda})^k \to (\mathcal{B}^\lambda_s)^k=B^k$ to $\Sigma_{\mathcal{A}_{s}}$ varies with $s\in S_\lambda(B)$, for a generic $B$ in the family $\mathcal{B}^{\lambda}$. This shows that if $\Sigma_{B}$ is $d$-dimensional for a very-general $(g-1)$-dimensional abelian variety $B$ with $0<d<g-1$, then $\Sigma_A$ is at most $(d-1)$-dimensional for a very general abelian variety $A$ of dimension $g$.\
She proceeds to show that the set $$\bigcup_{i=1}^k \text{pr}_i(|k\{0_A\}|)=\left\{a_1\in A: \exists a_2,\ldots, a_k\in A\text{ such that }\sum_{i=1}^k\{a_i\}= k\{0_A\}\in CH_0(A)\right\}$$ is contained in a naturally defined subset $A_k\subset A$ and that $\dim A_k\leq k-1$. In particular, $A_k\neq A$ if $\dim A\geq k$. It follows from the discussion above that $$\dim\bigcup_{i=1}^k \text{pr}_i(|k\{0_A\}|)=0$$ if $\dim A\geq 2k-1$ and $A$ is very general. An induction argument finishes the proof. Voisin’s results are very similar in spirit to those of [@AP] but a key difference is that the latter is concerned with subvarieties of $A^k$ and not of $A$. We will see that this has important technical consequences. While Lemma 1.5 in [@V] shows that the restriction of the projection $B\times E\to B$ to $\Sigma_{B\times E}$ is generically finite on its image, this becomes a serious sticking point in [@AP] (see lemmas 6.8 to 6.10 of [@AP]). One of the innovations of this article is a way to bypass this generic finiteness assumptions by using the inductive nature of the argument.
Aknowledgements {#aknowledgements .unnumbered}
---------------
This paper owes a lot to the work of Alzati, Pirola, and Voisin. I thank Madhav Nori and Alexander Beilinson for countless useful and insightful conversations as well as for their support. I would also like to extend my gratitude to Claire Voisin for bringing my attention to this circle of ideas by writing [@V], and for kindly answering some questions about the proof of Theorem 0.1 from that article.
Preliminaries
=============
In this section we fix some notation and establish some facts about positive dimensional orbits for rational equivalence on abelian varieties.\
A variety will mean a quasi-projective reduced scheme of finite type over $\mathbb{C}$. In what follows, $X$ is a smooth projective variety, $\mathcal{X}/S$ is a family of such varieties, $A$ is an abelian variety of dimension $g$ with polarization type $\theta$, and $\mathcal{A}/S$ is a family of such abelian varieties. Mostly we will be concerned with locally complete families of abelian varieties, namely families $\mathcal{A}/S$ such that the corresponding morphism from $S$ to the moduli stack of $g$-dimensional abelian varieties with polarization type $\theta$ is dominant. In an effort to simplify notation, we will often write $\mathcal{X}^k$ instead of $\mathcal{X}_S^k$ to denote the $k$-fold fiber product of $\mathcal{X}$ with itself over $S$. $\mathcal{Z}\subset \mathcal{A}^{k}$ will be a subvariety such that $\mathcal{Z}\to S$ is flat with irreducible fibers $\mathcal{Z}_{s}$. Finally, $CH_0(X)$ will denote the Chow group of zero-cycles of a smooth projective variety $X$ with $\mathbb{Q}$-coefficients.
\[convention\] In many of our arguments, we will have a family of varieties $\mathcal{X}\to S$ and a subvariety $\mathcal{Z}\subset \mathcal{X}$, such that $\mathcal{Z}\to S$ is flat with irreducible fibers. We will often need to base change by a generically finite morphism $S'\to S$. To avoid the growth of notation we will denote the base changed family by $\mathcal{X}\to S$ again. Moreover, if $S_\lambda\subset S$ is a Zariski closed subset, the base change of $S_\lambda$ under $S'\to S$ will also be denoted $S_\lambda$. Note that this applies also to the statement of theorems. For instance if we say a statement holds for a family $\mathcal{X}/S$ we mean that it holds for some $\mathcal{X}_{S'}/S'$, where $S'\to S$ is generically finite.
Instead of considering orbits for rational equivalence, one can consider subvarieties of orbits. This makes talking about families of orbits somewhat simpler.
\[CCS\] A constant cycle subvariety of degree $k$ $\text{CCS}_k$ of $X$ is a Zariski closed subset of $X^k$ contained in a fiber of $X^k\to CH_0(X)$. A Zariski closed subset $\mathcal{Z}\subset \mathcal{X}^k$ is a $\text{CCS}_k/S$ if $\mathcal{Z}_s$ is a $\text{CCS}_k$ of $\mathcal{X}_s^k$ for every $s\in S$.
The notion of $\text{CCS}_k$ above is closely related but not to be confused with constant cycle subvarieties in the sense of Huybrechts (see [@H]). Indeed a $CCS_1$ is exactly the analogue of a constant cycle subvariety as defined in [@H] for K3 surfaces. Nonetheless a $\text{CCS}_k$ of $X$ need not be a $CCS_1$ of $X^k$; in the first case we consider rational equivalence of cycles in $X$, and in the other rational equivalence of cycles in $X^k$. We will not only be interested in $\text{CCS}_k$ but in families of $\text{CCS}_k$ and subvarieties of $X^k$ foliated by $\text{CCS}_k$.
$\;$
1. An $(r+d)$-dimensional Zariski closed subset $Z\subset X^k$ is foliated by $d$-dimensional $\text{CCS}_k$ if for all $z\in Z$ we have $$\dim |z|\cap Z\geq d.$$
2. Similarly $\mathcal{Z}\subset \mathcal{X}^k$ is foliated by $d$-dimensional $\text{CCS}_k$ if $\mathcal{Z}_s$ is foliated by $d$-dimensional $\text{CCS}_k$ for all $s\in S$.\
3. An $r$-parameter family of $d$-dimensional $\text{CCS}_k$ of $X$ is an $r$-dimensional locally closed subset $D$ of a Chow variety of $X^k$ parametrizing dimension $d$ cycles with a fixed cycle class, such that for each $t\in D$ the corresponding cycle is a $\text{CCS}_k$.\
4. Similarly, an $r$-parameter family of $d$-dimensional $\text{CCS}_k$ of $\mathcal{X}/S$ is an $r$-dimensional locally closed subset $\mathcal{D}$ of the relative Chow variety of $\mathcal{X}^k$ parametrizing dimension $d$ cycles with a fixed cycle class, such that for each $t\in \mathcal{D}$ the fiber of the corresponding cycle over any $s\in S$ is a $\text{CCS}_k$.
Note that given $D$, an $r$-parameter family of $d$-dimensional $\text{CCS}_k$ of a variety $X$, and $\mathcal{Y}\to D$, the corresponding family of cycles, the set $\bigcup_{t\in D}\mathcal{Y}_t\subset X^k$ is foliated by $d$-dimensional $\text{CCS}_k$. Yet its dimension might be less than $(r+d)$. Conversely, any subvariety of $X^k$ foliated by positive dimensional $\text{CCS}_k$ arises in such a fashion from a family of $\text{CCS}_k$ after possibly passing to a Zariski open subset. Indeed, by work of Roĭtman (see [@R1]) the set $$\Delta_{rat}=\left\{((x_1,\ldots, x_k),(x_1',\ldots, x_k'))\in X^k\times X^k: \sum_{i=1}^k x_i=\sum_{i=1}^k x_i'\in CH_0(X)\right\}\subset X^k\times X^k$$ is a countable union of Zariski closed subsets. Consider $$\pi: \Delta_{rat}\cap Z\times Z\to Z.$$ Given an irreducible component $\Delta'$ of $\Delta_{rat}\cap Z\times Z$ that dominates $Z$ and has relative dimension $d$ over $Z$, we can consider the map from an open set in $Z$ to an appropriate Chow variety of $X^k$ taking $z\in Z$ to the cycle $[\Delta_z']$. Letting $D$ be the image of this morphism, we get a family of $\text{CCS}_k$ with the desired property.\
Given an abelian variety $A$, we denote by $A^r_{M}$ (or $A_M$, when $r=1$) the image of $A^r$ in $A^k$ under the embedding $i_M: A^r\to A^k$ given by $$(a_1,\ldots, a_r)\mapsto \left(\sum_{j=1}^r m_{1j}a_j,\sum_{j=1}^r m_{2j}a_j,\ldots, \sum_{j=1}^r m_{kj}a_j\right),$$ where $0\leq r\leq k$, and $M=(m_{ij})\in M_{k\times r}(\mathbb{Z})$ has rank $r$. We will use the same notation $V_M^r$, with $M\in M_{k\times r}({\mathbb{C}})$, for $V$ a vector space over ${\mathbb{C}}$. If $A$ is simple then all abelian subvarieties of $A^k$ are of this form. Let $\text{pr}_i: A^k\to A$ be the projections to the $i^{\text{th}}$ factor and, given a form $\omega\in H^0(A,\Omega^q)$, let $\omega_k:=\sum_{i=1}^k \text{pr}_i^*\omega$.\
For the sake of simplicity we mostly deal with $X^k$ rather than $\text{Sym}^kX$ and we take the liberty to call points of $X^k$ effective zero-cycles of degree $k$.
Given an abelian variety $A$, a zero-cycle $z=(z_1,\ldots, z_k)\in A^k$ is called normalized if $z_1+\ldots+z_k=0_A$. We write $A^{k,\,0}$ for the kernel of the summation map, i.e. the set of normalized effective zero-cycles of degree $k$.
In Corollary 3.5 of [@AP] the authors show the following generalization of Mumford’s result:
\[vanish\] Let $D$ be an an $r$-parameter family of $\text{CCS}_k$ of a variety $X$ with corresponding family of cycles $\mathcal{Z}\to D$. Denote by $g: \mathcal{Z}\to X^k$ the natural map. If $\omega\in H^0(A,\Omega^q)$ and $q>r$, then $$g^*(\omega_k)=0.$$
In proposition 3.2 of [@V], Voisin shows that if $A$ is an abelian variety, and $Z\subset A^k$ is such that $\omega_k|_{Z}=0$ for all $\omega\in H^0(A,\Omega^q)$ and all $q\geq 1$, then $\dim Z\leq k-1$. Along with the above proposition this gives:
\[1dimfam\] An abelian variety does not admit a one-parameter family of $(k-1)$-dimensional $\text{CCS}_k$.
This non-existence result along with our degeneration and projection argument will provide the proof of Theorem [\[no(k-1)\]](no(k-1)). For most other applications we will use non-existence results for large families of $\text{CCS}_k$ on surfaces $X$ with $p_g\neq 0$. In particular we have:
\[vanish2\] Let $X$ be a smooth projective surface with $p_g\neq 0$ and $\omega\neq 0\in H^0(X,\Omega^2_X)$. If $Z\subset X^k$ is a Zariski closed subset of dimension $m$ foliated by $d$-dimensional $\text{CCS}_k$ then $\omega_k^{\lceil(m-d+1)/2\rceil}$ restricts to zero on $Z$.
The set of points $z\in Z$ such that $z\in Z_{sm} \cap(|z|\cap Z)_{sm}$ is clearly Zariski dense. Thus it suffices to show that $\omega_k^{\lceil(m-d+1)/2\rceil}$ restricts to zero on $T_{Z,z}$ for such a $z$. Suppose that $m-d$ is odd (the even case is treated in the same way). Given any $(m-d+1)$-dimensional subspace of $T_{Z,z}$, it must meet the tangent space to $T_{|z|\cap Z,z}$ and so we can assume it admits a basis $v_1,\ldots, v_{m-d+1}$ with $v_1\in T_{|z|\cap Z,z}$. Hence $$\iota_{v_1,\ldots, v_{m-d+1}}{\omega_k^{(m-d+1)/2}}$$ will consist of a product of terms of the form $\iota_{v_i,v_j}{\omega_k}$. But $\iota_{v_1,v_j}{\omega_k}=0$ for any $j$ by Proposition [\[vanish\]](vanish). Indeed for any $j$ the space spanned by $v_1$ and $v_j$ is contained in the tangent space to a $(d+1)$-fold foliated by at least $d$-dimensional $\text{CCS}_k$.
\[surfbound\] Let $X$ and $\omega$ be as in the previous lemma. If $Z\subset X^k$ is such that $\omega_k^l$ restricts to zero on $Z$, then $\dim Z< k+l$.
Pick $z\in Z_{sm}$ and let $v_1,\ldots, v_m$ be a basis of $T_{Z,z}$. Complete it with $v_{m+1},\ldots, v_{2k}$ to a basis of $T_{X^k,z}$. Let $\omega$ be a symplectic form on $X$. Since $\omega_k^k$ is a volume form on $X^k$ we have $\iota_{v_1,\ldots, v_{2k}}\omega^k\neq 0$. If $m\geq k+l$ then $\iota_{v_{m+1},\ldots, v_{2k}}\omega_k^k\in (\omega_k^l)\subset H^0(X^k, \Omega_X^{\bullet})$. The non-vanishing stated above then implies $\omega_k^l$ cannot vanish on $Z$.
If $A$ is an abelian variety and $\omega\neq 0\in H^0(A,\Omega_A^2)$, then $$(i_M^*\omega_k)^{\wedge l}\neq 0 \in H^0(A_M^l,\Omega_{A_M^l}^{2l}).$$ In particular, if $\dim A=2$, the form $\omega_k$ restricts to a symplectic form on $A_M^l$.
Let $z,w$ be two coordinates on $A$ and $\omega=dz\wedge dw$. Let $z_{i},w_i$ be the corresponding coordinates on the $i^{\text{th}}$ factor of $A^l$. We have $$i_M^*\omega_k=\sum_{i=1}^k\left(\sum_{j=1}^lm_{ij}dz_{j}\wedge \sum_{j'=1}^lm_{ij'}dw_{j'}\right).$$ We claim that $$(i_M^*\omega_k)^{\wedge l}=l!\det \mathbf{G}\; dz_{1}\wedge dw_1\wedge\ldots \wedge dz_{l}\wedge dw_{l},$$ where $\mathbf{G}=(\langle M_i,M_j\rangle)_{1\leq i,j\leq l}$ is the Gram matrix of the columns $M_i$ of the matrix $M$. Since $M$ has maximal rank its Gram matrix has positive determinant. It follows that $(i_M^*\omega_k)^{\wedge l}\neq 0$. To prove the above claim we observe that $$i_M^*\omega_k=\sum_{i=1}^k\sum_{j,j'=1}^lm_{ij}m_{ij'}dz_j\wedge dw_{j'}=\sum_{j,j'=1}^l\langle M_j,M_{j'}\rangle dz_j\wedge dw_{j'}.$$ It follows that $$\begin{aligned}
(i_M^*\omega_k)^{\wedge l}&=\sum_{\sigma,\sigma'\in S_l}\prod_{j=1}^l\langle M_{\sigma(j)},M_{\sigma'(j)}\rangle dz_{\sigma(1)}\wedge dw_{\sigma'(1)}\wedge\ldots\wedge dz_{\sigma(l)}\wedge dw_{\sigma'(l)}\\
&=\sum_{\sigma,\sigma'\in S_l}\text{sgn}(\sigma)\text{sgn}(\sigma')\prod_{j=1}^l\langle M_{\sigma(j)},M_{\sigma'(j)}\rangle dz_1\wedge dw_1\wedge\ldots \wedge dz_l\wedge dw_l\\
&=\sum_{\sigma'\in S_l}\text{sgn}(\sigma')\sum_{\sigma'\sigma\in S_l}\text{sgn}(\sigma'\sigma)\prod_{j=1}^l\langle M_{\sigma'\sigma(j)},M_{\sigma'(j)}\rangle dz_1\wedge dw_1\wedge\ldots \wedge dz_l\wedge dw_l\\
&=\sum_{\sigma'\in S_l}\sum_{\sigma'\sigma\in S_l}\text{sgn}(\sigma)\prod_{j=1}^l\langle M_{\sigma(j)},M_{j}\rangle dz_1\wedge dw_1\wedge\ldots \wedge dz_l\wedge dw_l\\
&=l!\det\mathbf{G}\end{aligned}$$
In particular Lemma [\[vanish2\]](vanish2) implies:
\[notfoliated2\] If $Z\subset A^k$ is foliated by positive dimensional $\text{CCS}_k$ and $\dim A=2$ then it cannot be an abelian subvariety of the form $A^l_M$.
This corollary will play a crucial technical role in our argument. Using Lemma [\[surfbound\]](surfbound), we see that the proof of Lemma 2.5 also shows the following:
If $A$ is an abelian surface and $Z\subset A^{k,\,0}$ is such that $\omega_k^l$ vanishes on $Z$, then $$\dim Z< k+l-1.$$
\[normabsurfbound\] If $A$ is an abelian surface and $Z\subset A^{k,\,0}$ is foliated by $d$-dimensional $\text{CCS}_k$, then $$\dim Z\leq 2(k-1)-d.$$\[absurfbound\]
By Lemma [\[vanish2\]](vanish2) $\omega^{\lceil (\dim Z-d+1)/2\rceil}$ restricts to zero on $Z$. Then by Lemma [\[surfbound\]](surfbound) $$\dim Z<k+\lceil(\dim Z-d+1)/2\rceil-1.$$ This gives the stated bound for parity reasons.
One could instead seek existence results for subvarieties of $A^k$ foliated by $d$-dimensional $\text{CCS}_k$. Alzati and Pirola show in examples 5.2 and 5.3 of [@AP] that any abelian surface has a $2$-dimensional $\text{CCS}_3$ and a $2$-parameter family of normalized $\text{CCS}_1$. In particular, using the argument from Remark [\[trick\]](trick) we see that Corollary [\[normabsurfbound\]](normabsurfbound) is sharp for $d=0,1,2$.
\[HSL\] In [@HSL] Lin shows that Corollary [\[normabsurfbound\]](normabsurfbound) is sharp for every $d$.
The methods of [@HSL] can be used to show the following:
\[curvecase\] If an abelian variety $A$ of dimension $g$ is the quotient of the Jacobian of a smooth genus $g'$ curve $C$, then $A^k$ contains a $(g(k+1-g'-d)+d)$-dimensional subvariety foliated by $d$-dimensional normalized $\text{CCS}_k$ for all $k\geq g'+d-1$.
To simplify notation we identify $C$ with its image in $J(C)$. We can assume that $0_A\in C$. Recall that the summation map $\text{Sym}^lC\to J(C)$ has fibers $\mathbb{P}^{l-g'}$ for all $l\geq g'$. Moreover, if $(c_1,\ldots, c_l)$ and $(c_1',\ldots, c_l')$ are such that $\sum c_i=\sum c_i'\in J(C)$, then the zero cycles $\sum \{c_i\}$ and $\sum \{c_i\}'$ are equal in $CH_0(C)$ and thus in $CH_0(A)$.\
In light of Remark [\[trick\]](trick), it suffices to show that $A^{g'+d-1,\, 0}$ contains a $d$-dimensional $\text{CCS}_{g'+d-1}$. Consider the map $$\psi: C\times C^{g'+d-1}\to A^{g'+d-1}$$ given by $$(c_0,(c_1,\ldots, c_{g'+d-1}))\mapsto (c_1-c_0,\ldots, c_{g'+d-1}-c_0).$$ This morphism is generically finite on its image since the restriction of the summation map $A^2\to A$ to $C^2\subset A^2$ is. The intersection of the image of $\psi$ with $A^{g'+d-1,\, 0}$ is $d$-dimensional and we claim it is a $\text{CCS}_{g'-d+1}$. Indeed, given $$(c_1-c_0,\ldots, c_{g'+d-1}-c_0)\in \text{Im}(\psi)\cap A^{g'+d-1,\, 0},$$we have $$\sum_{i=1}^{g'+d-1} c_i=(g'+d-1)c_0$$ so that $$\sum_{i=1}^{g'+d-1} \{c_i\}=(g'+d-1)\{c_0\}\in CH_0(C).$$ It follows that $$\sum_{i=1}^{g'+d-1}\{c_i-c_0\}= (g'+d-1)\{0_A\}\in CH_0(A).$$
Since the Torelli morphism $\mathcal{M}_{3}\to \mathcal{A}_3$ is dominant the previous proposition provides a one-dimensional orbit of degree $3$ in a very general abelian $3$-fold. Our methods do not seem to provide any way to rule out the existence of a one-parameter family of one-dimensional $\text{CCS}_3$ on a very general abelian $3$-fold. Yet, the study of zero cycles on Jacobians does not seem to readily provide an example of such a family. This motivates the following:
Does a very general abelian $3$-fold admit a one-parameter family of normalized one-dimensional orbits of degree $3$?\
Degeneration and Projection
===========================
In this section we generalize Voisin’s method from Section 1 of [@V] to powers of abelian varieties. The key difference is that our generalization requires technical assumptions which are automatically satisfied in Voisin’s setting.\
Given $\mathcal{A}/S$ a locally complete family of abelian varieties of dimension $g$, and a positive integer $l< g$, let $S_{\lambda}\subset S$ denote loci along which $$\mathcal{A}_s\sim\mathcal{B}^\lambda_{s}\times \mathcal{E}^\lambda_{s},$$ where $\mathcal{B}^\lambda/S_\lambda$ and $\mathcal{E}^\lambda/S_\lambda$ are locally complete families of abelian varieties of dimension $l$ and $g-l$ respectively, and the index $\lambda\in \Lambda_l$ encodes the structure of the isogeny. Given a positive integer $l'< l$ we will also be concerned, inside each $S_{\lambda}$, with loci $S_{\lambda,\mu}$ along which $$\mathcal{B}_s^\lambda\sim \mathcal{D}_s^{\lambda,\mu}\times\mathcal{F}_s^{\lambda,\mu},$$ where $\mathcal{D}^{\lambda,\mu}/S_{\lambda,\mu}$ and $\mathcal{F}^{\lambda,\mu}/S_{\lambda,\mu}$ are locally complete families of abelian varieties of dimension $l'$ and $l-l'$ respectively, and the index $\mu\in \Lambda_{l'}^{\lambda}$ encodes the structure of the isogeny. For our applications we will mostly be concerned with $(l,l')=(g-1,2)$. Upon passing to a generically finite cover of $S_\lambda$ and $S_{\lambda,\mu}$ we can assume that we have projections
$$\begin{aligned}
p_{\lambda}: &\mathcal{A}_{S_{\lambda}}^k\to ({\mathcal{B}^{\lambda}})^k\\
p_{\mu}: &(\mathcal{B}^{\lambda}_{S_{\lambda,\mu}})^k\to (\mathcal{D}^{\lambda,\mu})^k,\\\end{aligned}$$
and we let $$p_{\lambda,\mu}:=p_{\mu}\circ p_{\lambda}$$ for $\mu\in \Lambda_{l'}^{\lambda}$. Note that, to keep an already unruly notation in check, we suppress the power $k$ from the notation of the projections. Given a subvariety $\mathcal{Z}\subset \mathcal{A}^k/S$ we consider the following subsets of $S$ and conditions on $\mathcal{Z}$ $$\begin{aligned}
R_{gf}&=\bigcup_{\lambda}\{s\in S_\lambda : p_{\lambda}|_{\mathcal{Z}_s}: \mathcal{Z}_s\to \mathcal{B}^{\lambda}_s\text{ is generically finite on its image}\},\\
R_{ab}&=\bigcup_{\lambda}\{s\in S_\lambda : p_{\lambda}(\mathcal{Z}_s)\text{ is not an abelian subvariety of }\mathcal{B}^{\lambda}_s\},\\
R_{st}&=\bigcup_{\lambda}\{s\in S_\lambda : p_{\lambda}(\mathcal{Z}_s) \text{ is not stabilized by an abelian subvariety of } \mathcal{B}^{\lambda}_s\},\end{aligned}$$ $$R_{gf}\subset S \text{ is dense} \tag{$*$},$$ $$R_{ab}\cap R_{gf}\subset S \text{ is dense} \tag{$**$},$$ $$R_{st}\cap R_{gf}\subset S \text{ is dense} \tag{$***$}.$$ Note that these sets and conditions depend on $\mathcal{Z}$ and $l$ and, while $\mathcal{Z}$ should ususally be clear from the context, we will say $(*)$ holds for a specified value of $l$. Moreover, we will always assume that $\mathcal{Z}\to S$ is irreducible and has irreducible fibers. Given an abelian variety $A$ we will denote by $T_A:=T_{A,0_A}$ the tangent space to $A$ at $0_A\in A$. We let $$\begin{aligned}
\mathscr{T}/S&:=T_{\mathcal{A}}/S,\\
G/S&:=\text{Gr}(g-l,{\mathscr{T}})/S,\\
G'/S&:=\text{Gr}(g-l',{\mathscr{T}})/S,\\
$$ and we consider the following sections $$\begin{aligned}
&\sigma_{\lambda}: S_\lambda\to G_{{S_\lambda}}=\text{Gr}(g-l,{\mathscr{T}}_{S_\lambda}),\;\qquad \qquad \qquad \sigma_{\lambda}(s):=T_{\ker(p_{\lambda,s})},\\
&\sigma_{\lambda,\mu}: S_{\lambda,\mu}\to G'_{{S_{\lambda,\mu}}}= \text{Gr}(g-l',{\mathscr{T}}_{S_{\lambda,\mu}}),\qquad \;\;\sigma_{\lambda,\mu}(s):=T_{\ker(p_{\lambda,\mu,s})}. \end{aligned}$$
Let $\mathcal{A}/S$ be a family of abelian varieties, and $\mathcal{Z}\subset \mathcal{A}$ a subvariety which is flat over the base. Then, the set of $s\in S$ such that $\mathcal{Z}_s$ is stabilized by a positive dimensional abelian subvariety of $\mathcal{A}_s$ is closed in $S$.
Consider the morphism $\mathcal{Z}\times_S\mathcal{A}\to \mathcal{A}$ given by $(z,a)\mapsto (z+a)$ and let $\mathcal{R}$ be the preimage of $\mathcal{Z}\subset \mathcal{A}$. Since $\mathcal{Z}\to S$ is flat, so is $\mathcal{Z}\times_S\mathcal{A}\to \mathcal{A}$. Since flat morphisms are open, the image of $\mathcal{Z}\times_S\mathcal{A}\setminus \mathcal{R}$ in $\mathcal{A}$ is open. The complement of this image is the closed subset $\mathcal{B}\subset \mathcal{A}$ which is the maximal abelian subscheme stabilizing $\mathcal{Z}$. Finally, the subset of $S$ over which $\mathcal{B}$ has positive dimensional fibers is closed by upper semi-continuity of fiber dimension.
$\bigcup_{\lambda\in \Lambda_l}\sigma_\lambda(S_{\lambda})\subset G$ is dense.
It suffices to consider the locus of abelian varieties isogenous to $E^g$ for some elliptic curve $E$. This locus is dense in $S$ and, given $s$ in this locus and any $M\in M_{k\times (g-l)}(\mathbb{Z})$ of rank $(g-l)$, the tangent space $T_{E^{g-l}_{M}}\in G_s$ is contained in $\sigma_\lambda(S_{\lambda})$ for some $\lambda\in \Lambda_l$. Since $$\{T_{E^{g-l}_{M}}: M\in M_{k\times (g-l)}(\mathbb{Z}),\; \text{rank} (M)=g-l\}\subset G_s$$ is dense in $G_s$, the result follows.
In the following $\mathcal{A}/S$ will be an almost complete family of abelian varieties.
If $\mathcal{Z}\subset \mathcal{A}^k$ is foliated by positive dimensional $\text{CCS}_k$ and $\dim A\geq 2$, then, for very general $s\in S$, the subset $\mathcal{Z}_s$ is not an abelian subvariety of the form $A^r_M$.
Consider the Zariski closed sets $$\{s\in S: \mathcal{Z}_s=(\mathcal{A}_{s})^r_M\}\subset S.$$ There are countably many such sets so it suffices to show that none of them is all of $S$. Suppose that $\mathcal{A}^r_M$ is foliated by positive dimensional $\text{CCS}_k$. By the previous lemma, there is a $\lambda\in \Lambda_2$ such that $p_\lambda((\mathcal{A}_{s})^r_M)=(\mathcal{B}^\lambda_s)^r_M$ is also foliated by positive dimensional orbits for generic $s\in S_\lambda$. This contradicts Corollary [\[notfoliated2\]](notfoliated2).
\[\*implies\*\*\] If $\mathcal{Z}\subset \mathcal{A}^k/S$ is foliated by positive dimensional orbits and $l\geq 2$, then $$(*)\implies (**).$$
First, observe that if $p_{\lambda}|_{\mathcal{Z}_s}: \mathcal{Z}_s\to \mathcal{B}_s^k$ is generically finite on its image, then its image is foliated by positive dimensional orbits. Moreover, if $R_{gf}\cap S_\lambda\neq \emptyset$, then $R_{gf}\cap S_\lambda$ is open in $S_\lambda$. Let $\lambda$ be such that $R_{gf}\cap S_\lambda\neq \emptyset$. For very general $s\in R_{gf}\cap S_\lambda$ the abelian variety $\mathcal{B}_s$ is simple. Thus, if the Zariski closed subset $p_\lambda(\mathcal{Z}_s)$ is an abelian subvariety of $\mathcal{B}_s^k$, it must be abelian subvariety of the form $\mathcal{B}_{M,t}^r$, contradicting Lemma [\[notfoliated2\]](notfoliated2). Hence if $R_{gf}\cap S_\lambda$ is non-empty then $R_{gf}\cap R_{ab}\cap S_\lambda$ is dense in $S_\lambda$.
We will also need the following Zariski closed subsets $$\begin{aligned}
S_{\lambda}(B)=&\{s\in S_{\lambda}: \mathcal{B}^{\lambda}_s\cong B\}\subset S_{\lambda}\\
S_{\lambda,\mu}(D,F)=&\{s\in S_{\lambda,\mu}: \mathcal{D}^{\lambda,\mu}_s\cong D,\mathcal{F}^{\lambda,\mu}_s\cong F\}\subset S_{\lambda,\mu}.\end{aligned}$$
The main result we will prove in this section is a generalization of Voisin’s method from [@V]. Recall that, given varieties $X,S$, a variety $\mathcal{Z}\subset X_S$ dominant over $S$ gives rise to a morphism from (an open in) $S$ to the Chow variety parametrizing cycles of class $[\mathcal{Z}_{s}]$ on $X$, where $s\in S$ is generic. We remind the reader of the notational convention of Remark [\[convention\]](convention), which allows us to remove the words in parenthesis from the previous sentence.
\[prop1\] Let $\mathcal{Z}\subset \mathcal{A}^k$ be a $d$-dimensional variety dominant over $S$, and satisfying $(*)$ and $(**)$. Then there exists a $\lambda\in \Lambda_l$ such that $$p_\lambda(\mathcal{A}_{S_{\lambda}(B)})\subset B_{S_{\lambda}(B)}=\mathcal{B}^\lambda_{S_\lambda(B)}$$ gives rise to a finite morphism from $S_{\lambda}(B)$ to the appropriate Chow variety.
From Lemma [\[\*implies\*\*\]](*implies**), if $\mathcal{Z}$ is foliated by $\text{CCS}_k$ then it satisfies $(**)$. Hence, the key assumption for our applications will be $(*)$. We propose to use these methods to prove Conjecture [\[Vweakconj\]](Vweakconj) in the following way: Assume that a very general abelian variety of dimension $2k-1$ has a one-dimensional $\text{CCS}_k$. This gives $\mathcal{Z}\subset \mathcal{A}^k/S$ of relative dimension $1$. It is easy to show that $(*)$ holds in this setting so we can use the previous proposition to get a one-parameter family of one-dimensional $\text{CCS}_k$ $$p_{\lambda}(\mathcal{Z}_{S_\lambda}(B))\subset (B_{S_\lambda(B)})^k$$ in a generic abelian variety $B$ of dimension $(g-1)$ with some polarization $\theta^\lambda$. This gives $$\mathcal{Z}'\subset ({\mathcal{B}}^{\lambda})^k/S_{\lambda}$$ which has relative dimension $2$ and such that $\mathcal{Z}'_s$ is foliated by positive dimensional orbits for generic $s\in S_\lambda$.\
We can hope to inductively apply Proposition [\[prop1\]](prop1) to $\mathcal{Z}'/S_\lambda$, eventually getting a large-dimensional subvariety of $B^k$ foliated by positive dimensional orbits, for an abelian surface $B$. Corollary [\[absurfbound\]](absurfbound) would then provide the desired contradiction. The key issue here is to ensure that condition $(*)$ is satisfied. At each step the dimension of the variety $\mathcal{Z}$ to which we apply Proposition [\[prop1\]](prop1) grows. Hence, it gets harder and harder to ensure generic finiteness of the projection. We have found a way around this using the fact that the variety $\mathcal{Z}$ to which we apply the proposition is obtained by successive degenerations and projections. We will introduce this argument in the following section.
We first reduce to the case where the condition $(***)$ holds. Consider $s_0\in R_{gf}\cap R_{st}$ such that $\mathcal{B}_{s_0}^{\lambda_0}$ is simple, where $s_0\in S_{\lambda_0}$ and $p_{\lambda_0}|_{\mathcal{Z}_{s_0}}$ is generically finite on its image. Suppose that $p_{\lambda_0}(\mathcal{Z}_0)$ is stabilized by $(\mathcal{B}_{s_0}^{\lambda_0})_M^r$ but not by any larger abelian subvariety of $\mathcal{B}_{s_0}^{\lambda_0}$. Then $$p_{\lambda_0}(\mathcal{Z}_{s_0})/(\mathcal{B}_{s_0}^{\lambda_0})_M^r\subset(\mathcal{B}_{s_0}^{\lambda_0})^k/ \mathcal(\mathcal{B}_{s_0}^{\lambda_0})_M^r$$ is not stabilized by any abelian subvariety of $(\mathcal{B}_{s_0}^{\lambda_0})^k/ \mathcal(\mathcal{B}_{s_0}^{\lambda_0})_M^r$. We have a diagram $$\begin{tikzcd}\label{lambdadiag1}
\mathcal{Z}_{S_{\lambda_0}}/(\mathcal{A}_{S_{\lambda_0}})_{M}^r \ar[r,dashed,"g"] \arrow{d}[swap]{p_{\lambda_0}}& \text{Gr}(d,{\mathscr{T}}_{S_{\lambda_0}}^k/T_{(\mathcal{A}_{S_{\lambda_0}})^r_M}) \ar[d,dashed,"\pi_{\sigma_{\lambda_0}(S_{\lambda_0})}"]\\
p_{\lambda_0}(\mathcal{Z}_{S_{\lambda_0}})/(\mathcal{B}_{S_{\lambda_0}})_M^r \ar[r,dashed,"g"]& \text{Gr}\left(d,\text{Gr}(d,{\mathscr{T}}_{S_{\lambda_0}}^k/[\sigma_{\lambda_0}(S_{\lambda_0})^k+T_{(\mathcal{B}^{\lambda_0})^r_M}]\right),
\end{tikzcd}$$ where $g$ denotes the Gauss map. Here $\pi_{\sigma_{\lambda_0}(S_{\lambda_0})}$ is the map induced by the quotient $${\mathscr{T}}_{S_{\lambda_0}}^k/T_{(\mathcal{A}_{S_{\lambda_0}})^r_M}\to {\mathscr{T}}_{S_{\lambda_0}}^k/[\sigma_{\lambda_0}(S_{\lambda_0})^k+T_{(\mathcal{B}^{\lambda_0})^r_{M}}].$$ We also denote by $p_{\lambda_0}$ the map induced by $p_{\lambda_0}$ on $ \mathcal{Z}_{S_{\lambda_0}}/(\mathcal{A}_{S_{\lambda_0}})^r_M$. Now, consider the base change by $G\to S$ $$\mathcal{Z}_{G}\subset \mathcal{A}^k_{G}.$$ The section $\sigma_{\lambda_0}$ is a closed immersion $S_{\lambda_0} \to G$ and $\mathcal{Z}_{S_{\lambda_0}}$ is the base change of $\mathcal{Z}_{G}$ under this immersion. The upper right corner of diagram ([\[lambdadiag1\]](lambdadiag1)) is the base-change by $\sigma_{\lambda_0}:S_{\lambda_0}\to G$ of the diagram
$$\label{diag1}
\begin{tikzcd}\mathcal{Z}_{G}/(\mathcal{A}_{M}^r)_{G}\ar[r,dashed, "g"] & \text{Gr}(d,{\mathscr{T}}_G^k/T_{(\mathcal{A}^r_{M})_G}) \ar[d,dashed, "\pi"]\\
& \text{Gr}(d, [{\mathscr{T}}_{G}/\mathcal{U}]^k/T_{(\mathcal{B}^r_{M})_G}),\\
\end{tikzcd}$$
where $\mathcal{U}\to G:=\text{Gr}(g-l,T)$ is the universal bundle and $\pi$ is the rational map induced by the quotient map. The composition $\pi\circ g$ is defined on a Zariski open in $\mathcal{Z}_G$ which meets $\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/\mathcal{A}_{M,\sigma_{\lambda_0}(s_0)}^r$ non-trivially. Indeed $g$ is defined on the smooth locus of $$\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/(\mathcal{A}_{\sigma_{\lambda_0}(s_0)})^r_M),$$ and the restriction of $p_{\lambda_0}$ to $\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/(\mathcal{A}_{\sigma_{\lambda_0}(s_0)})^r_M)$ is generically finite on its image by the following:
Consider an abelian variety $A\sim B\times E$, where $B$ and $E$ are abelian varieties of smaller dimension, and let $p$ be the projection $A^k\to {B}^k$. If $Z\subset A^k$ is such that $p|_{Z}: Z\to p(Z)$ is generically finite, then the projection $p: A^k/A_{M}^r\to {B}^k/{B}_{M}^r$ is such that $p|_{Z/A_M^r}$ is generically finite on its image.
Since $p|_{Z/A_M^r}$ is proper and locally of finite presentation, it suffice to show that it is quasi-finite. The fiber of $A^k\to {B}^k/{B}^r_M$ over $p(z)\in p(Z)/{B}^r_M$ for $z\in Z$ is the set of all $A^r_M$-translates of the fiber of $p$ over $p(z)$. Hence the fiber of $p|_{Z/A_M^r}$ over $p(z)$ is finite.
We deduce that $q:=g\circ \pi$ is defined in an open in the smooth locus of $\mathcal{Z}_{s_0}/(\mathcal{A}_{M,s_0}^r)$, so that $q$ is defined in an open in $\mathcal{Z}_G$ meeting $\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/(\mathcal{A}_{s_0})_M^r$ non-trivially.\
Since $p_{\lambda_0}(\mathcal{Z}_{s_0})/(\mathcal{B}_{s_0}^{\lambda_0})_M^r$ is not stabilized by any abelian subvariety of $(\mathcal{B}^{\lambda_0}_{s_0})^k$, the Gauss map $g$ at the bottom of diagram ([\[lambdadiag1\]](lambdadiag1)) is defined on the smooth locus of $p_{\lambda_0}(\mathcal{Z}_{s_0})/(\mathcal{B}^{\lambda_0}_{s_0})_M^r$ and generically finite by results of Griffiths and Harris (see (4.14) in [@GH]). The fact that the restriction of $p_{\lambda_0}$ to $\mathcal{Z}_{s_0}/(\mathcal{A}_{s_0})^r_M$ is generically finite on its image implies that the following composition is generically finite on its image $$\begin{tikzcd}g\circ p_{\lambda_0}: \mathcal{Z}_{s_0}/(\mathcal{A}_{s_0})_M^r\ar[r,dashed] & \text{Gr}\Big(d,{\mathscr{T}}^k/[\sigma_{\lambda_0}(s_0)^k+T_{(\mathcal{B}^{\lambda_0}_{s_0})^r_M}]\Big).
\end{tikzcd}$$ It follows that, restricting to an open of $G$ if needed, the map $q_t:=(\pi\circ g)_t$ is defined and is generically finite on its image for all $t\in G$. This generic finiteness statement implies that, for $s\in S_\lambda$ (such that $\sigma_\lambda(s)$ lies in an appropriate open in $G$), the Gauss map of $p_\lambda(\mathcal{Z}_{s})/(\mathcal{B}^\lambda_{s})_M^r$ is generically finite. Namely, for such an $s$, the variety $p_\lambda(\mathcal{Z}_s/(\mathcal{B}^\lambda_{s})_M^r)$ is not stabilized by an abelian subvariety. Moreover, if $\lambda\in \Lambda_l$ and $B$ in the family $\mathcal{B}^\lambda$ are such that the family $$p_{\lambda}(\mathcal{Z}_{S_{\lambda}(B)}/\mathcal{A}^r_{M})\subset ({B}^k/{B}^r_{M})_{S_\lambda(B)}$$ gives rise to a finite morphism from $S_{\lambda}(B)$ to the appropriate Chow variety of ${B}^k/{B}^{r}_M$, then the family $$p_{\lambda}(\mathcal{Z}_{S_{\lambda}(B)})\subset {B}^k_{S_\lambda(B)}$$ also gives rise to a finite morphism from $S_{\lambda}(B)$ to the appropriate Chow variety of $B^k$. Hence, replacing $\mathcal{Z}$ by $\mathcal{Z}/\mathcal{A}_{M}^r$ and $\mathcal{A}^k$ by $\mathcal{A}^k/\mathcal{A}^r_{M}$ we are reduced to the case where $(***)$ holds.\
Now consider the analogue of diagram ([\[lambdadiag1\]](lambdadiag1))
$$\begin{tikzcd}\label{diaglambda}
\mathcal{Z}_{S_{\lambda}} \ar[r,dashed,"g"] \arrow{d}[swap]{p_{\lambda}}& \text{Gr}(d,T^k) \ar[d,dashed,"\pi_{\sigma_{\lambda}(S_\lambda)}"]\\
p_{\lambda}(\mathcal{Z}_{S_{\lambda}})\ar[r,dashed,"g"]& \text{Gr}(d,[T/\mathcal{U}_{\sigma_{\lambda}(S_\lambda)}]^k)
\end{tikzcd}$$
as well as the analogue of diagram ([\[diag1\]](diag1))
$$\label{diag}
\begin{tikzcd}\mathcal{Z}_{G}\ar[r,dashed, "g"] & \text{Gr}(d,T_G^k) \ar[d,dashed, "\pi"]\\
& \text{Gr}(d, [T_{G}/\mathcal{U}]^k).\\
\end{tikzcd}$$
By the discussion above there is an open in $\mathcal{Z}_{G}$ on which the composition $q:=\pi\circ g$ is defined and generically finite on its image. The diagram ([\[diaglambda\]](diaglambda)) provides a factorization of $q$ along each $\sigma_\lambda(S_\lambda)$
$$\begin{tikzcd}\label{diagfact}
\mathcal{Z}_{\sigma_\lambda(S_\lambda)}\cong \mathcal{Z}_{S_\lambda} \ar[rr,dashed, "q:=\pi\circ g"] \ar[dr,swap,"p_\lambda"] &\; & \text{Gr}(d, [T_G/\mathcal{U}]^k)\\
& p_{\lambda}(\mathcal{Z}_{S_{\lambda}}).\ar[ur,swap,dashed,"g"]&
\end{tikzcd}$$
Hence (3.5) gives a factorization of $q$ on a Zariski dense subset of the base $G$.
\[cov1\] Let $\mathcal{Z}/S$ be a family with irreducible fibers and base, and $q: \mathcal{Z}/S\to \mathcal{X}/S$ be such that $q_s: \mathcal{Z}_s\to \mathcal{X}_s$ is generically finite for each $s\in S$. Consider $S'\subset S$, a Zariski dense subset, and suppose that for each $s'\in S'$ we have a factorization of $q_s$ $$\mathcal{Z}_s\xrightarrow{f_{s'}} f_{s'}(\mathcal{Z}_s)\xrightarrow{g_{s'}} \mathcal{X}_s.$$ Then there is a family $\mathcal{Z}'/S$, morphisms $p: \mathcal{Z}\to \mathcal{Z}'$ and $p': \mathcal{Z}'\to \mathcal{X}$, and a Zariski dense subset $S''\subset S'$ such that, for any $s''\in S''$, the morphisms $p_{s''}(\mathcal{Z}_{s''})$ and $f(\mathcal{Z}_{s''})$ are birational, and $p_{s''}$ and $p'_{s''}$ induce the same morphism on function fields as $f_{s''}$ and $g_{s''}$ respectively.
Restrict to a Zariski open subset of $\mathcal{X}$ (which we call $\mathcal{X}$ in keeping with remark [\[convention\]](convention)) over which $q$ is finite étale and such that $\mathcal{X}\to S$ is smooth. By work of Hironaka, we can find a compactification $\overline{\mathcal{X}}$ of $\mathcal{X}$ with simple normal crossing divisors at infinity. Restricting to an open in the base, we can assume that we have $\mathcal{X}/S\subset \overline{\mathcal{X}}/S$, such that $\overline{\mathcal{X}}_s\setminus \mathcal{X}_s$ is an snc divisor for any $s\in S$. One can use a version of Ehresmann’s lemma allowing for an snc divisor at infinity to see that $\mathcal{X}\to S$ is a locally-trivial fibration in the category of smooth manifolds.\
It follows that we get a diagram $$\begin{tikzcd}
\mathcal{Z}\arrow{r}{q} &\mathcal{X} \arrow{d} \\
&S,
\end{tikzcd}$$ where $q$ is a covering. Note that we renamed as $\mathcal{Z}$ an open subset of $\mathcal{Z}$ over which $q$ is étale. To complete the proof of Proposition [\[prop1\]](prop1) we will need the following Lemma.
\[cov2\] Given a diagram as above with $\mathcal{Z}_s$ connected for every $s$, and a factorization $$\begin{tikzcd}
\mathcal{Z}_{s_0}\ar[rr,"q"] \ar{dr}[swap]{f_{s_0}} & &\mathcal{X}_{s_0} \\
&f_{s_0}(\mathcal{Z}_{s_0}), \ar{ur} &
\end{tikzcd}$$ there is a factorization $$\begin{tikzcd}
\mathcal{Z}\ar[rr,"q"] \ar{dr}[swap]{f} & &\mathcal{X} \\
&f(\mathcal{Z}), \ar{ur} &
\end{tikzcd}$$ which identifies with the original factorization over $s_0\in S$.
Consider the Galois closure $\mathcal{Z}'\to \mathcal{X}$ of the covering $q: \mathcal{Z}\to \mathcal{X}$. Note that $\mathcal{Z}'_{s_0}$ is connected. Indeed there is a normal subgroup of the Galois group of $\mathcal{Z}'/\mathcal{X}$ corresponding to a deck transformation inducing the trivial permutation of the connected components of $\mathcal{Z}'_{s_0}$. This subgroup corresponds to a cover above $\mathcal{Z}$ since $\mathcal{Z}_{s_0}$ is connected. It follows that the map $\text{Gal}(\mathcal{Z}'/\mathcal{X})\to \text{Gal}(\mathcal{Z}'_{s_0}/\mathcal{X}_{s_0})$ is injective because a deck transformation which is the identity on the base and on fibers must be the identity. Since $\text{Gal}(\mathcal{Z}'/\mathcal{X})$ has order $$d:=\deg(\mathcal{Z}'/\mathcal{X})=\deg(\mathcal{Z}_{s_0}'/\mathcal{X}_{s_0}),$$ and $\text{Gal}(\mathcal{Z}'_{s_0}/\mathcal{X}_{s_0})$ has order at most $d$, this restriction morphism must be an isomorphism and $\mathcal{Z}_{s_0}'/\mathcal{X}_{s_0}$ is thus also Galois. One then has an equivalence of categories between the poset of intermediate coverings of $\mathcal{Z}'/\mathcal{X}$ and that of $\mathcal{Z}_{s_0}'/\mathcal{X}_{s_0}$, and hence between the poset of intermediate coverings of $\mathcal{Z}/\mathcal{X}$ and that of $\mathcal{Z}_{s_0}/\mathcal{X}_{s_0}$.
By the previous lemma, to each factorization $f_{s'}$ we can uniquely associate an intermediate cover $\mathcal{Z}\to \mathcal{Z}^{s'}$ of $\mathcal{Z}\to \mathcal{X}$, which agrees with $f_{s'}$ at $s'$. Since there are only finitely many intermediate covers of $\mathcal{Z}\to \mathcal{X}$, we get a partition of $S'$ according to the isomorphism type of the cover $\mathcal{Z}\to \mathcal{Z}^{s'}$. One subset $S''\subset S'$ of this partition must be dense in $S$. Let $f: \mathcal{Z}\to f(\mathcal{Z})$ be the corresponding intermediate cover.
For the rest of the proof of Proposition [\[prop1\]](prop1) we are back in the situation of diagrams ([\[diaglambda\]](diaglambda)), ([\[diag\]](diag)), and ([\[diagfact\]](diagfact)).
There is a variety $\mathcal{Z}'/G$, a morphism $p: \mathcal{Z}\to \mathcal{Z}'$, and a subset $\Lambda_{l,0}\subset \Lambda_{l}$ such that $$\bigcup_{\lambda\in \Lambda_{l,0}}\sigma_{\lambda}(S_{\lambda})\subset G$$ is dense, and such that $p_{\lambda}(\mathcal{Z}_t)$ and $p(\mathcal{Z}_t)$ are birational, and $p_t: \mathcal{Z}_t\to p(\mathcal{Z}_t)$ and $p_{\lambda,t}: \mathcal{Z}_t\to p_\lambda(\mathcal{Z}_t)$ induce the same map on function fields, for any $\lambda\in \Lambda_{l,0}$, $t\in \sigma_{\lambda}(S_\lambda)$.
This follows from the previous lemma and its proof once we observe that the intermediate covering of $q$ (or rather of an appropriate finite étale restriction of $q$ as above) associated to the factorization $p_{\lambda,t}: \mathcal{Z}_t\to p_{\lambda}(\mathcal{Z}_t)$ is independent of $t\in \sigma_{\lambda}(S_\lambda)$.
A technical point is that the maps $p_\lambda$ are a priori only defined after passing to a generically finite cover of $S_\lambda$. This does not cause problem as $p_{\lambda,s}$ is defined without passing to a generically finite cover and, given a generically finite cover $S_\lambda'\to S_\lambda$, the isomorphism type of the covering $p_{\lambda,s}: \mathcal{Z}_{s}\to p_{\lambda,s}(\mathcal{Z}_s)$ is the same as that of $p_{\lambda,s}: \mathcal{Z}^{\sharp} _{s}\to p_{\lambda,s}(\mathcal{Z}^\sharp_s)$, where $\mathcal{Z}^\sharp$ is obtained from the cover as prescribed in Remark [\[convention\]](convention).
To finish the proof of Proposition [\[prop1\]](prop1), consider desingularizations $$\widetilde{p}: \widetilde{\mathcal{Z}}/G\to \widetilde{\mathcal{Z}'}/G$$ with smooth fibers over $G$. We have the inclusion $$j: \mathcal{Z}/G\to \mathcal{A}^k/G$$ as well as $$\widetilde{j}: \widetilde{\mathcal{Z}}/G\to \mathcal{A}^k/G.$$ The morphism $\widetilde{j}$ gives rise to a pullback map $$\widetilde{j}^*: \text{Pic}^0(\mathcal{A}^k/G)\to \text{Pic}^0(\mathcal{\widetilde{Z}}/G).$$ Since $\widetilde{p}$ is generically finite on fibers we can consider the composition $$\widetilde{p}_*\circ \widetilde{j}^* : \text{Pic}^0(\mathcal{A}^k/G)\to \text{Pic}^0(\widetilde{\mathcal{Z}}/G)\to \text{Pic}^0(\widetilde{\mathcal{Z'}}/G).$$ This is a morphism of abelian schemes and we will show that it is non-zero along $\sigma_{\lambda}(S_\lambda)$, $\lambda\in \Lambda_{l,0}$. Consider $t\in \sigma_{\lambda}(S_\lambda)$. Then $\mathcal{A}_t$ is isogenous to $\mathcal{B}_t^{\lambda}\times \mathcal{E}_t^{\lambda}$. We have the following commutative diagram $$\begin{tikzcd}
\mathcal{Z}_t \arrow{r}{j} \arrow{d}[swap]{{p_\lambda}|_{\mathcal{Z}_t}} & \mathcal{A}_t^k \arrow{d}{p_{\lambda}}\\
p_{\lambda}(\mathcal{Z}_t) \arrow{r}{j'}& (\mathcal{B}_t^{\lambda})^k .
\end{tikzcd}$$ Consider a desingularization of $p_\lambda(\mathcal{Z}_t)$ and the induced map $$\widetilde{j}': \widetilde{p_\lambda(\mathcal{Z}_t)}\to (\mathcal{B}_t^{\lambda})^k.$$ Using the fact that $\widetilde{p}_t$ identifies birationally to $p_{\lambda,t}$, we get the following commutative diagram $$\begin{tikzcd}
\text{Pic}^0(\widetilde{\mathcal{Z}}_t) & \text{Pic}^0(\mathcal{A}_t^k) \arrow{l}[swap]{\widetilde{j}^*}\\
\text{Pic}^0(\widetilde{\mathcal{Z}}'_t)\cong \text{Pic}^0(\widetilde{p_\lambda(\mathcal{Z}_t)}) \arrow{u}{\widetilde{p}^*} & \text{Pic}^0({\mathcal{B}_t^{\lambda}}^k) \arrow{l}[swap]{\widetilde{j}'^*} \arrow{u}[swap]{p_{\lambda}^*} .
\end{tikzcd}$$ It follows that $$\begin{aligned}
\widetilde{p}_*\circ (\widetilde{j}^*\circ {p_{\lambda}}^*)=\widetilde{p}_*\circ (\widetilde{p}^*\circ \widetilde{j}'^*)=(\widetilde{p}_*\circ \widetilde{p}^*)\circ \widetilde{j}'^*=[\deg(\widetilde{p})]\circ \widetilde{j}'.\end{aligned}$$ Since $p_\lambda(\mathcal{\mathcal{Z}}_t)$ is positive dimensional, the morphism $\widetilde{j}'$ is non-zero and so $\widetilde{p}_*\circ \widetilde{j}^*$ is non-zero.\
Hence the kernel of $\widetilde{p}_*\circ \widetilde{j}^*$ is an abelian subscheme of $\mathcal{A}^k$ which is not all of $\mathcal{A}^k$. For very general $t\in G$ the abelian variety $\mathcal{A}_t$ is simple. Therefore, for such a $t$ the abelian subvariety $\ker(\widetilde{p}_*\circ \widetilde{j}^*)_t$ is of the form $(\mathcal{A}_{t})_M^r$, with $M\in M_{k\times r}(\mathbb{Z})$ of rank $r$, and $r\leq k-1$. Choosing $M$ and $r$ such that $$\{t\in G: \ker(\widetilde{p}_*\circ \widetilde{j}^*)_t=(\mathcal{A}_{t})_M^r\}\subset G$$ is dense, and observing that this set is closed, we see that $\ker(\widetilde{p}_*\circ \widetilde{j}^*)_t=(\mathcal{A}_{t})_M^r$ for all $t\in G$.\
Note that, for $t\in \sigma_{\lambda}(S_\lambda)$, we have $$\text{Pic}^0(\ker(p_{\lambda,t}))/\ker (\widetilde{p}_*\circ \widetilde{j}^*)_t\cap\ker(p_{\lambda,t})=\text{Pic}^0(\ker(p_{\lambda,t}))/(\mathcal{A}_{t})_M^r\cap\ker(p_{\lambda,t})\neq 0.$$ Now consider $\lambda\in \Lambda_l$ such that $\sigma_\lambda(S_\lambda)\neq \emptyset$, namely such that some point in $\sigma_{\lambda}(S_\lambda)$ has survived our various restriction to Zariski open subsets, and $B\in \mathcal{B}^\lambda$ such that $\sigma_\lambda(S_{\lambda}(B))\neq \emptyset$. Suppose that there is a curve $C\subset \sigma_\lambda(S_{\lambda}(B))\cong S_\lambda(B)$ such that $p_\lambda(\widetilde{Z}_t)=p_\lambda(\widetilde{Z}_{t'})$ for any $t,t'\in C$, namely such that $C$ is contracted by the morphism from $S_{\lambda}(B)$ to the Chow variety associated to the family $p_{\lambda}(\mathcal{Z}_{S_{\lambda}(B)})\subset B^k_{S_\lambda(B)}$. Now, since $\text{Pic}^0(\widetilde{Z}_t')\cong \text{Pic}^0(\widetilde{p_\lambda(\mathcal{Z}_t)})$ does not depend on $t\in C$, it must contain a variable abelian variety. This provides the desired contradiction and completes the proof of Proposition [\[prop1\]](prop1).
Salvaging generic finiteness and a proof of Voisin’s Conjecture
===============================================================
In this section we refine the results from the previous section in order to bypass assumption $(*)$ in the inductive application of Proposition [\[prop1\]](prop1). The idea is quite simple: In the last section we saw that we can degenerate to abelian varieties $A$ isogenous to $B\times E$ in such a way that, if we consider the restriction of the projection $A^k\to B^k$ to $Z\subset A^k$, the image of this projection varies with $E$. Here we want to degenerate to abelian varieties isogenous to $D\times F\times E$, where $E$ is an elliptic curve, and consider the restriction of the projections $A^k\to D^k$ and $A^k\to (D\times F)^k$ to $Z\subset A^k$. We can do this in such a way that the images of both of these projections vary with $E$. Hence, if we consider in $(D\times F)^k$ and $D^k$ the union of the image of these projections for every $E$, we get varieties $Z_1\subset (D\times F)^k$ and $Z_2\subset D^k$ of dimension $\dim Z+1$, and the restriction of the projection $(D\times F)^k\to D^k$ to $Z_1$ has image $Z_2$. It follows at once that this restriction is generically finite on its image. We spend this section making this simple idea rigorous and deducing a proof of Theorem [\[2k-2\]](2k-2).\
The families $\mathcal{B}^\lambda/S_\lambda$, $\lambda\in \Lambda_{(g-1)}$ introduced in the last section are families of $(g-1)$-dimensional abelian varieties with some polarization type $\theta^\lambda$. Hence, they give rise to a diagram $$\begin{tikzcd}
(\mathcal{B}^\lambda)^k \arrow{r}{\varphi_\lambda} \arrow{d}& \mathcal{A'}^k \arrow{d}\\
S_\lambda \arrow{r}{\psi_\lambda} & S',
\end{tikzcd}$$ where $\mathcal{A'}/S'$ is the universal family over the moduli stack of abelian varieties of dimension $(g-1)$ with polarization $\theta^\lambda$. Note that $\mathcal{A'}/S'$ depends of course on $\lambda$ but we suppress $\lambda$ from the notation. We will think of $\mathcal{A}'/S'$ as another locally complete family of abelian varieties. This family comes with its own set $\Lambda_{l'}'$ indexing loci $S_{\eta'}$ along which $\mathcal{A}'_s\sim \mathcal{B'}_s^\eta\times\mathcal{E'}^\eta_s$. Let $\varphi_{\lambda,\mu}$ be the composition of $\varphi_\lambda|_{(\mathcal{B}^\lambda_{S_{\lambda,\mu}})^k}$ with $$(\mathcal{D}^{\lambda,\mu})^k\to (\mathcal{D}^{\lambda,\mu}\times \mathcal{F}^{\lambda,\mu})^k\to (\mathcal{B}^{\lambda}_{S_{\lambda,\mu}})^k,$$ where the last map is the isogeny encoded by $\mu$. We get a diagram $$\begin{tikzcd}
(\mathcal{D}^{\lambda,\mu})^k \arrow{r}{\varphi_{\lambda,\mu}} \arrow{d}& (\mathcal{B'}^\eta)^k \arrow{d}\\
S_{\lambda,\mu} \arrow{r} & S'_{\eta},
\end{tikzcd}$$ where $\eta$ is some index in $\Lambda'_{l'}$, and $\psi_{\lambda}(S_{\lambda,\mu})=S_\eta'$. The main result of this section is:
\[mainprop\] Suppose that $\mathcal{Z}\subset \mathcal{A}^k$ satisfies $(*)$ and $(**)$ for $l'\geq 2$. Then there exists a $\lambda\in \Lambda_{(g-1)}$ such that $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))/S'$ satisfies $(*)$ for $l'$ and has relative dimension $\dim_S \mathcal{Z}+1$.
By Proposition 3.4, there is a $\lambda\in \Lambda_{(g-1)}$ such that $p_{\lambda}(\mathcal{Z}_{t})$ varies with $t\in S_\lambda(B)$, for generic $B$ in the family $\mathcal{B}^\lambda$ (alternatively such that $\dim_{S'} \varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))=\dim_S\mathcal{Z}+1$). The idea is to show that there is a subset $\Lambda_{l',0}^\lambda\subset \Lambda_{l'}^\lambda$ such that $$\bigcup_{\mu\in \Lambda_{l',0}^\lambda} S_{\lambda,\mu}\subset S_\lambda \text{ is dense}$$ and such that $p_{\lambda,\mu}(\mathcal{Z}_{t})$ varies with $t\in S_{\lambda,\mu}(D,F)$, for generic $D,F$ in the families $\mathcal{D}^{\lambda,\mu}, \mathcal{F}^{\lambda,\mu},$ and $\mu\in \Lambda_{l',0}^\lambda$. Indeed, we have a commutative diagram
$$\begin{tikzcd}
& (\mathcal{B}_{S_{\lambda,\mu}}^{\lambda})^k \ar{rr}{\varphi_\lambda} \ar{dd} \ar{dl}{p_{\lambda,\mu}} & & \mathcal{A'}^k \ar{dd} \ar{dl}{p_{\eta}} \\
( \mathcal{D}^{\lambda,\mu})^k \ar[crossing over]{rr}{\;\;\;\;\;\;\;\;\;\varphi_{\lambda,\mu}} \ar{dd} & & (\mathcal{B'}^{\eta})^k & \\
& S_{\lambda,\mu} \ar{rr} \ar[dl,equal] & & S'_{\eta} \ar[dl,equal] \\
S_{\lambda,\mu} \ar{rr} && S'_{\eta} \ar[from=uu,crossing over]\;, &
\end{tikzcd}$$ where $S'_{\eta}\subset S'$ is the image of $S_{\lambda,\mu}\to S_{\lambda}\to S'$. Consider the restriction of this diagram to $S_{\lambda,\mu}(D,F)$, for $D$ and $F$ in the families $\mathcal{D}^{\lambda,\mu}$ and $\mathcal{F}^{\lambda,\mu}$, namely
$$\begin{tikzcd}
& (\mathcal{B}_{S_{\lambda,\mu}(D,F)}^{\lambda})^k \ar{rr}{\varphi_\lambda} \ar{dd} \ar{dl}{p_{\lambda,\mu}} & & \mathcal{A'}^k \ar{dd} \ar{dl}{p_{\eta}} \\
( \mathcal{D}^{\lambda,\mu}_{S_{\lambda,\mu}(D,F)})^k \ar[crossing over]{rr}{\;\;\;\;\;\;\;\;\;\varphi_{\lambda,\mu}} \ar{dd} & & (\mathcal{B'}^{\eta})^k & \\
& S_{\lambda,\mu}(D,F) \ar{rr} \ar[dl,equal] & & S'_{\eta} \ar[dl,equal] \\
S_{\lambda,\mu}(D,F) \ar{rr} && S'_{\eta} \ar[from=uu,crossing over]\; . &
\end{tikzcd}$$ Now, if $p_{\lambda,\mu}(\mathcal{Z}_t)$ varies with $t\in S_{\lambda,\mu}(D,F)$, we have $$\dim \varphi_{\lambda}(p_{\lambda}(\mathcal{Z}_{S_{\lambda,\mu}(D,F)}))=\dim \varphi_{\lambda,\mu}(p_{\lambda,\mu}(\mathcal{Z}_{S_{\lambda,\mu}(D,F)}))=\dim_S\mathcal{Z}+1,$$ so that $p_{\eta}$ is generically finite. Hence, if there is a subset $\Lambda_{l',0}^\lambda$ such that $$\bigcup_{\mu\in \Lambda_{l',0}^\lambda} S_{\lambda,\mu}\subset S_\lambda$$ is dense, and $p_{\lambda,\mu}(\mathcal{Z}_{t})$ varies with $t\in S_{\lambda,\mu}(D,F)$ for $\mu\in \Lambda_{l',0}^\lambda$, then $\varphi_\lambda(\mathcal{Z}_{S_{\lambda}})\subset \mathcal{A'}^k$ satisfies condition $(*)$ for $l'$.\
Consider $$R_{st}':=\bigcup_{\lambda,\mu}\{s\in S_{\lambda,\mu} : p_{\lambda,\mu}(\mathcal{Z}_s) \text{ is not stabilized by an abelian subvariety}\}.$$ Following the same argument as in the proof of Proposition [\[prop1\]](prop1), we can assume that $R_{st}'$ is dense in $S$ (this is the analogue of condition $(***)$). Let $\mathcal{U}/G$ and $\mathcal{U'}/G'$ be the universal families over $G:=\mathbb{P}(T)$ and $G':=\text{Gr}(g-l',T)$. Consider the following diagrams analogous to ([\[diaglambda\]](diaglambda)) and ([\[diag\]](diag)): $$\begin{tikzcd}
\mathcal{Z}_{S_{\lambda,\mu}} \ar[r,dashed,"g"] \arrow{d}{p_{\lambda}}& \text{Gr}(d,{\mathscr{T}}_{S_{\lambda,\mu}}^k) \ar[d,dashed,"\pi_{\sigma_{\lambda}(S_{\lambda,\mu})}"]\\
p_{\lambda}(\mathcal{Z}_{S_{\lambda,\mu}})\ar[r,dashed,"g"] \arrow{d}{p_{\mu}}& \text{Gr}(d,[{\mathscr{T}}_{S_{\lambda,\mu}}/\mathcal{U}_{\sigma_{\lambda}(S_{\lambda,\mu})}]^k) \ar[d,dashed, "\pi_{\sigma_{\lambda,\mu}(S_{\lambda,\mu})}"]\\
p_{\lambda,\mu}(\mathcal{Z}_{S_{\lambda,\mu}}) \arrow[r,dashed,"g"]& \text{Gr}(d,[{\mathscr{T}}_{S_{\lambda,\mu}}/\mathcal{U'}_{\sigma_{\lambda,\mu}(S_{\lambda,\mu})}]^k),
\end{tikzcd}$$
$$\begin{tikzcd}\mathcal{Z}_{G'}\ar[r,dashed, "g"] & \text{Gr}(d,T_{G'}^k) \ar[d,dashed, "\pi"]\\
& \text{Gr}(d, [T_{G'}/\mathcal{U'}]^k).\\
\end{tikzcd}$$
Just as in the proof of Proposition [\[prop1\]](prop1), since $\mathcal{Z}$ satisfies $(*)$ and $(**)$ for $l'$, we can assume that $q_t:=\pi_t\circ g_t$ is generically finite on its image for any $t\in G'$, restricting to a Zariski open in $G'$ if necessary. Note that, for $t\in S_{\lambda,\mu}$, we have a factorization $$\begin{tikzcd}
\mathcal{Z}_t \arrow[rrr, dashed,"q"] \arrow{dr}{p_{\lambda}}& & & \text{Gr}(d,[T_{G'}/\mathcal{U}']^k)\\
&p_{\lambda}(\mathcal{Z}_t) \arrow{r}{p_\mu} & p_{\lambda,\mu}(\mathcal{Z}_t) \arrow[ur,dashed,"g"]. &
\end{tikzcd}$$ By Proposition [\[prop1\]](prop1), there is a $\lambda$ such that $\dim_{S'} \varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))/S'=\dim_S\mathcal{Z}+1$. One can consider analogues of Lemma [\[cov1\]](cov1) and [\[cov2\]](cov2) and see that there is a partition of $\Lambda_{l'}^{\lambda}$ in finitely many sets according to the isomorphism type of the covering $p_{\mu}$. Hence, since $$\bigcup_{\mu\in \Lambda_{l'}^{\lambda}}S_{\lambda,\mu}\subset S_{\lambda}$$ is dense, there is a subset $\Lambda_{l',0}^\lambda\subset \Lambda_{l'}^\lambda$, such that $$\bigcup_{\mu\in \Lambda_{l',0}^\lambda} S_{\lambda,\mu}\subset S_\lambda$$ is dense, and such that $p_{\lambda,\mu}(\mathcal{Z}_{t})$ varies with $t\in S_{\lambda,\mu}(D,F)$ for generic $D,F$ in the families $\mathcal{B}^{\lambda,\mu}, \mathcal{D}^{\lambda,\mu}$, and $\mu\in \Lambda_{l',0}^\lambda$.
Suppose that a very general abelian variety of dimension $g$ has a positive dimensional $\text{CCS}_k$. Then, for a very general abelian variety $A$ of dimension $(g-l)$ there is an $(l+1)$-dimensional subvariety of $A^k$ foliated by positive dimensional $\text{CCS}_k$.
Under the assumption of this corollary we get $\mathcal{Z}\subset \mathcal{A}^k/S$, a $\text{CCS}_k/S$, for $\mathcal{A}\to S$ a locally complete family of $g$-dimensional abelian varieties. Apply the previous proposition inductively. By Lemma 3.3 the condition $(**)$ follows from $(*)$.
\[conjpf\] Conjecture 1.3 holds: a very general abelian variety of dimension $\geq 2k-1$ has no positive dimensional $\text{CCS}_k$.
Note that any $\mathcal{Z}\subset \mathcal{A}^k$ of relative dimension $d$ satisfies $(*)$ for $l\geq d$ since $$\bigcup_{\lambda\in \Lambda_l}\sigma_\lambda(S_\lambda)\subset G$$ is dense. Indeed, if $V$ is a vector space and $W\subset V^k$ has dimension $d<\dim V$, then the restriction of the projection $V^k\to (V/H)^k$ to $W$ is an isomorphism onto its image for a generic $H\in \text{Gr}(\dim V-d,V)$. In particular if $\mathcal{Z}\subset \mathcal{A}^k$ has relative dimension $1$ then it satisfies $(*)$ for any $1\leq l\leq g-1$. Hence, if a very general abelian variety of dimension $2k-1$ has a positive dimensional $\text{CCS}_k$, then a very general abelian surface $B$ will be such that $B^{k,\, 0}$ contains a $(2k-2)$-dimensional subvariety foliated by positive dimensional $\text{CCS}_k$. This does not hold for any abelian surface, let alone generically, by Corollary [\[absurfbound\]](absurfbound).
\[2k-2\] For $k\geq 3$, a very general abelian variety of dimension at least $2k-2$ has no positive dimensional orbits of degree $k$, i.e. $\mathscr{G}(k)\leq 2k-2$.
Let $\mathcal{Z}\subset \mathcal{A}^{k,\, 0}$ be a one-dimensional normalized $\text{CCS}_k$, where $\dim_S\mathcal{A}=2k-2$. By the previous corollary, there is a $\lambda\in \Lambda_2$ such that $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))/S'$ has relative dimension $2k-3$. This was obtained by successive degenerations and projections. But the morphism from $S_\lambda(B)$ to an appropriate Chow variety of $B^{k,\, 0}$ given by $$s\mapsto [\varphi_\lambda(p_\lambda(\mathcal{Z}_s))]$$ is a $\binom{2k-3}{2}$-parameter family of $\text{CCS}_k$ on $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))$. Hence, $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))$ must be foliated by $\text{CCS}_k$ of dimension at least $2$. This contradicts Corollary [\[absurfbound\]](absurfbound).
\[gonalitybound\] For $k\geq 3$, a very general abelian variety of dimension $\geq 2k-2$ has gonality at least $k+1$. In particular Conjecture [\[Vweakconj\]](Vweakconj) holds.
\[no2dim\] A very general abelian variety of dimension $\geq 2k-4$ does not have a $2$-dimensional $\text{CCS}_k$ for $k\geq 4$.
Suppose $\mathcal{A}/S$ is a locally complete family of $(2k-4)$-dimensional abelian varieties with some polarization $\theta$, and that $\mathcal{Z}\subset \mathcal{A}^k$ is a $2$-dimensional $\text{CCS}_k/S$. Using the same argument as in the proof of Corollary [\[conjpf\]](conjpf), we see that $\mathcal{Z}$ satisfies $(*)$, and thus $(**)$. We now follow the proof of Theorem [\[2k-2\]](2k-2).
\[k+1\] A very general abelian variety $A$ of dimension at least $2k+2-l$ does not have a positive dimensional orbit of the form $|\sum_{i=1}^{k-l}\{a_i\}+l\{0_A\}|$, i.e. $\mathscr{G}_{l}(k)\leq 2k+2-l.$\
Moreover, if $A$ is a very general abelian variety of dimension at least $k+1$ the orbit $|k\{0_A\}|$ is countable, i.e. $\mathscr{G}_{k}(k)\leq k+1.$
By the results of [@V], it suffices to show that a very general abelian variety of dimension $2k+2-l$ has no positive dimensional orbits of the form $|\sum_{i=1}^{k-l}\{a_i\}+l\{0_A\}|$. If this were not the case, we could find $\mathcal{Z}\subset \mathcal{A}^k$, a one-dimensional $\text{CCS}_k$, where $\mathcal{A}/S$ is a locally complete family of $(2k+2-l)$-dimensional abelian varieties, and $$\{0_{\mathcal{A}_s}\}\times\ldots\times \{0_{\mathcal{A}_s}\}\times \mathcal{A}_s^{k-l}\cap \mathcal{Z}_s\neq \emptyset$$ for every $s\in S$. By Proposition [\[mainprop\]](mainprop) there is a $\lambda\in \Lambda_2$ such that $\varphi_\lambda( p_{\lambda}(\mathcal{Z}_{S_\lambda}))$ has relative dimension $2k+1-l$. Given a generic $B$ in the family $\mathcal{B}^\lambda$ and $\underline{b}=(a_1,\ldots, a_{k-l})
\in B^{k-l}$, consider $$S_{\lambda}(B,\underline{b}):=\{s\in S_\lambda(B): \underline{b}\in \phi_\lambda(p_\lambda(\mathcal{Z}_s))\}.$$ Clearly $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda(B,\underline{b})}))$ is a $\text{CCS}_k$. In particular, $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda(B)}))$ is foliated by $\text{CCS}_k$ of codimension at most $2(k-l)$. This contradicts Corollary [\[absurfbound\]](absurfbound). A similar argument shows $\mathscr{G}_k(k)\leq k+1$.
Applications to Other Measures of Irrationality
===============================================
We have seen how the minimal degree of a positive dimensional orbit gives a lower bound on the gonality of a smooth projective variety and used this to provide a new lower bound on the gonality of very general abelian varieties. In this section we show how one can use results about the maximal dimension of $\text{CCS}_k$ in order to give lower bounds on other measures of irrationality for very general abelian varieties. We finish by discussing another conjecture of Voisin from [@V] and its implication for the gonality of very general abelian varieties.\
Recall the definitions of some of the measures of irrationality of irreducible $n$-dimensional projective varieties: $$\begin{aligned}
\text{irr}(X)&:=\min \left\{\delta>0: \exists \text{ degree } \delta\text{ rational covering } X\to \mathbb{P}^n\right\}\\
\text{gon}(X)&:=\min \left\{c>0: \exists \text{ a non-constant morphism } C\to X, \text{ where } C \text{ has gonality c}\right\}.\end{aligned}$$ Additionally, we will consider the following measure of irrationality which interpolates between the *degree of irrationality* $\text{irr}(X)=\text{irr}_n(X)$ and the *gonality* $\text{gon}(X)=\text{irr}_1(X)$. $$\text{irr}_d(X):=\min \left\{\delta: \exists\text{ a } d\text{-dimensional irreducible subvariety } Z\subset X \text{ with }irr(Z)=\delta\right\}.$$
The methods of the previous section can be applied to get:
If $A$ is a very general abelian variety of dimension $\geq 2k-4$ and $k\geq 4$, then $$\text{\textup{irr}}_2(A)\geq k+1.$$
A surface with degree of irrationality $k$ in a smooth projective variety $X$ provides a $2$-dimensional $\text{CCS}_{k}$. The result then follows from Theorem [\[no2dim\]](no2dim).
Similarly, we can use bounds on the dimension of a $\text{CCS}_k$ to obtain bounds on the degree of irrationality of abelian varieties. To our knowledge, the best bound currently in the literature is the Sommese bound $\text{irr}(A)\geq \dim A+1$ (see [@BE] Section 4), for any abelian variety $A$. It is an interesting fact that this bound follows easily from Voisin’s Theorem [\[k-1\]](k-1). Indeed, a dominant morphism from $A$ to $\mathbb{P}^{\dim A}$ of degree at most $\dim A$ would provide a $(\dim A)$-dimensional $\text{CCS}_{\dim A}$. Note that Yoshihara and Tokunaga-Yoshihara ([@Y],[@TY]) provide examples of abelian surfaces $A$ with $\text{irr}(A)=3$, so that the Sommese bound is tight for $\dim A=2$. In fact, we do not know of a single example of an abelian surface $A$ with $\dim A>3$.\
Ou results allow us to show that Sommese’s bound is not optimal, at least for very general abelian varieties.
\[no(k-1)\] Orbits of degree $k$ on a very general abelian variety of dimension at least $k-1$ have dimension at most $k-2$, for $k\geq 4$.
\[sommese\] If $A$ is a very general abelian variety of dimension $g\geq 3$, then $$\text{\textup{irr}}(A)\geq g+2.$$
Suppose that we have $\mathcal{A}/S$, a locally complete family of $(k-1)$-dimensional abelian varieties, and $\mathcal{Z}\subset \mathcal{A}^{k,\, 0}/S$, a $(k-1)$-dimensional $\text{CCS}_k/S$. We claim that $\mathcal{Z}$ satisfies $(*)$ for $l=k-2$. Assuming this, for appropriate $\lambda\in \Lambda_{(k-2)}$ and $B$ in the family $\mathcal{B}^\lambda$, the subvariety $\phi_\lambda(p_{\lambda}(\mathcal{Z}_{S_\lambda(B)}))\subset B^{k,\, 0}$ is $k$-dimensional and foliated by $(k-1)$-dimensional $\text{CCS}_{k}$. This contradicts Corollary [\[1dimfam\]](1dimfam).\
To show that $\mathcal{Z}$ satisfies $(*)$ for $l=k-2$, we will need the following easy lemma which we give without proof:
Given $V$ is a $g$-dimensional vector space, and $W\subset V^r$ a $g$-dimensional subspace such that the restriction of $\pi_L: V^r\to (V/L)^r$ to $W$ is not an isomorphism for any $L\in \mathbb{P}(V)$, then $W=V^1_M$ for some $M\in \mathbb{P}({\mathbb{C}}^r)$.
Hence, if $\mathcal{Z}$ fails to satisfy $(*)$ for $l=k-2$, for any $s\in S$ and $z\in (\mathcal{Z}_s)_{sm}$ the tangent space $T_{\mathcal{Z}_s,z}$ must be of the form $(T_{\mathcal{A}_s})^1_M\subset T_{\mathcal{A}_s}^{k,\, 0}$ for $M\in \mathbb{P}({\mathbb{C}}^{k,\, 0})$. Here, given a vector space $V$ we use the notation $V^{r\, 0}$ for the kernel of the summation map $V^r\to V$. It follows that for each $s\in S$ we get a morphism $(\mathcal{Z}_s)_{sm}\to \mathbb{P}({\mathbb{C}}^{k,\, 0})$. For very general $s$, the abelian variety $\mathcal{A}_s$ is simple and so $\mathcal{Z}_s$ cannot be stabilized by an abelian subvariety of $\mathcal{A}_s^k$. Thus, the Gauss map of $\mathcal{Z}_s$ is generically finite on its image and so is the morphism $(\mathcal{Z}_s)_{sm}\to \mathbb{P}({\mathbb{C}}^{k,\, 0})$. It follows that the image of this morphism must contain an open in $\mathbb{P}({\mathbb{C}}^{k,\, 0})$. Any open in $\mathbb{P}({\mathbb{C}}^{k,\, 0})$ contains real points and if $M\in \mathbb{P}(\mathbb{R}^{k,\, 0})$, then $(T_{\mathcal{A}_s})^1_M$ cannot be totally isotropic for $\omega_k$ for any non-zero $\omega\in H^0(\mathcal{A}_s,\Omega^2)$. This provides the desired contradiction.
The previous corollary motivates the following:
Exhibit an abelian threefold $A$ with $d_r(A)\geq 4$.
We have reasons to believe that Corollary [\[sommese\]](sommese) is also not optimal. Indeed, the key obstacle to proving a stronger lower bound is the need for $(*)$ to be satisfied. A careful study of the Gauss map of cycles on very general abelian variety is likely to provide stronger results.\
Similarly, we believe that Theorem [\[2k-2\]](2k-2) can be improved. In fact, though Conjecture [\[Vweakconj\]](Vweakconj) is the main conjecture of [@V], it is not the most ambitious. Voisin proposes to attack Conjecture [\[Vweakconj\]](Vweakconj) by studying what she calls the locus $Z_A$ of positive dimensional normalized orbits of degree $k$ $$Z_A:=\left\{a_1\in A: \exists\; a_2,\ldots, a_{k-1}: \dim\left|\{a_1\}+\ldots+\{a_{k-1}\}+\{-\sum_{i=1}^k a_i\}\right|>0\right\}.$$ In particular she suggests to deduce Conjecture [\[Vweakconj\]](Vweakconj) from the following conjecture:
\[norm\] If $A$ is a very general abelian variety $$\dim Z_A\leq k-1.$$
Voisin shows that this conjecture implies Conjecture [\[Vweakconj\]](Vweakconj) but it in fact implies the following stronger conjecture:
A very general abelian variety of dimension at least $k+1$ does not have a positive dimensional orbit. i.e. $\mathscr{G}(k)\leq k+1$.\[Vstrongconj\]
\[trick\]The previous conjecture follows from Conjecture [\[norm\]](norm). Indeed, if a very general abelian variety of dimension $k$ has a positive dimensional orbit $$|\{a_1\}+\ldots+\{a_{k-1}\}|$$ of degree $k-1$, then for any $a\in A$ the orbit $$|\{(k-1)a\}+\{a_1-a\}+\ldots+\{a_{k-1}-a\}| \label{trick}$$ is positive dimensional. This was noticed by Voisin in Example 5.3 of [@V]. It follows that $Z_A=A$ and so $\dim Z_A=k>k-1$.
As mentioned above, the results of Pirola and Alzati-Pirola give $\mathscr{G}(2)\leq 3$ and $\mathscr{G}(3)\leq 4$. Our main theorem provides us with the bound $\mathscr{G}(4)\leq 6$. An interesting question is to determine if $\mathscr{G}(4)\leq 5$. This would provide additional evidence in favor of Conjecture [\[Vstrongconj\]](Vstrongconj).
Support of zero-cycles on abelian varieties
===========================================
In [@V] the author shows the following surprising proposition:
\[propV\] Consider an abelian variety $A$ and an effective zero-cycle $\sum_{i=1}^k \{x_i\}$ on A such that $$\sum_{i=1}^k \{x_i\}= k\{0_A\}\in CH_0(A).$$ Then for $i=1,\ldots, k$ $$(\{x_i\}-\{0_A\})^{*k}=0 \in CH_0(A),$$ where $*$ denotes the Pontryagin product.
Voisin defines a subset $A_k:=\{a\in A: (\{a\}-\{0\})^{*k}=0\}\subset A$ and shows that $\dim A_k\leq k-1$. Given a smooth projective variety $X$ and a zero-cycle of the form $z=\sum_{i=1}^k\{x_i\}\in Z_0(X)$ the support of $z$ is $$\text{supp}(z)=\{x_i: i=1,\ldots, k\}\subset X.$$ Similarly we will call the $k$-support of $z$ the following subset of $X$ $$\text{supp}_k(z)=\bigcup_{z'=\sum_{i=1}^k\{x_i'\}: z'\sim z}\text{supp}(z').$$ The previous proposition can then be rephrased as: $\text{supp}_k(k\{0_A\})\subset A_k$. Here we present a generalization of this result. Given $\underline{x}\in X^k$ we let $$A_{k,\underline{x}}:=\{a\in A: (\{a\}-\{x_1\})*\cdots*(\{a\}-\{x_k\})=0\in CH_0(A)\}.$$ One shows easily, using the same argument as Voisin, that $\dim A_{k,\underline{x}}\leq k-1$.
\[propA\] Consider an abelian variety $A$ and effective zero-cycles $\sum_{i=1}^k\{x_i\}$, $\sum_{i=1}^k \{y_i\}$ on $A$ such that $$\sum_{i=1}^k \{x_i\}= \sum_{i=1}^k \{y_i\}\in CH_0(A).$$ Then for $i=1,\ldots, k$ $$\prod_{j=1}^k(\{x_i\}-\{y_j\})=0\in CH_0(A),$$ where the product is the Pontryagin product.
Upon presenting this result to Nori he recognized it as a more effective reformulation of results of his from around 2005. They had been obtained in an attempt to understand work of Colombo-Van Geeman [@CVG] but left unpublished for lack of an application. We present here Nori’s proof as it is more elegant than our original proof. This proof was also suggested to Voisin by Beauville in the context of Proposition [\[propV\]](propV).\
Let $X$ be a smooth projective variety and consider the graded algebra $$\bigoplus_{n=1}^\infty CH_0(X^n).$$ The multiplication is given by extending by linearity $$\begin{aligned}
X^{m}\times X^n&\to X^{m+n}\\
((x_1,\ldots, x_m),(x_1',\ldots, x_n'))&\mapsto (x_1,\ldots, x_m,x_1',\ldots, x_n')\end{aligned}$$ to a product $$Z_0(X^m)\times Z_0(X^n)\to Z_0(X^{n+m}).$$ It is easy to see that the resulting product descends to rational equivalence on the components. Let $$R:=\bigoplus_{n=1}^\infty CH_0(X^n)/(ab-ba)=\bigoplus_{i=1}^\infty R_n$$ be the abelianization of this algebra. Recall that by $CH_0(X^n)$ we mean the Chow group of zero-cycles with rational coefficients.
\[Norilemma\] If $z=\sum_{i=1}^k\{x_i\}\in Z_0(X)$ and $y\in \text{supp}_k(z)$, then $$(\{y\}-\{x_1\})(\{y\}-\{x_2\})\cdots(\{y\}-\{x_k\})=0\in R,$$ where the product is taken in $R$ and we consider $\{y\}-\{x_i\}$ as elements of $R_1\subset R$.
Since $y\in \text{supp}_k(z)$, there is $\underline{y}=(y_1=y,y_2,\ldots, y_k)\in X^k$ such that $\sum_{i=1}^k \{y_i\}=\sum_{i=1}^k \{x_i\}$. Consider the diagonal embeddings $$\Delta_l: X\to X^l.$$ These give linear maps $${\Delta_l}_*: R_1\to R_l$$ such that $${\Delta_l}_*\left(\sum_{i=1}^k \{y_i\}\right)=\sum_{i=1}^k \{y_i\}^l\in R_l.$$ Since $$\sum_{i=1}^k \{y_i\}=\sum_{i=1}^k \{x_i\}$$ we get $$p_l(\underline{y})=\sum_{i=1}^k \{y_i\}^l=\sum_{i=1}^k \{x_i\}^l=p_l(\{\underline{x}\})\in R_l,$$ where $p_l$ is the $l^{th}$ Newton polynomial and $\{\underline{x}\}=(\{x_1\},\ldots, \{x_{k}\})$. On the other hand we have $$(\{y\}-\{x_1\})(\{y\}-\{x_2\})\ldots(\{y\}-\{x_k\})=\{y\}^k-e_1(\underline{x})\{y\}^{k-1}+\ldots+(-1)^ke_k(\underline{x})\in R_k,$$ where $e_l$ is the $l^{\text{th}}$ elementary symmetric polynomial. Since the elementary symmetric polynomials can be written as polynomials in the Newton polynomials and since $p_l(\underline{y})=p_l(\underline{x})$ for all $l\in \mathbb{N}$, we get $$\prod_{i=1}^k(\{y\}-\{x_i\})=\sum_{i=0}^k(-1)^i\{y\}^{k-i}e_i(\underline{x})=\sum_{i=0}^k(-1)^i\{y\}^{k-i}e_i(\underline{y})=\prod_{i=1}^k(\{y\}-\{y_i\})=0\in R_k.$$
If $X=A$ is an abelian variety we have a summation morphism $A^l\to A$ inducing maps $$CH_0(A^l)\to CH_0(A),$$ and so a map $$\sigma: R\to CH_0(A)$$ such that $$\sigma\left(\prod_{i=1}^k(\{y\}-\{x_i\})\right)=(\{y\}-\{x_1\})*\ldots* (\{y\}-\{x_k\})\in CH_0(A).$$
Lemma [\[Norilemma\]](Norilemma) in fact has many more interesting corollaries. Consider $p(t_1,\ldots, t_k)\in {\mathbb{C}}[t_1,\ldots, t_k]$ and the $S_k$ action on ${\mathbb{C}}[t_1,\ldots, t_k]$ given by permutation of the variables. Let $H_p\subset S_k$ be the subgroup stabilizing $p$.
Consider an abelian variety $A$ and effective zero-cycles $\sum_{i=1}^k\{x_i\}$, $\sum_{i=1}^k \{y_i\}$ on $A$. Then $$\sum_{i=1}^k \{x_i\}= \sum_{i=1}^k \{y_i\}\in CH_0(A)$$ if and only if $$\prod_{\sigma\in S_k/H_p}\big(p(\{y_1\},\ldots, \{y_k\})-(\sigma\cdot p)(\{x_1\},\ldots, \{x_k\})\big)=0\in CH_0(A)$$ for every $p\in {\mathbb{C}}[t_1,\ldots, t_k]$. Here the product is the Pontryagin product.
The if direction follows trivially from considering $p(t_1,\ldots, t_k)=t_1+\ldots+t_k$. The proof of Lemma [\[Norilemma\]](Norilemma) completes the argument.
The special case $p=t_1$ is Proposition [\[propA\]](propA). Another corollary of Lemma [\[Norilemma\]](Norilemma) is the following:
Given an effective zero-cycle $z=\sum_{i=1}^k\{x_i\}$ on an abelian variety $A$, and $y_1,\ldots, y_{k+1}\in \text{supp}_k(z)$, the following identity is satisfied $$\prod_{i<j}(\{y_i\}-\{y_j\})=0\in CH_0(A).$$
Let $e_l$ be the $l^{\text{th}}$ elementary symmetric polynomial. By Lemma [\[Norilemma\]](Norilemma) we have $$\{y_i\}^{k}-e_1(\{\underline{x}\})\{y_i\}^{k-1}+\ldots +(-1)^ke_k(\{\underline{x}\})=0\in R_k$$ for $i=1,\ldots, k+1$, where $\{\underline{x}\}=(\{x_1\},\ldots,\{x_k\})$. This gives a non-trivial linear relation between the rows of the Vandermonde matrix $(\{y_{i}\}^{j-1})_{1\leq i,j\leq k+1}$. It follows that the Vandermonde determinant vanishes. Using the morphism $\sigma: R\to CH_0(A)$ from the proof of [\[propA\]](propA) finishes the argument.
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H. Yoshihara. Degree of irrationality of a product of two elliptic curves, Proc. Am. Math. Soc. 124, no. 5, (1996), 1371-1375.
[^1]: For simplicity we will call both fibers of $X^k\to CH_0(X)$ and $\text{Sym}^k X\to CH_0(X)$ orbits of degree $k$ for rational equivalence.
[^2]: Note that $${\mathbb{Z}}_{\geq \mathscr{G}(k)}=\begin{cases}\begin{rcases}g\in \mathbb{Z}_{>0}: \text{a very general abelian variety of dimension } g\\ \text{ does not have a positive dimensional orbit of degree } k\end{rcases}\end{cases}.$$ To see this, observe that if a very general abelian variety $A$ of dimension $g$ has a positive dimensional orbit $Z\subset A^k$ of degree $k$, we can degenerate $A$ to an abelian variety isogenous to a product $B\times E$ in such a way that the restriction of the projection $p: (B\times E)^k\to B^k$ to $Z$ has a positive dimensional image.
[^3]: Consider abelian varieties of dimension $g$ isogenous to $B\times E$, where $B$ is a $(g-1)$-dimensional abelian variety and $E$ is an elliptic curve.
[^4]: See Definition [\[CCS\]](CCS).
|
---
abstract: 'In this paper, we investigate property testing whether or not a degree d multivariate polynomial is a sum of squares or is far from a sum of squares. We show that if we require that the property tester always accepts YES instances and uses random samples, $n^{\Omega(d)}$ samples are required, which is not much fewer than it would take to completely determine the polynomial. To prove this lower bound, we show that with high probability, multivariate polynomial interpolation matches arbitrary values on random points and the resulting polynomial has small norm. We then consider a particular polynomial which is non-negative yet not a sum of squares and use pseudo-expectation values to prove it is far from being a sum of squares.'
author:
- 'Aaron Potechin[^1]'
- 'Liu Yang[^2]'
bibliography:
- 'soda17.bib'
title: A Note on Property Testing Sum of Squares and Multivariate Polynomial Interpolation
---
.
Introduction
============
In recent years, property testing and the sum of squares hierarchy have both been fruitful areas of research. In property testing, we aim to find algorithms which only look at a small portion of the input. However, instead of requiring an exact answer, we only require that we can distinguish between a function which has a given property and a function which is far from having that property. Thus far, property testers have been found for many properties of boolean functions including monotonicity, dictatorships, juntas, and being low degree [@monotonicity; @ParnasRS02; @Blaisjuntas]. For a survey on results in property testing, see Oded Goldreich’s book [@Goldreichbook].
The sum of squares hierarchy, independently investigated by Nesterov [@nesterov], Shor [@Shor87], Parrilo [@Parrilo00], and Lasserre [@Lasserre01], is a hierarchy of semidefinite programs which has the advantages of being broadly applicable, powerful, and in some sense, simple. The sum of squares hierarchy is broadly applicable because it can be applied to any system of polynomial equations over the reals and most problems of interest can be put into this form. The sum of squares hierarchy is surprisingly powerful; it captures the best known algorithms for several problems including the Goemans-Williamson algorithm for maximum cut [@GoemansW95], the Geomans-Linial relaxation for sparsest cut (analyzed by Arora,Rao,Vazirani [@AroraRV09]), and the subexponential time algorithm found by Arora, Barak, and Steurer [@AroraBS10] for unique games. Finally, the sum of squares hierarchy is in some sense simple as all that it uses is the fact that squares must be non-negative over the real numbers. For a survey on the sum of squares hierarchy, see Barak and Steurer’s survey [@bd].
A central question in researching the sum of squares hierarchy is determining whether a given polynomial is non-negative and whether it is a sum of squares. In the setting where we know all the coefficients of the polynomial, we can determine whther it is a sum of squares in polynomial time using semidefinite programming while determining whether it is non-negative is NP-hard. In this paper, we consider the question of property testing whether a polynomial is a sum of squares on random samples. In this setting, rather than knowing the full polynomial, we only have its value on randomly sampled points. However, we only need to determine whether it is a sum of squares or is far from being a sum of squares.
This work is also related to research on the difference between non-negative polynomials and polynomials which are sum of squares. This research began with Hilbert [@Hilbert], who proved the existence of polynomials which are non-negative yet not a sum of squares. The first explicit example of such a polynomial was found by Motzkin [@Motzkin67]. More recently, Bleckherman [@Blekherman] showed that there are significantly more polynomials which are non-negative than polynomials which are sums of squares. That said, to the best of our knowledge these papers do not analyze the distance of these polynomials from being sums of squares.
Results and Outline
-------------------
Our main result is the following
For all $d \geq 2$ and all $\delta > 0$, there is an $\epsilon > 0$ such that for sufficiently large $n$, if we require that our property tester always accepts YES instances and use random samples then property testing whether a degree $2d$ polynomial is a sum of squares requires at least $n^{\frac{d}{2} - \delta}$ samples.
Along the way, we prove the following result for multivariate polynomial interpolation on random points:
For all $d$ and all $\delta > 0$, given points $p_1,\cdots,p_m \in \mathbb{R}^n$ randomly sampled from the multivariate normal distribution with covariance matrix $Id$, if $n$ is sufficiently large and $m \leq n^{d-\delta}$ then with very high probability, for all $v_1,\cdots,v_m$, there is a polynomial $g$ of degree $d$ such that
1. $\forall i, g(p_i) = v_i$
2. $||g||$ is $O(\frac{||v||}{n^{\frac{d}{2}}})$
3. If we further have that $m \leq n^{\frac{d}{2}-\delta}$ then $||g^2||$ is $\tilde{O}(\frac{||v||^4}{n^{2d}})$
This paper is organized as follows. In Section \[prelims\] we give definitions and conventions which we will use for the remainder of the paper. In Section \[nonnegativitysection\], as a warm-up we consider the question of property testing non-negativity. This question is non-trivial because of how distance is defined in our setting. In Section \[sostestersection\] we describe our tester for being a sum of squares which we will prove a lower bound against. In Section \[interpolationsection\], we prove our theorem on multivariate polynomial interpolation, showing that this tester will accept with high probability as long as the values it receives are non-negative and not too large. Finally, in Section \[pseudoexpectationconstruction\] we complete our lower bound by giving a non-negative function $f$ of norm $1$ and lower bounding its distance from being a sum of squares using pseudo-expectation values.
Preliminaries {#prelims}
=============
For our results, we consider randomly sampling bounded degree real-valued polynomials over the multivariate normal distribution. We use the following conventions
1. We take $d$ or $2d$ to be the degree of our polynomials and assume that $d$ is a constant.
2. We take $m$ to be the number of sampled points.
3. Often, we will not be precise with functions of $d$ or logarithmic factors, so we absorb such functions into an $\tilde{O}$.
We use the following definitions on the multivariate normal distribution.
\[normaldistdef\]
1. $\mathcal{N}(0,1)$ is the univariate normal distribution with probability density $\mu(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$.
2. $\mathcal{N}(0,Id)$ is the multivariate normal distribution with probability density $$\mu(x_1,\cdots,x_n) = \frac{1}{(2\pi)^{\frac{n}{2}}}e^{-\frac{||x||^2}{2}} = \prod_{i=1}^{n}{\mu(x_i)}$$
3. For real-valued functions $f,g$, we define the inner product $\langle{f,g}\rangle = \int_{\mathcal{N}(0,Id)}{fg}$. We define $||f|| = \sqrt{\langle{f,f}\rangle}$
4. Given a set of real valued functions $S$ and a function $f$, we define the distance of $f$ from $S$ to be $d(f,S) = \min_{g:g \in S}{\{||f-g||\}}$
\[distdefremark\] It should be noted that this definition of distance differs from the definition of distance commonly used in the property testing literature, which is $d(f,S) = \min_{g:g \in S}{\{\mu(\{x:f(x) \neq g(x)\})\}}$. We use this definition of distance as it is more suitable for analyzing polynomials; if two polynomials have almost identical coefficients they will be very close to each other under Definition \[normaldistdef\] but will be distance $1$ from each other using this definition.
To help us index monomials, we use the following definitions:
1. We use $I$ (and occasionally $J$) to denote a multi-set of elements in $[1,n]$.
2. We define $x_I = \prod_{i \in I}{x_i}$.
3. We define $|I|$ to be the total number of elements of $I$ (counting multiplicities)
4. Given an $I$ and a $k \in [1,n]$, we define $I_k$ to be the multiplicity of $k$ in $I$.
5. Given an $I$ and a $t \in [1,|I|]$, we define $I(t)$ to be the number such that $\sum_{j=1}^{I(t)-1}{I_j} < t$ but $\sum_{j=1}^{I(t)}{i_j} \geq t$. In other words, if we put the elements of $I$ in sorted order, $I(t)$ will be the tth element which appears.
Sometimes we will also attach subscripts to $I$. To distinguish between this and the notation above, we will only use the above notation with the letters $k$ and $t$ and in the case where both occur, we will put the subscript in parentheses on the inside. For example, if we want the multiplicity of $k$ in $I_j$ then we will write $(I_j)_k$
For our analysis, it will be extremely useful to work with the orthonormal basis of polynomials. For the multivariate normal distribution, this basis is the Hermite polynomials and we use the following definitions
1. We define $h_j(x)$ to be the jth Hermite polynomial normalized so that $||h_j|| = 1$
2. Given an $I$, we define $h_{I}(x_1,\cdots,x_n) = \prod_{k=1}^{n}{h_{I_k}(x_k)}$
The multivariate polynomials $h_I(x_1,\cdots,x_n)$ are an orthonormal basis over $\mathcal{N}(0,Id)$.
For any polynomial $g$ we can write $g = \sum_{I}{{c_I}{h_I}}$ where $c_I = \langle{g,h_I}\rangle$ and we have that $||g||^2 = \sum_{I}{c^2_I}$.
Our results can be generalized to different product distributions. To do this, simply replace the Hermite polynomials with the appropriate orthonormal basis in a single variable.
Finally, we need the concept of pseudo-expectation values, which is extremely useful for analyzing the sum of squares hierarchy. As we show below, pseudo-expectation values allow us to lower bound the distance of a degree $2d$ polynomial $f$ from being a sum of squares.
We define degree $2d$ pseudo-expectation values to be a linear map $\tilde{E}$ from polynomials of degree at most $2d$ to $\mathbb{R}$ which satisfies the following conditions:
1. $\tilde{E}[1] = 1$
2. $\forall g: deg(g) \leq d, \tilde{E}[g^2] \geq 0$
This second condition can be equivalently stated as follows
Given degree $2d$ pseudo-expectation values $\tilde{E}$, define the moment matrix $M$ to be the matrix with rows and columns indexed by monomials $\{x_I: |I| \leq d\}$ and entries $M_{IJ} = \tilde{E}[{x_I}x_J]$
The condition that $\forall g: deg(g) \leq d, \tilde{E}[g^2] \geq 0$ is equivalent to the condition that $M \succeq 0$.
We now show how pseudo-expectation values can be used to show a lower bound on how far a polynomial $f$ is from being a sum of squares of degree $\leq 2d$.
\[distlowerbound\] Given pseudo-expectation values $\tilde{E}$, if $\tilde{E}[f] < 0$ then for all $g$ of degree at most $2d$ such that $g$ is a sum of squares, $||f-g||^2 \geq \frac{(\tilde{E}[f])^2}{\sum_{I:|I| \leq 2d}{(\tilde{E}[h_I])^2}}$
Write $g-f = \sum_{I}{{c_I}h_I}$. Observe that $\tilde{E}[g-f] = -\tilde{E}[f] + \tilde{E}[g] \geq -\tilde{E}[f]$ which implies that $$\sum_{I}{{c_I}\tilde{E}[h_I]} \geq -\tilde{E}[f]$$ Using Cauchy-Schwarz, $$\sqrt{\sum_{I}{c^2_I}}\sqrt{\sum_{I}{(\tilde{E}[h_I])^2}} \geq \sum_{I}{{c_I}\tilde{E}[h_I]} \geq -\tilde{E}[f]$$ Since both sides are non-negative, the result follows by squaring both sides and dividing both sides by $\sum_{I}{(\tilde{E}[h_I])^2}$.
Property testing non-negativity of degree $d$ polynomials {#nonnegativitysection}
=========================================================
As a warm-up, in this section we consider the closely related question of property testing whether a degree $d$ polynomial is non-negative or far from being non-negative. While this question is trivial under the definition of distance in Remark \[distdefremark\], it is non-trivial with our norm-based definition of distance. We also note that to the best of our knowledge, this problem is open if we consider the distance from the smaller set of non-negative degree $d$ polynomials rather than the set of all non-negative functions.
The following property tester distinguishes with high probability between an $f$ which is a degree $d$ non-negative polynomial and an $f$ which is a degree $d$ polynomial that is $(\epsilon||f||)$-far from being non-negative.
1. Take $\frac{10B}{\epsilon}$ random samples where $B = e\left(4 + \ln{\left(\frac{1}{\epsilon}\right)}\right)^{2d}$
2. If any sample gives a negative value, return NO. Otherwise, return YES.
Normalize $f$ so that $||f|| = 1$. Let $f^{-}$ be the negative part of $f$ and let $f^{+}$ be the non-negative part of $f$. If $f$ is $\epsilon$-far from being non-negative yet $\frac{10B}{\epsilon}$ random samples fails to find a negative value of $f$ with high probability then we must have that $||f^{-}|| > \epsilon$ yet $f^{-}$ is supported on a set of measure at most $\frac{\epsilon}{B}$.
However, by a corollary of the hypercontractivity theorem (which applies in the Gaussian setting as well, see O’Donnell’s lecture notes on hypercontractivity [@LectureA]), for all $q$, $$||f^{-}||_q \leq ||f||_q \leq (\sqrt{q-1})^d||f||_2 = (\sqrt{q-1})^d$$ Given that $||f^{-}||_2 \geq \epsilon$ and $f^{-}$ is supported on a set of measure at most $\frac{\epsilon}{B}$, for $q > 2$ we minimize $||f^{-}||_q$ (over all functions, not just polynomials) by setting $f^{-}$ equal to $-\sqrt{B}$ on a set of measure $\frac{\epsilon}{B}$ and setting $f^{-} = 0$ elsewhere. This implies that $${\epsilon}B^{\frac{q}{2}-1} \leq ||f^{-}||^q_q < q^{\frac{dq}{2}}$$ This gives a contradition when $B \geq \left(\frac{1}{\epsilon}q^{\frac{dq}{2}}\right)^{\frac{2}{q-2}}$. Taking $q = 4 + \ln{(\frac{1}{\epsilon})}$, $$\left(\frac{1}{\epsilon}q^{\frac{dq}{2}}\right)^{\frac{2}{q-2}} \leq \left(\frac{1}{\epsilon}\right)^{\frac{1}{1 + \ln{(\frac{1}{\epsilon})}}}\left(4 + \ln{\left(\frac{1}{\epsilon}\right)}\right)^{2d}
\leq e\left(4 + \ln{\left(\frac{1}{\epsilon}\right)}\right)^{2d}$$ Thus, we have a contradiction as long as $B \geq e\left(4 + \ln{\left(\frac{1}{\epsilon}\right)}\right)^{2d}$.
If we instead consider the distance from non-negative degree $d$ polynomials, it is no longer clear whether any degree $d$ $f$ which is far from being a non-negative degree $d$ polynomial must be negative on a constant proportion of the inputs. We leave this as a question for further research.
Algorithm for testing SOS {#sostestersection}
=========================
In this section, we describe a tester for property testing whether a polynomial $f$ of degree $2d$ is a sum of squares of norm at most $1$ or is far from being a degree $2d$ sum of squares of norm at most $1$. This tester is optimal over all testers which always accept YES instances. Thus, to prove our lower bound it is sufficient to show that this tester fails with high probability.
Given data $\{f(p_i) = v_i, i \in [1,m]\}$, we can try to test whether a polynomial $f$ of degree at most $2d$ is a sum of squares as follows.
Given a coefficient matrix $M$ with rows and columns indexed by multi-sets $I$ of size at most $d$, define $f_{M} = \sum_{J}{\left(\sum_{I,I': I \cup I' = J}{M_{II'}}\right)x_J}$.
A polynomial $f$ can be written as a sum of squares if and only there exists a coefficient matrix $M$ such that $f_M = f$ and $M \succeq 0$.
Thus, we can search for a coefficient matrix $M$ such that
1. $\forall i \in [1,m], f_M(x_i) = v_i$
2. $M \succeq 0$
If such a coefficient matrix $M$ is found then we output YES, otherwise we output NO.
This algorithm outputs YES precisely when there is polynomial $f_M$ of degree at most $2d$ which is a sum of squares and matches the data. However, for all we know, $||f_M||$ could be very high. On the other hand, in multivariate polynomial interpolation, when the polynomial is underdetermined it is natural to minimize the norm of the polynomial. To take this into account, we instead consider the following property testing problem and algorithm: Assumption: One of the following cases holds:
1. $f$ has degree at most $2d$, $f$ is a sum of squares, and $||f|| \leq 1$.
2. For all $g$ such that $g$ has degree at most $2d$, $g$ is a sum of squares, and $||g|| \leq 1$, $||f-g|| > \epsilon$.
Algorithm: Search for a coefficient matrix satisfying the following conditions:
1. $\forall i \in [1,m], f_M(x_i) = v_i$
2. $||f_M|| \leq 1$
3. $M \succeq 0$
These conditions on $M$ are all convex, so this algorithm can be implemented with convex optimization.
This algorithm is optimal if we require that the property tester always accept YES instances, as it says YES precisely when there is a function $f_M$ which is a sum of squares, matches the data, and has norm at most $1$.
To prove our lower bound, it is necessary and sufficient to find a degree $2d$ polynomial $f$ of norm $1$ which is $(\epsilon)$-far from being a degree $2d$ sum of squares such that if we take $m$ randomly sampled points where $m \leq n^{\frac{d}{2} - \delta}$, this tester accepts $f$ with high probability.
Norm Bounds for Multivariate Polynomial Interpolation {#interpolationsection}
=====================================================
In polynomial interpolation, we are given points $p_1,\cdots,p_m$ and values $v_1,\cdots,v_m$ and we want to find a polynomial $g$ of a given degree $d$ such that $\forall i, g(p_i) = v_i$. Single variable polynomial interpolation is very well understood; it can be achieved preciasely when $m \leq d+1$. However, multivariable polynomial interpolation is much less well understood. In this section, we consider the case when the $p_i$ are random. In this case, interpolation is almost surely possible as long as $m \leq \sum_{i=0}^{d}{\binom{n+i-1}{i}}$, where $n$ is the number of variables. However, this does not say anything about the norm of the resulting polynomial. In this section, we show that for all $\delta > 0$, if $m \leq n^{d - \delta}$ and $n$ is sufficiently large the we can find a $g$ which matches all the data and has small norm. Moreover, if $m \leq n^{\frac{d}{2} - \delta}$ then $||g^2||$ has small norm as well. More precisely, we show the following theorem.
We define $C$ to be the expected value of $\sum_{I:0 < |I| \leq d}{h_I(p)^2}$ for a random point $p$.
\[polynomialinterpolationtheorem\] For all $d$ and all $\delta > 0$, given points $p_1,\cdots,p_m \in \mathbb{R}^n$ randomly sampled from the multivariate normal distribution with covariance matrix $Id$, if $n$ is sufficiently large and $m \leq n^{d-\delta}$ then with very high probability, for all $v_1,\cdots,v_m$, there is a polynomial $g$ of degree $d$ such that
1. $\forall i, g(p_i) = v_i$
2. $||g|| = (1 \pm o(1))\frac{||v||}{\sqrt{C}}$
3. If we further have that $m \leq n^{\frac{d}{2}-\delta}$ then $||g^2||$ is $\tilde{O}(\frac{||v||^4}{n^{2d}})$
This theorem shows that our tester will accept with high probability as long as $m \leq n^{\frac{d}{2} - \delta}$ and all our sampled points have non-negative values. To see this, note that given data $f(p_i) = v_i$, this theorem says that with high probability there is a $g$ such that $g(p_i) = \sqrt{v_i}$ and $||g^2||$ has small norm. Thus, $g^2$ matches the data and has small norm so the tester must accept.
Construction of the function $g$
--------------------------------
To construct our function $g$, we use the following strategy:
1. We construct a function $g_i$ of degree $d$ for each point $p_i$.
2. We take the matrix $M$ where $M_{ij} = g_j(p_i)$.
3. We take $x$ to be a solution to $Mx = v$.
4. We take $g = \sum_{j}{{x_j}g_j}$.
For all $i$, $g(p_i) = v_i$.
For all $i$, $g(p_i) = \sum_{j}{{x_j}g_j(p_i)} = \sum_{j}{M_{ij}x_j} = v_i$.
We now construct the functions $g_i$ of degree $d$ for each point $p_i$. These functions are constructed so that with high probability, for all $i$, $g_i(p_i) \approx 1$ and for all $i \neq j$, $|g_i(p_j)|$ is small.
Given a point $p_i = (v_1,\cdots,v_n)$, we define $g_i = \frac{\sum_{I:0 < |I| \leq d}{h_I(p_i)h_I}}{C}$
Analysis of the function $g$
----------------------------
To analyze the function $g$, it is useful to consider the following matrix $H$ which is closely related to $M$.
We define $H$ to be the matrix with rows indexed by $I$ where $0 < |I| \leq d$, columns indexed by $i$, and entries $H_{Ii} = h_I(p_i)$.
$M = \frac{{H^T}H}{C}$
Observe that $$\frac{1}{C}({H^T}H)_{ij} = \frac{1}{C}\sum_{I:0 <|I| \leq d}{(h_I(p_i)h_I)(p_j)} = \left(\frac{\sum_{I:0<|I| \leq d}{h_I(p_j)h_I}}{C}\right)(p_i) = g_j(p_i) = M_{ij}$$
$g = \frac{1}{C}\sum_{I}{{(Hv)_I}h_I}$
We now have that $$||g||^2 = \frac{1}{C^2}\sum_{I}{\left(\sum_{j=1}^{m}{{x_j}H_{Ij}}\right)^2} = \frac{1}{C^2}{x^T{H^TH}x} = \frac{1}{C}{x^T}Mx = \frac{1}{C}{v^T}M^{-1}v$$ In the next subsection, we will show that with high probability $M$ is very close to the identity which immediately implies that with high probability, $||g|| = (1 \pm o(1))\frac{||v||}{\sqrt{C}}$, as needed.
Analysis of $H$ and $M$
-----------------------
In this subsection, we analyze the matrices $H$ and $M$. We begin by analyzing $H$ in order to develop the necessary techniques.
\[Hnormboundtheorem\] For all $\delta > 0$ and all $d$, if $n$ is sufficiently large and $m \leq n^{d - \epsilon}$ then with high probability, $||H||$ is $\tilde{O}(n^{\frac{d}{2}})$
We can use the trace power method to probabilistically bound $||H||$. For this, we need to bound $$E\left[tr((HH^T)^q)\right] = \sum_{i_1,\cdots,i_q,I_1,\cdots,I_q: \forall j, 0 < |I_j| \leq d}{E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right]}$$ where we take $I_{q+1} = I_1$ and $i_{q+1} = i_1$. We partition this sum based on the intersection pattern $P$ of which of the $i_1,\cdots,i_q$ are equal to each other and how $I_1,\cdots,I_q$ interact with each other. We then analyze which intersection patterns give terms with nonzero expected value.
We define an intersection pattern $P$ to be the following data:
1. For all $j' \neq j$, $P$ has the equality $i_{j'} = i_j$ or the inequality $i_{j'} \neq i_j$
2. For all $j,j',t,t'$, $P$ has the equality $I_{j'}(t') = I_j(t)$ or the inequality $I_{j'}(t') \neq I_j(t)$
where these equalities and inequalities are consistent with each other (i.e. transitivity is satisfied for the equalities).
\[countingintersectionpatterns\] There are at most $(4dq)^{4dq}$ possible intersection patterns.
Choose an arbitrary ordering of the $i_j$ and an arbitrary ordering of the $I_j(t)$. To specify an intersection pattern, it suffices to specify which $i_j$ and $I_j(t)$ are equal to previous $i_j$ and $I_j(t)$ and if so, to specify one of the equalities which hold. The total number of choices is at most $(4dq)^{4dq}$.
\[xylemma\] For any intersection pattern $P$ which gives a nonzero expected value, letting $x = |\{k:\exists j: (I_{j})_k > 0\}|$ and letting $y$ be the number of distinct $i_j$, $y + \frac{x}{d} \leq q+1$
The key observation is that if we consider the multiset $$\{(i_j,k): (I_j)_k > 0\} \cup \{(i_j,k): (I_{j+1})_k > 0\} = \{(i_j,k): (I_j)_k > 0\} \cup \{(i_{j-1},k): (I_j)_k > 0\},$$ if any element of this multiset appears exactly once then $E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right] = 0$ over the random choices for the points $\{p_i\}$.
With this observation in mind, for each $k$, consider the graph fomed by the edges $\{(i_{j-1},i_j): (I_j)_k > 0\}$. In a term with nonzero expectation, for all $k$, every vertex in this graph with nonzero degree must have degree at least 2 (where we consider loops as adding 2 to the degree). Thus, these graphs must consist of loops/cycles and loops/cycles joined by paths. This implies the following upper bound on $x$
Let $G_y$ be the multi-graph consisting of the $q$ edges $\{i_{j-1},i_j\}$
Given a multi-graph $G$, we define $w(G)$ to be the maximum number such that $\exists G_1,\cdots,G_t$ and $w_1,\cdots,w_t$ satisfying the following conditions
1. $w(G) =\sum_{i}{w_i}$
2. $\forall i, V(G_i) = V(G), E(G_i) \subseteq E(G)$, $E(G_i)$ is nonempty, and no vertex of $G_i$ has degree exactly $1$ (where we consider loops as adding 2 to the degree).
3. $\forall i, w_i \geq 0$
4. $\forall j, \sum_{i: (i_{j-1},i_j) \in E(G_i)}{w_i} \leq 1$
\[weightupperboundlemma\] For any intersection pattern which gives a nonzero expected value, $x = |\{k:\exists j: (I_{j})_k > 0\}| \leq d \cdot w(G_y)$
For each $k:\exists j: (I_{j})_k > 0$, we construct the graph $G_k$ where $V(G_k) = V(G_y),E(G_k) = \{(i_{j-1},i_j): (I_{j})_k > 0\}$ and assign it weight $\frac{1}{d}$. From the above observation, no vertex of $G_k$ can have degree exactly $1$ (where we consider loops as adding 2 to the degree). Also, we have that the total weight on any edge $(i_{j-1},i_j)$ is at most $1$ as at most $d$ graphs $G_k$ contribute to it and each contribution is $\frac{1}{d}$. Thus, $\sum_{k:\exists j: (I_{j})_k > 0}{\frac{1}{d}} = \frac{x}{d} \leq w(G_y)$, as needed.
With this bound in mind, we now prove the following lemma which will immediately imply our result.
\[weightvertextradeofflemma\] For all connected multi-graphs $G$, $w(G) + |V(G)| \leq |E(G)|+1$
We first reduce to the case where every non-loop edge of $G$ has multiplicity at least two with the following lemma.
If $G$ is a multigraph which has a non-loop edge $e$ that appears with multiplicity $1$ and $G'$ is the graph formed by contracting this edge then $w(G) \leq w(G')$
Observe that if subgraphs $G_1,\cdots,G_t$ of $G$ all have no vertex of degree exactly $1$, then letting $G'_1,\cdots,G'_t$ be the graphs $G_1,\cdots,G_t$ formed by making the two endpoints of $e$ equal and removing $e$ (if present), $G'_1,\cdots,G'_t$ are subgraphs of $G'$ and have no vertices of degree exactly $1$. To see this, note that for any vertex $v'$ in $G'_i$ except for the vertex formed by making the two endpoints of $e$ equal, the number of edges incident to $v'$ is unaffected. For the $v'$ formed by making the two endpoints of $e$ equal, each of these endpoints must have had an edge besides $e$ incident with it, so the degree of this $v'$ is at least 2.
Using this lemma, if $G$ has a non-loop edge $e$ which appears with multiplicity $1$, $G'$ is the graph formed by contracting this edge, and $w(G') + |V(G')| \leq |E(G')|+1$ then $$w(G) + |V(G)| \leq w(G') + |V(G')| + 1 \leq |E(G')|+2 = |E(G)| + 1$$ Thus, it is sufficient to prove the lemma for $G'$. Applying this logic repeatedly, it is sufficient to prove the result for the case where every non-loop edge of $G$ has multiplicity at least two.
We define $E_{loop}(G)$ to be the multi-set of loops in $G$ and we define $E_{nonloop}(G)$ to be $E(G) \setminus E_{loop}(G)$.
For all $G$, $w(G) \leq |E_{loop}(G)| + \frac{|E_{nonloop}(G)|}{2}$
Let $G_1,\cdots,G_k$ and $w_1,\cdots,w_k$ be graphs and weights such that
1. $w(G) =\sum_{i=1}^{k}{w_i}$
2. $\forall i, V(G_i) = V(G), E(G_i) \subseteq E(G)$, $E(G_i)$ is nonempty, and no vertex of $G_i$ has degree exactly $1$ (where we consider loops as adding 2 to the degree).
3. $\forall i, w_i \geq 0$
4. $\forall j, \sum_{i \in [1,k]: (i_{j-1},i_j) \in E(G_i)}{w_i} \leq 1$
Observe that each $G_i$ must either have at least one loop or at least two non-loop edges. Thus, $$\begin{aligned}
|E_{loop}(G)| + \frac{|E_{nonloop}(G)|}{2} &\geq \sum_{j:(i_{j-1},i_j) \in E_{loop}(G)}{\sum_{i:(i_{j-1},i_j) \in E(G_i)}{w_i}} +
\frac{1}{2}\sum_{j:(i_{j-1},i_j) \in E_{nonloop}(G)}{\sum_{i:(i_{j-1},i_j) \in E(G_i)}{w_i}} \\
&= \sum_{i}{\left(\sum_{j:(i_{j-1},i_j) \in E(G_i) \cap E_{loop}(G)}{w_i} + \frac{1}{2}\sum_{j:(i_{j-1},i_j) \in E(G_i) \cap E_{nonloop}(G)}{w_i}\right)} \\
&\geq \sum_{i}{w_i} = w(G)\end{aligned}$$ as needed.
If $G$ is connected and every non-loop edge of $G$ has multiplicity at least 2 then $|V(G)| \leq \frac{|E_{nonloop}(G)|}{2} + 1$
Imagine building up $G$ from one isolated vertex. We can add loops for free, but every time we add a neighbor of an existing vertex, we must add at least two edges (as all edges have multiplicity at least two).
Putting these lemmas together, $|V(G)| + w(G) \leq |E_{loop}(G)| + |E_{nonloop}(G)| + 1 = |E(G)| + 1$ which proves Lemma \[weightvertextradeofflemma\].
Putting Lemmas \[weightupperboundlemma\] and \[weightvertextradeofflemma\] together, Lemma \[xylemma\] follows immediately. We have that $y + \frac{x}{d} \leq |V(G)| + w(G) \leq |E(G)| + 1 = q+1$, as needed.
We now consider the expression $$E\left[tr((HH^T)^q)\right] = \sum_{i_1,\cdots,i_q,I_1,\cdots,I_q: \forall j, 0 < deg(I_j) \leq d}{E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right]}$$ Lemma \[xylemma\] implies that for any intersection pattern which gives a nonzero expected value, there are at most $\max_{x,y:y\geq 1,y+\frac{x}{d} \leq q+1}{\{{m^y}n^x\}} \leq mn^{dq}$ choices for $i_1,\cdots,i_q,I_1,\cdots,I_q$. By Lemma \[countingintersectionpatterns\], there are at most $(4dq)^{4dq}$ possible intersection patterns. To complete our upper bound, we just need to show a bound on $E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right]$ for a particular $i_1,\cdots,i_q,I_1,\cdots,I_q$, which we do with the following lemma
For any $i_1,\cdots,i_q,I_1,\cdots,I_q$, $$E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right] \leq (4dq)^{4dq}$$
Observe that for any $i_1,\cdots,i_q,I_1,\cdots,I_q$, $E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right]$ is a product of expressions of the form $E\left[\left(\prod_{i=1}^{k}{h_{j_i}}\right)(x)\right]$ for some $j_1,\cdots,j_k$. For each such expression we have the following bound.
Letting $d' = \sum_{i=1}^{k}{j_i}$, $E\left[\left(\prod_{i=1}^{k}{h_{j_i}}\right)(x)\right] \leq (d')^{2d'}$
We use the fact that for all $j \geq 1$, the sum of the absolute values of the coefficients of $h_j$ is at most $j^j$. This implies that the sum of the absolute values of the coefficients of $\prod_{i=1}^{k}{h_{j_i}}$ is at most ${d'}^{d'}$. Over a normal distribution $E[x^{p}] = \prod_{i=1}^{\frac{p}{2}}{(2i-1)} \leq p^p$ if $p$ is even and is $0$ if $p$ is odd, which implies the result.
The total sum of all the degrees is at most $2dq$ as this is the maximum number of pairs $I_j(t),i_j$ and $I_{j+1}(t),i_j$. Thus, the product over all of the expressions which we have is at most $(4dq)^{4dq}$, as needed.
Putting everything together, $$E\left[tr((HH^T)^q)\right] = \sum_{i_1,\cdots,i_q,I_1,\cdots,I_q: \forall j, 0 < deg(I_j) \leq \frac{d}{2}}{E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right]}
\leq (4dq)^{8dq}mn^{dq}$$ We now apply Markov’s inequality. For all $q$ and all $\beta \geq 0$, $$\begin{aligned}
Pr\left[||H|| \geq \sqrt[2q]{n^{\beta}E\left[tr((HH^T)^q)\right]}\right] &= Pr\left[||H||^{2q} \geq n^{\beta}E\left[tr((HH^T)^q)\right]\right] \\
& \geq Pr\left[tr((HH^T)^q) \geq n^{\beta}E\left[tr((HH^T)^q)\right]\right] \leq \frac{1}{n^{\beta}}\end{aligned}$$ Applying this with $q \sim dq\beta\log{n}$, Theorem \[Hnormboundtheorem\] follows.
With the techniques we developed to prove Theorem \[Hnormboundtheorem\], we can now anaylze $M$.
\[Manalysistheorem\] For all $d$ and all $\delta > 0$, for sufficiently large $n$, if $m \leq n^{d - \delta}$ and we write $M = Id + M'$ then with high probability $||M'|| << 1$.
This theorem can be proved by considering the diagonal part and off-diagonal part of $M$. For the diagonal part of $M$, observe that $M_{ii} = \frac{1}{C}\sum_{I:0 <|I| \leq d}{h_I(p_i)^2}$. Since $C$ is the epected value of $\sum_{I:0 <|I| \leq d}{h_I(p)^2}$ for a random point $p$ and this value is tightly concentrated around its expectation, with high probability $M_{ii}$ will be $1 \pm o(1)$ for all $i$. For the off-diagonal part of $M$, we can use the trace power method to bound its norm. Let $M''$ be the off-diagonal part of $M$.
For all $d$ and all $\delta > 0$, for sufficiently large $n$, if $m \leq n^{d - \delta}$ then with high probability $||M''||$ is $\tilde{O}\left(\frac{\sqrt{m}}{n^{\frac{d}{2}}}\right)$
Observe that $$E\left[tr((M'')^q)\right] = \frac{1}{C^q}\sum_{i_1,\cdots,i_q,I_1,\cdots,I_q: \forall j, 0 < deg(I_j) \leq \frac{d}{2}, i_{j} \neq i_{j+1}}{E\left[\prod_{j=1}^{q}{H_{I_j{i_j}}H_{I_{j+1}{i_j}}}\right]}$$ Up to the $\frac{1}{C^q}$ factor, this is the same expression we had for $E\left[tr((HH^T)^q)\right]$ except that since we are restricting ourselves to the off-diagonal part of $M$ we additionally have the constraint that $i_{j} \neq i_{j+1}$ for all $j$. This constraint implies that for any term with nonzero expected value, there is no $k$ such that $(I_j)_k > 0$ and $(I_{j'})_{k} = 0$ for all $j \neq j'$. This in turn implies that we only need to consider intersection patterns with $x \leq \frac{dq}{2}$, which means that the maximum number of choices for a given intersection pattern is at most $(m^{(\frac{q}{2}+1)}n^{\frac{dq}{2}})$ rather than $mn^{dq}$. Thus, our final bound on $E\left[tr((M'')^q)\right]$ will be $$E\left[tr((M'')^q)\right] \leq (4dq)^{8dq}\frac{1}{C^q}m^{(\frac{q}{2}+1)}n^{\frac{dq}{2}}$$ Recalling that $C = \sum_{I:0 < |I| \leq d}{h_I(p)^2}$, $C$ is $\Theta(n^d)$ and the result can be shown in same the way as Theorem \[Hnormboundtheorem\] using Markov’s inequality (where we choose an even $q$).
In fact, our analysis of $M$ gives us improved norm bounds on $||H||$. In particular, with high probability $||H||$ is $\sqrt{C}(1 \pm o(1))$
Analysis of $||g^2||$
---------------------
In this subsection, we show how to probabilistically bound $||g^2||$.
We define the matrix $Q$ so that $Q_{J(I,I')}$ is the coefficient of $h_J$ in ${h_I}h_{I'}$.
We have that $$g^2 = \frac{1}{C^2}\sum_{J}{\left(Q_{J(I,I')}\sum_{I,I'}{\sum_{j=1}^{m}{\sum_{j' = 1}^{m}{{x_j}H_{Ij}x_{j'}H_{I'j}}}}\right)h_J}$$ Thus, $$\begin{aligned}
||g^2||^2 &= \frac{1}{C^4}\sum_{J}{\left(Q_{J(I,I')}\sum_{I,I'}{\sum_{j=1}^{m}{\sum_{j' = 1}^{m}{{x_j}H_{Ij}x_{j'}H_{I'j}}}}\right)^2} \\
&= \frac{1}{C^4}(x \otimes x)^T((H \otimes H)^TQ^TQ(H \otimes H))(x \otimes x)\end{aligned}$$
For all $d,\delta$ and all sufficiently large $n$, if $m \leq n^{\frac{d}{2} - \delta}$ then with high probability, for all vectors $x$, $||Q(H \otimes H)(x \otimes x)||^2$ is $\tilde{O}(n^{2d}||x||^4)$
We break $Q(H \otimes H)$ into two parts.
1. Let $A$ be the matrix such that $A_{J(j,j')} = (Q(H \otimes H))_{J(j,j')}$ if $j' = j$ and is $0$ otherwise.
2. Let $R$ be the matrix such that $R_{J{j,j'}} = (Q(H \otimes H))_{J(j,j')}$ if $j' \neq j$ and is $0$ otherwise.
For the first part, we observe that letting $A_{(j,j)}$ be the $(j,j)$ column of $A$, $$||A(x \otimes x)|| = ||\sum_{j}{{x^2_j}A_{jj}}|| \leq \left(\sum_{j}{x^2_j}\right)\max_{j}{\{||A_{jj}||\}} = ||x||^2\max_{j}{\{||A_{jj}||\}}$$ Thus, $||A(x \otimes x)||^2 \leq \left(\max_{j}{\{||A_{jj}||\}}\right)^2||x||^4$ and it is sufficient to probabilistically bound $\max_{j}{\{||A_{jj}||\}}$
With high probability, $\max_{j}{\{||A_{jj}||\}}$ is $\tilde{O}(n^d)$
Observe that for all $j$, $$||A_{jj}||^2 = \sum_{J,I_1,I_2,I_3,I_4}{Q_{J(I_1,I_2)}Q_{J(I_3,I_4)}H_{{I_1}j}H_{{I_2}j}H_{{I_3}j}H_{{I_4}j}}$$ The entries of $Q$ are $O(1)$ and with high probability the entries of $H$ are $\tilde{O}(1)$, so we just need to bound the number of $I_1,I_2,I_3,I_4$ which give a nonzero contribution. For this, observe that for any nonzero term,
1. $I_1 \Delta I_2 \subseteq J$ and $I_3 \Delta I_4 \subseteq J$ where $\Delta$ is the symmetric difference.
2. $J \subseteq I_1 \cup I_2$ and $J \subseteq I_3 \cup I_4$
Together, these observations imply that there cannot be a $k$ such that precisely one of $(I_1)_k,(I_2)_k,(I_3)_k,(I_4)_k$ is nonzero. In turn, this implies that there are $O(n^{2d})$ choices for $I_1,I_2,I_3,I_4$ which give a nonzero contribution and the result follows.
For the second part, we bound the norm of $R$.
\[Rnormboundlemma\] For all $d,\delta$ and all sufficiently large $n$, if $m \leq n^{\frac{d}{2} - \delta}$ then with high probability, $||R||$ is $\tilde{O}(n^{d})$.
This can be shown using the trace power method. We have that $$\begin{aligned}
&E\left[(R^TR)^q\right] = \\
&\sum_{\{j_1,j'_1,J_1,I_{11},I_{12},I_{13},I_{14}, \cdots, j_q,j'_q,J_q,I_{q1},I_{q2},I_{q3},I_{q4}\}}{E\left[\prod_{a=1}^{q}
{Q_{J_a(I_{a1},I_{a2})}Q_{J_a(I_{a3},I_{a4})}H_{{I_{a1}}j_a}H_{{I_{a2}}j'_a}H_{{I_{a3}}j_{a+1}}H_{{I_{a4}}j'_{a+1}}}\right]}\end{aligned}$$ Similar to before, we can partition this sum into intersection patterns and consider which patterns have nonzero expectation.
We take $x$ to be the number of distinct $k$ such that $(I_{ai}) > 0$ for some $a,i$ and we take $y$ to be the number of distinct $j_a$ and $j'_a$.
For any term with nonzero expected value, $y + \frac{2x}{d} \leq 4q + 2$
In any term with nonzero expected value, following the same logic as before, for each block $I_{a1},I_{a2},I_{a3},I_{a4}$ there cannot be a $k$ such that precisely one of $(I_{a1})_k,(I_{a2})_k,(I_{a3})_k,(I_{a4})_k$ is nonzero. If every $k$ which appears in a block appears in at least two blocks then $x \leq d$. We trivially have that $y \leq 2q$ so the result holds in this case.
If there is a $k$ which appears in only one block then this implies an equality between $j_a = j_{a+1}$ or an equality $j'_{a} = j'_{a+1}$. Roughly speaking, each such equality allows $d$ additional values $k$ to only appear in one block, decreasing $y$ by $1$ but increasing $x$ by $\frac{d}{2}$. This leaves $y + \frac{2x}{d}$ unchanged. To see why we have the $+2$, consider the extreme case when all the $j_a$ are equal and all the $j'_a$ are equal. In this case $y = 2$ and we can have $x = 2dq$.
With this lemma in hand, since $m \leq n^{\frac{d}{2} - \delta}$, for any intersection pattern which gives a nonzero expected value, the total number of choices for the $j_a,j'_a,I_{a1},I_{a2},I_{a3},I_{a4}$ is $O(m^2n^{dq})$. Lemma \[Rnormboundlemma\] can now be shown using the same techniques used to prove Theorem \[Hnormboundtheorem\].
Putting these results together, it follows that with high probability, for all vectors $x$, $||Q(H \otimes H)(x \otimes x)||^2$ is $\tilde{O}(n^{2d}||x||^4)$, as needed.
While Lemma \[Rnormboundlemma\] is essentially tight, it should be possible to obtain the same bound on $||g^2||^2$ for $m \leq n^{d - \delta}$ if we can effectively use the fact that we are dealing with $(x \otimes x)$ rather than an arbitrary vector, just as we did for $A$. We leave this as a question for further research.
A non-negative polynomial which is far from being a sum of squares {#pseudoexpectationconstruction}
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In this section, we complete our lower bound by giving a non-negative polynomial $f$ and showing that $f$ is far from being a sum of squares.
We take $f = \frac{r}{2}x^{r+2}y^{r} + \frac{r}{2}x^{r}y^{r+2} - (r+1)x^{r}y^{r} + (1+c)$ where $c \geq 0$, $r \geq 2$ is even, and we take $d = 2r+2$
This polynomial $f$ is a generalization of the Motzkin polynomial (which is the case $r = 2,c=0$).
$f \geq c$
Observe that $(x^{r+2}y^{r})^{\frac{r}{2r+2}}(x^{r}y^{r+2})^{\frac{r}{2r+2}}(1)^{\frac{2}{2r+2}} = x^{r}y^{r}$ and $\frac{r}{2r+2} + \frac{r}{2r+2} + \frac{2}{2r+2} = 1$. By the AM-GM inequality, $\frac{r}{2r+2}(x^{r+2}y^{r}) + \frac{r}{2r+2}(x^{r}y^{r+2}) + \frac{2}{2r+2} \geq x^{r}y^{r}$ and the result follows.
$f$ is $\left(\frac{1}{(d^3\sqrt[r]{2+c})^{2d^4}}\right)$-far from being SOS.
We take the following pseudo-expectation values.
Take $k > 1$ and take $B = (kd^3)^{(3d^3)}$. We split up the pseudo-expectation values $\tilde{E}[{x^a}{y^b}]$ into cases as follows
1. If $a > b$ then we take $\tilde{E}[{x^a}{y^b}] = \frac{(kd^3)^{(a^2 + (a+b)^2)}}{(kd^3)^{2d^2}}B^{a - \frac{r+2}{r}b}$.
2. If $b > a$ then we take $\tilde{E}[{x^a}{y^b}] = \frac{(kd^3)^{(b^2 + (a+b)^2)}}{(kd^3)^{2d^2}}B^{a - \frac{r+2}{r}b}$.
3. For all $a > 0$ we take $\tilde{E}[{x^a}{y^a}] = \frac{4^{(a^2)}}{4^{(r^2)}}{k^a}$
4. We have $\tilde{E}[1] = 1$.
5. We take $\tilde{E}[p(x,y)q(\text{other variables})] = \tilde{E}[p(x,y)]E[q]$. This guarantees that $\tilde{E}[h_{I}] = 0$ whenever $I$ contains a variable besides $x$ and $y$.
$\tilde{E}[f] = c+(r+1)(1-k^r)$
This follows immediately from the observations that $\tilde{E}[{x^{r+2}}{y^{r}}] = \tilde{E}[{x^{r}}{y^{r+2}}] = 1$ and $\tilde{E}[x^{r}y^{r}] = k^r$.
We need to show that these pseudo-expectation values give a PSD moment matrix.
For all $a,b$ such that $a+b \leq d$, $$\tilde{E}[{x^a}{y^b}] \geq \min{\left\{\frac{(kd^3)^{(a^2 + (a+b)^2)}}{(kd^3)^{2d^2}}B^{a - \frac{r+2}{r}b},
\frac{(kd^3)^{(b^2 + (a+b)^2)}}{(kd^3)^{2d^2}}B^{a - \frac{r+2}{r}b}\right\}}$$
For all $a_1,b_1,a_2,b_2$ such that $a_1 + b_1 \leq d$, $a_1 + b_2 \leq d$, $a_1 \neq a_2$ or $b_1 \neq b_2$, and $a_1 \neq b_1$ or $a_2 \neq b_2$, $$\tilde{E}[x^{a_1 + a_2}y^{b_1 + b_2}] \leq \frac{1}{2d^2}\sqrt{\tilde{E}[x^{2a_1}y^{2b_1}]\tilde{E}[x^{2a_2}y^{2b_2}]}$$
If $a_1 + a_2 > b_1 + b_2$ then we have that $$\sqrt{\tilde{E}[x^{2a_1}y^{2b_1}]\tilde{E}[x^{2a_2}y^{2b_2}]} \geq \frac{(kd^3)^{(2a_1^2 + 2(a_1+b_1)^2 + 2a_2^2 + 2(a_2+b_2)^2)}}{(kd^3)^{2d^2}}B^{a_1+a_2 - \frac{r+2}{r}(b_1+b_2)}$$ Thus, $$\frac{\tilde{E}[x^{a_1 + a_2}y^{b_1 + b_2}]}{\sqrt{\tilde{E}[x^{2a_1}y^{2b_1}]\tilde{E}[x^{2a_2}y^{2b_2}]}} \leq
(kd^3)^{-((a_1-a_2)^2 + (a_1+b_1 - a_2 - b_2)^2)}$$ Since we either have that $a_1 \neq a_2$ or $a_1 + b_1 \neq a_2 + b_2$, this is at most $\frac{1}{kd^3} \leq \frac{1}{2d^2}$.
If $b_1 + b_2 > a_1 + a_2$ then we can use a symmetrical argument. If $a_1 + a_2 = b_1 + b_2$ and $a_1 > b_1$ then $$\sqrt{\tilde{E}[x^{2a_1}y^{2b_1}]\tilde{E}[x^{2a_2}y^{2b_2}]} = \frac{(kd^3)^{(2a_1^2 + 2(a_1+b_1)^2 + 2b_2^2 + 2(a_2+b_2)^2)}}{(kd^3)^{2d^2}}B^{a_1+b_2 - \frac{r+2}{r}(b_1+a_2)}$$ Since $a_1 + b_2 \geq b_1 + a_2 + 2$ we can’t have $b_1 = a_2 = r$, $a_1+b_2 - \frac{r+2}{r}(b_1+a_2) \geq \frac{1}{r}$. Since $B \geq (kd^3)^{(3d^3)}$, $$\frac{(kd^3)^{(2a_1^2 + 2(a_1+b_1)^2 + 2b_2^2 + 2(a_2+b_2)^2)}}{(kd^3)^{2d^2}}B^{a_1+b_2 - \frac{r+2}{r}(b_1+a_2)} \geq
\frac{(kd^3)^{\frac{1}{r}(3d^3)}}{(kd^3)^{2d^2}} \geq (kd^3)^{d^2}$$ Thus, $$\frac{\tilde{E}[x^{a_1 + a_2}y^{b_1 + b_2}]}{\sqrt{\tilde{E}[x^{2a_1}y^{2b_1}]\tilde{E}[x^{2a_2}y^{2b_2}]}} \leq \frac{k^d}{(kd^3)^{d^2}} \leq \frac{1}{2d^2}$$
For all $a,b$ such that $a \neq b$, $a \leq r$, $b \leq r$, and $a+b \leq d$, $\tilde{E}[x^{a+b}y^{a+b}] \leq \frac{1}{4^{|a-b|}}\sqrt{\tilde{E}[x^{2a}y^{2a}]\tilde{E}[x^{2b}y^{2b}]}$
We have that $\sqrt{\tilde{E}[x^{2a}y^{2a}]\tilde{E}[x^{2b}y^{2b}]} \geq \frac{4^{2(a^2 + b^2)}}{4^{(r^2)}}{k^{a+b}}$ while $\tilde{E}[x^{a+b}y^{a+b}] = \frac{4^{(a+b)^2}}{4^{(r^2)}}{k^{a+b}}$. Thus, $$\frac{\tilde{E}[x^{a+b}y^{a+b}]}{\sqrt{\tilde{E}[x^{2a}y^{2a}]\tilde{E}[x^{2b}y^{2b}]}} \leq \frac{1}{4^{(a-b)^2}} \leq \frac{1}{4^{|a-b|}}$$
Combining all of these results, it can be shown that the moment matrix $M$ corresponding to $\tilde{E}$ is PSD. We now bound $\sum_{I:|I| \leq d}{(\tilde{E}[h_I])^2}$. By a large margin, the dominant terms will come from the leading coefficients of the degree $d$ Hermite polynomials for $x$ and $y$. The leading coefficient of the degree d Hermite polynomial is $\frac{1}{\sqrt{d!}}$ so we have that $$\sum_{I:|I| \leq d}{(\tilde{E}[h_I])^2} \leq \frac{2}{\sqrt{d!}}B^{\frac{d}{2}} \leq (kd^3)^{2d^4}$$ By Lemma \[distlowerbound\], for all $g$ of degree at most $d$ such that $g$ is a sum of squares, $$||f-g||^2 \geq \frac{(\tilde{E}[f])^2}{\sum_{I:|I| \leq d}{(\tilde{E}[h_I])^2}} = \frac{\left(c+(r+1)(1-k^r)\right)^2}{(kd^3)^{2d^4}}$$ Taking $k = \sqrt[r]{2+c}$, the right hand side is at least $\frac{1}{(d^3\sqrt[r]{2+c})^{2d^4}}$ and this completes the proof
In fact, we could have taken any polynomial $f$ on a constant number of variables which is non-negative but not a sum of squares and this covers the case of polynomials with degree $0$ mod $4$ (the above construction only gives us polynomials of degree $2$ mod $4$). However, it would be preferable to have an example which really depends on all its variables. We leave this as a question for future work. Also, the constant is a rapidly decaying function of $d$ and it would be very interesting to obtain a more reasonable constant.
Future Work
===========
In this paper, we have shown that property testing whether a polynomial is a sum of squares using random samples and a tester which always accepts YES instances is hard; we need $n^{\Omega(d)}$ samples, which is not much less than we would need to completely determine the polnomial. That said, this work raises a number of questions, including but not limited to the following:
1. What can be shown for adaptive sampling and/or testers which only need to accept YES instances with high probability?
2. What is the threshold at which polynomial interpolation is likely to result in a polynomial with high norm? In other words, what is threshold at which $M$ stops being close to the identity?
3. Can we obtain almost tight bounds on $||g^c||$ for $c \geq 2$ for polynomial interpolation on random points?
4. If a degree $d$ polynomial $f$ is far from being a degree $d$ non-negative polynomial, must it be negative on a constant proportion of inputs?
5. Can we find a degree $d$ polynomial which is non-negative, far from being a sum of squares, and is far from being a junta (even after a change in coordinates)? Can we property test whether there is some basis in which a polynomial $f$ is a junta?
6. For a given $d$, is there a constant $\epsilon \in (0,1)$ where there is a more efficient way to property test whether a polynomial $f$ of norm $1$ is a sum of squares or is $\epsilon$-far from being a sum of squares?
Example: 4-XOR polynomial
==========================
In this section, we briefly discuss an attempt at creating a polynomial based on 4-XOR which is far from being non-negative yet passes the property test and why it does not quite work. The polynomial is constructed as follows.
1. Randomly choose $n^{2 - \delta}$ equations of the form $x_I = b_I$ where $I$ consists of 4 distinct elements of $[1,n]$ and $b_I \in \{-1,+1\}$.
2. Take the polynomial $p = \sum_{I}{-{b_I}x_I}$
As shown by Grigoriev [@Grigoriev01b], later rediscovered by Schoenebeck [@Schoenebeck08], and explained in Boaz Barak’s lecture notes [@LectureB], we can construct pseudo-expectation values for a constant $d \geq 4$ as follows:
1. Start with the equations $x_I = b_I$ for every $I$ which was chosen.
2. As long as there are sets $I,J$ of size at most $d$ such that $|I \Delta J| \leq d$ and we have not yet set $b_{I \Delta J} = {b_I}b_J$, set $b_{I \Delta J} = {b_I}b_J$. If this gives a contradiction because we already set $b_{I \Delta J} = -{b_I}b_J$, halt and fail. However, with high probability this will not happen.
3. Once we are done, we define $\tilde{E}$ as follows.
1. For all sets $I$ of size at most $d$, take $\tilde{E}[x_I] = b_I$ if $b_I$ was set and take $\tilde{E}[x_I] = 0$ otherwise.
2. For all sets $I$ of size between $d+1$ and $2d$, we set $\tilde{E}[x_I] = 0$.
3. For all multisets $I$ such that $|I| \leq 2d$ and $I$ contains some $x_i$ with multiplicity 2, we take $\tilde{E}[x_I] = \tilde{E}[x_{I \setminus \{x_i,x_i\}}]$
Observe that $\tilde{E}[p] = -n^{2-\delta}$, so $p$ is very far from being a sum of squares. In fact, using Lemma \[distlowerbound\], it can be shown that with high probability, $p$ is $(1 - o(1))||p||$-far from being a sum of squares. If we could add a sum of squares polynomial $g$ to $p$ so that $||g + p||$ is $O(||p||)$ and with high probability polynomially many random samples of $g+p$ will all have nonnegative values, then this would give another example of a polynomial which passes our property tester with high probability yet is far from being a sum of squares. However, this may not be possible. As a special case, if we try taking $g = C$ for a constant $C$ then we would need $C >> ||p||$ in order to make it so that with high probability, polynomially many random samples of $C + p$ all have nonnegative values.
[^1]: Institute for Advanced Study. Supported by the Simons Collaboration for Algorithms and Geometry and by the NSF under agreement No. CCF-1412958. Part of this work was done while at Cornell University.
[^2]: Yale University
|
---
abstract: 'We identify SDSSJ121010.1+334722.9 as an eclipsing post-common-envelope binary, with an orbital period of ${\mbox{$P_{\mathrm{orb}}$}}\,=\,2.988\,\mathrm{h}$, containing a very cool, low-mass, DAZ white dwarf and a low-mass main-sequence star of spectral type M5. A model atmosphere analysis of the metal absorption lines detected in the blue part of the optical spectrum, along with the GALEX near-ultraviolet flux, yields a white dwarf temperature of ${\mbox{$T_{\mathrm{eff,WD}}$}}\,=\,6000\,\pm\,200$K and a metallicity value of $\log[\mathrm{Z/H}]\,=\,-2.0\,\pm\,0.3$. The [[Na[I]{}]{}$\lambda\lambda$8183.27,8194.81]{} absorption doublet is used to measure the radial velocity of the secondary star, ${\mbox{$K_{\mathrm{sec}}$}}\,=\,251.7\,\pm\,2.0\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ and [Fe[I]{}]{} absorption lines in the blue part of the spectrum provide the radial velocity of the white dwarf, ${\mbox{$K_{\mathrm{WD}}$}}\,=\,95.3\,\pm\,2.1\,{\mbox{$\mathrm{km\,s^{-1}}$}}$, yielding a mass ratio of $q\,=\,0.379\,\pm\,0.009$. Light curve model fitting, using the Markov Chain Monte Carlo (MCMC) method, gives the inclination angle as $i\,=\,(79.05^{\circ}\,-\,79.36^{\circ})\,\pm\,0.15^{\circ}$, and the stellar masses as ${\mbox{$M_{\mathrm{WD}}$}}\,=\,0.415\,\pm\,0.010\,{\mbox{$\mathrm{M}_{\odot}$}}$ and ${\mbox{$M_{\mathrm{sec}}$}}\,=\,0.158\,\pm\,0.006\,{\mbox{$\mathrm{M}_{\odot}$}}$. Systematic uncertainties in the absolute calibration of the photometric data influence the determination of the stellar radii. The radius of the white dwarf is found to be ${\mbox{$R_{\mathrm{WD}}$}}\,=\,(0.0157\,-\,0.0161)\,\pm\,0.0003\,{\mbox{$\mathrm{R}_{\odot}$}}$ and the volume-averaged radius of the tidally distorted secondary is ${\mbox{$R_{\mathrm{sec,vol.aver.}}$}}\,=\,(0.197\,-\,0.203)\,\pm\,0.003\,{\mbox{$\mathrm{R}_{\odot}$}}$. The white dwarf in is a very strong He-core candidate.'
author:
- |
S. Pyrzas$^{1}$[^1], B. T. Gänsicke$^{1}$, S. Brady$^{2}$, S. G. Parsons$^{1}$, T. R. Marsh$^{1}$, D. Koester$^{3}$,E. Breedt$^{1}$, C. M. Copperwheat$^{1}$, A. Nebot Gómez-Morán$^{4}$, A. Rebassa-Mansergas$^{5}$, M. R. Schreiber$^{5}$ and M. Zorotovic$^{5}$\
$^{1}$Department of Physics, University of Warwick, Coventry, CV4 7AL, UK\
$^{2}$AAVSO, 5 Melba Drive, Hudson, NH 03051, USA\
$^{3}$Institut für Theoretische Physik und Astrophysik, University of Kiel, 24098 Kiel, Germany\
$^{4}$Université de Strasbourg, CNRS, UMR7550, Observatoire Astronomique de Strasbourg, 11 Rue de l’Université, F-67000, Strasbourg, France\
$^{5}$Departamento de Física y Astronomía, Facultad de Ciencias, Universidad de Valparaíso, Avenida Gran Bretana 1111, Valparaíso, Chile\
bibliography:
- 'aamnem99.bib'
- 'thesis.bib'
date: 'Accepted 2011 August 31. Received 2011 August 26; in original form 2011 July 27'
title: 'Post Common Envelope Binaries from SDSS. XV: Accurate stellar parameters for a cool $0.4\,{\mbox{$\mathrm{M}_{\odot}$}}$ white dwarf and a $0.16\,{\mbox{$\mathrm{M}_{\odot}$}}$ M-dwarf in a 3h eclipsing binary'
---
\[firstpage\]
binaries: close - binaries: eclipsing - stars: fundamental parameters - stars: white dwarfs - stars: late-type - stars: individual: SDSS121010.1+334722.9
Introduction {#sec:intro}
============
Our understanding of stellar structure and evolution leads to the fundamental prediction that the masses and radii of stars obey certain mass-radius (M-R) relations. The calibration and testing of the M-R relations requires accurate and model-independent measurements of stellar masses and radii, commonly achieved with eclipsing binaries [e.g. @andersen91-1; @southworth+clausen07-1].
Among main-sequence (MS) stars, M-dwarfs of low mass ($<\,0.3{\mbox{$\mathrm{M}_{\odot}$}}$), are the most ubiquitous. However, few eclipsing low-mass MS+MS binaries are known (e.g. @lopez-morales07-1 [@moralesetal09-1; @cakirli+ibanoglu10-1; @irwinetal10-1; @dimitrov10-1] and references therein) and have accurate measurements of their masses and radii, affecting the calibration of the low-mass end of the MS M-R relation. To further complicate matters, existing measurements consistently result in radii up to 15% larger and effective temperatures 400K or more below the values predicted by theory [e.g. @ribas06-1; @lopez-morales07-1]. This is not only the case for low-mass MS+MS binaries [@bayless+orosz06-1], but it is also present in field stars [@bergeretal06-1; @moralesetal08-1] and the host stars of transiting extra-solar planets [@torres07-1].
The situation is similar for white dwarfs, the most common type of stellar remnant. Very few white dwarfs have model-independent measurements of their masses and radii [see @parsonsetal10-1], and eclipsing WD+WD binaries have only recently been discovered [@steifadtetal10-1; @parsonsetal11-1; @brownetal11-1]. Consequently, the finite temperature M-R relation of white dwarfs [e.g. @wood95-1; @paneietal00-2] remains largely untested by observations [@provencaletal98-1].
An alternative approach leading to accurate mass and radius measurements for WDs and MS stars is the study of eclipsing WD+MS binaries. Until recently, the population of eclipsing WD+MS binaries had stagnated with only seven systems known [see @pyrzasetal09-1 for a list], a direct result of the small number of the entire WD+MS binaries sample [$\sim\,30$ systems; @schreiber+gaensicke03-1].
However, in recent years, progress has been made thanks to the Sloan Digital Sky Survey [SDSS; @yorketal00-1]. A dedicated search for WD+MS binaries contained in the spectroscopic SDSS Data Release 6 [@adelmanetal08-1] and DR7 [@abazajianetal09-1] yielded more than 1600 systems [e.g. @rebassamansergasetal10-1], of which $\sim\,1/3$ are (short-period) post-common-envelope binaries (PCEBs) [@schreiberetal08-1]. The majority of these PCEBs contain low-mass, late-type M dwarfs [@rebassamansergasetal10-1], while a large percentage of the WD primaries are of low-mass as well [@rebassamansergasetal11-1].
A significant fraction of eclipsing systems should exist among this sample of PCEBs. Identifying and studying these eclipsing systems will substantially increase the observational constraints on the M-R relation of both WDs and MS stars. Therefore, we have begun the first dedicated search for eclipsing WD+MS binaries in the SDSS, and 5 new systems have already been published (@nebotgomezmoranetal09-1 [@pyrzasetal09-1], but see also @drakeetal10-1 for a complementary sample).
SDSSJ121010.1+334722.9 (henceforth ), the subject of this paper, is one of the new systems identified in this search. In what follows, we present our observations (Sec.\[sec:obsnred\]), determine the orbital period and ephemeris (Sec.\[sec:porbephem\]) and analyse the spectrum of the white dwarf (Sec.\[sec:wdanalysis\]). Radial velocity measurements (Sec.\[sec:radvel\]) combined with light curve fitting (Sec.\[sec:fitting\]) lead to the determination of the masses and radii of the binary components (Sec.\[sec:result\]). We also explore the past and future evolution of the system (Sec.\[sec:evolution\]).
Target infromation, observations and reductions {#sec:obsnred}
===============================================
was discovered by @rebassamansergasetal10-1 as a WDMS binary dominated by the flux of a low-mass companion with a spectral type M5V, suggesting that the white dwarf must be very cool. Inspecting the [[Na[I]{}]{}$\lambda\lambda$8183.27,8194.81]{} doublet in the six SDSS sub-spectra[^2] with exposure times of 15–30min taken over the course of three nights, we found large radial velocity variations that strongly suggested an orbital period of a few hours. We obtained time-series photometry of with a 16-inch telescope equipped with an ST8-XME CCD camera, with the aim to measure the orbital period from the expected ellipsoidal modulation, and immediately detected a shallow eclipse in the light curve. Enticed by this discovery, we scheduled for additional high-time resolution photometry, using RISE on the Liverpool Telescope (LT), with which a total of 9 eclipses were observed.
Table\[tab:coordmag\] lists the SDSS coordinates and magnitudes of and the three comparison stars used in the analysis presented in this paper, while Table\[tab:obslog\] summarises our photometric and spectroscopic observations. We note that has a GALEX [@morrisseyetal07-1] near-ultraviolet (NUV) detection, but no far-ultraviolet (FUV) detection.
Star RA Dec $u$ $g$ $r$ $i$ $z$ NUV
------ ----------- ---------- ------- ------- ------- ------- ------- --------
T 182.54221 33.78969 18.10 16.94 16.16 14.92 14.02 20.821
C1 182.55470 33.76832 17.72 15.98 15.33 15.11 15.02
C2 182.54229 33.73406 19.95 17.33 16.01 15.33 14.94
C3 182.62616 33.78141 16.85 15.80 15.46 15.34 15.34
: SDSS coordinates and $u,g,r,i,z$ magnitudes of the target and the comparison stars used in the analysis. We also provide the GALEX near-UV magnitude of .[]{data-label="tab:coordmag"}
Date Telescope Filter/Grating Exp.\[s\] Blocks Frames Eclipses
------------- ----------- ---------------- ----------- -------- -------- ----------
2009 Apr 01 LT V+R 5 1 708 1
2009 Apr 02 LT V+R 5 2 1416 0
2009 Apr 03 LT V+R 5 2 1416 1
2009 Apr 04 LT V+R 5 2 1416 1
2009 Apr 05 LT V+R 5 3 2124 1
2009 Apr 06 LT V+R 5 1 708 0
2009 Apr 29 WHT R600B/R1200R 900 - 1 -
2009 May 02 WHT R600B/R1200R 900 - 3 -
2010 Apr 23 WHT R600B/R1200R 600 - 1 -
2010 May 18 WHT R600B/R600R 900 - 12 -
2011 Feb 06 LT V+R 5 1 720 1
2011 Mar 02 LT V+R 5 1 720 1
2011 Apr 02 LT V+R 5 1 720 1
2011 May 08 LT V+R 5 1 720 1
2011 Jul 03 LT V+R 5 1 720 1
: Log of the photometric and spectroscopic observations. For the LT observations, we also provide the number of one-hour observing blocks per night.[]{data-label="tab:obslog"}
Photometry: LT/RISE {#subsec:photom}
-------------------
Photometric observations were obtained with the robotic 2.0m Liverpool Telescope (LT) on La Palma, Canary Islands, using the high-speed frame-transfer CCD camera RISE [@steeleetal04-1] equipped with a single wideband V+R filter [@steeleetal08-1]. Observations were carried out in one hour blocks, using a 2x2 binning mode with exposure times of 5 seconds.
The data were de-biased and flat-fielded in the standard fashion within the LT reduction pipeline and aperture photometry was performed using [sextractor]{} [@bertin+arnouts96-1] in the manner described in @gaensickeetal04-1.
A sample light curve is shown in Figure\[fig:samplelc\]. The out-of-eclipse variation is ellipsoidal modulation, arising from the tidally deformed secondary.
Spectroscopy: WHT/ISIS {#subsec:spectra}
----------------------
Time-resolved spectroscopy was carried out at the 4.2m William Herschel Telescope (WHT) on La Palma, Canary Islands, equipped with the double-armed Intermediate dispersion Spectrograph and Imaging System (ISIS). The spectrograph was used with a 1 slit, and an 600 lines/mm grating (R600B/R600R) on each of the blue and red arms, although a few spectra were obtained with a 1200 lines/mm grating on the red arm (R1200R). Both the EEV12 CCD on the blue arm and the REDPLUS CCD on the red arm were binned by three in the spatial direction and two in the spectral direction. This setup resulted in an average dispersion of 0.88Å per binned pixel over the wavelength range $3643 - 5137$Å (blue arm) and 0.99Å per binned pixel over the wavelength range $7691 - 9184$Å (red arm, R600R). From measurements of the full width at half maximum of arclines and strong skylines, we determine the resolution to be 1.4Å.
The spectra were reduced using the [starlink]{}[^3] packages [kappa]{} and [figaro]{} and then optimally extracted [@horne86-1] using the [pamela]{}[^4] code [@marsh89-1]. The wavelength scale was derived from Copper-Neon and Copper-Argon arc lamp exposures taken every hour during the observations, which we interpolated to the middle of each of the science exposures. For the blue arm the calibration was determined from a 5th order polynomial fit to 25 lines, with a root mean square (RMS) of 0.029Å. The red arm was also fitted with a 5th order polynomial, to 17 arclines. The RMS was 0.032Å.
Orbital period and ephemeris {#sec:porbephem}
============================
We determined the orbital period and ephemeris of through mid-eclipse timings. This was achieved as follows:
Mid-eclipse times were measured by mirroring the observed eclipse profile around an estimate of the eclipse centre and shifting the mirrored profile against the original until the best overlap was found. This method is particularly well-suited for the box-shaped eclipse profiles in (deeply) eclipsing PCEBs.
An initial estimate of the cycle count was then obtained by fitting eclipse phases $(\phi^{\rm{observed}}_{\rm{0}} - \phi^{\rm{fit}}_{\rm{0}})^{-2}$ over a wide range of trial periods. Once an unambiguous cycle count was established, a linear fit, of the form $T\,=\,T_0\,+\,{\mbox{$P_{\mathrm{orb}}$}}\times E$, was performed to the times of mid-eclipse versus cycle count, yielding a preliminary orbital ephemeris.
Subsequently, we phase-folded our data set using this preliminary ephemeris and proceeded with the light curve model fitting (see Sec.\[sec:fitting\]). Having an accurate model at hand, we re-fitted each light curve individually. This provides a robust estimate of the error on the mid-eclipse time, as our code includes the time of mid-eclipse $T_0$ as a free parameter.
Repeating the cycle count determination and the linear ephemeris fitting, as described above, we obtain the following ephemeris for ,
$$\label{eq:ephe}
\mathrm{MJD}\left(\mathrm{BTDB}\right)\,=\,54923.033\,686(6)\,+\,0.124\,489\,764(1)\,E$$
calculated on a Modified Julian Date-timescale and corrected to the solar system barycentre, with the numbers in parentheses indicating the error on the last digit. Thus, has an orbital period of ${\mbox{$P_{\mathrm{orb}}$}}\,=\,2.987\,754\,336(24)\,\mathrm{h}$. The mid-eclipse times, the observed minus calculated values (O-C) and their respective errors are given in Table\[tab:midecltim\]. Given the short baseline, there is as yet no evidence for period changes which are frequently seen in such binaries [e.g. @parsonsetal10-2].
Mid-Eclipse \[d\] Error \[d\] O-C \[s\] Error \[s\] Cycle
------------------- ------------- ----------- ------------- -------
54923.0336744 0.0000060 -1 1 0
54925.0255324 0.0000082 1 1 16
54926.1459281 0.0000069 -0 1 25
54927.1418460 0.0000087 -0 1 33
55599.1376175 0.0000061 3 1 5431
55623.0396100 0.0000056 -1 1 5623
55654.0375754 0.0000081 -0 1 5872
55690.0151216 0.0000063 1 1 6161
55745.9109933 0.0000069 -2 1 6610
: Times of mid-eclipse (and their errors), O-C values (and their errors) and cycle number for the ephemeris of . Mid-eclipse times and errors are in MJD(BTDB), O-C values and errors are in seconds.[]{data-label="tab:midecltim"}
Spectroscopic analysis {#sec:wdanalysis}
======================
Whereas the SDSS spectrum of remained inconclusive with respect to the nature of the white dwarf [@rebassamansergasetal10-1], our blue-arm WHT spectroscopy immediately revealed a host of narrow metal lines that exhibit radial velocity variations anti-phased with respect to those of the M-dwarf. The WHT spectra obtained in May 2010, averaged in the white dwarf restframe and continuum-normalised, are shown in Fig.\[f-lineid\] and illustrate the wealth of absorption lines from Mg, Al, Si, Ca, Mn, and Fe. Similar metal lines have been detected in the optical spectra of a few other cool PCEBs, e.g. RRCae [@zuckermanetal03-1] or LTT560 [@tappertetal07-1], and indicate accretion of mass via a wind from the M-dwarf.
We have analysed the blue WHT spectra using hydrogen-dominated but metal-polluted (DAZ) spectra calculated with the stellar atmosphere code described by @koester10-1. We fixed the surface gravity to $\log g=7.70$, as determined from the fits to the LT light curve (Sect.\[sec:fitting\]). The model grid covered effective temperatures $5400\,\mathrm{K}\,\le\,{\mbox{$T_{\mathrm{eff,WD}}$}}\,\le\,7400\,\mathrm{K}$ in steps of 200K and metal and He abundances of $\log[\mathrm{Z/H}]\,=\,-3.0, -2.3, -2.0, -1.3, -1.0$, with all relevant elements up to zinc included, and fixed their relative abundances ratios to the respective solar values. We then fitted the model spectra to the average WHT spectrum in the range 3645–3930Å, where the contribution of the M-dwarf is entirely negligible. A good fit is found for ${\mbox{$T_{\mathrm{eff,WD}}$}}\,\simeq\,6000$K and metal abundances at $\simeq\,0.01$ their solar values, however, the effective temperature and the metal abundances are strongly correlated (Fig.\[f-chi2\]).
This degeneracy is lifted by including the GALEX detection of , as the predicted near-UV flux is a strong function of the effective temperature. The uncertainty in the absolute flux calibrations of our WHT spectra and the GALEX observations introduces a small systematic uncertainty on the final result, and we settle for ${\mbox{$T_{\mathrm{eff,WD}}$}}\,=\,6000\,\pm\,200$K and $\log[\mathrm{Z/H}]\,=\,-2.0\,\pm\,0.3$ . Independently, the weakness of the Balmer lines in the WHT spectrum also requires that ${\mbox{$T_{\mathrm{eff,WD}}$}}\,\lesssim\,6400\,\rm{K}$. The spectral modelling of is illustrated on Fig.\[fig:specfit\].
Adopting the white dwarf radius from the light curve fit (Sect.\[sec:fitting\] and \[sec:result\]), ${\mbox{$R_{\mathrm{WD}}$}}\,=\,0.0159\,{\mbox{$\mathrm{R}_{\odot}$}}$, the flux-scaling factor of the best-fit spectral model implies a distance of $d\,\simeq\,50\,\pm\,5$pc, which is in good agreement with $d\,\sim\,66\,\pm\,34$pc estimated by @rebassamansergasetal10-1 from fitting the M-dwarf.
The detection of metals in the photosphere of the white dwarf allows an estimate of the accretion rate [e.g. @dupuisetal93-1; @koester+wilken06-1], as long as the system is in accretion-diffusion equilibrium. In cool, hydrogen-rich atmospheres, such as the one in , the diffusion time scales of the different metals detected in the WHT spectrum vary by a factor of $\sim\,2$ for a given temperature, and are, for ${\mbox{$T_{\mathrm{eff,WD}}$}}\,=\,6000\,\rm{K}$, in the range 30000-60000years[^5]. It is plausible to assume that the average accretion rate over the diffusion time scales involved is constant, as the binary configuration (separation of the two stars, Roche-lobe filling factor of the companion) changes on much longer time scales. Summing up the mass fluxes at the bottom of the convective envelope, and taking into account the uncertainties in and the metal abundances, gives $\dot M\,\simeq\,(5\,\pm\,2)\times10^{-15}\,\mathrm{{\mbox{$\mathrm{M}_{\odot}$}}\,yr^{-1}}$. There are now three PCEBs with similar stellar components that have measured accretion rates, RRCae ($\dot M\,\simeq\,4\times10^{-16}\,\mathrm{{\mbox{$\mathrm{M}_{\odot}$}}\,yr^{-1}}$; @debes06-1), LTT560 ($\dot M\,\simeq\,5\times10^{-15}\,\mathrm{{\mbox{$\mathrm{M}_{\odot}$}}\,yr^{-1}}$; @tappertetal11-2), and ($\dot M\,\simeq\,5\times10^{-15}\,\mathrm{{\mbox{$\mathrm{M}_{\odot}$}}\,yr^{-1}}$).
Whereas and LTT560 have similar orbital periods, the period of RRCae is roughly twice as long, suggesting that the efficiency of wind-accretion decreases as the binary separation and Roche-lobe size of the companion increase, as is expected. A more systematic analysis of the wind-loss rates of M-dwarfs and the efficiency of wind accretion in close binaries would be desirable, but will require a much larger sample of systems.
The spectroscopic orbit {#sec:radvel}
=======================
Radial velocities of the binary components have been measured from the [[Fe[I]{}]{}$\lambda\lambda$4045.813,4063.594,4071.737,4132.058,4143.869]{} absorption lines for the white dwarf and the [[Na[I]{}]{}$\lambda\lambda$8183.27,8194.81]{} absorption doublet for the secondary star.
The [Fe[I]{}]{} lines were simultaneously fitted with a second-order polynomial plus five Gaussians of common width and a separation fixed to the corresponding laboratory values. A sine fit to the radial velocities, phase-folded using the orbital ephemeris (Equation\[eq:ephe\]) yields ${\mbox{$K_{\mathrm{WD}}$}}\,=\,95.3\,\pm\,2.1\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ and ${\mbox{$\gamma_{\mathrm{WD}}$}}\,=\,24.2\,\pm\,1.4\,{\mbox{$\mathrm{km\,s^{-1}}$}}$.
The [Na[I]{}]{} doublet was fitted with a second-order polynomial plus two Gaussians of common width and a separation fixed to the corresponding laboratory value. A sine fit to the radial velocities, phase-folded using the orbital ephemeris yields ${\mbox{$K_{\mathrm{sec}}$}}\,=\,251.7\,\pm\,2.0\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ and ${\mbox{$\gamma_{\mathrm{sec}}$}}\,=\,12.2\,\pm\,0.9\,{\mbox{$\mathrm{km\,s^{-1}}$}}$.
Figure\[fig:rvorb\] shows the measured radial velocities phase-folded on the orbital period and the corresponding sine-fits.
Knowledge of both radial velocities allows us to obtain the mass ratio $q$ of the binary, namely $q\,=\,{\mbox{$K_{\mathrm{WD}}$}}/{\mbox{$K_{\mathrm{sec}}$}}\,=\,0.379\,\pm\,0.009$. We tentatively interpret the difference between and as the gravitational redshift of the white dwarf , which yields ${\mbox{$\mathrm{z_{WD,spec}}$}}\,=\,11.9\,\pm\,1.7\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ (see also Sec.\[sec:result\]).
Light curve modelling {#sec:fitting}
=====================
To obtain the stellar parameters of the binary components, light curve models were fitted to the data using [lcurve]{} (see @copperwheatetal10-1 for a description, as well as @pyrzasetal09-1; @parsonsetal10-1; @parsonsetal11-1 for further applications).
Code input
----------
The code computes a model based on input system parameters supplied by the user. The physical parameters defining the models are (i) the mass ratio $q = {\mbox{$M_{\mathrm{sec}}$}}/{\mbox{$M_{\mathrm{WD}}$}}$, (ii) the binary inclination $i$, (iii) the stellar radii scaled by the binary separation ${\mbox{$r_{\mathrm{WD}}$}}\,=\,{\mbox{$R_{\mathrm{WD}}$}}/a$ and ${\mbox{$r_{\mathrm{sec}}$}}\,=\,{\mbox{$R_{\mathrm{sec}}$}}/a$, (iv) the unirradiated stellar temperatures and , (v) the sum of the unprojected stellar orbital speeds ${\mbox{$V_{\mathrm{S}}$}}\,=\,\left({\mbox{$K_{\mathrm{WD}}$}}\,+\,{\mbox{$K_{\mathrm{sec}}$}}\right)/\mathrm{sin}\,i$, (vi) the time of mid-eclipse of the white dwarf $T_0$, (vii) limb- and gravity darkening coefficients and (viii) the distance $d$. The code accounts for the distance simply as a scaling factor that can be calculated very rapidly for any given model, and so it does not enter the optimisation process. All other parameters can be allowed to vary during the fit.
Free and fixed parameters
-------------------------
During the minimisation, we kept fixed at ${\mbox{$T_{\mathrm{eff,WD}}$}}\,=\,6000\,\mathrm{K}$. The gravity darkening of the secondary was also kept fixed at $0.08$ (the usual value for a convective atmosphere). Limb darkening coefficients were also held fixed. For the white dwarf we calculated quadratic limb darkening coefficients from a white dwarf model with ${\mbox{$T_{\mathrm{eff,WD}}$}}\,=\,6000\,\mathrm{K}$ and $\mathrm{log}\,g\,=\,7.70$, folded through the RISE filter profile. The corresponding values were found to be a$\,=\,0.174$ and $b\,=\,0.421$ for $I(\mu)/I(1)\,=\,1\,-\,$a$\left(1\,-\,\mu\right)\,-\,b\left(1\,-\,\mu\right)^{2}$, with $\mu$ being the cosine of the angle between the line of sight and the surface normal. For the secondary star we used the Tables of @claret+bloemen11-1. We interpolated between the values of V and R for a $T\,=\,3000\,\mathrm{K}$ and $\mathrm{log}\,g\,=\,5$ star, to obtain quadratic limb darkening coefficients a$'\,=\,0.62$ and $b'\,=\,0.273$. All other parameters were allowed to vary.
Minimisation
------------
Initial minimisation is achieved using the downhill-[simplex]{} and [levenberg-marquardt]{} methods [@press02-1], while the Markov Chain Monte Carlo (MCMC) method [@pressetal07-1] was used to determine the distributions of our model parameters [e.g. @ford06-1 and references therein].
The MCMC method involves making random jumps in the model parameters, with new models being accepted or rejected according to their probability computed as a Bayesian posterior probability (the probability of the model parameters, $\theta,$ given the data, D, $P\left(\theta|D\right)$). $P\left(\theta|D\right)$ is driven by a combination of $\chi^2$ and a prior probability, $P(\theta)$, that is based on previous knowledge of the model parameters.
In our case, the prior probabilities for most parameters are assumed to be uniform. The photometric data provide constraints for the radii and inclination angle, however, the photometry alone cannot constrain the masses, as the light curve itself is only weakly depended on $q$. To alleviate this, we can use our knowledge of and . At each jump, the model values and are calculated through $q$, $i$ and . $P(\theta)$ is then evaluated on the basis of the observed and , assuming a Gaussian prior probability $P(\mu,\sigma^{2})$, with $\mu$ and $\sigma$ corresponding to the measured values and errors of and .
A crucial practical consideration of MCMC is the number of steps required to fairly sample the parameter space, which is largely determined by how closely the distribution of parameter jumps matches the true distribution. We therefore built up an estimate of the correct distribution starting from uncorrelated jumps in the parameters, after which we computed the covariance matrix from the resultant chain of parameter values. The covariance matrix was then used to define a multivariate normal distribution that was used to make the jumps for the next chain. At each stage the actual size of the jumps was scaled by a single factor set to deliver a model acceptance rate of $\approx 25$% [@robertsetal97-1]. After 3 such cycles, the covariance matrix showed only small changes, and at this point we carried out the long “production runs” during which the covariance and scale factor which define the parameter jumps were held fixed.
Stellar parameters
------------------
Using the following set of equations, the stellar and binary parameters are obtained directly from the posterior distribution of the model parameters, as outputed from the MCMC minimisation.
The binary separation is obtained from the model parameter through
$$\label{eq:binsep}
a\,=\,\frac{{\mbox{$P_{\mathrm{orb}}$}}}{2\,\pi}\,{\mbox{$V_{\mathrm{S}}$}}$$
The white dwarf and secondary masses are obtained from the model parameters $q$ and as
$$\label{eq:wdmass}
{\mbox{$M_{\mathrm{WD}}$}}\,=\,\frac{{\mbox{$P_{\mathrm{orb}}$}}}{2\,\pi\,\mathrm{G}}\,\frac{1}{1\,+\,q}\,{\mbox{$V_{\mathrm{S}}$}}^{3}$$
and
$$\label{eq:secmass}
{\mbox{$M_{\mathrm{sec}}$}}\,=\,\frac{{\mbox{$P_{\mathrm{orb}}$}}}{2\,\pi\,\mathrm{G}}\,\frac{q}{1\,+\,q}\,{\mbox{$V_{\mathrm{S}}$}}^{3}$$
The stellar radii are directly obtained from the model parameters and and Eq.\[eq:binsep\] and the surface gravity of the white dwarf is of course given by
$$\label{eq:logg}
\mathrm{log}\,g\,=\,\rm{log}\left(\frac{G{\mbox{$M_{\mathrm{WD}}$}}}{R^{2}_{\rm{WD}}}\right)$$
Intrinsic data uncertainties
----------------------------
The acquisition of high-precision absolute photometry on the LT in service mode is somewhat difficult to achieve. Each observing block individually covered only a third of the orbital phase and the blocks were obtained over many nights, under varying conditions (seeing, sky brightness, extinction, airmass). The data are sensitive to changes in conditions, as they have been obtained through the very broad and non-standard V+R filter of RISE. In the absence of a flux standard, the photometry cannot be calibrated in absolute terms. When phase-folding the LT data, significant scatter is found at orbital phases where individual observing blocks with discrepant calibrations contribute. This affects both the shape of the eclipse, mainly the steepness of the WD ingress/egress and, to a lesser extend, the eclipse duration, and the out-of-eclipse variation, i.e. the profile of the ellipsoidal modulation. As a result, there is an unavoidable systematic uncertainty in the photometric accuracy of our data, which will influence the determination of the stellar parameters.
To gauge the effect of the systematic uncertainties we worked in the following fashion: each observing block has been reduced thrice, each time using one of the three comparison stars reported in Table\[tab:coordmag\]. C1 has a $g-r$ colour index comparable to , C2 is fairly red, while C3 is fairly blue. The data of each reduction were then phase-folded together and two light curves were produced: one containing all the photometric points and one where (2-3) observing blocks with an obviously large intrinsic scattering were omitted. Thus, we ended up with six phase-folded light curves. A dedicated MCMC optimisation was calculated for each light curve. We will use the following notation when refering to these chains: C1A denotes a light curve produced with comparison star C1 and all data points, C2E denotes a light curve produced with comparison star C2 excluding observing blocks, and so on.
Results {#sec:result}
=======
{width="\textwidth"}
The results of the six MCMC processes are summarised in Table\[tab:mcmcresult\]. The quoted values and errors are purely of statistical nature and represent the mean and RMS of the posterior distribution of each parameter. The radius of the secondary, as determined by and $a$, is measured along the line connecting the centres of the two stars and, due to the tidal distortion, its value is larger than the average radius. Therefore, on Table\[tab:mcmcresult\] we also report the more representative value of the volume-averaged radius.
Parameter C1A C1E C2A C2E C3A C3E
------------------------ ----------------------- ----------------------- ----------------------- ----------------------- ----------------------- -----------------------
q $0.380\,\pm\,0.010$ $0.380\,\pm\,0.010$ $0.381\,\pm\,0.010$ $0.380\,\pm\,0.010$ $0.378\,\pm\,0.010$ $0.379\,\pm\,0.010$
i \[$^{\circ}$\] $79.05\,\pm\,0.15$ $79.28\,\pm\,0.15$ $79.03\,\pm\,0.15$ $79.13\,\pm\,0.15$ $79.36\,\pm\,0.18$ $79.29\,\pm\,0.16$
\[\] $0.415\,\pm\,0.010$ $0.414\,\pm\,0.010$ $0.415\,\pm\,0.010$ $0.415\,\pm\,0.010$ $0.414\,\pm\,0.010$ $0.414\,\pm\,0.010$
\[\] $0.0157\,\pm\,0.0003$ $0.0159\,\pm\,0.0003$ $0.0161\,\pm\,0.0003$ $0.0159\,\pm\,0.0003$ $0.0138\,\pm\,0.0003$ $0.0150\,\pm\,0.0003$
WD $\mathrm{log}\,g$ $7.664\,\pm\,0.015$ $7.652\,\pm\,0.016$ $7.641\,\pm\,0.015$ $7.649\,\pm\,0.017$ $7.773\,\pm\,0.023$ $7.700\,\pm\,0.019$
\[\] $0.158\,\pm\,0.006$ $0.157\,\pm\,0.006$ $0.158\,\pm\,0.006$ $0.158\,\pm\,0.006$ $0.156\,\pm\,0.007$ $0.157\,\pm\,0.006$
\[\] $0.217\,\pm\,0.003$ $0.212\,\pm\,0.003$ $0.217\,\pm\,0.003$ $0.215\,\pm\,0.003$ $0.210\,\pm\,0.004$ $0.211\,\pm\,0.003$
\[\] $0.202\,\pm\,0.003$ $0.199\,\pm\,0.003$ $0.203\,\pm\,0.003$ $0.201\,\pm\,0.003$ $0.197\,\pm\,0.003$ $0.198\,\pm\,0.003$
\[$\rm{K}$\] $\sim\,2530$ $\sim\,2550$ $\sim\,2530$ $\sim\,2550$ $\sim\,2500$ $\sim\,2550$
Binary separation \[\] $0.871\,\pm\,0.008$ $0.870\,\pm\,0.008$ $0.871\,\pm\,0.008$ $0.871\,\pm\,0.008$ $0.869\,\pm\,0.008$ $0.870\,\pm\,0.008$
To illustrate the achieved quality of the fits, we plot models C1A and C1E in Figure\[fig:fitfig\]. While the overall quality of the fit is very satisfactory, the model seems to slightly overpredict the flux at the “wings” of the ellipsoidal modulation profile (phases $\sim\,0.05\,-\,0.15$ and $\sim\,0.85\,-\,0.95$). This discrepancy could be data related, due to the intrinsic scattering of points; system related, e.g. due to the presence of starspots affecting the modulation; model related, as the treatment of stellar temperatures is based on blackbody spectra, for one specific wavelength; or due to a combination of these factors.
With regard to the binary and stellar parameters, the MCMC results indicate the following: as expected for a detached system, the light curves depend very weakly on $q$ and its value is well constrained by the radial velocitites. All six chains give inclination angle values just above $79^{\circ}$, consistent with each other within the errors. There is a slight shift upwards when excluding blocks from the phase-folded light curve.
The tight spectroscopic constraints, mean that the component masses are largely independent of the model/data set used. Thus, the white dwarf in has a mass of ${\mbox{$M_{\mathrm{WD}}$}}\,=\,0.415\,\pm\,0.010\,{\mbox{$\mathrm{M}_{\odot}$}}$ and the secondary star a mass of ${\mbox{$M_{\mathrm{sec}}$}}\,=\,0.158\,\pm\,0.006\,{\mbox{$\mathrm{M}_{\odot}$}}$.
The quantity most seriously affected by systematics is the white dwarf radius. This is especially evident when considering models C3A and C3E. However, such a discrepancy is expected, since C3 is considerably bluer than and is more susceptible to airmass/colour effects, leading to large intrinsic scattering. The values for as obtained from C1A, C1E, C2A and C2E are consistent within their errors, indicating a systematic uncertainty comparable to the statistical one. This is illustrated in Figure\[fig:wdmr\].
The secondary star radius is affected in a similar, albeit less pronounced, way. All six models lead to values broadly consistent within their statistical errors and a systematic uncertainty of the same order as the statistical one. Figure\[fig:secmr\] shows the six different values of the volume-averaged secondary star radius overplotted on a M-R relation for MS stars. Taken at face value, the results of the MCMC optimisation indicate that the secondary is $\sim\,10$ percent larger than theoretically predicted. As can be seen in Figure\[fig:secmr\], this discrepancy drops to $\sim\,5$ percent, if magnetic activity of the secondary is taken into account. With regard to the secondary temperature, we note again that due to the blackbody approximation, the value of does not necessarily represent the true temperature of the star, it is effectively just a flux scaling factor.
The gravitational redshift predicted by the light curve models (Table\[tab:mcmcresult\]), correcting for the redshift of the secondary star, the difference in transverse Doppler shifts and the potential at the secondary star owing to the white dwarf, are ${\mbox{$\mathrm{z_{WD}}$}}\,=\,15.9\,\pm\,0.4\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ from C1E and ${\mbox{$\mathrm{z_{WD}}$}}\,=\,15.8\,\pm\,0.4\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ from C2E, where the errors are purely statistical and have been derived in the same manner as the other quantities reported in Table\[tab:mcmcresult\]. The systematic uncertainties in our photometric data might still be influencing the result, as the inclination angle and the stellar radii enter the calculation of . Comparing with the spectroscopically determined value of ${\mbox{$\mathrm{z_{WD,spec}}$}}\,=\,{\mbox{$\gamma_{\mathrm{WD}}$}}\,-\,{\mbox{$\gamma_{\mathrm{sec}}$}}\,=\,11.9\,\pm\,1.7\,{\mbox{$\mathrm{km\,s^{-1}}$}}$ we find that they are consistent within $\sim\,2\,\sigma$. The systemic velocities and are determined from spectroscopic observations obtained using a dual-arm spectrograph, with the white dwarf velocity measured in the blue arm and that of the secondary measured in the red arm (Sec.\[sec:obsnred\], \[sec:radvel\]). The observations in both arms are independently wavelength-calibrated and the RMS of $\sim\,$0.03Å (Sec.\[sec:obsnred\]) corresponds to an accuracy of the zero-point of $\sim\,1\,-\,2\,{\mbox{$\mathrm{km\,s^{-1}}$}}$. The potential of an offset in the calibrations of the two arms enters the determination of as an additional systematic uncertainty.
Past and future evolution of {#sec:evolution}
=============================
Considering its short orbital period, must have formed through common-envelope evolution [@paczynski76-1; @webbink08-1]. As shown by @schreiber+gaensicke03-1, if the binary and stellar parameters are known, it is possible to reconstruct the past and predict the future evolution of PCEBs for a given angular momentum loss prescription. Here, we assume classical disrupted magnetic braking [@verbunt+zwaan81-1]. In this context, given the low mass of the secondary, the only angular momentum loss mechanism for is gravitational radiation. Based on the temperature and the mass of the white dwarf we interpolate the cooling tracks of @althaus+benvenuto97-1 and obtain a cooling age of $t_{\rm{cool}}\,=\,3.5\,\rm{Gyr}$. This corresponds to the time that passed since the binary left the common envelope. We calculate the period it had when it left the common envelope to be $P_{\mathrm{CE}}\,=\,4.24$h. Following the same method as in @zorotovicetal10-1 and based on their results we reconstructed the initial parameters of the binary using a common-envelope efficiency of $\alpha_{\mathrm{CE}}=0.25$ and the same fraction of recombination energy [see @zorotovicetal10-1 for more details]. We found an initial mass of $M_{\rm{prog}}\,=\,1.33\,{\mbox{$\mathrm{M}_{\odot}$}}$ for the progenitor of the white dwarf, which filled its Roche lobe when its radius was $R_{\rm{prog}}\,=\,91.3\,{\mbox{$\mathrm{R}_{\odot}$}}$. At that point, the orbital separation was $a\,=\,162.7\,{\mbox{$\mathrm{R}_{\odot}$}}$, and the age of the system was $t_{\rm{sys}}\,=\,4.4\,\rm{Gyr}$, since the time it was formed. Using the radius of the secondary[^6] we calculate that the system will reach a semi-detached configuration and become a cataclysmic variable (CV) at an orbital period of $P_{\rm{sd}}\,\sim\,2\,$h in $t_{\rm{sd}}\,=\,1.5\,\rm{Gyr}$.
Given that the current places right at the upper edge of the CV orbital period gap[^7], and that the calculated $P_{\rm{sd}}$, when will start mass-transfer, is right at the lower edge of the period gap, we are tempted to speculate whether is in fact a detached CV entering (or just having entered) the period gap. @davisetal08-1 have shown that a large number of detached WD+MS binaries with orbital periods between 2-3 hours are in fact CVs that have switched off mass-transfer and are crossing the period gap. This could in principle explain the apparently over-sized secondary in , as expected from the disrupted magnetic braking theory [e.g. @rappaportetal83-1]. However, the temperature of the WD in seems to be uncomfortably low for a WD that has recently stopped accreting [@townsley+gaensicke09-1].
Discussion and Conclusions {#sec:conclusion}
==========================
In this paper, we have identified as an eclipsing PCEB containing a very cool, low-mass, DAZ white dwarf and a low-mass main-sequence companion.
Using combined constraints from spectroscopic and photometric observations we have managed to measure the fundamental stellar parameters of the binary components. Systematic uncertainties in the absolute calibration of our photometric data, influence the determination of the stellar radii. The stellar masses, however, remain unaffected and were measured to a $1\%$ accuracy. The (formal) statistical uncertainties in all binary parameters indicate the level of precision that can be achieved in this system. All parameters are summarised in Table\[tab:allparam\].
Parameter Value
---------------------------- ------------------------------------
\[d\] 0.124489764(1)
$q$ $0.379\,\pm\,0.009$
$a$\[\] $0.870\,\pm\,0.008$
Inclination \[$^{\circ}$\] $(79.05\,-\,79.36)\,\pm\,0.15$
\[\] $0.415\,\pm\,0.010$
\[\] $(0.0157\,-\,0.0161)\,\pm\,0.0003$
$\mathrm{log}\,g$ $7.65\,\pm\,0.02$
\[$\rm{K}$\] $6000\,\pm\,200$
\[\] $95.3\,\pm\,2.1$
\[\] $0.158\,\pm\,0.006$
\[\] $(0.210\,-\,0.217)\,\pm\,0.003$
\[\] $(0.197\,-\,0.203)\,\pm\,0.003$
\[\] $251.7\,\pm\,2.0$
: Adopted stellar and binary parameters for .[]{data-label="tab:allparam"}
With a mass of ${\mbox{$M_{\mathrm{WD}}$}}\,=\,0.415\,\pm\,0.010\,{\mbox{$\mathrm{M}_{\odot}$}}$ and a temperature of ${\mbox{$T_{\mathrm{eff,WD}}$}}\,\sim\,6000\,\mathrm{K}$, the DAZ white dwarf in pushes the boundaries in a hitherto unexplored region of the WD parameter space. The M-R results from the four Chains C1 and C2 are consistent with a He-core WD, assuming a hydrogen layer of $M(\rm{H})/{\mbox{$M_{\mathrm{WD}}$}}\,=\,3\times10^{-4}$. However, due to lack of observational constraints for the H-layer thickness and the uncertainty in the radii, we will defer identifying the WD as a definite He-core and simply emphasise the strong candidacy.
The secondary star, with a mass of ${\mbox{$M_{\mathrm{sec}}$}}\,=\,0.158\,\pm\,0.006\,{\mbox{$\mathrm{M}_{\odot}$}}$, illustrates once more the excellent opportunity that PCEBs give us for testing and calibrating the M-R relations of low-mass stars. Taking the radius measurements at face value, the secondary star seems to be $\sim\,10$ percent larger than the theoretical values, although this drops to $\sim\,5$ percent, if magnetic activity is taken into consideration. In this context, the magnetic activity present in the secondary can lead to the formation of stellar (dark) spots on the surface. The effect of these spots is to block the outgoing heat flux, reducing $T_{\rm{eff}}$ and, as a result, the secondary expands to maintain thermal equilibrium [@chabrieretal07-1; @moralesetal10-1]. @krausetal11-1 found that low-mass stars in short period binaries appear to be overinflated (although their analysis was restricted to ${\mbox{$M_{\mathrm{sec}}$}}\,>\,0.3\,{\mbox{$\mathrm{M}_{\odot}$}}$), which seems to be the case for . We should note however, that the mass and radius of the secondary star in the eclipsing PCEB NNSer (with ${\mbox{$M_{\mathrm{sec}}$}}\,=\,0.111\,\pm\,0.004\,{\mbox{$\mathrm{M}_{\odot}$}}$ and comparable orbital period to ) is consistent with theoretical M-R predictions, even though it is heavily irradiated by the hot WD primary [@parsonsetal10-1].
We have speculated whether is in fact a detached CV entering the period gap, which could explain the large radius of the secondary. This hypothesis could be tested by measuring the rotational velocity of the white dwarf. This can be achieved through high-resolution spectroscopy of the [Fe[I]{}]{} absorption lines in the WD photosphere [see e.g. @tappertetal11-2].
In any case, it is highly desirable to improve the measurement of the stellar radii in SDSS1210 to the comparable precision to the masses presented here. This will require high-precision photometry in standard filters, such as e.g. delivered by ULTRACAM [@dhillonetal07-1].
Acknowledgements {#acknowledgements .unnumbered}
================
We thank the anonymous referee for a prompt report. BTG, TRM, EB and CMC are supported by an STFC Rolling Grant. MRS and ARM acknowledge financial support from FONDECYT in the form of grants 1100782 and 3110049. MZ acknowledges support from Gemini/CONICYT (grant 32100026). Based in part on observations made with the William Herschel Telescope operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias and on observations made with the Liverpool Telescope operated on the island of La Palma by Liverpool John Moores University in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias with financial support from the UK Science and Technology Facilities Council.
\[lastpage\]
[^1]: E-mail: [email protected]
[^2]: The sub-exposures that are co-added to produce one SDSS spectrum of a given object
[^3]: Maintained and developed by the Joint Astronomy Centre and available from http://starlink.jach.hawaii.edu/starlink
[^4]: Available from http://www.warwick.ac.uk/go/trmarsh
[^5]: For completeness, we note that because we have adopted solar abundance ratios for the metals, these small differences in diffusion time scales imply slightly non-solar ratios in the accreted material. In principle, the individual metal-to-metal ratios can be determined from the observed spectrum of the white dwarf, and hence allow to infer the abundances of the companion star, however, this requires data with substantially higher spectral resolution to resolve the line blends.
[^6]: We assume a representative value of ${\mbox{$R_{\mathrm{sec,vol.aver.}}$}}\,=\,0.2\,{\mbox{$\mathrm{R}_{\odot}$}}$ for the volume-averaged radius of the secondary
[^7]: The orbital period range where only a small number of CVs are found.
|
---
abstract: 'We have obtained observations of the ultraviolet spectrum of AM CVn, an ultra-short-period helium cataclysmic variable, using the Space Telescope Imaging Spectrograph (STIS) aboard the Hubble Space Telescope (HST). We obtained data in time-tag mode during two consecutive orbits of HST, covering 1600–3150 Å and 1140–1710 Å, respectively. The mean spectrum is approximately flat in $f_\nu$. The absorption profiles of the strong lines of , , , , and are blue-shifted and in some cases asymmetric, evidencing a wind that is partly occulted by the accretion disk. There is weak red-shifted emission from and . The profiles of these lines vary mildly with time. The light curve shows a decline of $\sim$20% over the span of the observations. There is also flickering and a 27 s (or 54 s) “dwarf nova oscillation”, revealed in a power-spectrum analysis. The amplitude of this oscillation is larger at shorter wavelengths. We assemble and illustrate the spectral energy distribution (s.e.d.) of AM CVn from the ultraviolet to the near-infrared. Modeling the accretion phenomenon in this binary system can in principle lead to a robust estimate of the mass accretion rate on to the central white dwarf, which is of great interest in characterizing the evolutionary history of the binary system. Inferences about the mass accretion rate depend strongly on the local radiative properties of the disk, as we illustrate. Uncertainty in the distance of AM CVn and other parameters of the binary system presently limit the ability to confidently infer the mass accretion rate.'
author:
- 'Richard A. Wade, Michael Eracleous[^1], and Hélène M. L. G. Flohic'
title: 'New Ultraviolet Observations of AM CVn [^2]'
---
Introduction {#sect:intro}
============
AM CVn (HZ 29; $\alpha_{2000}= 12^h 34^m 54\fs 6$, $\delta_{2000}= +37^\circ 37\arcmin 43\arcsec$, $l=140^\circ$, $b=+79^\circ$, $V\approx$14.1) is the type star of the small class of helium cataclysmic variables (HeCVs). These are also called interacting binary white dwarfs, although the mass donor star which fills its Roche lobe and transfers mass via a gas stream and accretion disk to the mass-gaining white dwarf may not itself be fully degenerate. Hydrogen lines are absent from the spectra of these objects. A recent review of the class and a summary of possible evolutionary pathways leading to the AM CVn stars can be found in Nelemans (2005).
The orbital period of AM CVn is $P_{\rm orb} = 1029$ s (Nelemans et al. 2001). A signal is found at this period in the power spectrum of time-series photometry, and a second signal is found near $P_{\rm sh}
= 1051$ s. This latter signal is thought to represent a “permanent superhump”, at the beat period between the orbital period and the precession period of a non-circular disk. From the fractional excess of $P_{\rm sh}$ over $P_{\rm orb}$, it is thought possible to estimate the mass ratio of the binary system, $q= M_2/M_1$, where $M_2$ is the mass of the donor star (see §\[sect:obsdsed\]). A recent paper describes a kinematic measurement of $q$ (Roelofs et al. 2006).
The accretion disk in AM CVn appears always to be in a stable “high” state, consistent with the relatively high mass transfer rate that is expected of the HeCVs at the short end of the observed range of periods. (Smak 1983; Tsugawa & Osaki 1997; Deloye et al. 2005). Mass transfer is thought to be driven ultimately by gravitational wave radiation of orbital angular momentum, combined with the expansion of the donor star as mass is lost. In this model, the orbital period evolves to larger values with time, and the mass transfer rate to smaller values. A [*measurement*]{} of the mass accretion rate can help constrain the mass ratio and/or the donor star mass, from which it may be possible to reconstruct the prior history of the binary (Deloye et al. 2005). Inferring the mass transfer rate from observation involves measuring the distance of the system and estimating the bolometric flux, or accurately modeling the spectral energy distribution (s.e.d.) of the accretion disk, also taking into account light from the two stars. Either method involves measuring the emitted spectrum over a broad range of wavelengths.
The visible spectrum of AM CVn has been studied extensively, and the visible s.e.d. has been modeled on the assumption that it is dominated by the accretion disk (e.g., El-Khoury & Wickramasinghe 2000; Nasser et al. 2001; Nagel et al. 2004). A reasonable degree of success has been achieved in these studies in accounting for the profiles of the absorption lines, although different authors have inferred (or imposed!) different properties of the disk such as the mass transfer rate, the disk size, and the disk inclination. The ultraviolet (UV) spectrum and the overall s.e.d.have received less attention in recent years but offer the possibility, when used together with the lines, of more powerfully constraining the parameters of the disk model.
With the above considerations in mind, we obtained observations of the UV spectrum of AM CVn using the Space Telescope Imaging Spectrogram (STIS) aboard the Hubble Space Telescope (HST). One goal was to model the UV absorption line spectrum of the accretion disk in some detail and thus infer the mass transfer rate $dM/dt$ and inclination angle $i$ of the disk rather directly. We also hoped to learn more about the chemical composition of the material being transferred. Instead of showing an absorption spectrum that arises purely from a steady disk in local hydrostatic equilibrium, however, the STIS observations show evidence of a disk wind, which obliterates many of the stronger diagnostic lines and blends that are key to such an analysis. Moreover, the existence of a wind calls into question some of the standard assumptions underlying models of accretion disk atmospheres.
The STIS data greatly exceed, in quality, reliability, spectral resolution, and time resolution, most of the UV observations of AM CVn that were made with the International Ultraviolet Explorer (IUE) in the period 1978-1991 (which nevertheless hinted at the presence of wind line profiles). The STIS data exceed in wavelength coverage some earlier HST observations of AM CVn, made using the Goddard High Resolution Spectrograph (GHRS). They have revealed some behavior not previously reported, including oscillations in light level with a period of $\sim$27 s.
We defer a detailed modeling effort and in this contribution focus on describing the UV spectrum and its behavior and on presenting the overall s.e.d. of AM CVn. We describe the STIS observations and the major features of the spectrum, emphasizing the wind line profiles, their variation with time, and the time variation and power spectrum of the measured fluxes. We assemble the UV-visible-infrared s.e.d.of AM CVn, which forms one basis for modeling the accretion process in this binary. We briefly discuss some practicalities and uncertainties of such modeling, which lead to uncertainties in the inferred mass accretion rate. In particular, we illustrate the critical role played by the radiative properties of the gas in the accretion disk. We also emphasize the importance of an accurate knowledge of the binary mass ratio and distance to AM CVn, as they relate to inferring $dM/dt$.
The paper is organized as follows. In §2 we describe the observations with STIS, then present our findings concerning mean and time-dependent behavior of the flux level and line profiles. In §3 we discuss the apparent variation in UV flux level, identify the [27-s]{} oscillations as “dwarf nova oscillations”, compare the “wind” line profiles with theoretical expectations and observations of other systems, and present and discuss the s.e.d. of AM CVn. We briefly summarize our findings and discussion in §4.
Observations and Findings {#sect:obsns}
=========================
We observed AM CVn with STIS on 2002 February 21, during two consecutive orbits of the HST. The target was observed in time-tag mode, in the first orbit with the NUV-MAMA and grating G230L for 32 minutes, and in the second orbit with the FUV-MAMA and grating G140L for 40 minutes.[^3] Thus we have complete orbital coverage of the binary star with each detector/disperser setup, but we do not have coverage of the 13.4 h precessional period of the eccentric disk which gives rise to a 1051 s superhump period (Patterson et al. 1993).
The NUV observation started at 09:16:57 UT and lasted for 1920 s, with wavelength coverage 1577–3161 Å. The FUV observation started at 10:35:48 UT and lasted for 2400 s, thus ending almost 2 hours after the start of the NUV exposure. The FUV wavelength coverage was 1121–1717 Å, overlapping the NUV coverage. The “52X0.1” aperture ($0\farcs1$ wide, long slit) was used. An Acquisition/Peak-up sequence of exposures was done prior to the NUV observations to center the star accurately in the slit. During the spectrum exposures, the r.m.s. jitter of HST on the V2 and V3 axes was less than about 8.5 milli-arcseconds, with no recenterings and no losses of lock.
Trend of Flux Level with Time {#sect:trend}
-----------------------------
The light curve of AM CVn (1650–1700 Å integrated flux) during the [*HST*]{} observation is shown in Figure \[fig:lcurve\]. This wavelength range was observed using both instrument setups and does not include any strong lines. There is an apparent, more-or-less steady decline in flux, which corresponds to a change of 0.21 mag over the course of 2 hours.
A portion of the apparent decline can possibly be accounted for in terms of a small inconsistency of calibration between the NUV/G230L and FUV/G140L observing modes. Three active galactic nuclei, observed by one of the authors with STIS using these same modes in the same back-to-back fashion (albeit with the 52X0.2 aperture rather than the 52X0.1 aperture) are suitable as a test of this, since their UV fluxes do not vary significantly over the course of a day. The 1650–1700 Å fluxes differ by about 6–7 percent, in the same sense as observed for AM CVn. For AM CVn, the change in the [*mean*]{} NUV/G130L 1650–1700 Å flux to the [*mean*]{} FUV/G140L flux is 15 percent.
Within each observation (NUV or FUV), a declining straight line fit to the full-band fluxes binned in 30-second intervals gives a much smaller $\chi^2$ statistic compared with a constant-flux fit, the slopes being $\sim$0.1 percent per minute in each case. Taking the 1650–1700 Å fluxes at face value, and considering the NUV and FUV data as a single series gives an average decline of 0.17 percent per minute. Presumptively “correcting” the FUV flux relative to the NUV flux by 7 percent, as suggested by the AGN data, would give an average rate of decline for the combined data sets of $\sim$0.09 percent per minute, similar to the individually estimated slopes. There is, of course, structure in the light curves on top of straight-line behavior (discussed in §\[variations\] below), so these exercises are only suggestive, not probative.
We have considered whether there is a likely artifactual origin for the apparent decline in flux, and find nothing plausible. Thus we regard the apparent decline as probably real, perhaps modified by a $\sim7$% correction to the relative calibration of the FUV and NUV data. (We do not apply any such correction further in this paper, however.) In §\[sect:uvflux\], we discuss other published observations of AM CVn, as they bear on possible similar flux variations in UV or visible light.
The Mean Ultraviolet Spectrum {#sect:meanspec}
-----------------------------
The mean short-wavelength (FUV-MAMA, G140L) spectrum of AM CVn is shown in Figure \[fig:shortspec\]. The spectrum covers 1150–1715 Å with a resolution of 1.4 Å ($\approx 300~{\rm
km~s^{-1}}$ at mid-range), sampled at $\approx 0.6$ Å pixel$^{-1}$. This spectral region is characterized by a roughly flat continuum (in $f_\nu$) upon which are superposed strong absorption lines of , , , and . The and features are flanked by emission on the “red” side, and all the absorption components of the lines are blue-shifted. We interpret these to be wind lines and discuss their profiles further below. These lines are labeled in Figure \[fig:shortspec\]. Table \[table:windlines\] summarizes the identifications and laboratory wavelengths of these features. An unmarked, broad feature centered near 1300 Å is likely a blend of numerous (subordinate) lines; these and the numerous other weaker features observable in the spectrum probably arise in the accretion disk and would be kinematically blended depending on the projected orbital speed of the gas at their respective radii of formation.
Several sharp interstellar lines are also evident, indicated by tick marks above the spectrum, and their observed wavelengths, equivalent widths (EWs), and identifications are given in Table \[table:islines\]. The average velocity offset of these measured lines from their adopted laboratory values is $-1~{\rm
km~s^{-1}}$ with a standard deviation of $\sigma(\Delta v) = 36~{\rm
km~s^{-1}}$.
The “Lyman-$\alpha$” feature (rest wavelength 1215.67 Å) is probably a mixture of an interstellar component and an intrinsic component, which in this hydrogen-deficient object is likely the (2-4) transition at 1215.15 Å. Like the other strong absorption lines, this feature is asymmetric, with the blue absorption wing being broader than the red wing. This feature is included in Table \[table:islines\]; the tabulated equivalent width is derived from a Gaussian fit to the core and red side of the line; if the extended blue wing is included, the EW is 4.26 Å. Given the low resolution offered by the G140L grating, we do not attempt to decompose this feature or derive a interstellar column density.
The mean long-wavelength (NUV-MAMA, G230L) spectrum of AM CVn is shown in Figure \[fig:longspec\]. The spectrum covers 1600–3150 Å with a resolution of 2.8 Å ($\approx 350~{\rm km~s^{-1}}$ at mid-range), sampled at $\approx 1.6$ Å pixel$^{-1}$. This spectral region is also characterized by a roughly flat continuum. Interstellar lines (see Table \[table:islines\]) are marked above the spectrum as before; the intrinsic lines of and (see Table \[table:windlines\]) are marked and labeled. (The line sits in the overlap region common to both the G140L and G230L spectra.) The local flux minimum in the $\sim$1800–2000 Å region (Fig. 3) is not of interstellar origin. The pattern of flux variation here is similar to that seen in the spectra of early B stars (B0.5 – B3, especially luminosity classes II and I; cf. Wu et al. 1991). Important contributors likely include and (see the atlases by Rountree & Sonneborn 1993; and Walborn et al. 1995).
Mean profiles of the strong absorption lines from , , , , and are shown in detail in Figure \[fig:profiles\], on a common velocity scale. The velocity zero point in each panel is set at the nominal (rest) wavelength of the line, or the unweighted average wavelength in the case of doublets. (The zero-velocity wavelengths for each feature are listed in Table \[table:windlines\].) The tick marks show the nominal location of each doublet component in the adopted velocity frame. All of the observed line cores appear to be blue-shifted by 500–800 km s$^{-1}$, and and have red-shifted emission components (P-Cygni profiles), suggesting the presence of an outflowing wind. The terminal velocities of the blue-shifted absorption wings are in the range 2000–3000 km s$^{-1}$.
![Normalized profiles of some strong absorption lines are shown on a common velocity scale. The velocity zero point is set at the nominal wavelength of a line, or the average wavelength in the case of doublets; the zero-velocity wavelengths are given in Table \[table:windlines\]. The tick marks show the nominal location of the doublet components in the adopted velocity frame. Normalization is with respect to the local continuum. The vertical scale varies from panel to panel. \[fig:profiles\]](f4.eps)
Short-Term Variations in the Time Domain\[variations\]
------------------------------------------------------
Because the STIS observations of AM CVn were obtained in time-tag mode, we were able to search for time variations in the spectrum, such as rapid variations in the line profiles, or short-period periodic or quasi-periodic variations in the light level. We discuss first variations in the line profiles, then variations in brightness.
We binned the time-tagged observations for both the and FUV/G140L NUV/G230L data sets, using both 30-second and 3-second bins. From the 30-second binned data, we prepared trailed spectrograms for each region. Inspection of the trailed spectrograms (not illustrated) did not reveal any marked time-dependent behavior. There are, however, some subtle variations in the profiles of the wind lines, which became more evident when the mean spectrum was subtracted from the 30-second binned data. The pattern of variation comprises an interval of a few hundred seconds during which both red and blue “edges” of the wind absorption features are located at longer wavelengths than in the mean spectrum, followed by an interval of similar length in which the features are shifted to shorter than average wavelengths. The total displacement corresponds to a few hundred km s$^{-1}$. The Ly$\alpha$ feature and other interstellar features do not show this variation. The pattern of red-shift followed by blue-shift is repeated about 900 s later, in the second half of the NUV/G230L observation set. This pattern is illustrated in Figure \[fig:varyprofiles\], where spectra averaged over two 480 s intervals are shown for the (mainly interstellar) Ly$\alpha$ line and the , , , and profiles. The two intervals are separated by a 120 s gap. (We did not attempt to optimize the size and spacing of the intervals chosen for illustration, so 480 s should not be interpreted as the duration of either the “red” or “blue” phase.) The wind from AM CVn thus seems to be mildly unsteady. Further interpretation of the wind lines is offered in §\[sect:lineinterp\].
![Comparison of the wind line profiles at two epochs. The histogram-like lines represent the average spectrum of AM CVn during 480 s (spectra \#10–25 of a sequence of eighty 30-s NUV/G140L binned spectra), while continuous lines represent the average spectrum during a later 480 s period (spectra \#30–45). The gap between the two intervals is 120 s. The zero-velocity wavelengths used for each multiplet are as in Table \[table:windlines\]. The (mainly interstellar) “Ly$\alpha$” profile is also shown; its red edge does not participate in the velocity shifts shown by the wind features. \[fig:varyprofiles\]](f5.eps)
The 30-second and 3-second binned light curves of AM CVn for the short-wavelength FUV/G140L data (1150–1715 Å) are shown in Figure \[fig:short\_lc\]. There is evident structure (flickering) in the light curve, which is also manifested as low-frequency noise in the power spectrum. The power spectrum of the G140L data is shown in Figure \[fig:short\_power\]. The mean level was subtracted from the 3-second binned light curve, prior to computing the Fourier transform. The Nyquist period is 6 seconds, and the power spectrum is sampled at 513 frequencies. Each point in the spectrum is statistically independent of its neighbors, that is, a pure monochromatic signal would show power in a (centered) single bin. Normalized power spectra are shown. The significance of a signal is given in terms of the probability, $N \exp({-p/\langle p \rangle})$, that an isolated peak of power $p$ could arise by chance in a white-noise spectrum containing $N$ bins with mean power $\langle p
\rangle$). See Eracleous et al. (1991) for further details.
In addition to the low-frequency noise, there is a peak at $f =
37.0$ mHz (${1/\nu}=27.0\pm0.3$ s). The peak is slightly broadened: a Gaussian fit shows the FWHM to be 0.61 mHz. The estimated coherence of this oscillation is thus $Q \approx 60$, where $Q$ is the ratio of the peak frequency to the FWHM of the peak. Pure-frequency (sinusoidal) test signals of various periods injected into the time series always resulted in power spectrum peaks with $Q \gtrsim 100$.
The 30-second and 3-second binned light curves for the long-wavelength NUV/G230L data (1600–3150 Å) are shown in Figure \[fig:long\_lc\]. As with the short-wavelength data, flickering is evident over a range of time scales. The normalized power spectrum of the G230L data is shown in Figure \[fig:long\_power\]. As before, the mean level was subtracted from the light curve, prior to computing the Fourier transform. The Nyquist period is 6 seconds, and the power spectrum is sampled at 513 frequencies. The power spectrum shows low-frequency noise, corresponding to the flickering in the light curve. The prominent peak at a period of 27 s that is seen in the short-wavelength power-spectrum is not as obvious here. While a peak at that period is discernible, its power is not significantly above the noise level.
The shape and amplitude of the oscillation vary with wavelength, as illustrated by the pulse profiles in Figure \[fig:pulseprof\]. The data are folded on a 54-second pulse period and shown in eighteen 3-second bins. About forty-four cycles are averaged in the short-wavelength data, and thirty-five cycles in the long-wavelength data. The pulse is strongest and the signal-to-noise ratio highest in the bluest band (bottom panel), with both quantities decreasing toward increasing wavelength. The relatively blue spectrum of the pulsed light explains why the pulses are easily detected in the power spectrum of the FUV light but are absent in the power spectrum of the NUV light. The pulse is somewhat asymmetric, indicating that the true period may be 54 seconds rather than 27 seconds. Further discussion of the oscillations is presented in §\[sect:oscns\].
![Pulse profiles in four different bands, folded on a 54-second period. The dynamic range is similar in all frames to facilitate comparison of the relative pulse amplitude between bands. The pulse is strongest and the signal-to-noise ratio highest in the bluest band. The pulse is somewhat asymmetric, indicating that the true period may be 54 seconds rather than 27 seconds. \[fig:pulseprof\]](f10.eps)
Discussion {#sect:discuss}
==========
The Ultraviolet Flux Level {#sect:uvflux}
--------------------------
We noted in §\[sect:trend\] the apparent decline in AM CVn’s UV flux level during our STIS observations. Our tentative conclusion is that this decline is larger than can easily be accounted for by instrumental or calibration effects, and should be regarded as real, especially as it seems to be at least marginally present [*within*]{} each of the FUV and NUV observations separately as well as when they are considered together.
Noting that an excursion of similar amplitude was seen in 1981 March, but in the visible band and lasting only 500 s (Elsworth et al.1982), we investigate whether other brightness changes of similar rate, duration, or amplitude have been observed in AM CVn before, either in visible or UV light.
R. Kent Honeycutt has kindly communicated to us the results of synoptic monitoring of AM CVn in the $V$ band over the time period 1990 Nov – 1996 Aug. These observations were made using the Roboscope (Honeycutt et al. 1994). One to four magnitude measurements were made per night, with 506 measurements in total spread over 417 nights. On nights with multiple observations (usually separated by a few hours) the intranight variations in brightness are consistent with the measured errors of observation, which are typically at or below the one per cent level. On a few occasions brightness changes of a few per cent from one night to the next are noted, although they are of marginal statistical significance. Slow drifts in the average $V$ magnitude of a similar amplitude are also noted over the course of an observing season. No “outbursts” are seen, nor any rapid and persistent changes in brightness such as those seen in the STIS data of Figure \[fig:lcurve\]. The median magnitude is $V=14.11$.
Skillman et al. (1999) also report from high-speed photometry that the visual brightness of AM CVn is quite robust, always between $V=14.10$ and 14.20 and never varying more than 0.07 mag during a night.
Taking the various calibrated fluxes at face value, we find the UV flux from AM CVn to vary somewhat over [*long*]{} intervals, as shown by the mean UV spectrum of AM CVn obtained in 1995 with the GHRS in program 6085. The s.e.d. is flat ($f_\nu \approx$ constant) over the wavelength range 1260-1560 Å, with a mean continuum flux level of about 16.5 mJy, or about 15% higher than our nominal FUV/G140L mean flux level.
We have also compared the UV brightness of AM CVn found in our STIS observations with the flux level measured by IUE at several epochs between 1978 and 1988. (See Boggess et al. 1978 for a description of IUE instrumentation.) We considered only IUE observations made in the LOW dispersion mode using the LARGE aperture. We retrieved NEWSIPS (Final Archive) fluxes in ASCII form from the Browse facility at MAST[^4] and computed average flux densities for each observation, in 60 Å (100 Å) intervals centered at 1450 Å or 2700 Å, depending on the camera used. At 1450 Å, the IUE fluxes range between 2.1 and $2.4 \times
10^{-13}~{\rm erg~cm^{-2}~s^{-1}}$ Å$^{-1}$, and the corresponding STIS measurement is 2.0 in the same units. At 2700 Å, the IUE fluxes lie between 5.5 and $6.0 \times 10^{-14}~{\rm
erg~cm^{-2}~s^{-1}}$ Å$^{-1}$, and the STIS measurement is 5.1 in the same units. Thus the STIS observations show AM CVn to be somewhat less bright in 2002 February than was recorded by IUE, although by only $\sim$10%, comparable to the range of variation observed with IUE. Massa & Fitzpatrick (2000) point out some concerns about the NEWSIPS absolute calibration of IUE fluxes, at the 10% level, so our comparison of IUE and STIS fluxes is preliminary only.
Ramsay et al. (2005) present near-UV observations of AM CVn with the Optical Monitor aboard the XMM-Newton observatory, spanning an interval of about 12000 s. The effective wavelength was 2910 Å (UVW1 filter, range 2400–3400 Å). Excepting a few outliers, count rates in 120 s bins do not show deviations from the mean rate that are larger than about 3%. This rate is not easily converted to absolute flux units for comparison with other instruments.
In physical terms, it is not difficult to imagine that variations in UV brightness at the $\sim$10% level might occur, driven by variations in accretion rate through the disk and onto the central white dwarf. One might expect larger variations at UV wavelengths than in visible light, and indeed the higher amplitude at shorter wavelengths of the 27-s oscillations seen in the STIS data are an example (§\[variations\]). Other cataclysmic variables (CVs) in a persistent high-luminosity state have also been observed to vary in the UV. Hartley et al. (2002) document $\sim$30% variations in the continuum level of V3885 Sgr and less certain variations at the factor-of-two level in IX Vel, between observations with STIS spaced weeks or months apart.
To summarize, AM CVn’s brightness in the UV [*sometimes*]{} appears to be more strongly variable than in the visible on time scales of hours or longer, although the data are too sparse to allow full characterization of this variation, and some calibration uncertainties persist at the few percent level. Without further time-series observations in the UV, we cannot say whether or not this behavior (hours-long trends) happens frequently.
The oscillations {#sect:oscns}
----------------
Our power spectrum analysis of the UV light curve of AM CVn (§\[variations\]) revealed a slightly broadened peak at 37 mHz, corresponding to an oscillation period of 27.0 s. Folding the light curve shows a waveform that is approximately sinusoidal, although alternating maxima have slightly different heights suggesting the underlying period may be $\sim$54 s. This is only a 2–$\sigma$ effect, however, and we discuss the oscillation in terms of a 27 s period. The amplitude (half of the peak-to-peak variation) is $\sim$1% in the shorter wavelengths, diminishing to about half this in the longest observed wavelengths (Figure \[fig:pulseprof\]).
Patterson et al. (1979) reported a “transient coherent or quasi-coherent periodicity at 26.3 s” in visible-light time series photometry of AM CVn. Patterson et al. (1992) showed the power spectrum of this oscillation. The amplitude was $\sim$0.01 mag, uncertainly measured because of the poor coherence of the signal; they referred to the signal as a “quasi-periodic oscillation” and revised the mean period to 26.2 s. In a recharacterization, Skillman et al.(1999) stated that “the period and coherence of this signal are quite plausible for ‘dwarf nova oscillations’.” We adopt the point of view that the signal seen by Patterson et al. (1979) and by us at different times and in different wavebands arises from the same cause.
Are these variations “dwarf nova oscillations” (DNO), or are they “quasi-periodic oscillations” (QPO)? According to the review by Warner (2004), DNOs are moderately coherent ($Q_W \equiv |dP/dt|^{-1}
\gtrsim 10^3$), nearly-sinusoidal signals with periods typically in the range 8 to 40 s. QPOs are less coherent, with periods typically $\sim$10 times longer than the corresponding DNOs for a CV. DNOs and QPOs may appear simultaneously or not. DNOs are characteristically observed during the outbursts of dwarf novae, but are also seen in some luminous “novalike” CVs. In an outburst, the period of the DNO varies with time, inversely related to the luminosity of the dwarf nova. In addition to this luminosity-linked variation, the DNO period may show sudden small jumps, best studied by means of an amplitude and phase analysis, in which short segments of the light curve are fitted directly in a sliding window using a sinusoidal model for the variation. Given the similarity of the DNO period to the Keplerian period in the inner disk, DNOs are interpreted as having their origin at or near the surface of the accreting white dwarf or in the inner disk. Warner (1995) argues that QPOs, with their longest periods corresponding to the orbital timescale in the outer disk, may be linked to oscillations of the disk, with a mixture of periods corresponding to different radial zones.
If the 27-s (37 mHz) signal from AM CVn is a DNO, we might look for a QPO near 4–5 mHz. The power spectrum shows a “grassy” structure at these frequencies, which may indicate the presence of a QPO or may simply be part of poorly-defined “1/f noise”, characteristic of low-level “flickering”. More data would be required to develop a well-averaged power spectrum that would show whether a broad QPO “bump” lies on top of a 1/f “continuum”.
We performed a simple analysis of amplitude and phase for the FUV light curve of AM CVn, using a moving window 270 s (10 cycles) long. We fitted a model consisting of a linear baseline and a pure sinusoid with fixed period. The form of the baseline is $A + Bt$, and $A$ and $B$ vary as the window is shifted along, to take out some of the low-frequency wandering in the light curve. We note well-defined “DNO-like” phase drifts of duration several hundred seconds, indicating a period that is changing around the mean period, as in Warner (2004; his Fig. 2). We also note that the amplitude of the fitted sinusoid is typically higher in the first half of the time series than later (in our Figure \[fig:short\_lc\], there is high amplitude around Relative Time = 16 minutes, and low amplitude around 26 minutes, as extreme examples). The width (FWHM) of the spike in the power spectrum partly arises from this amplitude modulation (effectively, only half of the time series is useful in establishing the frequency of the signal), and partly arises from the small drifts of period during the half of the observation where the oscillation is strong.
Despite the difference in chemical composition of AM CVn’s disk compared with the usual hydrogen-rich CV, the temperature of the inner disk is likely similar to a dwarf nova in outburst. It would therefore not be surprising to observe DNOs from AM CVn in the same period range as normally observed. The amplitude of a DNO in the UV would be high relative to that in visible light, since the UV light from the inner disk is not diluted by light from the outer disk. We therefore adopt the interpretation of Skillman et al. (1999) in characterizing the 27-s oscillation as a DNO. We note, however, that the defining test of whether the period changes with system luminosity cannot be carried out, since AM CVn does not undergo “dwarf nova” outbursts. Labeling the oscillation as a DNO does not tell us the cause of the oscillation, which seems to be still very open to debate. Different points of view as to the physical mechanisms involved are represented by Warner & Woudt (2006), Mauche (2002), and Piro & Bildsten (2005).
Since the “outer disk” in AM CVn is less extensive than in longer-period CVs, one is observing the inner disk with relatively low dilution by lower temperature gas, even in visible light. It might therefore be regarded as slightly puzzling that the $\sim$26 s oscillation has not been reported more frequently. But the source is relatively faint, and attention historically has been focused on the “orbital/superhump” modulation at 1029/1051 s, both factors tending to the use of integration times that are long compared with 26 s. In the ultraviolet, the same is true. The analysis by Solheim et al.(1997) of time-resolved UV spectra, obtained with the GHRS, made use of data binned in $\sim50$ s intervals (with gaps). It thus is not sensitive to the frequency range that includes the signal we report here. Ramsay et al. (2005) report a power spectrum of the integrated UV light from AM CVn, from their observations using the XMM-Newton Optical Monitor. It shows significant power near periods of 996, 529, and 332 s. Their analysis does not extend to periods as short as 27 or 54 s, however.
The wind lines {#sect:lineinterp}
--------------
The blue-shifted broad absorption features in the UV spectrum of AM CVn (Table 1 and Figures 4 & 5), with red-shifted emission apparent only in and perhaps , are typical of the wind-formed lines seen in the spectra of all luminous CVs seen at low or moderate inclination. In AM CVn, the deepest lines are and . The terminal velocity of the absorption trough is near $-3000~{\rm km~s^{-1}}$, most cleanly seen in .
Time-resolved UV spectra of AM CVn, obtained in 1995 with the GHRS, have apparently not been published in detail, although a mean spectrum and some time-series analysis are reported in Solheim et al. (1997). A superficial comparison of the mean GHRS and STIS spectra in the region of overlap shows that they are very similar, apart from the difference in overall brightness level mentioned earlier.
Proga (2005) reviews the modeling of CV winds, while Froning (2005) reviews the observations and highlights some of the difficulties that models have in reproducing the diversity of behavior that is seen in the lines. Kinematic models of CV wind lines (e.g., Knigge & Drew 1996) explain them as being formed mainly by scattering in a non-spherically symmetric, or “bi-conical” outflow. The winds lines are stronger in higher luminosity CVs, suggesting that they are driven by radiation pressure. Two-dimensional, time-dependent hydrodynamic models of radiation-driven winds indeed produce a slow, dense “equatorial” wind bounded on the pole-ward side by a faster more tenuous flow (Proga, Stone & Drew 1998, 1999). The winds are predicted to be stronger for a higher luminosity system. The terminal velocities are a few times the escape velocity from the white dwarf. If the disk luminosity is higher than that of the central white dwarf, the outflow tends to become unstable and clumpy. Some qualitative success has been achieved at matching predicted line profiles with observations, although problems remain, and a unique matching of a model to a data set is elusive (e.g., Proga et al.2002; Proga 2003; Long & Knigge 2002; Froning 2005).
We have made a qualitative comparison of the mean profiles from Figure 4 with illustrative computed profiles presented in Proga et al.(2002), Long & Knigge (2002), and Proga (2003). We concentrate mainly on the profile. AM CVn is not an eclipsing binary system, so an intermediate inclination angle is inferred. Consistent with this, the overall observed shape in (terminal velocity, concavity, and location of maximum depth close to zero velocity) matches reasonably well to model profiles calculated for intermediate inclinations ($i=30^\circ$ or $55^\circ$). The fixed parameters of these models (white dwarf mass and radius, disk radius, etc.) are not tuned to AM CVn, and the sampling of the variable parameters (inclination, wind mass loss rate) is too coarse to allow more quantitative conclusions to be drawn, but the explanation of the observed line profiles in terms of a bi-conical wind seems to be secure.
It is reassuring that CV wind theory seems to be applicable in the case of hydrogen-deficient, “ultracompact” binaries such as AM CVn. Nor is this a surprising result, if the wind originates in the inner disk and is viewed against continuum light from the inner disk. The outer disk is “missing”, but seems to have little effect on the shaping of the [*absorption*]{} part of the line profile, and the UV opacities in the UV-forming inner disk are affected at only the factor-of-two level by the absence of hydrogen. The shape of the ionizing continuum from a hydrogen-deficient disk is expected to be different, however, so the interpretation of the strength of a wind line profile in terms of a wind mass loss rate may be expected to differ from the hydrogen-dominated case. (The strength of the line depends on the population of the ion, hence on the ionization state of the gas as well as the overall density of the wind).
The shifting positions of the resonance line absorption features shown in Figure \[fig:varyprofiles\] is a small but real effect, likely showing that conditions in the outflow are not steady. In AM CVn, the profiles as a whole seem to shift back and forth, maintaining their shapes. Certain other luminous CVs show much more “wild” behavior in the wind lines, e.g., V603 Aql (Prinja et al. 2000a) and BZ Cam (Prinja et al. 2000b), in which the shapes of the lines and their equivalent widths change drastically, and individual “blobs” or clumps may be traced as their velocities change with time. The theory of radiation-driven winds suggests that the outflow becomes unstable and clumpy if the disk luminosity (vertically directed) dominates over the radially-directed luminosity from the central star (Proga et al.1998). There is thus some hope that this aspect of disk wind theory can be confronted by the AM CVn observations, where the instability seems to be marginal, if the disk and white dwarf luminosities can be determined. In the case of RW Sex, on the other hand, the time variations in the profiles of far-UV wind lines seem to be related to orbital modulation, and any fluctuations of density or speed in the wind may occur on scales small enough that they do not appear with adequate contrast in the observed line profiles (Prinja et al. 2003). We noted in §\[variations\] a rough time scale for the observed line variations of $\sim$900 s, comparable to the orbital period of AM CVn (1029 s), but only further observations could establish whether there is a strict correspondence. (Interest is added, owing to the fact that the secondary star in AM CVn, although small, is much closer to the source of the wind and thus might present a means of modulating the outflow on the orbital period.) At present, our only firm conclusion is that there are variations present in the wind line profiles, with the cause yet to be determined.
The observed spectral energy distribution {#sect:obsdsed}
-----------------------------------------
We show in Figure \[fig:sed\] the observed s.e.d. of AM CVn, assembled from several different sources. Data at the shortest wavelengths come from FUSE observations (epoch 2004 Dec 26), as read by eye from “preview” spectra provided at the MAST website[[^5]]{}; the points represent estimated continuum fluxes at 1000, 1050, 1100, and 1150 Å. Next are the STIS data reported earlier in this paper (“G140L” and “G230L”). Optical and near-infrared data (“MCSP”) are from the Multichannel Spectrophotometer formerly in use at the Hale 5m telescope. These data are described in Oke & Wade (1982), and can be made available in machine-readable form on request; we suggest that these quasi-monochromatic flux density measurements be used in place of broadband $UBVRI$ magnitudes when modeling the s.e.d. Finally the infrared (IR) broadband fluxes from 2MASS (Skrutskie et al. 2006) are given, using the absolute calibration and zero-point offsets from Cohen et al. (2003). We summarize these data here: isophotal wavelengths of the 2MASS $J$, $H$, and $K_s$ bands are 1.235, 1.662, and 2.159 microns respectively; the corresponding isophotal monochromatic flux densities are 2.57, 1.74, and 0.91 mJy with random errors of 3%, 4.5%, and 6.6%. No correction for interstellar extinction has been made in Figure \[fig:sed\].
![Spectral energy distribution of AM CVn, assembled from various sources at various epochs (see §\[sect:obsdsed\]). No correction for interstellar extinction has been made. Also shows are example calculations of the s.e.d. from a (non-irradiated) steady-state accretion disk specified by parameters suggested by Roelofs et al. (2006); see §\[sect:obsdsed\] for details. The upper (dotted) curve shows a model that uses blackbody spectra to represent the local emission from the disk surface; the lower (solid) line uses interpolated spectra from pure helium model atmospheres instead, neglecting limb darkening. No other sources of light are included. \[fig:sed\]](f11.eps)
The GHRS fluxes (1260–1540 Å) from 1995 lie about 15% (0.06 dex) higher than the STIS/G140L fluxes shown in the figure. The GHRS fluxes would more smoothly connect the STIS/G230L and FUSE data sets. This points up a general problem with modeling the s.e.d. of AM CVn, namely what fluxes are to be modeled. In view of the mixed nature of the data — some broadband, some quasi-monochromatic; obtained at different epochs with evidence of variability; etc. — an attempt to find an “optimum” fit to the s.e.d. should give careful consideration to the figure of merit that is used to determine goodness of fit, including, e.g., how the various parts of the spectrum are weighted.
The spectrum of AM CVn can be modeled to infer $dM/dt$ and perhaps other quantities such as $M_1$, $M_2$ (the mass of the donor star), the distance $d$, and the inclination $i$ of the disk orbital plane to the line of sight. An accurate knowledge of $dM/dt$ is needed to (1) confirm that the disk in AM CVn satisfies the thermal stability criterion for disks in a permanent “high” state (Smak 1983; Tsugawa & Osaki 1997); (2) assess the thermal and evolutionary properties of the donor star (e.g., Deloye et al. 2005); and (3) properly partition the observed flux between the disk and the other contributors to the radiation. Item (3) is necessary for deriving abundances from modeling of the absorption line spectrum in the UV, where some of the light may come from the central white dwarf and some from the disk.
The library of local disk spectra that is used to construct a model s.e.d. of the entire disk is of paramount importance. Blackbody spectra, for example, differ greatly from the s.e.d.s of helium atmospheres. We illustrate in Figure \[fig:sed\] the difference between a blackbody disk (BBD) and a helium-atmosphere disk (HeD), computed with the same model parameters. Our model disk is the usual steady-state, geometrically thin, optically thick, non-irradiated accretion disk. We choose $M_1= 0.69~{\rm M_\sun}$, $R_1 = R_{\rm wd}
= 7.8 \times 10^8$ cm, $R_d = 7.6 \times 10^9$ cm, $dM/dt = 6.7 \times
10^{-9}~{\rm M_\sun~yr^{-1}}$, $d= 600$ pc, and $i= 43\degr$, similar to the model for AM CVn proposed by Roelofs et al. (2006). Here $R_1
= R_{\rm wd}$ is the inner radius of the disk, taken to be the radius of the white dwarf; and $R_d$ is the outer radius of the disk. For this illustration we have used angle-averaged fluxes from Wesemael (1981) and D. Koester, private communication. No sources of light other than the disk are included. The UV-optical-IR model fluxes from the HeD are a factor of $\sim1.6$ lower than those from the BBD, owing to different bolometric corrections in the two cases (the “missing” flux is emitted in the extreme ultraviolet). With the chosen parameters, the overall fit of the HeD model is worse than that of the BBD. By increasing $dM/dt$, the HeD model can be made to better match the normalization of the observed s.e.d.
Independent constraints on $d$, $M_1$, $R_1$, $M_2$, and $i$, if they are trustworthy, are very important for reducing the number of correlated parameters or the degree of correlation in a model of the accretion disk. In the AM CVn system, two constraints are especially significant for inferring $dM/dt$ from the s.e.d. The first is the system’s mass ratio $q$, for which estimates range from $q=0.087$ (Nelemans et al. 2001) to $q=0.22$ (Pearson 2003). Roelofs et al.(2006) have put forward a spectroscopically determined $q=0.18 \pm
0.01$. Once $q$ is determined, $M_1$ and $M_2$ cannot be varied independently. Varying $M_1$ affects both temperatures and orbital speeds (line smearing) in the disk, as well as affecting the fundamental solid angle $(R_1/d)^2$; hence $dM/dt$ as inferred from observation is also affected. Meanwhile $dM/dt$ is tied to $M_2$ through the theory of mass transfer driven by angular momentum loss.
The second constraint is the distance. The often-used value $d =
235$ pc is based on an unpublished parallax, $\pi_{\rm abs} = 4.25 \pm
0.43$ mas from the USNO parallax team (C. Dahn, private communication 2003 & 2006; 62 observations over 8.1 y; seven stars in the astrometric reference frame). Recently a new parallax determination using the Fine Guidance Sensor aboard the HST, $\pi = 1.65 \pm
0.30$ mas ($d \approx 606$ pc), has been published (Roelofs et al.2007). This study has a duration of 1.6 y and relies on three astrometric reference stars, all to the south of AM CVn. In Figure \[fig:sed\], reducing $d$ from 600 pc to a smaller value would raise the predicted fluxes of the model for a given $dM/dt$. A substantially smaller distance such as $d = 235$ pc would therefore mandate a reduction in the model $dM/dt$ in order to match the observed flux level, leading also to a cooler and “redder” model disk; in the case of the HeD model, the model would better match the observed relative s.e.d. The formal difference between the two parallax measurements is statistically very significant, and $d^2$ differs by a factor of $\approx$7. Choosing to apply one of the advertised distance estimates over the other, as a modeling constraint, has a huge impact on whether any particular model for AM CVn can successfully account for both the s.e.d. and spectroscopic observations of that system. We are not in a position to resolve this discrepancy.
The strong linkages among $q$, $d$, $dM/dt$, and other system parameters are illustrated in the analysis of several AM CVn-type systems by Roelofs et al. (2007).
Summary {#sect:summary}
=======
We have presented new, time-resolved STIS observations of AM CVn, which show a UV spectrum that is approximately flat in $f_\nu$. The absorption profiles of $\lambda$1240, $\lambda$1398, $\lambda$1549, $\lambda$1640, and $\lambda$1718 are asymmetric and blue-shifted, evidencing a wind that is partly occulted by the accretion disk. There is also weak emission at and . These features are consistent with profiles predicted for scattering in a bi-conical wind from an accretion disk, viewed at intermediate inclination. The profiles of these wind lines vary mildly with time, showing shifts in the observed wavelengths of both red and blue absorption edges. Sharp (interstellar) absorption lines are also seen. The Lyman-$\alpha$ feature is presumably interstellar, but may be blended with the (2-4) transition arising in AM CVn itself. Numerous weaker spectral features of various widths are found, probably arising in the accretion disk and kinematically blended.
The UV light curve of AM CVn from the STIS observations shows an apparent, relatively steady decline by $\sim$20% over the span of the observations. Only a portion of the nominal decline can be attributed to possible calibration uncertainties. We summarize data that suggest AM CVn’s brightness varies by a larger amount in the UV than in the optical. There are also short-term “white light” variations, including a 27-s DNO that is stronger at shorter wavelengths. The true period of the DNO may be 54 s.
We have assembled the UV-visible-IR s.e.d. of AM CVn by combining the STIS observations with data from FUSE, the Palomar MCSP instrument, and 2MASS. Successful models of the accretion process in AM CVn, accounting for the shape and normalization of the entire observed s.e.d., may give a robust estimate of the mass accretion rate $dM/dt$ and other parameters of the system. The mass accretion rate is of great interest to understanding the origin and subsequent evolution of HeCVs. Inferences about $dM/dt$ depend strongly on the local radiative properties of the gas that makes up the accretion disk. We have illustrated this by explicit computation of the s.e.d. from a blackbody disk and a Helium-atmosphere disk, using the same example specification of $M_1$, $R_1$, $R_d$, $dM/dt$, $d$, and $i$ in both cases. The results are quite different. Other key factors in inferring $dM/dt$ include what is assumed about the mass of the accreting star (strongly influenced by the adopted mass ratio $q$) and the distance to the system.
We are grateful to R. K. Honeycutt for communicating to us the results of Roboscope synoptic observations of AM CVn, to M. Rogers for assistance in the review of literature, and to C. Dahn for some correspondence about the USNO parallax result. D. Koester kindly provided some model spectra for pure helium atmospheres. We benefitted from helpful comments by an anonymous referee. M.E. acknowledges the hospitality of the Astrophysics Department at the American Museum of Natural History.
This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Some of the data presented in this paper were obtained from the Multimission Archive at the Space Telescope Science Institute (MAST). STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA Office of Space Science via grant NAG5-7584 and by other grants and contracts. Support for HST GO program \#8159 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555.
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[^1]: The Center for Gravitational Wave Physics, The Pennsylvania State University, Pennsylvania State University, University Park, PA 16802
[^2]: B
[^3]: See http://www.stsci.edu/hst/stis/ for the STIS Instrument Handbook.
[^4]: Multimission Archive at the Space Telescope Science Institute,\
http://archive.stsci.edu/iue/
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abstract: 'We adopt the perspective of an aggregator, which seeks to coordinate its *purchase* of demand reductions from a fixed group of residential electricity customers, with its *sale* of the aggregate demand reduction in a two-settlement wholesale energy market. The aggregator procures reductions in demand by offering its customers a uniform price for reductions in consumption relative to their predetermined baselines. Prior to its realization of the aggregate demand reduction, the aggregator must also determine how much energy to sell into the two-settlement energy market. In the day-ahead market, the aggregator commits to a forward contract, which calls for the delivery of energy in the real-time market. The underlying aggregate demand curve, which relates the aggregate demand reduction to the aggregator’s offered price, is assumed to be affine and subject to unobservable, random shocks. Assuming that both the parameters of the demand curve and the distribution of the random shocks are initially unknown to the aggregator, we investigate the extent to which the aggregator might dynamically adapt its offered prices and forward contracts to maximize its expected profit over a time window of $T$ days. Specifically, we design a dynamic pricing and contract offering policy that resolves the aggregator’s need to learn the unknown demand model with its desire to maximize its cumulative expected profit over time. In particular, the proposed pricing policy is proven to incur a *regret* over $T$ days that is no greater than $O(\log(T)\sqrt{T})$.'
address: 'School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA.'
author:
- Kia Khezeli
- Eilyan Bitar
bibliography:
- 'references.bib'
title: An Online Learning Approach to Buying and Selling Demand Response
---
Demand response,dynamic pricing,online learning,electricity markets.
=0mu plus 1mu
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---
abstract: 'We present the MAssive ClusterS and Intercluster Structures (MACSIS) project, a suite of 390 clusters simulated with baryonic physics that yields realistic massive galaxy clusters capable of matching a wide range of observed properties. MACSIS extends the recent BAHAMAS simulation to higher masses, enabling robust predictions for the redshift evolution of cluster properties and an assessment of the effect of selecting only the hottest systems. We study the observable-mass scaling relations and the X-ray luminosity-temperature relation over the complete observed cluster mass range. As expected, we find the slope of these scaling relations and the evolution of their normalization with redshift departs significantly from the self-similar predictions. However, for a sample of hot clusters with core-excised temperatures $k_{\rm{B}}T\geq5\,\rm{keV}$ the normalization and slope of the observable-mass relations and their evolution are significantly closer to self-similar. The exception is the temperature-mass relation, for which the increased importance of non-thermal pressure support and biased X-ray temperatures leads to a greater departure from self-similarity in the hottest systems. As a consequence, these also affect the slope and evolution of the normalization in the luminosity-temperature relation. The median hot gas profiles show good agreement with observational data at $z=0$ and $z=1$, with their evolution again departing significantly from the self-similar prediction. However, selecting a hot sample of clusters yields profiles that evolve significantly closer to the self-similar prediction. In conclusion, our results show that understanding the selection function is vital for robust calibration of cluster properties with mass and redshift.'
author:
- |
David J. Barnes$^1$[^1], Scott T. Kay$^1$, Monique A. Henson$^1$, Ian G. McCarthy$^2$, Joop Schaye$^3$ and Adrian Jenkins$^4$\
$^{1}$Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, UK\
$^{2}$Astrophysics Research Institute, Liverpool John Moores University, 146 Brownlow Hill, Liverpool L3 5RF, UK\
$^{3}$Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, the Netherlands\
$^{4}$Institute for Computational Cosmology, Department of Physics, University of Durham, South Road, Durham DH1 3LE, UK
bibliography:
- 'ms.bib'
date: 'Accepted XXX. Received YYY; in original form ZZZ'
title: The redshift evolution of massive galaxy clusters in the MACSIS simulations
---
\[firstpage\]
galaxies: clusters: general - galaxies: clusters: intracluster medium - X-rays: galaxies: clusters - galaxies: evolution - methods: numerical - hydrodynamics
Introduction {#sec:intro}
============
Galaxy clusters form from large primordial density fluctuations that have collapsed and virialised by the present epoch, with more massive clusters forming from larger and rarer fluctuations. This makes them especially sensitive to fundamental cosmological parameters, such as the matter density, the amplitude of the matter power spectrum and the equation of state of dark energy [see @Voit2005; @AllenEvrardMantz2011; @KravtsovBorgani2012; @Weinberg2013]. The observable properties of a galaxy cluster result from a non-trivial interplay between gravitational collapse and astrophysical processes. The diverse range of formation histories of the cluster population leads to scatter in the observable-mass scaling relations and, as surveys select clusters based on an observable, this can lead to a biased sample of clusters, resulting in systematics when using them as a cosmological probe [e.g. @Mantz2010]. Many previous studies have shown that the relationship between a cluster observable, such as its temperature or X-ray luminosity, and a quantity of interest for cosmology, e.g. its mass, has a smaller scatter for more massive, dynamically relaxed objects [@EkeNavarroFrenk1998; @Kay2004; @Crain2007; @NagaiVikhlininKravtsov2007a; @Planelles2013]. Therefore, the fundamental requirement when probing cosmological parameters with galaxy clusters is a sample of relaxed, massive clusters with well calibrated mass-observable scaling relations.
However, galaxy clusters are rare objects, becoming increasingly rare with increasing mass, and to observe a sample large enough to be representative of the underlying population requires a survey with significant size and depth. Currently ongoing and impending observational campaigns, such as the Dark Energy Survey [@DEScol2005], *eRosita* [@Merloni2012], *Euclid* [@Laureijs2011], <span style="font-variant:small-caps;">SPT-3G</span> [@Benson2014] and Advanced ACTpol [@Henderson2016], will be the first to have sufficient volume to yield significant samples of massive clusters. Due to their rarity, the majority of these massive clusters will be at high redshift and it is therefore critical to understand how the cluster observables and their associated scatter evolve. Additionally, the most massive clusters will be the brightest and easiest to detect objects at high redshift, making it vital to understand the selection function of the chosen cluster observable and whether the most massive clusters are representative of the underlying cluster population. Theoretical modelling of the formation of clusters and their observable properties is required to understand these issues and to further clusters as probes of cosmology. Due to the range of scales involved in cluster formation, the need to incorporate astrophysical processes and to self-consistently predict observable properties, cosmological hydrodynamical simulations are the only viable option.
Recent progress in the modelling of large-scale structure formation has been driven mainly by the inclusion of supermassive black holes and their associated Active Galactic Nucleus (AGN) feedback, which has been shown to be critical for reproducing many cluster properties [@Bhattacharya2008; @PuchweinSijackiSpringel2008; @McCarthy2010; @Fabjan2010]. A number of independent simulations are now able to produce realistic clusters that simultaneously reproduce many cluster properties in good agreement with the observations [@LeBrun2014; @Pike2014; @Planelles2014]. Results from the recent BAryons and HAloes of MAssive Systems (BAHAMAS) simulations [@McCarthy2016] have shown that by calibrating the subgrid model for feedback to match a small number of key observables, in this case the global galaxy stellar mass function and the gas fraction of clusters, simulations of large-scale structure are now able to reproduce many observed scaling relations and their associated scatter over two decades in halo mass. However, full gas physics simulations of large-scale structure formation, with sufficient resolution, are still computationally expensive. This has limited previous studies to either small samples with $<50$ objects or to volumes of $596\,\mathrm{Mpc}$, all of which are too small to contain the representative sample of massive clusters that is required for cosmological studies above $z=0$.
This paper introduces the Virgo consortium’s MACSIS project, a sample of $390$ massive clusters selected from a large volume dark matter simulation and resimulated with full gas physics to enable self-consistent observable predictions. The simulations extend the BAHAMAS simulations to the most massive clusters expected to form in a $\Lambda\rm{CDM}$ cosmology. In this paper we study the cluster scaling relations and their evolution. We combine the MACSIS and BAHAMAS simulations to produce a sample that spans the complete mass range and that can be studied to high redshift, using the progenitors of the MACSIS sample. We also select the hottest clusters from the combined sample and a relaxed subset of them to examine the impact of such selections on the scaling relations and their evolution. We then study the gas profiles to further understand the differences between the samples.
This paper is organised as follows. In Section \[sec:MACsamp\] we introduce the MACSIS sample and discuss the parent dark matter simulation from which the sample was selected, the selection criteria used, the model used to resimulate the haloes, how we produced the observable quantities and the three samples we use in this work. In Section \[sec:screlations\] we investigate how the scaling relations evolve and how this evolution changes when a hot cluster sample or relaxed, hot cluster sample is selected. We then study the hot gas profiles to understand the differences in the evolution of the relations for the different samples in Section \[sec:gasprofs\]. Finally, in Section \[sec:sad\] we discuss our results and summarise our main findings.
Parent simulation and sample selection {#sec:MACsamp}
======================================
In this section we describe the parent simulation, the selection of the MACSIS sample, the baryonic physics used in the resimulation of the sample and the calculation of the observable properties of the resimulated clusters. Additionally, we describe how MACSIS and BAHAMAS clusters were selected to produce the combined sample and the cuts made to yield a hot sample and its relaxed subset.
The parent simulation
---------------------
To obtain a population of massive clusters we require a simulation with a very large volume $(> 1\,\rm{Gpc}^{3})$. With current computational resources it is unfeasible to simulate such a volume with hydrodynamics and the required gas physics, such as radiative cooling, star formation and feedback, at a resolution high enough to accurately capture the cluster properties. An alternative option is to apply the zoomed simulation technique to a representative sample of objects from a larger volume. Therefore, we select a sample of massive haloes from a dark matter only simulation that has sufficient volume to yield a population of massive clusters and the resolution to ensure they are well characterized. We label this simulation the ‘parent’ simulation.
The parent simulation is a periodic cube with a side length of $3.2\,\rm{Gpc}$. Its cosmological parameters are taken from the Planck 2013 results combined with baryonic acoustic oscillations, WMAP polarization and high multipole moments experiments [@Planck2014I] and are $\Omega_{\rm{b}}=0.04825$, $\Omega_{\rm{m}}=0.307$, $\Omega_{\Lambda}=0.693$, $h\equiv H_0/(100\,\rm{km}\,\rm{s}^{-1}\,\rm{Mpc}^{-1})=0.6777$, $\sigma_{8}=0.8288$, $n_{\rm{s}}=0.9611$ and $Y=0.248$. We note that there are minor differences between these values and the Planck-only cosmology used for the BAHAMAS simulations, but this has negligible impact on the results presented here. The simulation contained $N=2520^3$ dark matter particles that were arranged in an initial glass-like configuration and then displaced according to second-order Lagrangian perturbation theory $(\mathrm{2LPT})$ using the <span style="font-variant:small-caps;">ic\_2lpt\_gen</span> code [@Jenkins2010] and the public Gaussian white noise field *Panphasia* [@Jenkins2013; @JenkinsBooth2013]. [^2] The particle mass of this simulation is $m_{\rm{DM}}=5.43\times10^{10}\,\mathrm{M}_{\rm{\odot}}/h$ and the comoving gravitational softening length was set to $40\,\mathrm{kpc}/h$. The simulation was evolved from redshift $z=127$ using a version of the Lagrangian TreePM-SPH code <span style="font-variant:small-caps;">p-gadget3</span> [last described in @Springel2005]. Haloes were identified at $z=0$ using a *Friends-of-Friends* (FoF) algorithm with a standard linking length of $b=0.2$ in units of the mean interparticle separation [@Davis1985].
![Comparison of the *Friends-of-Friends* mass function of the parent simulation against those from @Jenkins2001, @Angulo2012, @Watson2013 and @Heitmann2015 (top) with the residual differences (bottom). We find good agreement with @Heitmann2015, but for values of $\ln(\sigma^{-1})>0.4$ we find a growing discrepancy between the parent simulation and the other simulations. This is likely due to our use of $2^{\mathrm{nd}}$ order Lagrangian perturbation theory when generating the initial conditions of the parent simulation and cosmic variance for the rarest haloes.[]{data-label="fig:massfunc"}](Parent_Mass_Function.png){width="\columnwidth"}
We plot the FoF mass function of the parent simulation at $z=0$ in Fig. \[fig:massfunc\]. We compare it to the published relations of @Jenkins2001, @Angulo2012, @Watson2013 and @Heitmann2015. We plot the scaled differential mass function $$f(\sigma) = \frac{M}{\bar{\rho}}\frac{dn}{d\ln\sigma^{-1}}(M,z)\:,$$ where $M$ is halo mass, $\bar{\rho}$ is the mean density of the Universe at $z=0$, $n$ is the number of haloes per unit volume, and $\sigma^{2}$ is the variance of the linear density field when smoothed with a top-hat filter. We plot the mass function as a function of the variable $\ln(\sigma^{-1})$ as it is insensitive to cosmology [@Jenkins2001]. For $\ln(\sigma^{-1}) < 0.3$ we find that all of the mass functions show reasonable agreement with differences of $\sim5-10\%$ between them, with the small differences likely due to the mass function not being exactly universal [@Tinker2008; @Courtin2011]. However, for larger values the mass functions begin to diverge, as the parent simulation has an excess of massive clusters compared to the other simulations. This is likely due to two effects. First, the MACSIS simulation is the only one to use $\mathrm{2LPT}$ when generating the initial conditions. It has been shown that not using $\mathrm{2LPT}$ results in a significant underestimation of the abundance of the rarest objects [@Crocce2006; @Reed2013]. The second effect is simply statistics: even in a very large volume there are still low numbers of the rarest and most massive clusters, where there is likely to be significant variance between the simulation volumes.
The MACSIS sample
-----------------
To select the MACSIS sample, all haloes with $M_{\rm{FoF}}>10^{15}\,\mathrm{M}_{\odot}$ were grouped in logarithmically spaced bins , with $\Delta\log_{10}\,M_{\rm{FoF}}=0.2$. If a bin contained less than one hundred haloes then all of the objects in that bin were selected. For bins with more than one hundred objects the bin was then further subdivided into bins of 0.02 dex and ten objects from each sub-bin were then selected at random. The subdividing of the bins ensured that our random selection was not biased to low masses by the steep slope of the mass function. This selection procedure results in a sample of $390$ haloes that is mass limited above $10^{15.6}\,\rm{M}_{\odot}$ and randomly sampled below this limit. Table \[tab:smpcomp\] shows the fraction of haloes selected from the parent simulation in each mass bin. We have compared the properties of the selected haloes with those of the underlying population and found the MACSIS sample to be representative. Additionally, in Appendix \[app:seleff\] we demonstrate that selecting by a halo’s FoF mass does not bias our results when binning clusters by their $M_{500}$.
-------------------------------------------- --------- --------- ----------
Mass Bin Fraction
Selected
$15.0 \leq \log_{10}(M_{\rm{FoF}}) < 15.2$ $100~~$ $7084~$ $0.01$
$15.2 \leq \log_{10}(M_{\rm{FoF}}) < 15.4$ $100~~$ $2095~$ $0.05$
$15.4 \leq \log_{10}(M_{\rm{FoF}}) < 15.6$ $100~~$ $485~$ $0.21$
$15.6 \leq \log_{10}(M_{\rm{FoF}}) < 15.8$ $83~~$ $83~$ $1.00$
$15.8 \leq \log_{10}(M_{\rm{FoF}})$ $7~~$ $7~$ $1.00$
-------------------------------------------- --------- --------- ----------
: Table showing the fraction of haloes from the parent simulation that are part of the MACSIS sample for the selection mass bins. The sample is complete above $M_{\rm{FoF}} > 10^{15.6}\,\mathrm{M}_{\odot}$. The parent simulation contains $9754$ haloes with $M_{\rm{FoF}} > 10^{15.0}\,\mathrm{M}_{\odot}$ at $z=0$.
\[tab:smpcomp\]
{width="\textwidth"}
Due to current computational constraints the BAHAMAS simulations are limited to periodic cubes with a side length of $596\,\mathrm{Mpc}$. There are very few clusters with a mass greater than $10^{15}\,\mathrm{M}_{\odot}$ in a volume of this size, and those that are present may be affected by the loss of power from large-scale modes that are absent due to their wavelengths being greater than the box size. The zoom simulations of the MACSIS project provide an extension to the BAHAMAS periodic simulations. They provide the most massive clusters and allow the mass-observable scaling relations to be studied across the complete cluster mass range.
We use the zoomed simulation technique [@KatzWhite1993; @Tormen1997] to re-simulate the chosen sample at increased resolution. We perform both DM only and full gas physics re-simulations. The Lagrangian region for every cluster was selected so that its volume was devoid of lower resolution particles beyond a cluster centric radius of $5r_{200}$.[^3] The resolution of the Lagrangian region was increased such that the particles in the DM only simulations had a mass of $m_{\rm{DM}}=5.2\times10^9\,\mathrm{M}_{\odot}/h$ and in the hydrodynamic re-simulations the dark matter particles had a mass of $m_{\rm{DM}}=4.4\times10^9\,\mathrm{M}_{\odot}/h$ and the gas particles had an initial mass of $m_{\rm{gas}}=8.0\times10^8\,\mathrm{M}_{\odot}/h$. In all simulations the Plummer equivalent gravitational softening length for the high-resolution particles was fixed to $4\mathrm{kpc}/h$ in comoving units for $z>3$ and in physical coordinates thereafter. The smoothed particle hydrodynamics interpolation used $48$ neighbours and the minimum smoothing length was set to one tenth of the gravitational softening. A schematic view of the zoom approach is shown in Fig. \[fig:zoomed\].
The resolution and softening of the zoom re-simulations were deliberately chosen to match the values of the periodic box simulations of the BAHAMAS project [@McCarthy2016], which is a calibrated version of the OWLS code [@Schaye2010], which was also used for cosmo-OWLS [@LeBrun2014]. The subgrid models for feedback from star formation and AGN used in the BAHAMAS simulations was calibrated to obtain a good fit to the observed galaxy stellar mass function and the amplitude of the gas fraction-total mass relation, respectively, at $z=0$. Without any further tuning, the simulations then produce a population of groups and clusters that shows excellent agreement with the observations for a range of galaxy-halo, hot gas-halo and galaxy-hot gas relations.
Baryonic physics
----------------
The BAHAMAS simulations were run with a version of <span style="font-variant:small-caps;">p-gadget3</span> that has been heavily modified to include new subgrid physics as part of the OWLS project [@Schaye2010]. We now briefly describe the subgrid physics, but refer the reader to @Schaye2010, @LeBrun2014 and @McCarthy2016 for greater detail, including the impact of varying the free parameters in the model and the calibration strategy. Radiative cooling is calculated on an element-by-element basis following @WiersmaSchayeSmith2009, interpolating the rates as a function of density, temperature and redshift from pre-computed tables generated with <span style="font-variant:small-caps;">cloudy</span> [@Ferland1998]. It accounts for heating and cooling due to the primary cosmic microwave background and a @HaardtMadau2001 ultra-violet/X-ray background. The background due to reionization is assumed to switch on at $z=9$.
Star formation is modelled stochastically in a way that by construction reproduces the observations, as discussed in @SchayeDallaVecchia2008. Lacking the resolution and physics to correctly model the cold interstellar medium, gas particles with a density $n_{\rm{H}} > 0.1\,\rm{cm}^{-3}$ follow an imposed equation of state with $P\propto\rho^{4/3}$. These gas particles then form stars at a pressure-dependent rate that reproduces the observed Kennicutt-Schmidt law [@Schmidt1959; @Kennicutt1998]. Stellar evolution and the resulting chemical enrichment are implemented using the model of @Wiersma2009, where $11$ chemical elements (H, He, C, N, O, Ne, Mg, Si, S, Ca and Fe) are followed. The mass loss rates are calculated assuming Type Ia and Type II supernovae and winds from massive and asymptotic giant branch stars. Stellar feedback is implemented via the kinetic wind model of @DallaVecchiaSchaye2008. The BAHAMAS simulations used the calibrated mass-loading factor of $\eta_{\rm{w}}=2$ and wind velocity $v_{\rm{w}}=300\,\rm{km/s}$. This corresponds to $20$ percent of available energy from Type II supernovae, assuming a @Chabrier2003 IMF, and yields an excellent fit to the observed galaxy mass function.
The seeding, growth and feedback from supermassive black holes (BH) is implemented using the prescription of @BoothSchaye2009, a modified version of the method developed by @SpringelDiMatteoHernquist2005. A FoF algorithm is run on-the-fly and BH seed particles, with $m_{\mathrm{BH}}=10^{-3}m_{\mathrm{gas}}$, are placed in haloes that contain at least $100$ DM particles, which corresponds to a halo mass of $\sim5\times10^{11}\,\rm{M}_{\odot}$. BHs grow via Eddington-limited accretion of gas at the Bondi-Hoyle-Littleton rate, with a boost factor that is a power-law of the local density for gas above the star formation density threshold. They also grow by direct mergers with other BHs. A fraction, $\epsilon$, of the rest mass energy of the accreted gas is then used to heat $n_{\rm{heat}}$ neighbour particles by increasing their temperature by $\Delta T_{\rm{heat}}$. Changes to these parameters have a significant impact on the hot gas properties of clusters. The calibrated values of these parameters in the BAHAMAS simulations are $n_{\rm{heat}}=20$ and $\Delta T_{\rm{heat}}=10^{7.8}\,\rm{K}$. The feedback efficiency $\epsilon=\epsilon_{\rm{r}}\epsilon_{\rm{f}}$, where $\epsilon_{\rm{r}}=0.1$ is the radiative efficiency and $\epsilon_{\rm{f}}=0.15$ is the fraction of $\epsilon_{\rm{r}}$ that couples to the surrounding gas. The choice of the efficiency, assuming it is non-zero, is generally of little consequence as the feedback establishes a self-regulating scenario, but determines the black hole masses [@BoothSchaye2009].
Calculating observable properties
---------------------------------
Previous studies have shown that there can be significant biases in the observable properties of clusters due to issues such as multi-temperature structures and gas inhomogeneities [e.g. @NagaiVikhlininKravtsov2007b; @Khedekar2013]. Therefore, when investigating cluster properties it is critical that, as far as possible, we make a like-with-like comparison with the observations. Following @LeBrun2014, we do this by producing synthetic observational data for each cluster and analysing it in a manner similar to what is done for real data. Using the particle’s temperature, density and metallicity, where the metallicity is smoothed over a particle’s neighbours, we first compute a rest-frame X-ray spectrum in the $0.05-100.0\,\rm{keV}$ band for all gas particles, using the Astrophysical Plasma Emission Code [<span style="font-variant:small-caps;">apec</span>; @Smith2001] via the <span style="font-variant:small-caps;">pyatomdb</span> module with atomic data from <span style="font-variant:small-caps;">atomdb</span> v3.0.2 [last described in @Foster2012]. A particle’s spectrum is a sum of the individual spectra for each chemical element tracked by the simulations, scaled by the particle’s elemental abundance. We ignore particles with a temperature lower than $10^5\,\rm{K}$ as they make a negligible contribution to the total X-ray emission.
We then estimate the density, temperature and metallicity of the hot gas in $25$ logarithmically spaced radial bins by fitting a single temperature <span style="font-variant:small-caps;">apec</span> model, with a fixed metallicity, to the summed spectra of all particles that fall within that radial bin. We then scale the spectra by the relative abundance of the heavy elements as the fiducial spectra assume solar abundance [@AndersGrevesse1989]. The spectra have an energy resolution of $150\,\rm{eV}$ in the range $0.05-10.0\,\rm{keV}$ and are logarithmically spaced between $10.0-100.0\,\rm{keV}$. To get a closer match to the observations, we multiply the spectra by the effective area of *Chandra*. To derive temperature and density profiles of a cluster, we fit the spectrum in the range $0.5-10.0\,\rm{keV}$ for each radial bin with a single temperature model using a least-squares approach.
The temperature and density profiles derived from the X-ray spectra are then used to perform a hydrostatic mass analysis of the cluster. The profiles are fit with the density and temperature models proposed by @Vikhlinin2006 to produce a hydrostatic mass profile. We then derive various mass and radius estimates, such as $M_{500}$ and $r_{500}$, from the hydrostatic mass profiles. With these estimates we calculate quantities, such as $M_{\rm{gas}}$ or $Y_{\rm{SZ}}$, by summing the properties of the particles that fall within the set. Core-excised quantities are calculated in the radial range $0.15-1.0$ of the aperture. Luminosities are calculated by integrating the spectra of all particles within the aperture in the requisite energy band, for example, bolometric luminosities are calculated in the range $0.05-100.0\,\rm{keV}$. Averaged X-ray temperatures are calculated by fitting a single temperature model to the sum of the spectra of all particles within the aperture. We repeat this analysis for all clusters in the combined sample at all redshifts of interest. All quantities derived in this manner are labelled with the sub-script ‘spec’.
Cluster sample selection
------------------------
{width="\textwidth"}
We select clusters from MACSIS and BAHAMAS to form a ‘combined’ sample with which we can investigate the cluster scaling relations. We perform our analysis at $z=0.0,0.25,0.5,1.0$ and $1.5$. We create this sample at each redshift by selecting all clusters with a mass $M_{500,\rm{spec}}\geq10^{14}\,\rm{M}_{\odot}$. Additionally, we introduce a mass cut at every redshift below which we remove any MACSIS clusters. For example, at $z=0$ ($z=1$) this cut is made at $M_{500,\mathrm{spec}}=10^{14.78}\,\rm{M}_{\odot}$ ($M_{500,\mathrm{spec}}=10^{14.3}\,\rm{M}_{\odot}$). This removes a tail of clusters with low $M_{500,\mathrm{spec}}$, but have high $M_{\mathrm{FoF}}/M_{500,\mathrm{spec}}$ ratios (see Appendix \[app:seleff\]). For the luminosity-temperature relation, we use the temperature-mass relation of the combined sample to convert the mass cut into a temperature cut. At $z=0$ this results in a sample of $1294$ clusters, containing $1098$ clusters from BAHAMAS and $196$ MACSIS clusters, and at $z=1$ a sample of $225$ clusters, $99$ from BAHAMAS and $126$ from MACSIS.
The MACSIS clusters enable the investigation of the behaviour of the most massive clusters at low redshift. These clusters are commonly selected in cosmological analyses because their deep potentials are expected to reduce the impact of non-gravitational processes and as the brightest clusters they require shorter exposures. We select a hot, and therefore massive, cluster sample by selecting all clusters in the combined sample with a core-excised X-ray temperature greater than $5\,\rm{keV}$. At $z=0$ ($z=0.5$) this yields a sample of $244$ ($186$) clusters, with $190$ ($173$) coming from the MACSIS sample. Finally, we examine the impact of selecting a relaxed subset of the hot cluster sample. Theoretically, there are many ways to define a relaxed halo [see @Neto2007; @Duffy2008; @Klypin2011; @DuttonMaccio2014; @Klypin2016]. For this study we use the following criteria $$X_{\rm{off}} < 0.07\,;~f_{\rm{sub}} < 0.1~\rm{and}~\lambda < 0.07\,, \nonumber$$ where $X_{\rm{off}}$ is distance between the cluster’s minimum gravitational potential and centre of mass, divided by its virial radius; $f_{\rm{sub}}$ is the mass fraction within the virial radius that is bound to substructures; and $\lambda$ is the spin parameter for all particles inside $r_{200}$. These criteria are not designed to select a small subset that comprises the most relaxed objects, but to simply remove those clusters that are significantly disturbed. This results in a subsample at $z=0$ ($z=0.5$) that contains $213$ ($117$) clusters, with $177$ ($111$) coming from the MACSIS sample.
The scaling relations of massive clusters {#sec:screlations}
=========================================
In this section we present our main results, measuring the scaling relations of our cluster samples across a range of redshifts.
Comparison to observational data
--------------------------------
Fig. \[fig:observations\] shows the gas mass, $M_{\rm{gas},500,\rm{spec}}$, the integrated Sunyaev-Zel’dovich (SZ) signal, $Y_{\rm{SZ}}$, measured in a $5r_{500,spec}$ aperture as a function of estimated total mass, $M_{500,\rm{spec}}$, (at $z=0$ and $z=1$) and the core-excised bolometric X-ray luminosity, $L^{\rm{X,ce}}_{500,\rm{spec}}$, as a function of core-excised X-ray temperature, $T^{\rm{X,ce}}_{500,\rm{spec}}$, for the combined sample. We compare the sample to the relevant observational data. At all redshifts the MACSIS sample provides a consistent extension to the BAHAMAS clusters with similar scatter. At low redshift, @McCarthy2016 have shown that the BAHAMAS sample shows good agreement with the observed median relations and shows similar intrinsic scatter. The MACSIS sample continues this agreement to observed high-mass clusters, though there are significantly fewer clusters to compare against. In detail, it appears that the $M_{500,\mathrm{gas,spec}}$-$M_{500,\mathrm{spec}}$ and $L^{\mathrm{X,ce}}_{500,\mathrm{spec}}$-$T^{\mathrm{X,ce}}_{500,\mathrm{spec}}$ relations are slightly steeper than observed. However, we would exercise caution as we have not applied the same selection criteria as was used for the observational X-ray analyses.
At high redshift observational data becomes sparse and currently only SZ surveys have detected a reasonable number of clusters. At $z=1$ these clusters are all significantly more massive than any cluster in the BAHAMAS volume. However, the progenitors of the very massive MACSIS clusters provide a sample that can be compared with these observations. We find that the median relation shows good agreement with the observations and the intrinsic scatter of the clusters about the median relation is consistent with the scatter in the observations. Overall, we find that all quantities computed in a like-with-like manner show good agreement with the observations.
[l r C C r C C r C C]{} Scaling relation & & &\
& & & & & & & & &\
$L^{\rm{X,ce}}_{500}-M_{500}$ & $44.50^{+0.01}_{-0.01}$ & $1.88^{+0.03}_{-0.05}$ & $0.15^{+0.01}_{-0.02}$ & $44.71^{+0.02}_{-0.02}$ & $1.36^{+0.08}_{-0.07}$ & $0.12^{+0.01}_{-0.02}$ & $44.69^{+0.03}_{-0.03}$ & $1.43^{+0.13}_{-0.09}$ & $0.11^{+0.01}_{-0.01}$\
$k_{\rm{B}}T^{\rm{X,ce}}_{500}-M_{500}$ & $0.68^{+0.01}_{-0.01}$ & $0.58^{+0.01}_{-0.01}$ & $0.048^{+0.003}_{-0.003}$ & $0.71^{+0.01}_{-0.01}$ & $0.51^{+0.04}_{-0.04}$ & $0.05^{+0.01}_{-0.01}$ & $0.70^{+0.01}_{-0.01}$ & $0.55^{+0.06}_{-0.03}$ & $0.04^{+0.01}_{-0.01}$\
$M_{\rm{gas},500}-M_{500}$ & $13.67^{+0.01}_{-0.01}$ & $1.25^{+0.01}_{-0.03}$ & $0.07^{+0.01}_{-0.01}$ & $13.77^{+0.01}_{-0.01}$ & $1.02^{+0.03}_{-0.03}$ & $0.06^{+0.01}_{-0.01}$ & $13.75^{+0.01}_{-0.01}$ & $1.05^{+0.04}_{-0.04}$ & $0.05^{+0.01}_{-0.01}$\
$Y_{\rm{X},500}-M_{500}$ & $14.33^{+0.01}_{-0.01}$ & $1.84^{+0.02}_{-0.05}$ & $0.12^{+0.01}_{-0.01}$ & $14.47^{+0.02}_{-0.02}$ & $1.51^{+0.07}_{-0.08}$ & $0.11^{+0.01}_{-0.01}$ & $14.45^{+0.02}_{-0.02}$ & $1.59^{+0.12}_{-0.06}$ & $0.08^{+0.01}_{-0.01}$\
$Y_{\rm{SZ},500}-M_{500}$ & $-4.51^{+0.01}_{-0.01}$ & $1.88^{+0.02}_{-0.03}$ & $0.10^{+0.01}_{-0.01}$ & $-4.39^{+0.02}_{-0.02}$ & $1.60^{+0.07}_{-0.05}$ & $0.10^{+0.01}_{-0.02}$ & $-4.42^{+0.02}_{-0.02}$ & $1.69^{+0.07}_{-0.07}$ & $0.09^{+0.01}_{-0.01}$\
$L^{\rm{X,ce}}_{500}-T^{\rm{X,ce}}_{500}$ & $44.80^{+0.02}_{-0.01}$ & $3.01^{+0.04}_{-0.04}$ & $0.14^{+0.01}_{-0.01}$ & $44.93^{+0.01}_{-0.01}$ & $2.41^{+0.12}_{-0.12}$ & $0.11^{+0.01}_{-0.01}$ & $44.89^{+0.02}_{-0.02}$ & $2.53^{+0.12}_{-0.13}$ & $0.10^{+0.01}_{-0.01}$\
\[tab:z0bestfit\]
Modelling cluster scaling relations
-----------------------------------
As a baseline for understanding how the scaling relations evolve as a function of mass and redshift, we adopt the following self-similar scalings $$M_{\rm{gas},\Delta}\propto M_{\Delta}\,,$$ $$\label{eq:T-M}
T_{\Delta}\propto M^{2/3}_{\Delta}E^{2/3}(z)\,,$$ $$Y_{\rm{X},\Delta}\propto M_{\Delta}^{5/3}E^{2/3}(z)\,,$$ $$Y_{\rm{SZ},\Delta}\propto M_{\Delta}^{5/3}E^{2/3}(z)\,,$$ $$\label{eq:L-M}
L_{\Delta}^{\rm{X,bol}}\propto M^{4/3}_{\Delta}E^{7/3}(z)\,,$$ $$L_{\Delta}^{\rm{X,bol}}\propto T^{2}E(z)\,,$$ where $E(z)\equiv H(z)/H_0=\sqrt{\Omega_{m}(1+z)^3+\Omega_{\Lambda}}$, $\Delta$ is the chosen overdensity relative to the critical density and $Y_{\rm{X}}$ is the X-ray analogue of the integrated SZ effect. These are derived in Appendix \[app:ssr\]. Although shown to be too simplistic by the first X-ray studies of clusters [@Mushotzky1984; @EdgeStewart1991; @David1993], the self-similar relations allow us to investigate if astrophysical processes are less significant in more massive clusters or at higher redshift. To enable a comparison with the self-similar predictions, and previous work, we fit the scaling relations of our samples at each redshift. We derive a median relation by first binning the clusters into bins of log mass (width $0.1$ dex) or log temperature (width $0.07$ dex) and then computing the median in each bin with more than ten clusters. We also remove the evolution in normalization predicted by self-similar relations. The medians of the bins are then fit with a power-law of the form $$E^{\beta}(z)Y=10^A\left(\frac{X}{X_0}\right)^{\alpha}\,,
\label{eq:plfit}$$ where $A$ and $\alpha$ describe the normalization and slope of the best fit respectively, $\beta$ removes the expected self-similar evolution with redshift, $X$ is either the total mass or temperature and $Y$ is the observable quantity ($M_{\rm{gas}}$, $L^{\rm{X,bol}}$, etc.). $X_0$ is the pivot point, which we set to $4\times10^{14}\,\rm{M}_{\odot}$ for observable-mass relations and to $6\,\rm{keV}$ for observable-temperature relations. We note that we fix the pivot for all samples and all redshifts. Fitting to the medians of bins, rather than individual clusters, prevents the fit from being dominated by low-mass objects, which are significantly more abundant due to the shape of the mass function. For the hot sample and its relaxed subset there are too few bins with ten or more clusters to reliably derive a best-fit relation at $z\geq1$. By limiting our sample to systems with $M_{500}\geq10^{14}\,\rm{M}_{\odot}$ we avoid any breaks in the powerlaw relations that have been seen both observationally and in previous simulation work [@LeBrun2016].
We compute the scatter about the best-fit relation at each redshift by calculating the root mean squared (*rms*) dispersion in each bin according to $$\sigma_{\log_{10}Y}=\sqrt{\frac{1}{N}\sum_{i=1}^N\left[\log_{10}(Y_i)-\log_{10}(Y_{\rm{BF}})\right]^2}\,,$$ where $i$ runs over all clusters in the bin, $Y_{\rm{BF}}$ is the best fit relation for a cluster with a value $X_i$ and we note that $\sigma_{\ln Y}=\ln(10)\sigma_{\log_{10}Y}$. We obtain the uncertainties for our fit parameters by bootstrap re-sampling the clusters $10,000$ times. The best-fit values of all the scaling relations considered for the three samples (combined, hot and relaxed) at $z=0$ are summarized in Table \[tab:z0bestfit\] and other redshifts are listed in Appendix \[app:fitpar\]. We now discuss each relation in turn.
Gas Mass-Total Mass {#ssec:Mg-M}
-------------------
{width="\textwidth"}
We plot the hot gas mass-total mass scaling relation for the three samples in Fig. \[fig:MgMsr\]. The best-fit normalization for the combined sample shows significant evolution with redshift, with clusters of a fixed mass containing $25\%$ more hot gas at $z=1$ than at $z=0$. With the inclusion of star formation, radiative cooling and feedback from supernovae and AGN, the departure from self-similarity is not unexpected. The increasing normalization with redshift is due to either the impact of AGN feedback or the conversion of gas to stars. As the normalization of the baryonic mass exhibits a similar trend, this evolution is being driven by AGN feedback. A plausible explanation is as follows. The mean density of the Universe increases with redshift and cluster potentials at a fixed mass get deeper with increasing redshift. This reduces the efficiency with which AGN expel gas from the cluster with increasing redshift, leading to a higher gas mass at higher redshift for clusters at a fixed mass. In addition, AGN have less time to act on and expel gas from clusters that form at higher redshifts. The AGN breaks the self-similar assumption of a constant gas fraction, resulting in the normalization of the gas mass-total mass relation increasing with increasing redshift. However, we note that this behaviour appears to be dependent on the implementation of the subgrid physics. @LeBrun2016 use the same subgrid implementation, but with different parameters, and obtain similar behaviour. However, @Planelles2013 see a constant baryon fraction with redshift suggesting that feedback is not expelling gas beyond $r_{500}$.
The bottom left panel of Fig. \[fig:MgMsr\] shows that the normalizations of the best-fit relations for the hot sample of clusters and for the relaxed subset of hot clusters are higher at $z=0$ than the normalization of the combined sample and evolve less with redshift. This is because hotter clusters are generally more massive and have deeper potential wells, reducing the amount of gas the AGN can permanently expel from the cluster during its formation. This flattens the slope of the relation leading to a higher normalization at the pivot.
The bottom right panel of Fig. \[fig:MgMsr\] shows that the slope of the best-fit relation of the combined sample is significantly steeper than the self-similar prediction of unity. At a given redshift AGN feedback has expelled more gas from lower mass clusters, due to their shallower potentials, leading to a tilt in the relation. We find a slope of $\alpha=1.25^{+0.01}_{-0.03}$. Our slope is mildly shallower than found in previous a simulation work, where @LeBrun2016 find a slope of $1.32$ for their AGN8.0 simulation, but consistent with observations, where @Arnaud2007 found a slope of $1.25\pm0.06$ for a sample of clusters observed with *XMM*. We find negligible evolution in the slope of the relation for the combined sample.
The hot cluster sample and the relaxed subset have best-fit slopes that are consistent with the self-similar prediction. The increased depth of the potential well in massive clusters means that their gas mass is approximately a constant fraction of their total mass. Specifically, we find that most massive clusters have a median gas fraction $f_{\mathrm{gas}}=0.89\pm0.09$ of the universal baryon fraction at $z=0$. This results in slopes of $\alpha=1.02\pm0.03$ and $1.05\pm0.04$ for the hot cluster sample and the relaxed subset respectively. We find good agreement with the slope of $1.05\pm0.05$ found by @Mantz2016 and the self-similar slope found by @Vikhlinin2009 for relaxed cluster samples. The slope of the best-fit relation for both samples shows no significant evolution with redshift.
The top right panel of Fig. \[fig:MgMsr\] shows that the scatter about the best-fit relation is independent of both mass and redshift. Averaged over all mass bins it has a value of $\sigma_{\log_{10}Y}=0.07$ at $z=0$. The scatter reduces slightly for the hot cluster sample, with a value of $0.06$, and further still for the relaxed subset, with a value of $0.05$. The scatter is in reasonable agreement with the scatter of $0.04$ found by @Arnaud2007 for a sample of clusters observed with *XMM*.
{width="\textwidth"}
X-ray Temperature-Mass {#ssec:Tx-M}
----------------------
The evolution of the core-excised spectroscopic temperature-total mass scaling relations, and their scatter, for the three samples are shown in Fig. \[fig:TxMsr\]. The normalization of the best-fit relation of the combined sample shows a minor evolution with redshift, being $15\%$ lower at $z=1$ compared to $z=0$ (bottom left panel). In the self-similar model the temperature of the ICM is related to the depth of the gravitational potential of the cluster, under the assumption of hydrostatic equilibrium. Previous simulation work has shown that the non-thermal pressure in mass-limited samples grows with redshift due to the increasing importance of mergers and resulting incomplete thermalisation [@Stanek2010; @LeBrun2016]. Therefore, clusters increasingly violate the assumption of hydrostatic equilibrium with redshift and require a lower temperature at a fixed mass to balance gravitational collapse, which leads to a normlization that decreases with redshift compared to self-similar. The effective temperature of the non-thermal pressure can be estimated via $$T_{\rm{kin}}=\left(\frac{\mu m_{\rm{p}}}{k_{\rm{B}}}\right)\sigma_{\rm{gas}}^2\,$$ where $\sigma_{\rm{gas}}$ is the 1D velocity dispersion of the gas particles, $\mu=0.59$ is the mean molecular weight, $m_{\rm{p}}$ is the mass of the proton and $k_{\rm{B}}$ is the Boltzmann constant. Fig. \[fig:TxTkinMsr\] shows the evolution of the temperature-mass normalization once this effective kinetic temperature has been added to the spectral temperature. For all three samples the addition of the kinetic temperature results in a normalization that shows significantly reduced evolution with respect to self-similar.
![Evolution of the normalization of the spectroscopic temperature-total mass relation when the effective non-thermal support temperature is included. All three samples show negligible evolution with redshift relative to self-similar once non-thermal pressure support is included.[]{data-label="fig:TxTkinMsr"}](Tx+Tkin-M_Norm_spec.png){width="\columnwidth"}
The normalizations of the best-fit relations for the hot cluster and the relaxed hot samples are slightly higher than for the combined sample, but they show a similar trend with redshift that is removed when the kinetic temperature is included. The higher normalization occurs because, again, the hot sample has a flatter slope with mass. This flatter slope is driven by two processes. First, non-thermal pressure support becomes more important in higher mass clusters at a fixed redshift, as they have had less time to thermalise, and this lowers their temperatures. Second, we find that the bias between the spectroscopic and mass-weighted temperatures increases mildly with mass. This does not appear to be caused by cold clumps due to the SPH method, but is due to the presence of cooler gas in the outskirts of massive clusters that is hotter than the $0.5\,\rm{keV}$ lower limit, contributing to the X-ray spectrum, and biasing the measured temperature low for the most massive clusters. Fig. \[fig:Tx\_bias\] shows the fractional difference between the spectroscopic and mass-weighted core-excised temperatures as a function of mass. Similar to @Biffi2014, we find that for low-mass clusters the spectroscopic temperature estimate agrees well with the mass-weighted estimate at $z=0$. However, as cluster mass increases we find that the spectroscopic estimate is increasingly biased low compared to the mass-weighted estimate. This will also impact the hydrostatic mass estimate of the cluster and we refer the reader to @Henson2016 for a more in-depth study. Both of these effects lead to a flattening of the slope with mass and a higher normalization for the hot samples. We note that removing the most disturbed clusters produces a marginal decrease in the normalization of the relation, which is due to the steeper slope yielding a lower normalization at the pivot point.
![Plot of fractional difference between the spectroscopic and mass-weighted temperature estimates as a function of $M_{500}$ for the combined sample at $z=0$ (blue squares) and $z=1$ (red triangles). Error bars show $68\%$ of the population.[]{data-label="fig:Tx_bias"}](Tx_bias.png){width="\columnwidth"}
We find the slope of the best-fit relation for the combined sample to be $\alpha=0.58\pm0.01$ at $z=0$. This is in good agreement with the slope found by previous simulation work, where values of $0.55\pm0.01$ [@Short2010], $0.576\pm0.002$ [@Stanek2010], $0.54\pm0.01$ [@Planelles2014], $0.56\pm0.03$ [@Biffi2014], $0.60\pm0.01$ [@Pike2014] and $0.58$ [@LeBrun2016] were found. All of these are in agreement with the observed temperature-total mass relation found for volume-limited samples, with values of $0.58\pm0.03$ for a sample of clusters observed with *XMM* [@Arnaud2007] and $0.56\pm0.07$ for a sample of low-redshift clusters [@Giles2015]. We note that a caveat to these comparisons is the differing mass ranges will alter the slope as the relation is not a perfect power law. All of these relations are slightly flatter than the predicted self-similar slope of $2/3$ due to non-thermal pressure support and temperature bias.
{width="\textwidth"}
Selecting only hot clusters produces a best-fit relation with a slope of $0.51\pm0.04$, flatter than the combined relation. The best-fit slope of $0.55^{+0.06}_{-0.03}$ for the relaxed subset, is compatible with the combined sample. The slope of the relaxed subset is compatible with the slope found by @Mantz2016 of $0.66\pm0.05$ and the slope of $0.65\pm0.04$ found by @Vikhlinin2009 for relaxed clusters. However, we note that our relaxation criteria only remove the most disturbed objects, as opposed to the criteria of @Mantz2015 which select the most relaxed objects. Therefore, we would likely recover a steeper slope with stricter relaxation criteria. Both samples are equally affected by the spectroscopic temperature being biased low. The slopes of the hot sample and the relaxed subset show no clear trend with redshift.
The temperature-mass scaling relations shows very low scatter, which is independent of both mass and redshift. The average scatter across all mass bins is $\sigma_{\log_{10}Y}=0.046$, $0.045$ and $0.039$ for the combined sample, hot sample and relaxed subset, respectively, at $z=0$. These values are consistent with the values found by both observations and previous simulations [@Arnaud2007; @Giles2015; @Stanek2010; @Short2010].
$Y_{\rm{X}}$- Mass
------------------
The power-law fits to the X-ray analogue of the integrated SZ effect-total mass relations for the three samples, and their scatter, are shown in Fig. \[fig:YxMsr\]. The X-ray analogue signal, $Y_{\rm{X}}$, is the product of the core excised spectral temperature and the gas mass, and the relation should reflect the combination of the two previously presented relations. We indeed find this to be the case. For the combined sample, the decreasing temperature-total mass normalization with increasing redshift offsets the increasing gas mass-total mass normalization, producing almost no evolution of the normalization for the $Y_{\rm{X}}$-total mass relation. The same trend was found by @LeBrun2016. Therefore, the normalization evolves in a close to self-similar manner.
Selecting a sample of hot clusters or a relaxed subset of them leads to higher overall normalization of the best-fit relation. This is mainly due to the reduced impact of AGN feedback on the gas mass-total mass relation, which flattens the relation and leads to a higher normalization at the pivot. Both samples agree very well with the predicted self-similar evolution of the normalization of the relation, with the normalization of the relaxed subset changing by less than one percent between $z=0$ and $z=0.5$.
The slope of the $Y_{\rm{X}}$-total mass relation is simply the sum of the slopes of the temperature-mass and gas mass-total mass relations and for the combined sample the slope is significantly steeper than the $5/3$ value predicted by self-similar theory. We find a value of $\alpha=1.84^{+0.02}_{-0.05}$ at $z=0$. The slope of our best-fit relation is consistent with those of previous simulations, who found values of $1.78\pm0.01$ [@Short2010], $1.73\pm0.01$ [@Planelles2014] and $1.89$ [@LeBrun2016]. Our result is also in agreement with the observational value found by @Arnaud2007 of $1.82\pm0.1$ using the REXCESS cluster sample. The physical reason for the steeper slope is that gas is preferentially removed from lower mass clusters by feedback. In response to gas expulsion the remaining gas increases in temperature, offsetting some of the losses, but the loss of gas dominates and steepens the relation. The value of the slope for the best-fit relation is approximately constant with redshift, within the uncertainty of the fits.
{width="\textwidth"}
Selecting a sample of hot clusters leads to a significant flattening of the slope of the relation, slightly flatter than the self-similar prediction of $5/3$. With the gas mass-total mass relations of the hot sample and relaxed subset being very close to self-similar, the shallower than self-similar slope is due to the temperature-mass relation. The best-fit slope of both samples shows no significant trend with redshift.
The scatter about the best-fit relation is independent of both mass and redshift for all three samples, but it is noisy. We find an average value of $0.12$ at $z=0$ for the scatter for the combined sample, $0.11$ for the hot cluster sample and $0.08$ for the relaxed subset. These values are larger than those found previously for both simulations, where values of $0.04$ [@Short2010], $0.08$ [@Planelles2014] and $0.04$ [@LeBrun2016] were found, and observations, where a value of $0.04$ was found for a sample of clusters observed with *XMM* [@Arnaud2007].
$Y_{\rm{SZ}}$-Total Mass
------------------------
The integrated SZ effect-total mass relations for the three samples are shown in Fig. \[fig:YszMsr\]. Both the integrated SZ signal and its X-ray analogue measure the total energy of the hot gas in the ICM, however the SZ signal depends on the mass-weighted temperature rather than the X-ray spectral temperature. Our best-fit relation for the combined sample shows a mild evolution with redshift, with clusters at $z=1$ yielding an integrated signal that is $27\%$ higher than clusters at $z=0$ for a fixed mass. The evolution reflects the evolution in the gas mass-total mass relation. The increased evolution of its normalization compared to its X-ray analogue suggests that the normalization of the mass-weighted temperature evolves more self-similarly then the spectroscopic X-ray temperature and is indeed confirmed by the study of the mass-weighted temperature-total mass relation.
Selecting a sample of hot clusters or a relaxed subset of them significantly reduces the evolution in the normalization. The normalization of both samples, within the uncertainty of the fits, evolves in agreement with the self-similar prediction. Selecting a hot sample leads to a $25\%$ higher normalization than the combined sample at $z=0$, due to the flatter slope of the gas mass-total mass relation yielding a flatter $Y_{\rm{SZ}}$ slope and a higher normalization at the pivot point.
The best-fit relation for the combined sample produces a slope of $\alpha=1.88^{+0.02}_{-0.04}$ at $z=0$, which is significantly steeper than the $5/3$ value predicted by the self-similar model. The value for the slope of the relation is consistent with previous values from both simulations, where values of $1.825\pm0.003$ [@Stanek2010], $1.71\pm0.03$ [@Battaglia2012], $1.74\pm0.01$ [@Planelles2014], $1.70\pm0.02$ [@Pike2014], $1.68\pm=0.05$ [@Yu2015] and $1.94$ [@LeBrun2016] have been found, and observations, where $1.79\pm0.08$ was found for the Planck clusters [@Planck2014XX] and $\alpha=1.77\pm0.35$ was found for the clusters in the $2500\,\rm{deg}^2$ SPT survey. The steeper than self-similar slope is the result of the gas mass-total mass relation having a steeper slope. We find that the slope of the relation is independent of redshift.
{width="\textwidth"}
The best-fit slopes of the hot cluster sample and the relaxed subset are consistent with the slope predicted by self-similar theory. The slopes of both samples are consistent with no evolution.
The scatter of the clusters about the best-fit relation shows no trend with either mass or redshift for all three samples. We find an average scatter of $\sigma_{\log_{10}Y}=0.10$, $0.10$ and $0.09$ for the combined, hot and relaxed samples, respectively, at $z=0$. This is larger than the scatter reported by previous simulations, where @Battaglia2012, @Pike2014, @Planelles2014 and @LeBrun2016 found values of $0.06$, $0.03$, $0.07$ and $0.04$ respectively, but in reasonable agreement with the values of $0.12\pm0.03$ and $0.08$ observed by @Yu2015 and @Planck2014XX respectively.
Bolometric X-ray Luminosity-Total Mass
--------------------------------------
Fig. \[fig:LxMsr\] shows the core-excised bolometric X-ray luminosity-total mass scaling relations for the three samples and their evolution with redshift. The normalization of the best-fit relation for the combined sample shows significant evolution with redshift, being $80\%$ higher at $z=1$ compared to $z=0$. The same physics driving the gas mass-total mass relation, increased binding energy, is driving the departure from self-similar. The X-ray emission of a cluster is particularly sensitive to the thermal structure of the ICM, which depends on processes such as radiative cooling and feedback. Therefore, it is not surprising that the luminosity-mass relation shows significantly more evolution than other observable-mass relations.
Selecting a sample of hot clusters significantly reduces the evolution in the normalization. Both the hot sample and the relaxed subset have a normalization that is $\approx60\%$ larger at $z=0$ compared to the combined sample. The deeper potentials of more massive clusters reduces the impact of the AGN feedback and flattens the relation. This flattening leads to a higher luminosity at the pivot point. The normalizations of the best-fit relations for both the hot sample and its relaxed subset show very minor evolution, which is consistent with the self-similar prediction.
{width="\textwidth"}
The slope of the best-fit relation for the combined sample is significantly steeper than the $4/3$ slope predicted by self-similar theory. At $z=0$ we find a slope of $\alpha=1.88^{+0.03}_{-0.05}$ for the combined sample. This steepening is driven by AGN feedback being more effective in lower mass clusters. The slope at $z=0$ is in reasonable agreement with the slopes found in volume-limited observational samples, such as @Pratt2009 who found a slope of $1.80\pm0.05$ for the REXCESS sample and @Giles2015 who found a slope of $2.14\pm0.21$ for a sample of $34$ low-redshift clusters. Previous simulation work by @Short2010, using the semi-analytic feedback model of the Millennium Gas project, found a bolometric luminosity-total mass slope of $1.77\pm0.03$ and @Stanek2010, using the preheating model of the Millennium Gas project, found a slope of $1.87\pm0.01$. @Biffi2014 found a slope of $1.45\pm0.05$ for the MUSIC simulations. The slope of the best-fit relation for the combined sample is approximately independent of redshift, with a very mild steepening of the slope with redshift occurring due to the reduction in fitting range with increasing redshift.
The slopes of the best-fit relation follow the same trend as the gas mass-total mass relation, with the hot sample and its relaxed subset producing shallower slopes that are in much better agreement with self-similar theory. Our best-fit slope is consistent with the observational result of @Mantz2016, who found a self-similar slope for the core-excised luminosity-total mass relation for a sample of $40$ relaxed clusters with $k_{\rm{B}}T\geq\,5\,\rm{keV}$.
The scatter about the best fit relation is approximately independent of both mass and redshift for all three samples, although it is relatively noisy. Averaging the scatter for the combined sample across all mass bins produces a value of $\sigma_{\log_{10}Y}=0.15$. This is in reasonable agreement with the scatter found in low-redshift observational samples, where @Pratt2009 find a value of $0.17\pm0.03$ and @Giles2015 find a value of $0.22\pm0.03$. Selecting hot clusters and a relaxed subset produces a small reduction in scatter about the best-fit relation with values of $0.12$ and $0.11$ respectively.
X-ray Luminosity-Temperature
----------------------------
Finally, we study the redshift evolution of the X-ray luminosity-spectroscopic temperature relation. Both quantities of the luminosity-temperature scaling relation are observable, with the temperature tracing the depth of the potential of the cluster. This makes it a useful relation to study the impact of non-gravitational physics. In Fig. \[fig:LxTxsr\] we plot the bolometric X-ray luminosity-spectroscopic temperature scaling relation for the three samples of clusters. The normalization of the best-fit relation for the combined sample shows significant evolution with redshift relative to self-similar. Clusters with a temperature of $6\,\rm{keV}$ at $z=1$ have a luminosity $94\%$ greater than clusters with the same temperature at $z=0$. This evolution can be thought of as being due to a combination of the evolution of the temperature-mass and luminosity-mass relations. The decreasing temperature-mass normalization and increasing luminosity-mass normalization with redshift combine to yield a significant evolution of the luminosity-temperature normalization relative to self-similar.
Selecting a sample of hot clusters, or a relaxed subset of them, reduces the evolution, but there is still a mild evolution in the normalization. Hot clusters at a fixed temperature at $z=0.5$ are $\approx15\%$ more luminous than those at $z=0$. Combining equations (\[eq:T-M\]) and (\[eq:L-M\]), but allowing the slope of the relations to vary from their self-similar values yields $$\label{eq:Lx-Txevo}
L_{\rm{X},\Delta}^{\rm{bol}}\propto T^{\alpha_{\rm{LM}}/\alpha_{\rm{TM}}}E^{7/3-2\alpha_{\rm{LM}}/3\alpha_{\rm{TM}}}(z)\,,$$ where $\alpha_{\rm{LM}}$ and $\alpha_{\rm{TM}}$ are the slopes of the luminosity-mass and temperature-mass relations respectively. Hence, deviations of their slopes from self-similar leads to evolution of the normalization of the luminosity-temperature relation that is not self-similar. With the luminosity-mass relation being self-similar for the hot cluster sample and its relaxed subset, the evolution of the normalization is being driven by the flatter than self-similar slope of the temperature-mass relation, which is due to the increased importance of non-thermal pressure support and the increasingly biased spectroscopic temperatures of more massive clusters.
We find a slope of $\alpha=3.01\pm0.04$ for the best-fit relation at $z=0$. This is significantly steeper than the slope of $2$ predicted by self-similar theory. However, this value is reasonable agreement with previous simulation work, $3.30\pm0.07$ [@Short2010], and those found by observations, $2.95\pm0.15$ for the REXCESS sample [@Pratt2009] and $\alpha=3.63\pm0.27$ for a sample of $114$ clusters observed with *Chandra* [@Maughan2012]. It is clear from equation (\[eq:Lx-Txevo\]) that the slope of the relation depends on the slopes of the luminosity-mass and temperature-mass relations. The steeper than expected slope for the combined sample is due to the combined effects of AGN feedback on the luminosity slope and non-thermal pressure support and temperature bias on the temperature slope, both of which lead to a steepening of the relation compared to the self-similar prediction. We find that the best-fit relation steepens slightly with redshift, increasing to $3.35\pm0.07$ at $z=1$. This evolution is due to the removal of high-mass objects with redshift.
The best-fit slope of the hot cluster sample and the relaxed subset are flatter than the combined relation with slopes of $2.41\pm0.12$ and $2.53\pm0.13$. This is still significantly steeper than the slope predicted by self-similar theory, but in good agreement with the slope of $2.44\pm0.43$ observed by @Maughan2012 for their relaxed cool core cluster sample. With both samples exhibiting self-similar slopes for the luminosity-mass relations, the deviation from self-similarity is being driven by their temperature-mass relations.
The scatter about the best-fit relation demonstrates a trend with both temperature and redshift. Although somewhat noisy, the scatter appears to increase with decreasing temperature. The average scatter at $z=0$ for the combined sample is $\sigma_{\log_{10}Y}=0.14$ . This scatter is consistent with the simulations of @Short2010, who found a scatter $0.10$, and the intrinsic observational scatter of $0.12$ found by @Pratt2009. However, it is significantly lower than the scatter of $0.29$ found by @Maughan2012. The scatter reduces for the hot cluster sample and the relaxed subset to $0.11$ and $0.10$ respectively.
Summary
-------
Overall, the scaling relations of the combined sample show good agreement with previous work, both simulations and observations. Departures from self-similarity are driven by the increased efficiency of gas expulsion by AGN feedback in clusters with shallower potentials, due to being less massive or forming at a lower redshift; the increased contribution of non-thermal pressure that supports the ICM against gravity in more massive clusters or those at higher redshifts; and the increase in the spectroscopic temperature bias for the most massive clusters. The MACSIS sample enabled the scaling relations to be studied to higher redshifts, as their progenitors are still clusters at high redshift, and the examination of the impact of selecting a sample of hot clusters on the evolution of the scaling relations. This demonstrated that massive clusters are more self-similar and evolve more self-similarly with redshift compared to the overall cluster population, as the efficiency of gas expulsion by AGN feedback is reduced due to their deeper potentials. However, it also highlighted that non-thermal pressure support becomes more important in these clusters and that their spectroscopic temperatures are biased low.
Evolution of gas profiles {#sec:gasprofs}
=========================
Most of the scaling relations of hot, and therefore massive, clusters evolve in a way that is consistent with the predictions of the self-similar model. However, the combined sample showed significant deviations from the self-similar model due to the impact of non-gravitational processes. To further understand the differences between the samples in the evolution of their scaling relations, we now examine the gas profiles of the different cluster samples. To enable a quantitative comparison with the observational data requires us to compare like-with-like. Therefore, we restrict the mass range of the combined sample to $2.0\times10^{14}\,\mathrm{M}_{\odot}\leq M_{500,\mathrm{spec}}\leq1.0\times10^{15}\,\mathrm{M}_{\odot}$, yielding a sample with a median mass of $2.44\times10^{14}\,\mathrm{M}_{\odot}$. We compare this to the REXCESS cluster sample which has a median mass of $2.68\times10^{14}\,\mathrm{M}_{\odot}$ and a sample of clusters from @Giles2015 with a median mass of $5.43\times10^{14}\,\mathrm{M}_{\odot}$. Although this mass matching does not account for selection effects, it should allow for a quantitative comparison. We do not alter the hot sample or the relaxed subset. We factor out the expected self-similar evolution in the profiles by dividing by the appropriate quantity, e.g. $\rho_{\rm{crit}}$, $k_{\rm{B}}T_{500}$, $P_{500}$ or $K_{500}$. We define these quantities as $$\rho_{\rm{crit}}(z)\equiv E^2(z)\frac{3H^2_0}{8\pi G}\,,$$ $$k_{\rm{B}}T_{500}=\frac{GM_{500}\mu m_{\rm{p}}}{2r_{500}}\,,$$ $$P_{500}=500f_{\rm{b}}k_{\rm{B}}T_{500}\frac{\rho_{\rm{crit}}}{\mu m_{\rm{p}}}\,,$$ $$K_{500}=\frac{k_{\rm{B}}T_{500}}{\left(500f_{\rm{b}}(\rho_{\rm{crit}}/\mu_{\rm{e}}m_{\rm{p}})\right)^{2/3}}\,,$$ where $H_0$ is the Hubble constant, $G$ is the gravitational constant, $\mu_{\rm{e}}$ is the mean atomic weight per free electron and $f_{\rm{b}}=\Omega_{\rm{b}}/\Omega_{\rm{m}}$ is the universal baryon fraction. Therefore, any changes in the profiles are due to non-gravitational physics, such as AGN feedback or non-thermal pressure support.
Density profiles
----------------
![Median gas density profiles for the combined (grey dash-dot), hot (dark red dashed) and relaxed hot (red solid) samples at $z=0$ (top panel) and $z=1$ (middle panel), scaled by $(r/r_{500,\rm{spec}})^2$ to reduce dynamic range. The grey hatched region shows the $16^{\rm{th}}$ to $84^{\rm{th}}$ percentiles of the combined sample profile. Overlaid as black squares, triangles and circles are the median observed profiles from the REXCESS sample [@Croston2008], a sample of low redshift clustes observed with *Chandra* [@Giles2015] and a high-redshift, SPT-selected sample [@McDonald2013; @McDonald2014] respectively, with the error bars showing the $16^{\rm{th}}$ and $84^{\rm{th}}$ percentiles. The bottom panel shows the $\log_{10}$ of the ratio of the profiles at $z=0$ and $z=1$ for each sample.[]{data-label="fig:gas_prof"}](Density_spec.png){width="\columnwidth"}
The three-dimensional dimensionless density profiles for the three cluster samples at $z=0$ and $z=1$ are shown in Fig. \[fig:gas\_prof\]. We have scaled the profiles by $r^2$ to reduce the dynamic range. At $z=0$, we compare the median profile of the combined sample with the observed median profiles from @Croston2008 for the REXCESS sample and @Giles2015 for a sample of low-redshift clusters observed with *Chandra*. The combined sample shows good agreement with the observed profiles and has similar intrinsic scatter. Beyond a radius of $0.15r_{500,\rm{spec}}$ the median profiles of the hot sample and its relaxed subset have a similar shape as the combined sample, but the densities are higher as they are on average more massive clusters. Inside this radius the profiles of both samples have a shallower gradient compared to the combined sample. This is caused by the accretion of low-entropy, high-density gas that sinks to the centre of the cluster potential, becoming increasingly important below $z=1$ [@Power2014]. This effect is not offset in massive clusters by the AGN feedback and so their density profiles have a shallower gradient in the core. We note that this effect can potentially impact the relations we presented in section \[sec:screlations\]. However, we presented core-excised temperatures and luminosities, which should minimise any bias introduced by the accretion of poorly mixed gas.
At $z=1$, we compare the median density profiles of the three samples to the observed profile from @McDonald2013, which has been derived from a sample of $40$ clusters with a mean redshift of $z=0.82$. These clusters were selected from the SPT $2500\,\rm{deg}^2$ survey catalogue and observed with *Chandra*. There is a reasonable agreement between the combined sample’s median profile and the observations, but the observations are higher between $0.2-1.0r_{500}$. The observed profile is in better agreement with the median profiles of the hot sample and its relaxed subset. This suggests that the observed clusters are more representative of more massive objects at $z=1$. There is better agreement between the density profiles of the three samples at $z=1$ because the mass cut of $M=10^{14}\,\rm{M}_{\odot}$ causes the samples to converge with increasing redshift. Selecting relaxed hot clusters leads to a median profile that is slightly more centrally concentrated than for all hot clusters.
In the bottom panel of the plot we show the $\log_{10}$ of the ratio of the median density profile at $z=0$ and the median profile at $z=1$ for each sample. For the hot cluster sample and the relaxed subset the profiles have evolved in a self-similar way beyond $0.2\,r_{500}$, showing very little change. Inside of this radius the impact of accreting low-entropy, high-density gas that sinks to the centre of the cluster is apparent as an increase in the density profiles from $z=1$ to $z=0$. For the combined sample the difference between the two profiles shows the increase of the depth of the potential with redshift. This leads to higher densities at $z=1$ and a negative change density profile at all radii with decreasing redshift.
Temperature profiles
--------------------
![Median temperature profiles for the three samples. The details are the same as for Fig. \[fig:gas\_prof\], except that the REXCESS data was taken from @Arnaud2010.[]{data-label="fig:temp_prof"}](Temperature_spec.png){width="\columnwidth"}
Fig. \[fig:temp\_prof\] shows the three-dimensional temperature profiles divided by the predicted self-similar temperature. At $z=0$ the profiles all have a similar shape, but the normalization of the combined sample is somewhat higher than those of the hot sample and its relaxed subset. This is due to the lower gas density of the combined sample, which requires a higher temperature to balance gravitational collapse. Also, there is likely to be a small effect due to the mass dependence of non-thermal pressure support, with more massive clusters having more non-thermal support and lower temperatures. The accretion of low-entropy, cold gas that sinks to the cluster core produces a steeper temperature gradient in the central profiles of the hot sample and its relaxed subset. Overlaid are the observed median temperature profiles from two cluster samples, the REXCESS sample [@Arnaud2010] and a sample of clusters observed with *Chandra* [@Giles2015]. The median profile of the combined sample and its intrinsic scatter show good agreement with the observed temperature profiles and their scatter.
At $z=1$ all samples have a similar profile shape, but the hot sample has a lower normalization compared to the combined and relaxed hot sample. This is because non-thermal pressure support becomes increasingly important in clusters of a fixed mass with redshift, leading to a lower temperature in hot clusters. The relaxed sample removes the most disturbed objects with greatest level of non-thermal support, producing a higher median temperature profile. We compare to the observed median profile of @McDonald2014. The median profiles of the combined sample and the relaxed hot sample slightly under predict the observations at $0.3\,\rm{r}_{500}$ and over predict the observations at large radii, but the observed profile is within the scatter of the combined sample.
Within $r_{500,\rm{spec}}$ the median temperature profiles show significantly less evolution between the two redshifts than the density profiles. The combined and hot samples deviate from self-similarity and show an increase in temperature from $z=1$ to $z=0$ at all radii, consistent with the decreasing temperature-mass normalization with increasing redshift found in Section \[ssec:Tx-M\]. This is because non-thermal pressure support decreases with increasing redshift. Therefore, as clusters thermalise their temperatures must increase to balance gravitational collapse, resulting in a hotter temperature profile at $z=0$ compared to $z=1$. Selecting a relaxed subset reduces the non-thermal pressure support and the median profile changes significantly less from $z=1$ to $z=0$ inside $r_{500}$.
Pressure profiles
-----------------
![Median pressure profiles for the three samples. The details are the same as for Fig. \[fig:gas\_prof\], except that the REXCESS data was taken from @Arnaud2010. The green curve shows the best-fit pressure profile from @Planck2013.[]{data-label="fig:pres_prof"}](Pressure_spec.png){width="\columnwidth"}
The dimensionless pressure profiles, scaled by $r^{3}$, of the three cluster samples are shown in Fig. \[fig:pres\_prof\]. The increased mass of the hot sample and its relaxed subset lead to median pressure profiles that are higher in the centre at $z=0$ due to their higher densities. We compare the median profiles to the observed median pressure profiles from @Arnaud2010 and @Giles2015 and the best-fit profile from @Planck2013. We note that the *Planck* result is based on the stacked profile of nearby systems. For @Giles2015 we have combined their published density and temperature profiles to produce a pressure profile for each cluster. There is good agreement between the combined sample and the observed profiles, with a slight over prediction at large radii. For comparison to the Planck best-fit parameters we fit the mean profiles of our clusters at both redshifts with a generalised Navarro-Frenk-White pressure profile [@NFW1997; @NagaiVikhlininKravtsov2007a] of the form $$P(x)=\frac{P_{0}}{(c_{500}x)^{\gamma}\left[1+(c_{500}x)^{\alpha}\right]^{(\beta-\gamma)/\alpha}}\:.
\label{gNFWeq}$$ We fit a four parameter model with $\gamma=0.31$ fixed. The results are shown in table \[tab:Ppro\_fit\].
$z$ Sample $c_{500}$ $\alpha$ $\beta$
----- ------------- --------- ----------- ---------- ---------
*Planck* $6.41$ $1.81$ $1.33$ $4.13$
$0$ Combined $8.80$ $1.56$ $1.09$ $4.01$
Hot $20.66$ $0.52$ $0.70$ $6.69$
Relaxed Hot $24.01$ $0.54$ $0.69$ $6.79$
$1$ Combined $6.96$ $0.99$ $1.26$ $5.84$
Hot $6.44$ $0.51$ $1.14$ $9.44$
Relaxed Hot $9.28$ $1.97$ $1.61$ $4.11$
: Table showing the best-fit generalised Navarro-Frenk-White pressure profile parameters (see eq. \[gNFWeq\]) for the combined, hot and relaxed hot samples of clusters present in this work. We fix $\gamma=0.31$.
\[tab:Ppro\_fit\]
At $z=1$ the median profiles of the three samples are in closer agreement with each other, because the minimum mass limit of $M=10^{14}\,\rm{M}_{\odot}$ causes the samples to converge at high redshift. We compare our median pressure profiles with the observed profile of @McDonald2014. They find a median pressure profile that is in good agreement with the median profiles, but it is most consistent with the relaxed hot sample of massive clusters.
The pressure profile of the relaxed subset shows very little evolution between $z=1$ and $z=0$, except for the core where the increasing density leads to an increased pressure with decreasing redshift. The hot sample shows an increased pressure in the core with decreasing redshift, due to the increased density, but a negative change in pressure from $z=1$ to $z=0$ at larger radii. The combined sample shows a negative pressure change between $z=1$ and $z=0$ at all radii. The decreased pressure with decreasing redshift is caused by the decrease in density from $z=1$ to $z=0$.
Entropy Profiles
----------------
![Median entropy profiles for the three samples. The details are the same as Fig. \[fig:gas\_prof\], except that the REXCESS data was taken from @Pratt2010. We also show the prediction from non-radiative simulations for $z=0$ [@VoitKayBryan2005].[]{data-label="fig:enty_prof"}](Entropy_spec.png){width="\columnwidth"}
The median entropy profiles are shown in the bottom right panel of Fig. \[fig:enty\_prof\] and they have been normalized by the predicted self-similar entropy. We note that we define entropy as $$K_{\Delta}\equiv\frac{k_{\rm{B}}T_{\Delta}}{n^{2/3}_{e,\Delta}}\,$$ where $n_e$ is the electron number density and $\Delta$ is the chosen overdensity relative to the critical density of the Universe. At $z=0$ the the combined sample shows a higher normalization compared to the hot sample and its relaxed subset. This is due to it lower density profile and higher temperature profile. The gradients of the hot sample and the relaxed subset profiles steepen in the centre due to the accretion of low entropy gas. We compare with the observed median profiles of @Pratt2010 and @Giles2015, and the baseline profile of @VoitKayBryan2005 derived from non-radiative SPH simulations. The combined sample is in good agreement with the observations and tends to the non-radiative predictions at large radii.
At $z=1$ the three samples are in reasonable agreement with each other, all having a similar shape with the hot sample showing a marginally lower normalization. This change from $z=0$ is in agreement with the evolution in their density and temperature profiles. We compare the profiles to the observations of @McDonald2014. The combined and relaxed hot sample show good agreement with the observed profile for $r<0.5r_{500,\rm{spec}}$, but over predict the entropy at larger radii. In contrast the median profile of the hot sample is consistent with the observations at large radii, but under predicts the entropy in the centre of the cluster.
The departure from self-similarity for the three samples is due to a combination of the evolution in their temperature and density profiles. The relaxed hot sample shows a mild increase in entropy from $z=1$ to $z=0$ at large radii, due to change in its temperature profile, and a decrease in entropy in the core due to the increase in density at $z=0$. The increased normalization of the hot sample’s temperature profile at $z=0$ compared to $z=1$ leads to an increased entropy profile with decreasing redshift, except in the core. The combined sample shows an increase in entropy at all radii at $z=0$ compared to $z=1$ and is produced by the decreased density and increased temperature with decreasing redshift.
Summary & Discussion {#sec:sad}
====================
In this work we have presented the MACSIS clusters, a sample of 390 zoomed simulations of the most massive and rarest clusters run with the state-of-the-art, calibrated baryonic physics model from the BAHAMAS project [@McCarthy2016] that yields realistic clusters. Such massive clusters are absent from the BAHAMAS simulation volumes of $596\,\mathrm{Mpc}$ as the simulated volume is too small. After introducing the selection of the sample from the parent $3.2\,\mathrm{Gpc}$ volume simulated with the *Planck* 2013 cosmology, and demonstrating the agreement of the properties of our massive cluster sample with the properties of observed massive clusters, we examined the evolution of the cluster scaling relations and the evolution of the cluster gas profiles.
By combining the MACSIS sample with the clusters in the BAHAMAS volume, we were able to examine the cluster scaling relations over the full observed mass range for the first time. Additionally, the MACSIS clusters enabled the study of the evolution of the cluster scaling relations to unprecedentedly high redshifts. Finally, the MACSIS sample enabled clusters to be selected in ways which mimic a cosmological study, such as selecting the hottest clusters, to examine if the scaling relations of such objects evolve differently from the underlying cluster population. Our main results are:
- As shown in Fig. \[fig:observations\], the MACSIS simulations yield realistic massive clusters at low redshift and their progenitors are in good agreement with the limited observational data that is available at high redshift (i.e. $z=1$).
- Scaling relations for the combined sample that spans the full observed cluster mass range show significant deviations from the simple self-similar theory (see Figs. \[fig:MgMsr\]-\[fig:LxTxsr\]). Both the slope of the relations and the redshift evolution of the normalization are significantly affected by non-gravitational physics. The low redshift relations are in good agreement with observations and with most previous simulation work.
- The main drivers of non-self-similar evolution are AGN feedback, non-thermal pressure support and a mild mass dependence of the spectroscopic temperature bias. Shallower potentials of clusters that are less massive or form at lower redshifts allows feedback from AGN to eject more gas. Non-thermal pressure lowers a cluster’s temperature for a given potential and is more important in more massive clusters that have had less time to thermalise. We found that the spectroscopic temperature bias increases for the most massive clusters.
- With the exception of the luminosity-temperature relation, we found the scatter about the best-fit scaling relations is insensitive to mass and redshift for all of the cluster samples.
- Selecting a hot cluster sample, i.e. core-excised spectroscopic temperatures $k_{\rm{B}}T^{\mathrm{X,ce}}_{500\mathrm{spec}}\geq5\,\rm{keV}$, significantly alters the scaling relations and their evolution. Excluding the spectroscopic temperature-total mass relation, we find that the scaling relations of the hot cluster sample evolve in a much more self-similar manner. After accounting for the expected self-similar evolution with redshift, we find that the normalizations are consistent with no evolution. The slopes of the best-fit relations at each redshift are also broadly consistent with the slopes predicted by self-similar theory. However, the spectroscopic temperature-total mass relation of the hot sample deviates further from self-similarity than the combined sample. Selecting hot clusters removes the less massive clusters from the sample, so the hot sample is dynamically younger than the combined sample as more massive clusters form later in the hierarchical merger scenario. This increases the average level of non-thermal support in the hot sample, leading to a flatter spectroscopic temperature-total mass relation. Additionally, the spectroscopic temperature bias flattens the relation for the most massive clusters and this has a larger impact in a sample of only hot clusters.
- Selecting a relaxed subset of hot clusters, where the most dynamically disturbed objects are removed, leads to a small reduction in the scatter for most scaling relations. Removing the most disturbed objects also leads to a reduction in the level of non-thermal support in the sample compared to the complete hot sample. This leads to steeper slope of the spectroscopic temperature-total mass relation compared to the hot sample and a value that is closer to the self-similar prediction.
- The median hot gas profiles of the combined sample in general shows good agreement with observed radial profiles. The low redshift data is in very good agreement, while the data at $z=1$ shows reasonable agreement with the relaxed hot sample.
- Comparison of the hot gas profiles at $z=0$ and $z=1$ show evolution different from self-similar prediction (see Figs. \[fig:gas\_prof\]-\[fig:enty\_prof\]). The combined sample shows a decreasing density profile with decreasing redshift, suggesting the impact of AGN feedback. Selecting a sample of hot clusters produces a median density profile that evolves in much more self-similar manner. The combined and hot samples have a median temperature profile that increases with decreasing redshift. This is likely driven by decreasing importance of non-thermal pressure support with decreasing redshift. Selecting relaxed hot cluster sample produces a median profile that evolves in better agreement with the self-similar prediction.
MACSIS enables the study of the observable properties of the most massive and rarest galaxy clusters. We have demonstrated that their progenitors provide a good match to the currently limited observational data at high redshift and that their observable properties evolve in a significantly more self-similar manner than for lower-mass and less-relaxed clusters. We have shown how the selection function can impact the derived scaling relations and radial profiles.
The size of the parent simulation enables the creation of synthetic lightcones with an area comparable to currently ongoing surveys. This will allow the impact of selection biases to be fully examined and the covariance of observable properties to be studied. Another route for future work is to improve our understanding of structure in the ICM, as the limited resolution and traditional SPH scheme used in this work limits our ability resolve structures and understand their impact on observable properties.
Acknowledgements {#acknowledgements .unnumbered}
================
This work used the DiRAC Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BIS National E-infrastructure capital grant ST/K00042X/1, STFC capital grants ST/H008519/1 and ST/K00087X/1, STFC DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the National E-Infrastructure. DJB and STK acknowledge support from STFC through grant ST/L000768/1. MAH is supported by an STFC quota studentship. IGM is supported by a STFC Advanced Fellowship. The research was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement 278594-GasAroundGalaxies. ARJ acknowledges support from STFC through grant ST/L00075X/1.
Selection Effects {#app:seleff}
=================
{width="\textwidth"}
The selection of the MACSIS sample was done using the $M_{\mathrm{FoF}}$ mass of a halo in the parent simulation, but the scaling relations are presented using the $M_{500}$ cluster mass. This could potentially lead to a selection bias that impacts on the scaling relations presented in this work. Therefore, we make a mass cut to remove those clusters with large $M_{\rm{FoF}}/M_{500}$ ratio. To assess the impact of our sample selection we plot the log of the ratio of four cluster observables: the core-excised bolometric X-ray luminosity, the core-excised spectroscopic temperature, the gas mass and the integrated SZ signal, against their expected value from the best-fit relation as a function of the log of the ratio of the cluster’s $M_{\rm{FoF}}$ and $M_{500}$ for the combined sample of clusters, with the cut included. Fig. \[fig:SelEff\] shows the result of these plots. Any differences in the correlations of these ratios between MACSIS and BAHAMAS could indicate that selection effects were impacting the scaling relations. We have calculated the Spearman’s rank correlation coefficients for both samples for all four observable ratios and find that only the gas mass shows a significant $(>2\sigma)$ difference between the two samples. However, all of the quantities show only weak correlations. We therefore conclude that the cut to remove clusters with extremely high $M_{\rm{FoF}}/M_{500}$ ratios has minimised any bias due to selection by $M_{\mathrm{FoF}}$.
Self-similar relations {#app:ssr}
======================
If galaxy clusters were to form from a purely monolithic gravitational collapse, astrophysical processes were negligible and they were virialised, then we would expect them to be self-similar objects. This would mean that their properties would depend only on their mass [@WhiteRees1978; @Kaiser1986]. The critical density of the Universe is defined as $$\rho_{\rm{crit}}(z)\equiv E^2(z)\frac{3H^2_0}{8\pi G}\,,$$ where $H_0$ is the Hubble constant, $G$ is the gravitational constant and $$E(z)\equiv\frac{H(z)}{H_0}=\sqrt{\Omega_{m}(1+z)^3+\Omega_{\Lambda}}\,.$$ A cluster can then be defined as an overdensity with mass, $M$, inside a sphere of radius, $r$, with some average density, $\Delta$, relative to the critical density $$M_{\Delta}\propto\Delta\rho_{\rm{crit}}(z)r^3_{\Delta}\propto E^{2}(z)\,r^3_{\Delta}.$$ As gas collapses into the potential, $\Phi$, of the cluster it is heated and, assuming that it is a collapsed iosthermal sphere, it will reach a temperature, $T$, of $$k_{B}T_{\Delta}\equiv\frac{1}{2}\Phi=\frac{GM_{\Delta}\mu m_{p}}{2r_{\Delta}}\,,$$ where $k_{B}$ is the Boltzmann constant, $m_{p}$ is the mass of the proton and $\mu$ is the mean molecular weight. Therefore, the self-similar temperature of the cluster is proportional to its mass via $$\label{eq:Tx-M}
T_{\Delta}\propto M^{2/3}_{\Delta}E^{2/3}(z)\,.$$ Under the assumption that main cooling mechanism of the cluster is thermal bremsstrahlung, the cluster gas will emit X-rays and its bolometric emission is is proportional to $$L_{\rm{X},\Delta}^{\rm{bol}}\propto\rho^2\Lambda(T)r_{\Delta}^3\propto\rho^2T^{1/2}r_{\Delta}^3\propto M^{4/3}_{\Delta}E^{7/3}(z)\,,$$ where the cooling function $\Lambda(T)\propto T^{1/2}$ for the bolometric case [e.g. @Sarazin1986]. Using equation (\[eq:Tx-M\]), we can derive the self-similar prediction for the X-ray luminosity-temperature relation $$L_{\rm{X},\Delta}^{\rm{bol}}\propto T^{2}E(z)\,.$$ Assuming a constant gas fraction, the integrated Sunyaev-Zel’dovich signal, $Y_{\rm{SZ}}$, and its X-ray analogue, $Y_{\rm{X}}$, of the cluster can be predicted by $$Y_{\rm{SZ},\Delta}\propto Y_{\rm{X},\Delta}\equiv M_{\Delta}T_{\Delta}\,,$$ and the self-similar relations are $$Y_{\rm{SZ},\Delta}\propto M_{\Delta}^{5/3}E^{2/3}(z)\,,$$ $$Y_{\rm{X},\Delta}\propto M_{\Delta}^{5/3}E^{2/3}(z)\,.$$
Fit parameters {#app:fitpar}
==============
The tables \[tab:Lx-Mtab\]-\[tab:Lx-Txtab\] below list the parameter values for the best-fit relations of the scaling relations presented in this paper. For $z>1$ there are too few clusters in too many bins to reliably measure a bit-fit relation for the hot cluster sample and the relaxed subset and these values are not presented.
[l C C C C C C C C C]{} Redshift & & &\
& $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$\
$0.00$ & $44.50^{+0.01}_{-0.01}$ & $1.88^{+0.03}_{-0.05}$ & $0.15^{+0.01}_{-0.02}$ & $44.71^{+0.02}_{-0.02}$ & $1.36^{+0.08}_{-0.07}$ & $0.12^{+0.01}_{-0.02}$ & $44.69^{+0.03}_{-0.03}$ & $1.43^{+0.13}_{-0.09}$ & $0.11^{+0.01}_{-0.01}$\
$0.25$ & $44.60^{+0.01}_{-0.02}$ & $1.98^{+0.03}_{-0.05}$ & $0.12^{+0.01}_{-0.02}$ & $44.74^{+0.03}_{-0.03}$ & $1.42^{+0.14}_{-0.13}$ & $0.12^{+0.02}_{-0.01}$ & $44.69^{+0.03}_{-0.03}$ & $1.58^{+0.10}_{-0.13}$ & $0.09^{+0.01}_{-0.01}$\
$0.50$ & $44.63^{+0.01}_{-0.01}$ & $1.91^{+0.03}_{-0.04}$ & $0.11^{+0.01}_{-0.01}$ & $44.74^{+0.02}_{-0.02}$ & $1.32^{+0.10}_{-0.11}$ & $0.12^{+0.01}_{-0.02}$ & $44.73^{+0.01}_{-0.01}$ & $1.44^{+0.13}_{-0.09}$ & $0.10^{+0.01}_{-0.03}$\
$1.00$ & $44.75^{+0.08}_{-0.06}$ & $2.02^{+0.19}_{-0.14}$ & $0.12^{+0.01}_{-0.02}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
$1.50$ & $44.98^{+0.19}_{-0.12}$ & $2.13^{+0.32}_{-0.21}$ & $0.13^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
\[tab:Lx-Mtab\]
[l C C C C C C C C C]{} Redshift & & &\
& $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$\
$0.00$ & $0.68^{+0.00}_{-0.00}$ & $0.58^{+0.01}_{-0.01}$ & $0.05^{+0.003}_{-0.003}$ & $0.71^{+0.01}_{-0.01}$ & $0.51^{+0.04}_{-0.04}$ & $0.05^{+0.01}_{-0.002}$ & $0.70^{+0.01}_{-0.01}$ & $0.55^{+0.06}_{-0.03}$ & $0.04^{+0.003}_{-0.010}$\
$0.25$ & $0.67^{+0.00}_{-0.01}$ & $0.60^{+0.01}_{-0.01}$ & $0.04^{+0.004}_{-0.001}$ & $0.69^{+0.01}_{-0.01}$ & $0.50^{+0.07}_{-0.05}$ & $0.04^{+0.005}_{-0.002}$ & $0.68^{+0.01}_{-0.01}$ & $0.58^{+0.05}_{-0.06}$ & $0.04^{+0.005}_{-0.005}$\
$0.50$ & $0.64^{+0.01}_{-0.01}$ & $0.57^{+0.02}_{-0.01}$ & $0.04^{+0.002}_{-0.001}$ & $0.66^{+0.01}_{-0.01}$ & $0.46^{+0.07}_{-0.05}$ & $0.05^{+0.008}_{-0.012}$ & $0.66^{+0.01}_{-0.01}$ & $0.51^{+0.05}_{-0.07}$ & $0.03^{+0.009}_{-0.004}$\
$1.00$ & $0.61^{+0.02}_{-0.02}$ & $0.58^{+0.04}_{-0.05}$ & $0.05^{+0.001}_{-0.003}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
$1.50$ & $0.60^{+0.03}_{-0.03}$ & $0.61^{+0.06}_{-0.06}$ & $0.04^{+0.001}_{-0.002}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
\[tab:Tx-Mtab\]
[l C C C C C C C C C]{} Redshift & & &\
& $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$\
$0.00$ & $13.67^{+0.01}_{-0.01}$ & $1.25^{+0.01}_{-0.03}$ & $0.07^{+0.01}_{-0.01}$ & $13.77^{+0.01}_{-0.01}$ & $1.02^{+0.03}_{-0.03}$ & $0.06^{+0.01}_{-0.01}$ & $13.75^{+0.01}_{-0.01}$ & $1.05^{+0.04}_{-0.04}$ & $0.05^{+0.01}_{-0.01}$\
$0.25$ & $13.72^{+0.00}_{-0.01}$ & $1.29^{+0.01}_{-0.02}$ & $0.06^{+0.01}_{-0.01}$ & $13.79^{+0.01}_{-0.01}$ & $1.04^{+0.04}_{-0.06}$ & $0.06^{+0.01}_{-0.01}$ & $13.77^{+0.01}_{-0.01}$ & $1.09^{+0.04}_{-0.04}$ & $0.04^{+0.01}_{-0.01}$\
$0.50$ & $13.73^{+0.01}_{-0.01}$ & $1.25^{+0.03}_{-0.02}$ & $0.07^{+0.01}_{-0.01}$ & $13.80^{+0.01}_{-0.01}$ & $0.92^{+0.06}_{-0.05}$ & $0.05^{+0.01}_{-0.01}$ & $13.79^{+0.01}_{-0.01}$ & $0.97^{+0.08}_{-0.06}$ & $0.04^{+0.01}_{-0.01}$\
$1.00$ & $13.77^{+0.04}_{-0.03}$ & $1.29^{+0.09}_{-0.07}$ & $0.07^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
$1.50$ & $13.85^{+0.05}_{-0.08}$ & $1.31^{+0.11}_{-0.14}$ & $0.06^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
\[tab:Mg-Mtab\]
[l C C C C C C C C C]{} Redshift & & &\
& $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$\
$0.00$ & $14.33^{+0.01}_{-0.01}$ & $1.84^{+0.02}_{-0.05}$ & $0.12^{+0.01}_{-0.01}$ & $14.47^{+0.02}_{-0.02}$ & $1.51^{+0.07}_{-0.08}$ & $0.11^{+0.01}_{-0.01}$ & $14.45^{+0.02}_{-0.02}$ & $1.59^{+0.12}_{-0.06}$ & $0.08^{+0.01}_{-0.01}$\
$0.25$ & $14.38^{+0.01}_{-0.01}$ & $1.91^{+0.02}_{-0.04}$ & $0.11^{+0.01}_{-0.01}$ & $14.47^{+0.02}_{-0.02}$ & $1.57^{+0.09}_{-0.12}$ & $0.10^{+0.01}_{-0.01}$ & $14.45^{+0.02}_{-0.02}$ & $1.67^{+0.09}_{-0.08}$ & $0.07^{+0.01}_{-0.01}$\
$0.50$ & $14.37^{+0.01}_{-0.01}$ & $1.85^{+0.04}_{-0.04}$ & $0.11^{+0.01}_{-0.01}$ & $14.47^{+0.02}_{-0.01}$ & $1.35^{+0.10}_{-0.09}$ & $0.10^{+0.01}_{-0.01}$ & $14.45^{+0.02}_{-0.01}$ & $1.45^{+0.15}_{-0.11}$ & $0.07^{+0.01}_{-0.01}$\
$1.00$ & $14.39^{+0.07}_{-0.05}$ & $1.89^{+0.15}_{-0.12}$ & $0.12^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
$1.50$ & $14.48^{+0.08}_{-0.06}$ & $1.98^{+0.18}_{-0.13}$ & $0.10^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
\[tab:Yx-Mtab\]
[l C C C C C C C C C]{} Redshift & & &\
& $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$\
$0.00$ & $-4.51^{+0.01}_{-0.01}$ & $1.88^{+0.02}_{-0.03}$ & $0.10^{+0.01}_{-0.01}$ & $-4.39^{+0.02}_{-0.02}$ & $1.60^{+0.07}_{-0.05}$ & $0.10^{+0.01}_{-0.02}$ & $-4.42^{+0.02}_{-0.02}$ & $1.69^{+0.07}_{-0.07}$ & $0.09^{+0.01}_{-0.01}$\
$0.25$ & $-4.46^{+0.01}_{-0.01}$ & $1.94^{+0.02}_{-0.03}$ & $0.10^{+0.01}_{-0.01}$ & $-4.36^{+0.02}_{-0.02}$ & $1.62^{+0.10}_{-0.11}$ & $0.10^{+0.01}_{-0.01}$ & $-4.40^{+0.02}_{-0.03}$ & $1.74^{+0.09}_{-0.09}$ & $0.08^{+0.01}_{-0.01}$\
$0.50$ & $-4.45^{+0.01}_{-0.01}$ & $1.88^{+0.03}_{-0.03}$ & $0.10^{+0.01}_{-0.01}$ & $-4.37^{+0.02}_{-0.02}$ & $1.48^{+0.10}_{-0.10}$ & $0.10^{+0.01}_{-0.01}$ & $-4.38^{+0.02}_{-0.01}$ & $1.59^{+0.17}_{-0.14}$ & $0.08^{+0.01}_{-0.01}$\
$1.00$ & $-4.41^{+0.07}_{-0.05}$ & $1.91^{+0.15}_{-0.11}$ & $0.11^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
$1.50$ & $-4.29^{+0.05}_{-0.06}$ & $2.04^{+0.09}_{-0.12}$ & $0.10^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
\[tab:Ysz-Mtab\]
[l C C C C C C C C C]{} Redshift & & &\
& $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$ & $A$ & $\alpha$ & $\langle\sigma_{\log_{10}Y}\rangle$\
$0.00$ & $44.80^{+0.02}_{-0.01}$ & $3.01^{+0.04}_{-0.04}$ & $0.14^{+0.01}_{-0.01}$ & $44.93^{+0.01}_{-0.01}$ & $2.41^{+0.12}_{-0.12}$ & $0.11^{+0.01}_{-0.01}$ & $44.89^{+0.02}_{-0.02}$ & $2.53^{+0.12}_{-0.13}$ & $0.10^{+0.01}_{-0.01}$\
$0.25$ & $44.89^{+0.01}_{-0.01}$ & $3.15^{+0.03}_{-0.04}$ & $0.12^{+0.01}_{-0.01}$ & $44.95^{+0.02}_{-0.01}$ & $2.82^{+0.16}_{-0.21}$ & $0.11^{+0.01}_{-0.01}$ & $44.94^{+0.02}_{-0.02}$ & $2.67^{+0.16}_{-0.17}$ & $0.09^{+0.01}_{-0.01}$\
$0.50$ & $44.94^{+0.01}_{-0.01}$ & $3.19^{+0.03}_{-0.03}$ & $0.11^{+0.01}_{-0.01}$ & $44.99^{+0.01}_{-0.01}$ & $2.67^{+0.12}_{-0.19}$ & $0.10^{+0.01}_{-0.01}$ & $44.97^{+0.01}_{-0.02}$ & $2.62^{+0.27}_{-0.17}$ & $0.08^{+0.01}_{-0.01}$\
$1.00$ & $45.08^{+0.02}_{-0.02}$ & $3.36^{+0.05}_{-0.08}$ & $0.13^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
$1.50$ & $45.19^{+0.13}_{-0.11}$ & $3.45^{+0.37}_{-0.31}$ & $0.12^{+0.01}_{-0.01}$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$\
\[tab:Lx-Txtab\]
\[lastpage\]
[^1]: Contact e-mail: <[email protected]>
[^2]: The phase descriptor for this volume is \[Panph1, L14, (2152, 5744, 757), S3, CH1814785143, EAGLE\_L3200\_VOL1\].
[^3]: We define $r_{200}$ as the radius at which the enclosed average density is two hundred times the critical density of the Universe.
|
---
abstract: 'In this paper we consider a general problem set-up for a wide class of convex and robust distributed optimization problems in peer-to-peer networks. In this set-up convex constraint sets are distributed to the network processors who have to compute the optimizer of a linear cost function subject to the constraints. We propose a novel fully distributed algorithm, named *cutting-plane consensus*, to solve the problem, based on an outer polyhedral approximation of the constraint sets. Processors running the algorithm compute and exchange linear approximations of their locally feasible sets. Independently of the number of processors in the network, each processor stores only a small number of linear constraints, making the algorithm scalable to large networks. The cutting-plane consensus algorithm is presented and analyzed for the general framework. Specifically, we prove that all processors running the algorithm agree on an optimizer of the global problem, and that the algorithm is tolerant to node and link failures as long as network connectivity is preserved. Then, the cutting plane consensus algorithm is specified to three different classes of distributed optimization problems, namely (i) *inequality constrained problems*, (ii) *robust optimization problems*, and (iii) *almost separable optimization problems with separable objective functions and coupling constraints*. For each one of these problem classes we solve a concrete problem that can be expressed in that framework and present computational results. That is, we show how to solve: position estimation in wireless sensor networks, a distributed robust linear program and, a distributed microgrid control problem.'
author:
- 'Mathias B[ü]{}rger, Giuseppe Notarstefano, and Frank Allg[ö]{}wer [^1] [^2] [^3]'
bibliography:
- 'Decomposition\_long.bib'
title: |
A Polyhedral Approximation Framework for\
Convex and Robust Distributed Optimization
---
Introduction
============
The ability to solve optimization problems by local data exchange between identical processors with small computation and communication capabilities is a fundamental prerequisite for numerous decision and control systems. Algorithms for such distributed systems have to work within the following specifications [@Bullo2009]. All processors running the algorithm are exactly identical and each processor has only a small memory available. The data assigned by the algorithm to a processor should be independent of the overall network size or only slowly growing with the degree of the processor node in the network. None of the processors has global information or can solve the problem independently.
This paper addresses a class of optimization problems in distributed processor networks with asynchronous communication. Distributed, or peer-to-peer, optimization is related to parallel [@Bertsekas1997] or large-scale optimization [@Lasdon2002], but has to meet further requirements, such as asynchronous communication and lack of shared memory or coordination units. Distributed optimization has gained significant attention in the last years. Initially major attention was given to asynchronous distributed subgradient methods [@Tsitsiklis1986], [@AN-AO:09]. Asynchronous distributed primal and dual subgradient algorithms are important tools in network utility maximization and have been intensively studied from a communication networks perspective, see [@Low1999], [@Low2002]. Combined with projection operations, subgradient methods can also solve constrained optimization problems [@Nedic2010], [@Low1999]. In the last years, the research scope has been widened and now several different distributed algorithms are explored, each suited for particular optimization problems. Distributed Newton methods are proposed for Network Utility Maximization [@Zargham2011], [@Zargham2011a], or unconstrained strongly convex problems [@Zanella2011]. Distributed variants of Alternating Direction Method of Multipliers (ADMM) have been proposed for distributed estimation [@Schizas2008], and in the wider context of machine learning [@Boyd2010]. The ADMM shows often a good convergence rate. However, one structural difference between ADMM and distributed algorithms such as subgradient methods or the novel method proposed in this paper has to be emphasized. In a centralized implementation ADMM requires a coordination step. Distributed ADMM replaces this central coordination with a consensus algorithm. However, this requires a *synchronization* between all processors in the network, i.e., all processors have to switch synchronously between local computations and the consensus algorithm. Fully distributed algorithms, as the one proposed here, work *asynchronously* and every processor can switch between local computations and communications at its own pace. An alternative research direction was established in [@GN-FB:06z] and [@Notarstefano2009], where distributed abstract optimization problems were considered. A similar approach was explored in [@Burger2011a], [@Burger2011b] for a distributed simplex algorithm that solves degenerate linear programs and multi-agent assignment problems. Some results on the use of distributed cutting-plane methods for robust optimization have been presented in [@Burger2012]. The results of [@Burger2012] are presented in the present paper in the wider context of general distributed optimization using cutting-plane methods.
**The contributions of this paper are as follows.** Motivated by several important applications, we consider a general distributed optimization framework in which each processor has knowledge of a convex constraint set and a linear cost function has to be optimized over the intersection of these constraint sets. It is worth noting that linearity of the cost function is not a limitation and that strict convexity of the optimization problem is not required. A novel fully distributed algorithm named *cutting plane consensus* is proposed to solve this class of distributed problems. The algorithm uses a polyhedral outer-approximation of the constraint set. Processors performing the algorithm generate and exchange a *small and fixed* number of linear constraints, which provide a polyhedral approximation of the original optimization problem. Then, each processor updates its local estimate of the globally optimal solution as the minimal 2-norm solution of the approximate optimization problem. We prove the correctness of the algorithm in the sense that all processor asymptotically agree on a globally optimal solution. We show that the proposed algorithm satisfies all requirements of peer-to-peer processor networks. In particular, it requires only a strictly bounded local memory and the communication is allowed to be asynchronous. We also prove that the algorithm has an inherent tolerance against the failure of single processors.
To highlight the generality of the proposed polyhedral approximation method, we show how it can solve three different representations of the general distributed convex program. First, we consider constraint sets defined by nominal convex *inequality constraints*. Second, we discuss the method for a class of *uncertain* or *semi-infinite constraints*. We show that the novel algorithm is capable of computing robust solutions to uncertain problems in peer-to-peer networks. Finally, we show that *almost separable convex programs*, i.e., convex optimization problems with separable objective functions and coupling constraints, can be formulated in the general framework when their dual representation is considered. Applied to this problem class the [[Cutting-Plane Consensus ]{}]{}algorithm can be seen as a fully distributed version of the classical Dantzig-Wolfe decomposition, or column generation method, with no central coordinating master program.
The general algorithm derived in the paper applies directly to each of the three problem classes, and all convergence guarantees remain valid. We present for each problem class a relevant decision problem, which can be solved by the novel algorithm. In particular, it is shown that localization in sensor networks, robust linear programming and distributed control of microgrids can be solved by the algorithm. Additionally, computational studies are presented which show that the novel algorithm has an advantageous time complexity.
**Relation to other optimization methods:** The general problem formulation of this paper is similar to the formulation considered in [@Nedic2010]. However, while the approach in [@Nedic2010] requires a projection operation, which might be computationally expensive for some constraint classes, our approach requires only the knowledge of a polyhedral approximation. Additionally, our method works on general time-varying directed graphs, and does not require a balanced communication.
The almost separable optimization problem setup studied in Section \[sec.SeparableCost\] is the classical setup for large scale optimization. Dual decomposition methods decompose these problems into a master program and several subproblems. Cutting-plane methods can be used to solve the master program, leading to algorithms that originate in the classical Dantzig-Wolfe decomposition [@Dantzig1961], [@Lasdon2002]. The algorithm we propose differs significantly from classical decomposition methods. Indeed, our algorithm performs in asynchronous peer-to-peer networks with identical processors, without any central or coordinating master program.
A distributed ADMM implementation [@Boyd2010] uses an average consensus algorithm to replace the update of the central master program. While this allows to perform all computations decentralized, it requires a *synchronization* between the processors. Additionally, preforming a consensus algorithm repeatedly might require many communication steps between the processors. In contrast, our method requires neither synchronized communication nor a repeated averaging.
**The remainder of the paper is organized as follows.** The optimization problem and the processor network model are introduced in Section \[sec.ProblemFormulation\]. In Section \[sec.OuterApproximation\] the ideas of polyhedral outer-approximation and minimal norm linear programming are reviewed. The main contribution of this paper, the [[Cutting-Plane Consensus ]{}]{}algorithm, is presented in a general form in Section \[sec.Algorithm\], where also the correctness of the algorithm and its fault-tolerance are proven. The application of the algorithm to inequality constrained problems and to a localization problem in sensor networks is presented in Section \[sec.Inequality\]. In Section \[sec.Robustness\] it is shown how the algorithm can be used to solve distributed robust optimization problems, and a computational study is presented, which compares the completion time of the novel algorithm to an ADMM algorithm. The application of the [[Cutting-Plane Consensus ]{}]{}algorithm to almost separable convex optimization problems and to distributed microgrid control is discussed in Section \[sec.SeparableCost\]. Finally, a concluding discussion is given in Section \[sec.Conclusions\].
Problem Formulation and Network Model {#sec.ProblemFormulation}
=====================================
We consider a set of processors $V = \{1,\ldots,n\}$, each equipped with communication and computation capabilities. Each processor $i$ has knowledge of a convex and closed constraint set ${\mathcal}{Z}_{i} \subset \mathbb{R}^{d}$. The processors have to agree on a decision vector $z \in \mathbb{R}^{d}$ maximizing a linear objective over the intersection of all sets ${\mathcal}{Z}_{i}$. That is, the processors have to solve the distributed convex optimization problem $$\begin{aligned}
\begin{split} \label{prob.Basic}
{\text{maximize}}&\quad c^{T}z \\
{\text{subject to}}&\quad z \in \bigcap_{i=1}^{n} {\mathcal}{Z}_{i}.
\end{split}\end{aligned}$$ We denote the feasible set in the following as ${\mathcal}{Z} :=
\bigcap_{i=1}^{n}{\mathcal}{Z}_{i}$. We assume that ${\mathcal}{Z}$ is non-empty and that has a finite optimal solution.
The communication between the processors is modeled by a directed graph (digraph) ${\mathcal{G}_{c}}=(V,E)$, named *communication graph*. The node set $V = \{1,\ldots,n\}$ is the set of processor identifiers, and the edge set $E \subset \{1,\ldots,n\}^{2}$ characterizes the communication among the processors. If the edge-set does not change over time, the graph is called static otherwise it is called time-varying. We model the communication with time-varying digraphs of the form ${\mathcal{G}_{c}}(t)=(V,E(t))$, where $t \in \mathbb{N}$ represents a slotted universal time. A graph ${\mathcal{G}_{c}}(t)$ models the communication in the sense that at time $t$ there is an edge from node $i$ to node $j$ if and only if processor $i$ transmits information to processor $j$ at time $t$. The time-varying set of outgoing (incoming) *neighbors* of node $i$ at time $t$, i.e., the set of nodes to (from) which there are edges from (to) $i$ at time $t$, is denoted by ${\mathcal{N}_O}(i,t)$ (${\mathcal{N}_I}(i,t)$). In a static directed graph, the minimum number of edges between node $i$ and $j$ is called the *distance* from $i$ to $j$ and is denoted by ${\operatorname{dist}}(i,j)$. The maximum ${\operatorname{dist}}(i,j)$ taken over all pairs $(i,j)$ is the *diameter* of the graph ${\mathcal{G}_{c}}$ and is denoted by ${\operatorname{diam}}({\mathcal{G}_{c}})$. A static digraph is said to be *strongly connected* if for every pair of nodes $(i,j)$ there exists a path of directed edges that goes from $i$ to $j$. For the time-varying communication graph we rely on the concept of a jointly strongly connected graph.
\[ass.PeriodicConnectivity\] For every time instant $t \in \mathbb{N}$, the union digraph ${\mathcal{G}_{c}}^{\infty}(t):=\cup_{\tau = t}^{\infty} {\mathcal{G}_{c}}(\tau)$ is strongly connected.
In this paper we develop a distributed, asynchronous algorithm solving problem according to the network model described above. Each processor stores a small set of data and transmits at each time instant these data to its out-neighbors ${\mathcal{N}_O}(i,t)$. It is worth noting that in general it is impossible to encode the convex set ${\mathcal}{Z}_{i}$ with finite data. Thus, the information about the sets ${\mathcal}{Z}_{i}$ cannot be explicitly exchanged among the processors.
Polyhedral Approximation and Minimal Norm Linear Programming {#sec.OuterApproximation}
============================================================
We start recalling some important concepts form convex and linear optimization. We will work in the following with half-spaces of the form $ {\mathit{h}}:= \{ z: a^{T}z
- b \leq 0 \}, $ where $a \in \mathbb{R}^{d}$ and $b \in \mathbb{R}$. A half-space is called a *cutting-plane* if it satisfies the following properties. Given a closed convex set ${\mathcal}{S} \subset \mathbb{R}^{d}$ and a query point $z_{q} \notin {\mathcal}{S}$, a cutting-plane $h(z_{q})$ separates $z_{q}$ from ${\mathcal}{S}$, i.e., $a(z_{q}) \neq 0$ and $$\begin{aligned}
\label{eqn.CuttingPlaneBasic} a^{T}(z_{q})z \leq b(z_{q}) \quad
\mbox{for\;all}\; z \in {\mathcal}{S}, \quad \mbox{and} \quad a^{T}(z_{q})z_{q} -
b(z_{q}) = s(z_{q}) > 0.\end{aligned}$$ The concept of cutting-planes leads to the first algorithmic primitive, the cutting-plane oracle.
> **Cutting-Plane Oracle** ${\operatorname{ORC} }(z_{q},{\mathcal}{S})$: queried at $z_{q} \in
> \mathbb{R}^{d}$ for the set ${\mathcal}{S}$. If (i) $z_{q} \notin {\mathcal}{S}$ then it returns a cutting-plane ${\mathit{h}}(z_{q})$, separating $z_{q}$ and ${\mathcal}{S}$, otherwise (ii) it asserts that $z_{q} \in {\mathcal}{S}$ and returns an empty ${\mathit{h}}$.
We make the following assumption on the cutting plane oracle, following the general cutting-plane framework of [@Eaves1971].
\[ass.Separator\] The cutting-plane oracle ${\operatorname{ORC} }(z_{q},{\mathcal}{S})$ is such that (i) $\|a(z_{q})\|_{2} < \infty$ and (ii) $z_{q}(t) \rightarrow \bar{z}$ and $s(z_{q}(t)) \rightarrow 0$ implies that $\bar{z} \in {\mathcal}{S}.$
Note that this assumption is not very restrictive and holds for many important problem formulations. In fact, we discuss three important problem classes for which the assumption holds.
Given a collection of cutting-planes $H = \cup_{k=1}^{m} {\mathit{h}}_{k}$, the polyhedron induced by these cutting-planes is ${\mathcal}{H} = \{ z : A_{H}^{T}z \leq
b_{H}\}$, with the matrix $A_{H} \in \mathbb{R}^{d \times m}$ as $A_{H}
=[a_{1},\ldots,a_{m}]$, and the vector $b_{H} = [b_{1},\ldots,b_{m}]^{T}$.
We refer to both a half-space $h$ and the data inducing the half space with a small italic letter. A collection of cutting-planes is denoted with italic capital letters, e.g., $H =
\bigcup_{k=1}^{m} h_{k}$. For a collection of cutting-planes, we denote the induced polyhedron with capital calligraphic letters, e.g., ${\mathcal}{H}$. Please note the following notational aspect. A collection of cutting-planes $B$ that is a subset of the cutting-planes contained in $H$ is denoted as $B
\subset H$, while the induced polyhedra satisfy ${\mathcal}{B} \supseteq
{\mathcal}{H}$.
Assume that each cutting-plane $h_{i}$ is generated as a separating hyperplane for some set ${\mathcal}{Z}_{j}$, and let $H$ be a collection of cutting-planes. The linear *approximate program* $$\begin{aligned}
\begin{split} \label{prob.BasicApproxLP} \max_{z} &\quad c^{T}z \quad
\mathrm{s.t.} \; A_{H}^{T}z \leq b_{H}
\end{split}\end{aligned}$$ is then a relaxation of the original optimization problem since the polyhedron ${\mathcal}{H} = \{z:A_{H}^{T}z \leq b_{H}\}$ is an outer approximation of the original constraint set ${\mathcal}{Z} =
\bigcap_{i=1}^{n}{\mathcal}{Z}_{i}$. We denote in the following the optimal value of as $
\gamma_{H} $, i.e., $\gamma_{H} := \max_{z \in {\mathcal}{H}} c^{T}z$. The linear program has in general several optimizers, and we denote the set of all optimizers of with $$\begin{aligned}
\Gamma_{H} := \{z \in {\mathcal}{H} : \; c^{T}z \geq c^{T}v, \forall v \in {\mathcal}{H}
\}. \label{eqn.Gamma}\end{aligned}$$ It is a standard result in linear programming that $\Gamma_{H}$ is always a polyhedral set. We consider throughout the paper the unique optimal solution to which has the minimal 2-norm, i.e., we aim to compute $$\begin{aligned}
z_{H}^{*} = \mathrm{arg} \min_{z \in \Gamma_{H}}
\|z\|_{2}. \label{prob.LPprojection}\end{aligned}$$ Finding a minimal norm solution to a linear program is a classical problem and various solution methods are proposed in the literature. Starting from the early reference [@Mangasarian1983] research on this topic is still actively pursued [@Zhao2002]. The minimal 2-norm solution can be efficiently computed as the solution of a quadratic program.
\[prop.2normComputation\] Let $u^{*} \in
\mathbb{R}^{|H|},\alpha^{*} \in \mathbb{R}, l^{*}\in \mathbb{R}^{d}$ be the optimal solution to $$\begin{aligned}
\begin{split} \label{prob.2NormDual}
\min_{u, \alpha, l} \quad & \frac{1}{2} (A_{H}u + \alpha c)^{T}(A_{H}u + \alpha c) + b_{H}^{T}u + c^{T}l \\
\mathrm{s.t.} \quad & A_{H}^{T}l - \alpha b \geq 0, \; u \geq 0
\end{split}
\end{aligned}$$ then $ z^{*}_{H} = -A_{H}u^{*} - \alpha^{*}c $ solves .
The proof of this result is presented in Appendix A. The minimal 2-norm solution has the important property that it always maximizes a strongly concave cost function.
\[prop.QuadraticObjective\] Let a set of cutting-planes define the polyhedron ${\mathcal}{H}$ and let $z^{*}_{H}$ be the minimal 2-norm solution to . Consider the quadratically perturbed linear objective $$J_{\epsilon}(z) = c^{T}z - \frac{\epsilon}{2}\|z\|^{2}_{2}$$ parametrized with a constant $\epsilon>0$. Then there exists a $\bar{\epsilon} >
0$ such that for any $\epsilon \in [0,\bar{\epsilon}]$ $$\begin{aligned}
z^{*}_{H} = \mathrm{arg} \max_{z \in {\mathcal}{H}} \;
J_{\epsilon}(z). \label{prob.QuadraticPerturbation}\end{aligned}$$
The proof of this result is very similar to the classical proof presented in [@Mangasarian1979]. However, since the considered set-up is slightly different and the result is fundamental for the methodologies developed in the paper, we present the proof in Appendix B.
Any solution to a (feasible) linear program of the form is fully determined by at most $d$ constraints. This is naturally also true for the minimal 2-norm solution of a linear program. We formalize this property with the notion of *basis*. Given a collection of cutting-planes $H$, we say that a subset $B \subseteq H$ is a basis of $H$ if the minimal 2-norm solution to the linear program defined with the constraint set $B$, say $z_{B}^{*}$, is identical to the minimal 2-norm solution of the linear program defined with the constraint set $H$, say $z_{H}^{*}$, i.e., $z_{B}^{*} = z_{H}^{*}$, while for any strict subset of cutting-planes $B'
\subset B$, it holds that $z_{B'}^{*} \neq z_{B}^{*}$. For a feasible problem, the cardinality of a basis is bounded by the dimension of the problem, i.e., $|B| \leq d$. Throughout this paper, a basis is always considered to be a basis with respect to the 2-norm solution of the linear program and a basis computation requires to compute the solution to problem . Note, however, that the active constraints at an optimal point $z_{H}^{*}$ are always a superset of a basis at this point and are exactly a basis if the problem is not degenerate. Therefore, in most cases it will be sufficient to find the active constraints, which are easy to detect.
The Cutting-Plane Consensus Algorithm {#sec.Algorithm}
=====================================
For a network of processors, we propose and analyze the [[Cutting-Plane Consensus ]{}]{}algorithm to solve distributed convex optimization problems of the form .
The Distributed [[Cutting-Plane Consensus ]{}]{}Algorithm {#the-distributed-cutting-planeconsensusalgorithm .unnumbered}
---------------------------------------------------------
The algorithm to solve general distributed optimization problems is as follows.\
> **Cutting-Plane Consensus:** Processors store and update collections of cutting-planes. The cutting-planes stored by agent $i$ at iteration $t$ are always a basis of a corresponding linear approximate program , and are denoted by $B^{[i]}(t)$. A processor initializes its local collection of cutting-planes $B^{[i]}_{0}$ with a set of cutting-planes chosen such that ${\mathcal}{B}^{[i]}_{0} \supset
> {\mathcal}{Z}_{i}$ and $\max_{z \in {\mathcal}{B}^{[i]}_{0}} c^{T}z < \infty$. Each processor repeats then the following steps:
>
> 1. it transmits its current basis $B^{[i]}(t)$ to all its out-neighbors ${\mathcal{N}_O}(i,t)$ and receives the basis of its in-neighbors $Y^{[i]}(t) = \bigcup_{j \in {\mathcal{N}_I}(i,t)} B^{[j]}(t)$;
>
> 2. it defines $H_{tmp}^{[i]}(t) = B^{[i]}(t) \cup Y^{[i]}(t)$, and computes (i) a query point $z^{[i]}(t)$ as the minimal 2-norm solution to the approximate program , i.e., $$z^{[i]}(t) = \mathrm{arg} \min_{z \in \Gamma_{H_{tmp}^{[i]}(t)}} \|z\|_{2}$$ and (ii) a minimal set of active constraints $B_{tmp}^{[i]}(t)$;
>
> 3. it calls the cutting-plane oracle for the constraint set ${\mathcal}{Z}_{i}$ at the query point $z^{[i]}(t)$, $$h(z^{[i]}(t)) = {\operatorname{ORC} }(z^{[i]}(t),{\mathcal}{Z}_{i});$$
>
> 4. it updates its collection of cutting planes as follows: if $z^{[i]}(t) \in {\mathcal}{Z}_{i}$ then $B^{[i]}(t+1) = B_{tmp}^{[i]}(t)$, otherwise $B^{[i]}(t+1)$ is set to the minimal basis of $B_{tmp}^{[i]}(t) \cup
> h(z^{[i]}(t)) $.\
>
The four steps of the algorithm can be summarized as communication (S1), computation of the query point (S2), generation of cutting-plane (S3) and dropping of all inactive constraints (S4). The [[Cutting-Plane Consensus ]{}]{}algorithm is explicitly designed for the use in processor networks. We want to emphasize here the following four aspects of the algorithm.\
Each processor can initialize the local constraint sets as a basis of the artificial constraint set $\{z \in
\mathbb{R}^{d} : -M{\boldsymbol{1}}\leq z \leq M{\boldsymbol{1}}\}$ for some $M \gg 1$. If $M \in
\mathbb{R}_{> 0}$ is chosen sufficiently large, the artificial constraints will be dropped during the evolution of the algorithm.\
Each processor stores and transmits at most $(d+1)d$ numbers at a time. In particular, processors exchange bases of , which are defined by not more than $d$ cutting-planes. Each cutting-plane is fully defined by $d+1$ numbers.\
Each processor has to compute locally the 2-norm solution to a linear program with $d(|{\mathcal{N}_I}(i,t)|+1)$ constraints.\
The [[Cutting-Plane Consensus ]{}]{}algorithm does not require a time-synchronization. Each processor can perform its local computations at any speed and update its local state whenever it receives data from some of its in-neighbors.\
Due to these properties, the [[Cutting-Plane Consensus ]{}]{}algorithm is particularly well suited for optimization in large networks of identical processors.
Technical Analysis of the [[Cutting-Plane Consensus ]{}]{}Algorithm {#sec.Analysis .unnumbered}
-------------------------------------------------------------------
Before starting the proof of the algorithm correctness, we point out three important technical properties related to its evolution:
- The linear constraints stored by a processor form always a *polyhedral outer-approximation* of the globally feasible set ${\mathcal}{Z}$.
- The cost-function of each processor is monotonically non-increasing over the evolution of the algorithm.
- If the communication graph ${\mathcal{G}_{c}}$ is a strongly connected *static* graph, then after ${\operatorname{diam}}({\mathcal{G}_{c}})$ communication rounds, all processors in the network compute a query-point with a cost not worse than the best processor at the initial iteration.
These properties provide an intuition about the functionality of the algorithm and the line we will follow to prove its correctness. They are formalized and proven rigorously in Lemma \[prop.TechIssues\] in Appendix \[sec.AppendixProofs\]. We are ready to establish the correctness of the [[Cutting-Plane Consensus ]{}]{}algorithm. We start by formalizing two auxiliary results which are also interesting on their own. The first result states the convergence of the query points to the locally feasible sets.
\[prop.Convergence\] Assume Assumption \[ass.Separator\] holds. Let $z^{[i]}(t)$ be the query point generated by processor $i$ performing the [[Cutting-Plane Consensus ]{}]{}algorithm. Then, the sequence $\{z^{[i]}(t)\}_{t \geq 0}$ has a limit point in the set ${\mathcal}{Z}_{i}$, i.e., there exists $\bar{z}^{[i]} \in {\mathcal}{Z}_{i}$ such that $$\begin{aligned}
\lim_{t\rightarrow \infty} \; \| z^{[i]}(t) - \bar{z}^{[i]}\|_{2}
\rightarrow 0. \eqoprocend
\end{aligned}$$
The second result shows that all processors in the network will reach an agreement.
\[prop.Agreement\] Assume the communication network ${\mathcal{G}_{c}}(t)$ is jointly strongly connected. Let $z^{[i]}(t)$ be query points generated by the [[Cutting-Plane Consensus ]{}]{}algorithm, then $$\begin{aligned}
\lim_{t \rightarrow \infty} \; \|z^{[i]}(t) -z^{[j]}(t)\|_{2} \rightarrow 0,
\quad \mbox{for\; all\;} i,j \in \{1,\ldots,n\}. \eqoprocend
\end{aligned}$$
The correctness of the algorithm is summarized in the following theorem.
\[thm.AsymptoticConvergence\] Let ${\mathcal{G}_{c}}(t)$ be a jointly strongly connected communication network with processors performing the [[Cutting-Plane Consensus ]{}]{}algorithm, and let Assumption \[ass.Separator\] hold. Let $z^{*}$ be the unique optimizer to with minimal 2-norm, then $$\begin{aligned}
\lim_{t \rightarrow \infty} \|z^{[i]}(t) - z^{*}\|_{2} \rightarrow 0 \quad
\mbox{for\;all}\; i \in \{1,\ldots,n\}. \eqoprocend
\end{aligned}$$
For the clarity of presentation, the technical proofs of Lemma \[prop.Convergence\], Lemma \[prop.Agreement\], and Theorem \[thm.AsymptoticConvergence\] are presented in Appendix \[sec.AppendixProofs\].
A major advantage for using the [[Cutting-Plane Consensus ]{}]{}algorithm in distributed systems is its inherent fault-tolerance. The requirements on the communication network are very weak and the algorithm can well handle disturbances in the communication like, e.g., packet-losses or delays. Additionally, the algorithm has an inherent tolerance against processor failures. We say that a processor fails if it stops at some time $t_{f}$ to communicate with other processors.
Suppose that processor $l$ fails at time $t_{f}$, and that the communication network remains jointly strongly connected after the failure of processor $l$. Let $z^{[l]}(t_{f})$ be the last query point computed by processor $l$ and define $\gamma^{[l]}(t_{f})=c^{T}z^{[l]}(t_{f})$. Then the query-points computed by all processors converge, i.e., $ \lim_{t \rightarrow \infty}
\|z^{[i]}(t) - \bar{z}_{-l}\| \rightarrow 0,$ with $\bar{z}_{-l}$ satisfying $$\bar{z}_{-l} \in \left( \bigcap_{i \neq l} {\mathcal}{Z}_{i} \right) \quad
\mbox{and} \quad c^{T}\bar{z}_{-l} \leq \gamma^{[l]}(t_{f}).$$
Consider the evolution of the algorithm starting at time $t_{f}$. With Lemma \[prop.Convergence\] and Lemma \[prop.Agreement\] one can conclude that for all processors $i \neq l$, the query points will converge to the set $\left( \bigcap_{i \neq l} {\mathcal}{Z}_{i} \right)$. Additionally, the out-neighbors of the failing processor $l$ have received a basis $B^{[l]}(t_{f})$ such that the optimal value of the linear approximate program is $\gamma^{[l]}(t_{f})$. Any query point $z^{[i]}(t), t \geq t_{f}$, subsequently computed by the out-neighbors of processor $l$ as the solution to must therefore be such that $c^{T}z^{[i]}(t) \leq \gamma^{[l]}(t_{f})$ for all $t \geq t_{f}$.
This last result provides directly a paradigm for the design of fault-tolerant systems.
Suppose that for all $l \in V$, $ \bigcap_{i=1,i \neq l}^{n} {\mathcal}{Z}_{i} =
{\mathcal}{Z}$. Then for all $l \in V$, $\bar{z}_{-l} = z^{*}$ with $z^{*}$ the optimal solution to .
The abstract problem formulation and the [[Cutting-Plane Consensus ]{}]{}algorithm provide a *general framework for distributed convex optimization*. We show in the following that a variety of important representations of the constraint sets are covered by this set-up. Depending on the formulation of the local constraint sets ${\mathcal}{Z}_{i}$ different cutting-plane oracles must be defined, leading to different realizations of the algorithm. We specify in the following the [[Cutting-Plane Consensus ]{}]{}algorithm to three important problem classes. We want to stress that the correctness proofs established here for the general set-up will hold directly for the three specific problem formulations discussed in the remainder of the paper.
Convex Optimization with Distributed Inequality Constraints {#sec.Inequality}
===========================================================
As first concrete setup, we consider the most natural realization of the general problem formulation with the local constraint set defined by a convex inequality, i.e., $$\begin{aligned}
\label{prob.Inequality} {\mathcal}{Z}_{i} = \{z : f_{i}(z) \leq 0\}.\end{aligned}$$ The functions $f_{i}: \mathbb{R}^{d} \mapsto \mathbb{R}$ are assumed to be convex, but not necessarily differentiable. Thus, the set-up includes also the case in which processor $i$ is assigned more that one constraint, say ${\mathcal}{Z}_{i} = \{z :
f_{i1}(z) \leq 0, f_{i2}(z) \leq 0, \ldots, f_{ik}(z) \leq 0 \}$. In fact, one can define $f_{i}(z) := \max_{j \in \{1,\ldots k\}} f_{ij}(z)$ and directly obtain the formulation .
To define a cutting-plane oracle for constraints of the form , we use the concept of subdifferential. Given a query-point $z_{q} \in \mathbb{R}^{d}$, the subdifferential of $f_{i}$ at $z_{q}$ is $$\begin{aligned}
\partial f_{i}(z_{q}) = \{g_{i} \in \mathbb{R}^{d} : f_{i}(z) - f_{i}(z_{q})
\geq g_{i}^{T}(z - z_{q}), \; \forall z \in \mathbb{R}^{d} \}.\end{aligned}$$ An element $g_{i} \in \partial f_{i}(z_{q})$ is called a subgradient of $f_{i}$ at $z_{q}$. If the function $f_{i}$ is differentiable, then its gradient $\nabla
f_{i}(z_{q})$ is a subgradient. A cutting-plane oracle for constraints of the form is now as follows, see, e.g., [@Kelley1960].
> **Cutting-plane Oracle:** If a query point $z_{q}$ is such that $f_{i}(z_q) > 0$, then $$\begin{aligned}
> f_{i}(z_{q}) + g_{i}^{T}(z - z_{q}) \leq
> 0, \label{eqn.CuttingPlaneSubgradient}
> \end{aligned}$$ for some $g_{i} \in \partial f_{i}(z)$, is returned, .
Note also that Assumption \[ass.Separator\] is satisfied, since $s(z_{q}) =
f_{i}(z_{q}) + g_{i}^{T}(z_{q} - z_{q}) = f(z_{q})$, and $f(z_{q}) = 0$ implies $z_{q} \in {\mathcal}{Z}_{i}$. If $f_{i}(z) := \max_{j \in \{1,\ldots k\}} f_{ij}(z)$, then $\partial
f_{i}(z_{q}) = \mathbf{Co} \cup \{\partial f_{ij}(z_{q}) : f_{ij}(z_{q}) =
f_{i}(z_{q}) \}$, where $\mathbf{Co}$ denotes the convex hull. Thus, the method is applicable for constraints where subgradients can be obtained.
An important class of constraints are *semi-definite* constraints of the form $ {\mathcal}{Z}_{i} = \{ z : F_{i}(z):=F_{i0} + z_{1}F_{i1} + \cdots +
z_{d}F_{id} \leq 0 \}, $ where $F_{ij}$ are real symmetric matrices, and $'\leq 0'$ denotes negative semi-definite. The semi-definite constraint can be formulated as inequality constraint $$\begin{aligned}
f_{i}(z) := \lambda_{\max}(F_{i}(z)) \leq 0,
\end{aligned}$$ with $\lambda_{\max}$ the largest eigenvalue of $F(z)$. It is discussed, e.g., in [@Scherer], that given a query point $z_{q}$ and a normalized eigenvector $v_{q}^{*}$ of $F_{i}(z_{q})$ corresponding to $\lambda_{\max}(F_{i}(z_{q}))$, then the vector $g_{i} =
[v_{q}^{*T}F_{1}v_{q}^{*}, \ldots, v_{q}^{*T}F_{d}v_{q}^{*}]^{T}$ is a subgradient of $f_{i}(z)$. The [[Cutting-Plane Consensus ]{}]{}algorithm can thus handle semi-definite constraints and has to be seen in the context of the recent work on cutting-plane methods for semi-definite programming [@Krishnan2006], [@Konno2003].
The [[Cutting-Plane Consensus ]{}]{}algorithm is directly applicable to problems where processors are assigned convex, possibly non-differentiable, inequality constraints. Such distributed problems appear in various important application. For example, the distributed position estimation problem in wireless sensor networks can be formulated in the form with convex inequality and semi-definite constraints available only locally to (some of) the sensor nodes.
Application Example: Convex Position Estimation in Wireless Sensor Networks {#application-example-convex-position-estimation-in-wireless-sensor-networks .unnumbered}
---------------------------------------------------------------------------
Wide-area networks of cheap sensors with wireless communication are envisioned to be key elements of modern infrastructure systems. In most applications, only few sensors are equipped with localization tools, and it is necessary to estimate the position of the other sensors, see [@Bachrach2005].
In [@Doherty2001] the sensor localization problem is formulated as a convex optimization problem, which is then solved by a central unit using semidefinite programming. The semi-definite formulation proposed in [@Doherty2001] has been later extended in the literature. We formulate the distributed position estimation problem given in [@Doherty2001] in the general distributed convex optimization framework and show that the general [[Cutting-Plane Consensus ]{}]{}algorithm can be used for a fully distributed solution, using only local message passing between the sensors.
Let in the following $\mathbf{v}_{i} \in \mathbb{R}^{2}$ denote the known position of sensor $i \in \{1,\ldots,n\}$. We want to estimate the unknown position of an additional sensor $z \in \mathbb{R}^{2}$. In [@Doherty2001], two different estimation mechanisms are considered: (i) laser transmitters at nodes which scan through some angle, leading to a cone set, which can be expressed by three linear constraints of the form $f(z) :=
a_{i}^{T}z -b_{i} \leq 0,$ $a_{i} \in \mathbb{R}^{2\times 1}$ and $b_{i} \in
\mathbb{R}$, two bounding the angle and one bounding the distance and (ii) the range of the RF transmitter, leading to circular constraints of the form $ \|z -
\mathbf{v}_{i}\|_{2}^{2} \leq r_{i}^{2}$. Using the Schur-complement, the quadratic constraint can be formulated as a semi-definite constraint of the form $$F_{i}(z) := (-1)\begin{bmatrix}r_{i}I_{2} & (z - \mathbf{v}_{i}) \\ (z-\mathbf{v}_{i})^{T} & r_{i} \end{bmatrix} \leq 0,$$ where $I_ {2}$ is the $2 \times 2$ identity matrix.
Each sensor $i$ can bound the position of the unknown sensor to be contained in the convex set ${\mathcal}{Z}_{i}$, which is, depending on the available sensing mechanism, a disk represented by a semi-definite constraint ${\mathcal}{Z}_{i} = \{z :
F_{i}(z) \leq 0\}$, a cone ${\mathcal}{Z}_{i} = \{z : f_{ij}(z) \leq 0, j=1,2,3 \}$, or a quadrant, ${\mathcal}{Z}_{i} = \{z : F_{ij}(z) \leq 0, f_{ij} \leq 0, j=1,2,3
\}$.
The sensing nodes can now compute the smallest box $\{ z \in \mathbb{R}^{2} :
[z^{l}_{x}, z^{l}_{x}]^{T} \leq z \leq [z^{u}_{x}, z^{u}_{y}]^{T}\}$ that is guaranteed to contain the unknown position using the [[Cutting-Plane Consensus ]{}]{}algorithm. As proposed in [@Doherty2001], the minimal bounding box can be computed by solving four optimization problems with linear objectives. To compute, for example, $z^{u}_{x}$ one defines the objective $c_{x} = [1,
0]^{T}$ and solves $ z^{u}_{x} := \max\; c_{x}^{T}z, \; \mbox{s.t.} \; z \in
\bigcap_{i=1}^{n}\mathcal{Z}_{i}. $ In the same way $z^{l}_{x}, z^{l}_{y},
z^{u}_{y}$ can be determined. Figure \[fig.localization\] illustrates a configuration where four nodes estimate the position of one node. A linear version of such a distributed estimation problem, i.e., with all constraints being linear inequalities, has been considered in the previous work [@Notarstefano2009].
Robust Optimization with Uncertain Constraints {#sec.Robustness}
==============================================
The general formulation covers also distributed robust optimization problems with uncertain constraints. The [[Cutting-Plane Consensus ]{}]{}algorithm can therefore be used to solve a class of *robust optimization problems in peer-to-peer processor networks*.
In particular, we consider constraint sets with parametric uncertainties of the form $$\begin{aligned}
\label{prob.RobustOptimization} {\mathcal}{Z}_{i} = \{ z : \;
f_{i}(z,\theta) \leq 0,\; \mbox{for\; all\;} \theta \in \Omega_{i} \}\end{aligned}$$ where $\theta_{i}$ is an uncertain parameter, taking values in the compact convex set $\Omega_{i}$. We assume that $f_{i}$ is convex in $z$ for any fixed $\theta$. If additionally $f_{i}$ is concave in $\theta$ and $\Omega_{i}$ is a convex set, we say that the resulting optimization problem is convex [@Lopez2007]. As we will see later on, the first condition is cruicial for the application of the algorithm. The second condition will ensure that the problem can be solved exactly by our algorithm.
The problem with constraints of the form is a *distributed deterministic robust* [@BenTal2009] or distributed *semi-infinite* optimization problem [@Lopez2007]. Each processor has knowledge of an infinite number of constraints, determined by the parameter $\theta$ and the uncertainty set $\Omega_{i}$. Obviously, uncertain constraints as appear frequently in distributed decision problems. Here we focus on a deterministic worst-case optimization problem, where a solution that is feasible for any possible representation of the uncertainty is sought. A comprehensive theory for robust optimization in centralized systems has been developed and is presented, e.g., in [@BenTal2009].
Nowadays, mainly two different approaches are pursued in robust optimization. In one research direction infinite, uncertain constraints are replaced by a finite number of “sampled" constraints. Sampling methods select a finite number of parameter values and provide bounds for the expected violation of the uncertain constraints [@Calafiore2010]. In a distributed setup, a sampling approach has been explored in [@Carlone2012a]. The other research direction aims at formulating robust counterparts of the uncertain constraints , leading often to nominal semi-definite problems (see, e.g., [@BenTal2009]). Handling the uncertain constraint from a semi-infinite optimization point of view , allows also to apply exchange methods [@Reemtsen1994], where the sampling point is chosen as the solution of a finite approximation of the optimization problem. Recently, cutting-plane methods have been considered in the context of centralized robust optimization [@Mutapcic2009]. Robust optimization in processor networks is a relatively new problem. Robust optimization for communication networks using dual decomposition is considered in [@Yang2008]. We connect the robust optimization problem with uncertain constraints to our general distributed optimization framework, and show that the [[Cutting-Plane Consensus ]{}]{}algorithm can solve the problem in processor networks. In fact, the novel [[Cutting-Plane Consensus ]{}]{}algorithm is related to the exchange and cutting-plane methods [@Reemtsen1994], [@Mutapcic2009]. We define the cutting-plane oracle for the distributed robust optimization problem as follows.
> **Pessimizing Cutting-Plane Oracle:** Given a query point $z_{q}$, the worst-case parameter value $\theta_{q}^{*}$ is the maximizer of the optimization problem $$\begin{aligned}
> \label{prob.PessimizingStep} \max_{\theta} &\;
> f_{i}(z_{q},\theta) \; \quad \mathrm{s.t.}\; \theta \in \Omega_{i}.
> \end{aligned}$$ The query point $z_{q}$ is contained in ${\mathcal}{Z}_{i}$ if and only if the value of is smaller or equal to zero. If $z_{q} \notin
> {\mathcal}{Z}_{i}$, then cutting-plane is generated as $$\begin{aligned}
> \label{eqn.RobustCuttingPlane} f_{i}(z_{q},\theta_{q}^{*}) +
> g_{i}^{T}(z - z_{q}) \leq 0
> \end{aligned}$$ where $g_{i}^{T} \in \partial f_{i}(z_{q},\theta_{q}^{*})$ is a subgradient of $f_{i}$.
To see that is a cutting-plane, note that a query point $z_{q} \notin {\mathcal}{Z}_{i}$ is cut off, since $f_{i}(z_{q},\theta_{q}^{*}) +
g_{i}^{T}(z_{q} - z_{q}) = f_{i}(z_{q},\theta_{q}^{*}) > 0$. Additionally, for any point $z \in {\mathcal}{Z}_{i}$, we have $0 \geq f_{i}(z_{i},\theta)$ for all $\theta \in \Omega_{i}$, and in particular $0 \geq f_{i}(z_{i},\theta_{q}^{*})
\geq f_{i}(z_{q},\theta_{q}^{*}) + g_{i}^{T}(z - z_{q})$. Note that Assumption \[ass.Separator\] is satisfied since $f_{i}(z_{q},\theta_{q}^{*})=0$ implies that $z_{q} \in {\mathcal}{Z}_{i}$.
The oracle of the robust optimization problem requires to solve an additional optimization problem for determining the worst case parameter . Following [@Mutapcic2009], we call this the *pessimizing step*. For the practical applicability of our algorithm it is important to stress that the pessimizing steps are performed in parallel on different processors.
The pessimizing step can in general be performed by numerical tools. It can be solved exactly if the problem is convex, i.e., $f_{i}$ is concave in the uncertain parameter.
However, even if the convexity condition is not satisfied it might still be possible to find an exact solution. Reference [@Mutapcic2009] provides a formal discussion about when the pessimizing step can be solved exactly or even analytically. We review here parts of the discussion. Assume, e.g., that $f_{i}$ is convex in $\theta_{i}$ for all $z$, and $\Omega_{i}$ is a bounded polyhedron, with the extreme points $\{\theta_{i}^{1},\ldots,\theta_{i}^{k}\}$. The maximum of $f_{i}(z,u)$ is then the maximum of $f_{i}(z,\theta_{i}^{1}),\ldots, f_{i}(z,\theta_{i}^{k})$, and can be solved exactly by evaluating and comparing a finite number of functions. Furthermore, if $f_{i}(z,\theta_{i})$ is an affine function in $\theta_{i}$, i.e., $f_{i}(z,\theta_{i}) = \alpha_{i}(z)\theta_{i} + \beta_{i}(z)$ and the uncertainty set is an ellipsoid, i.e., $\Omega_{i}= \{\theta : \theta =
\bar{\theta}_{i} + P_{i}u, \; \|u\|_{2} \leq 1 \}$ for some nominal parameter value $\bar{\theta}_{i}$ and a positive definite matrix $P_{i}$, then the worst-case parameter value can be computed analytically as $$\begin{aligned}
\label{eqn.EllipsoidalWorstCase} \theta_{i}^{*} = \bar{\theta}_{i}
+ \frac{P_{i}P_{i}^{T}\alpha_{i}(z)}{\| P_{i} \alpha_{i}(z)\|_{2}}.\end{aligned}$$ Finally, if $f_{i}$ is affine in the uncertain parameter and the uncertainty set is a polyhedron, the pessimizing step becomes a linear program.
Computational Study: Robust Linear Programming {#computational-study-robust-linear-programming .unnumbered}
----------------------------------------------
We evaluate in the following the time complexity of the algorithm in a computational study for distributed robust linear programming. We follow here [@Ben1999] and consider robust linear programs in the form with linear uncertain constraints $$\begin{aligned}
\label{prob.RobustLP} a_{i}^{T}z \leq b_{i},\quad a_{i} \in
{\mathcal}{A}_{i}, \quad i \in \{1,\ldots,n\}.\end{aligned}$$ The data of the constraints is only known to be contained in a set, i.e., $a_{i}
\in {\mathcal}{A}_{i}$. Although our algorithm can in principle handle any convex uncertainty set ${\mathcal}{A}_{i}$, we restrict us for this computational study to the important class of *ellipsoidal uncertainties* ${\mathcal}{A}_{i} = \{a_{i} :
a_{i} = \bar{a}_{i} + P_{i}u_{i}, \|u_{i}\|_{2} \leq 1 \}.$ The uncertainty ellipsoids are centered at the points $\bar{a}_{i}$ and their shapes are determined by the matrices $P_{i} \in \mathbb{R}^{d\times d}$. It is known in the literature that the centralized problem can be solved as a nonlinear *conic quadratic program* [@Ben1999] $$\begin{aligned}
\label{prob.RobustLP_SOCP}
\begin{split}
\max &\; c^{T}z, \quad \mbox{s.t.} \quad \bar{a}_{i}^{T}z + \|P_{i} z \|_{2}
\leq b_{i}, \quad i \in \{1,\ldots,n\}.
\end{split}\end{aligned}$$ We will apply our algorithm directly to the uncertain problem model and use the nonlinear problem formulation only as a reference for the computational study. For the particular problem the pessimizing step can be performed analytically. Note that $ \sup_{a_{i} \in {\mathcal}{A}_{i}} a_{i}^{T}z_{q} =
\bar{a}_{i}^{T}z_{q} + \sup_{\|u\|_{2} \leq 1} \{u^{T}P_{i}^{T}z_{q}\} =
\bar{a}_{i}^{T}z_{q} + \|P_{i}^{T}z_{q}\|_{2}. $ The worst-case parameter is therefore given by $$\begin{aligned}
a^{*}_{i} = \bar{a}_{i} + \frac{ P_{i}P_{i}^{T} z_{q} }{\|P_{i}z_{q}\|_{2}}.\end{aligned}$$ A cutting-plane defined according to takes simply the form $ a_{i}^{*}z \leq b_{i}, $ i.e., the linear constraint with the worst case parameter value.
For the computational study, we generate random linear programs in the following way. The nominal problem data $a_{i} \in \mathbb{R}^{d}$ and $c \in \mathbb{R}^{d}$ are independently drawn from a Gaussian distribution with mean $0$ and standard deviation $10$. The coefficients of the vector $b$ are then computed as $b_{i} =
\left( a_{i}^{T}a_{i} \right)^{1/2}$. This random linear program model has been originally proposed in [@Dunham1977]. The matrices $P_{i}$ are generated as $P_{i}=M_{i}^{T}M_{i}$ with the coefficients of $M_{i} \in \mathbb{R}^{d \times d}$ chosen randomly according to a normal distribution with mean $0$ and standard deviation $1$. All simulations are done with dimension $d = 10$. We consider the number of communication rounds required until the query points of all processors are close to the optimal solution $z^{*}$, i.e., we stop the algorithm centrally if for all $i \in V$, $\|z^{[i]}(t)-z^{*}\|_{2} \leq 0.1$. In Figure \[fig.RobustLP\], the completion time for two different communication graphs is illustrated. We compare random Erd[ő]{}s-R[é]{}nyi graphs, with edge probability $p=1.2 \frac{\log(n)}{n}$, and circulant graphs with $5$ out-neighbors for each processor. It can be seen in Figure \[fig.RobustLP\] that the number of communication rounds grows with the network size for the circulant graph, which have a growing diameter, but remains almost constant for the random Erd[ő]{}s-R[é]{}ny graphs, which have always a small diameter. The simulations suggests, that the completion time depends primarily on the *diameter* of the communication graph.
We consider for a comparison the ADMM algorithm combined with a dual-decomposition, as described, e.g., in [@Boyd2010 pp. 48], to solve the nominal conic quadratic problem representation of the robust optimization problem.[^4] In one iteration of the ADMM algorithm, all processors must update their local variables synchronously and then compute the average of all decision variables. Figure \[fig.RobustLP\] (right axis) shows the number of iterations of the ADMM to compute the solution to the random linear programs with the same precision as the [[Cutting-Plane Consensus ]{}]{}algorithm. Note that the ADMM algorithm requires almost three times more iterations than the [[Cutting-Plane Consensus ]{}]{}algorithm requires communication rounds. Note also that the ADMM algorithm requires for each iteration an averaging of the local solutions, which can be done by a consensus algorithm. Taking into account that the number of communication rounds required to compute an average by a consensus algorithm is lower bounded by $\Omega\left(n^{2}\log(\frac{1}{\delta})\right)$, where $\delta$ is the desired precision [@Olshevsky2009], it is obvious that processors running the ADMM algorithm need to communicate significantly more often than processors running the [[Cutting-Plane Consensus ]{}]{}algorithm. Although the simulations do not compare the time-complexity of the algorithms in terms of computation units, they clearly suggest that the [[Cutting-Plane Consensus ]{}]{}algorithm is advantageous for applications where communication is costly or time consuming.
Separable Cost Optimization with Distributed Column Generation {#sec.SeparableCost}
==============================================================
The general convex problem set-up covers also the very important class of *almost separable optimization problems*, i.e., problems where each processor is assigned local decision variables with a local objective function and the local variables are coupled by a coupling constraint. We sketch here the application of the [[Cutting-Plane Consensus ]{}]{}algorithm to convex problems with separable costs and linear coupling constraints of the form $$\begin{aligned}
\begin{split} \label{prob.SeparableCost}
\min & \quad \sum_{i=1}^{n} f_{i}(x_{i}) \\
\mathrm{s.t.} &\quad \sum_{i=1}^{n} G_{i}x_{i} = \mathbf{h}, \quad x_{i} \in
{\mathcal}{X}_{i},
\end{split}\end{aligned}$$ where $x_{i} \in \mathbb{R}^{m_{i}}$ is the decision vector assigned to processor $i$, $f_{i}: \mathbb{R}^{m_{i}} \mapsto \mathbb{R}$ is a convex objective function processor $i$ aims to minimize, and ${\mathcal}{X}_{i} \subset
\mathbb{R}^{m_{i}}$ is a convex set, defining the feasible region for the decision vector $x_{i}$. For the clarity of presentation, we assume here that all sets ${\mathcal}{X}_{i}$ are bounded, although this assumption can be relaxed. The local decision variables $x_{i}$ are all coupled by a linear separable constraint with a right-hand side vector $\mathbf{h} \in \mathbb{R}^{r}$. The coupling linear constraint is of dimension $r$, and we assume here that $r$ is small compared to the number of decision variables, i.e., $r \ll
\sum_{i=1}^{n}m_{i}$. The problem formulation is the standard formulation considered for large scale optimization with decomposition methods [@Lasdon2002]. Standard large-scale optimization methods for exploit the separable structure of the dual problem, and define a coordinating master program and several sub-problems, leading to a structure as shown in Figure \[fig.DecompositionStructures\](a). In contrast, we are seeking an optimization method without a master problem using only asynchronous message-passing between neighboring processors, as visualized in Figure \[fig.DecompositionStructures\](b).
The method we propose here is strongly related to the classical *Dantzig-Wolfe (DW) decomposition* or *column generation* [@Dantzig1961], [@Lasdon2002]. The DW decomposition is dual to the cutting-plane method, see e.g., [@Eaves1971]. We exploit this duality relation here. Once again we want to stress that the DW decomposition requires a coordinating master problem, which is not required for our algorithm. In [@Burger2011b] we proposed a similar algorithm for purely linear programs taking only the primal perspective on the problem.
The problem can be formulated in the general framework , when its dual is considered. Let $\pi \in
\mathbb{R}^{r}$ be the dual variable corresponding to the coupling constraint. The dual problem to can then be written as $$\begin{aligned}
\begin{split}
\max_{\pi} \; -\mathbf{h}^{T}\pi + \sum_{i=1}^{n} \left\{ \min_{x_{i} \in
{\mathcal}{X}_{i}} f_{i}(x_{i}) + \pi^{T}G_{i}x_{i} \right\}.
\end{split}\end{aligned}$$ One can now define a new variable $u_{i} := \min_{x_{i} \in {\mathcal}{X}_{i}}
f_{i}(x_{i}) + \pi^{T}G_{i}x_{i},$ leading to the alternative representation of the dual as $$\begin{aligned}
\label{prob.ColumnDual}
\begin{split}
\max_{\pi,u_{i} } \; &-\mathbf{h}^{T}\pi + \sum_{i=1}^{n} u_{i} \\
&(\pi,u) \in \{(\pi,u): u_{i} \leq f_{i}(x_{i}) + \pi^{T}G_{i}x_{i},\;
\forall x_{i} \in {\mathcal}{X}_{i} \}.
\end{split}\end{aligned}$$ This problem is explicitly in the form with $z = [\pi^{T},
u_{1},\ldots,u_{n}]^{T} \in \mathbb{R}^{r+n}$, $c = [-\mathbf{h}^{T}, {\boldsymbol{1}}^{T}_{n}
]^{T}$ and $ {\mathcal}{Z}_{i} := \{ (\pi,u_{i}) : u_{i} \leq f_{i}(x_{i}) +
\pi^{T}G_{i}x_{i}, \forall x_{i} \in {\mathcal}{X}_{i} \}. $ The cutting-plane oracle can now be defined as follows. A query point is denoted as $z_{q} =
[\pi_{q}^{T}, u_{q,1}, \ldots, u_{q,n}]^{T}$ and is contained in the set ${\mathcal}{Z}_{i}$ if and only if $$\begin{aligned}
u_{q,i} \leq f_{i}(x_{i}) + \pi_{q}^{T}G_{i}x_{i},\quad \forall x_{i} \in
{\mathcal}{X}_{i}.\end{aligned}$$
> **Constraint Generating Oracle:** Let $\bar{x}_{i}$ denote the optimal solution vector to $$\begin{aligned}
> \label{prob.SubProblem} \min_{x_{i}} \; f_{i}(x_{i}) +
> \pi_{q}^{T}G_{i}x_{i},\quad \mathrm{s.t.\;} x_{i} \in {\mathcal}{X}_{i}
> \end{aligned}$$ and let $\gamma_{i}^{*}$ be the optimal value of . If $u_{q,i} > \gamma_{i}^{*} $ then $z_{q} \notin {\mathcal}{Z}_{i}$. A cutting plane separating $z_{q}$ and ${\mathcal}{Z}_{i}$ is then $$\begin{aligned}
> u_{i} - f_{i}(\bar{x}_{i}) - \pi^{T}G_{i}\bar{x}_{i} \leq 0.
> \end{aligned}$$
Clearly, $u_{q,i} - f_{i}(\bar{x}_{i}) - \pi_{q}^{T}G_{i}\bar{x}_{i} >
0$ for $ (\pi_{q}, u_{q}) \notin {\mathcal}{Z}_{i}$ and $u_{q,i} - f_{i}(\bar{x}_{i}) -
\pi_{q}^{T}A_{i}\bar{x}_{i} \leq 0$ for all $(\pi,u) \in {\mathcal}{Z}_{i}$. Also, Assumption \[ass.Separator\] holds since $s(z_{q}) = u_{q,i} -
f_{i}(\bar{x}_{i}) - \pi_{q}^{T}G_{i}\bar{x}_{i}$ and $s(z_{q}) \rightarrow 0$ implies $(\pi_{q},u_{q}) \in {\mathcal}{Z}_{i}$. The proposed procedure of constructing a constraint is known as “constraint generation" or, taking the primal perspective, as “column generation". We name the *local subproblem* $SP_{i}$, since it corresponds to the subproblem of the DW decomposition. The approximate linear program formed by each processor is called here *local master problem* $MP_{i}$, since it is a local version of the master program of the DW-decomposition.
It is worth noting that here $z = [\pi^{T}, u_{1},\ldots,u_{n}]^{T}$ and thus the dimension of the problem, $d = r+ n$, is no longer independent of the number of processors. Additionally, the set-up considered in this section requires a unique identifier to be assigned to each processor. These two additional restrictions have to be taken into account for an implementation of the algorithm.
\[fig.DecompositionStructures\_DW\] \[fig.DecompositionStructures\_CPC\]
The [[Cutting-Plane Consensus ]{}]{}algorithm is applied here to the dual problem and will compute the dual solution to , i.e., $$\lim_{t\rightarrow \infty}
\|\pi^{[i]}(t) -\pi^{*}\|_{2} \rightarrow 0.$$ If all $f_{i}(\cdot)$ in are strictly convex, the solutions of the local subproblems of each processor will converge to the optimal solution, i.e., $\lim_{t \rightarrow \infty}
\|\bar{x}_{i}^{[i]}(t) - x_{i}^{*}\| \rightarrow 0, \; \mbox{for\;all\;} i \in
V, $ where $x^{*}=[x_{1}^{*},\ldots,x_{n}^{*}]$ is the optimal primal solution to , and $\bar{x}_{i}^{[i]}(t)$ is the solution to computed by processor $i$ at time $t$. However, this is not true if some $f_{i}(\cdot)$ are only convex but not strongly convex. Then recovering a primal optimal solution from the dual solution can be done using the method known from DW decomposition. We assume that each processor stores the points at which a constraint is generated, $\bar{x}^{[i]}(\tau)$, where the index $i$ indicates which processor computed at time $\tau$ the point $\bar{x}_{i}(\tau)$ as solution to . Define $\bar{G}_{i\tau} := G_{i}\bar{x}_{i}(\tau)$ and $\bar{f}_{i\tau} :=
f_{i}(\bar{x}_{i}(\tau))$. The scalar inequalities of the approximate linear program are all of the form $$\begin{aligned}
\label{eqn.GeneratedConstraint} u_{i} -\bar{G}_{i\tau}^{T} \pi
\leq \bar{f}_{i\tau}.\end{aligned}$$ One can now formulate the linear programming dual to the approximate program . Let $\lambda_{j\tau} \in \mathbb{R}_{\geq 0}$ be the Lagrange multiplier to the constraint , the linear programming dual to is a linear program with the following structure: $$\begin{aligned}
\label{prob.ApproxDualLP}
\begin{split}
\min_{\lambda_{i\tau} \geq 0}\; &\sum_{i=1}^{n} \sum_{\tau} \bar{f}_{i\tau} \lambda_{i\tau} \\
&\sum_{i=1}^{n} \sum_{\tau} \bar{G}_{i\tau}\lambda_{i\tau} = \mathbf{h},
\quad \sum_{\tau} \lambda_{i\tau} = 1, \; i \in \{1,\ldots,n\}.
\end{split}\end{aligned}$$
We assume in the following that all processors have the same set of constraints as their basis. Please note that this can be achieved by halting the algorithm at some time and running a suitable agreement mechanism, such as the one proposed in [@Notarstefano2009]. A processor can now reconstruct its component of the solution vector as the convex combination $ x_{i}^{*} = \sum_{\tau}
\bar{x}_{i}(\tau)\lambda_{i\tau}^{*},$ where $\lambda_{i\tau}^{*}$ solves . The resulting solution vector $x^{*} = [ x_{1}^{*},\ldots, x_{n}^{*}]$ is globally feasible since $ \sum_{i=1}^{n} \sum_{\tau}
\bar{G}_{i\tau}\lambda_{i\tau}^{*} = \sum_{i=1}^{n} G_{i} \left(\sum_{\tau}
\bar{x}_{i}(\tau)\lambda_{i\tau}^{*} \right) = \sum_{i=1}^{n}G_{i}x_{i}^{*}=
\mathbf{h}. $ Additionally, if all processors have computed the globally optimal solution to , then the recovered $x_{i}^{*}=\sum_{k}
\bar{x}_{i}(\tau)\lambda_{i\tau}$ is also the optimal primal solution to . To see this note that strong duality implies that the optimal value of is equivalent to the value of the linear approximate problem , which we denote with $f^{*}$. Thus, $f^{*}=
\sum_{i=1}^{n} \sum_{\tau=1} f_{i}(\bar{x}_{i}(\tau))\lambda_{i\tau}^{*}$. Convexity of $f_{i}(\cdot)$ and $\sum_{\tau} \lambda_{i\tau} = 1$ implies that $
f^{*} = \sum_{i=1}^{n} \sum_{\tau} f_{i}(\bar{x}_{i}(\tau))\lambda_{i\tau}^{*}
\geq \sum_{i=1}^{n} f_{i}(\sum_{\tau} \bar{x}_{i}(\tau) \lambda_{i\tau}^{*}) =:
\sum_{i=1}^{n} f_{i}(x_{i}^{*}). $ Since $x^{*}=[x_{1}^{*},\ldots,x_{n}^{*}]$ is a feasible solution it must hold that $\sum_{i=1}^{n} f_{i}(x_{i}^{*}) =
f^{*}$. Please note that the proposed method requires each processor to store its own local solutions $\bar{x}_{i}^{[i]}(t)$ to generated during the evolution of the algorithm, but does not require that the processors exchange those solutions. For a more explicit discussion on the reconstruction of the feasible solution, we refer the reader to the literature on nonlinear DW-decomposition [@Lasdon2002] or our recent paper [@Burger2011b].
Application Example: Distributed Microgrid Control {#application-example-distributed-microgrid-control .unnumbered}
---------------------------------------------------
The previous discussion shows that the [[Cutting-Plane Consensus ]{}]{}algorithm is applicable for many important control problems, such as for example distributed microgrid control. Microgrids are local collections of distributed energy sources, energy storage devices and controllable loads. Most existing control strategies still use a central controller to optimize the operation [@Zamora2010], while for several reasons, detailed, e.g., in [@Zamora2010], distributed control strategies, which do not require to collect all data at a central coordinator, are desirable. We consider the following optimization model of the microgrid, described recently in [@Kraning2012]. A microgrid consists of several generators, controllable loads, storage devices and a connection to the main grid over which power can be bought or sold. In the following, we use the notational convention that energy generation corresponds to positive variables, while energy consumption corresponds to negative variables. A *generator* generates power $p_{gen}(t), t \in [0,T]$ within the absolute bounds $\underline{p}(t) \leq p_{gen}(t) \leq \bar{p}(t)$ and the rate constraints $\underline{r}(t) \leq p_{gen}(t+1) - p_{gen}(t) \leq
\bar{r}(t)$. The cost to produce power by a generator is modeled as a quadratic function $f_{gen}(t) = \alpha p_{gen}(t) + \beta p^2_{gen}(t)$. A *storage device* can store or release power $p_{st}(t), t \in [0,T]$ within the bounds $-d_{st} \leq p_{st}(t) \leq c_{st}$. The charge level of the storage device is then $ q_{st}(t) = q_{st, init} +
\sum_{\tau=0}^{t}p_{st}(\tau)$ and must be maintained between $0 \leq q_{st}(t)
\leq q_{max}$. Note that $p_{st}(t)$ takes negative values if the storage device is charged and positive values if it is discharged. A *controllable load* has a desired load profile $l_{cl}(t)$ and incorporates a cost if the load is not satisfied, i.e., $f_{cl}(t) =
\alpha(l_{cl}(t)-p_{cl}(t))_{+}$, where $(z)_{+} = \max\{0,z\}$. Finally, the microgrid has a single control unit, which coordinates the connection to the main grid and can trade energy. The maximal energy that can be traded is $|p_{tr}| \leq E$. The cost to sell or buy energy is modeled as $
f_{tr} = -c^{T}p_{tr} + \gamma^{T} |p_{tr}|$ where $c$ is the price vector and $\gamma$ is a general transaction cost. The power demand $D(t)$ in the microgrid is predicted over a horizon $T$. The control objective is to minimize the cost of power generation while satisfying the overall demand. This control problem can be directly formulated as in the form , with the local objective functions $f_{i} = \sum_{t = 0}^{T}f_{i}(t)$, the right-hand side vector of the coupling constraint as the predicted demand $\mathbf{h}= [D(1), \ldots, D(T)]^{T}$ and $\mathcal{X}_{i}$ as the local constraints of each unit.
The [[Cutting-Plane Consensus ]{}]{}algorithm can solve this problem in a distributed way. Note that the objective functions $f_{i}$ considered here are all convex, but not strictly convex. If all objective functions were strictly convex, one could use the distributed Newtons method [@Zargham2011], which has locally a quadratic convergence rate. However, the distributed Newton method does not apply to this problem formulation. The [[Cutting-Plane Consensus ]{}]{}algorithm does not require strict convexity of the cost functions.
We present simulation results for an example set-up with $n=101$ decision units, i.e., 60 generators, 20 storage devices, 20 controllable loads and one connection to the main grid. A random demand is predicted for 15 minute time intervals over a horizon of three hours, based on a constant off-set, a sinusoidal growth and a random component. The algorithm is initialized with each processor computing a basis out of the box-constraint set $\{z : -10^{5} \cdot {\boldsymbol{1}}\leq z \leq 10^{5}\cdot {\boldsymbol{1}}\}$, leading to a very high initial objective value.
Figure \[fig.Microgrid\] shows the largest objective value over all processors, relative to the best solution found as the algorithm is continued to perform. The evolution of the objective value is shown for three different $k$-regular graphs. It can be clearly seen that the convergence speed depends strongly on the structure of the communication graph. The convergence for a network with a 2-regular communication structure is significantly slower than for a network with a higher regular graph. We also want to emphasize the observation that the difference in the convergence speed between $k=8$ and $k=32$ is not as big as the increased communication would let one expect. This shows that the improvement obtained from more communication between the processors becomes smaller with more communication. A good performance of the algorithm can also be obtained with little communication between the processors. Please note that for all communication graphs the [[Cutting-Plane Consensus ]{}]{}algorithm requires only few communication rounds to converge to a fairly good solution. Although the convergence to an exact optimal solution might take more iterations, a good sub-optimal solution can be found after very few communication rounds. This property makes the [[Cutting-Plane Consensus ]{}]{}attractive for control and decision applications.
Discussion and Conclusions {#sec.Conclusions}
==========================
We proposed a framework for distributed convex and robust optimization using a polyhedral approximation method. As a general problem formulation, we consider problems where convex constraint sets are distributed to processors, and the processors have to compute the optimizer of a linear objective function over the intersection of the constraint sets. We proposed the novel [[Cutting-Plane Consensus ]{}]{}algorithm as an asynchronous algorithm performing in peer-to-peer networks. The algorithm is well scalable to large networks in the sense that the amount of data each processor has to store and process is small and independent of the network size.
The appealing property of the considered outer-approximation method lies in the fact that it imposes very little requirements on the structure of the constraint sets. Merely the only requirement is that a cutting-plane oracle exists. We have presented oracles for various formulations of the constraint sets, in particular, inequality and convex uncertain or semi-infinite constraints. Also, we showed that, as the dual problem formulation is considered, also almost separable convex optimization problems can be formulated in the proposed framework. We showed for each of the proposed problem formulations how the cutting-plane oracle can be defined.
Finally, we illustrated that the proposed set-up is of interest for various decision and control problems. These include the localization problem in sensor networks. They include also less obvious problems as, e.g., distributed microgrid control, where the novel algorithm can be applied to the dual problem formulation. In this context we showed that the application of the algorithm to the dual problem has the major advantage that a feasible solution can be found in a fully distributed way even before the algorithm has converged to an optimal solution.
Proofs of Section \[sec.OuterApproximation\]
--------------------------------------------
### Proof of Proposition \[prop.2normComputation\]
The minimal 2-norm solution is the solution to $$\begin{aligned}
\label{prob.MinNormQP}
\min_{z,y} \frac{1}{2}z^{T}z, \; \mbox{s.t.} \; A_{H}^{T}z \leq b_{H}, \; A_{H}y = c, \; c^{T}z - b_{H}^{T}y = 0,\; y \geq 0,\end{aligned}$$ where the constraints represent the linear programming optimality conditions (KKT-conditions). The Lagrangian of can be directly determined to be $$\begin{aligned}
\mathcal{L}(z,y,u,l,\alpha) = \frac{1}{2}z^{T}z + u^{T}(A_{H}^{T}z -b_{H}) + l^{T}(A_{H}y-c) + \alpha(c^{T}z-b_{H}^{T}y),\; y,u \geq 0.\end{aligned}$$ It follows now that $y^{*} = \mbox{arg}\min_{y\geq 0} \; \mathcal{L}(z,y,u,l,\alpha) = 0$ if $ A_{H}^{T}l - \alpha b_{H} \geq 0.$ From $ z^{*} = \mbox{arg} \min_{z}\;, \mathcal{L}(z,y,u,l,\alpha) $ follows that $z^{*} = -A_{H}u - \alpha c.$ The problem stated in the proposition is now $\min_{u \geq 0, l, \alpha} \; -\mathcal{L}(z^{*},y^{*},u,l,\alpha)$. $\hfill \blacksquare$
### Proof of Lemma \[prop.QuadraticObjective\]
The minimal 2-norm solution $z_{H}^{*}$ is the unique minimizer of $$\begin{aligned}
\begin{split}
\min_{z} \; & \frac{1}{2}\|z\|^{2},\quad \mathrm{s.t.}\; c^{T}z \geq \gamma_{H},\; A_{H}^{T}z \leq b_{H}.
\end{split}\end{aligned}$$ and satisfies therefore the feasibility conditions $
c^{T}z^{*}_{H} = \gamma_{H}$ and $A_{H}^{T}z^{*}_{H} \leq b_{H}.$ Since $z_{H}^{*}$ is an optimal solution, there exist multipliers $\mu^{*} \in \mathbb{R}$ and $\lambda^{*} \in \mathbb{R}^{|H|}_{\geq 0}$, such that the KKT conditions are satisfied, i.e., $$\begin{aligned}
z^{*}_{H} - \mu^{*}c + A_{H} \lambda^{*} &= 0 \label{eq.FOC_QP_1} \\
\lambda^{*T}A_{H}^{T}z^{*}_{H} - \lambda^{*T}b_{H} &= 0. \label{eq.FOC_QP_2} \end{aligned}$$ Since $z^{*}_{H}$ is also a solution to the original linear program , there also exists a multiplier vector $y^{*} \in \mathbb{R}^{|H|}_{\geq 0}$ satisfying the linear programming optimality conditions $$\begin{aligned}
-c + A_{H}y^{*} &= 0 \label{eq.FOC_LP_1}\\
y^{*T}A_{H}^{T}z^{*}_{H} - b^{T}_{H}y^{*} &= 0. \label{eq.FOC_LP_2}\end{aligned}$$
We have to show now that the existence of $z^{*}_{H},\mu^{*},\lambda^{*}$ and $y^{*}$ imply, for a sufficiently small $\epsilon$, the existence of a multiplier vector $\pi^{*}$ satisfying the optimality conditions of , which are $$\begin{aligned}
\begin{split}
-c + \epsilon z^{*}_{H} + A_{H}\pi^{*} &= 0\quad \mbox{and} \quad
\pi^{*T}A_{H}^{T}z^{*}_{H} - \pi^{*}b_{H} = 0. \label{eq.FOC_Perturbed}
\end{split}\end{aligned}$$
We distinguish now the two cases $\mu^{*} > 0$ and $\mu^{*} = 0$. First, assume $\mu^{*} > 0$. We can multiply with $\frac{t}{\mu^{*}}$, for arbitrary $t \in (0,1]$, and add to this , multiplied by $(1-t)$ to obtain $$\begin{aligned}
\frac{t}{\mu^{*}}z^{*}_{H} - c + A_{H}(\frac{t}{\mu^{*}}\lambda^{*} + (1-t)y^{*}) = 0. \label{eq.MUpos_1}\end{aligned}$$ The same steps can be repeated with and to obtain $$\begin{aligned}
(\frac{t}{\mu^{*}}\lambda^{*T} + (1-t)y^{*T})(A_{H}^{T}z^{*}_{H} - b_{H}) = 0. \label{eq.MUpos_2}\end{aligned}$$ With and , for any $\epsilon \leq \frac{1}{\mu^{*}}$, one can define $t_{\epsilon} = \epsilon \mu^{*}$. Then $
\pi^{*} = \frac{t_{\epsilon}}{\mu^{*}}\lambda^{*T} + (1-t_{\epsilon})y^{*T}
$ solves . In the second case $\mu^{*}=0$, one can pick an arbitrary $\epsilon >0$, multiply (and , respectively) with $\epsilon$ and add (or , respectively) to obtain $
\epsilon z^{*}_{H} - c + A_{H}(\epsilon \lambda^{*} +y^{*}) = 0$ and $
(\epsilon \lambda^{*} +y^{*})(A_{H}^{T}z^{*}_{H} - b_{H}) = 0.$ Now, $\pi^{*} := (\epsilon \lambda^{*} +y^{*})$ solves . $\hfill \blacksquare$
Proofs of Section IV: Correctness of the Algorithm {#sec.AppendixProofs}
--------------------------------------------------
Some technical properties of the algorithm are formalized in the following result.
\[prop.TechIssues\] Let $z^{[i]}(t)$ be the query point and $B^{[i]}(t)$ the corresponding basis. Let ${\mathcal{B}}^{[i]}(t)
\subset \mathbb{R}^{d}$ be the feasible set induced by $B^{[i]}(t)$. Then,
1. ${\mathcal{B}}^{[i]}(t) \supset {\mathcal}{Z}$ for all $i \in \{1\ldots,n\}$ and $ t \geq
0$;
2. $\lim_{t \rightarrow \infty} z^{[i]}(t) = \bar{z}$ and $\bar{z} \in
{\mathcal}{Z}$ implies $\bar{z}$ is a minimizer of ;
3. there exists $\underline{\epsilon} > 0$ such that for all $i \in
\{1,\ldots,n\}$ and all $t \geq 0$, the query points $z^{[i]}(t)$ maximize the objective function $${J_{\epsilon}}(z) := c^{T}z - \frac{\epsilon}{2}\|z\|^{2}_{2}$$ over the set of constraints $B^{[i]}(t) \cup Y^{[i]}(t)$ (as defined in (S2)) for all $\epsilon \in [0,\underline{\epsilon}]$;
4. ${J_{\underline{\epsilon}}}(z^{[i]}(t+1)) \leq {J_{\underline{\epsilon}}}(z^{[i]}(t))$ for all $i \in \{1,\ldots,n\}$ and all $t\geq 0$;
5. if ${\mathcal}{G}_{c}$ is a strongly connected *static* graph, then ${J_{\underline{\epsilon}}}(z^{[j]}(t+{\operatorname{diam}}({\mathcal}{G}_{c})) ) \leq {J_{\underline{\epsilon}}}(z^{[i]}(t))$ for all $i,j \in \{1,\ldots,n\}$ and all $t\geq0$.
To see (i), note that any cut $h_{k}$ generated by the oracle of processor $i$, ${\operatorname{ORC} }(\cdot,{\mathcal}{Z}_{i})$ is such that the half-space $h_{k}$ contains ${\mathcal}{Z}_{i}$, and in particular $h_{k}$ contains ${\mathcal}{Z} =
\bigcap_{i=1}^{n}{\mathcal}{Z}_{i}$. Thus any collection of cuts $H = \bigcup_{k}
h_{k}$, generated by arbitrary processors is such that ${\mathcal}{H} \supset
\bigcap_{i=1}^{n} {\mathcal}{Z}_{i} = {\mathcal}{Z}$, and in particular ${\mathcal}{B}^{[i]}(t)
\supset {\mathcal}{Z}$. The claim (ii) follows since $z^{[i]}(t)$ is computed as a maximizer of the linear cost $c^{T}z $ over the collection of cutting-planes $H^{[i]}_{tmp}(t)$. The induced polyhedron is such that ${\mathcal{H}}_{tmp}^{[i]}(t)
\supset {\mathcal}{Z}$. Therefore, we can conclude that $c^{T}z^{[i]}(t) \geq
c^{T}z^{*}$, where $z^{*}$ is an optimizer of . By continuity of the linear objective function, we have that $c^{T}\bar{z}\geq
c^{T}z^{*}$ On the other hand, $c^{T}z \leq c^{T}z^{*}$ for all $z \in
{\mathcal}{Z}$. This proves the statement. The statement (iii) follows from Lemma \[prop.QuadraticObjective\]. For any approximate program defined by processor $i$ at time $t$, there exists a constant $\bar{\epsilon}_{it} > 0$ such that $z^{[i]}(t)$ is the unique maximizer of the family of strictly concave objective functions $ J_{\epsilon}(z) := c^{T}z -
\frac{\epsilon}{2}\|z \|^{2}, \quad \epsilon \in [0,\bar{\epsilon}_{it} ], $ over the set of constraints $B^{[i]}(t) \cup Y^{[i]}(t)$. One can now always find $\underline{\epsilon} > 0$ such that $\underline{\epsilon} \leq \bar{\epsilon}_{it}$ for all $i\in \{1,\ldots,n\}$ and $t\geq 0$. To see claim (iv), note that adding cutting-planes, either by receiving them from neighbors (S2) or by generating them with the oracle (S3), can only decrease the value of the strictly concave objective function ${J_{\underline{\epsilon}}}(\cdot)$ and the basis computation in (S4) keeps, by its definition, the value of ${J_{\underline{\epsilon}}}(\cdot)$ constant. Finally, (v) can be seen as follows. Starting at any time $t$ at some processor $i$, at time $t+1$ all processors in $l \in {\mathcal{N}_I}(i,t)$ received the basis of processor $i$, and compute a query point that satisfies ${J_{\underline{\epsilon}}}(z^{[l]}(t+1)) \leq {J_{\underline{\epsilon}}}(z^{[i]}(t))$ for all $l \in {\mathcal{N}_I}(i,t)$. This argument can be repeatedly applied to see that, in the static, strongly connected communication graph ${\mathcal}{G}_{c}$, at least after ${\operatorname{diam}}({\mathcal}{G}_{c})$ iterations, all processors in the network have an objective value smaller than ${J_{\underline{\epsilon}}}(z^{[i]}(t))$.
Next we present the proof of Lemma \[prop.Convergence\], Lemma \[prop.Agreement\], and Theorem \[thm.AsymptoticConvergence\]. The following proofs use the parameterized cost function ${J_{\underline{\epsilon}}}(\cdot)$. However, for the clarity of presentation we will simplify our notation in the following proofs and write simply $J(\cdot)$ instead of ${J_{\underline{\epsilon}}}(\cdot)$.
### Proof of Lemma \[prop.Convergence\]
All $z^{[i]}(t)$ are computed as maximizers of the common strictly concave objective function $J(\cdot)$ (Lemma \[prop.TechIssues\] (iii)) and $J(\cdot)$ is monotonically non-increasing over the sequence of query points computed by a processor (Lemma \[prop.TechIssues\] (iv)). Any sequence $\{ J(z^{[i]}(t)) \}_{t \geq 0}$, $i \in \{1,\ldots,n\}$, has therefore a limit point, i.e., $\lim_{t \rightarrow \infty} J(z^{[i]}(t))
\rightarrow \bar{J}^{[i]}$. Since the sequence is convergent, it holds that $ \lim_{t\rightarrow \infty}
\left(J(z^{[i]}(t)) - J(z^{[i]}(t+1)) \right) \rightarrow 0.$ By strict concavity of $J(\cdot)$ follows that $J(z^{[i]}(t)) - J(z^{[i]}(t+1)) >
\sigma \|z^{[i]}(t) - z^{[i]}(t+1)\|^{2}_{2} $ for some $\sigma > 0$. Consequently, $\lim_{t\rightarrow \infty} \|z^{[i]}(t) - z^{[i]}(t+1)\|_{2}
\rightarrow 0$ and the sequence of query points has a limit point, i.e., $
\lim_{t \rightarrow \infty} \|z^{[i]}(t)- \bar{z}^{[i]}\|_{2} \rightarrow 0$. Suppose now, to get a contradiction, that $\bar{z}^{[i]} \notin {\mathcal}{Z}_{i}$. Then there exists $\delta > 0$ such that all $z$ satisfying $\|z -
\bar{z}^{[i]}\|_{2} < \delta$ are not contained in ${\mathcal}{Z}_{i}$. Since $\lim_{t\rightarrow \infty} \|z^{[i]}(t) - \bar{z}^{[i]}\|_{2}
\rightarrow 0$, there exists a time instant $T_{\delta}$ such that $\|z^{[i]}(t) - \bar{z}^{[i]}\|_{2} < \delta$ for all $t \geq T_{\delta}$, and thus $z^{[i]}(t) \notin {\mathcal}{Z}_{i}$ for $t \geq T_{\delta}$. But now, for all $t \geq T_{\delta}$ the oracle ${\operatorname{ORC} }(z^{[i]}(t),{\mathcal}{Z}_{i})$ will generate a cutting-plane according to , cutting off $z^{[i]}(t)$. According to , it must hold that $a^{T}(z^{[i]}(t))z^{[i]}(t)-b(z^{[i]}) = s(z^{[i]}(t)) > 0$ and $a^{T}(z^{[i]}(t))z^{[i]}(t+1)-b(z^{[i]}) \leq 0$. This implies that $a^{T}(z^{[i]}(t))\left( z^{[i]}(t) - z^{[i]}(t+1) \right)
\geq s(z^{[i]}(t))$ and consequently $ \| z^{[i]}(t) - z^{[i]}(t+1)\|_{2} \geq (\| a(z^{[i]}(t))\|_{2})^{-1}
s(z^{[i]}(t)). $ By Assumption \[ass.Separator\] (i) holds $\| a(z^{[i]}(t))\|_{2} < \infty$ and thus $\lim_{t\rightarrow \infty} s(z^{[i]}(t)) \rightarrow 0$. As a consequence of Ass. \[ass.Separator\] (ii) follows directly that $\bar{z}^{[i]} \in {\mathcal}{Z}_{i}$, providing the contradiction. $ \hfill \blacksquare$\
### Proof of Lemma \[prop.Agreement\]:
Let $\bar{J}^{[i]}:= J(\bar{z}^{[i]})$ be the objective value of the limit point $\bar{z}^{[i]}$ of the sequence $\{z^{[i]}(t)\}_{t\geq 0}$ computed by processor $i$. We show first that the limiting objective values $\bar{J}^{[i]}$ are identical for all processors. Suppose by contradiction that there exist two processors, say $i$ and $j$, such that $ \bar{J}^{[i]} < \bar{J}^{[j]}. $ Pick now $\delta_{0} > 0$ such that $\bar{J}^{[j]} - \bar{J}^{[i]} > \delta_{0}. $ The sequences $\{J(z^{[i]}(t)) \}_{t \geq 0}$ and $\{J(z^{[j]}(t)) \}_{t
\geq 0}$ are monotonically increasing and convergent. Thus, for every $\delta > 0$ there exists a time $T_{\delta}$ such that for all $t \geq
T_{\delta}$, $ J(z^{[i]}(t)) - \bar{J}^{[i]} \leq \delta$ and $
J(z^{[j]}(t)) - \bar{J}^{[j]} \leq \delta.$ This implies that there exists $T_{\delta_{0}}$ such that for all $t \geq
T_{\delta_{0}}$, $$J(z^{[i]}(t)) \leq \delta_{0} + \bar{J}^{[i]} < \bar{J}^{[j]}.$$ Additionally, since the objective functions are non-increasing, it follows that for any time instant $t' \geq 0$, $J(z^{[j]}(t')) \geq \bar{J}^{[j]}$. Thus, for all $t \geq T_{\delta_{0}}$ and all $t' \geq 0$, $$\begin{aligned}
\label{eqn.Agreement.Cond1} J(z^{[i]}(t)) < J(z^{[j]}(t')).\end{aligned}$$
Pick now $t_{0} \geq T_{\delta_{0}}$. For all $\tau \geq 0$ define now an index set $I_{\tau}$ as follows: Set $I_{0} = \{i\}$ and for any $\tau \geq 0$ define $I_{\tau}$ by adding to $I_{\tau-1}$ all indices $k$ for which there exist some $l \in I_{\tau-1}$ such that $(k,l) \in E(t_{0}+\tau)$. Since, by assumption ${\mathcal}{G}_{c}^{\infty}(t_{0})$ is strongly connected, the set $I_{\tau}$ will eventually include all indices $1,\ldots,n$, and in particular there is $\tau^{*}$ such that $j \in I_{\tau^{*}}$. The algorithm is such that for all $l \in I_{\tau}$, $ J(z^{[l]}(t_{0}+\tau))
\leq J(z^{[i]}(t_{0})) $ and thus $$\begin{aligned}
\label{eqn.Agreement.Cond2}
J(z^{[j]}(t_{0}+\tau^{*})) \leq
J(z^{[i]}(t_{0})).\end{aligned}$$ But contradicts , proving that $ \bar{J}^{[i]} = \bar{J}^{[2]} = \cdots = \bar{J}^{[n]} =:
\bar{J}.$ Thus, it must hold that for all $i,j \in \{1,\ldots,n\}$, $\lim_{t\rightarrow \infty} | J(z^{[i]}(t)) - J(z^{j}(t))| \rightarrow 0$. From the strict concavity of $J(\cdot)$ follows that $|J(z^{[i]}(t)) -
J(z^{[j]}(t))| > \sigma \|z^{[i]}(t) - z^{[j]}(t)\|_{2}^{2},$ for some $\sigma
> 0$. Therefore, $\lim_{t\rightarrow \infty }\|z^{[i]}(t) - z^{[j]}(t)\|_{2}
\rightarrow \infty$, which proves the theorem. $ \hfill \blacksquare$\
### Proof of Theorem \[thm.AsymptoticConvergence\]
It follows from Lemma \[prop.Agreement\] that the query points of all processors converge to the same query point, i.e., $\bar{z}^{[i]} = \bar{z}$ for all processors $i$. Now, we can conclude from Lemma \[prop.Convergence\] that $\bar{z} \in {\mathcal}{Z}_{i}$ for all $i$ and thus $\bar{z} \in {\mathcal}{Z}$. It follows now from Lemma \[prop.TechIssues\], part (ii), that $\bar{z}$ is an optimal solution to . It remains to show that $\bar{z}$ is the optimal solution with minimal 2-norm. Let $z^{*}$ be the optimal solution with minimal 2-norm. Then there exists an $\epsilon > 0$ such that the parameterized objective function satisfies ${J_{\epsilon}}(z^{*}) > {J_{\epsilon}}(z)$ for all $z \in
{\mathcal}{Z}$ and ${J_{\epsilon}}(z^{[i]}(t)) \geq {J_{\epsilon}}(z^{*})$ for all $t$. With the same argumentation used for Lemma \[prop.TechIssues\], part (ii), we conclude that $\bar{z}$ is the unique solution maximizing ${J_{\epsilon}}(\cdot)$ over ${\mathcal}{Z}$, i.e., $\bar{z}$ is the optimal solution to with minimal 2-norm. $\hfill \blacksquare$\
[^1]: M. B[ü]{}rger and F. Allg[ö]{}wer thank the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. The research of G. Notarstefano has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 224428 (CHAT).
[^2]: M. B[ü]{}rger and F. Allg[ö]{}wer are with the Institute for Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 70550 Stuttgart, Germany, ` {mathias.buerger, frank.allgower}@ist.uni-stuttgart.de`.
[^3]: Giuseppe Notarstefano is with the Department of Engineering, University of Lecce, Via per Monteroni, 73100 Lecce, Italy, `[email protected].`
[^4]: We use in the simulations a step-size $\rho = 200$, see [@Boyd2010 Chapter 7] for the notation. Please note that the choice of the step-size of the ADMM method has to be done heuristically. We have selected the step-size as the best step-size we found experimentally for the smallest problem scenario $n=20$. Although the convergence speed of the ADMM method might improve with another step-size, in our experience most heuristic choices led to a significant deterioration of the performance.
|
---
abstract: 'In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMM-like algorithm that we refer to as relaxed ADMM (R-ADMM) via a generalized *Peaceman-Rachford Splitting* operator on the Lagrange dual formulation of the original optimization problem. This specific algorithm depends on two parameters, namely the averaging coefficient $\alpha$ and the augmented Lagrangian coefficient $\rho$. We show that by setting $\alpha=1/2$ we recover the standard ADMM algorithm as a special case of our algorithm. Moreover, by properly manipulating the proposed R-ADMM, we are able to provide two alternative ADMM-like algorithms that present easier implementation and reduced complexity in terms of memory, communication and computational requirements. Most importantly the latter of these two algorithms provides the first ADMM-like algorithm which has guaranteed convergence even in the presence of lossy communication under the same assumption of standard ADMM with lossless communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over cycle graphs and random geometric graphs subject to i.i.d. random packet losses.'
author:
- 'N. Bastianello$^\dagger$, M. Todescato$^\ddagger$, R. Carli$^\dagger$, L. Schenato$^\dagger$ [^1] [^2]'
bibliography:
- './IEEEabrv.bib'
- './references.bib'
title: '**Distributed Optimization over Lossy Networks via Relaxed Peaceman-Rachford Splitting: a Robust ADMM Approach** '
---
distributed optimization, ADMM, operator theory, splitting methods, Peaceman-Rachford operator
Introduction {#sec:intro}
============
From classical control theory to more recent Machine Learning applications, many problems can be cast as optimization problems [@Slavakis:14] and, in particular, as large-scale optimization problems given the advent of Internet-of-Things we are witnessing with its ever-increasing growth of large-scale cyber-physical systems. Hence, stemming from classical optimization theory, in order to break down the computational complexity, parallel and distributed optimization methods have been the focus of a wide branch of research [@BD:TP:1989]. Within this vast topic, typical applications, going under the name of *distributed consensus optimization*, foresee distributed computing nodes to communicate in order to achieve a desired common goal. More formally, the distributed nodes seek to $$\min_{x}\sum_{i=1}^Nf_i(x)$$ where, usually, each $f_i$ is owned by one node only. Toward this application among very many different optimization algorithms explored in past as well as in current literature, e.g. subgradient methods [@BJ:MR:MJ:2010], the well known Alternating Direction Method of Multipliers (ADMM), first introduced in [@glowinski1975approximation] and [@gabay1976dual], is recently receiving an ever-increasing interest because of its numerical efficiency and its natural structure which makes it well-suited for distributed and parallel computing. In particular, the relatively recent monograph [@boyd2011distributed] reveals the ADMM in detail presenting a broad set of selected applications to which ADMM is suitably applied. For a wider set of applications together with some convergence results we refer the interested reader to [@fukushima1992application; @eckstein1994some; @eckstein1992douglas; @EG:AT:ES:MJ:2015].\
While ADMM can be proficiently applied to distributed setups, rigorous convergence results are usually provided only in scenarios characterized by synchronous updates and lossless communications.However, practically it is rarely possible and often difficult to ensure synchronization and communication reliability among computing nodes. And even when this is possible via specific communication protocols, it is clear how the impossibility to deal with asynchronous and lossy updates would majorly compromise the algorithm applicability.\
Hence, an extensive body of work has been devoted to overcoming this limitation by adapting the ADMM to operate in an asynchronous fashion. Among the first steps in this direction, [@iutzeler2013asynchronous] proves convergence when only one randomly selected coordinate is updated at each iteration. Similarly, [@wei20131] suggests to update only the variables related to a subset of constraints randomly selected at each iteration, showing convergence of the algorithm with a rate of $O(1/k)$. To deal with asynchronous updates, a master-slave architecture is proposed in [@zhang2014asynchronous]. The more recent [@bianchi2016coordinate] extends the formulation introduced in [@iutzeler2013asynchronous] to allow the update of a subset of coordinates at each instant. In [@chang2016asynchronous], in view of large-scale optimization, the convergence rate of a partially asynchronous ADMM – i.e., subject to a maximum allowed delay – is studied. Finally [@peng2016arock] defines a framework for asynchronous operations used to solve a broad class of optimization problems and showing how to derive an asynchronous ADMM formulation.\
Conversely to the above works that deal with asynchronous updates, to the best of our knowledge, no work explicitly focuses on the robustness of ADMM to packet losses. Yet, to set the stage for the analysis of robustness of the ADMM algorithm to losses in the communication, we resort to a different body of literature on operator theory. Here, the underlying idea is to convert optimization problems into the problem of finding the fixed points of suitable nonexpansive operators [@bauschke2011convex]. However, the mere application of the so-called *proximal point algorithm* (PPA) – introduced in [@rockafellar1976monotone] and the later [@parikh2014proximal] – to look for the fixed points can be unwieldy in complex optimization problems. Hence, particular credits have been given to *splitting methods* which exploit the problem’s structure to break it in smaller and more manageable pieces. It is in the framework of splitting operators, and in particular the well recognized Peaceman-Rachford (PRS) [@peaceman1955numerical] and Douglas-Rachford (DRS) [@douglas1956numerical; @lions1979splitting] splitting, that the ADMM comes into place. Indeed, the classical formulation of the ADMM naturally arises as application of the DRS to the Lagrange dual problem of the original optimization problem [@eckstein2012augmented]. For further details on a variety of splitting operators and their application in asynchronous setups we refer to [@davis2016convergence] and [@hannah2016unbounded], respectively.\
In this paper we present and analyze different formulation for the ADMM algorithm. We are particularly interested to the broad class of distributed consensus optimization problems. Our final goal is to present a novel robustness result in scenarios characterized by synchronous but possibly lossy updates among distributed nodes. To achieve our result we start by considering a prototypical optimization problem assuming reliable loss-free communication. In this case, by leveraging the general framework arising from the Krasnosel’skii-Mann (KM) iteration for averaging operators [@krasnosel1955two; @mann1953mean], we derive a relaxed version of the ADMM (R-ADMM). Next, we draw the attention to the problem of interest, i.e., distributed consensus optimization. We first present the natural algorithmic implementation of the R-ADMM tailored for the problem. Then we propose two different implementations which are particularly favorable for storage and communication purposes. Moreover, the latter turns out extremely advantageous and yet robust in the presence of lossy communication. As natural byproduct we obtain a comprehensive and self-contained overview on the algorithm and a plethora of possible practical implementations.\
The remainder of the paper is organized as follows. Section \[sec:operators-background\] presents the necessary background on splitting operators. Section \[sec:ADMMandRADMM\] reviews the classical ADMM algorithm and its generalized version. Section \[sec:distributed\_consensus\] focuses on the analysis of distributed consensus optimization. Section \[sec:simulation\] collects some numerical simulations and Section \[sec:conclusions\] concludes the paper. The technical proofs can be found in the Appendices.
Background on Splitting Operators {#sec:operators-background}
=================================
This Section introduces some background on operator theory on Hilbert spaces and, in particular, on nonexpansive operators. The interest for operator theory stems from the fact that a convex optimization problem can be cast into the problem of finding the fixed point(s) of a suitable nonexpansive operator $T$ [@davis2016convergence; @bauschke2011convex], that is the points $x^*$ such that $Tx^*=x^*$.
Definitions and Properties {#subsec:definitions}
--------------------------
Let $\mathcal{X}$ be a Hilbert space, an operator $T:\mathcal{X}\rightarrow\mathcal{X}$ is said to be *nonexpansive* if it has unitary Lipschitz constant, *i.e.* it verifies $\|Tx-Ty\|\leq\|x-y\|$ for any two $x,y\in\mathcal{X}$.
Let $\mathcal{X}$ be a Hilbert space, $T:\mathcal{X}\rightarrow\mathcal{X}$ a nonexpansive operator and $\alpha\in(0,1)$. We define the *$\alpha$-averaged operator* $T_\alpha$ as $T_\alpha=(1-\alpha)I+\alpha T$, where $I$ is the identity operator on $\mathcal{X}$.
Notice that $\alpha$-averaged operators are also nonexpansive, indeed nonexpansive operators are $1$-averaged. Moreover, the $\alpha$-averaged operator $T_\alpha$ has the same fixed points of $T$ [@bauschke2011convex].
Let $\mathcal{X}$ be a Hilbert space and $f:\mathcal{X}\rightarrow\mathbb{R}\cup\{+\infty\}$ be a closed, proper and convex function. We define the *proximal operator* of $f$ with penalty $\rho>0$, $\operatorname{prox}_{\rho f}:\mathcal{X}\rightarrow\mathcal{X}$, as $$\operatorname{prox}_{\rho f}(y)=\operatorname*{arg\,min}_{x\in\mathcal{X}}\left\{f(x)+\frac{1}{2\rho}\|x-y\|^2\right\}.$$ Moreover, we define the relative *reflective operator* as $\operatorname{refl}_{\rho f}=2\operatorname{prox}_{\rho f}-I$.
It can be seen that the proximal operator is $1/2$-averaged and the reflective operator is nonexpansive [@bauschke2011convex].
Finding the Fixed Points of Nonexpansive Operators {#subsec:fixpoints}
--------------------------------------------------
One of the prototypical algorithm for finding the fixed points of $T$ is the Krasnosel’skii-Mann (KM) iteration [@bauschke2011convex] $$\label{eq:km-iteration}
x(k+1)=T_\alpha x(k)=(1-\alpha)x(k)+\alpha Tx(k)$$ where in general the step size $\alpha$ can be time-varying. Notice that the KM iteration is equivalent to $x(k+1)=x(k)-\alpha Sx(k)$, where $S=I-T$, that is, finding the fixed points of $T$ coincides with finding the zeros of $S$.\
Now, consider the general unconstrained problem $$\label{eq:MinProblem}
\min_{x\in\mathcal{X}} \{f(x)+g(x)\},$$ where $f,g$ are closed proper and convex not necessarily smooth functions. Further, assume that simultaneous minimization of $f+g$ is unwieldy while minimizing $f$ and $g$ separately is manageable. In this case, to compute the solution of we can apply the KM iteration to the *Peaceman-Rachford Splitting operator*, defined as (see [@davis2016convergence; @peng2016arock]), $$T_{PRS}=\operatorname{refl}_{\rho f}\circ\operatorname{refl}_{\rho g}.$$ As show in [@bauschke2011convex], the iteration $$\label{eq:KM_T_PRS}
x(k+1)=(1-\alpha)x(k)+\alpha T_{PRS}x(k)$$ can be conveniently implemented by the following updates $$\begin{aligned}
\psi(k)&=\operatorname{prox}_{\rho g}(z(k))\label{eq:prs-1}\\
\xi(k)&=\operatorname{prox}_{\rho f}(2\psi(k)-z(k))\label{eq:prs-2}\\
z(k+1)&=z(k)+2\alpha(\xi(k)-\psi(k))\label{eq:prs-3}\end{aligned}$$ where $\psi, \xi, z$ are suitable auxiliary variables while the optimal solution $x^*$ to is recovered from the limit $z^*$ of the iterate $z(k)$ by computing $x^*=\operatorname{prox}_{\rho g}(z^*)$. This algorithm goes under the name of *relaxed Peaceman-Rachford splitting* (R-PRS), where “relaxed” denotes the fact that the KM iteration is $\alpha$-averaged. In case $\alpha=1$ we recover the classic *Peaceman-Rachford splitting* introduced in [@peaceman1955numerical], and in case $\alpha=1/2$ we recover the *Douglas-Rachford splitting* [@douglas1956numerical].\
The important feature of splitting schemes such as the R-PRS is that they divide the computational load of iterate into smaller subproblems that can be solved more efficiently.
From the ADMM to the Relaxed ADMM {#sec:ADMMandRADMM}
=================================
In this Section, we first review the popular ADMM algorithm [@gabay1976dual; @boyd2011distributed], then we introduce the more general *relaxed* ADMM (R-ADMM) algorithm, and compare the two methods.
The ADMM Algorithm {#subsec:ADMM}
------------------
Consider the following optimization problem $$\begin{aligned}
\label{eq:primal-problem}
\begin{split}
&\min_{x\in\mathcal{X},y\in\mathcal{Y}} \{f(x)+g(y)\}\\
&\text{s.t.}\ Ax+By=c
\end{split}\end{aligned}$$ where $\mathcal{X}$ and $\mathcal{Y}$ are Hilbert spaces, $f:\mathcal{X}\rightarrow\mathbb{R}\cup\{+\infty\}$ and $g:\mathcal{Y}\rightarrow\mathbb{R}\cup\{+\infty\}$ are closed, proper and convex functions[^3].\
To solve problem via the ADMM algorithm, we first define the *augmented Lagrangian* as $$\begin{aligned}
\label{eq:augmented-lagr}
\begin{split}
\mathcal{L}_{\rho}(x,y;w)&=f(x)+g(y)-w^\top\left(Ax+By-c\right)\\&+\frac{\rho}{2}\|Ax+By-c\|^2
\end{split}\end{aligned}$$ where $\rho>0$ and $w$ is the vector of Lagrange multipliers. The ADMM algorithm consists in keeping alternating the following update equations $$\begin{aligned}
y(k+1)&=\operatorname{arg\,min}_y \mathcal{L}_\rho(x(k),y;w(k))\label{eq:admm-1}\\
w(k+1)&=w(k)-\rho(Ax(k)+By(k+1)-c)\label{eq:admm-2}\\
x(k+1)&=\operatorname{arg\,min}_x \mathcal{L}_\rho(x,y(k+1);w(k+1))\label{eq:admm-3}.\end{aligned}$$ Notice that the above formulation is equivalent to the one proposed in [@boyd2011distributed] except for a change in the order of the updates which however does not affect the convergence properties of the algorithm. Moreover, the ADMM algorithm is provably shown to converge to the optimal solution of for any $\rho>0$ assuming that $\mathcal{L}_0$ has a saddle point [@boyd2011distributed].\
We conclude this section by stressing the following fact. While the ADMM in its classical form – is typically presented as an augmented Lagrangian method computed with respect to the primal problem , the algorithm naturally arises from the application of the DRS to the Lagrange dual of problem . This will be made clear in the next section.
The Relaxed ADMM {#subsec:R-ADMM}
----------------
The Relaxed ADMM algorithm can be derived applying the R-PRS method described in Section \[sec:operators-background\] to the Lagrange dual of problem , that is to $$\label{eq:dual-problem}
\min_{w\in\mathcal{W}}\left\{d_f(w)+d_g(w)\right\}$$ where $$\begin{aligned}
d_f(w)&=f^*(A^\top w)\\
d_g(w)&=g^*(B^\top w)-w^\top c,\end{aligned}$$ and $f^*$, $g^*$ are the convex conjugates of $f$ and $g$[^4]. The derivation of problem can be found in [@davis2016convergence; @peng2016arock].\
Observe that, given the structure of problem (i.e., proper closed and convex functions and linear constraints) there is no duality gap and, in turn, the optimal solutions of and of attain the same optimal value.\
The motivation for dealing with the Lagrange dual problem relies on the fact that the minimization in is performed over a single variable, thus allowing for the use of the R-PRS algorithm described in , and .\
Lemma $11$ in [@davis2016convergence] shows that the update and the update , applied to the dual problem, can be conveniently computed by, respectively, $$\begin{aligned}
y(k)&=\operatorname*{arg\,min}_y\left\{g(y)-z^\top(k)(By-c)+\frac{\rho}{2}\|By-c\|^2\right\}\nonumber\\
\psi(k)&=z(k)-\rho(By(k)-c)\label{eq:psi-update}
\end{aligned}$$ and $$\begin{aligned}
x(k)&=\operatorname*{arg\,min}_x\left\{f(x)-(2\psi(k)-z(k))^\top Ax+\frac{\rho}{2}\|Ax\|^2\right\}\nonumber\\
\xi(k)&=2\psi(k)-z(k)-\rho Ax(k)\label{eq:xi-update}\end{aligned}$$ The so called *relaxed ADMM* algorithm (in short R-ADMM) consists in applying iteratively the set of five equations given by the two equations in , the two equations in and equation .\
It is worth stressing a fundamental difference regarding the auxiliary variables $z$ in – and those used in ,. Indeed, when implementing the R-PRS –, the KM iteration is applied directly to the primal problem . Hence $z$ has the same dimension of the primal variable $x$. Conversely, the R-ADMM as in , and , is derived on the dual problem . Hence, in this case $z$ has dimension of the constraints. This will become more clear later when dealing with distributed consensus optimization problems.\
Next, we derive a more compact formulation of the R-ADMM, that shows clearly the relation with the popular ADMM algorithm described in , and .\
First of all, by adding $\psi(k)$ on both sides of the second equation in , we obtain $$2\psi(k)-z(k)=\psi(k)-\rho(By(k)-c)\label{eq:equality-1}$$ and substituting this equation in we get $$\xi(k)=\psi(k)-\rho(Ax(k)+By(k)-c)\label{eq:equality-2}.$$ Getting $z(k)$ from and, substituting back into the first equation in we obtain $$\begin{aligned}
\begin{split}
x(k)&=\operatorname*{arg\,min}_x\{f(x)-\psi^\top(k)(Ax+By(k)-c)\\&+\frac{\rho}{2}\|Ax+By(k)-c\|^2\}.
\end{split}\end{aligned}$$ where terms independent of $x$ were added. By substituting and in we get $$z(k+1)=\psi(k)-\rho Ax(k)-\rho(2\alpha-1)(Ax(k)+By(k)-c)\label{eq:z-update}$$ and by plugging into the first equation in and adding some terms that do not depend on $y$, we get $$\begin{aligned}
\begin{split}
y&(k+1)=\operatorname*{arg\,min}_y\{g(y)\\
&-\psi^\top(k)(Ax(k)+By-c)+\rho\|By-c\|^2\\
&+\rho\left[Ax(k)+(2\alpha-1)(Ax(k)+By(k)-c)\right]^\top(By-c)\}.
\end{split}\end{aligned}$$ Finally, recalling the defintion of augmented Lagrangian and renaming $\psi$ as $w$, we arrive to the three updates that represent the R-ADMM algorithm $$\begin{aligned}
\begin{split}
y(k+1)&=\operatorname*{arg\,min}_y\{\mathcal{L}_\rho(x(k),y;w(k))\\&+\rho(2\alpha-1)\langle By,(Ax(k)+By(k)-c)\rangle\}\label{eq:r-admm-1}
\end{split}\\
\begin{split}
w(k+1)&=w(k)-\rho(Ax(k)+By(k+1)-c)\\&-\rho(2\alpha-1)(Ax(k)+By(k)-c)\label{eq:r-admm-2}
\end{split}\\
x(k+1)&=\operatorname*{arg\,min}_x\mathcal{L}_\rho(x,y(k+1);w(k+1)).\label{eq:r-admm-3}\end{aligned}$$ The ADMM algorithm in – can be recovered from this formulation of the R-ADMM by setting $\alpha=1/2$, which cancels the additional terms weighted by $2\alpha-1$.\
It is of notice that the R-ADMM has two tunable parameters, $\rho$ and $\alpha$, against the only one of the ADMM, $\rho$, which is the cause of the greater reliability of the R-ADMM.
Figure \[fig:relationships-algorithms\] depicts the relationships between the splitting operators derived from the relaxed Peaceman-Rachford, and the classic and relaxed ADMM.
(r-prs) at (0,0) [R-PRS]{}; (prs) at (2.5,2.5) [PRS]{}; (drs) at (2.5,-2.5) [DRS]{}; (r-admm) at (-2.5,2.5) [R-ADMM]{}; (admm) at (-2.5,-2.5) [ADMM]{};
(r-prs) edge node\[right\] [$\alpha=1/2$]{} (drs) (r-prs) edge node\[right\] [$\alpha=1$]{} (prs) (r-prs) edge node\[align=center\] [applied to\
Lagrange\
dual of ]{} (r-admm) (drs) edge node\[align=center\] [applied to Lagrange\
dual of ]{} (admm) (r-admm) edge node\[left\] [$\alpha=1/2$]{} (admm);
Distributed Consensus Optimization {#sec:distributed_consensus}
==================================
This Section introduces the distributed consensus convex optimization problem that we are interested in, and the solutions obtained by applying the R-ADMM algorithm.
Problem Formulation {#subsec:distributed_problem}
-------------------
Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ be a graph, with $\mathcal{V}$ the set of $N$ vertices, labeled $1$ through $N$, and $\mathcal{E}$ the set of undirected edges. For $i \in \mathcal{V}$, by $\mathcal{N}_i$ we denote the set of neighbors of node $i$ in $\mathcal{G}$, namely, $$\mathcal{N}_i =\left\{j \in V \,:\, (i,j) \in \mathcal{E} \right\}.$$ We are interested in solving the following optimization problem $$\begin{aligned}
\label{eq:opt_problem}
\begin{split}
&\min_{x}\sum_{i=1}^Nf_i(x)
\end{split}\end{aligned}$$ where $f_i:\mathbb{R}^n \rightarrow\mathbb{R}\cup\{+\infty\}$ are closed, proper and convex functions and where $f_i$ is known only to node $i$. In the following we denote by $x^*$ the optimal solution of .\
Observe that can be equivalently formulated as $$\begin{aligned}
\label{eq:distributed-primal}
\begin{split}
&\min_{x_i,\forall i}\sum_{i=1}^Nf_i(x_i)\\
&\text{s.t.}\ x_i=x_j,\ \forall (i,j)\in\mathcal{E}
\end{split}\end{aligned}$$ By introducing for each edge $(i,j)\in\mathcal{E}$ the two *bridge variables* $y_{ij}$ and $y_{ji}$, the constraints in can be rewritten as $$\begin{aligned}
\begin{split}
& x_i=y_{ij}\\
& x_j=y_{ji}\\
& y_{ij}=y_{ji}
\end{split}\ \ \ \forall (i,j)\in\mathcal{E}.\end{aligned}$$ Defining $\mathbf{x}=[x_1^\top,\ldots,x_N^\top]^\top$, $f(\mathbf{x})=\sum_if_i(x_i)$, and stacking all bridge variables in $\mathbf{y} \in \mathbb{R}^{n |\mathcal{E}|}$, we can reformulate the problem as $$\begin{aligned}
& \min_{\mathbf{x}} f(\mathbf{x})\\
& \text{s.t.}\ \ A\mathbf{x}+\mathbf{y}=0\\
& \mathbf{y}=P\mathbf{y}\end{aligned}$$ for a suitable $A$ matrix and with $P$ being a permutation matrix that swaps $y_{ij}$ with $y_{ji}$. Making use of the indicator function $\iota_{(I-P)}(\mathbf{y})$ which is equal to 0 if $(I-P)\mathbf{y}=0$, and $+\infty$ otherwise, we can finally rewrite problem as $$\begin{aligned}
\label{eq:primal-indicator-f}
\begin{split}
& \min_{\mathbf{x},\mathbf{y}}\left\{f(\mathbf{x})+\iota_{(I-P)}(\mathbf{y})\right\}\\
& \text{s.t.}\ \ A\mathbf{x}+\mathbf{y}=0.
\end{split}\end{aligned}$$ In next Section we apply the R-ADMM algorithm described in Section \[subsec:R-ADMM\] to the above problem.
R-ADMM for Convex Distributed Optimization {#subsec:distributed_R-ADMM}
------------------------------------------
In this section we employ , and to solve problem . To do so we introduce the dual variables $w_{ij}$ and $w_{ji}$ which are associated to the constraints $x_i=y_{ij}$ and $x_j=y_{ji}$, respectively. The resulting algorithm is described in Algorithm \[alg:r-admm-three-eqs\]. Observe that R-ADMM applied to is amenable of a *distributed* implementation, in the sense that node $i$ stores in memory only the variables $x_i$, $y_{ij}$, $w_{ij}, j\in\mathcal{N}_i$, and updates these variables exchanging information only with its neighbors, i.e, with nodes in $\mathcal{N}_i$.
$k\leftarrow0$
Notice that, in the update of $x_i$, the term $w_{ij}(k+1)-\rho y_{ij}(k+1)$ can be rewritten, using the previous updates, as a function of the variables computed at time $k$ only. Therefore, only one round of transmissions is necessary.\
The above implementation of the R-ADMM is quite straightforward and popular but very unwieldy due to the fact that, depending on the number of neighbors, there might be nodes which need to store, update and transmit a large number of variables. The derivation of Algorithm \[alg:r-admm-three-eqs\] is reported in Appendix \[app:derivation-alg-3eqs\].
In the following we provide an alternative algorithm which is derived directly from the application of the set of five equations in , and to the dual of problem . Notice, that since the vector $z$ has the same dimension of the vector $w$, this implies the presence of also the variables $z_{ij}$ and $z_{ji}$ for any $(i,j) \in \mathcal{E}$.\
We have the following Proposition, which is proved in Appendix \[app:proof-prop-1\].
\[pr:r-admm-five-eqs\] The implementation of the R-ADMM algorithm described in the set of five equations given in , and applied to the dual of problem , reduces to alternating between the following two updates $$\begin{aligned}
\label{eq:x-update-distributed}
& x_i(k)=\operatorname*{arg\,min}_{x_i}\left\{f_i(x_i)-\left(\sum_{j\in\mathcal{N}_i}z_{ji}^\top(k)\right) x_i\right.\\
&\,\,\,\qquad\qquad\qquad \qquad \qquad \qquad\qquad \Biggl.+\,\frac{\rho}{2}|\mathcal{N}_i|\|x_i\|^2 \Biggr\},\nonumber
$$ for all $i \in V$, and $$\begin{aligned}
\label{eq:z-update-distributed}
\begin{split}
& z_{ij}(k+1)=(1-\alpha)z_{ij}(k)-\alpha z_{ji}(k)+2\alpha\rho x_i(k)\\
& z_{ji}(k+1)=(1-\alpha)z_{ji}(k)-\alpha z_{ij}(k)+2\alpha\rho x_j(k)
\end{split}\end{aligned}$$ for all $(i,j) \in \mathcal{E}$.
Observe that the reformulation of , and as in Proposition \[pr:r-admm-five-eqs\] is possible for the particular structure of Problem and, in particular, for the structure of the constraints $Ax+y=0$. In general, given a set of constraints $Ax+By=c$ being $A$, $B$ and $c$ generic matrices and vector, such reformulation might not be possible.
The previous proposition naturally suggests an alternative distributed implementation of the R-ADMM Algorithm \[alg:r-admm-three-eqs\], in which each node $i$ stores in its local memory the variables $x_i$ and $z_{ij},j\in\mathcal{N}_i$. Then, at each iteration of the algorithm, each node $i$ first collects the variables $z_{ji},j\in\mathcal{N}_i$; second, updates $x_i$ and $z_{ij}$ according to and the first of , respectively; finally, it sends $z_{ij}$ to $j\in\mathcal{N}_i$.\
Differently to the natural implementation just briefly described, we present a slightly different implementation building upon the observation that each node $i$, to update $x_i$ as in requires the variables $z_{ji}$ rather than $z_{ij}$ for $j\in\mathcal{N}_i$. Consequently, we assume node $i$ stores in its memory and is in charge for the update of $z_{ji},j\in\mathcal{N}_i$. The implementation is described in Algorithm \[alg:smart-distributed-r-admm\].
$k\leftarrow0$
As we can see, both Algorithms \[alg:r-admm-three-eqs\] and \[alg:smart-distributed-r-admm\] need a single round of transmissions at each time $k$. However, they differ for the number of packets that each node has to transmit and for the number of variables that a node has to update. Table \[tab:variables-counts\] reports the comparison between the two algorithms.
Alg. \[alg:r-admm-three-eqs\] Alg. \[alg:smart-distributed-r-admm\]
----------------- ------------------------------- ---------------------------------------
Update and Send $2|\mathcal{N}_i|+1$ $|\mathcal{N}_i|+1$
Store $3|\mathcal{N}_i|$ $|\mathcal{N}_i|$
: Comparison of R-ADMM implementations.[]{data-label="tab:variables-counts"}
Therefore, exploiting the auxiliary $z$ variables we have obtained an algorithm with smaller memory and computational requirements.\
We conclude this section by stating the convergence properties of Algorithms \[alg:r-admm-three-eqs\], \[alg:smart-distributed-r-admm\]. The proof can be found in Appendix \[app:proof-convergence\].
\[prop:convergence\] Consider Algorithm \[alg:smart-distributed-r-admm\]. Let $(\alpha, \rho)$ be such that $0<\alpha <1$ and $\rho >0$. Then, for any initial conditions, the trajectories $k \to x_i(k)$, $i \in V$, generated by Algorithm \[alg:smart-distributed-r-admm\], converge to the optimal solution of , i.e., $$\lim_{k \to \infty} x_i(k) = x^*, \qquad \forall i \in \mathcal{V},$$ for any $x_i(0)$ and $z_{ji}(0)$, $j \in \mathcal{N}_i$. The same result holds true also for Algorithms \[alg:r-admm-three-eqs\].
Distributed R-ADMM over lossy networks {#sec:robustADMM}
======================================
The distributed algorithms illustrated in the previous section work under the standing assumption that the communication channels are reliable, that is, no packet losses occur. The goal of this section is to relax this communication requirement and, in particular, to show that Algorithm \[alg:smart-distributed-r-admm\] still converges, under a probabilistic assumption on communication failures which is next stated.
\[ass:lossy\] During any iteration of Algorithm \[alg:smart-distributed-r-admm\], the communication from node $i$ to node $j$ can be lost with some probability $p$.
In order to describe the communication failure more precisely, we introduce the family of independent binary random variables $L_{ij}(k)$, $k=0,1,2,\ldots$, $i \in \mathcal{V}$, $j \in \mathcal{N}_i$, such that[^5] $$\mathbb{P}\left[L_{ij}=1\right]=p, \qquad \mathbb{P}\left[L_{ij}=0\right]=1-p.$$ We emphasize the fact that independence is assumed among all $L_{ij}(k)$ as $i, j$ and $k$ vary. If the packet transmitted, during the $k$-th iteration by node $i$ to node $j$ is lost, then $L_{ij}(k)=1$, otherwise $L_{ij}(k)=0$.\
In this lossy scenario, Algorithm \[alg:smart-distributed-r-admm\] is modified as shown in Algorithm \[alg:robust-smart-distributed-r-admm\].
$k\leftarrow0$
In this case, at $k$-th iteration node $i$ updates $x_i$ as in . Then, for $j \in \mathcal{N}_i$, it computes $q_{i \to j}$ as in and transmits it to node $j$. If node $j$ receives $q_{i \to j}$, then it updates $z_{ij}$ as $z_{ij}(k+1)=(1-\alpha)z_{ij}(k)+ \alpha q_{i \to j}$, otherwise $z_{ij}$ remains unchanged, i.e., $z_{ij}(k+1)=z_{ij}(k)$. This last step can be compactly describes as $$\begin{aligned}
z_{ij}(k+1)&=L_{ij}(k)z_{ij}(k) + \\
&\qquad +\left(1-L_{ij}(k)\right) \,\left( (1-\alpha)z_{ij}(k)+ \alpha q_{i \to j}\right)\end{aligned}$$ We have the following Proposition, whose proof is reported in Appendix \[app:convergence-lossy\].
\[prop:convergence\_lossy\] Consider Algorithm \[alg:robust-smart-distributed-r-admm\] working under the scenario described in Assumption \[ass:lossy\]. Let $(\alpha, \rho)$ be such that $0<\alpha <1$ and $\rho >0$. Then, for any initial conditions, the trajectories $k \to x_i(k)$, $i \in \mathcal{V}$, generated by Algorithm \[alg:robust-smart-distributed-r-admm\], converge almost surely to the optimal solution of , i.e., $$\lim_{k \to \infty} x_i(k) = x^*, \qquad \forall i \in \mathcal{V},$$ with probability one, for all $i \in \mathcal{V}$, for any $x_i(0)$ and $z_{ji}(0)$, $j \in \mathcal{N}_i$.
We stress that the underling idea behind the result of Proposition \[prop:convergence\_lossy\] relies on rewriting Algorithm \[alg:robust-smart-distributed-r-admm\] as a stochastic KM iteration and then to resort to a different set of methodological tools from probabilistic analysis [@bianchi2016coordinate].
We have restricted the analysis to the case of synchronous communication since we were mainly interested in investigating the algorithm performance in the presence of packet losses. The practically more appealing asynchronous scenario will be the focus of future research.
Interestingly, while the robustness result that we provide in the lossy scenario holds true for Algorithm \[alg:robust-smart-distributed-r-admm\], we cannot prove the same for Algorithms \[alg:r-admm-three-eqs\] which, in the case of synchronous and reliable communications, is instead characterized by the same convergent behavior despite of the different communication and memory requirements.
Observe that Proposition \[prop:convergence\], for the case of reliable communications, and Proposition \[prop:convergence\_lossy\], regarding the lossy scenario, share exactly the same region of convergence in the space of the parameters. This means that Algorithm \[alg:smart-distributed-r-admm\] remains provably convergent if $0<\alpha<1$ and $\rho>0$ in both cases. However, observe that the result is not *necessary and sufficient* and, in particular, the convergence might hold also for value of $\alpha\geq 1$. Indeed, in the simulation Section \[sec:simulation\] we show that, for the case of quadratic functions $f_i,\ i\in\mathcal{V}$, the region of attraction in parameter space is larger. Moreover, despite what suggested by the intuition, the larger the packet loss probability $p$, the larger the region of convergence. However, this increased region of stability is counterbalanced by a slower convergence rate of the algorithm.
Simulations {#sec:simulation}
===========
In this section we provide some experimental simulations to test the proposed R-ADMM Algorithm \[alg:robust-smart-distributed-r-admm\] to solve distributed consensus optimization problems . We are particularly interested in showing the algorithm performances in the presence of packet losses in the communication among neighboring nodes. To simplify the numerical analysis we restrict to the case of quadratic cost functions of the form $$f_i(x_i)=a_ix_i^2 + b_ix_i + c_i$$ where, in general, the quantities $a_i,b_i,c_i\in\mathbb{R}$ are different for each node $i$. In this case the update of the primal variables becomes linear and, in particular, Eq. reduces to $$\begin{aligned}
&x_i(k)=\frac{\sum_{j\in\mathcal{N}_i}z_{ji}(k)-b_i}{2a_i+\rho|\mathcal{N}_i|}\, .
$$ We consider the family of random geometric graphs with $N=10$ and communication radius $r=0.1$\[p.u.\] in which two nodes are connected if and only if their relative distance is less that $r$. We perform a set of 100 Monte Carlo runs for different values of packet losses probability $p$, step size $\alpha$ and penalty parameters $\rho$.\
First of all, for different values of packet loss probability $p$ and for fixed values of step size $\alpha=1$ and penalty $\rho=1$, Figure \[fig:evolution\_different\_losses\] shows the evolution of the relative error $$\log\frac{\|x(k)-x^*\|}{\|x^*\|}$$ computed with respect to the unique minimizer $x^*$ and averaged over 100 Monte Carlo runs. As expected, the higher the packet loss probability, the smaller the rate of convergence. Indeed, failures in the communication among neighboring nodes negatively affect the computations.\
![Evolution, in log-scale, of the relative error of Alg. \[alg:robust-smart-distributed-r-admm\] computed w.r.t. the unique optimal solution $x^*$ as function of different values of packet loss probability $p$ for step size $\alpha=1$ and penalty $\rho=1$. Average over 100 Monte Carlo runs.[]{data-label="fig:evolution_different_losses"}](ErrorEvolution_DifferentPacketLoss_RandGeomGraph){width="\columnwidth"}
Figure \[fig:randgeom\_stability\_boundaries\] plots the stability boundaries of the R-ADMM Algorithm \[alg:robust-smart-distributed-r-admm\] as function of step size $\alpha$ and penalty $\rho$ for different packet loss probabilities $p$. More specifically, each curve in Figure \[fig:randgeom\_stability\_boundaries\] represents the numerical boundary below which the algorithm is found to be convergent and above which, conversely, the algorithm diverges. In this case the results turn out extremely interesting. Indeed, given $\alpha$ and $\rho$, for increasing packet loss probability $p$, the stability region enlarges. This means that the higher the loss probability is, the more robust the algorithm is. The numerical findings are perfectly in line with the result of Proposition \[prop:convergence\_lossy\], telling us that for $\alpha\in (0,1)$ the algorithm converges for any value of $\rho$. However, it suggests the additional interesting fact that the theory misses to capture a larger area – in parameters space and depending on $p$ – for which the algorithm still converges. This will certainly be a direction of future investigation.
![Stability boundaries of Alg. \[alg:robust-smart-distributed-r-admm\] as function of the step size $\alpha$ and the penalty $\rho$ for different values of loss probability $p$ for the family of random geometric graphs. Average over 100 Monte Carlo runs.[]{data-label="fig:randgeom_stability_boundaries"}](StabilityBoundariesDifferentLosses_RandGeomGraph){width="\columnwidth"}
Finally, Figure \[fig:randgeom\_different\_stepsizes\] reports the evolution of the error as a function of different values of the step-size $\alpha$. Notice that to values of $\alpha$ that are larger than $1/2$ correspond faster convergences. Recalling that setting $\alpha=1/2$ yields the standard ADMM, then it is clear that the use of the R-ADMM can speed up the convergence, which motivates its use against the use of the classic ADMM.
![Evolution, in log-scale, of the relative error of Alg. \[alg:robust-smart-distributed-r-admm\] computed w.r.t. the unique optimal solution $x^*$ as function of different values of the step size $\alpha$, with fixed packet loss probability $p=0.6$ and penalty $\rho=1$. Average over 100 Monte Carlo runs.[]{data-label="fig:randgeom_different_stepsizes"}](ErrorEvolution_DifferentStepSize_RandGeomGraph){width="\columnwidth"}
Conclusions and Future Directions {#sec:conclusions}
=================================
In this paper we addressed the problem of distributed consensus optimization in the presence of synchronous but unreliable communications. Building upon results in operator theory on Hilbert spaces, we leveraged the relaxed Peaceman-Rachford Splitting operator to introduced what is referred to R-ADMM, a generalization of the well known ADMM algorithm. We started by drawing some interesting connections with the classical formulation as typically presented. Then, we introduced several algorithmic reformulations of the R-ADMM which differs in terms of computational, memory and communication requirements. Interestingly the last implementation, besides being extremely light from both the communication and memory point of views, turns out the be provably robust to random communication failures. Indeed, we rigorously proved how, in the lossy scenario, the region of convergence in parameters space remains unchanged compared to the case of reliable communication; yet, we numerically showed that the region of convergence is positively affected by a larger packet loss probability. The drawback lies in a slower convergence rate of the algorithm.\
There remain many open questions paving the paths to future research directions such as analysis of the asynchronous case and generalization of the results to more general distributed optimization problems.
Derivation of Algorithm \[alg:r-admm-three-eqs\] {#app:derivation-alg-3eqs}
================================================
First of all we derive the augmented Lagrangian for problem , and obtain $$\begin{aligned}
\label{eq:augmented-lagr-distributed}
\begin{split}
\mathcal{L}_\rho(x,y;w)=\sum_{i=1}^Nf_i(x_i)&+\iota_{(I-P)}(y)+\\&-w^\top(Ax+y)+\frac{\rho}{2}\|Ax+y\|^2,
\end{split}\end{aligned}$$ where $\|Ax+y\|^2=\|Ax\|^2+\|y\|^2+2\langle Ax,y\rangle$. We can now proceed to derive equations – for the problem at hand.
### Equation
By and discarding the terms that do not depend on $y$ we get $$\begin{aligned}
y(k+1)=\operatorname*{arg\,min}_y&\Big\{\iota_{(I-P)}(y)-w^\top(k)y+\frac{\rho}{2}\|y\|^2\\&+2\alpha\rho\langle Ax(k),y\rangle+\rho(2\alpha-1)\langle y,y(k)\rangle\Big\}\end{aligned}$$ where we summed the terms with the inner product $\langle Ax(k),y\rangle$. Therefore we need to solve the problem $$\begin{aligned}
y(k+1)=\operatorname*{arg\,min}_{y=Py}&\Big\{-w^\top(k)y+\frac{\rho}{2}\|y\|^2 \\
&+2\alpha\rho\langle Ax(k),y\rangle+\rho(2\alpha-1)\langle y,y(k)\rangle\Big\}\end{aligned}$$ that for simplicity we can write as $$\label{eq:problem-y}
y(k+1)=\operatorname*{arg\,min}_{y=Py}\{h_{\alpha,\rho}(y;x(k),w(k))\}.$$\
We apply now the Karush-Kuhn-Tucker (KKT) conditions [@boyd2004convex] to problem and obtain the system $$\begin{aligned}
&\nabla\Big[h_{\alpha,\rho}(y;x(k),w(k))-\nu^\top(I-P)y\Big|_{y(k+1),\nu^*}=0\label{eq:kkt-1}\\
&y(k+1)=Py(k+1)\label{eq:kkt-2}\end{aligned}$$ where $\nu^*$ is the optimal value of the Lagrange multipliers of the problem.\
By computing the gradient in we obtain $$\begin{aligned}
\label{eq:kkt-1-bis}
\begin{split}
y(k+1)=\frac{1}{\rho}\big[w(k)&-2\alpha\rho Ax(k)\\&-\rho(2\alpha-1)y(k)+(I-P)\nu^*\big].
\end{split}\end{aligned}$$ We substitute this formula for $y(k+1)$ in the right-hand side of which results in $$\begin{aligned}
\label{eq:kkt-2-bis}
\begin{split}
y(k+1)=\frac{1}{\rho}\big[P&w(k)-2\alpha\rho PAx(k)\\&-\rho(2\alpha-1)Py(k)-(I-P)\nu^*\big]
\end{split}\end{aligned}$$ for the fact that $P^2=I$ and hence $P(I-P)=-(I-P)$.\
We sum now equations and and obtain $$\begin{aligned}
\label{eq:y-update-final}
\begin{split}
y(k+1)=\frac{1}{2\rho}(I+P)\big[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)\big].
\end{split}\end{aligned}$$ Finally noting that, given a vector $t$ of dimension equal to that of $y$, the $ij$-th element of $(I+P)t$ is equal to $t_{ij}+t_{ji}$, then the update for $y_{ij}(k+1)$ follows.
### Equation
By equation and we can write $$\begin{aligned}
w(k+1)=&w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)+\\
&-\frac{1}{2}(I+P)[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)]\\
=&\frac{1}{2}(I-P)[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)]\end{aligned}$$ and by the definition of $I-P$ we get the update equation for $w_{ij}(k+1)$ stated in Algorithm \[alg:r-admm-three-eqs\].
### Equation
Finally we apply equation to the problem at hand, which means that we need to solve $$\begin{aligned}
x(k+1)=&\operatorname*{arg\,min}_x\Bigg\{\sum_{i=1}^Nf_i(x_i)+\\&-\Big(w(k+1)-\rho y(k+1)\Big)^\top Ax+\frac{\rho}{2}\|Ax\|^2\Bigg\}.\end{aligned}$$ We know that each variable $x_i$ appears in $|\mathcal{N}_i|$ constraints and therefore $\|Ax\|^2=\sum_{i=1}^N|\mathcal{N}_i|\|x_i\|^2$. Moreover, given a vector $t$ with the same size as $y$, we have $$\begin{aligned}
t^\top Ax&=
\begin{bmatrix}
\cdots & t_{ji}^\top & \cdots & t_{ji}^\top & \cdots
\end{bmatrix}
\begin{bmatrix}
\vdots\\-x_i\\ \vdots\\-x_j\\ \vdots
\end{bmatrix}\\
&=\sum_{(i,j)\in\mathcal{E}}\left(t_{ji}^\top x_i+t_{ij}^\top x_j\right)\\
&=\sum_{i=1}^N\left(\sum_{j\in\mathcal{N}_i}t_{ji}^\top\right)x_i.\end{aligned}$$ and we get the update equation for $x_i(k+1)$ substituting $\Big(w(k+1)-\rho y(k+1)\Big)$ to $t$. Notice that by the results obtained above we have $$\begin{aligned}
\Big(w(k+1)-&\rho y(k+1)\Big)=\\&=-P[w(k)-2\alpha\rho Ax(k)-\rho(2\alpha-1)y(k)]\end{aligned}$$ which means that $x(k+1)$ can be computed as a function of the $x$, $y$ and $w$ variables at time $k$ only.
Proof of Proposition \[pr:r-admm-five-eqs\] {#app:proof-prop-1}
===========================================
### Equations
The following derivation shares some points with the derivation described in the section above. Indeed, applying the first equation of to the problem at hand requires that we solve $$y(k)=\operatorname*{arg\,min}_{y=Py}\left\{-z^\top(k)y+\frac{\rho}{2}\|y\|^2\right\},$$ which can be done by solving the system of KKT conditions of the problem as performed above. The result is $$\label{eq:y-update-proof}
y(k)=\frac{1}{2\rho}(I+P)z(k).$$ It easily follows from that $\psi(k)=\frac{1}{2}(I-P)z(k)$.
### Equations
First of all we have $(2\psi(k)-z(k))=-Pz(k)$, hence according to the same reasoning employed above to derive the expression for $x(k+1)$ we find . Moreover, we have $\xi(k)=-Pz(k)-\rho Ax(k)$.
### Equation
By the results derived above we can easily compute $$z(k+1)=(1-\alpha)z(k)-\alpha Pz(k)-2\alpha\rho Ax(k)$$ which gives equations .
Notice that to compute the variables $y(k)$, $\psi(k)$, $x(k)$ and $\xi(k)$ we need only the variables $z(k)$. Moreover, to update $z$ we require only $z(k)$ and $x(k)$. Hence the five update equations reduce to the updates for $x$ and $z$ only.
Proof of Proposition \[prop:convergence\] {#app:proof-convergence}
=========================================
To prove convergence of the R-ADMM in the two implementations of Algorithms \[alg:r-admm-three-eqs\] and \[alg:smart-distributed-r-admm\], we resort to the following result, adapted from [@bauschke2011convex Corollary 27.4].
\[pr:convergence-deterministic\] Consider problem and assume that it has solution; let $\alpha\in(0,1)$, $\rho>0$, and $x(0)\in\mathcal{X}$. Assume to apply equations – to the problem. Then there exists $z^*$ such that
- $x^*=\operatorname{prox}_{\rho g}(z^*)\in\operatorname*{arg\,min}_x\{f(x)+g(x)\}$, and
- $\{z(k)\}_{k\in\mathbb{N}}$ converges weakly to $z^*$.
We need to show now that this result applies to the dual problem of problem . First of all, by formulation of the problem we have that $f$ is convex and proper (and also closed). Moreover, by [@bauschke2011convex Example 8.3] we know that the indicator function of a convex set is convex (and, by definition, proper). But the set of vectors $y$ that satisfy $(I-P)y=0$ is indeed convex, hence also $g$ is convex and proper.\
Now [@rockafellar2015convex Theorem 12.2] states that the convex conjugate of a convex and proper function is closed, convex and proper. Therefore both $d_f$ and $d_g$ are closed, convex and proper, which means that we can apply the convergence result in Proposition \[pr:convergence-deterministic\] to the dual problem of .\
Therefore we have that $w^*=\operatorname{prox}_{\rho d_g}(z^*)$ is indeed a solution of the dual problem and $\{z(k)\}_{k\in\mathbb{N}}$ converges to $z^*$. But since the duality gap is zero, then when we attain the optimum of the dual problem we have obtained that of the primal as well.
Proof of Proposition \[prop:convergence\_lossy\] {#app:convergence-lossy}
================================================
In order to prove the convergence of Algorithm \[alg:robust-smart-distributed-r-admm\] we need to introduce a probabilistic framework in which to reformulate the KM update. For this stochastic version of the KM iteration we can state a convergence result adapted from [@bianchi2016coordinate Theorem 3] and show that indeed Algorithm \[alg:robust-smart-distributed-r-admm\] is represented by this formulation.
We are therefore interested in altering the standard KM iteration in order to include a stochastic selection of which coordinates in $\mathcal{I}=\{1,\ldots,M\}$ to update at each instant. To do so we introduce the operator $\hat{T}^{(\xi)}:\mathcal{X}\rightarrow\mathcal{X}$ whose $i$-th coordinate is given by $\hat{T}^{(\xi)}_ix=T_ix$ if the coordinate is to be updated ($i\in \xi$), $\hat{T}^{(\xi)}_ix=x_i$ otherwise ($i\not\in \xi$). In general the subset of coordinates to be updated changes from one instant to the next. Therefore, on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, we define the random i.i.d. sequence $\{\xi_k\}_{k\in\mathbb{N}}$, with $\xi_k:\Omega\rightarrow 2^\mathcal{I}$, to keep track of which coordinates are updated at each instant. The stochastic KM iteration is finally defined as $$\label{eq:stochastic-km}
x(k+1)=(1-\alpha)x(k)+\alpha\hat{T}^{(\xi_{k+1})}x(k)$$ and consists of the $\alpha$-averaging of a stochastic operator.
The stochastic iteration satisfies the following convergence result, which is particularized from [@bianchi2016coordinate] using the fact that a nonexpansive operator is $1$-averaged, and a constant step size.
Let $T$ be a nonexpansive operator with at least a fixed point, and let the step size be $\alpha\in(0,1)$. Let $\{\xi_k\}_{k\in\mathbb{N}}$ be a random i.i.d. sequence on $2^\mathcal{I}$ such that $$\forall i\in\mathcal{I},\ \exists I\in2^\mathcal{I}\ \text{s.t.}\ i\in I\ \text{and}\ \mathbb{P}[\xi_1=I]>0.$$ Then for any deterministic initial condition $x(0)$ the stochastic KM iteration converges almost surely to a random variable with support in the set of fixed points of $T$.
We turn now to the distributed optimization problem, in which the stochastic KM iteration is performed on the auxiliary variables $z$. In particular we assume that the packet loss occurs with probability $p$, and that in the case of packet loss the relative variable is not updated. As shown in the main paper, this update rule can be compactly written as $$\label{eq:operator-packet-loss}
\hat{T}^{(\xi_{k+1})}z(k)=L_kz(k)+(I-L_k)Tz(k)$$ where $L_k$ is the diagonal matrix with elements the realizations of the binary random variables that model the packet loss at time $k$. Recall that these variables take value 1 if the packet is lost.\
Substituting now the operator into we get the update equation $$\label{eq:stochastic-km-order-1}
z(k+1)=(1-\alpha)z(k)+\alpha\left[L_kz(k)+(I-L_k)Tz(k)\right]$$ which conforms to the stochastic KM iteration for which the convergence result is stated.\
Finally, notice that in the main article the $\alpha$-averaging is applied before the stochastic coordinate selection, that is the update is given by $$\label{eq:stochastic-km-order-2}
z(k+1)=L_kz(k)+(I-L_k)\left[(1-\alpha)z(k)+\alpha Tz(k)\right].$$ However it can be easily shown that and do indeed coincide, hence proving the convergence of our update scheme.
[^1]: $^\dagger$ Department of Information Engineering (DEI), University of Padova, Italy. [[email protected], \[carlirug|schenato\]@dei.unipd.it]{}.
[^2]: $^\ddagger$ Bosch Center for Artificial Intelligence. Renningen, Germany. [[email protected]]{}. The work was carried out during the author’s postdoctoral fellowship at DEI.
[^3]: A function $f:\mathcal{X}\rightarrow\mathbb{R}\cup\{+\infty\}$ is said to be *closed* if $\forall a \in\mathbb{R}$ the set $\{x\in\operatorname{dom}(f)\ |\ f(x)\leq a\}$ is closed. Moreover, $f$ is said to be *proper* if it does not attain $-\infty$ [@boyd2011distributed].
[^4]: The *convex conjugate* of a function $f$ is defined as $f^*(y)=\sup_{x\in\mathcal{X}}\{\langle y,x\rangle-f(x)\}$.
[^5]: We highlight that the results of this section can be extended to the case where the loss probability is different for edge.
|
---
abstract: 'The semi-classical approximation to black hole partition functions is not well-defined, because the classical action is unbounded and the first variation of the uncorrected action does not vanish for all variations preserving the boundary conditions. Both problems can be solved by adding a Hamilton-Jacobi counterterm. I show that the same problem and solution arises in quantum mechanics for half-binding potentials.'
address: |
Center for Theoretical Physics, Massachusetts Institute of Technology,\
77 Massachusetts Ave., Cambridge, MA 02139, USA\
E-mail: [email protected]
author:
- 'D. GRUMILLER'
title: |
PATH INTEGRAL FOR HALF-BINDING POTENTIALS\
AS QUANTUM MECHANICAL ANALOG FOR\
BLACK HOLE PARTITION FUNCTIONS
---
Introduction and statement of the problem {#se:1}
=========================================
Path integrals have illuminated many aspects of quantum mechanics and quantum field theory [@Dresden:2007], but there remain some challenges to path integral formulations of quantum theories [@Jackiw:2007bc]. In this proceedings contribution I describe a problem arising for quantum mechanical potentials that are ‘half-binding’ (the definition of this term will be given below). I shall demonstrate that the naive semi-classical approximation to the path integral breaks down for two reasons: the leading contribution to the partition function is singular and the first variation of the action does not vanish for all variations preserving the boundary conditions. I discuss how both issues can be resolved by adding an appropriate (Hamilton-Jacobi) counterterm as boundary term to the action. Moreover, I shall point out formal similarities to black hole (BH) partition functions, so in that sense these quantum mechanical systems may serve as toy models to elucidate certain aspects of BH physics. For sake of clarity I focus on a specific Hamiltonian [@deAlfaro:1976je], $$\label{eq:dresden1}
H(q,p)=\frac{p^2}{2}+V(q)\,,\qquad V(q)=\frac{1}{q^2}\,,$$ where $q$ is restricted to positive values. If $q$ is small the Hamiltonian rises without bound, like for a binding potential. If $q$ is large the potential is negligible, and the asymptotic dynamics is dominated by free propagation. I refer to a potential with these properties as ‘half-binding’. \[The conformal properties [@deAlfaro:1976je] of will not play any role in this discussion.\]
Consistency of the variational principle based on the Lagrangian action, $$\label{eq:dresden2}
I[q]=\int\limits_{t_i}^{t_f}\!dt \,\Big(\frac{\dot{q}^2}{2}-\frac{1}{q^2}\Big) \,,$$ requires to fix the initial and final value of $q$ at $t_i$ and $t_f$, respectively. I am interested here mostly in the limit $t_f\to\infty$, which implies that $q|^{t_f}=\infty$ is the appropriate asymptotic boundary condition. The initial time is set to zero, $t_i=0$, without loss of generality. The Lagrangian path integral, $$\label{eq:dresden3}
{\mathcal{Z}}= \int {\mathcal{D}}q \, \exp\Big(-\frac{1}{\hbar}\,I[q] \Big)\,,$$ consists of a coherent sum over all field configurations consistent with the boundary data. Even though is exactly soluble, it is illustrative to consider the semi-classical expansion of the action, $$\label{eq:dresden4}
I[q_{\rm cl} + \delta q] = I[q_{\rm cl}] + \delta I\big|_{\rm EOM} + {\cal O}(\delta q^2) \,,$$ and of the partition function $$\label{eq:dresden5}
{\mathcal{Z}}= \exp\Big(-\frac{1}{\hbar}\,I[q_{\rm cl}]\Big) \, \int {\mathcal{D}}\delta q \, \exp\Big(-\frac{1}{\hbar}\,{\cal O}(\delta q^2) \Big) \,.$$ The semi-classical approximation is well-defined only if the on-shell action is bounded, $|I[q_{\rm cl}]|<\infty$, and only if the first variation of the action vanishes on-shell, $\delta I|_{\rm EOM} = 0$, for all field configurations preserving the boundary conditions. I demonstrate now that neither is the case for the example .
The on-shell action diverges because asymptotically the propagation is essentially free, and because of the assumption $t_f\to\infty$. This is an idealization of situations where boundary conditions are imposed at late times, $t_f\sim 1/\epsilon$, with $\epsilon\ll 1$. In that case also $q_f\sim 1/\epsilon$ classically. However, the path integral does not only take into account classical contributions, but also samples nearby field configurations whose asymptotic behavior is $q \sim q_f [1 + \epsilon \, \Delta q + {\cal O}(\epsilon^2)]$, where $\Delta q$ is finite. Therefore, the first variation of the action, evaluated on-shell, is given by the boundary term $$\label{eq:dresden5.5}
\dot{q}\,\delta q|^{t_f} - \dot{q}\,\delta q|^{t_i=0} = \dot{q}\,\delta q|^{t_f} \sim \lim_{\epsilon\to 0} [\dot{q}\, \Delta q + {\cal O}(\epsilon)]\big|^{t_f}\neq 0\,.$$ The inequality emerges, because arbitrary finite variations $\delta q|^{t_f}$ certainly preserve the boundary condition $q|^{t_f}=\infty$.[^1] The two problems described here spoil the semi-classical approximation to the partition function.
Hamilton-Jacobi counterterm for half-binding potentials {#se:2}
=======================================================
Both problems can be solved by considering an improved action[^2] $$\label{eq:dresden6}
\Gamma[q] =\int\limits_{0}^{t_f}\! dt \,\Big(\frac{\dot{q}^2}{2}-\frac{1}{q^2}\Big) - {S}(q,t)\Big|^{t_f}_{0}\,,$$ which differs from by a boundary counterterm depending solely on quantities that are kept fixed at the boundary. The variation of , $$\label{eq:dresden7}
\delta \Gamma|_{\rm EOM} = \Big(\dot{q}-\frac{\partial {S}}{\partial q}\Big)\delta q\Big|_{0}^{t_f} = \Big(\dot{q}-\frac{\partial {S}}{\partial q}\Big)\delta q\Big|^{t_f}\,,$$ does not necessarily suffer from the second problem if $\partial {S}/\partial q$ asymptotically behaves like $\dot{q}$, i.e., like the momentum $p$.
The method [@deBoer:1999xf; @Grumiller:2007ju] that I am going to review does not involve the subtraction of the action evaluated on a specific field configuration, but rather is intrinsic. Moreover, the amount of guesswork is minimal: Hamilton’s principal function is a well-known function of the boundary data with the property $\partial{S}/\partial q= p$. Therefore it is natural to postulate that ${S}$ in solves the Hamilton-Jacobi equation, $$\label{eq:dresden8}
H\Big(q,\frac{\partial {S}}{\partial q}\Big)+\frac{\partial {S}}{\partial t} = 0\,.$$
The complete integral [@Kamke] $$\label{eq:dresden9}
{S}(q,t)= c_0 -Et+\sqrt{2(Eq^2-1)}+\sqrt{2}\,\arctan{\frac{1}{\sqrt{Eq^2-1}}}$$ allows to construct the enveloping solution[^3] $$\label{eq:dresden10}
{S}(q,t)=\frac{q^2}{2t} \left(\sqrt{4\Delta_+-8t^2/q^4}-\Delta_+\right)+\sqrt{2}\,\arctan{\frac{1}{\sqrt{q^4\Delta_+/(2t^2)-1}}}\,,$$ where $\Delta_+ := \frac12 (1+\sqrt{1-8t^2/q^4})$. The asymptotic expansion ${S}=q^2/(2t)+{\cal O}(t/q^2)$ is consistent with the intuitive idea that the asymptotically free propagation is the source of all subtleties. But the expression contains a great deal of additional (non-asymptotic) information, which can be physically relevant, as mentioned in the next Section.
Let me now come back to the two problems. Since asymptotically $\dot{q}|_{\rm EOM}=v=\rm const.$, the on-shell action $$\label{eq:dresden11}
\Gamma\big|_{\rm EOM} = \frac{v^2}{2}\int\limits^{t_f}_0 \!dt - \frac{v^2}{2} t^f + {\cal O}(1) = {\cal O}(1)$$ evidently is finite. The terms of order of unity entail the information about the potential $V(q)$. The first variation $$\label{eq:dresden12}
\delta\Gamma\big|_{\rm EOM} = \Big(\underbrace{\dot{q}-\frac{q}{t}}_{{\cal O}(1/t)}+ \,{\cal O}(1/t^2)\Big)\delta q\Big|^{t_f} = {\cal O}(1/t)\underbrace{\delta q}_{\rm finite}\Big|^{t_f} = 0$$ vanishes for all variations preserving the boundary conditions. The two problems mentioned in the previous Section indeed are resolved by the improved action with .
The considerations above apply in the same way to the Hamiltonian with a more general class of half-binding potentials $V(q)$. In particular, the (manifestly positive) potential $V(q)$ is required to be monotonically decreasing, and to vanish faster than $1/q$ for large $q$. Going through the same steps as above is straightforward. Other generalizations, e.g. to non-monotonic potentials or potentials with Coulomb-like behavior, may involve technical refinements, but the general procedure is always the same: one has to solve the Hamilton-Jacobi equation to obtain the correct counterterm ${S}$ in .
Comparison with black hole partition functions {#se:3}
==============================================
The same issues as in the previous Section arise when evaluating BH partition functions. Probably the simplest non-trivial model is 2-dimensional dilaton gravity (cf. e.g. [@Grumiller:2002nm] for recent reviews), $$\begin{gathered}
\label{Action}
I[g,X] = - \frac{1}{16\pi G_2}\,\int_{{\mathcal{M}}} \!\! d^{\,2}x \,\sqrt{g}\, \left( X\,R - U(X)\,
\left(\nabla X\right)^2 - 2 \, V(X) \raisebox{12pt}{~}\right) \\
- \frac{1}{8\pi G_2}\, \int_{{\partial {\mathcal{M}}}} {\!\!\!}dx \, \sqrt{\gamma}\,X\,K ~.\end{gathered}$$ An explanation of the notation can be found in [@Grumiller:2007ju]. The boundary term in is the dilaton gravity analog of the Gibbons-Hawking-York boundary term. The latter arises in quantum mechanics if one converts the action $I=\int dt[-q\dot{p}-H(q,p)]$ into standard form, but it is [*not*]{} related to the Hamilton-Jacobi counterterm. It was shown first (second) in the second (first) order formulation [@Grumiller:2007ju] ([@Bergamin:2007sm]) that the improved action is given by $$\label{ActionConclusion}
\Gamma[g,X] = I[g,X] + \frac{1}{8\pi G_2}\, \int_{{\partial {\mathcal{M}}}} {\!\!\!}dx \,\sqrt{\gamma} \, {S}(X) \,,$$ with the solution of the Hamilton-Jacobi equation ($V(X)\leq 0$) $$\label{eq:C}
{S}(X) = \Big(-2 e^{-\int^X \!dy \,U(y)}\int^X \!\!dy\,V(y)\,e^{\int^y\!dz\,U(z)}\Big)^{1/2}\,.$$ The lower integration constant in the integrals over the function $U$ is always the same and therefore cancels; the lower integration constant in the remaining integral represents the ambiguity mentioned in footnote \[fn:1\]. [^4]
The BH partition function based upon the improved action , $$\label{PartitionFunction2}
{\mathcal{Z}}= \int {\mathcal{D}}g \,{\mathcal{D}}X \, \exp\Big(- \frac{1}{\hbar}\,\Gamma[\,g,X]\Big) \approx \exp\Big(- \frac{1}{\hbar}\,\Gamma[\,g_{\rm cl},X_{\rm cl}]\Big)\,,$$ by standard methods establishes the BH free energy. The asymptotic part of the counterterm leads to the correct asymptotic charges for BHs with (essentially) arbitrary asymptotic behavior, and to consistency with the first law of thermodynamics (which is non-trivial [@Davis:2004xiNote]). The finite part of the counterterm allows a quasi-local description of BH thermodynamics [@Grumiller:2007ju].
Perhaps one might exploit the formal analogy between BH partition functions and quantum mechanical partition functions described in this work to construct interesting condensed matter analogs [@Novello:2002qg] mimicking thermodynamical aspects of BHs.
Acknowledgments {#acknowledgments .unnumbered}
===============
I am grateful to Roman Jackiw and Robert McNees for enjoyable discussions. I thank Luzi Bergamin and Ren[é]{} Meyer for reading the manuscript and Wolfhard Janke for valuable administrative help.
This work is supported in part by funds provided by the U.S. Department of Energy (DoE) under the cooperative research agreement DEFG02-05ER41360 and by the project MC-OIF 021421 of the European Commission under the Sixth EU Framework Programme for Research and Technological Development (FP6). Part of my travel expenses where covered by funds provided by the conference “Path integrals – New Trends and Perspectives” at the Max-Planck Institute PKS in Dresden.
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[^1]: Canonical transformations can shift the problem, but of course they cannot solve it. For instance, with $Q=1/q$ and $P=-pq^2$ the correct asymptotic boundary condition is $Q|^{t_f}=0$ and therefore also $\delta Q|^{t_f}=0$. But now the momentum $P$ (and thus $\dot{Q}$) diverges at the boundary, so that the expression $\dot{Q}\,\delta Q|^{t_f}$ becomes undefined and does not necessarily vanish for all variations preserving the boundary conditions.
[^2]: There exists a variety of subtraction methods in quantum mechanics [@Kleinert], in General Relativity (and generalizations thereof) [@Regge:1974zd] and in holographic renormalization within the context of AdS/CFT [@Henningson:1998gx]. Many of them have ad-hoc elements and require the subtraction of the action evaluated on a specific field configuration (like the ground state solution); in some cases there are several “natural” candidates, in others there is none, and in at least one example the “natural” guess even turned out to be wrong [@Davis:2004xiNote].
[^3]: \[fn:1\] One is forced to take the enveloping solution, since ${S}$ is part of the definition of the improved action and therefore cannot depend on constants of motion. The energy $E$ is eliminated from by solving $\partial{S}/\partial E=0$ for $E$. The other constant, $c_0$, is set to zero by hand, but other choices are possible. Such an ambiguity always remains in this (and any other) approach. It reflects the freedom to shift the free energy of the ground state.
[^4]: Solving for $V$ yields $V=-1/2 \,[({S}^2)^\prime+{S}^2 U]$, which reveals that the Hamilton-Jacobi counterterm ${S}(X)$ is the supergravity pre-potential (up to a numerical factor) [@Grumiller:2002nm]. A similar (pseudo-)supersymmetric story exists in quantum mechanics [@Townsend:2007aw]. Cf. also [@deBoer:1999xf].
|
---
abstract: 'In this paper we investigate the geometry of a discrete Bayesian network whose graph is a tree all of whose variables are binary and the only observed variables are those labeling its leaves. We obtain a full semialgebraic geometric description of these models which is given by polynomial equations and inequalities. Our analysis is based on combinatorial results generalizing the notion of cumulants so that they apply to the models under analysis. The geometric structure we obtain links to the notion of a tree metric considered in phylogenetic analysis and to some interesting determinantal formulas involving the hyperdeterminant of $2\times 2\times 2$ tables.'
address:
- |
Piotr Zwiernik\
University of Warwick\
Department of Statistics\
CV7AL, Coventry, UK.
- |
Jim Q. Smith\
University of Warwick\
Department of Statistics\
CV7AL, Coventry, UK.
author:
- '[Piotr]{} [Zwiernik]{}'
- '[Jim Q.]{} [Smith]{}'
bibliography:
- '../!bibliografie/algebraic\_statistics.bib'
title: The geometry of hidden tree Markov models for binary data
---
Introduction {#sec:introduction}
============
A Bayesian network whose graph is a tree all of whose inner nodes represent variables which are not directly observed lie in an important class of models, containing phylogenetic tree models and hidden Markov models. Inference for this model class tends to be challenging and often needs to employ fragile numerical algorithms. In [@pwz2010-identifiability] we established a useful new coordinate system for such models when all of the variables are binary. This analysis enabled us not only to address various identifiability issues but also helped us to derive exact formulas for the maximum likelihood estimators given that the sample proportions were in this model class.
The application of this new coordinate system reaches far beyond understanding the identifiability and it can be used to analyze the global structure of these tree models. For example [@cavender1997443] gave an intriguing correspondence between, on the one hand, a correlation system on tree models and on the other distances induced by trees where the length between two nodes in a tree is given as a sum of the length of edges in the path joining them. Our new coordinate system for tree models enables us to explore in detail this relationship between probabilistic tree models (also called the tree decomposable distributions in [@pearl_tarsi86]) and tree metrics.
It has been known for some time that the constraints on possible distances between any two leaves in the tree imply some additional inequality constraints on the possible covariances between the binary variables represented by the leaves. These inequalities, given in (\[eq:suffic-ineq2\]), follow from the four-point condition ([@semple2003pol], Definition 7.1.5) together with some other simple non-negativity constraints. In this paper we show that these two types of inequality constraints cannot be sufficient. Thus any probability distribution in the model class must satisfy certain additional constraints involving higher order moments. We provide the full set of the defining constraints in Theorem \[th:parameters0\]. This is given by a list of polynomial equations and inequalities which describe the set of all probability distributions in the model.
Our approach here is founded in a geometric study of tree models through the method of phylogenetic invariants first introduced by Lake [@lake1987rit], and Cavender and Felsenstein [@cavender1987ips]. These invariant algebraic relationships are expressed as a set of polynomial equations over the observed probability tables which must hold for a given phylogenetic model to be valid. We note that these algebraic techniques have also been embraced by computational algebraic geometers [@allman2008pia][@eriksson2004pag][@sturmfels2005tip] enhancing the statistical and computational analyses of such models [@casanellas2007pni] (see also [@allman_invariants2007] and references therein).
The main technical deficiency of using phylogenetic invariants in this way is that they do not give a *full* geometric description of the statistical model. This is important for example in any subsequent Bayesian analysis of the class of tree models. The additional inequalities obtained as the main result of this paper complete this description. Where and how these inequality constraints can helpfully supplement an analysis based on phylogenetic invariants is illustrated by the simple example given below.
\[ex:1\] Let $T$ be the tripod tree in Figure \[fig:tripod\] where we use the convention that observed nodes are depicted by black nodes.
![The graphical representation of the tripod tree model.[]{data-label="fig:tripod"}](tripod)
The inner node represents a binary hidden variable $H$ and the leaves represent binary observable variables $X_1, X_2, X_3$. The model is given by all probability distributions $p_{\a}$ for $\a\in\{0,1\}^{3}$ such that $$p_{\a}=\theta^{(H)}_{0}\prod_{i=1}^{3}\theta^{(i)}_{\a_{i}|0}+\theta^{(H)}_{1}\prod_{i=1}^{3}\theta^{(i)}_{\a_{i}|1},$$ where $\theta^{(H)}_{i}=\P(H=i)$ for $i=0,1$ and $\theta^{{(i)}}_{j|k}=\P(X_{i}=j|H=k)$ for $i=1,2,3$ and $j,k=0,1$. The model has full dimension over the observed margin $(X_{1},X_{2},X_{3})$ and consequently there are no equations defining it. However, it is not a saturated model since not all the marginal probability distributions over the observed vector $(X_{1},X_{2},X_{3})$ lie in the model class. For example Lazarsfeld [@lazarsfeld1968lsa Section 3.1] showed that the second order moments of the observed distribution must satisfy $${\rm Cov}(X_1,X_2){\rm Cov}(X_1,X_3){\rm Cov}(X_2,X_3)\geq 0.$$ This constraint, which clearly impacts the inferences we might want to make, is not acknowledged through the study of phylogenetic invariants. Therefore inference based solely on these invariants is incomplete. In particular naive estimates derived through these methods can be infeasible within the model class in a sense illustrated later in this paper.
This example motivated the closer investigation of the semi-algebraic features associated with the geometry of binary tree models with hidden inner nodes. The main problem with the geometric analysis of these models is that in general it is hard to obtain the inequality constraints defining a model even for very simple examples (see [@drton2007asm Section 4.3], [@garcia2005agb Section 7]). Despite this, some results can be found in the literature. Thus in the case of a binary naive Bayes model a somewhat complicated solution was given by Auvray et al. [@auvray2006sad]. In the binary case there are also some partial results for general tree structures given by Pearl and Tarsi [@pearl_tarsi86] and Steel and Faller [@steel2009mls]. The most important applications in biology involve variables that can take four values. Recently Matsen [@matsen2007fti] gave a set of inequalities in this case for group-based phylogenetic models (additional symmetries are assumed) using the Fourier transformation of the raw probabilities. Here we provide a simpler and more statistically transparent way to express the constrained space.
The paper is organized as follows. In Section \[sec:models\_trees\] we briefly introduce general Markov models. We then proceed to describe a convenient change of coordinates for these models given in [@pwz2010-identifiability]. In the new coordinate system the parametrization of the model has an elegant product form. This is then used in Section \[sec:semi-tripod\] to obtain the full semi-algebraic description of a simple naive Bayes model. In Section \[sec:metrics\] we state the main result of the paper given by Theorem \[th:parameters0\] and provide some necessary constraints on the probability distributions in the model class using a correspondence with tree metrics. In Section \[sec:quartet\] we discuss these results for a simple quartet tree model. We prove our main theorem in Appendix \[sec:proof\]. Finally, in Appendix \[sec:invariants\] we show that the equations given in Theorem \[th:parameters0\] are equivalent to equations given earlier by Allman and Rhodes [@allman2008pia].
Tree models and tree cumulants {#sec:models_trees}
==============================
\[sec:correlations\]
In this paper we always assume that random variables are binary taking values either $0$ or $1$. We consider models with *hidden* variables, i.e. variables whose values are never directly observed. The vector $Y$ has as its components all variables in the graphical model, both those that are observed and those that are hidden. The subvector of $Y$ of observed variables is denoted by $X$ and the subvector of [hidden]{} variables by $H$.
A (*directed*) *tree* $T=(V,E)$, where $V$ is the set of vertices (or nodes) and $E\subseteq V\times V$ is the set of edges of $T$, is a connected (*directed*) graph with no cycles. A *rooted tree* is a directed tree that has one distinguished vertex called the *root*, denoted by the letter $r$, and all the edges are directed away from $r$. A rooted tree is usually denoted by $T^{r}$. For each $v\in V$ by $\pa(v)$ we denote the node preceding $v$ in $T^{r}$. In particular $\pa(r)=\emptyset$. A vertex of $T$ of degree one is called a *leaf*. A vertex of $T$ that is not a leaf is called an *inner node*.
Let $T$ denote an undirected tree with $n$ leaves and let $T^{r}=(V,E)$ denote $T$ rooted in $r\in V$. A Markov process on a rooted tree $T^r$ is a sequence $\{Y_v:\,v\in V\}$ of random variables such that for each $\a=(\a_v)_{v\in V}\in \{0,1\}^{V}$ its joint distribution satisfies $$\label{eq:p_albar}
p_\a(\theta)=\theta^{(r)}_{\a_{r}}\prod_{v\in V\setminus r} \theta^{(v)}_{\a_{v}|\a_{\pa(v)}},$$ where $\theta^{(r)}_{\a_{r}}=\P(Y_{r}=\a_{r})$ and $\theta^{(v)}_{\a_{v}|\a_{\pa(v)}}=\P(Y_v={\a}_v|Y_{\pa(v)}=\a_{\pa(v)})$. Since $\theta^{(r)}_{0}+\theta^{(r)}_{1}=1$ and $\theta^{(v)}_{0|i}+\theta^{(v)}_{1|i}=1$ for all $v\in V\setminus\{r\}$ and $i=0,1$ then the set of parameters consists of exactly $2|E|+1$ free parameters: we have two parameters: $\theta^{(v)}_{1|0}$, $\theta^{(v)}_{1|1}$ for each edge $(u,v)\in E$ and one parameter $\theta^{(r)}_{1}$ for the root. We denote the parameter space by $\Theta_{T}=[0,1]^{2|E|+1}$ and the Markov process on $T^{r}$ by $\widetilde{\cM}_{T}$.
\[rem:rootings\] The reason to omit the root $r$ in the notation is that this model does not depend on the rooting and is equivalent to the undirected graphical model given by global Markov properties on $T$. To prove this note that $T^{r}$ is a perfect directed graph and hence by [@lauritzen:96 Proposition 3.28] parametrisation in (\[eq:p\_albar\]) is equivalent to factorisation with respect to $T$. Since $T$ is decomposable this factorisation is equivalent to the global Markov properties by [@lauritzen:96 Proposition 3.19].
Let $\Delta_{2^n-1}=\{p\in\R^{2^n}: \sum_\beta p_\beta=1, p_\beta \geq 0\}$ with indices $\beta$ ranging over $\{0,1\}^n$ be the probability simplex of all possible distributions of $X=(X_1,\ldots, X_n)$ represented by the leaves of $T$. We assume now that all the inner nodes represent hidden variables. Equation (\[eq:p\_albar\]) induces a polynomial map $f_T:\Theta_{T}\rightarrow \Delta_{2^n-1}$ obtained by marginalization over all the inner nodes of $T$ $$\label{eq:p_albar2}
p_{\beta}(\theta)=\sum_\cH \theta^{(r)}_{\a_{r}}\prod_{v\in V\setminus r}\theta^{(v)}_{\a_{v}|\a_{\pa(v)}},$$ where $\cH$ is the set of all $\a\in \{0,1\}^{V}$ such that the restriction to the leaves of $T$ is equal to $\beta$. We let $\cM_{T}=f_{T}(\Theta_{T})$ denote the *general Markov model* over the set of observable random variables (c.f. [@semple2003pol Section 8.3]).
A *semialgebraic set* in $\R^{d}$ is any space given by a finite number of polynomial equations and inequalities. Since $\Theta_{T}$ is a [semialgebraic set]{} and $f_{T}$ is a polynomial map then by [@bochnak1998rag Proposition 2.2.7] $\cM_{T}$ is a semialgebraic set as well. Moreover, if $f$ is a polynomial isomorphism from $\Delta_{2^{n}-1}$ to another space then $f(\cM_{T})$ is also a semialgebraic set. The semialgebraic description of $f(\cM_{T})$ in $f(\Delta_{2^{n}-1})$ gives the semialgebraic description of $\cM_{T}$.
In [@pwz2010-identifiability] we described a convenient change of coordinates for directed tree models as a function of the usual parametrization (\[eq:p\_albar2\]) which is expressed in terms of the probabilities. The idea was to define a polynomial isomorphism $f_{p\kappa}$ from $\Delta_{2^{n}-1}$ to the space of new parameters called tree cumulants $\cK_{T}$. We defined a partially ordered set (poset) of all the partitions of the set of leaves induced by removing edges of the given tree $T$. Then tree cumulants are given as a function of probabilities induced by a Möbius function on the poset. The details of this change of coordinates are given Appendix \[app:change\] and are illustrated below.
The tree cumulants are given by $2^{n}-1$ coordinates: $n$ means ${\lambda}_{i}=\E X_{i}$ for all $i=1,\ldots,n$ and a set of real-valued parameters $\{\kappa_{I}:\,I\subseteq[n]\mbox{ where } |I|\geq 2\}$. Each formula for $\kappa_{I}$ is expressed as a function of the higher order central moments of the observed variables. These formulas are given explicitly in equation (\[eq:kappa-in-rho\]) of Appendix \[app:change\]. Since $f_{p\kappa}$ is a polynomial isomorphism then by [@bochnak1998rag Proposition 2.2.7] $\cM_{T}^{\kappa}=f_{p\kappa}(\cM_{T})$ is a semialgebraic set. In this paper we provide the full semialgebraic description of $\cM_{T}^{\kappa}$, i.e. the complete set of polynomial equations and inequalities involving the tree cumulants which describes $\cM_{T}^{\kappa}$ as the subset of $\cK_{T}$.
\[ex:quartet1\] Consider the quartet tree model, i.e. the general Markov model given by the graph in Figure \[fig:quartet\].
![A quartet tree[]{data-label="fig:quartet"}](graph1)
The tree cumulants are given by $15$ coordinates: ${\lambda}_{i}$ for $i=1,2,3,4$ and $\kappa_{I}$ for $I\subseteq[4]$ such that $|I|\geq 2$. Denoting $U_{i}=X_{i}-\E X_{i}$ we have $\kappa_{ij}=\E U_{i}U_{j}={\rm Cov}(X_{i},X_{j})$ for $1\leq i<j\leq 4$ and $$\kappa_{ijk}=\E \left(U_{i}U_{j}U_{k}\right)$$ for all $1\leq i<j<k\leq 4$ which we note is a third order central moment. However in general tree cumulants of higher order cannot be equated with corresponding central moments but only expressed as functions of them. These functions are obtained by performing an appropriate Möbius inversion. Thus for example from equation (\[eq:kappa-in-rho\]) in Appendix \[app:change\] we have that $$\kappa_{1234}=\E \left(U_{1}U_{2}U_{3}U_{4}\right)-\E \left(U_{1}U_{2}\right)\E \left(U_{3}U_{4}\right).$$ Note that since the observed higher order central moments can be expressed as functions of probabilities, tree cumulants can also be expressed as functions of these probabilities.
Let $T^{r}=(V,E)$ and let $\Omega_{T}$ denote the set of parameters with coordinates given by $\bar{\mu}_{v}$ for $v\in V$ and $\eta_{u,v}$ for $(u,v)\in E$. Define a reparametrization map $f_{\theta\omega}: \Theta_{T}\rightarrow \Omega_{T}$ as follows: $$\label{eq:uij}
\begin{array}{ll}
\eta_{u,v}=\theta^{(v)}_{1|1}-\theta^{(v)}_{1|0} & \mbox{for every $(u,v)\in E$ and}\\
\bar{\mu}_v=1-2\lambda_v & \mbox{for each } v\in V,
\end{array}$$ where we claim that $\lambda_{v}=\E Y_{v}$ is a polynomial in the original parameters $\theta$. To see this let $r,v_{1},\ldots, v_{k},v$ be a directed path in $T$. Then $$\label{eq:means-from-prob}
\lambda_{v}=\P(Y_{v}=1)=\sum_{\a\in\{0,1\}^{k+1}} \theta^{(v)}_{1|\a_{k}}\theta^{(v_{k})}_{\a_{k}|\a_{k-1}}\cdots \theta^{(r)}_{\a_{r}}.$$ It can be easily checked that if ${\rm Var}(Y_{u})>0$ then $\eta_{u,v}={\rm Cov}(Y_{u},Y_{v})/{\rm Var}(Y_{u})$. Hence $\eta_{u,v}$ is just the regression coefficient of $Y_{v}$ with respect to $Y_{u}$, namely $\E(Y_{v}-\E Y_{v}|Y_{u})=\eta_{u,v}(Y_{u}-\E Y_{u})$. This provides a clear statistical interpretation for the new parameters. The parameter space $\Omega_{T}$ is given by the following constraints: $$\label{eq:constraints}
\begin{array}{l}
-1\leq \bar{\mu}_{r}\leq 1, \qquad\mbox{and for each } (u,v)\in E\\
-(1+\bar{\mu}_{v}) \leq (1-\bar{\mu}_{u})\eta_{u,v} \leq (1-\bar{\mu}_{v})\\
-(1-\bar{\mu}_{v}) \leq (1+\bar{\mu}_{u})\eta_{u,v} \leq (1+\bar{\mu}_{v}).
\end{array}$$ In Appendix \[app:change\] we show that there is a polynomial isomorphism between $\Theta_{T}$ and $\Omega_{T}$ giving the following diagram, where the dashed arrow denotes the induced parameterization. $$\label{eq:diagram}
\xymatrixcolsep{4pc}\xymatrix{
\Theta_{T} \ar@<1ex>[d]^{f_{\theta\omega}} \ar[r]^{f_{T}} &\Delta_{2^{n}-1}\ar@<1ex>[d]^{f_{p\kappa}}\\
\Omega_{T}\ar@<1ex>[u]^{f_{\omega\theta}} \ar@{-->}[r]^{\psi_{T}} &\cK_{T}\ar@<1ex>[u]^{f_{\kappa p}}}$$ One motivation behind this change of coordinates is that the induced parametrization $\psi_{T}:\Omega_{T}\rightarrow \cK_{T}$ has a particularly elegant form.
\[prop:monomial\]Let $T$ be an undirected tree with $n$ leaves all of whose inner nodes have degree at most three. Let $T^{r}=(V,E)$ be $T$ rooted in $r\in V$. Then $\cM_T^\kappa$ is parametrized by the map $\psi_{T}:\Omega_{T}\rightarrow\cK_{T}$ given as $\lambda_{i}=\frac{1}{2}(1-\bar{\mu}_{i})$ for $i=1,\ldots, n$ and $$\label{eq:kappa_def_general}
\kappa_{I}=\frac{1}{4}\left(1-\bar{\mu}_{r(I)}^{2}\right) \prod_{v\in {\rm int}(V(I))} \bar{\mu}_{v}^{\deg(v)-2}\prod_{(u,v)\in E(I)} \eta_{u,v}\quad\mbox{ for } I\subseteq[n], |I|\geq 2$$ where the degree is taken in $T(I)=(V(I),E(I))$; ${\rm int}(V(I))$ denotes the set of inner nodes of $T(I)$ and $r(I)$ denotes the root of $T^{r}(I)$.
For a simple example how this proposition works see Section 6 in [@pwz2010-identifiability].
Proposition \[prop:monomial\] has been formulated for trivalent trees. However it can be easily extended to a more general case. For a given tree a *contraction of an edge* $(u,v)$ results in another tree obtained from the original tree by identifying the nodes $u$ and $v$ and removing the edge $(u,v)$. Let $T$ be a tree and let ${T^{*}}$ be any trivalent tree such that $T$ is obtained from ${T^{*}}$ by edge contractions. Then $\cM_{T}^{\kappa}\subseteq \cM_{{T^{*}}}^{\kappa}\subset\cK_{{T^{*}}}$ and by Lemma 4.2 in [@pwz2010-identifiability] the parameterization in (\[eq:kappa\_def\_general\]) remains valid for $T$ but expressed in the coordinates of $\cK_{{T^{*}}}$.
Let $T$ be a star tree with four leaves, i.e. a tree with one inner node $r$ and four leaves connected to $r$ by edges $(r,i)$ for $i=1,2,3,4$. This tree can be obtained from the quartet tree $T^{*}$ in Figure \[fig:quartet\] by contracting $(r,a)$. The model of the star tree can be realized as a subset of $\cK_{{T^{*}}}$, i.e. the space of tree cumulants for the quartet tree. The coordinates of $\cK_{{T^{*}}}$ are obtained in Example \[ex:quartet1\] and the parametrization of $\cM_{T}^{\kappa}$ is given for example by $$\kappa_{1234}^{*}=\frac{1}{4}(1-\bar{\mu}_{r}^{2})\bar{\mu}_{r}^{2}\eta_{r,1}\eta_{r,2}\eta_{r,3}\eta_{r,4},$$ where edges $(r,3)$, $(r,4)$ in $T$ correspond to edges $(a,3)$, $(a,4)$ in $T^{*}$.
Note however that this star tree may be obtained from many different trivalent trees by edge contraction. It follows that there exist many ways to embed the model and retain the parametrization.
The semialgebraic description of the tripod tree model {#sec:semi-tripod}
======================================================
In this section we obtain the full semialgebraic description of the tripod tree model. This result is not new (see e.g. [@auvray2006sad], [@settimi1998gbg] and a special case given by [@pearl1986fusion Theorem 3.1]). However it is convenient to give a new proof of this result both to unify notation and to introduce the strategy which is used to attack the general case later. We begin with a definition.
\[def:hyperdet\] Let $A$ be a $2\times 2\times 2$ table. The hyperdeterminant of $A$ as defined by Gelfand, Kapranov, Zelevinsky [@gelfand1994dra Chapter 14] is given by $$\begin{aligned}
{\rm Det} \,A&=&(a_{000}^{2}a_{111}^{2}+a_{001}^{2}a_{110}^{2}+a_{010}^{2}a_{101}^{2}+a_{011}^{2}a_{100}^{2})\\
&-&2(a_{000}a_{001}a_{110}a_{111}+a_{000}a_{010}a_{101}a_{111}+a_{000}a_{011}a_{100}a_{111}\\
&+&a_{001}a_{010}a_{101}a_{110}+a_{001}a_{011}a_{110}a_{100}+a_{010}a_{011}a_{101}a_{100})\\
&+&4(a_{000}a_{011}a_{101}a_{110}+a_{001}a_{010}a_{100}a_{111}).\end{aligned}$$
If $\sum a_{ijk}=1$ then treating all entries formally as joint cell probabilities (without positivity constraints) we can simplify this formula using the change of coordinates to central moments. The reparameterizations in Appendix A are well defined for this extended space of probabilities and we have that $$\label{eq:hyperdet}
{\rm Det} \,A=\mu_{123}^{2}+4\mu_{12}\mu_{13}\mu_{23},$$ which can be verified by direct computations. We note in passing that a similar idea of treating moments formally lies behind the umbral calculus [@rota_cumulants].
From the construction of tree cumulants (c.f. Appendix \[app:change\]) it follows that for all $I\subseteq[n]$ such that $2\leq |I|\leq 3$. Henceforth, for clarity, these lower order tree cumulants will be written as their more familiar corresponding central moments.
\[lem:semi\_tripod\] Let $\cM_{T}$ be the general Markov model on a tripod tree $T$ rooted in any node of $T$. Let $P$ be a $2\times 2\times 2$ probability table for three binary random variables $(X_{1}, X_{2}, X_{3})$ with central moments $\mu_{12}, \mu_{13},\mu_{23}$, $\mu_{123}$ (equivalent to the corresponding tree cumulants) and $\lambda_{i}=\E X_{i}$, for $i=1,2,3$. Then $\cM_{T}^{\kappa}$ is given by $\lambda_{i}=\frac{1}{2}(1-\bar{\mu}_{i})$ for $i=1,2,3$ and $$\label{eq:star}\begin{array}{l}
\mu_{ij}=\frac{1}{4}(1-\bar{\mu}_{h}^2)\eta_{h,i}\eta_{h,j}\mbox{ for all } i\neq j\in \{1,2,3\} \mbox{ and }\\[.3cm]
\mu_{123}=\frac{1}{4}(1-\bar{\mu}_{h}^2)\bar{\mu}_{h} \eta_{h,1}\eta_{h,2}\eta_{h,3},
\end{array}$$ subject to constraints in (\[eq:constraints\]). Moreover, $P\in\cM_{T}$ if and only if $K=f_{p\kappa}(P)\in\cK_{T}=\cC_{3}$ satisfies the following inequalities $$\label{eq:series-ineq-tripod}
\mu_{12}\mu_{13}\mu_{23}\geq 0,$$ $$\label{eq:series-ineq-tripod1}
\mu_{12}^{2}\mu_{13}^{2}+\mu_{12}^{2}\mu_{23}^{2}+\mu_{13}^{2}\mu_{23}^{2}\leq {\rm Det} \,P \leq \min_{1\leq i<j\leq 3} \mu_{ij}^{2}$$ and $$\label{eq:series-ineq-tripod2}
{{\rm Det}\, P}\leq \left((1\pm \bar{\mu}_{i})\mu_{jk}\mp\mu_{123}\right)^{2},$$ for all $i=1,2,3$ where by $j,k$ we denote elements of $\{1,2,3\}\setminus i$.
By Remark \[rem:rootings\] $\cM_{T}^{\kappa}$ does not depend on the rooting and hence we can assume that $T$ is rooted in $h$. In this case the parameterization in (\[eq:star\]) follows from Proposition \[prop:monomial\].
Denote by $\cM$ the subset of $\cK_{T}$ given by inequalities in (\[eq:series-ineq-tripod\]), (\[eq:series-ineq-tripod1\]) and (\[eq:series-ineq-tripod2\]). We need to show that $\cM=\cM_{T}^{\kappa}$. First we prove that $\cM_{T}^{\kappa}\subseteq \cM$. Let $K=\psi_{T}(\omega)$ for some $\omega\in\Omega_{T}$ with coordinates given by $\bar{\mu}_{h}$ and $\bar{\mu}_{i}$, $\eta_{h,i}$ for $i=1,2,3$. From (\[eq:star\]) $$\label{eq:prod-mus}
\mu_{12}\mu_{13}\mu_{23}=\left(\frac{1}{4}(1-\bar{\mu}_{h}^{2})\right)^{3}(\eta_{h,1}\eta_{h,2}\eta_{h,3})^{2}.$$ Since by (\[eq:constraints\]) $\bar{\mu}_{h}\in[-1,1]$, this implies the inequality in (\[eq:series-ineq-tripod\]). Moreover, we have $$\label{eq:det-in-model}
{\rm Det} \,P=\mu_{123}^{2}+4\mu_{12}\mu_{13}\mu_{23}=\frac{1}{16}(1-\bar{\mu}_{h}^{2})^{2}(\eta_{h,1}\eta_{h,2}\eta_{h,3})^{2}.$$ Multiplying both sides by $\bar{\mu}_{h}^{2}$ and applying the second equation in (\[eq:star\]) gives that $$\label{eq:s-tripod}
\bar{\mu}_{h}^{2}\,{{\rm Det} \,P}={\mu_{123}^{2}},\quad (1-\bar{\mu}_{h}^{2})\,{{\rm Det} \,P}={4\mu_{12}\mu_{13}\mu_{23}}.$$ On the other hand (\[eq:star\]) and (\[eq:det-in-model\]) imply also that $$\label{eq:eta-tripod}
\eta_{h,i}^{2}\,\mu_{jk}^{2}={\rm Det} \,P\quad\mbox{ for all } i=1,2,3.$$ Again by substituting $\mu_{ij}$ for $\frac{1}{4}(1-\bar{\mu}_{h}^{2})\eta_{h,i}\eta_{h,j}$ and rearranging we obtain $$\label{eq:sum-of-squares}
\mu_{12}^{2}\mu_{13}^{2}+\mu_{12}^{2}\mu_{23}^{2}+\mu_{13}^{2}\mu_{23}^{2}=\frac{1}{16}(1-\bar{\mu}_{h}^{2})^{2}(\eta_{h,1}^{2}+\eta_{h,2}^{2}+\eta_{h,3}^{2}){\rm Det} P.$$
Since necessarily $\eta_{h,i}^{2},\bar{\mu}_{h}^{2}\in[0,1]$ then (\[eq:s-tripod\]), (\[eq:eta-tripod\]) and (\[eq:sum-of-squares\]) imply that $$\mu_{12}^{2}\mu_{13}^{2}+\mu_{12}^{2}\mu_{23}^{2}+\mu_{13}^{2}\mu_{23}^{2}\leq {\rm Det} \,P \leq \min_{i,j} \mu_{ij}^{2}.$$ To show that $K$ satisfies (\[eq:series-ineq-tripod2\]) note that if $\mu_{jk}=0$ then it is trivially satisfied since in this case both sides of (\[eq:series-ineq-tripod2\]) are equal to $\mu_{123}^{2}$. If $\mu_{jk}\neq 0$ then (\[eq:series-ineq-tripod2\]) can be equivalently rewritten as $$\label{eq:stare-nowe}
|\mu_{jk}|\sqrt{{\rm Det}\,P}\pm\mu_{123}\mu_{jk}\leq (1\pm \bar{\mu}_{i})\mu_{jk}^{2}.$$ Now simply use (\[eq:star\]) to substitute for the corresponding moments. After trivial reductions we then obtain $$|\eta_{h,i}|\pm \bar{\mu}_{h}\eta_{h,i}\leq (1\pm \bar{\mu}_{i})$$ which is equivalent to (\[eq:constraints\]). Therefore since by hypothesis (\[eq:constraints\]) holds we also have that $\cM_{T}^{\kappa}\subseteq \cM$.
Now we show that $\cM\subseteq {\cM_{T}^{\kappa}}$ by proving that for $K\in\cM$ a parameter $\omega$ in (\[eq:star\]) exists which satisfies constraints defining $\Omega_{T}$ and $K=\psi_{T}(\omega)$. Let $P=f_{p\kappa}^{-1}(K)$ then from (\[eq:series-ineq-tripod\]) we know that ${\rm Det} \,P\geq 0$. So consider separately two situations: first when ${\rm Det} \,P=0$ and second when ${\rm Det} \,P>0$. In the first case again from (\[eq:series-ineq-tripod\]) necessarily $\mu_{123}=0$. Moreover, the inequality (\[eq:series-ineq-tripod1\]) implies that at least two covariances are zero. If all the covariances are zero then taking $\eta_{h,1}=\eta_{h,2}=\eta_{h,3}=0$ and $\bar{\mu}_{h}^{2}=1$ we obtain a valid choice of parameters in (\[eq:star\]) and their values satisfy (\[eq:constraints\]). When one covariance is non-zero, say $\mu_{12}\neq 0$, then if a choice of parameters exists it must satisfy $\bar{\mu}_{h}^{2}\neq 1$, $\eta_{h,1},\eta_{h,2}\neq 0$ and $\eta_{h,3}=0$. Such a choice of parameters will exist if we can ensure that $\mu_{12}=(1-\bar{\mu}_{h}^{2})\eta_{h,1}\eta_{h,2}$. This follows from Corollary 2 in [@gilula1979svd] which states that if only $\mu_{12}\neq 0$ then there always exists a choice of parameters for model $X_{1}\indep X_{2}|H$, where $H$ is hidden.
Assume now that ${\rm Det} \,P>0$. By (\[eq:series-ineq-tripod1\]) this implies that $\mu_{ij}\neq 0$ for each $i<j=1,2,3$. Set $\bar{\mu}_{h}^{2}=\frac{\mu_{123}^{2}}{{{\rm Det} \,P}}$ and $\eta_{h,i}^{2}=\frac{{{\rm Det} \,P}}{\mu_{jk}^{2}}$ for $i=1,2,3$. It follows that $(\frac{1}{4}(1-\bar{\mu}_{h}^{2}))^{2}\eta_{h,i}^{2}\eta_{h,j}^{2}=\mu_{ij}^{2}$ for $i,j=1,2,3$ and $(\frac{1}{4}(1-\bar{\mu}_{h}^{2}))^{2} \bar{\mu}_{h}^{2} \eta_{h,1}^{2}\eta_{h,2}^{2}\eta_{h,3}^{2}=\mu_{123}^{2}$. This coincides with (\[eq:star\]) modulo the sign. It can be easily shown that $\mu_{12}\mu_{13}\mu_{23}>0$ implies that there exist a choice of signs for $\eta_{h,i}$ for $i=1,2,3$ such that $$\frac{1}{4}(1-\bar{\mu}_{h}^{2})\eta_{h,i}\eta_{h,j}=\mu_{ij}$$ for all $1\leq i<j\leq 3$ as in (\[eq:star\]). For example set ${\rm sgn}(\eta_{h,i})={\rm sgn}(\mu_{jk})$ and use the fact that by our assumption ${\rm sgn}(\mu_{ij})={\rm sgn}(\mu_{ik}){\rm sgn}(\mu_{jk})$. This choice of signs already determines the sign of $\bar{\mu}_{h}$ so that $$\frac{1}{4}(1-\bar{\mu}_{h}^{2}) \bar{\mu}_{h} \eta_{h,1}\eta_{h,2}\eta_{h,3}=\mu_{123}$$ holds.
It remains to show that parameters set in this way satisfy the constraints defining $\Omega_{T}$. First note that since $0\leq 4\mu_{12}\mu_{13}\mu_{23}\leq {\rm Det} \,P$ then $\bar{\mu}_{h}^{2}\in [0,1]$ as required. From Appendix D in [@pwz2010-identifiability] we know that if $(\eta_{h,1},\eta_{h,2},\eta_{h,3},\bar{\mu}_{h})$ is one choice of parameters then there exists only one alternative choice and it is $(-\eta_{h,1},-\eta_{h,2},-\eta_{h,3},-\bar{\mu}_{h})$. For a fixed $i=1,2,3$ it is easily checked that $(\eta_{h,i},\bar{\mu}_{h})$ satisfies (\[eq:constraints\]) if and only if $(-\eta_{h,i},-\bar{\mu}_{h})$ does. Therefore we can assume that $\eta_{h,i}=\frac{\sqrt{{\rm Det} P}}{|\mu_{jk}|}>0$. In this case $\bar{\mu}_{h}=s({j,k})\frac{\mu_{123}}{\sqrt{{\rm Det}P}}$ where $s({j,k})={\rm sgn}(\mu_{jk})$. It follows that (\[eq:constraints\]) is satisfied if and only if $$\label{eq:pm1}
-(1\mp\bar{\mu}_{i})\mu_{jk}^{2}\leq \sqrt{\mu_{jk}^{2}{\rm Det} \,P} \pm\mu_{123}\mu_{jk}\leq (1\pm\bar{\mu}_{i})\mu_{jk}^{2}.$$ However, by (\[eq:series-ineq-tripod\]), $\sqrt{\mu_{jk}^{2}{\rm Det} \,P} \pm\mu_{123}\mu_{jk}\geq 0$ and hence the first inequality in (\[eq:pm1\]) is satisfied. The second inequality in (\[eq:pm1\]) is exactly (\[eq:stare-nowe\]) which is equivalent to (\[eq:series-ineq-tripod2\]).
A connection with tree metrics {#sec:metrics}
==============================
Now let $T$ be a general undirected tree with $n$ leaves and $T^{r}=(V,E)$ is the $T$ rooted in $r\in V$. Before stating the main theorem of the paper we first show how to obtain an elegant set of necessary constraints on $\cM_{T}$. In this section we assume that $\bar{\mu}_{r}^{2}\neq 1$ and $\eta_{u,v}\neq 0$ for all $(u,v)\in E$. By the Remark 4.3 in [@pwz2010-identifiability] this implies that $\bar{\mu}_{v}^{2}\neq 1$ for all $v\in V$. Since ${\rm Var}(Y_{u})=\frac{1}{4}(1-\bar{\mu}_{u}^{2})$ the correlation between $Y_{u}$ and $Y_{v}$ is defined as $\rho_{uv}=\frac{4\mu_{uv}}{\sqrt{(1-\bar{\mu}_{u}^{2})(1-\bar{\mu}_{v}^{2})}}$. This gives $$\label{eq:rhouv}
\rho_{uv}=\eta_{u,v}\sqrt{\frac{1-\bar{\mu}_{u}^{2}}{1-\bar{\mu}_{v}^{2}}}=\eta_{v,u}\sqrt{\frac{1-\bar{\mu}_{v}^{2}}{1-\bar{\mu}_{u}^{2}}}.$$
\[lem:prodrho\] For any $i,j\in [n]$ let $E({ij})$ be the set of edges on the unique path joining $i$ and $j$ in $T$. Then $$\label{eq:prod_rij}
\rho_{ij}=\prod_{(u,v)\in E({ij})}\rho_{uv}$$ for each probability distribution in ${\cM}_T^{\kappa}$ such that all the correlations are well defined.
By (\[eq:kappa\_def\_general\]) applied to $T(ij)$ we have $\mu_{ij}=\frac{1}{4}(1-\bar{\mu}_{r}^{2})\prod_{(u,v)\in E(ij)}\eta_{u,v}$, where $r$ is the root of the path between $i$ and $j$ and hence $$\rho_{ij}=\sqrt{\frac{1-\bar{\mu}_{r}^{2}}{1-\bar{\mu}_{i}^{2}}}\sqrt{\frac{1-\bar{\mu}_{r}^{2}}{1-\bar{\mu}_{j}^{2}}}\prod_{(u,v)\in E(ij)}\eta_{u,v}.$$ Now apply (\[eq:rhouv\]) to each $\eta_{u,v}$ in the product above to show (\[eq:prod\_rij\]).
The above equation allows us to demonstrate an interesting reformulation of our problem in term of tree metrics (c.f. [@semple2003pol Section 7]) which we explain below (see also Cavender [@cavender1997443]).
\[def:treemetric\] An arbitrary function $\delta: \,[n]\times [n]\rightarrow\R$ is called a *tree metric* if there exists a tree $T=(V,E)$ with the set of leaves given by $[n]$ and with a positive real-valued weighting $w: E\rightarrow \R_{>0}$ such that for all $i,j\in [n]$ $$\delta(i,j)=\left\{\begin{array}{ll}
\sum_{e\in E(ij)} w(e),&\mbox{ if } i\neq j,\\
0, & \mbox{otherwise}.
\end{array}\right.$$
Let now $d: V\times V\rightarrow\R$ be a map defined as $$d(k,l)=\left\{\begin{array}{ll}
-\log(\rho_{kl}^{2}), & \mbox{for all } k,l\in V \mbox{ such that } \rho_{kl}\neq 0,\\
+\infty, & \mbox{otherwise}
\end{array}\right.$$ then $d(k,l)\geq 0$ because $\rho_{kl}^{2}\leq 1$ and $d(k,k)=0$ for all $k\in V$ since $\rho_{kk}=1$. If $K\in\cM_T^\kappa$ then by (\[eq:prod\_rij\]) $\rho_{ij}^{2}=\prod_{e\in E(ij)}\rho_{e}^{2}$ and we can define map $d_{(T;K)}: [n]\times [n]\rightarrow \R$ $$\label{eq:metric}
-\log (\rho_{ij}^{2})=d_{(T;K)}(i,j)=\left\{\begin{array}{ll}
\sum_{(u,v)\in E(ij)} d(u,v), & \mbox{if } i\neq j,\\
0, & \mbox{otherwise.}
\end{array}\right.$$ This map is a tree metric by Definition \[def:treemetric\]. In our case we have a point in the model space defining all the second order correlations and $d_{(T;K)}(i,j)$ for $i,j\in [n]$. The question is: What are the conditions for the “distances” between leaves so that there exists a tree $T$ and edge lengths $d(u,v)$ for all $(u,v)\in E$ such that (\[eq:metric\]) is satisfied? Or equivalently: What are the conditions on the absolute values of the second order correlations in order that $\rho_{ij}^{2}=\prod_{e\in E_{ij}}\rho_{e}^{2}$ (for some edge correlations) is satisfied? We have the following theorem.
\[th:3metric\] A function $\delta:\, [n]\times [n]\rightarrow\R$ is a tree metric on $[n]$ if and only if for every four (not necessarily distinct) elements $i,j,k,l\in[n]$, $$\delta(i,j)+\delta(k,l)\leq \max\left\{\delta(i,k)+\delta(j,l),\delta(i,l)+\delta(j,k) \right\}.$$ Moreover, a tree metric defines the tree uniquely.
Since $\delta(i,j)=\log(-\rho_{ij})$ the constraints in Theorem \[th:3metric\] translate in terms of correlations to $$-\log(\rho_{ij}^{2}\rho_{kl}^{2})\leq -\min \{\log(\rho_{ik}^{2}\rho_{jl}^{2}),\log(\rho_{il}^{2}\rho_{jk}^{2}) \} .$$ Since $\log$ is a monotone function we obtain $$\label{eq:suffic-ineq1}
\min\left\{\frac{\rho_{ik}^{2}\rho_{jl}^{2}}{\rho_{ij}^{2}\rho_{kl}^{2}},\frac{\rho_{il}^{2}\rho_{jk}^{2}}{\rho_{ij}^{2}\rho_{kl}^{2}} \right\}=\min\left\{\frac{\mu_{ik}^{2}\mu_{jl}^{2}}{\mu_{ij}^{2}\mu_{kl}^{2}},\frac{\mu_{il}^{2}\mu_{jk}^{2}}{\mu_{ij}^{2}\mu_{kl}^{2}} \right\}\leq 1$$ for all not necessarily distinct leaves $i,j,k,l\in [n]$. Hence using the relation between correlations and tree metrics given in [@cavender1997443] we managed to provide a set of simple semialgebraic constraints on the model. Furthermore, later in Theorem \[th:parameters0\] we show that these constraints are not the only active constraints on the model $\cM_{T}$. Before we present this theorem it is helpful to make some simple observations about the relationship between correlations and probabilistic tree models.
Since $\rho_{uv}$ can have different signs we define a signed tree metric as a tree metric with an additional sign assignment for each edge of $T$. There are additional natural constraints which assure that there exists a choice of signs for edge correlations such that (\[eq:prod\_rij\]) is satisfied.
\[lem:inequalities2\] Let $T$ be a tree with $n$ leaves. Suppose that we have a map $\sigma:[n]\times[n]\rightarrow \{-1,1\}$. Then there exists a map $s_{0}:E\rightarrow \{-1,1\}$ such that for all $i,j\in [n]$ $$\label{eq:sij}
\sigma(i,j)=\prod_{(u,v)\in E(ij)}s_{0}(u,v)$$ if and only if for all triples $i,j,k\in[n]$ $\sigma(i,j)\sigma(i,k)\sigma(j,k)=1$.
First assume that the map $s_{0}:E\rightarrow \{-1,1\}$, given in the statement of the lemma, exists. This induces a map $s:V\times V\rightarrow \{-1,1\}$ such that $s(k,l)=\prod_{(u,v)\in E(kl)}s_{0}(u,v)$. For any triple $i,j,k$ there exists a unique inner node $h$ which is the intersection of all three paths between $i,j,k$. By the above equation the choice of signs for all $(u,v)\in E$ gives $s(i,h),s(j,h)$ and $s(k,h)$. Since $s(i,j)=s(i,h)s(j,h)$ and the same for the two other pairs, we get that $s(i,j)s(i,k)s(j,k)=s^2(i,h)s^2(j,h)s^2(k,h)=1$ and the result follows since by construction $\sigma(i,j)=s(i,j)$ for all $i,j\in [n]$.
Now we prove the converse implication. Whenever there is a path $E(uv)$ in $T$ such that all its inner nodes have degree two then a sign assignment satisfying (\[eq:sij\]) exists if and only if there exists a sign assignment for the same tree but with $E(uv)$ contracted to a single edge $(u,v)$. Hence we can assume that the degree of each inner node is at least three.
We use an inductive argument with respect to number of hidden nodes. First we will show that the theorem is true for trees with one inner node (star trees). In this case we will use induction with respect to number of leaves. It can easily be checked directly that the theorem is true for the tripod tree. Assume it works for all star trees with $k\leq m-1$ leaves and let $T$ be a star tree with $m$ leaves. By assumption for any three leaves $i,j,k$: $\sigma(i,j)\sigma(i,k)\sigma(j,k)=1$. If we consider a subtree with $(1,h)$ deleted then by induction assumption we can find a consistent choice of signs for all remaining edge correlations. A choice of a sign for $(1,h)$ consistent with (\[eq:sij\]) exists if for all $i\geq 2$ $\sigma(1,i)=s_{0}{(1,h)}s_{0}(i,h)$. This is true if either $\sigma(1,i)s_{0}(i,h)=1$ for all $i$ or $\sigma(1,i)s_{0}(i,h)=-1$ for all $i$. Assume it is not true, i.e. there exist two leaves $i,j$ such that $\sigma(1,i)s_{0}(i,h)=1$ and $\sigma(1,j)s_{0}(j,h)=-1$. Then in particular since $\sigma(i,j)=s_{0}(i,h)s_{0}(j,h)$ we would have that $\sigma(1,i)\sigma(1,j)\sigma(i,j)=-1$ which contradicts our assumption.
If the number of the inner nodes is greater than one then pick an inner node $h$ adjacent to exactly one inner node. Let $h'$ be the inner node adjacent to $h$ and let $I$ be a subset of leaves which are adjacent to $h$. Choose one $i\in I$ and consider a subtree $T'$ obtained by removing all leaves in $I$ and the incident edges apart from the node $i$ and the edge $(h,i)$. By the induction, since $h$ has degree two in the resulting subtree, we can find signs for all edge correlations of $T'$. Set $s_{0}(h,h')=1$ then $s_{0}(h,i)=s(h',i)$ which identifies $s_{0}(h,i)$. The result follows since the choice of $i\in I$ was arbitrary.
The lemma implies that for all $i,j,k\in [n]$ necessarily $\rho_{ij}\rho_{ik}\rho_{jk}\geq 0$ or equivalently that for all $i,j,k\in [n]$ necessarily $\mu_{ij}\mu_{ik}\mu_{jk}\geq 0$. This in particular implies that $\frac{\mu_{ik}\mu_{jl}}{\mu_{ij}\mu_{kl}}\geq 0$ for all $i,j,k,l\in[n]$. By taking the square root in (\[eq:suffic-ineq1\]) these constraints can be combined and rearranged to give the inequalities $$\label{eq:suffic-ineq2}
0\leq \min\left\{\frac{\mu_{ik}\mu_{jl}}{\mu_{ij}\mu_{kl}},\frac{\mu_{il}\mu_{jk}}{\mu_{ij}\mu_{kl}} \right\}\leq 1$$ for all (not necessarily distinct) $i,j,k,l\in [n]$. In this way we obtain a set of elegant semialgebraic constraints on the model space. In Theorem \[th:parameters0\] we show that (\[eq:suffic-ineq2\]) provides the complete set of inequality constraints on $\cM_{T}$ that involve only second order moments in their expression.
The fact that additional constraints involving higher order moments exist is illustrated in the following simple example.
Consider the tripod tree model in Lemma \[lem:semi\_tripod\]. Let $K$ be a point in $\cK_{T}$ given by ${\lambda}_{i}=0.15$ for $i=1,2,3$, $\mu_{ij}=0.0625$ (or equivalently $\rho_{ij}=0.49$) for each $i<j$ and $\mu_{123}= 0.0526$. This point lies in the space of tree cumulants $\cK_{T}$ which can be checked by mapping back the central moments to probabilities, since the resulting vector $[p_{\a}]$ lies in $\Delta_{7}$.
Clearly $K$ satisfies all the tree metric constraints in (\[eq:suffic-ineq2\]). The equation (\[eq:prod\_rij\]) is satisfied with $\rho_{hi}=0.7$ for each $i=1,2,3$. We now show that despite this $K\notin\cM_{T}^{\kappa}$. For if $K\in\cM_{T}^{\kappa}$ then we could find $\bar{\mu}_{h}$ and $\eta_{h,i}$ satisfying constraints in (\[eq:constraints\]) so that (\[eq:star\]) held. Using the formulas in Corollary 5.5 in [@pwz2010-identifiability] it is easy to compute that $\bar{\mu}_{h}=0.86$ and $\eta_{h,i}\approx 0.98$. However, $K$ is not in the model since these parameters do not lie in $\Omega_{T}$. Indeed, $$(1+\bar{\mu}_{h})\eta_{h,i}\approx 1.8228 > (1+\bar{\mu}_{i})=1.7$$ and hence (\[eq:constraints\]) is not satisfied.
The consequence of the fact that the parameters do not lie in $\Omega_{T}$ is that this parametrization does not lead to a valid assignment of conditional probabilities to the edges of the tree. For example with the values given above we can calculate that the induced marginal distribution for $(X_{i},H)$ would have to satisfy which is obviously not a consistent assignment for a probability model. Thus there must exist other constraints involving observed higher order moments that need to hold for a probability model to be valid. We note that for the tripod tree these were given by Lemma \[lem:semi\_tripod\].
In Appendix \[sec:proof\] we prove the following theorem which gives the complete set of constraints which have to be satisfied by tree cumulants to lie in $\cM_{T}$ in the case when $T$ is a trivalent tree. Let $P\in\Delta_{2^{n}-1}$ be the probability distribution of the vector $(X_{1},\ldots, X_{n})$ then for any $i,j,k\in[n]$ let $P^{ijk}$ denote the $2\times 2\times 2$ table of the marginal distribution of $(X_{i},X_{j},X_{k})$.
\[th:parameters0\] Let $T=(V,E)$ be a trivalent tree with $n$ leaves. Let $\cM_{T}\subseteq\Delta_{2^{n}-1}$ be the model defined as an image of the parametrization in (\[eq:p\_albar2\]) and $\cM_{T}^{\kappa}=f_{p\kappa}(\cM_{T})$. Suppose $P$ is a joint probability distribution on $n$ binary variables and $K=f_{p\kappa}(P)$. Then $K\in\cM_{T}^{\kappa}$ (or equivalently $P\in\cM_T$) if and only if the following five conditions hold:
(C1)
: For each edge split $A|B$ (c.f. Definition \[def:edge-part\]) of the set of leaves of $T$ whenever we have four nonempty subsets (not necessarily disjoint) $I_{1},I_{2}\subseteq A$, $J_{1},J_{2}\subseteq B$ then $$\kappa_{I_{1}J_{1}}\kappa_{I_{2}J_{2}}-\kappa_{I_{1}J_{2}}\kappa_{I_{2}J_{1}}=0.$$
(C2)
: For all $1\leq i<j<k\leq n$ we have $$\mu_{ij}\mu_{ik}\mu_{jk}\geq 0$$ and $$\label{eq:ineq-series}
(\mu_{ij}^{2}\mu_{ik}^{2}+\mu_{ij}^{2}\mu_{jk}^{2}+\mu_{ik}^{2}\mu_{jk}^{2})\leq {\rm Det} \,P^{ijk}\leq \min_{l,l'\in\{i,j,k\}} \mu_{l l'}^{2} ,$$
(C3)
: for all $1\leq i<j<k\leq n$ $$\label{eq:ineqI}
\begin{array}{l}
{\rm Det} P^{ijk}\leq \left((1\pm \bar{\mu}_{\sigma(i)})\mu_{\sigma(j)\sigma(k)}\mp \mu_{ijk}\right)^{2},
\end{array}$$ for all three permutations $\sigma$ of ${\{i,j,k\}}$ such that $\sigma(j)<\sigma(k)$.\
(C4)
: for all $I\subseteq[n]$ if there exist $i,j\in I$ such that $\mu_{ij}=0$ then $\kappa_{I}=0$\
(C5)
: for any $ i,j,k,l\in[n]$ such that there exists $e\in E$ inducing a split $(A)(B)$ such that $i,j\in A$ and $k,l\in B$ we have $$\label{eq:ineqII}
\begin{array}{l}
(2\mu_{ik}\mu_{jl})^{2}\leq (\sqrt{\mu_{jl}^{2}{\rm Det} \,P^{ijk}}\pm\mu_{jl}\mu_{ijk})(\sqrt{{\rm Det} \,P^{ikl}}\mp\mu_{ikl}).
\end{array}$$
Example: The quartet tree model {#sec:quartet}
===============================
We can check that (modulo the numerical error) the point $K\in\cK_{T}$ provided in Table 1 in [@pwz2010-identifiability] satisfies all the constraints in Theorem \[th:parameters0\]. To check (C1) note for example that $$\kappa_{13}\kappa_{24}-\kappa_{14}\kappa_{23}=0.0160\cdot 0.0128-0.0160\cdot 0.0128=0,$$ $$\kappa_{123}\kappa_{134}-\kappa_{1234}\kappa_{13}=(-0.00384)\cdot (-0.00256)-0.0006144
\cdot 0.016=1.6941\cdot 10^{-21}\approx 0.$$ The last equation shows that due to the limited precision of numerical software typically the equations in (C1) will not be satisfied exactly even if a point lies in the model class.
To check (C2) verify for example that ${\rm Det} P^{123}\approx 4.096\cdot 10^{-5}$, $\min \{\mu_{12}^{2},\mu_{13}^{2},\mu_{23}^{2}\}\approx 1.6384\cdot 10^{-4}$ and $\mu_{12}^{2}\mu_{13}^{2}+\mu_{12}^{2}\mu_{23}^{2}+\mu_{13}^{2}\mu_{23}^{2}\approx 4.7186\cdot 10^{-7}$. For (C3) again we check only one of all the constraints. One has $$\left((1\pm \bar{\mu}_{1})\mu_{23}\mp \mu_{123}\right)^{2}=\{1.3271\cdot 10^{-4}, 1.9825\cdot 10^{-4}\}$$ $$\left((1\pm \bar{\mu}_{2})\mu_{13}\mp \mu_{123}\right)^{2}=2.5600\cdot 10^{-4}$$ $$\left((1\pm \bar{\mu}_{3})\mu_{12}\mp \mu_{123}\right)^{2}=\{9.4372\cdot 10^{-4}, 0.0011\}$$ and hence $${\rm Det} P^{123}\approx 4.096\cdot 10^{-5} \leq \min\left\{\left((1\pm \bar{\mu}_{\sigma(i)})\mu_{\sigma(j)\sigma(k)}\mp \mu_{ijk}\right)^{2}\right\}\approx 1.3271\cdot 10^{-4}$$ is satisfied. We can check (C5) in a similar way.
From the point of view of the original motivation a different scenario is of an interest. Imagine that we have $K\in\cK_{T}$ such that all the equations in (C1) are satisfied, i.e. all the phylogenetic invariants hold. If one of the constraints in (C2)-(C5) does not hold then $K\notin\cM_{T}^{\kappa}$. This shows that the method of phylogenetic invariants as it is commonly used may lead to spurious results. For example consider sample proportions and the corresponding tree cumulants as in Table \[tab:SEMquartet2\].
$$\begin{array}{ccccc}
\a & I & p_{\a} & \lambda_{I} & \kappa_{I}\\
\hline
0000 & \emptyset &0.0755& 1.0000& 1.0000\\
0001 & 4& 0.0483& 0.5800 & 0\\
0010 & 3& 0.0483& 0.5800 & 0\\
0011 & 34& 0.0579& 0.3700 & 0.0336 \\
0100 & 2& 0.0479& 0.6200 & 0\\
0101 & 24& 0.0399& 0.3724 & 0.0128 \\
0110 & 23& 0.0399& 0.3724 & 0.0128 \\
0111 & 234& 0.0623 & 0.2422 & -0.0020 \\
1000 & 1& 0.0171& 0.5800 & 0\\
1001 & 14& 0.0315& 0.3716 & 0.0352 \\
1010 & 13& 0.0315& 0.3716 & 0.0352 \\
1011 & 134& 0.0699& 0.2498 & -0.0056 \\
1100 & 12& 0.0695& 0.4300 & 0.0704 \\
1101 & 124& 0.0903& 0.2702 & -0.0084 \\
1110 & 123& 0.0903& 0.2702 & -0.0084 \\
1111 & 1234& 0.1799& 0.1799 & 0.0014 \\
\end{array}$$
It can be checked that for this point all the equations in (C1) are satisfied. However it is not in the model space. Using the formulas in Corollary 5.5 [@pwz2010-identifiability] it is simple to confirm that the point mapping to $K$ satisfies $\theta^{(1)}_{1|0}=-0.3$. This cannot therefore be a probability and so $\theta\notin \Theta_{T}$.
Discussion
==========
The new coordinate system proposed in [@pwz2010-identifiability] provides a better insight into the geometry of phylogenetic tree models with binary observations. The elegant form of the parameterization is useful and has already enabled us to obtain the full geometric description of the model class. One of the interesting implications of this result for phylogenetic tree models is that we could consider different simpler model classes containing the original one in such a way that the whole evolutionary interpretation in terms of the tree topologies remains valid. If we are interested only in the tree we could consider the model defined only by a subsets of constraints in Theorem \[th:parameters0\] involving only covariances. The cost of this reduction is that the conditional independencies induce by the original model do not hold anymore which in turn affects the interpretation of the model. We note that this approach is in a similar spirit to that employed to motivate the MAG model class introduced in [@spirtes97heuristic].
This work has encouraged us to use this reparametrization to estimate models within Bayesian framework. When the sample proportions lie in the model class then we have already noted that the MLEs are given by formulas in Corollary 5.5 in [@pwz2010-identifiability]. This may help to answer some questions about analytic solutions for maximum likelihood problems in phylogenetics (see e.g. [@yang2000complexity], [@chor2000mml]). In a later paper we prove various formal methods for incorporating the semialgebraic geometry in a model to improve the prior specification of the tree model and hence enhance the estimation of the model parameters. In particular we discuss how samplers can be constructed which have better properties than standard ones when applied in this context.
Acknowledgements {#acknowledgements .unnumbered}
================
Diane Maclagan and John Rhodes contributed substantially to this paper. We would also like to thank Bernd Sturmfels for a stimulating discussion at the early stage of our work and Lior Pachter for pointing out reference [@cavender1997443].
|
---
abstract: 'The well-known Gumbel-Max Trick for sampling elements from a categorical distribution (or more generally a nonnegative vector) and its variants have been widely used in areas such as machine learning and information retrieval. To sample a random element $i$ (or a Gumbel-Max variable $i$) in proportion to its positive weight $v_i$, the Gumbel-Max Trick first computes a Gumbel random variable $g_i$ for each positive weight element $i$, and then samples the element $i$ with the largest value of $g_i+\ln v_i$. Recently, applications including similarity estimation and graph embedding require to generate $k$ independent Gumbel-Max variables from high dimensional vectors. However, it is computationally expensive for a large $k$ (e.g., hundreds or even thousands) when using the traditional Gumbel-Max Trick. To solve this problem, we propose a novel algorithm, *FastGM*, that reduces the time complexity from $O(kn^+)$ to $O(k \ln k + n^+)$, where $n^+$ is the number of positive elements in the vector of interest. Instead of computing $k$ independent Gumbel random variables directly, we find that there exists a technique to generate these variables in descending order. Using this technique, our method FastGM computes variables $g_i+\ln v_i$ for all positive elements $i$ in descending order. As a result, FastGM significantly reduces the computation time because we can stop the procedure of Gumbel random variables computing for many elements especially for those with small weights. Experiments on a variety of real-world datasets show that FastGM is orders of magnitude faster than state-of-the-art methods without sacrificing accuracy and incurring additional expenses.'
author:
- 'Yiyan Qi$^{1}$, Pinghui Wang$^{2,1,*}$, Yuanming Zhang$^{1}$, Junzhou Zhao$^{1,*}$, Guangjian Tian$^{3}$,'
- 'Xiaohong Guan$^{2,1,4}$'
bibliography:
- 'COPH.bib'
- 'randpe.bib'
- 'ctstream.bib'
- 'simcar.bib'
- 'albitmap.bib'
- 'dynamic.bib'
title: 'Fast Generating A Large Number of Gumbel-Max Variables'
---
[^1]
<ccs2012> <concept> <concept\_id>10002950.10003648.10003671</concept\_id> <concept\_desc>Mathematics of computing Probabilistic algorithms</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10002951.10003317.10003338.10003342</concept\_id> <concept\_desc>Information systems Similarity measures</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10003752.10003809.10010055.10010057</concept\_id> <concept\_desc>Theory of computation Sketching and sampling</concept\_desc> <concept\_significance>500</concept\_significance> </concept> </ccs2012>
Acknowledgment {#acknowledgment .unnumbered}
==============
The research presented in this paper is supported in part by National Key R&D Program of China (2018YFC0830500), Shenzhen Basic Research Grant (JCYJ20170816100819428), National Natural Science Foundation of China (61922067, U1736205, 61902305), MoE-CMCC “Artifical Intelligence” Project (MCM20190701), Natural Science Basic Research Plan in Shaanxi Province of China (2019JM-159), Natural Science Basic Research Plan in ZheJiang Province of China (LGG18F020016).
[^1]: $^*$Corresponding Author.
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abstract: 'We present a 2-dimensional cellular automaton model for the simulation of pedestrian dynamics. The model is extremely efficient and allows simulations of large crowds faster than real time since it includes only nearest-neighbour interactions. Nevertheless it is able to reproduce collective effects and self-organization encountered in pedestrian dynamics. This is achieved by introducing a so-called *floor field* which mediates the long-range interactions between the pedestrians. This field modifies the transition rates to neighbouring cells. It has its own dynamics (diffusion and decay) and can be changed by the motion of the pedestrians. Therefore the model uses an idea similar to chemotaxis, but with pedestrians following a virtual rather than a chemical trace.'
author:
- Andreas Schadschneider
title: 'Cellular Automaton Approach to Pedestrian Dynamics - Theory'
---
Introduction
============
Efficient computer simulations of large crowds consisting of hundreds or thousands of individuals require simple models which nevertheless provide an accurate description of reality. One such class of models, so-called cellular automata (CA), has been studied in statistical physics for a long time [@Wolfram; @stauca].
In CA space, time and state variables are discrete which makes them ideally suited for high-performance computer simulations. However, CA modelling differs in several respects from continuum models. These are usually based on coupled differential equations which often can not be treated analytically. One has to solve them numerically and therefore the equations have to be discretized. In general, only space and time variables become discrete whereas the state variable is still continuous. One important point is now that CA are discrete from the beginning and that this discreteness is already taken into account in the definition of the model and its dynamics. This allows to obtain the desired behaviour in a much simpler way. On the other hand, the numerical solution of (discretized) differential equations is only accurate in the limit $\Delta x$, $\Delta t \to 0$. This is different in the CA where $\Delta x$ and $\Delta t$ are finite and accurate results can be obtained since the rules (dynamics) are designed such that the discreteness is an important part of the model.
In order to achieve complex behaviour in a simple fashion one often resorts to a stochastic description. A realistic situation seldomly can be described completely by a deterministic approach. Already minor events can lead to a very different behaviour due to the complexity of the interactions involved. For the problem of pedestrian motion this becomes evident e.g. in the case of a panic where the behaviour of people seems almost unpredictable. But also for “normal” situations a stochastic component in the dynamics can lead to a more accurate description of complex phenomena since it takes into account that we usually do not have full knowledge about the state of the system and its dynamics. Here one has to keep in mind that in general some sort of average over different realizations of the process (e.g. different sequences of random numbers) has to be taken. Even if there are single realizations which yield unrealistic behaviour the average process will be a good description of the real process. Furthermore a stochastic description allows to answer questions like “What is the probability that the evacuation of this building will take longer than 3 minutes ?” in a natural way.
In the following we will present a detailed description of the model and the basic philosophy of our approach. Applications are presented in Part II [@part2].
Other Modelling Approaches
==========================
During the last decade considerable research has been done on the topic of highway traffic using methods from physics [@juelich; @tgf97; @tgf99; @helb; @chowd; @nagel99; @dhrev]. Cellular automata inspired by the pioneering works [@NagelS; @BML] compose by now an important class of models. Most studies have been devoted to one-dimensional systems, where several analytic approaches exist to calculate or approximate the stationary state.
On the other hand, pedestrian dynamics has not been studied as extensively as vehicular traffic, especially using a cellular automata approach. One reason is probably its generically two-dimensional nature. In recent years, continuum models have been most successful in modelling pedestrian dynamics. An important example are the [*social force models*]{} (see e.g. [@helb; @dhrev; @social] and references therein). Here pedestrians are treated as particles[^1] subject to long-ranged forces induced by the social behaviour of the individuals. This leads to (coupled) equations of motion similar to Newtonian mechanics. There are, however, important differences since, e.g., in general the third law (“actio = reactio”) is not fulfilled.
In contrast to the social force models our approach is closer in spirit to the general strategy of modelling (elementary) forces on a microscopic level by the exchange of mediating particles which are bosons. It is therefore similar to [*active walker models*]{} [@activewalker; @trail] used so far mainly to describe trail formation, chemotaxis (see [@benjacob] for a review) etc. Here the walker leaves a trace by modifying the underground on his path. This modification is real in the sense that it could be measured in principle. For trail formation, vegetation is destroyed by the walker and in chemotaxis he leaves a chemical trace. In contrast, in our model the trace is virtual. Its main purpose is to transform effects of long-ranged interactions (e.g. following people walking some distance ahead) into a local interaction (with the “trace”). This allows for a much more efficient simulation on a computer.
Cellular automata for pedestrian dynamics have been proposed in [@fukui; @nagatani; @hubert]. These models can be considered as generalizations of the Biham-Middleton-Levine model for city traffic [@BML]. Most works have focussed on the occurrence of a jamming transition as the density of pedestrians is increased. All models have only nearest-neighbour interactions, except for the generalization proposed in [@hubert] which is used for analyzing evacuation processes on-board passenger ships. The other models use a kind of “sublattice-dynamics” which distinguishes between different types of pedestrians according to their preferred walking direction. Such an update is not easy to generalize to more complex situations where the walking direction can change. To our knowledge so far most other collective effects encountered empirically [@helb; @dhrev; @CrowdFluids; @CrowdFluids2; @weidmann; @panic] have not been reproduced using these models. Another discrete model has been proposed earlier by Gipps and Marksjös [@gipps]. This model is somewhat closer in spirit to our model than the cellular automata approaches of [@fukui; @nagatani; @hubert] since the transitions are determined by the occupancies of the neighbouring cells. However, this model can not reproduce all collective effects either. In [@bolay] a discretized version of the social force model has been introduced. The repulsive potentials by the pedestrians are stored in a global potential, with pedestrians reacting to the gradients of this global potential. Although this model is able to reproduce collective effects it is not flexible enough to treat individual reactions to other pedestrians, and collision-avoidance is not always guaranteed for velocities greater than 1.
Basic Principles of the Model {#sec_principles}
=============================
First we discuss some general principles we took into account in the development of our model [@ourpaper]. The implementation of the interactions between the pedestrians uses an idea similar to chemotaxis. The pedestrians leave a virtual trace which then influences the motion of other pedestrians. This allows for a very efficient implementation on a computer since now all interactions are local. The transition probabilities for all pedestrians just depend on the occupation numbers and strength of the virtual trace in his neighbourhood, i.e. we have translated the long-ranged spatial interaction into a local interaction with “memory”. The number of interaction terms in other long-ranged models, e.g.the social-force model, grows proportionally to the square of the number of particles whereas in our model it grows only linearly.
The idea of a virtual trace can be generalized to a so-called [*floor field*]{}. This floor field includes the virtual trace created by the pedestrians as well as a static component which does not change with time. The latter allows to model e.g. preferred areas, walls and other obstacles. The pedestrians then react to both types of floor fields.
To keep the model simple, we want to provide the particles with as little intelligence as possible and to achieve the formation of complex structures and collective effects by means of self-organization. In contrast to older approaches we do not make detailed assumptions about the human behaviour. Nevertheless the model is able to reproduce many of the basic phenomena.
The key feature to substitute individual intelligence is the floor field. Apart from the occupation number each cell carries an additional quantity (field) which can be either discrete or continuous. This field can have its own dynamics given by diffusion and decay coefficients.
Interactions between pedestrians are repulsive for short distances. One likes to keep a minimal distance to others in order to avoid bumping into them. In the simplest version of our model this is taken into account through hard-core repulsion which prevents multiple occupation of the cells. For longer distances the interaction is often attractive. E.g. when walking in a crowded area it is usually advantageous to follow directly behind the predecessor. Large crowds may also be attractive due to curiosity.
With two particle species moving in opposite directions, each with its own floor field, effects can be observed which are so far only achieved by continuous models [@social]: lane formation and oscillation of the direction of flow at doors. We consider this model to be another example for the ability of cellular automata to create complex behaviour out of simple rules and the great applicability of this approach to all kinds of traffic flow problems.
In contrast to vehicular traffic the time needed for acceleration and braking is negligible. The velocity distribution of pedestrians is sharply peaked [@CrowdFluids]. These facts naturally lead to a model where the pedestrians have a maximal velocity ${\ensuremath v_{max}}=1$, i.e. only transitions to neighbour cells are allowed. Furthermore, a greater ${\ensuremath v_{max}}$, i.e. pedestrians are allowed to move more than just one cell per timestep, would be harder to implement in two dimensions, especially when combined with parallel dynamics, and reduce the computational efficiency. The number of possible target cells increases quadratically with the interaction range. Furthermore one has to check whether the path is blocked by other pedestrians. This might even be ambigious for diagonal motion and crossing trajectories. Also higher velocity models lead to timescales which are much too small (see Sec. \[sec\_def\]).
Definition of the Model and its Dynamics {#sec_def}
========================================
The area available for pedestrians is divided into cells of approximately $40\times 40~cm^2$. This is the typical space occupied by a pedestrian in a dense crowd [@weidmann]. Each cell can either be empty or occupied by exactly one particle (pedestrian). For special situations it might be desirable to use a finer discretization, e.g. such that each pedestrian occupies four cells instead of one.
The update is done in parallel for all particles. This introduces a timescale into the dynamics which can roughly be identified with the reaction time $t_{\rm reac}$. In the deterministic limit, corresponding to the maximal possible walking velocity in our model, a single pedestrian (not interacting with others) moves with a velocity of one cell per timestep, i.e. $40~cm$ per timestep. Empirically the average velocity of a pedestrian is about $1.3~m/s$ [@weidmann]. This gives an estimate for the real time corresponding to one timestep in our model of approximately $0.3~sec$ which is of the order of the reaction time $t_{\rm reac}$, and thus consistent with our microscopic rules. It also agrees nicely with the time needed to reach the normal walking speed which is about $0.5~sec$. This corresponds to at least $v_{max}$ timesteps if the pedestrian can only accelerate by one unit per timestep. Therefore in models with large $v_{max}$ a timestep would correspond to a real time shorter than the smallest relevant timescale. This makes the model more complicated than necessary and reduces the efficiency of simulations.
Basic Rules {#sec_rules}
-----------
Each particle is given a preferred walking direction. From this direction, a $3 \times 3$ *matrix of preferences* is constructed which contains the probabilities for a move of the particle. The central element describes the probability for the particle not to move at all, the remaining 8 correspond to a move to the neighbouring cells (see Fig. \[fig\_prefs\]). The probabilities can be related to the velocity and the longitudinal and transversal standard deviations (see [@ourpaper; @diplom] for details). So the matrix of preferences contains information about the preferred walking direction and speed. In principle, it can differ from cell to cell depending on the geometry and aim of the pedestrians. In the simplest case the pedestrian is allowed to move in one direction only without fluctuations and in the corresponding matrix of preference only one element is one and all others are zero. In the following it is assumed that a matrix of preferences is given at every timestep for each pedestrian. These can e.g. be obtained from some model for route selection which assigns certain routes to each pedestrian.
This ansatz can easily be extended by fixing the direction of preference for each cell separately, e.g. to handle structures inside buildings. Then the particles would use the matrix belonging to the cell they occupy at a given step. However, a similar effect can be obtained much simpler by introducing a second floor field (see Sec. \[sec\_floor\]).
In each update step for each particle a desired move is chosen according to these probabilities. This is done in parallel for all particles. If the target cell is occupied, the particle does not move. If it is not occupied, and no other particle targets the same cell, the move is executed. If more than one particle share the same target cell, one is chosen according to the relative probabilities with which each particle chose their target. This particle moves while its rivals for the same target keep their position (see Fig. \[fig\_conflict\]).
The rules presented up to here are a straightforward generalization of the CA rules used so far for the description of traffic flow [@fukui; @nagatani]. The main difference is that in principle transitions in all directions are possible and each pedestrian $j$ might have her own preferred direction of motion characterized by a matrix of preferences $M^{(j)}$. The only interaction between particles taken into account so far is hard-core exclusion.
Floor Field {#sec_floor}
-----------
In order to reproduce certain collective phenomena it is necessary to introduce further longer-ranged interactions. In some continuous models this is done using the idea of a social force [@helb; @dhrev; @social]. Here we present a different approach. Since we want to keep the model as simple as possible we try to avoid using a long-range interaction explicitly. Instead we introduce the concept of a [*floor field*]{} which is modified by the pedestrians and which in turn modifies the transition probabilities. This allows to take into account interactions between pedestrians and the geometry of the system (building) in a unified and simple way without loosing the advantages of local transition rules. The floor field modifies the transition probabilities in such a way that a motion into the direction of larger fields is preferred.
The floor field can be thought of as a second grid of cells underlying the grid of cells occupied by the pedestrians. It can be discrete or continuous. As already explained in Sec. \[sec\_principles\] we distinguish two types of fields which will be called static and dynamic floor fields, respectively.
The [*dynamic floor field*]{} $D$ is just the virtual trace left by the pedestrians (see Sec. \[sec\_principles\]). It is modified by the presence of pedestrians and has its own dynamics, i.e. diffusion and decay. Usually the dynamic floor field is used to model a (“long-ranged”) attractive interaction between the particles. Each pedestrian leaves a “trace”, i.e. the floor field of occupied cells is increased. Explicit examples where such an interaction is relevant are given in Part II [@part2]. The dynamic floor field is also subject to diffusion and decay which leads to a dilution and finally the vanishing of the trace after some time.
The [*static floor field*]{} $S$ does not evolve with time and is not changed by the presence of pedestrians. Such a field can be used to specify regions of space which are more attractive, e.g. an emergency exit (see the example in [@part2]) or shop windows. This has an effect similar to a position-dependent matrix of preference, but is much easier to realize.
The [*transition probability*]{} $p_{ij}$ in direction $(i,j)$ (see Fig. \[fig\_prefs\]) now depends on four contributions:
- the matrix of preference $M_{ij}$ which contains the information about the aim and average velocity of the pedestrian.
- the value $D_{ij}$ of the dynamic floor field at the target cell. This contribution takes into account the effects of the motion of the other pedestrians. In many applications (see [@part2]) it is attractive to “follow the crowd”, i.e. transitions in directions $(i,j)$ with a large value of the dynamic floor field are preferred.
- the value $S_{ij}$ of the static floor field. It allows to model effects of the geometry. E.g. in a corridor it is usually less attractive to walk close to the walls. Such an effect can be incorporated in a static floor field which decreases near the walls.
- the occupation number $n_{ij}$ of the target cell. A motion in direction $(i,j)$ is only allowed if the target cell is empty ($n_{ij}=0$) and forbidden if it is already occupied ($n_{ij}=1$).
One simple possibility to take into account all contributions (i)–(iv) is to define the transition probability in direction $(i,j)$ by $$p_{ij}=NM_{ij}D_{ij}S_{ij}(1-n_{ij}).
\label{transprob}$$ $N$ is a normalization factor to ensure $\sum_{(i,j)}p_{ij}=1$ where the sum is over the nine possible target cells. There are also slightly more general forms of the transition probabilities which have been studied in [@ourpaper; @diplom].
Since the total transition probability is proportional to the dynamic floor field it becomes more attractive to follow in the footsteps of other pedestrians. This effect competes with the preferred walking direction specified by $M_{ij}$ and the effects of the geometry encoded in $S_{ij}$. The relative influence of the contributions (i)–(iii) is controlled by coupling parameters. These depend on the situation to be studied. Consider for example a situation where people want to leave a large room (see [@part2]). Normal circumstances, where everybody is able to see the exit, can be modelled by solely using a static floor field which decreases radially with the distance from the door. Since transitions in the direction of larger fields are more likely this will automatically guarantee that everybody is walking in the direction of the door. If, however, the exit can not be seen by everybody, e.g. in a smoke-filled room or in the case of failing lights, people will try to follow others hoping that they know the location of the exit. In this case the coupling to the dynamic floor field is much stronger and the static field has a considerable influence only in the vicinity of the door. This example will be studied in more detail in [@part2].
Dynamics of the Floor Field {#dynfloor}
---------------------------
In contrast to the static floor field $S$ the dynamic floor field $D$ is changed by the motion of pedestrians. Furthermore it is subject to diffusion and decay. Its dynamics consists of three steps:
- If a pedestrian leaves a cell $(x,y)$ the dynamic floor field $D_{xy}$ corresponding to this cell is increased by $\Delta D_{xy}$. The increment $\Delta D_{xy}$ is a parameter of the model and can either be discrete or continuous.
- To model the diffusion, a certain amount of the field is distributed among the neighbouring cells.
- To model the decay of the field, the field strength is reduced by a decay constant $\delta$.
In (a) the virtual trace left by the motion of the pedestrians is created. (b) is necessary because pedestrians do not necessarily follow exactly in the footsteps of others. Diffusion leads to broadening and dilution of the trace. (c) implies that the lifetime of the trace is finite and that it will vanish after some time. Diffusion and decay of the dynamic field lead to an effective interaction strength between the pedestrians which decays exponentially with the distance [@diplom].
In [@ourpaper] we have introduced two variants of the floor field, a discrete and a continuous one. In the discrete case the field strength $D_{xy}$ can be interpreted as the number of bosonic particles (“bosons”) at the cell $(x,y)$. In (a) the number of bosons is increased by one. In (b) bosons can move with probability $\gamma$ to neighbouring cells and in (c) bosons are removed with probability $\alpha$. In the continuous case the dynamics in (b) and (c) is described by a diffusion-decay equation $$\frac{\partial D}{\partial t} = d \cdot \Delta D - \delta \cdot D
\label{eq_diffu}$$ where $d$ is the diffusion constant and $\delta$ the decay constant. Details can be found in [@ourpaper; @diplom].
Summary of the Update Rules
---------------------------
The update rules of the full model including the interaction with the floor fields then have the following structure:
1. The dynamic floor field $D$ is modified according to its diffusion and decay rules (see Sec. \[dynfloor\]).
2. For each pedestrian, the transition probabilities $p_{ij}$ for a move to an unoccupied neighbour cell $(i,j)$ are determined by the matrix of preferences and the local dynamic and static floor fields, e.g. $p_{ij}\propto M_{ij}D_{ij}S_{ij}$ (see Sec. \[sec\_floor\])..
3. Each pedestrian chooses a target cell based on the probabilities of the transition matrix $P=(p_{ij})$.
4. The conflicts arising by any two or more pedestrians attempting to move to the same target cell are resolved, e.g. using the procedure described in Sec. \[sec\_rules\].
5. The pedestrians which are allowed to move execute their step.
6. The pedestrians alter the dynamic floor field $D_{xy}$ of the cell $(x,y)$ they occupied before the move (see Sec. \[dynfloor\]).
These rules have to be applied to all pedestrians at the same time (parallel dynamics). This introduces a timescale into the dynamics which corresponds to approximately $0.3~sec$ of real time. This allows e.g. to translate evacuation times measured in computer simulations into real times.
Conclusions
===========
We have introduced a stochastic cellular automaton to simulate pedestrian behaviour. We focused here on the general concept. The effects which can be observed with the basic approach will be presented in [@part2] together with simple applications.
The key mechanism is the introduction of a floor field which acts as a substitute for pedestrian intelligence and leads to collective phenomena. This floor field makes it possible to translate spatial long-ranged interactions into non-local interactions in time. The latter can be implemented much more efficiently on a computer. Another advantage is an easier treatment of complex geometries. In models with long-range interactions, e.g. the social-force models, one always has to check explicitly whether pedestrians are separated by walls in which case there should be no interaction between them.
The general idea in our model is similar to chemotaxis. However, the pedestrians leave a virtual trace rather than a chemical one. This virtual trace has its own dynamics (diffusion and decay) which e.g. restricts the interaction range (in time). It is realized through a dynamical floor field which allows to give the pedestrians only minimal intelligence and to use local interactions. Together with the static floor field it offers the possibility to take different effects into account in a unified way, e.g.the social forces between the pedestrians or the geometry of the building.
In Part II [@part2] we will demonstrate that the approach indeed is able to reproduce the known collective effects and self-organization phenomena. Therefore the model is a good starting point for realistic applications [@ourpaper; @part2].
The model can also be applied to more complex geometries and various characteristics of a crowd can be simulated without major changes. So it should be possible to study the effects of panic (see [@panic] and references therein). In [@part2] we show results for simple evacuation simulations.
The description of pedestrians using a cellular automaton approach allows for very high simulation speeds. Therefore, we have the possibility to extract the complete statistical properties of our model using Monte Carlo simulations.
Finally it should be emphasized that we have presented only the basic ideas of the approach. For realistic applications modifications might be appropriate, e.g. smaller cell sizes etc. One can also introduce more than just one species of pedestrians (e.g. two groups moving in opposite directions). In this case each species interacts with its own floor field. In the simplest case these fields are independent from each other.
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[^1]: In the following we use “pedestrian” and “particle” interchangeably.
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---
author:
- |
Oren Katzir\
Tel-Aviv University\
Dani Lischinski\
Hebrew University of Jerusalem\
Daniel Cohen-Or\
Tel-Aviv University\
title: 'Cross-Domain Cascaded Deep Feature Translation'
---
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---
abstract: 'This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in [@Vesku], to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.'
address:
- 'Benny Avelin, Aalto University, Institute of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland'
- 'Benny Avelin, Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden'
- 'Vesa Julin, Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 Jyväskylä, Finland'
author:
- Benny Avelin
- Vesa Julin
title: 'A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term'
---
Introduction
============
In this paper we study the boundary behavior of solutions of the following non-homogeneous, fully nonlinear equation $$\label{thePDE} F(D^2u, Du, x) = 0\,.$$ The operator $F$ is assumed to be elliptic in the sense that there are $0< \lambda \leq \Lambda$ such that $$\label{ellipticity} \lambda \text{Tr}(Y) \leq F(X, p, x) - F(X +Y, p, x)\leq \Lambda \text{Tr}(Y)\,, \quad \forall (x,p) \in \rn \times \rn$$ for every pair of symmetric matrices $X, Y$ where $Y$ is positive semidefinite. We assume that $F$ has a drift term which satisfies the following growth condition $$\label{non-homogeneity} |F(0, p, x) \big| \leq \phi(|p|)\,, \quad \forall (x,p) \in \rn \times \rn$$ where $\phi: [0,\infty) \to [0,\infty)$ is continuous, increasing, and satisfies the structural conditions from [@Vesku] (see ). Note that the function $F(\cdot, p ,x)$ is $1$-homogeneous while $F(0,\cdot,x)$ in general is not. In the case there is no drift term, i.e., $\phi=0$, we say that the equation \[thePDE\] is homogeneous.
The problem we are interested in is the so-called Carleson estimate [@Carl]. The Carleson estimate can be stated for the Laplace equation in modern notation as follows. Let $\Omega \subset \rn$ be a sufficiently regular bounded domain and $x_0 \in\partial \Omega$ . If $u$ is a non-negative harmonic function in $ B(x_0,4R) \cap \Omega$ which vanishes continuously on $\partial \Omega \cap B(x_0,4R)$, then $$\label{lapcarl} \sup_{B(x_0,R/C) \cap \Omega} u \leq C u(A_R)\,,$$ where the constant $C$ depends only on $\partial \Omega$ and $N$, and where $A_R \in B(x_0,R/C) \cap \Omega $ such that $d(A_R,\partial \Omega) > R/C^2$ ($A_R$ is usually called a corkscrew point). For $\Omega$ to be sufficiently regular it is enough to assume that $\Omega$ is e.g., an NTA-domain, see [@JK]. The Carleson estimate is very important and useful when studying the boundary behavior and free boundary problems for linear elliptic equations [@CFMS; @CFS; @JK; @K], for $p$-Laplace type elliptic equations [@ALuN; @ALuN1; @AN1; @LLuN; @LN1; @LN5; @LN6], for parabolic $p$-Laplace type equations [@A1; @AGS], and for homogeneous fully nonlinear equations [@Fe; @FS; @FS1].
In this paper we deal with either Lipschitz or $C^{1,1}$ domains and assume that they are locally given by graphs in balls centered at the boundary with radius up to $R_0 > 0$ which unless otherwise stated satisfies $R_0 \leq 16$. For a given Lipschitz domain with Lipschitz constant $l$ we denote $L = \max\{l,2 \}$. The main result of this paper is the sharp Carleson type estimate for non-negative solutions of \[thePDE\]. Due to the non-homogeneity of the equation it is easy to see that \[lapcarl\] cannot hold. Instead the Carleson estimate takes a similar form as the generalized interior Harnack inequality proved in [@Vesku] (see ). Our main result reads as follows.
\[mainthm\] Assume that $\Omega$ is a Lipschitz domain such that $0 \in\partial \Omega$ and assume $u \in C(B_{4R}\cap \overline{\Omega})$, with $R \in (0,R_0/4]$, is a non-negative solution of \[thePDE\]. Let $A_R \in B_{R/2L} \cap \Omega $ be a point such that $d(A_R, \partial \Omega) > R/(4L^2)$, and assume that $u = 0$ on $\partial \Omega \cap B_{4 R}$. There exists a constant $C > 1$ which is independent of $u$ and of the radius $R$ such that $$\int_{u(A_R)}^{M} \frac{dt}{R^2 \phi(t/R)+t} \leq C\,,$$ where $ M = \sup_{B_{R/C} \cap \Omega} u$.
This result is sharp since already the interior Harnack estimate is sharp. The novelty of is that the constant does not depend on the solution itself.
Let us point out a few consequences of . Let us assume that $u$ is as in the theorem. First, if $\phi$ satisfies $$\label{osgood2} \int_1^{\infty} \frac{dt}{\phi(t)} = \infty$$ then there is an increasing function $f_R$ such that the maximum $M$ is bounded by the value $f_R(u(A_R))$. The function $f_R$ is defined implicitly by the estimate in the theorem. If $\phi$ does not satisfy \[osgood2\] then the maximum $M$ may take arbitrary large values (see [@Vesku]). However, even if $\phi$ does not satisfy \[osgood2\] we may still deduce that if $u(A_R) \leq 1$ then the maximum $M$ is uniformly bounded assuming that the radius $R$ is small enough. This follows from the fact that $R^2 \phi(t/R) \to 0$ locally uniformly under our growth assumption on $\phi$. Second if $\phi$ satisfies the Osgood condition $$\label{osgood1} \int_0^1 \frac{dt}{\phi(t)} = \infty$$ then $u(A_R) = 0$ implies that $u$ is zero everywhere. In other words \[osgood1\] implies the strong minimum principle. If $\phi$ does not satisfy \[osgood1\] then the strong minimum principle does not hold. Finally, in the homogeneous case $\phi = 0$ reduces to the classical Carleson estimate.
In the homogeneous case, perhaps the most flexible proof of the Carleson estimate is due to [@CFMS] and has been adapted to many situations, see e.g., [@AdLu; @A1; @AGS; @CNP; @FaSa1; @FaSa2]. This proof relies on two basic estimates:
1. \[hold\] A decay estimate up to the boundary (Hölder continuity), sometimes denoted by the oscillation lemma.
2. \[blow\] An upper estimate of the blow-up rate for singular solutions.
The point is that the rate of blow-up dictated by \[blow\] does not need to be sharp, this is because it only needs to match the geometric decay dictated by \[hold\].
Let us make some notes regarding the proof of \[hold\] and \[blow\] in the homogeneous case. In the context of divergence form equations, the proof of \[hold\] is standard and follows e.g., from the flexible methods developed by De Giorgi [@DeG], and is thus valid for very general domains (outer density condition). However, in the context of non-divergence form equations, this is far from trivial if the domain is irregular. In fact, in Lipschitz domains it is basically only known for linear equations, and the proof relies on the classical result by Krylov and Safonov [@KS1; @KS2]. For more regular domains the approach is usually via flattening, symmetry and iterating the Harnack inequality. If the Harnack inequality for a non-negative solution in $B_{2R}$ holds, i.e., $$\nonumber \sup_{B_R} u \leq C \inf_{B_R} u\,,$$ for a constant $C$ independent of $u$ and $R$, then a well known proof of \[blow\] consists of iterating the Harnack inequality in a dyadic fashion up to the point of singularity.
Due to the non-homogeneity of our equation the classical Harnack inequality no longer holds, and we will instead use the generalized Harnack inequality, which states that a non-negative solution $u \in C(B_{2R})$ of \[thePDE\] with $R \leq 1$ satisfies $$\label{px new harnack} \int_{m}^{M} \frac{dt}{R^2\phi(t/R)+ t} \leq C,$$ where $m = \inf_{B_R} u$, $M = \sup_{B_R} u$ and $C$ is a constant which is independent of $u$ and $R$. To continue our discussion it is important to note that the standard Harnack inequality for harmonic functions can be written as $$\nonumber \int_m^M \frac{dt}{t} \leq C.$$ As such, the term $R^2 \phi(t/R)$ in \[px new harnack\] is the non-homogeneous correction term which compensates the effect of \[non-homogeneity\]. When using \[px new harnack\] the ”contest” between the correction term and the base term $t$ becomes evident. When we study the blow-up rate \[blow\] for solutions of \[thePDE\] () our goal is to show that for every solution there exists a critical threshold level where the correction term becomes small and stays small, all the way up to the singularity. This means that the asymptotic behavior after the critical level is the same as in the homogeneous case. This argument strongly relies on the structural assumptions on $\phi$ which imply that $\phi(t)= \eta(t)t$ for a slowly increasing function $\eta$. Similarly when we prove the Hölder continuity estimate () we show that there is a critical radius such that below it the oscillation of the solution reduces in a geometric fashion. Again the point is to quantify the critical radius.
First application: Boundary Harnack inequality
----------------------------------------------
In the last section of this paper, we consider the boundary Harnack inequality. Our contribution in this direction is the same as for the Carleson estimate, i.e., we derive an estimate where the constant does not depend on the solution. The proof is based on a barrier function estimate and this requires the domain to be $C^{1,1}$-regular.
\[the boundary Harnack\] Assume that $\Omega$ is a $C^{1,1}$-regular domain such that $0 \in \partial \Omega$. Let $u,v \in C(B_{4R} \cap \overline{\Omega})$, with $R \in (0,R_0/4]$, be two positive solutions of \[thePDE\]. Let $A_R \in B_{R/2L} \cap \Omega $ be such that $d(A_R,\partial \Omega) > R/(4L^2)$ and assume that $v(A_R)= u(A_R)>0$ and $v = u = 0$ on $\partial \Omega \cap B_{4R}$. There exists a constant $C$, which is independent of $u,v$ and of the radius $R$, and numbers $\mu_0, \mu_1 \in [0,\infty]$ such that $\mu_0 \leq u(A_R) \leq \mu_1$, $$\sup_{x \in B_{R/C} \cap \Omega} \frac{v(x)}{u(x)} \leq \frac{\mu_1}{\mu_0} \,,$$ and $$\int_{\mu_0}^{\mu_1} \frac{dt}{R^2 \phi(t/R)+t} \leq C\,.$$
In the homogeneous case reduces to the classical boundary Harnack inequality, i.e., the ratio $v/u$ is bounded by a uniform constant. If $\phi$ satisfies the Osgood conditions \[osgood2\] and \[osgood1\] then implies that the ratio $v/u$ is bounded. In the general case when $\phi$ does not satisfy \[osgood2\] and \[osgood1\] the ratio $v/u$ can be unbounded. Note that we allow $\mu_0 = 0$ and $\mu_1 = \infty$. In this case, arguing as in the case of , we may still conclude that if $u(A_R)=1$ then the ratio $v/u$ is bounded when the radius $R$ is small enough. At the end of the paper we give an example which shows that in the model case $\phi(t)= (|\log t|+1)t$ is essentially sharp.
Consequences for the theory of the $p(x)$-Laplacian
---------------------------------------------------
Consider the $p(x)$-Laplace equation $$\label{px}- \text{div} (|\nabla u|^{p(x)-2} \nabla u) = 0, \quad 1 < p(x) < \infty\,.$$ Let us make the assumption that $p(\cdot)$ is continuously differentiable. In non-divergence form this equation is of the form \[thePDE\] and has a drift term which satisfies \[non-homogeneity\] with $\phi(t)= C(|\log t|+1)t$ (see [@Vesku; @JLP]). Solutions of \[px\] are called $p(x)$-harmonic functions.
Let us return to the previous outline of the proof of the Carleson estimate. In [@Alk; @HKL; @Wo] it was proved that a non-negative $p(x)$-harmonic function $u$ in $B_{2R}$ satisfies the following Harnack type estimate $$\sup_{B_R} u \leq C (\inf_{B_R}u+R)$$ for a constant $C$ depending on the solution $u$. In [@AdLu], Adamowicz and Lundström used the above estimate to prove a version of \[lapcarl\] with a constant depending on the solution. From our perspective provides an improvement over this. Specifically, calculating the integral in in the context of the $p(x)$-Laplacian we obtain the following corollary.
Let $\Omega$ be as in and $p \in C^1(\rn)$ such that $1 < p_- \leq p(x) \leq p_+ < \infty$. Assume that $u \in C(B_{4R}\cap \overline{\Omega})$, with $R \in (0,R_0/4]$, is a non-negative $p(x)$-harmonic function. Let $A_R \in B_{R/2L} \cap \Omega $ be a point such that $d(A_R,\partial \Omega) > R/(4L^2)$, and assume that $u = 0$ on $\partial \Omega \cap B_{4 R}$. There exists a constant $C(N,p_-,p_+,\|p\|_{C^1},L) > 1$ which is independent of $u$ and $R$ such that $$\nonumber \sup_{B_{R/C} \cap \Omega} u \leq C \max \left \{ u(A_R)^{1+C R},u(A_R)^{\frac{1}{1+CR}} \right \}\,.$$
Let us now turn our attention to . An immediate corollary for $p(x)$-harmonic functions is.
\[p(x) bHp\] Assume that $\Omega$ is $C^{1,1}$-regular domain such that $0 \in \partial \Omega$. Let $u,v \in C(B_{4R} \cap \overline{\Omega})$, with $R \in (0,R_0/4]$, be two positive $p(x)$-harmonic functions. Let $A_R \in B_{R/2L} \cap \Omega $ be such that $d(A_R,\partial \Omega) > R/(4L^2)$, and assume that $v(A_R)= u(A_R)>0$ and $v = u = 0$ on $\partial \Omega \cap B_{4R}$. There exists a constant $C(N,p_-,p_+,\|p\|_{C^1},L) > 1$ which is independent of $u,v$ and $R$ such that $$\sup_{x \in B_{R/C} \cap \Omega} \frac{v(x)}{u(x)} \leq C \max \left \{ u(A_R)^{CR}, u(A_R)^{-CR} \right \} .$$
The above corollary is similar to the boundary Harnack inequality proved in [@AdLu], but in the constants does not depend on the solution. As we already mentioned we provide an example in that shows that is essentially sharp.
Organization of the paper
-------------------------
In we list all the assumptions on $\phi$ in \[non-homogeneity\] and recall the definition of a Reifenberg flat domain. In we prove the sharp Hölder continuity estimate up to the boundary in Reifenberg flat domains (). By this we mean that we give the sharp Hölder norm in terms of the maximum of the solution. In we study the blow-up rate of a solution near the boundary in NTA-domains (). These results are crucial in the proof of the Carleson estimate but are of independent interest. In we give the proof of the Carleson estimate (). In we prove the boundary Harnack estimate ().
Acknowledgment {#acknowledgment .unnumbered}
==============
The first author was supported by the Swedish Research Council, dnr: 637-2014-6822. The second author was supported by the Academy of Finland grant 268393.
Preliminaries {#secprel}
=============
Throughout the paper $B(x,r)$ denotes the open ball centered at $x$ with radius $r$. When the ball is centered at the origin we simply write $B_r$. Given a point $x \in \rn$ and a set $E \subset \rn$ we denote their distance by $d(x,E):= \inf_{y \in E}|x-y|$.
We recall the definition of a viscosity solution.
\[visco\_def\] We call a lower semicontinuous function $u: \Omega \to \er$ a *viscosity supersolution* of \[thePDE\] in $\Omega$ if the following holds: if $x_ 0 \in \Omega$ and $\varphi \in C^2(\Omega)$ are such that $u- \varphi$ has a local minimum at $x_0$ then $$F(D^2\varphi(x_0), D\varphi(x_0),x_0) \geq 0.$$ An upper semicontinuous function $u: \Omega \to \er$ is a viscosity subsolution of \[thePDE\] in $\Omega$ if the following holds: if $x_ 0 \in \Omega$ and $\varphi \in C^2(\Omega)$ are such that $u- \varphi$ has a local maximum at $x_0$ then $$F(D^2\varphi(x_0), D\varphi(x_0),x_0) \leq 0.$$ Finally a continuous function is a viscosity solution if it is both a super- and a subsolution.
As mentioned in the introduction we assume that $F$ in \[thePDE\] has a drift term which satisfies the growth condition $$|F(0, p, x) \big| \leq \phi(|p|)$$ for every $(x,p) \in \rn \times \rn$, where $\phi: [0,\infty) \to [0,\infty)$ is continuous, increasing, and satisfies the following structural conditions from [@Vesku]. For $t > 0$ we write $\phi(t)$ as $$\nonumber \phi(t) = \eta(t)\, t$$ and assume the following.
1. \[P1\] $\phi: [0, \infty) \to [0, \infty)$ is locally Lipschitz continuous in $(0, \infty)$ and $\phi(t)\geq t$ for every $t \geq 0$. Moreover, $\eta: (0, \infty) \to [1,\infty)$ is non-increasing on $(0,1)$ and nondecreasing on $ [1, \infty) $.
2. \[P2\] $\eta$ satisfies $$\lim_{t \to \infty} \frac{t \eta'(t)}{\eta(t)} \log(\eta(t)) = 0.$$
3. \[P3\] There is a constant $\Lambda_0$ such that $$\eta(st) \leq \Lambda_0 \eta(s) \eta(t)$$ for every $s,t \in (0, \infty)$.
The assumption \[P2\] implies that $\eta$ is a slowly increasing function [@BGT]. We will repeatedly use the fact that for every ${\varepsilon}>0$ there is a constant $C_{\varepsilon}$ such that $\eta(t) \leq C_{\varepsilon}t^{{\varepsilon}}$ for every $t \geq 1$, see again [@BGT]. We explicitly note that our assumptions \[P1,P2,P3\] do not rule out the possibility that $\phi(0)>0$, that $\phi$ is non-Lipschitz at $0$, and that the maximum/comparison-principle does not hold. Moreover the assumptions \[P1,P2,P3\] do not imply that $\phi$ satisfies the Osgood conditions \[osgood2\] and \[osgood1\].
We may replace the equation \[thePDE\] by two inequalities which follow from the ellipticity condition and the modulus of continuity of the drift term \[non-homogeneity\]. In other words if $u$ is a solution of \[thePDE\] then it is a viscosity supersolution of $$\label{model1} \mathcal{P}_{\lambda, \Lambda}^+(D^2 u) = -\phi(|Du|)$$ and a viscosity subsolution of $$\label{model2} \mathcal{P}_{\lambda, \Lambda}^-(D^2 u) = \phi(|Du|)$$ in $\Omega$. Here $\mathcal{P}_{\lambda, \Lambda}^-, \mathcal{P}_{\lambda, \Lambda}^+$ are the usual Pucci operators, which are defined for a symmetric matrix $X \in \mathbb{S}^{n \times n}$ with eigenvalues $e_1, e_2, \dots, e_n$ as $$\mathcal{P}_{\lambda, \Lambda}^+(X):= -\lambda \sum_{e_i \geq 0} e_i - \Lambda \sum_{e_i<0} e_i \qquad \text{and} \qquad \mathcal{P}_{\lambda, \Lambda}^-(X):= -\Lambda \sum_{e_i \geq 0} e_i - \lambda \sum_{e_i<0} e_i .$$ We note that all the results of this paper hold if we instead of assuming that $u$ is a solution of \[thePDE\] we only assume that it is a supersolution of \[model1\] and a subsolution of \[model2\]. We recall the result from [@Vesku].
\[weakHarnack\] Assume that $u\in C(B(x_0,2r))$, with $r \leq 1$, is a non-negative solution of \[thePDE\]. Denote $m:= \inf_{B(x_0,r)}u$ and $M:= \sup_{B(x_0,r)}u$. There is a constant $C$ which is independent of $u$ and $r$ such that $$\int_{m}^{M} \frac{dt}{r^2\phi(t/r)+ t} \leq C.$$
To describe the kind of domains we will be considering we first recall the definition of Reifenberg flat domains.
\[def:hyperplane:approximable\] Let $ \Omega \subset \rn $ be a bounded domain. Then $\partial \Omega $ is said to be uniformly $ ( \delta, r_0 )$-approximable by hyperplanes, provided there exists, whenever $ w \in \partial \Omega$ and $ 0 < r < r_0, $ a hyperplane $ \pi $ containing $w$ such that $$h ( \partial \Omega \cap B(w,r), \pi \cap B(w,r) ) \leq \delta r\,.$$ Here $h ( E, F ) = \max ( \sup \{ d ( y, E ) : y \in F \}, \sup \{ d ( y, F ) : y \in E \} )$ is the Hausdorff distance between the sets $ E, F \subset \rn$ .
We denote by $ {\mathcal F} ( \delta, r_0 ) $ the class of all domains $ \Omega$ which satisfy . Let $ \Omega \in {\mathcal F} ( \delta, r_0 )$, $w\in \partial\Omega$, $0<r<r_0,$ and let $\pi$ be as in . We say that $\partial\Omega$ separates $ B_r(w), $ if $$\label{eqn:separating} \{ x \in \Omega \cap B(w,r) : d ( x, \partial \Omega ) \geq 2 \delta r \} \subset \mbox{ one component of } \rn \setminus \pi.$$
\[def:riefenberg:flat\] Let $ \Omega\subset\rn$ be a bounded domain. Then $ \Omega$ and $ \partial \Omega $ are said to be $ (\delta, r_0 )$-Reifenberg flat provided $ \Omega \in { \mathcal F} ( \delta, r_0 ) $, $\delta < 1/8$ and provided \[eqn:separating\] holds whenever $ 0 < r < r_0, w \in \partial \Omega.$
For short we say that $ \Omega $ and $\partial\Omega$ are $ \delta $-Reifenberg flat whenever $ \Omega $ and $\partial\Omega$ are $ ( \delta, r_0 ) $-Reifenberg flat for some $ r_0 > 0. $ We note that an equivalent definition of Reifenberg flat domains is given in [@KT].
Next we recall the following definition of NTA-domains.
\[def:NTA\] A bounded domain $\Omega$ is called non-tangentially accessible **(NTA)** if there exist $L \geq 2$ and $r_0$ such that the following are fulfilled:
1. \[NTA1\] **corkscrew condition:** for any $ w\in
\partial\Omega, 0<r<r_0,$ there exists a point $a_r(w) \in \Omega $ such that $$\nonumber L^{-1}r<|a_r(w)-w|<r, \quad d(a_r(w),
\partial\Omega)>L^{-1}r,$$
2. \[NTA2\] $\rn \setminus \Omega$ satisfies \[NTA1\],
3. \[NTA3\] **uniform condition:** if $ w \in
\partial \Omega, 0 < r < r_0, $ and $ w_1, w_2 \in B ( w, r) \cap \Omega, $ then there exists a rectifiable curve $ \gamma: [0, 1] \to \Omega $ with $ \gamma ( 0 ) = w_1,\, \gamma ( 1 ) = w_2, $ such that
1. $H^1 ( \gamma ) \, \leq \, L \, | w_1 - w_2 |,$
2. $\min\{H^1(\gamma([0,t])), \, H^1(\gamma([t,1]))\, \}\, \leq \, L \, d ( \gamma(t),
\partial \Omega)$, for all $t \in [0,1]$.
In , $ H^1 $ denotes length or the one-dimensional Hausdorff measure. We note that \[NTA3\] is different but equivalent to the usual Harnack chain condition given in [@JK] (see [@BL], Lemma 2.5). Moreover, using [@KT Theorem 3.1] we see that there exists $\hat\delta=\hat\delta(N)>0$ such that if $\Omega\subset\rn$ is a $(\delta,r_0)$-Reifenberg flat domain and if $0<\delta\leq\hat\delta$, then $\Omega$ is an NTA-domain in the sense described above with constant $L=L(N)$. In the following we assume $0<\delta\leq\hat\delta$ and we refer to $L$ as the NTA constant of $\Omega$.
\[SmallLip\] Let $\Omega$ be a Lipschitz domain with constant $ l < 1/8$ then $\Omega$ is $\delta$-Reifenberg flat with constant $$\nonumber \delta = \sin(\arctan(l)) = \frac{l}{\sqrt{l^2+1}}\,.$$ Moreover note that $\delta < l$ and that any Lipschitz domain is also an NTA-domain.
Reduction argument {#ssec reduction}
------------------
### Reduction to small Lipschitz constant {#reduction-to-small-lipschitz-constant .unnumbered}
First we observe that we may assume that the domain $\Omega$ in is Reifenberg flat with small $\delta$. Indeed assume $\Omega$ is an $l$-Lipschitz domain, and the equation \[thePDE\] has ellipticity constants $\lambda$ and $\Lambda$. We may stretch the domain by a linear map $\mathcal{T}$ such that $\Omega'= \mathcal{T}(\Omega)$ is an $\hat{l}$-Lipschitz domain with $\hat{l} < \frac{1}{100}$. Moreover, if $u$ is a solution of \[thePDE\] in $\Omega$ then $v(x) = u(\mathcal{T}^{-1}(x))$ is a solution of a similar equation with ellipticity constants $\tilde{\lambda}$ and $\tilde{\Lambda}$. Thus we may consider the case when $\Omega$ an $l$-Lipschitz domain, with $l \leq 1/100$. In particular, by we may assume that $\Omega$ is Reifenberg flat with constant $1/100$.
### Reduction to a canonical scale {#reduction-to-a-canonical-scale .unnumbered}
In the proof of we prefer to scale the radius $R$ to one and $R_0 = 16$. In this way we do not get confused by the many radii which appear in the proof. Let us assume that $u$ is as in the theorem. By rescaling $u_R(x) := \frac{u(Rx)}{R}$ we obtain a function $u_R$ which is a solution to the equation $$\label{rescaledPDE} F_R(D^2 u_R, D u_R,x)=0$$ where
$$\label{rescaled-non-homogeneity} | F_R(0,p,\cdot)| \leq {\phi_R}(|p|) := R \phi(|p|).$$
Note that \[rescaledPDE\] is of type \[thePDE\], satisfying \[ellipticity\] with the same constants and with nonlinearity ${\phi_R}$.
Since ${\phi_R}$ does not satisfy \[P1,P2,P3\] we need to rephrase in our new scale as we cannot use it directly for $u_R$ (see ). With this in mind we prove the Hölder regularity estimates ( and ) and the blow-up estimate () assuming that we have a solution of \[rescaledPDE,rescaled-non-homogeneity\].
The simplifying point is that if we denote $$\label{phi R} {\Phi_R}(t) := {\phi_R}(t) + t \geq {\phi_R}\,,$$ then we see that ${\Phi_R}$ satisfies \[P1,P2\] with $\eta_R(t) = R\eta(t)+1$ and instead of \[P3\] it satisfies
1. \[P3’\] $$\eta_R(st) \leq \Lambda_0 \eta(s) \eta_R(t), \qquad \text{for every $s,t \in (0, \infty)$,}$$
with the $\Lambda_0$ from \[P3\] for $\phi$. Rephrasing in terms of $u_R$ we see that if we denote $M_R = \sup_{B_1 \cap \Omega} u_R$ and $m_R = \frac{u(A_R)}{R}$, then becomes $$\label{mainthmrescaled} \int_{m_R}^{M_R} \frac{dt}{{\Phi_R}(t)} \leq C \,.$$ Thus our aim will be to prove that for a solution of \[rescaledPDE,rescaled-non-homogeneity\], \[mainthmrescaled\] holds.
\[corHarnack\] Assume that $u\in C(B(x_0,2r))$, is a non-negative solution of \[rescaledPDE,rescaled-non-homogeneity\]. Denote $m:= \inf_{B(x_0,r)}u$ and $M:= \sup_{B(x_0,r)}u$. Let $\alpha_0 \in (0,1)$, then there exists a constant $C(\alpha_0) > 1$ which is independent of $u$, $r$ and $R$ such that $$\int_{m}^{M} \frac{dt}{ r^{\alpha} {\Phi_R}(t)+ t} \leq C, \quad \forall \alpha \in [0,\alpha_0]\,.$$
We define $v \in C(B(x_0,2\rho))$, where $\rho = rR \leq 1$, by rescaling $v(y) = Ru(y/R)$. Then $v$ is a solution of \[thePDE\] with non-homogeneity $\phi$ and implies $$\label{scaling weak harnack} C \geq \int_{Rm}^{RM} \frac{ds}{\rho^2\phi(s/\rho)+ s} = \int_{m}^{M} \frac{dt}{R r^2\phi(t/r)+ t}.$$ Since $\eta$ is slowly increasing function we have $\eta(t) \leq C_\ep t^{\ep}$ for all $t>1$ and for any $\ep$. It is now easy to see that if ${\varepsilon}\geq {\varepsilon}_0$ for some fixed ${\varepsilon}_0 \in (0,1)$ then $\eta(t) \leq C_{\ep_0} t^\ep$. Therefore by \[P3\] we deduce that for any $\alpha \in [0,\alpha_0]$ we have $$Rr^2\phi(t/r) = r^2 R\eta(t/r) \frac{t}{r} \leq \Lambda_0 r \, \eta(1/r) R\eta(t)t \leq \Lambda_0 C_{1-\alpha_0} r r^{\alpha-1} \, R\phi(t) \leq C r^{\alpha}\, {\Phi_R}(t)\,,$$ for a constant $C(\alpha_0) > 1$. Plugging this into \[scaling weak harnack\] gives the result.
Hölder continuity estimates {#seccontblow}
===========================
In this section we prove interior and boundary Hölder continuity estimates ( and ) when $\Omega$ is Reifenberg flat. We note that solutions of \[thePDE\] are known to be Hölder continuous [@Si]. The point of the following results is to derive the sharp Hölder norm with respect to the $L^\infty$-norm of the solution. As we mentioned in the previous section we assume that $u$ solution of \[rescaledPDE,rescaled-non-homogeneity\].
\[holder cont\] Let $u\in C(B(x_0, 2r))$, with $r \leq 1$, be a solution of \[rescaledPDE\], \[rescaled-non-homogeneity\], and denote $M = \sup_{B(x_0,r)}|u|$. Then $u$ is Hölder continuous, i.e. for every $\rho \leq r$ the following holds $$\operatorname*{osc}_{B(x_0,\rho)} u \leq C_1 M \left(\frac{\rho}{r} \right)^\alpha + C_1{\Phi_R}(M) \rho^{1/4} r^{1/4}$$ for some $C_1$ and $\alpha \in (0,1/4)$, which are independent of $u, r$ and $R$. We also have the following oscillation decay, there exist a constant $C=C(\Lambda/\lambda,N)>0$ and $\tau = \tau(\Lambda/\lambda,N) \in (0,1)$ such that $$\label{oscdec} \operatorname*{osc}_{B(x_0,\rho/2)} u \leq \tau \operatorname*{osc}_{B(x_0,\rho)} u + C {\Phi_R}(M) \sqrt{\rho}, \quad \rho \in (0,r).$$
Without loss of generality we may assume that $x_0 = 0$. For every $\rho \leq r$ we denote $M_{\rho} = \sup_{B_\rho} u$ and $m_{\rho} = \inf_{B_\rho} u$. Define functions $v(x) = M_{\rho}-u$ and $w(x)= u(x) - m_{\rho}$ which are non-negative in $B_\rho$. Denote $M_{v,\rho/2} = \sup_{B_\rho/2} v$, $m_{v,\rho/2} = \inf_{B_\rho/2} v$ and $M_{w,\rho/2}$ and $m_{w,\rho/2}$ for the supremum and infimum of $w$. Note that $M_{v,\rho/2}, M_{w,\rho/2} \leq 2M$. Since $v$ is a solution of \[rescaledPDE,rescaled-non-homogeneity\] we obtain from with $\alpha = 1/2$ that $$\int_{m_{v,\rho/2}}^{M_{v,\rho/2}} \frac{dt}{ \sqrt{\rho} {\Phi_R}(M) + t} \leq C.$$ By integrating this we deduce $$M_{v,\rho/2} \leq C m_{v,\rho/2} + C {\Phi_R}(M)\sqrt{\rho}.$$ This in turn implies $$\label{from harnack v} M_\rho - m_{\rho/2} \leq C (M_\rho - M_{\rho/2}) + C {\Phi_R}(M)\sqrt{\rho}.$$ Similar argument applied to $w$ yields $$\label{from harnack w} M_{\rho/2} - m_{\rho} \leq C ( m_{\rho/2} - m_{\rho}) + C {\Phi_R}(M)\sqrt{\rho}.$$
Denote $\omega(\rho) = \operatorname*{osc}_{B_\rho} u$. Adding \[from harnack v\] and \[from harnack w\] gives $$\nonumber \omega(\rho/2) \leq \tau \omega(\rho) + C {\Phi_R}(M) \sqrt{\rho}$$ for every $\rho \leq r \leq 1$ where $\tau = \frac{C-1}{C+1}<1$. This is \[oscdec\]. Moreover by [@GT Lemma 8.23] the following holds $$\omega(\rho) \leq C\omega(r) \left( \frac{\rho}{r}\right)^{\alpha} + C {\Phi_R}(M) \rho^{1/4} r^{1/4}$$ for some $\alpha >0$.
We will turn our attention to the Hölder continuity up to the boundary for solutions vanishing at the boundary. The boundary regularity does not follow directly from the interior regularity. There is an additional difficulty due to the fact that the comparison principle does not hold for \[thePDE\]. In fact, even the maximum principle in general is not true. We need two lemmas which allow us to overcome the lack of comparison principle.
The first lemma shows the existence of the maximal solution of the equation \[model2\] with a given Dirichlet boundary data. Here we do not need the assumption \[P3\] so we may state the result for $\phi$ instead of ${\Phi_R}$.
\[maximal comparison function\] Let $\Omega$ be a bounded Lipschitz domain. Assume that $u \in C(\overline{\Omega})$ is a subsolution of \[model2\] in $\Omega$. Then there exists a function $v \in C(\overline{\Omega})$ which is a solution of \[model2\] in $\Omega$ such that $ u=v$ on $
\partial \Omega$ and $u \leq v$ in $ \Omega$.
\[maximal comparison function remark\] Assume that we have a subsolution of \[model2\] with nonlinearity ${\phi_R}$. Then by scaling to the original scale as in the proof of we get a subsolution to \[model2\] with nonlinearity $\phi$, consequently we can apply and subsequently scale back to the canonical scale to obtain that also holds for subsolutions of \[model2\] with nonlinearity ${\phi_R}$.
For ${\varepsilon}>0$ small we define $$\phi_{\varepsilon}(t):= (1+{\varepsilon})\max\{\phi(t), \phi({\varepsilon}) \},$$ where $\phi$ is from \[non-homogeneity\]. It is straightforward to check that $\phi_{{\varepsilon}}$ satisfies the assumptions \[P1,P2,P3\]. We claim that there exists a solution $v_{\varepsilon}\in C(\overline{\Omega})$ of $$\label{bar eq}
\begin{cases}
&\mathcal{P}^-_{\lambda, \Lambda}(D^2v) = \phi_{\varepsilon}(|Dv|) \quad \text{in }\, \Omega,\\
&v=u \qquad \text{on }\,
\partial \Omega.
\end{cases}$$
The existence of $v_{\varepsilon}$ follows from [@Si]. We need to check that \[bar eq\] satisfies the assumptions in [@Si]. First we write \[bar eq\] as $$\mathcal{P}^-_{\lambda, \Lambda}(D^2v) - \phi_{\varepsilon}(|Dv|) + (1+{\varepsilon}) \phi({\varepsilon}) = (1+{\varepsilon})\phi({\varepsilon})$$ and denote $F(X,p) = \mathcal{P}^-_{\lambda, \Lambda}(X) - \phi_{\varepsilon}(|p|) + (1+{\varepsilon})\phi({\varepsilon})$. Then it holds that $$F(0,0) = 0.$$ Since the Pucci operator is uniformly elliptic [@CC] we only have to check that $$\label{check sira} \phi_{\varepsilon}(t)- \phi_{\varepsilon}(s) \leq C_1(t+s)|t-s| + C_2|t-s|$$ holds for every $s,t \geq 0$. Note that we allow the constants above to depend on ${\varepsilon}$. Since $\phi$ is locally Lipschitz and satisfies the condition \[P2\] we have for every $t \geq {\varepsilon}$ that $$\phi'(t) = \eta'(t)t+ \eta(t) \leq C \eta(t) \leq C t,$$ where the last inequality follows from the $\eta(t) \leq Ct$ for every $t \geq 1$. Thus we have $$\phi_{\varepsilon}(t)- \phi_{\varepsilon}(s) \leq \sup_{{\varepsilon}\leq \xi \leq t+s+ {\varepsilon}} \phi'(\xi)\, |t-s| \leq C\sup_{\xi \leq t+s+ {\varepsilon}} |\xi| |t- s| \leq C(t+s + {\varepsilon})|t-s|.$$ Hence we have \[check sira\] and the existence $v_\epsilon$ follows.
Let $0< {\varepsilon}_1 < {\varepsilon}_2$ and let $v_{{\varepsilon}_1}$ and $v_{{\varepsilon}_2}$ be solutions of the corresponding equations \[bar eq\]. Let us show that the solutions are monotone with respect to ${\varepsilon}$, i.e., $$\label{comparison claim} v_{{\varepsilon}_1}(x) \leq v_{{\varepsilon}_2}(x) \qquad \text{for every }\, x \in \Omega.$$
The claim \[comparison claim\] follows from the standard comparison principle for semicontinuous functions and we only give the sketch of the argument. For more details and for the notation see [@CIL Section 3]. Assume that the claim does not hold. Then we conclude that there exists points $x,y \in \Omega$, a vector $p \in \rn$ and symmetric matrices $X,Y$ such that $X \leq Y$ and the pair $(p,X)$ belongs to the semi-jet $\bar{D}^{2,+}v_{{\varepsilon}_1}(x)$ and $(p,Y)$ belongs to the semi-jet $\bar{D}^{2,-}v_{{\varepsilon}_2}(y)$. If the functions $v_{{\varepsilon}_1}, v_{{\varepsilon}_2}$ were $C^2$ regular this would mean that $p = Dv_{{\varepsilon}_1}(x) = Dv_{{\varepsilon}_2}(y)$, $X = D^2v_{{\varepsilon}_1}(x)$ and $Y = D^2v_{{\varepsilon}_2}(y)$. Since $v_{{\varepsilon}_1}$ is a subsolution of \[bar eq\] we have $$\mathcal{P}^-_{\lambda, \Lambda}(X) - \phi_{{\varepsilon}_1}(|q|) \leq 0$$ and since $v_{{\varepsilon}_2}$ is a supersolution of \[bar eq\] we have $$\mathcal{P}^-_{\lambda, \Lambda}(Y) - \phi_{{\varepsilon}_2}(|q|) \geq 0.$$ On the other hand, it follows from $X \leq Y$ and $\phi_{{\varepsilon}_1}(|q|)< \phi_{{\varepsilon}_2}(|q|)$ that $$0 \geq \mathcal{P}^-_{\lambda, \Lambda}(X) - \phi_{{\varepsilon}_1}(|q|) >\mathcal{P}^-_{\lambda, \Lambda}(Y) - \phi_{{\varepsilon}_2}(|q|) \geq 0$$ which is a contradiction. Thus we have \[comparison claim\].
We note that repeating the above argument we get that $u(x) \leq v_{{\varepsilon}}(x)$ for every $x \in \Omega$ and for every ${\varepsilon}>0$. Hence we have that $v_{\varepsilon}(x) \to v(x)$ point-wise in $\Omega$ and $v \geq u$. It follows from the interior Hölder regularity that $v_{\varepsilon}$ are locally uniformly Hölder continuous in $\Omega$. Therefore $v$ is continuous in $\Omega$ and by a standard viscosity convergence argument it is a solution of \[model2\] in $\Omega$. Moreover, it follows from \[comparison claim\] that for every ${\varepsilon}\in (0, {\varepsilon}_0)$ the following holds $$u(x) \leq v_{{\varepsilon}}(x) \leq v_{{\varepsilon}_0}(x) \qquad \text{for every }\, x \in \overline{\Omega}.$$ Since $u= v_{{\varepsilon}_0}$ on $
\partial \Omega$ we conclude that $v \in C(\overline{\Omega})$ and $v =u$ on $
\partial \Omega$.
The next result shows that in small balls the equation \[model2\] almost satisfies the maximum principle.
\[almost maximum princ\] Let $v \in C(\overline{B}_r)$ be a subsolution of \[model2\] in $B_r$ with non-homogeneity ${\Phi_R}$ and $r\leq 1$ such that $v \leq M$ on $
\partial B_r$ and let $\sigma>1$. There is a constant $c_0>0$, which depends on $\sigma$, such that if $r \leq \frac{c_0}{\eta_R(M)^2}$ then $$\sup_{B_r}v \leq \sigma M.$$ Furthermore if \[osgood1\] holds then the maximum principle holds, i.e., $$\sup_{B_r}v \leq M.$$
For every small ${\varepsilon}>0$ we define $$\phi_{\varepsilon}(t):= (1+{\varepsilon})\max\{{\Phi_R}(t), {\Phi_R}({\varepsilon}) \},$$ We first let $\varepsilon_0 = 1/2$ and consider $r_0$ so small that $$\nonumber \lambda \int_0^1 \frac{ds}{\phi_{\varepsilon_0}(s)} > 2r_0,$$ where $\lambda>0$ is the ellipticity constant of the Pucci operator $\mathcal{P}^-_{\lambda,\Lambda}$. Clearly we may choose $r_0$ such that it does not depend on $R\leq1$. Note that this implies $$\nonumber \lambda \int_0^1 \frac{ds}{\phi_{\varepsilon}(s)} > 2r, \qquad \text{for all } \, \varepsilon \in [0,\varepsilon_0],\, r \in (0,r_0]\,.$$ When $0 < r \leq r_0$ we may define a function $f_{\varepsilon}:[0,r] \to [0,\infty)$ by the implicit function theorem as $$t = \lambda \int_0^{f_{\varepsilon}(t)} \frac{ds}{\phi_{\varepsilon}(s)}$$ In particular, we have $f_{{\varepsilon}} < 1$ by the choice of $r_0$. Define a function $g_{\varepsilon}:[0,r] \to [0,\infty)$ as $$g_{\varepsilon}(t) := \int_0^t f_{\varepsilon}(s)\, ds.$$ Then $g_{\varepsilon}$ is increasing and satisfies $g_{\varepsilon}'' =\lambda^{-1} \phi_{{\varepsilon}}(g_{\varepsilon}') $. We define a radial function $w_{\varepsilon}: \overline{B}_r \to \er$ as $$w_{\varepsilon}(x) =g_{\varepsilon}(r) + M - g_{{\varepsilon}}(|x|).$$ Now note that $\phi_{\varepsilon}(s)$ is constant for $s \leq {\varepsilon}$ and thus $g_{\varepsilon}(s) = c s^2$ close to $0$. It is therefore straightforward to see that $w_{\varepsilon}\in C^2(B_r)$ and a calculation shows that $$\mathcal{P}^-_{\lambda, \Lambda}(D^2w) \geq \phi_{\varepsilon}(|Dw|) \qquad \text{in }\, B_r$$ and $w_{\varepsilon}= M \geq v$ on $
\partial B_r$. It follows from the fact that $v$ is a subsolution of \[model2\] with ${\Phi_R}$ instead of $\phi$, and from $\phi_{\varepsilon}> {\Phi_R}$ that $v - w_{\varepsilon}$ does not attain local maximum in $B_r$ (see the proof of ). Hence we have $w_{\varepsilon}\geq v$ in $B_r$.
Now if $\phi$ satisfies the Osgood condition $$\label{osgo} \int_0^{1} \frac{ds}{\phi(s)} = \infty$$ then we easily see that $w_{\varepsilon}\to M$ and we get the maximum principle, i.e., $$\nonumber \sup_{B_r}w \leq M.$$
With the above in mind let us assume that \[osgo\] does not hold. In this case it is easy to see that $f_{\varepsilon}\to f$ uniformly and for every $t \in (0,r)$ it holds that $$\label{def of f} t = \lambda \int_0^{f(t)} \frac{ds}{{\Phi_R}(s)}.$$ Moreover $g_{\varepsilon}\to g$ uniformly with $g(t) = \int_0^t f(s)\, ds$ and $w_{\varepsilon}\to w$ with $$\nonumber w(x) = g(r) + M - g(|x|).$$ We still have $w \geq v$ in $B_r$ and $w = M$ on $
\partial B_r$. Thus it is enough to show that $$\sup_{B_r}w \leq \sigma M \,,$$ which by the definition of $w$ is equivalent to $$g(r) \leq (\sigma-1) M =: \mu M.$$
We argue by contradiction and assume $g(r) > \mu M$. Since $g(0)= 0$ it follows from the mean value theorem that there exists $\xi < r$ such that $g'(\xi) = f(\xi)= \frac{\mu M}{r}$. Therefore it follows from \[def of f\] that $$\label{almost max long}
\begin{split}
r > \xi = \lambda \int_0^{f(\xi)} \frac{ds}{{\Phi_R}(s)} &= \lambda \int_0^{ \frac{\mu M}{r}} \frac{ds}{{\Phi_R}(s)} \geq \lambda \int_{\frac{\mu M}{2r}}^{\frac{\mu M}{r}} \frac{ds}{\eta_R(s) s} \\
&\geq \lambda \left( \sup_{\frac{\mu M}{2r} <t < \frac{\mu M}{r}} \eta_R(t) \right)^{-1} \int_{\frac{\mu M}{2r}}^{\frac{\mu M}{r}} \frac{ds}{s}\\
&\geq \log 2 \, \lambda \left( \sup_{\frac{\mu M}{2r} <t < \frac{\mu M}{r}} \eta_R(t) \right)^{-1}.
\end{split}$$ By the assumption \[P3’\] we obtain $$\sup_{\frac{\mu M}{2r} <t < \frac{\mu M}{r}} \eta_R(t) \leq \Lambda_0^3 \eta(\mu) \eta(1/r) \eta_R(M) \leq C \eta_R(M) r^{-1/2},$$ where the last inequality follows from $\eta(t) \leq C\sqrt{t}$ for $t \geq 1$. Hence by \[almost max long\] we have $$\sqrt{r} >\frac{c}{\eta_R(M)}\,,$$ which contradicts the assumption $$r \leq \frac{c_0}{\eta_R(M)^2}$$ when $c_0>0$ is small enough. Moreover since $\eta_R \geq 1$ we see that $$\nonumber \frac{c_0}{\eta_R(M)^2} \leq c_0 \leq r_0$$ if $c_0 > 0$ is again small enough.
We will now prove the boundary Hölder continuity.
\[bdry holder cont\] Let $\Omega \subset \rn$ be a $\delta$-Reifenberg flat domain, and assume $x_0 \in
\partial \Omega$. Let $u \in C(B(x_0,r) \cap \overline \Omega )$, with $0 < r \leq 1$, be a non-negative solution of \[rescaledPDE,rescaled-non-homogeneity\] such that $u=0$ on $
\partial \Omega$ and denote $M = \sup_{B(x_0,r) \cap \Omega} u$. Then if $\delta \leq \frac{1}{100}$, $u$ is Hölder continuous and for every $0 < \rho \leq r$ the following holds $$\sup_{B(x_0, \rho) \cap \Omega } u \leq C_1 M \left(\frac{\rho}{r} \right)^\alpha + C_1{\Phi_R}(M) \rho^{2\alpha} r^{2\alpha}$$ for some $C_1$ and $\alpha \in (0,1/8)$.
We may assume that $\delta = \frac{1}{100}$, since the proof does not improve with smaller value of $\delta$. We may also assume that $x_0 = 0$. In this proof we will proceed with a comparison construction which is more or less standard when dealing with $\delta$-Reifenberg domains.
First let $\tau \in (0,1)$ be from \[oscdec\] and let $\kappa \in (0,1/4)$ be a number which we choose later. Let us fix $\sigma := \frac{\tau+1}{2 \tau}$ and let $c_0(\sigma)$ be the constant from .
For the forthcoming iteration argument we wish to prove the following. Assume that we are given a radius $0 < \rho_0 \leq r$ and an upper bound $M_0$ which satisfy $$\label{osciassump} \sup_{B_{\rho_0} \cap \Omega} u \leq M_{0} \leq \frac{1}{\sigma} M \qquad \text{and}\qquad \rho_0 \leq \frac{c_0}{\eta_R(M_0)^2}.$$ Let us define the following quantities, $$\begin{aligned}
\rho_1 &= \kappa \rho_0, \quad \tau_1 = \frac{\tau+1}{2}, \\
M_1 &= \min\{\tau_1 M_0 + C {\Phi_R}(M) \sqrt{\rho_0},M_0\}.
\end{aligned}$$ We will show that when $\kappa>0$ is chosen small enough then \[osciassump\] implies $$\label{osci} \sup_{B_{\rho_1} \cap \Omega} u \leq M_{1} \qquad \text{and}\qquad \rho_1 \leq \frac{c_0}{\eta_R(M_1)^2}.$$ Moreover the choice of $\kappa$ is independent of $u, r$ and $R$.
We begin by proving the first claim of \[osci\]. To do this we will fix the scale $\rho_0$ in and obtain a hyperplane $\pi$ and a set of local coordinates such that after rotation, the normal of the plane is in the $e_N$ direction and $\Omega \cap B_\rho \subset \{x_N > -2 \delta \rho_0 \}$. Let us choose a point $z_0 = (0,-2\delta \rho_0) \in \er^{N-1}\times \er$ and notice that $$\begin{aligned}
\label{ball inclusion} B_{\rho_1} \subset B(z_0, \rho_0/4) \subset B(z_0, \rho_0/2) \subset B_{\rho_0}.
\end{aligned}$$ Denote $D := B(z_0, \rho_0/2) \cap \{x_N > -2\delta \rho_0\}$. First note that in the domain $D \cap \Omega$ $u$ is a subsolution of \[model2\] and vanishes continuously on $D \cap
\partial \Omega$. It is easy to see that if we extend $u \equiv 0$ to $D \setminus \overline{\Omega}$ then $u$ is a subsolution of \[model2\] in $D$ with nonlinearity ${\phi_R}$. By there exists $v \in C(\overline{D})$ which is a solution of the following Dirichlet problem $$\label{Q1comparison}
\begin{cases}
\mathcal{P}_{\lambda, \Lambda}^-(D^2 v) = {\phi_R}(|Dv|) &\text{ in } D \\
v = 0 & \text{ on }
\partial D \setminus \overline{\Omega} \\
v = u & \text{ on }
\partial D \cap \Omega
\end{cases}$$ and $u \leq v$ in $D$. Since $v$ solves \[model2\] we can (again by extending as zero) use \[osciassump\] and to get that $$\sup_{D} v \leq \sigma M_0 =: \hat M_0\,.$$ Moreover, since the equation in \[Q1comparison\] is of the type \[rescaledPDE,rescaled-non-homogeneity\] we can via reflection get a signed solution of an equation of the same type in $B(z_0, \rho_0/2) $ and therefore use the estimate \[oscdec\] in to get $$\sup_{ B(z_0, \rho_0/4) \cap \Omega} v \leq \tau \hat M_0 + C {\Phi_R}(\hat M_0) \sqrt{\rho_0}.$$ Since $u \leq v$ in $D$ we have by \[ball inclusion\] and \[osciassump\] that $$\sup_{B_{\rho_1} \cap \Omega} u \leq M_{1}$$ which implies the first claim of \[osci\].
To prove the second claim of \[osci\], the aim is to choose $\kappa$ such that it holds. First note that $\tau M_0 \leq M_1 \leq M_0$, which can be rephrased as $$\nonumber \tau \leq \frac{M_1}{M_0} \leq 1\,.$$ Hence using \[P3’\] and the fact that on $(0,1)$ $\eta$ is non-increasing, we get $$\nonumber \eta_R(M_1) \leq \Lambda_0 \eta(M_1/M_0) \eta_R(M_0) \leq \Lambda_0 \eta(\tau) \eta_R(M_0)\,.$$ From \[osciassump\] together with the above we get $$\nonumber \frac{c_0}{\eta_R(M_1)^2} \geq \frac{c_0}{\Lambda_0^2 \eta(\tau)^2 \eta_R(M_0)^2} \geq \frac{c_0}{\Lambda_0^2 \eta(\tau)^2} \rho_0\,.$$ Thus by choosing $\kappa = \min\{\frac{c_0}{\Lambda_0^2 \eta(\tau)^2},1/8\}$ we have proved the second claim of \[osci\].
Let us now choose $M_0 := \frac{1}{\sigma} M$ and a radius $\rho_0 \leq r$ such that $$\label{starting radii} \rho_0 := \min \bigg\{ \frac{c_0}{\eta_R(M_0)^2},r \bigg \}.$$ Define the radii $$\nonumber \rho_j := \kappa^j \rho_0$$ and the quantities $$\nonumber M_j := \min \big \{\tau_1 M_{j-1} + C {\Phi_R}(M) \sqrt{\rho_{j-1}},M_j\big \}.$$ Iterating the implication \[osciassump\] $\implies$ \[osci\] we obtain that $$\label{oscdecay} \operatorname*{osc}_{B_{\rho_j}} u \leq M_j, \quad j=1,\ldots.$$ Consider now the function $\omega(\rho):[0,\rho_0]$ such that $\omega(0)=0$ and $$\nonumber \omega(\rho_j) = M_j, \quad \omega(t \rho_{j+1} + (1-t) \rho_j) = M_{j+1} + t(M_{j}-M_{j+1}) \qquad \text{ for $t \in (0,1)$}.$$ Fix a radius $\rho \in (0,\rho_0]$. Let $k$ be such that $\rho \in [\rho_{k+1},\rho_k]$. Then $\kappa^2 \rho \in [\rho_{k+3},\rho_{k+2}]$. Let us estimate $$\label{iteration inequality} \omega(\kappa^2 \rho) \leq \omega(\rho_{k+2}) \leq \tau_1 \omega(\rho_{k+1}) + C {\Phi_R}(M) \sqrt{\rho_{k+1}} \leq \tau_1 \omega(\rho) + C {\Phi_R}(M) \sqrt{\rho}\,.$$ Thus as in , we may use \[iteration inequality\] and [@GT Lemma 8.23 (with $\tau = \kappa^2,\, \mu=1/2,\, \gamma = \tau_1$)] to get an $\tilde \alpha \in (0,1/4)$ such $$\label{iteration estimate1} \omega(\rho) \leq C \left (\frac{\rho}{\rho_0} \right )^{\tilde \alpha}\omega(\rho_0) + C {\Phi_R}(M) \rho^{ \tilde \alpha} \rho_0^{ \tilde \alpha}, \quad \rho \in [0,\rho_0]\,.$$ Using \[oscdecay\], \[iteration estimate1\], the definition of $\omega$ and arguing as before, we get for a new constant $C > 1$ $$\label{iteration estimate2} \operatorname*{osc}_{B_\rho \cap \Omega} u \leq C \left (\frac{\rho}{\rho_0} \right )^{\tilde \alpha} M + C {\Phi_R}(M) \rho^{\tilde \alpha} \rho_0^{\tilde \alpha}, \quad \rho \in [0,\rho_0]\,.$$
We have thus proved the oscillation decay for possibly small radii $\rho \leq \rho_0$. To finish the proof let us show that \[iteration estimate2\] implies the result. First if $\rho_0 = r$ we are done. Let us assume that $\rho_0 < r$. If $\rho \in (\rho_0,r)$ then it follows from $\tilde \alpha \in (0,1/4)$ and \[starting radii\] that $\rho_0^{2} \geq \frac{c}{\eta_R(M)}$ and therefore $$\label{iteration estimate3a} \operatorname*{osc}_{B_\rho \cap \Omega} u \leq M \leq C(\rho_0^{2} \eta_R(M))M \leq C {\Phi_R}(M) \rho^{\tilde \alpha} r^{\tilde \alpha} .$$ On the other hand if $\rho \leq \rho_0$ we get from Young’s inequality that $$\label{iteration estimate3} M \left( \frac{\rho}{\rho_0} \right )^{\tilde \alpha} \leq M \left ( \frac{\rho}{r} \right )^{ \tilde \alpha} + \rho^{ \tilde \alpha} r ^{ \tilde \alpha} \rho_0^{-2\tilde \alpha} M \leq M \left ( \frac{\rho}{r} \right )^{ \tilde \alpha} + C{\Phi_R}(M) \rho^{\tilde \alpha} r ^{ \tilde \alpha}$$ where the last inequality follows from $\rho_0^{2} \geq \frac{c}{\eta_R(M)}$. Therefore by the above inequality together with \[iteration estimate2\] and by \[iteration estimate3a\] we get $$\operatorname*{osc}_{B_\rho \cap \Omega} u \leq C \left (\frac{\rho}{r} \right )^{\tilde \alpha} M + C {\Phi_R}(M) \rho^{\tilde \alpha} r^{\tilde \alpha}, \quad \rho \in [0,r]\,.$$ Denoting $\alpha = \tilde \alpha/2$ finishes the proof.
Blow up estimates {#ssecblow}
=================
In this section we study the blow-up profile of non-negative solutions of \[thePDE\] near the boundary of an NTA-domain. The goal is to prove that every solution grows first with a growth-rate prescribed by the non-homogeneity of the equation. Then we show that there exists a critical value, which we are able to control in a quantitative way, such that after the critical value the solution blow-up as the solutions of the homogeneous equation.
We begin by describing this critical value of the solutions. As in we may rescale such that if we have an $(L,r_0)$-NTA-domain, with $L\geq 2,$ we may assume instead that we have an $(L,2 L^3)$-NTA-domain (the NTA constant $L\geq 2$ is independent of scaling), $\Omega \subset \rn$, and by translation we can assume that $0 \in\partial \Omega$. From [@Jones] (see also [@JK Theorem 3.11]) we conclude that there exists an $(L',r_0)$-NTA-domain $\Omega'$ such that $$\label{capdefinition} \Omega \cap B_{2} \subset \Omega' \subset \Omega \cap B_{2 L^2}\,,$$ where the constants $L',r_0$ depend only on $L \geq 2$ and the dimension $N$. As in [@JK] we call such a domain a *cap*. Let us also define the joint boundary of the domain and the cap, $\Gamma :=\partial \Omega' \cap\partial \Omega$ and the retracted caps $$\label{B plus} \Omega'_{s} := \Omega' \cap \{x: d(x,\Gamma) \geq s \}.$$ The reason for introducing the cap $\Omega'$ is the following useful property.
\[geometric lemma1\] Let $x \in \Omega'_{s} \setminus \Omega'_{2s}$ for $s \leq \tilde s := r_0/(2L')$. Then there is a constant $C = C(L') > 1$ such that $$\nonumber d(x,\Omega'_{2s}) \leq C s\,.$$
Let $y \in \Gamma$ such that $d(x,y)=d(x,\Gamma)$, since $\Omega'$ is an NTA-domain there exists an interior corkscrew point $z = a_{2 L' s}(y) \in \Omega'$ such that $$\nonumber d(z,
\partial \Omega') > 2s\,,$$ i.e. $z \in \Omega'_{2s}$. Furthermore by the triangle inequality $$\nonumber |z-x| \leq |z-y|+|y-x| \leq 2L's + 2s \equiv C(L')s,$$ which proves the lemma.
For the next lemma we denote $$M_s := \sup_{\Omega'_{s}} u,$$ where the retracted cap $\Omega'_{s}$ is defined in \[B plus\]. As in the previous section we assume that $u$ is a solution of \[rescaledPDE,rescaled-non-homogeneity\]. Again we write ${\Phi_R}(t) = \eta_R(t)t$ with $\eta_R(t)= R \eta(t) +1$.
\[Existence of a critical value\] \[blow up A\] Let $\Omega$ and $\Omega'$ be as above, and denote by $A = a_1(0)$ a corkscrew point for the origin $0 \in
\partial \Omega$. Assume that $u\in C(\Omega \cap B_{2 L^3})$ is a non-negative solution of \[rescaledPDE\], \[rescaled-non-homogeneity\] and denote $M = \sup_{\Omega'} u$. For every $\delta>0$ and $\alpha \in (0,1)$ there are $S \in (0,\tilde s]$ ($\tilde s$ is from ) and a constant $C$ depending on $\delta$ and $\alpha$ but not on $u$ and $R$ such that either $$\int_{u(A)}^{M} \frac{dt}{{\Phi_R}(t)} \leq C$$ or $$\label{alt2} \int_{u(A)}^{M_S} \frac{dt}{{\Phi_R}(t)} \leq C \qquad \text{and} \qquad S^{\alpha} \, \eta_R (M_S) \leq \delta\,.$$ If \[osgood2\] holds then we always have \[alt2\].
Before the proof we would like to point out that the properties in \[alt2\] for $M_S$ are exactly what we want for the critical value. The second inequality in \[alt2\] says that the non-homogeneous part in the Harnack inequality in is as small as we want. The first inequality says that we are still able to control the critical value $M_S$ in a precise way.
For the rest of this proof denote $\Omega_L = \Omega \cap B_{2 L^3}$. Let us fix $\delta>0$. To be able to use , we define for every $s \in [0,\tilde s]$ (where $\tilde s$ is from ) $$M_s := \sup_{\Omega'_{s}} u\,.$$
Let us first assume that there exists $s \in (0,\tilde s]$ such that $s^{\alpha}\, \eta_R\left( M_s\right) \leq \delta$ and prove that this implies the second statement of the lemma. We define $S \in (0,\tilde s]$ as $$S:= \sup \Big\{ s \in [0,\tilde s] \, : \, s^{\alpha}\, \eta_R\left( M_s\right) \leq \delta \Big\}.$$ We need to show that there exists a constant such that $$\label{bound for S} \int_{u(A)}^{M_S} \frac{dt}{{\Phi_R}(t)} \leq C.$$ To this aim let $K \in \en$ be such that $2^{-K-1}\tilde s \leq S \leq 2^{-K}\tilde s$. For every $k \leq K$ we define $s_k = 2^{-k}\tilde s$ and $s_{K}:= S$. Moreover we denote $$M_k := \sup_{\Omega'_{s_{k}}} u.$$ We claim that for every $k \leq K-1$ the following holds $$\label{for k less than K} \int_{M_{k}}^{M_{k+1}} \frac{dt}{{\Phi_R}(t)} \leq C 2^{-k \alpha}$$ for a constant $C > 1$ depending on the NTA constant $L$. Note that $M_S = M_{K}$.
Let us fix $k \leq K-1$. Let $x_{k+1}$ be a point in $\overline{\Omega'}_{s_{k+1}}$ such that $u(x_{k+1})=M_{k+1}$. By we deduce that there exists a point $\tilde x \in \Omega'_{s_{k}}$ such that $$\label{closest point NTA} |x_{k+1} - \tilde x| \leq C s_{k+1}$$ for a constant $C$ depending on $L$. Moreover $d(x_{k+1},\Gamma) \geq s_{k+1}$. It can be seen from (with respect to $\Omega$) that we may construct a sequence of equi-sized balls $B^j$, $j=1,\ldots,n$ with radii $\rho \approx s_k$ such that $2 B^j \subset \Omega_L$, pairwise intersecting and $$\nonumber x_{k+1} \in B^1, \quad \tilde x \in B^n\,,$$ where $n$ depends only on the NTA constants of $\Omega$. Let us now use in each ball, and denoting $$\nonumber \bar m_j = \inf_{B^j} u, \quad \bar M_j = \sup_{B^j} u, \quad j=1,\ldots, n\,,$$ we get $$\nonumber \sum_{j=1}^n \int_{\bar m_j}^{\bar M_j} \frac{dt}{\rho^\alpha {\Phi_R}(t) +t} \leq n\, C\,.$$ Since the balls are pairwise intersecting we get $$\nonumber \int_{\min_j \bar m_j}^{\max_j \bar M_j} \frac{dt}{\rho^\alpha {\Phi_R}(t)+t} \leq n\, C\,.$$ Now note two things. First, $M_k < \max_j \bar M_j$ since $x_{k+1} \in B^1$ and second, $M_{k-1} > \min_j \bar m_j$ since $\tilde x \in B^n \cap \Omega'_{s_k}$. Thus $$\nonumber \int_{M_{k}}^{M_{k+1}} \frac{dt}{\rho^\alpha {\Phi_R}(t)+t} \leq \int_{\min_j \bar m_j}^{\max_j \bar M_j} \frac{dt}{\rho^\alpha {\Phi_R}(t)+t} \leq n\, C\,.$$ Consequently for a constant $C_0 > 1$ depending only on $\alpha$ and on the NTA constants of $\Omega$ we have $$\label{from weak harnack} \int_{M_{k}}^{M_{k+1}} \frac{dt}{ (s_{k}^\alpha\eta_R(t) +1)t } \leq C_0.$$
Let us next show that $$\label{joku juttu} s_{k}^\alpha\, \eta_R (t) \geq \delta \qquad \text{for all }\, M_{k} < t < M_{k+1}.$$ Indeed, by the definition of $S$ we know that the following holds $$s^\alpha\, \eta_R ( M_s ) \geq \delta \qquad \text{for all }\, s_{k+1} < s < s_{k}.$$ Let us fix $t \in (M_k,M_{k+1})$. By continuity $t = M_s$ for some $s \in (s_{k+1},s_k)$. Therefore $$s_k^{\alpha} \eta_R(t) \geq s^\alpha \eta_R(M_s) \geq \delta$$ which proves \[joku juttu\].
Finally we have by \[from weak harnack\] and \[joku juttu\] that $$\left(\frac{\delta}{2 } \right) \int_{M_{k}}^{M_{k+1}} \frac{dt}{s_{k}^\alpha \eta_R(t) t} \leq \int_{M_{k}}^{M_{k+1}} \frac{dt}{ (s_{k}^\alpha \eta_R(t) + 1)t} \leq C_0.$$ This proves \[for k less than K\] since ${\Phi_R}(t)= \eta_R(t)t$ and $s_k \leq 2^{-k}$.
Summing \[for k less than K\] over $k = 0, \dots, K-1$ we conclude that there is a constant $C$ such that $$\int_{M_0}^{M_S} \frac{dt}{{\Phi_R}(t)} = \sum_{k=1}^K \int_{M_{k-1}}^{M_{k}}\frac{dt}{{\Phi_R}(t)} \leq C \sum_{k=1}^K 2^{-\alpha k} \leq C.$$ Recall that by definition $M_0 =\sup_{\Omega'_{\tilde s}} u$ where $\Omega'_{\tilde s} := \Omega' \cap \{x: d(x,\Gamma) \geq \tilde s \}$. Therefore it follows from the fact that $\Omega$ is an NTA-domain together with repeated use of the interior Harnack ( with equi-sized balls) as before (staying inside $\Omega_L$) that $$\label{the first step} \int_{u(A)}^{M_0} \frac{dt}{{\Phi_R}(t)} \leq C$$ and \[bound for S\] follows.
We need to treat the case when $s^{\alpha}\, \eta_R\left( M_s\right) > \delta$ for all $s \in (0,1]$. We show that this implies the first claim of the lemma. We define $M_k$ as before but now $K = \infty$. In this case we argue exactly as above and observe that \[for k less than K\] holds for every $k \in \en$. Since $M = \sup_{\Omega'} u = \lim_{k \to\infty }M_k$ we obtain $$\int_{M_{0}}^{M} \frac{dt}{{\Phi_R}(t)} = \sum_{k=0}^\infty \int_{M_{k}}^{M_{k+1}} \frac{dt}{{\Phi_R}(t)} \leq C \sum_{k=0}^\infty 2^{-\alpha k} \leq C.$$ Hence we have $$\label{second alternative} \int_{u(A)}^{M} \frac{dt}{{\Phi_R}(t)} \leq C$$ by \[the first step\].
Finally we note that in the above case the assumption $s^{\alpha}\, \eta_R\left( M_s\right) > \delta$ for all $s \in (0,1]$ necessarily implies $M = \lim_{k \to\infty }M_k = \infty$. Therefore if $\phi$ satisfies \[osgood2\] then \[second alternative\] provides a contradiction and we are never in the case that $s^{\alpha}\, \eta_R\left( M_s\right) > \delta$ for all $s \in (0,1]$.
Next we show that if $\delta$ in \[alt2\] is chosen small enough the solution $u$ will blow-up as the solution of the homogeneous equation. The reason for this is that we may choose $\delta$ so small that the non-homogeneous term in the Harnack inequality in stays small all the way up to the boundary. This follows from the assumption that $\phi$ is of the form $\phi(t)= \eta(t)t$, where $\eta$ is a slowly increasing function We continue to use the notation $M_s := \sup_{\Omega'_{s}} u$, where the retracted cap $\Omega'_{s}$ is defined in \[B plus\], ${\Phi_R}(t) = R\phi(t) +t$ and use the notation ${\Phi_R}(t) = \eta_R (t)t$, i.e., $\eta_R(t)= R \eta(t)+1$.
\[Blow up rate after the critical value\] \[blow up b\] Let $\Omega$, $\Omega'$, $u, \alpha$ and $S$ be as in . There exists a $\delta>0$ such that if for some $S \in (0,\tilde s]$ the following holds $$S^{\alpha} \, \eta_R (M_S) \leq \delta,$$ then for every $s \leq S $ $$\label{blow up b:a} M_s \leq \frac{C}{s^\gamma} M_S \qquad \text{and} \qquad s^\alpha \, \eta_R(M_s) \leq C$$ holds for some $\gamma>1$, where $\delta, \gamma$ and $C$ depends on $\alpha$ but not on $u$ and $R$.
First, let us choose $\delta_1>0$ such that $$\label{choice of delta_1} \Lambda_0 \eta (e^{2 C_0}) < \frac{1}{\delta_1} .$$ Here $\Lambda_0$ is the constant from the assumption \[P3\] and $C_0$ is from \[from weak harnack\]. Second, we choose $\delta_2>0$ such that $$\label{choice of delta_2} \Lambda_0 \, \sup_{k \in \en} 2^{-\alpha k} \, \eta (e^{2C_0k}) < \frac{1}{\delta_2}.$$ This is possible since $\eta$ is slowly increasing which implies $\eta (t) \leq C_\ep t^{\ep}$ for $t \geq 1$ for all $\ep>0$. We choose $\delta>0$ as $$\delta := \delta_1 \delta_2$$ and assume that $S^{\alpha} \, \eta_R (M_S) \leq \delta $.
Denote $$s_k := 2^{-k} S \qquad \text{ and} \qquad M_k := \sup_{\Omega'_{s_k}} u.$$ First we prove that for every $k \in \en$ the following holds: $$\label{bound for maximum} s_k^{\alpha} \eta_R (M_k) \leq \delta_1 \qquad \text{implies} \qquad s_{k}^{\alpha} \eta_R(t) < 1 \,\, \text{for all } \, t \in [M_k, M_{k+1}].$$ We argue by contradiction and assume that the implication \[bound for maximum\] is not true. Let $T \in (M_k, M_{k+1}]$ be the first number for which $$\label{conradiction 1} s_{k}^{\alpha} \eta_R (T) = 1.$$ Since $\eta \geq 1$ is non-increasing on $(0,1)$ then necessarily $T\geq 1$. Moreover, since we assume $s_k^{\alpha} \eta_R (M_k) \leq \delta_1$ then we have $s_{k}^{\alpha} \eta_R (t) \leq 1$ for all $M_k < t <T$. As in the proof of we choose $x_{k+1} \in \overline{\Omega'}_{s_{k+1}}$ such that $M_{k+1} = u(x_{k+1})$ and let $\tilde x \in \Omega'_{s_{k}}$ be such that \[closest point NTA\] holds (see ). Then we can proceed as in \[from weak harnack\] in to conclude that $$C_0 \geq \int_{M_{k}}^{M_{k+1}} \frac{dt}{(s_{k}^\alpha \eta_R(t) +1)\, t} \geq \int_{M_k}^{T} \frac{dt}{2t}\,.$$ This implies $T \leq e^{2 C_0}M_k$. Since $T\geq 1$ and since $\eta_R$ is non-decreasing on $[1,\infty)$, we have by the assumptions \[P3’\] on $\eta_R$ that $$s_{k}^{\alpha} \eta_R(T) \leq s_k^{\alpha} \eta_R(e^{2 C_0}M_k) \leq \Lambda_0 \, \eta(e^{2 C_0}) \, s_k^{\alpha} \eta_R(M_k) <1$$ by \[choice of delta\_1\]. This contradicts \[conradiction 1\] and therefore \[bound for maximum\] holds.
Recall that by our notation $M_S = M_0$ and $S = s_0$. We prove the first estimate in \[blow up b:a\] by induction and claim that for every $k \in \en$ it holds that $$\label{induction 1} M_k \leq e^{2 C_0k} M_0.$$ Clearly \[induction 1\] holds for $k=0$. Let us make the induction assumption that $$\label{indass1} \text{\cref{induction 1} holds true for $k > 0$ .}$$ First, by the assumptions on $u$ we have $$s_0^{\alpha} \, \eta_R(M_0) \leq \delta.$$ Let us show that we have $$\label{induction 2} s_{k}^{\alpha}\eta_R(t) <1, \qquad \text{for all } \, t \in [M_k, M_{k+1}].$$ If $M_k < 1$ then since $\eta_R$ is non-increasing on $(0,1)$ we have $$s_{k}^{\alpha}\eta_R(M_k) \leq s_{0}^{\alpha}\eta_R(M_{0}) \leq \delta < \delta_1$$ and \[induction 2\] follows from \[bound for maximum\]. If $M_k \geq 1$ then by the induction assumption and by the assumptions \[P3’\] on $\eta_R$ we have $$\begin{split}
s_k^\alpha \eta_R(M_k) &\leq s_k^\alpha\eta_R(e^{2 C_0k} M_0 ) \\
&\leq \Lambda_0 2^{-\alpha k} \eta(e^{2 C_0k} ) \, s_0^\alpha \eta_R(M_0) \\
&\leq \Lambda_0 2^{-\alpha k} \eta(e^{2 C_0k} ) \, \delta \leq \delta_1,
\end{split}$$ where the last inequality follows from \[choice of delta\_2\] and from the choice of $\delta$. Hence \[induction 2\] follows from \[bound for maximum\].
We need to show $$\label{induction 3} M_{k+1} \leq e^{2 C_0(k+1)} M_0.$$ Again arguing by iterating as in \[from weak harnack\] we may conclude that $$\int_{M_k}^{M_{k+1}} \frac{dt}{(s_{k}^{\alpha} \eta_R(t) + 1)\, t} \leq C_0.$$ Therefore by \[induction 2\] we have $$\int_{M_k}^{M_{k+1}} \frac{dt}{2 t} \leq C_0.$$ We integrate the above inequality and use the induction assumption \[indass1\] to deduce $$M_{k+1} \leq e^{2 C_0}M_k \leq e^{2C_0(k+1)}M_0$$ which proves \[induction 3\]. Thus we have showed that \[indass1\] implies \[induction 2\] and \[induction 3\] for $k+1$ and thus \[induction 1\] holds for all $k \geq 0$, which implies the first estimate in \[blow up b:a\]. The second estimate in \[blow up b:a\] follows from \[induction 2\].
Using the $\delta \in (0,1)$ given by in we get the following result.
\[blow up 2\] Let $\Omega$ and $\Omega'$ be as in the beginning of the section, and denote by $A = a_1(0)$ the corkscrew point for the origin $0 \in
\partial \Omega$. Assume that $u\in C(\Omega \cap B_{2 L^3})$ is a non-negative solution of \[rescaledPDE\], \[rescaled-non-homogeneity\] and denote $M = \sup_{\Omega'} u$. Let $\alpha \in (0,1)$, then there is a constant $C_2(\alpha) > 1$ such that either $$\label{S0} \int_{u(A)}^{M} \frac{dt}{{\Phi_R}(t)} \leq C_2,$$ or there is an $S \in (0,\tilde s]$ such that $$\label{S1} \int_{u(A)}^{M_S} \frac{dt}{{\Phi_R}(t)} \leq C_2 \,,$$ $$\label{S2} M_s \leq \frac{C_2}{s^\gamma} M_S \qquad \text{for every }\, s \in (0,S) \,,$$ and $$\label{S3} s^{\alpha}\, \eta_R(M_s) \leq C_2 \qquad \text{for every }\, s \in (0,S)\,.$$ However, if $\phi$ satisfies \[osgood2\], then \[S1\], \[S2\], and \[S3\] always hold.
Proof of Theorem \[mainthm\] {#secproof}
============================
This section is devoted to the proof of the main theorem.
### Reduction {#reduction .unnumbered}
As discussed in we may assume that $\Omega$ is a Lipschitz domain with constant $0 < l < 1$ small enough so that $\Omega$ is Reifenberg flat with $\delta \leq \frac{1}{100}$. Assume that $0 \in\partial \Omega$. Furthermore, again alluding to we will assume that $u \in C(B_{16} \cap \overline{\Omega})$ is a non-negative solution of \[rescaledPDE,rescaled-non-homogeneity\]. Due to the above assumption that $\Omega$ is a Lipschitz domain with constant $l$ and the assumption on scale (), we conclude that $\Omega$ is a $(2,16)$-NTA-domain.
### Setup {#setup .unnumbered}
Let us assume that we are in the case in that \[S1\], \[S2\] and \[S3\] hold. Indeed, if we have \[S0\] then the claim is trivially true.
Denote $M := \sup_{B_1 \cap \Omega} u$. Let $C_2$ be the constant from as well as the values $S, M_S$, and the cap $\Omega'$. Note that $B_2 \cap \Omega \subset \Omega'$ by \[capdefinition\]. We wish to prove that there is $\hat C>1$ such that $M \leq \hat C\, M_S$. This will prove the claim since ${\Phi_R}(t) \geq t$ and therefore $$\int_{u(A)}^{M} \frac{dt}{{\Phi_R}(t)} \leq \int_{u(A)}^{M_S} \frac{dt}{{\Phi_R}(t)} + \int_{M_S}^{\hat C M_S} \frac{dt}{t} \leq C_2 + \log \hat C.$$
### Contradiction argument {#contradiction-argument .unnumbered}
Assume that there exists a point $P_1 \in B_1 \cap \Omega$ such that $$\label{counterassumption} u(P_1) > \hat C\, M_S.$$ We will in the following use the short notation $d(x) = d(x,\partial \Omega) \leq d(x,\Gamma)$ where $\Gamma=\partial \Omega \cap\partial \Omega'$. By the definition of $M_S = \sup_{\Omega'_{S}} u$ we have $d(P_1) < S$. Therefore yields $$\label{P1 blow up} u(P_1) \leq \frac{C_2}{d(P_1)^\gamma} M_S \qquad \text{and} \qquad d(P_1)^{\alpha}\, \eta_R\left(u(P_1)\right) \leq C_2,$$ where $\alpha$ is from . To show the second statement above, note that by continuity there is an $s_1$ such that $M_{s_1} = u(P_1)$ and $d(P_1) \leq s_1$. Thus for this particular $s_1$ we get from that $$\nonumber s_1^{\alpha}\, \eta_R\left(M_{s_1}\right) \leq C_2$$ which gives the statement. From \[counterassumption\] we conclude $$\label{mainthm step1} d(P_1) \leq \left( \frac{C_2}{\hat C} \right)^{\frac{1}{\gamma}}=: d_1 \qquad \text{and} \qquad d(P_1)^{\alpha}\, \eta_R\left(u(P_1)\right) \leq C_2.$$
Let $k > C_1$ be a number such that $$\label{first holder small} C_1 \, k^{-\alpha} < 2^{-\gamma-1},$$ where $C_1$ and $\alpha$ are from and $\gamma>1$ is from . Moreover, by choosing $\hat C$ in \[counterassumption\] large enough we may assume that $d_1$ is so small that $$\label{second holder small} C_1\, C_2\, \Lambda_0 \, 2^{-\alpha}\,\phi(2^\gamma) d_1^{\alpha}< 2^{-\gamma-1}$$ and $$\label{kd1 small} k d_1 < \frac{1}{2},$$ where $C_2$ is from and $\Lambda_0$ is from the assumption \[P3\]. Let $\hat P_1$ be a point on $\partial \Omega$ such that $|P_1- \hat P_1 | = d(P_1)$. Let us show that there is a point $P_2 \in B_{kd_1}(\hat P_1)\cap \Omega$ such that $$\label{to show main} u(P_2) \geq 2^{\gamma} u(P_1), \qquad d(P_2) \leq \frac{d_1}{2}, \qquad \text{and} \qquad d(P_2)^{\alpha}\, \eta_R\left(u(P_2)\right) \leq C_2.$$
First we use in $B_{kd_1}(\hat P_1) \cap \Omega$ with $r = kd_1$ and $\rho = d(P_1) \leq d_1$, and conclude that there is a point $P_2 \in B_{kd_1}(\hat P_1) \cap \Omega$ such that $u(P_2)\geq u(P_1)$ and by \[kd1 small\] we have $$\begin{aligned}
\label{from Holder cont} \sup_{B_\rho(\hat P_1) \cap \Omega} u &\leq C_1 u(P_2) k^{-\alpha} + C_1{\Phi_R}( u(P_2)) \, (k d_1)^{2 \alpha} d(P_1)^{2\alpha} \\
&\leq C_1 u(P_2) k^{-\alpha} + C_1{\Phi_R}( u(P_2)) \, 2^{-\alpha} d(P_1)^{2\alpha}\,. \nonumber \end{aligned}$$
Let us show the first claim in \[to show main\], i.e., $$u(P_2) \geq 2^{\gamma} u(P_1).$$ We argue by contradiction and assume that $$u(P_2) <2^{\gamma} u(P_1).$$ Then we have by the assumption \[P3’\] on $\eta$ that $$\begin{split}
{\Phi_R}(u(P_2)) \leq {\Phi_R}(2^{\gamma} u(P_1)) &= 2^{\gamma} \eta_R(2^{\gamma} u(P_1)) \, u(P_1)\\
&\leq \Lambda_0 (2^{\gamma}\eta(2^{\gamma})) \, \eta_R( u(P_1))u(P_1) \\
&= \Lambda_0 \phi(2^{\gamma}) \, u(P_1) \, \eta_R( u(P_1)) .
\end{split}$$ Therefore by \[from Holder cont\], the above inequality, \[first holder small\] and \[mainthm step1\], and finally by \[second holder small\] we conclude $$\begin{split}
u(P_1) \leq \sup_{ B_\rho(\hat P_1) \cap \Omega} u &\leq C_1 u(P_2) k^{-\alpha} + C_1 2^{-\alpha} {\Phi_R}(u(P_2)))d(P_1)^{2\alpha} \\
&\leq C_1 u(P_2) k^{-\alpha} + C_1 \Lambda_0 2^{-\alpha}\phi(2^{\gamma})d_1^\alpha\, u(P_1)\, \eta_R( u(P_1)) d(P_1)^\alpha\\
&\leq 2^{-\gamma-1} u(P_2)+ C_1C_2 \Lambda_0 2^{-\alpha}\phi(2^\gamma) d_1^{\alpha} \, u(P_2) \\
&\leq 2^{-\gamma} \,u(P_2).
\end{split}$$ This contradicts $u(P_2) <2^{\gamma} u(P_1)$ and thus the first claim in \[to show main\] is proved. To continue we use the same argument as in \[P1 blow up\] by applying to get $$u(P_2) \leq \frac{C_2}{d(P_2)^\gamma}M_S, \qquad \text{and} \qquad d(P_2)^{\alpha}\, \eta_R\left(u(P_2)\right) \leq C_2,$$ which proves the third claim. Since $u(P_2) \geq 2^{\gamma} u(P_1)$ we deduce $$\hat C M_S \leq u(P_1) \leq 2^{-\gamma} u(P_2) \leq 2^{-\gamma} \, \frac{C_2}{d(P_2)^\gamma}M_S.$$ This implies $$d(P_2) \leq \frac{1}{2} \left( \frac{C_2}{\hat C} \right)^{\frac{1}{\gamma}} = \frac{d_1}{2}.$$ Hence we have proved \[to show main\].
We may repeat the argument for \[to show main\] we find a sequence of points $(P_i)$ such that $P_{i} \in B_{k d_{i-1}}(\hat P_{i-1}) \cap \Omega$, $$\label{u blows up} u(P_{i}) \geq 2^{\gamma} u(P_{i-1}) \qquad \text{and} \qquad d(P_i) := \dist(P_i,
\partial \Omega) \leq 2^{-i+1}d_1 =: d_{i}$$ for every $i = 2,3,\dots$. By construction for every $l \geq 2$ we have $$|P_{l} - P_1| \leq \sum_{i = 1}^{l-1} |P_{i+1} - P_i| \leq \sum_{i = 1}^{l-1} k d_{i} = 2 k d_1 \sum_{i = 1}^{l-1} 2^{-i} \leq 2 k d_1 <1.$$ Since $P_1 \in B_1 \cap \Omega$ we have $P_i \in B_2 \cap \Omega$ for every $i \in \en$. Moreover $$\lim_{i \to \infty} \dist(P_i,
\partial \Omega) = 0.$$ By \[u blows up\] we deduce that $$\lim_{i \to \infty} u(P_i) = \infty$$ which contradicts the fact that $u$ vanishes continuously on $\partial \Omega$.
The Boundary Harnack Principle {#secbhi}
==============================
In this section we use the Carleson estimate to prove a boundary Harnack principle for two non-negative solutions which vanish on the boundary (). The proof is based on barrier function estimate and this requires the boundary of the domain to satisfy exterior and interior ball condition, i.e., the boundary has to be $C^{1,1}$-regular.
*Proof of Theorem \[the boundary Harnack\]*
-------------------------------------------
Since $\Omega$ is a $C^{1,1}$-domain we may, by flattening the boundary, rescaling (see ) and translating, assume that $0 \in\partial \Omega$ and $\Omega \cap B(0,16C) = \rn_+ \cap B(0,16C)$, where $C$ is the constant in , and $u,v$ are solutions of \[rescaledPDE,rescaled-non-homogeneity\].
It is enough to show that $$\label{ratio bounded} \sup_{ t \in (0,1) } \frac{v(z+te_N)}{u(z+te_N)} \leq \frac{\mu_1}{\mu_0}$$ for every $z \in B(0,C) \cap \{ x_n = 1 \}$ for numbers $\mu_0$ and $\mu_1$ which satisfy the bound stated in the theorem. In fact, it is enough to show \[ratio bounded\] for $z = e_N$.
Denote $x_0 = -e_N$, $x_1 = 2e_N$ and $M_v = \sup_{B_3^+ } v$, $m_u = \inf_{B(x_1,1)}u$ where $B_3^+ = B_3 \cap \rn_+$. In particular, $m_u \leq M_v$. First, by we deduce $$\label{estimate m_u} \int_{m_u}^{u(A)} \frac{ds}{{\Phi_R}(s)} \leq C.$$ Second, by we have $$\label{estimate M_v} \int_{v(A)}^{M_v} \frac{ds}{{\Phi_R}(s)} \leq C.$$ We divide the proof in two cases. First we assume that $$\label{assume m_u} \int_0^{m_u/3} \frac{ds}{{\Phi_R}(s)} \geq 4\tilde{C}$$ and $$\label{assume M_v} \int_{M_v}^{\infty} \frac{ds}{{\Phi_R}(s)} \geq 2\tilde{C}$$ holds, where $\tilde{C}$ is a large constant which we choose later.
We construct two $C^2$-regular barrier functions $w_1, w_2$ such that $$\label{model1.1} \mathcal{P}_{\lambda, \Lambda}^-(D^2 w_2 ) \geq 2 {\Phi_R}(|Dw_2|), \qquad \text{in } V:= B(x_0, 3)\setminus \bar{B}(x_0,1)$$ and $$\label{model2.1} \mathcal{P}_{\lambda, \Lambda}^+(D^2 w_1) \leq - 2{\Phi_R}(|Dw_1|), \qquad \text{in } U:= B(x_1, 2)\setminus \bar{B}(x_1, 1).$$ Moreover $w_1, w_2$ are such that their gradient do not vanish and they have boundary values $w_2 \geq 0$ on $\partial B(x_0,1)$ and $w_2 = M_v$ on $\partial B(x_0, 3)$, and $w_1 = 0$ on $\partial B(x_1, 2)$ and $w_1 = m_u$ on $\partial B(x_1, 1)$. Hence we have that $w_2 \geq v$ on $\partial V$ and $w_1 \leq u$ on $\partial U$. Since $v$ is a viscosity subsolution of \[model2\] and since $|Dw_2|>0$ it follows from \[model1.1\] and the definition of viscosity subsolution that $v-w_2$ does not attain local maximum in $V$. Therefore we deduce that $w_2 \geq v$ in $V$. Similarly we have $w_1 \leq u$ in $U$. Thus it is enough to bound the ratio $$\sup_{ t \in (0,1) } \frac{w_2(te_N)}{w_1(te_N)}.$$
To construct $w_1$ we define $g:(0,1) \to \er$ such that $$\label{construct g} t = \int_{\mu_0}^{g(t)} \frac{ds}{\tilde{C} {\Phi_R}(s) } \quad \text{for }\, t \in (0,1)$$ where $\tilde{C}>1$, which is the constant in \[assume m\_u\] and \[assume M\_v\], and $0 < \mu_0 \leq m_u$ are constants which we choose later. Note that $g$ is well defined by the implicit function theorem due to \[assume M\_v\] (recall that $m_u \leq M_v$). Then we have $g(0)= \mu_0$ and $g' =\tilde{C} {\Phi_R}(g)$. We define the lower barrier $w_1: U \to \er$ by $$w_1(x):= \int_0^{2-|x-x_1|} g(t)\, dt.$$ Then $w_1 = 0$ on $\partial B(x_1, 2)$. If we choose $\mu_0 = 0$ in \[construct g\] we deduce from \[assume m\_u\] that $g(t) \leq m_u/3$ for all $t \in (0,1)$. This implies $w_1(x) \leq m_u/3$ for all $x \in\partial B(x_1, 1)$. On the other hand, by choosing $\mu_0 = m_u$ in \[construct g\] yields $g(t) > m_u$ for all $t \in (0,1)$, which implies $w_1(x) > m_u$ for all $x \in\partial B(x_1, 1)$. Hence, by continuity we may choose $0 < \mu_0 < m_u$ such that $w_1 = m_u$ on $\partial B(x_1, 1)$. Finally it follows from the construction that $$\nonumber \label{} \inf_{U} |Dw_1| \geq \inf_{t \in (0,1)} g(t) \geq \mu_0 >0\,.$$
After a straightforward calculation we see that since $g' =\tilde{C} {\Phi_R}(g)$, we may choose the constant $\tilde C>2$ in \[construct g\] large enough such that $w_1$ satisfies the following inequality in $U$ $$\begin{split}
\mathcal{P}_{\lambda, \Lambda}^+(D^2 w_1(x)) &= -\lambda \tilde{C} {\Phi_R}(g(2-|x-x_1|)) +\frac{n-1}{|x-x_1|}\Lambda\, g(2-|x-x_1|) \\
&\leq -2 {\Phi_R}(g(2-|x-x_1|)) \\
&=- 2 {\Phi_R}(|Dw_1(x)|).
\end{split}$$ The inequality above follows from ${\Phi_R}(t)\geq t$.
The upper barrier function $w_2$ is constructed similarly by defining first for every $\mu_1 \geq M_v/3$ a function $f :(0,3) \to \er$ as $$t = \int_{f(t)}^{\mu_1} \frac{ds}{\tilde{C}{\Phi_R}(s)} \quad \text{for }\, t \in (0,2).$$ This is well defined due to \[assume m\_u\]. For $x \in V$ we define $w_2(x)$ by $$w_2(x) := \int_0^{|x-x_0|-1} f(t)\, dt.$$ Then $w_2 = 0$ on $\partial B(x_0,1)$. By choosing $\mu_1 = M_v/3$ gives $w_2(x) <M_v$ for all $x \in\partial B(x_0,3)$. Therefore, by continuity we may choose $\mu_1 \geq M_v/3$ such that $w_2 = M_v$ on $\partial B(x_0,3)$. Finally we choose $\tilde{C}$ so large that $w_2$ satisfies \[model2.1\] in $V$. Note that it follows from \[assume m\_u\] that, $ \inf_{t \in (0,2)} f >0$. Hence we have $$\nonumber \label{} \inf_{V} |Dw_2| \geq \inf_{t \in (0,2)} f >0\,.$$
To prove the claim we will show that $$\label{bdry hr 1} \sup_{ t \in (0,1) } \frac{w_2(te_N)}{w_1(te_N)} \leq \frac{\mu_1}{\mu_0}$$ and that $$\label{bdry hr 2} \int_{\mu_0}^{\mu_1} \frac{dt}{{\Phi_R}(t)} \leq C.$$ Let us study the functions $\tilde{w}_1(t) =w_1(te_N)$ and $\tilde{w}_2(t) =w_2(te_N)$ for $t \in [0,1]$. By construction we have that $\tilde{w}_1'(t) =g(t)$ and $\tilde{w}_2'(t) =f(t)$. Since $g' \geq 2{\Phi_R}(g)$ and $f' \leq - 2{\Phi_R}(f)$ we conclude that $\tilde{w}_1$ is convex and $\tilde{w}_2$ is concave. In particular, for every $t \in (0,1)$ we have $$\tilde{w}_1(t) \geq \tilde{w}_1'(0)\, t = \mu_0 \, t\qquad \text{and} \qquad \tilde{w}_2(t) \leq \tilde{w}_2'(0)\, t = \mu_1 \,t.$$ In particular, we have $$\sup_{ t \in (0,1) } \frac{\tilde{w}_2(t)}{\tilde{w}_1(t)} \leq \frac{\mu_1}{\mu_0}\,,$$ which is \[bdry hr 1\].
Recall that $\tilde{w}_1(0) = 0$ and $\tilde{w}_1(1) = m_u$. By the mean value theorem there exists $\xi \in (0,1)$ such that $m_u = \tilde{w}'_1(\xi) = g(\xi)$. By \[construct g\] we have $$1 \geq \xi = \int_{\mu_0}^{g(\xi)} \frac{dt}{\tilde{C}{\Phi_R}(t)} = \int_{\mu_0}^{m_u} \frac{dt}{\tilde{C}{\Phi_R}(t)}.$$ Similarly we deduce that $$\nonumber \int_{M_v}^{\mu_1} \frac{dt}{{\Phi_R}(t)} \leq 3\tilde{C}.$$ Since $u(A) = v(A)$ the estimate \[bdry hr 2\] follows from the previous two inequalities, \[estimate m\_u,estimate M\_v\].
We need to deal the case when either \[assume m\_u\] or \[assume M\_v\] does not hold. In this case the result is almost trivial since we do not claim that the ratio $v/u$ is bounded. Assume first that \[assume m\_u\] does not hold. Then we simply choose $\mu_0 = 0$ and $\mu_1 = u(A)$. The estimate \[bdry hr 2\] follows from \[estimate m\_u\] as follows $$\int_{0}^{u(A)} \frac{dt}{{\Phi_R}(t)} \leq \int_{0}^{m_u/3} \frac{dt}{{\Phi_R}(t)} + \int_{m_u/3}^{m_u} \frac{dt}{t} + \int_{m_u}^{u(A)} \frac{dt}{{\Phi_R}(t)} \leq 4\tilde{C}+ \log 3 + C.$$ If \[assume M\_v\] does not hold, we choose $\mu_0 = v(A)$ and $\mu_1 = \infty$. Then by \[estimate M\_v\] we have $$\int_{v(A)}^\infty \frac{dt}{{\Phi_R}(t)} \leq \int_{v(A)}^{M_v} \frac{dt}{{\Phi_R}(t)} + \int_{M_v}^{\infty} \frac{dt}{t} \leq 2\tilde{C}+ C.$$
Example for the sharpness of the boundary Harnack principle
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Here we discuss the sharpness of . We will only consider the case of the $p(x)$-Laplace equation and show that is sharp. To simplify the argument we construct the example for cubes in the plane. To this aim we construct two non-negative $p(x)$-harmonic functions in the cube $Q = (0,1)^2 \subset \er^2$ such that $v(x_c) \leq u(x_c)$ at the center point $x_c =(1/2,1/2)$ and $u,v = 0$ at the bottom of the cube $(0,1)\times \{0\}$. We will show that the ratio in a smaller cube $Q' = (1/8,7/8)\times (0,1/2)$ $$\sup_{x \in Q'} \frac{v(x)}{u(x)}$$ is not uniformly bounded, but it depends on the value $u(x_c)$ as in the statement of .
First let us choose $p(\cdot) \in C^\infty(Q)$ to be $$p(x) = 3 - x_1, \qquad \text{where $x = (x_1,x_2)$}\,.$$ Then the $p(x)$-Laplace equation \[px\] in non-divergence form for smooth functions with non-vanishing gradient reads as $$\label{px equation} -\Delta w - (1-x_1)\Delta_\infty w = -\log (|\nabla w|) \, w_{x_1},$$ where $\Delta_\infty w = \big \langle D^2w \frac{Dw}{|Dw|}, \frac{Dw}{|Dw|} \big \rangle$ denotes the infinity Laplacian. The point is that the equation is homogeneous for functions of type $u(x) = f(x_2)$ and non-homogeneous for $u(x)= g(x_1)$.
Let $H_{min} > 10^4$ be a constant to be fixed, and consider $H \geq H_{min}$. We define $u \in C^2(Q)$ simply to be $$u(x) = 2 H x_2.$$ Then $u$ is a solution of \[px equation\] and satisfies $u(x_c) = H$ at the center point $x_c =(1/2,1/2)$ and $u(x) = 0$ when $x_2 = 0$. Let us construct a solution $v$ such that $v(x) = 0$ when $x_2 = 0$, $$\label{condition 1 for v} v(x_c) \leq H$$ and at a point $\hat{x} = (7/8,1/2) \in \overline{Q'}$ the following holds $$\label{condition 2 for v} v(\hat{x}) \geq H^{\gamma}$$ for some $\gamma>1$. This will prove that the ratio satisfies $$\sup_{x \in Q'} \frac{v(x)}{u(x)} \geq H^{\gamma-1}$$ since $u(\hat{x}) = H$, this implies that the power-like behavior observed in is sharp.
To this aim we choose $K>1$ to be the number which satisfies $$\label{choice fo K:M} H= \int_0^{1/2}e^{e^{K+s/2}}\,ds, \qquad \text{and define} \qquad M := \int_0^{1} e^{e^{K+s/2}}\,ds.$$ Note that since $e^{e^{K+s/2}}$ is increasing we have $$\label{estimate for H} H \leq e^{e^{K+1/4}}\,.$$ We will need the following easy estimate.
\[retarded estimate\] Let ${\varepsilon}\in (0,2^{-2})$ be fixed, then there exists a $\hat K({\varepsilon})$ such that for all $K \geq \hat K$ the following holds $$\nonumber \int_0^{1} e^{e^{K+s/2}}\,ds \geq e^{e^{K+1/2-{\varepsilon}}}$$
First let $K > 1$, and note that $f(s) = e^{e^{K+s/2}}$ is a strictly increasing function. Then from the mean value theorem we get $$\nonumber \int_{1-2{\varepsilon}}^{1-{\varepsilon}} e^{e^{K+s/2}} ds \geq {\varepsilon}e^{e^{K+1/2-{\varepsilon}}}\,.$$ It is now enough to show that $$\nonumber \frac{1}{{\varepsilon}} \int_{1-2{\varepsilon}}^{1-{\varepsilon}} e^{e^{K+s/2}} ds \leq \int_{1-{\varepsilon}}^{1} e^{e^{K+s/2}} ds.$$ A change of variables leads to $$\nonumber \frac{1}{{\varepsilon}} \int_{1-2{\varepsilon}}^{1-{\varepsilon}} e^{e^{K+s/2}} ds \leq \int_{1-2{\varepsilon}}^{1-{\varepsilon}} e^{e^{K+s/2+{\varepsilon}/2}} ds.$$ We now see that it is enough to prove the much stronger inequality $$\nonumber \frac{1}{{\varepsilon}} \leq \left [e^{e^{K}} \right ]^{e^{{\varepsilon}/2}-1} \,,$$ which is obviously true for a large enough $K({\varepsilon})$ since $e^{{\varepsilon}/2}-1 > 0$.
Let us denote $\Gamma = \{1\}\times (0,1) \subset\partial Q$. We choose $v$ to be the solution of the Dirichlet problem $$\label{veq}
\begin{cases}
&-\Delta v - (1-x_1)\Delta_\infty v = -\log (|\nabla v|) \, v_{x_1}, \\
&v = M \,\, \text{on} \,\,\Gamma, \\
&v = 0 \,\, \text{on} \,\,
\partial Q \setminus \Gamma.
\end{cases}$$ Let us show that $v$ satisfies \[condition 1 for v\] and \[condition 2 for v\].
To show \[condition 1 for v\] we construct a barrier function $\varphi$ such that $\varphi \geq v$ in $Q$ and $\varphi(x_c) = H$. We choose $\varphi(x) = F(x_1)$ where $F$ is an increasing and convex function which is a solution of $$F'' = \frac{1}{2}\log (F') \, F'$$ with $F(0)= 0$. We may solve the above equation explicitly by $$F(t) = \int_0^{t}e^{e^{K+s/2}}\,ds$$ for any $K \in \er$. When we choose $K$ as in \[choice fo K:M\] we get $\varphi(x_c)= F(1/2)=H$. Moreover by \[choice fo K:M\] it holds that $F(1)=M$ and therefore $\varphi \geq v$ on $\partial Q$. It is easy to see that $\varphi \in C^2(Q)$ satisfies $$-\Delta \varphi - (1-x_1)\Delta_\infty \varphi >-\log (|\nabla \varphi|) \, \varphi_{x_1} \qquad \text{in }\, Q.$$ Since $v$ is a solution of \[veq\] $v-\varphi$ does not attain a local maximum. Thus from $\varphi \geq v$ on $\partial Q$ it follows that $\varphi \geq v$ in $Q$. Hence we have \[condition 1 for v\].
To show \[condition 2 for v\] we construct a barrier function $\psi$ in $D:= Q \cap B(x_b,1/2)$, where $x_b = (5/4,1/2)$, such that $\psi \leq v$ in $D$ and $\psi(\hat{x}) \geq H^\gamma$. To this aim we define $\psi(x) = G(|x-x_b|)$ where $G :[1/4,1/2] \to\er$ is decreasing, non-negative and convex function such that $G(1/2)=0$, $G(1/4)=M$ and $$\label{equation for G} G'' \geq \log |G'| \, |G'| + 4|G'|.$$ Let us for a moment assume that such a function exists. The inequality \[equation for G\] implies that $\psi \in C^2(D)$ satisfies $$\begin{split}
-\Delta \psi(x) - (1-x_1)\Delta_\infty \psi(x) &= -(2-x_1)G''(|x- x_b|) -\frac{G'(|x- x_b|)}{|x- x_b|} \\
&< -G''(|x- x_b|) +4|G'(|x- x_b|)| \\
&\leq -\log |G'(|x- x_b|)| \, |G'(|x- x_b|)| \\
&\leq -\log |D\psi(x)| \, \psi_{x_1}(x)
\end{split}$$ for every $x \in D$. Therefore $v-\psi$ does not attain local minimum in $D$. From the conditions $G(1/2)=0$ and $G(1/4)=M$ it follows that $\psi \leq v$ on $\partial D$. Therefore $\psi \leq v$ in $D$.
To find $G$, we denote $G' =g$ and define $$g(t):=-e^{e^{R-(1+{\varepsilon})t}},$$ where the large parameter $R \in \er$ and the small parameter ${\varepsilon}\in(0,1)$ are chosen later. When $R>1$ is large $G' =g$ satisfies \[equation for G\]. To see this note that $$\nonumber g' = (1+{\varepsilon})e^{R-(1+{\varepsilon})t}|g|.$$ Thus \[equation for G\] becomes $$\nonumber (1+{\varepsilon})e^{R-(1+{\varepsilon})t}|g| \geq \log(|g|)|g|+4|g|.$$ After rewriting, this becomes $$\nonumber {\varepsilon}e^{R-(1+{\varepsilon})t} \geq 4,$$ which is true if $R > \hat R({\varepsilon})$.
We define $$G(t) = M + \int_{1/4}^t g(s) \, ds\,.$$ Let us first choose $H_{min}({\varepsilon})$, and thus $M$, large enough such that $$M > \int_{1/4}^{1/2} e^{e^{\hat R-(1+{\varepsilon})s}}\, ds > 0$$ and choose $R\geq \hat R(\epsilon)$ such that $$G(1/2) = M- \int_{1/4}^{1/2} e^{e^{R-(1+{\varepsilon})s}}\, ds = 0.$$ Then we have $G(1/2)= 0$ and $G(1/4)=M$ as wanted.
We need yet to show that $\psi(\hat{x}) \geq H^\gamma$. Recall that $\hat{x} = (7/8,1/2)$ and $x_b = (5/4,1/2)$. Hence $|\hat{x}-x_b|= 3/8$ and therefore $\psi(\hat{x}) = G(3/8)$. We use . \[choice fo K:M\], the definition of $G$ and the condition $G(1/2)=0$ to estimate $$\label{estimate for R} e^{e^{K+1/2-{\varepsilon}}} \leq \int_0^{1} e^{e^{K+s/2}}\,ds= M = \int_{1/4}^{1/2} e^{e^{R-(1+{\varepsilon})s}}\, ds \leq e^{e^{R-(1+{\varepsilon})/4}}$$ by possibly enlarging $H_{min}({\varepsilon})$. Therefore we may estimate the value of $\psi$ at $\hat{x}$ by \[estimate for R\] and \[estimate for H\] and get $$\begin{split}
\psi(\hat{x}) = G(3/8) -G(1/2) &= \int_{3/8}^{1/2} e^{e^{R-(1+{\varepsilon})s}}\, ds \geq \int_{3/8}^{7/16} e^{e^{R-(1+{\varepsilon})s}}\, ds \\
&\geq c e^{e^{R-7/16(1+{\varepsilon})}} \geq c \left(e^{e^{R-(1+{\varepsilon})/4}} \right)^{e^{-3/16-{\varepsilon}}}\\
&\geq c \left(e^{e^{K+1/2-{\varepsilon}}} \right)^{e^{-3/16-{\varepsilon}}} \qquad (\text{by \cref{estimate for R}}) \\
&= c \left(e^{e^{K+1/4 }} \right)^{e^{1/16-2{\varepsilon}}}\\
&\geq c H^{e^{1/16-2{\varepsilon}}} \qquad (\text{by \cref{estimate for H}}).
\end{split}$$ We define $\gamma= e^{1/16-2 {\varepsilon}}$ which is bigger than one by choosing ${\varepsilon}>0$ small enough, hence also fixing $H_{min}$. This shows \[condition 2 for v\].
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abstract: 'We use the grid consisting of bits of $3^n$ to motivate the definition of $2$-adic numbers. Specifically, we exhibit diagonal stripes in the bits of $3^{2^n}$, which turn out to be the first in an infinite sequence of such structures. Our observations are explained by a $2$-adic power series, providing some regularity among the disorder in the bits of powers of $3$. Generally, the base-$p$ representation of $k^{p^n}$ has these features.'
address: |
Mathematics Department\
Tulane University\
New Orleans, LA 70118, USA
author:
- 'Eric S. Rowland'
date: 'October 31, 2009'
title: |
Regularity versus complexity\
in the binary representation of $3^n$
---
Several mysteries {#mysteries}
=================
The binary representation of a number $m$ can be thought of as encoding the unique set of distinct powers of $2$ that sum to $m$. For example, $$81 = 1010001_2 = 1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0 = 2^6 + 2^4 + 2^0.$$ We will display binary representations graphically by rendering $0$ and $1$ respectively as ${\square}$ and ${\blacksquare}$. For reasons that will become clear, the convention in this paper when displaying binary representations graphically is to reverse the order of the digits relative to the standard ordering, so that higher indices are to the right. For example, we write $81 = {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}= 2^0 + 2^4 + 2^6$, where the dot identifies the $2^0$ position (somewhat like a decimal point).
The binary representations of the first several powers of $3$ grow steadily in length: $$\begin{aligned}
3^0 &= {\text{\d{${\blacksquare}$}}} \\
3^1 &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\blacksquare}\\
3^2 &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}\\
3^3 &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}\\
3^4 &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}\end{aligned}$$ Figure \[3n\] displays a grid in which the $n$th row contains the binary digits of $3^n$. Pictures like this were considered by Stephen Wolfram [@NKS page 119]. Small triangles and other local features can be seen, but overall we get an impression of uniform disorder. There is no global structure evident, aside from the right boundary of the pattern, which has slope $\log_2 3$.
![Powers of $3$ in base $2$. The $n$th row consists of the binary digits of $3^n$, in order of increasing exponents.[]{data-label="3n"}](3n)
There are some global regularities, however. In particular, every column is eventually periodic. This is because there are only finitely many (in fact $2^a$) states that can be assumed by the first $a$ columns taken together, so eventually the first $a$ columns return to a state that they have reached before, at which point they become periodic.
In fact, each column is not just eventually periodic but periodic from the start. This is because each row has a unique predecessor, namely the integer obtained by dividing by $3$. Put algebraically, $3$ is invertible modulo $2^a$ for every $a \geq 1$, so from a given row we may compute the previous row to as many bits as we want.
What if we try to compute $a$ bits of “row $-1$” — the predecessor to the initial condition? Certainly we can do this, and the result simply maintains the periodicity of the columns. We can iteratively compute predecessors and thereby uniquely continue the picture up the page. Figure \[3n-history\] shows the end of the unique infinite “history” leading up to the initial condition.
![Part of the history obtained by periodically continuing each column up the page.[]{data-label="3n-history"}](3n-history)
For those readers familiar with cellular automata, we mention that the ability to evolve the system backward in time is analogous to the same ability in a class of cellular automata whose local rules are bijective functions in the rightmost position. As with multiplying each row by $3$ to form the next, information only propagates to the right and is not lost in these automata, and consequently they are reversible under the condition that the left half of each row is determined (say, all white) [@rowland06]. Indeed, several themes of the present paper can be carried over to such cellular automata.
But what do the bits in such rows mean? Rows $n < 0$ in the history illustrated in Figure \[3n-history\] do not represent integers, since they contain $1$s in positions arbitrarily far to the right: The sum used to compute the value of such a row diverges. For example, row $-1$ represents the “infinite integer” $$\begin{aligned}
\cdots{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}\cdots
&= 2^0 + 2^1 + 2^3 + 2^5 + 2^7 + 2^9 + \cdots \\
&= 1 + \sum_{i=0}^\infty 2^{2i+1}.\end{aligned}$$ However, formally applying the geometric series formula to this divergent series produces $$1 + 2 \sum_{i=0}^\infty 4^i \stackrel{?}{=} 1 + \frac{2}{1 - 4} = \frac{1}{3} = 3^{-1},$$ which is certainly a natural object for row $n=-1$ to correspond to. Similarly, row $-2$ represents $$\begin{aligned}
\cdots{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}\cdots
&= 2^0 + \sum_{i=0}^\infty \left( 2^{6i+3} + 2^{6i+4} + 2^{6i+5} \right) \\
&\stackrel{?}{=} 1 + \frac{2^3}{1 - 2^6} + \frac{2^4}{1 - 2^6} + \frac{2^5}{1 - 2^6} \\
&= \frac{1}{9} = 3^{-2}.\end{aligned}$$ This is our first mystery, and in fact for each $n < 0$ there is a divergent series which produces $3^n$ under invalid applications of the geometric series formula.
\[divergent\] Each power $3^n$ for $n < 0$ is the “sum” of a divergent series.
In order to resolve this mystery we must first encounter several additional mysteries — all related to the first — regarding the binary representation of $3^n$.
Since there are only two cell values (${\square}$ and ${\blacksquare}$) in Figure \[3n\], the period length of each column is a power of $2$. (In fact for $a \geq 3$ the period length of $3^n \mod 2^a$ is $2^{a-2}$.) Therefore, another consequence of the column periodicity is that row $2^n$ resembles the initial condition in several bits, since the periods of the first several columns will have just started over. Brenton Bostick brought this “local nestedness” to my attention at the Midwest NKS Conference in 2005. For example $3^{2^2} = 81 = {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}$ agrees with the initial condition $3^0 = {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}\cdots$ to three places to the right of the $2^0$ position. Other terms in the subsequence $3^{2^n}$ agree with the initial condition to more places: $$\begin{aligned}
3^{2^0} &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\blacksquare}\\
3^{2^1} &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}\\
3^{2^2} &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}\\
3^{2^3} &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}\\
3^{2^4} &= {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}\end{aligned}$$ Figure \[32n\] shows many more rows, each row truncated at $600$ bits; the image can be efficiently produced with the following *Mathematica* code.
The large triangular region of white cells indicates a sort of convergence to the initial condition: The farther down we go in this image, the more columns have stabilized to the first bit in their period — the bit in row $0$. As Figure \[32n\] shows, each column (except the leftmost column) eventually becomes white, because row $0$ is simply $\cdots{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}\cdots = 1 = 3^0$. In other words, $3^{2^n}$ “converges” bitwise to $1$ as $n \to \infty$.
![The subsequence $3^{2^n}$. The $n$th row contains the first $600$ bits of $3^{2^n}$.[]{data-label="32n"}](32n)
This can be proven by Euler’s theorem, which states that if $k$ is coprime to $b$ then $k^{\phi(b)} \equiv 1 \mod b$, where the Euler totient function $\phi(b)$ is the number of integers $1 \leq x \leq b$ that are relatively prime to $b$. It is not difficult to convince oneself that if $p$ is prime then $\phi(p^{n+1}) = \frac{p-1}{p} \cdot p^{n+1} = (p-1) p^n$. In our case, letting $k=3$ and $b=2^{n+1}$ gives $3^{2^n} \equiv 1 \mod 2^{n+1}$. Letting $n \to \infty$ shows that every bit in $3^{2^n}$ eventually approaches the corresponding bit of $1$.
Perhaps we feel a little uneasy about giving much credence to this convergence, because certainly $3^{2^n}$ gets very large and far away from $1$ as $n$ gets large. Thus we record it as another mystery.
\[limit\] $\displaystyle{\lim_{n \to \infty} 3^{2^n} = 1}$.
In Figure \[32n\] we see additional structure as well — surprising diagonal lines above the white triangular region. More diagonals are filled in as we go down the page, so there appears to be another bitwise-convergent sequence here. To change the diagonal lines into vertical lines, we make the first column white (for uniformity) and shear the image (shifting each row one position left relative to the row above it). The result is shown as Figure \[32n-sheared\]. Indeed these (shifted) rows are converging bitwise to something — the row $$c_1 = \cdots{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\text{\d{${\square}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}\cdots.$$ The shearing operation can also be effected by shifting the $n$th row left by $n$ bits; in other words, divide the $n$th row by $2^n$. We may therefore write $$c_1 = \lim_{n \to \infty} \frac{3^{2^n} - 1}{2^n}.$$
![Bits of $3^{2^n}$, sheared so that the diagonal lines are now vertical.[]{data-label="32n-sheared"}](32n-sheared)
In Figure \[32n-sheared\] we also observe some secondary diagonal structures that were not easily visible before. They are not as demarcated as the first set and seem to be interacting with the complex background. In order to make the secondary diagonals vertical we would like to perform the same shearing operation. However, first we need to subtract the limiting pattern $c_1$ from each row. But subtract it how? The limit is a divergent “infinite integer”, but forming an integer from the first $a$ bits of $c_1$ and subtracting this integer from each row clears all the corresponding equal bits. Once we have subtracted the limit, we divide by $2^n$ to remove the $n$ bits of $0$s on row $n$. This produces Figure \[32n-shearedagain\], in which the secondary diagonals are no longer muddied by the background but produce a clear limiting pattern themselves. The new limit is $$\begin{aligned}
c_2 &= \lim_{n \to \infty} \frac{\frac{3^{2^n} - 1}{2^n} - c_1}{2^n} \\
&= \cdots{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\text{\d{${\square}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}\cdots.\end{aligned}$$
![Bits of $\left((3^{2^n} - 1)/2^n - c_1\right)/2^n$, obtained from $3^{2^n}$ by twice subtracting the limit and shearing.[]{data-label="32n-shearedagain"}](32n-shearedagain)
It is natural to let $c_0 = \lim_{n \to \infty} 3^{2^n} = 1$ be the first limit. If we continue to iterate this subtract-and-shear operation we continue to find convergent sequences of rows. This means that, despite the apparent complexity in bits of $3^{2^n}$, every region can be decomposed into a sum of simple periodic regions.
The next limit $$c_3 = \cdots{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\text{\d{${\square}$}}}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}{\mspace{-2.76mu}}{\square}\cdots$$ satisfies $$\lim_{n \to \infty} \frac{\frac{\frac{3^{2^n} - c_0}{2^n} - c_1}{2^n} - c_2}{2^n} - c_3 = 0.$$ Let us take this expression and unravel it to see the structure better. We find $$3^{2^n} - (c_0 + c_1 2^n + c_2 2^{2n} + c_3 2^{3n}) \to 0$$ as $n \to \infty$. Replacing $2^n$ with $x$ reveals that this is a power series: $$3^x = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + O(x^4).$$ Of course, we know a power series for $3^x$, namely $$3^x = e^{x \log 3} = 1 + x \log 3 + \frac{1}{2!} (x \log 3)^2 + \frac{1}{3!} (x \log 3)^3 + \frac{1}{4!} (x \log 3)^4 + \cdots,$$ so we might conjecture that $c_i = (\log 3)^i / i!$.
For $i=0$ we indeed have $(\log 3)^0 / 0! = 1 = c_0$. But for $i=1$ the conjecture seems to fail, because $c_1$ is not a real number but an “infinite integer”. (In any case, the bits of $c_1$ don’t resemble the binary representation of the real number $\log 3 = 1.00011001001111101010\cdots_2$.)
\[log3\] “$\log 3$” is not the real number $\log 3$.
A final observation we make is that the direction of convergence is opposite that of real numbers. All the sequences we have seen approach their limits by filling in bits from low indices to high indices, which is toward the right in our graphical convention of reversing the digits of integers. A convergent sequence of real numbers, on the other hand, fills in bits from high indices to low indices. Take the sequence $(1 + 1/n)^n$, for example. Some terms of this sequence (as real numbers) are shown in Figure \[e\]. The convergence proceeds from left to right, which is the same graphical direction but opposite numerical direction as the convergence of the sequence $3^{2^n}$ in Figure \[32n\].
![Binary representations of $(1 + 1/n)^n$ as real numbers, with most significant bits on the left. The terms are slowly converging to $e = 10.10110111111000010101\cdots_2$.[]{data-label="e"}](e)
In the setting of bits of $3^{2^n}$, then, the low indices of a number are somehow stronger than the higher indices. Therefore we should really think of the “tail” of numbers as being backward from the normal sense: In this mode of convergence, two numbers are close to each other if their leftmost bits agree — if their difference is divisible by a large power of $2$. This is why we have chosen the convention that higher indices are to the right.
\[close\] Two numbers are close if their difference is highly divisible by $2$.
$2$-adic numbers {#2-adics}
================
Our four mysteries suggest that there is a notion of number presenting itself through the binary representation of $3^{2^n}$ that is quite different from the real numbers. From Mystery \[close\] we must conclude that in some sense $2^i$ gets small as $i$ gets large, and the other mysteries support this conclusion. Let us therefore make this a *definition* instead of a mystery and introduce a new notion of “size” to make this precise.
Every rational number $r \neq 0$ has a representation $r = 2^\alpha \frac{n}{d}$ for integers $\alpha$, $n$, and $d$, where $n$ and $d$ are not divisible by $2$. Moreover, $\alpha$ is unique. We want $|r|_2$ to be large when $\alpha$ is small and small (but positive) when $\alpha$ is large. A natural choice is to let $|r|_2 = 2^{-\alpha}$; this is called the *$2$-adic norm* of $r$. For example, $|64|_2 = 1/64$ and $|-691/2730|_2 = 2$. Since $0$ is very highly divisible by $2$, let us define $|0|_2 = 0$.
Since large powers $2^i$ are small in the $2$-adic norm, a rational number can be a sum of arbitrarily large powers of $2$ when thought of $2$-adically, just as it can be a sum of arbitrarily large powers of $1/2$ when thought of as a real number. For example, $$\frac{1}{3} = 1 + \sum_{i=0}^\infty 2^{2i+1}.$$
In fact, every rational number has the representation $\sum_{i=N}^\infty c_i 2^i$ for some integer $N$ and $c_i \in \{0,1\}$. For example, $-1$ is rendered $2$-adically as $$-1 = \frac{1}{1 - 2} = \sum_{i=0}^\infty 2^i = 2^0 + 2^1 + 2^2 + 2^3 + \cdots = {\text{\d{${\blacksquare}$}}}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}{\mspace{-2.76mu}}{\blacksquare}\cdots,$$ which illustrates that the $2$-adic representation of a number is really a “limit” of its representations modulo $2^a$ as $a \to \infty$. For finite $a$, the representation modulo $2^a$ of course coincides with its two’s complement representation.
In general, $a$ bits of the $2$-adic representation of a rational number $r = 2^\alpha \frac{n}{d}$ can be found by computing the inverse $d^{-1} \mod 2^a$, which is an integer, and multiplying by $2^\alpha n$.
One can check that the $2$-adic norm induces a metric $d(x,y) = |x-y|_2$ on the rational numbers, akin to the usual metric induced by the absolute value. In particular, it satisfies the triangle inequality $|x-y|_2 + |y-z|_2 \geq |x-z|_2$.
There are some strange properties of this metric, however. Perhaps the most immediate is that every triangle is isosceles: If $|x-y|_2 = |y-z|_2$, then the triangle is isosceles by definition. On the other hand, if $|x-y|_2 \neq |y-z|_2$, then $$|x-z|_2 = \max(|x-y|_2, |y-z|_2).$$ For example, if $x-y = 20$ and $y-z = 6$ then $|x-y|_2 = 1/4 \neq 1/2 = |y-z|_2$ and $|x-z|_2 = |26|_2 = 1/2$.
Naturally, we write $x_n \to x$ if the sequence of $2$-adic norms $|x - x_n|_2$ approaches $0$ as $n \to \infty$. So indeed $3^{2^n} \to 1$ in the $2$-adic metric.
Of course, when we take a limit of rational numbers we may not get another rational number. Traditionally, the real numbers can be constructed by taking limits of rationals with respect to the real metric; each real number has an expansion $\sum_{i=N}^\infty c_i 2^{-i}$ for $c_i \in \{0, 1\}$. Similarly, we can take limits of rationals with respect to the $2$-adic metric and get a different completion of the rationals. This completion is called the *set of $2$-adic numbers*, and each $2$-adic number has a representation $\sum_{i=N}^\infty c_i 2^i$, where again $c_i \in \{0, 1\}$. Like the real numbers, this set is complete — it contains all its limit points.
It turns out that our power series $3^x = \sum_{i=0}^\infty (\log 3)^i x^i / i!$ is correct, but it must be interpreted not as a real power series but as a $2$-adic power series. This means that “$\log 3$” is not the real number $\log 3$ but the $2$-adic number $\log 3$. How do we compute it? The function $\log (1-x)$ has a $2$-adic power series that coincides with its real power series: $$\log (1 - x) = -\sum_{i=1}^\infty \frac{x^i}{i}.$$ Of course, in the real metric this power series diverges at $x = -2$, so it cannot be used to compute the real $\log 3$. But $2$-adically this series converges at $x = -2$ to the $2$-adic $c_1 = \log 3$. Similarly, $c_2 = (\log 3)^2 / 2!$, $c_3 = (\log 3)^3 / 3!$, and so on.
To be precise, one must of course establish the standard objects of calculus over the $2$-adic numbers — derivatives, power series, tests for convergence, etc. We do not undertake this task here but refer the reader to texts on the subject. The book of Gouvêa [@Gouvea] serves as a solid introduction, and Koblitz [@koblitz] provides a more advanced treatment.
Generalizations
===============
The results of the previous section can be generalized in several directions, and we discover that the power series structure we have seen is quite common.
Euler’s theorem tells us that there is nothing particularly special about $3^{2^n}$, and in fact $5^{2^n}$ and $7^{2^n}$ have exactly analogous structures in binary, as shown in the first row of Figure \[otherpowers\]. In general, if $k$ is odd then $k^x$ has a $2$-adic power series $1 + x \log k + \cdots$. Gouvêa [@Gouvea Section 4.5] discusses the region of convergence of such power series.
![Powers $k^{p^n}$ in base $p$.[]{data-label="otherpowers"}](otherpowers)
What about other bases $b > 2$? The second and third rows of Figure \[otherpowers\] show several examples. To address these cases we briefly generalize the discussion to $p$-adic numbers for prime $p$.
Of course we may define $|x|_b$ for general $b$ (prime or composite) in the analogous way. For primes $p$, $|x|_p$ is a norm on the set of rational numbers. For composite $b$ it is not since in general $|x \cdot y|_b \neq |x|_b \cdot |y|_b$; for example, $|4|_4 = 1/4 \neq 1 = |2|_4 \cdot |2|_4$. In fact, it is a theorem of Ostrowski that the $p$-adic norms and the real norm are (up to equivalence) the only nontrivial norms on the set of rational numbers.
Evidently $4^{3^n} \to 1$ in the $3$-adic metric. However, the $3$-adic limit of $2^{3^n}$ is not $1$ but $2 + 2 \cdot 3^1 + 2 \cdot 3^2 + \cdots = -1$. For general $k$ relatively prime to $p$, Euler’s theorem provides that $k^{(p-1) p^n} \equiv 1 \mod p^{n+1}$. In the limit, then, $k^{p^n}$ approaches a $(p-1)$th root of unity $1^{1/(p-1)}$. This root of unity is congruent to $k$ modulo $p$ and is called the *Teichmüller representative* of $k$. This accounts for the vertical stripes in the base-$5$ digits of $2^{5^n}$ and $3^{5^n}$; the $5$-adic fourth roots of unity congruent to $2$ and $3$ modulo $5$ are irrational. Note also that $2^{5^n} + 3^{5^n} \to 0$.
The $p$-adic power series of functions $f(x)$ other than $k^x$ are also evident in the base-$p$ digits of $f(p^n)$. Let $F_n$, $C_n$, $M_n$, and $B_n$ be the sequences of Fibonacci, Catalan, Motzkin, and Bell numbers. The sequences $C_{2^n}$ and $M_{2^n}$ have $2$-adic limits. The sequences $F_{2^n}$ and $B_{2^n}$ do not have $2$-adic limits, but $F_{2^{2n}}$, $F_{2^{2n+1}}$, $B_{2^{2n}}$, and $B_{2^{2n+1}}$ do, giving some indication of the ubiquity of $p$-adic convergence in combinatorial sequences.
Finally, consider the factorial function $x!$. The terms of the sequence $2^n!$, of course, become highly divisible by $2$, so $2^n! \to 0$ in the $2$-adic norm. However, a theorem of Legendre implies that $|2^n!|_2 = 1/2^{2^n-1}$, and it turns out that $$\frac{2^n!}{2^{2^n-1}}$$ has a (nonzero) $2$-adic limit.
[9]{}
Fernando Gouvêa, *$p$-adic Numbers: An Introduction* second edition, Universitext, Springer–Verlag, Berlin, 1997.
Neil Koblitz, *$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions* second edition, Springer–Verlag, New York, 1984.
Eric Rowland, Local nested structure in rule 30, *Complex Systems* **16** (2006) 239–258.
Stephen Wolfram, *A New Kind of Science*, Wolfram Media, Champaign, IL, 2002.
|
---
abstract: 'We present upper estimates for the number of negative eigenvalues of two-dimensional Schrödinger operators with potentials generated by Ahlfors regular measures of arbitrary fractional dimension $\alpha \in (0, 2]$. The estimates are given in terms of integrals of the potential with a logarithmic weight and of its $L\log L$ type Orlicz norms. In the case $\alpha =1$, our results are stronger than the known ones about Schrödinger operators with potentials supported by Lipschitz curves.'
author:
- 'Martin Karuhanga[^1] and Eugene Shargorodsky[^2]'
title: 'On negative eigenvalues of two-dimensional Schrödinger operators with singular potentials'
---
[**Keywords**]{}: Negative eigenvalues; Schrödinger operators; Singular potentials.
Introduction
============
Given a non-negative function $V \in L^1_{\textrm{loc}}(\mathbb{R}^d)$, consider the Schrödinger operator on $L^2(\mathbb{R}^d)$ $$\label{1}
H_V := -\Delta - V, \;\;\;\;\;\;\;\; V \geq 0,$$where $\Delta := \sum^d_{k = 1}\frac{\partial^2}{\partial x^2_k}$. This operator is defined by its quadratic form $$\begin{aligned}
&& \mathcal{E}_{V, \mathbb{R}^d}[u] = \int_{\mathbb{R}^d}|\nabla u(x)|^2\,dx - \int_{\mathbb{R}^d}V(x)|u(x)|^2\,dx ,\\
&& \mathrm{Dom}(\mathcal{E}_{V, \mathbb{R}^d}) = \left\{u\in W^1_2(\mathbb{R}^d)\cap L^2(\mathbb{R}^d, V(x)dx)\right\}.\end{aligned}$$ Denote by $N_-(\mathcal{E}_{V, \mathbb{R}^d})$ the number of negative eigenvalues of $H_V$ counted according to their multiplicity. An estimate for $N_-(\mathcal{E}_{V, \mathbb{R}^d})$ in the case $d \ge 3$ is given by the celebrated Cwikel-Lieb-Rozenblum inequality: $$\label{CLR}
N_-(\mathcal{E}_{V, \mathbb{R}^d})\le C_d\int_{\mathbb{R}^d}V(x)^{d/2}\,dx$$ (see, e.g., [@BE; @BEL; @Roz] and the references therein). If $V \in L^{d/2}(\mathbb{R}^d)$, then this estimate implies that $$\label{O}
N_-(\mathcal{E}_{\lambda V, \mathbb{R}^d}) = O\left(\lambda^{d/2}\right) \ \mbox{ as } \ \lambda \to +\infty .$$ The estimate is optimal in the sense that implies that $V \in L^{d/2}(\mathbb{R}^d)$ (see, e.g., [@RSIV (127)]).
It is well known that does not hold for $d = 2$. In this case, the Schrödinger operator has at least one negative eigenvalue for any nonzero $V \ge 0$, and no estimate of the type $$N_-(\mathcal{E}_{V, \mathbb{R}^2}) \le \mbox{const} + \int_{\mathbb{R}^2} V(x) W(x)\, dx$$ can hold, provided the weight function $W$ is bounded in a neighborhood of at least one point (see [@Grig]). Most known upper estimates for $N_-(\mathcal{E}_{V, \mathbb{R}^2})$ involve terms of two types: integrals of $V$ with a logarithmic weight and $L\log L$ type (or $L_p$, $p > 1$) Orlicz norms of $V$ (see [@Grig; @LapSolo; @MV; @MV1; @Eugene; @Sol] and the references therein). The following inequality is an example of such estimates $$N_-(\mathcal{E}_{V, \mathbb{R}^2}) \le 1 + \mbox{const} \left(\int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, dx +
\|V\|_{\mathcal{B}, \mathbb{R}^2}\right) , \ \ \
\forall V\ge 0 ,$$ where $\|\cdot\|_{\mathcal{B}, \mathbb{R}^2}$ denotes the Orlicz norm , . It was proved in [@Eugene], where it was also shown to be equivalent to the estimate conjectured in [@KMW] and weaker than the one obtained in [@Sol] (see [@Eugene] for stronger estimates). Ideally, one would like to have an optimal estimate of the type $$\label{ideal}
N_-(\mathcal{E}_{V, \mathbb{R}^2}) \le 1 + \Xi(V) ,$$ where $\Xi$ is a combination of certain norms, $\Xi(\lambda V) = O(\lambda)$ as $\lambda \to +\infty$, and, most importantly, $$\label{O2}
N_-(\mathcal{E}_{\lambda V, \mathbb{R}^2}) = O\left(\lambda\right) \ \mbox{ as } \ \lambda \to +\infty$$ implies that $\Xi(V) < \infty$. Unfortunately, even the strongest known estimates for $d = 2$ are not optimal in this sense (see [@Eugene]). Finding an optimal estimate of type seems to be a difficult problem. The estimates for $N_-(\mathcal{E}_{V, \mathbb{R}^2})$ with $V$ supported by Lipschitz curves obtained in [@Kar; @Eugene1] show that may hold for singular potentials supported by lower-dimensional sets. We believe that a better understanding of Schrödinger operators with such singular potentials (supported by fractal sets) might shed some additional light on the above problem. This was the main motivation for the present work, although the results obtained here might be of some relevance to the study of fractal antennae, apertures, screens, and transducers (see, e.g, [@CWH; @CWHM; @CWHMB; @GSK; @MW; @WG] and the references therein), especially in the case of impedance (Robin) boundary conditions (see [@HB; @Ne1; @Ne2; @Ne3]).
In this paper, we deal with the operator $$\label{2}
H_{V\mu} := -\Delta - V\mu\,,\,\,\,V \ge 0,$$ on $L^2(\mathbb{R}^2)$, where $V\in L^1_{\textrm{loc}}(\mathbb{R}^2, \mu)$ and $\mu$ is a $\sigma$-finite positive Radon measure on $\mathbb{R}^2$ that is Ahlfors regular of dimension $\alpha \in (0, 2]$ (see ). We provide a unified treatment of potentials locally integrable with respect to the Lebesgue measure on $\mathbb{R}^2$ ($\alpha = 2$), potentials supported by curves ($\alpha = 1$), and potentials supported by sets of fractional dimension $\alpha \in (0, 1)\cup(1,2)$. In the case $\alpha = 2$, we get the same estimate as in [@Eugene Theorem 6.1], which is stronger than most other known estimates that use isotropic norms. (Anisotropic norms like the ones used in [@Eugene Section 7] and [@LN] are not available in the case $\alpha < 2$ and hence are not treated here.) In the case $\alpha = 1$, our Theorem \[mainthm\] and Corollary \[maincor\] are stronger than the results obtained in [@Kar] and [@Eugene1] as we are now able to cover Ahlfors regular curves rather than just Lipschitz ones. In the case $\alpha \in (0, 1)\cup(1,2)$, our results seem to be completely new.
The proof of our main result, Theorem \[mainthm\], follows the same blueprint as in [@Sol] and [@Eugene], but dealing with measures supported by sets of fractional dimension causes quite a few difficulties. Some of them are listed below.\
1) One of the key technical ingredients in [@Sol] (and in [@Eugene]) was a result saying that the Orlicz norm of the potential over a square of the side length $t > 0$ with a fixed centre is a continuous function of $t$. This is no longer true for potentials of the form $V\mu$ (see ) if the measure $\mu$ is supported by an $\alpha$-dimensional set with $\alpha \in (0, 1]$ and hence can charge the sides of the square. Lemma \[direction\] allows one to choose the directions of the sides of the square in such a way that this difficulty is avoided (see Lemma \[measlemma2\]).\
2) The Birman-Laptev-Solomyak method (see Section \[variational\]) used in this paper (and in [@Sol], [@Eugene]) splits the problem into the radial and non-radial parts. The former is essentially a one-dimensional problem and is usually easier to handle than the latter. If the measure $\mu$ is supported by an $\alpha$-dimensional set with $\alpha \in (0, 2)$, then the radial operator corresponding to is a one-dimensional Schrödinger operator whose potential is a measure that may be supported by a set of a fractional dimension and may even have atoms if $\alpha \in (0, 1]$. Hence one needs to extend to such operators appropriate estimates known for Schrödinger operators with potentials locally integrable with respect to the one-dimensional Lebesgue measure ([@Sol2]). This has been carried out in [@KS].\
3) The Birman-Laptev-Solomyak method allows one to obtain spectral estimates for the non-radial part of the problem mentioned above by splitting $\mathbb{R}^2\setminus\{0\}$ into homothetic annuli centred at $0$, getting an estimate for one of those annuli, and then extending it by scaling to all other ones. Getting an estimate for an annulus usually involves covering it by carefully chosen squares, and an additional difficulty in the case of operator is that one has to distinguish between squares that are centred in the support of the mesure $\mu$ and those that are not. Obviously, this complication does not arise in the standard case where $\mu$ is the two-dimensional Lebesgue measure. Extending an estimate to all annuli by scaling is also not entirely trouble free for operator as the measure $\mu$ does not have to be homogeneous. Scaling leads to a change of measure, and one needs explicit information on how the constants in the estimates depend on the underlying measure. More precisely, one needs to show that those constants depend only on $c_1/c_0$ and $\alpha$ from . Again, it is clear that this complication does not arise in the case where $\mu$ is the two-dimensional Lebesgue measure.
The paper is organised as follows. Auxiliary results on Orlicz spaces and measures are collected in Section \[App\]. The main results are stated in Section \[mainresult\]. In Section \[variational\], we describe the Birman-Laptev-Solomyak method and then apply it in Section \[proof\] to the proof of Theorem \[mainthm\]. Corollary \[maincor\] is proved in Section \[corproof\]. The (non)optimality of our main estimate is discussed in Section \[remark\]. We show that $$N_-(\mathcal{E}_{\lambda\, V\!\mu, \mathbb{R}^2}) = O\left(\lambda\right) \ \mbox{ as } \ \lambda \to +\infty$$ implies that the first sum in the right-hand side of is finite. Unfortunately, this is not the case for the second sum. However, we show that the Orlicz $L\log L$ norm, the $\mathcal{B}$ norm (see ) to be more precise, cannot be substituted with a weaker Orlicz norm. Finally, we prove in Appendix some simple asymptotic results that are needed to justify the applicability of a suitable endpoint trace theorem ([@Maz Theorem 11.8]; see Theorem \[measthm2\] below) in our setting (see the proof of Lemma \[meascor\]).
Auxiliary material {#App}
==================
We start by recalling some notions and results from the theory of Orlicz spaces (see, e.g., [@Ad Ch. 8], [@KR], [@RR]). Let $(\Omega, \Sigma, \mu)$ be a measure space and let $\Psi : [0, +\infty) \rightarrow [0, +\infty)$ be a non-decreasing function. The Orlicz class $K_{\Psi}(\Omega, \mu)$ is the set of all of measurable functions $f : \Omega \rightarrow \mathbb{C}\;( \textrm{or}\;\mathbb{R})$ such that $$\label{orliczeqn}
\int_{\Omega}\Psi(|f(x)|)d\mu(x) < \infty\,.$$ If $\Psi(t) = t^p,\; 1\le p < \infty$, this is just the $L^p(\Omega, \mu)$ space.
A continuous non-decreasing convex function $\Psi : [0, +\infty) \rightarrow [0, +\infty)$ is called an $N$-function if $$\underset{t \rightarrow 0+}\lim\frac{\Psi (t)}{t} = 0 \;\;\; \textrm{and }\;\;\;\underset{t \rightarrow \infty}\lim\frac{\Psi (t)}{t} = \infty.$$ The function $\Phi : [0, +\infty) \rightarrow [0, +\infty)$ defined by $$\Phi(t) := \underset{s\geq 0}\sup\left(st - \Psi(s)\right)$$ is called complementary to $\Psi$.
Examples of complementary functions include: $$\begin{aligned}
\label{calB}
&&\Psi(t) = \frac{t^p}{p},\;\;1 < p < \infty,\;\;\;\;\Phi(t) = \frac{t^q}{q}, \;\;\frac{1}{p} + \frac{1}{q} = 1, \nonumber \\
&&\mathcal{A}(s) = e^{|s|} - 1 - |s| , \ \ \ \mathcal{B}(s) = (1 + |s|) \ln(1 + |s|) - |s| , \ \ \ s \in \mathbb{R} .\end{aligned}$$
We will use the following notation $a_+ := \max\{0, a\}$, $a \in \mathbb{R}$.
[([@Eugene Lemma 2.2])]{}\[elem\] $\frac12\, s\ln_+ s \le \mathcal{B}(s) \le s + 2s\ln_+ s$, $\forall s \ge 0$.
An $N$-function $\Psi$ is said to satisfy the global $\Delta_2$-condition if there exists a positive constant $k$ such that for every $t \geq 0$, $$\label{global}
\Psi(2t)\le k\Psi(t).$$ Similarly $\Psi$ is said to satisfy the $\Delta_2$-condition near infinity if there exists $t_0 > 0$ such that holds for all $t \geq t_0$.
A pair $(\Psi, \Omega)$ is called $\Delta$-regular if either $\Psi$ satisfies a global $\Delta_2$-condition, or $\Psi$ satisfies the $\Delta_2$-condition near infinity and $\mu(\Omega) < \infty$.
[([@Ad Lemma 8.8])]{} $K_{\Psi}(\Omega, \mu)$ is a vector space if and only if $(\Psi, \Omega)$ is $\Delta$-regular.
The $L_{\Psi}(\Omega, \mu)$ is the linear span of the Orlicz class $K_{\Psi}(\Omega, \mu)$, that is, the smallest vector space containing $K_{\Psi}(\Omega, \mu)$.
Consequently, $K_{\Psi}(\Omega, \mu) = L_{\Psi}(\Omega, \mu)$ if and only if $(\Psi, \Omega)$ is $\Delta$-regular.\
\
Let $\Phi$ and $\Psi$ be mutually complementary $N$-functions, and let $L_\Phi(\Omega,\mu)$, $L_\Psi(\Omega, \mu)$ be the corresponding Orlicz spaces. We will use the following norms on $L_\Psi(\Omega, \mu)$ $$\label{Orlicz}
\|f\|_{\Psi, \mu} = \|f\|_{\Psi, \Omega, \mu} = \sup\left\{\left|\int_\Omega f g d\mu\right| : \
\int_\Omega \Phi(|g|) d\mu \le 1\right\}$$ and $$\label{Luxemburg}
\|f\|_{(\Psi, \mu)} = \|f\|_{(\Psi, \Omega, \mu)} = \inf\left\{\kappa > 0 : \
\int_\Omega \Psi\left(\frac{|f|}{\kappa}\right) d\mu \le 1\right\} .$$ These two norms are equivalent $$\label{Luxemburgequiv}
\|f\|_{(\Psi, \mu)} \le \|f\|_{\Psi, \mu} \le 2 \|f\|_{(\Psi, \mu)}\, , \ \ \ \forall f \in L_\Psi(\Omega),$$(see, e.g., [@KR (9.24)]).\
Note that $$\label{LuxNormImpl}
\int_\Omega \Psi\left(\frac{|f|}{\kappa_0}\right) d\mu \le C_0, \ \ C_0 \ge 1 \ \ \Longrightarrow \ \
\|f\|_{(\Psi)} \le C_0 \kappa_0$$ (see [@Eugene]). Indeed, since $\Psi$ is convex and increasing on $[0, +\infty)$, and $\Psi(0) = 0$, we get for any $\kappa \ge C_0 \kappa_0$, $$\label{LuxProof}
\int_{\Omega} \Psi\left(\frac{|f|}{\kappa}\right) d\mu \le
\int_{\Omega} \Psi\left(\frac{|f|}{C_0 \kappa_0}\right) d\mu \le
\frac{1}{C_0} \int_{\Omega} \Psi\left(\frac{|f|}{\kappa_0}\right) d\mu \le 1 .$$ It follows from with $\kappa_0 = 1$ that $$\label{LuxNormPre}
\|f\|_{(\Psi, \mu)} \le \max\left\{1, \int_{\Omega} \Psi(|f|) d\mu\right\} .$$ We will need the following equivalent norm on $L_\Psi(\Omega, \mu)$ with $\mu(\Omega) < \infty$, which was introduced in [@Sol]: $$\label{OrlAverage}
\|f\|^{\rm (av)}_{\Psi, \mu} = \|f\|^{\rm (av)}_{\Psi, \Omega, \mu} = \sup\left\{\left|\int_\Omega f g d\mu\right| : \
\int_\Omega \Phi(|g|) d\mu \le \mu(\Omega)\right\} .$$
[([@KR Theorem 9.3])]{} For any $f\in L_{\Psi}(\Omega, \mu)$ and $g\in L_{\Phi}(\Omega, \mu)$ $$\label{Holder}
\left|\int_{\Omega}fg\,d\mu\right| \le \|f\|_{\Psi,\Omega, \mu}\|g\|_{\Phi,\Omega, \mu}.$$ In particular, $fg\in L^1(\Omega, \mu)$.
The above is called the Hölder inequality for Orlicz spaces. The following is referred to as the strengthened Hölder inequality: $$\label{h2}
\left|\int_{\Omega}fg\,d\mu\right| \le \|f\|_{(\Psi,\Omega, \mu)}\|g\|_{\Phi,\Omega, \mu}\,,$$ for all $f\in L_{\Psi}(\Omega, \mu)$ and $g\in L_{\Phi}(\Omega, \mu)$ (see [@KR (9.27)]).
\[lemma7\][([@Sol Lemma 3])]{} For any finite collection of pairwise disjoint subsets $\Omega_k$ of $\Omega$ $$\label{bsr1}
\sum_k\|f\|^{(av)}_{\Psi,\Omega_k, \mu} \le \|f\|^{(av)}_{\Psi,\Omega, \mu}.$$
Let $$\label{OrlAverage*}
\|f\|^{\rm (av),\tau}_{\Psi,\Omega, \mu} = \sup\left\{\left|\int_\Omega f \varphi d\mu\right| : \
\int_\Omega \Phi(|\varphi|) d\mu \le \tau\mu(\Omega)\right\},\;\;\;\tau > 0 .$$
\[lemma8\]For any $\tau_1,\;\tau_2 > 0$ $$\label{bsr2}
\min\left\{1, \frac{\tau_2}{\tau_1}\right\} \|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu}
\le \|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu} \le \max\left\{1, \frac{\tau_2}{\tau_1}\right\} \|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu}.$$
Let $$X_1:=\left\{\varphi\; :\;\int_{\Omega}\Phi(|\varphi|)d\mu \le \tau_1\mu(\Omega)\right\},\;\;\;X_2
:= \left\{\varphi\; :\;\int_{\Omega}\Phi(|\varphi|)d\mu \le \tau_2\mu(\Omega)\right\}.$$ Suppose that $\tau_1\le \tau_2$. Then, it is clear that $\|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu}\le \|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu}$. Now, since $\Phi$ is convex and $\Phi(0) = 0$, then $$\varphi\in X_2 \Rightarrow\;\;\frac{\tau_1}{\tau_2}\varphi\in X_1\,, \;\;\;(\textrm{cf}.\,\eqref{LuxProof}).$$ Hence, $$\|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu} = \underset{\varphi\in X_2}\sup\left|\int_{\Omega}f\varphi d\mu\right| \le \underset{\phi\in X_1}\sup\left|\int_{\Omega}f.\left(\frac{\tau_2}{\tau_1}\phi\right)d\mu\right| = \frac{\tau_2}{\tau_1}\|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu}.$$ On the other hand, suppose that $\tau_1 \geq \tau_2$. Then $$\|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu}\le \|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu} \le \frac{\tau_1}{\tau_2}\|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu}.$$Hence, $$\min\left\{1, \frac{\tau_2}{\tau_1}\right\} \|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu} \le \|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu}$$ and $$\|f\|^{\rm (av),\tau_2}_{\Psi,\Omega, \mu} \le \max\left\{1, \frac{\tau_2}{\tau_1}\right\} \|f\|^{\rm (av),\tau_1}_{\Psi,\Omega, \mu}.$$
As a result of the above Lemma, we have the following:
\[avequiv\][([@Eugene Lemma 2.1])]{} $$\min\{1, \mu(\Omega)\}\, \|f\|_{\Psi, \Omega, \mu} \le \|f\|^{\rm (av)}_{\Psi, \Omega, \mu}
\le \max\{1, \mu(\Omega)\}\, \|f\|_{\Psi, \Omega, \mu}.$$
Let $(\Omega_1, \Sigma_1)$ and $(\Omega_2, \Sigma_2)$ be a pair of measurable spaces and $\xi :(\Omega_1, \Sigma_1) \to (\Omega_2, \Sigma_2)$ be an isomorphism, i.e. let $\xi$ be a bijection such that both $\xi$ and $\xi^{-1}$ are measurable. Let $\mu$ be a finite measure on $(\Omega_2, \Sigma_2)$ and $V: (\Omega_2, \Sigma_2) \to \mathbb{C}$ be a measurable function. Then $\tilde{V} := V\circ \xi$ is a measurable function on $(\Omega_1, \Sigma_1)$ and $\tilde{\mu} := \mu\circ \xi$, $$\tilde{\mu}(E) = \mu(\xi(E)) , \ \ \ E \in \Sigma_1$$ is a mesure on $(\Omega_1, \Sigma_1)$. For any $c > 0$ and any mutually complementary $N$-functions $\Phi$ and $\Psi$, one gets using and the change of variable formula (see, e.g., [@strok Lemma 5.0.1]) $$\begin{aligned}
\label{scale}
\|V\|^{(\textrm{av})}_{\Psi, \Omega_2, \mu} &=&\textrm{sup}\left\{\left|\int_{\Omega_2}V f\,d\mu\right|
\;:\;\int_{\Omega_2} \Phi(|f|)\,d\mu \le \mu(\Omega_2)\right\}\nonumber\\
&=& \textrm{sup}\left\{\frac{1}{c}\left|\int_{\Omega_1}\tilde{V} g\,d(c\tilde{\mu})\right| \;:\;\int_{\Omega_1}\Phi(|g|)\,d(c\tilde{\mu}) \le
c\tilde{\mu}(\Omega_1)\right\}\nonumber\\
&=& \frac{1}{c}\left\|\tilde{V}\right\|^{(\textrm{av})}_{\Psi, \Omega_1, c\tilde{\mu}}\;.\end{aligned}$$ Hence, by Corollary \[avequiv\] $$\label{maz7}
\left\|\tilde{V}\right\|_{\Psi, \Omega_1, c\tilde{\mu}} \le
\frac{1}{\min\{1, c\tilde{\mu}(\Omega_1)\}} \left\|\tilde{V}\right\|^{(\textrm{av})}_{\Psi, \Omega_1, c\tilde{\mu}}
= \frac{c}{\min\{1, c\tilde{\mu}(\Omega_1)\}} \|V\|^{(\textrm{av})}_{\Psi, \Omega_2, \mu}\;.$$
\[L1Orl\] $$\|f\|_{L_1(\Omega, \mu)} \le \Psi^{-1}(1) \|f\|^{\rm (av)}_{\Psi, \Omega, \mu} .$$
Clearly, one only needs to consider the case $0 < \mu(\Omega) < \infty$. Let $\mu_1 := \frac{1}{\mu(\Omega)}\, \mu$. Then $\mu_1(\Omega) = 1$, and using , [@KR (9.11)], and (with $c = \frac{1}{\mu(\Omega)}$, $(\Omega_1, \Sigma_1) = (\Omega_2, \Sigma_2) = (\Omega, \Sigma)$, and $\xi(x) \equiv x$), one gets $$\begin{aligned}
&& \int_{\Omega} |f(x)|\,d\mu(x) = \mu(\Omega) \int_{\Omega} |f(x)|\,d\mu_1(x) \le
\mu(\Omega) \|f\|_{\Psi, \Omega, \mu_1} \|1\|_{\Phi, \Omega, \mu_1} \\
&& = \mu(\Omega) \|f\|^{\rm (av)}_{\Psi, \Omega, \mu_1} \Psi^{-1}(1)
= \mu(\Omega) \|f\|^{\rm (av)}_{\Psi, \Omega, \frac{1}{\mu(\Omega)}\, \mu} \Psi^{-1}(1)
= \|f\|^{\rm (av)}_{\Psi, \Omega, \mu} \Psi^{-1}(1) .\end{aligned}$$
\[avequivB\][([@Eugene Lemma 2.5])]{} Let $\mu(\Omega) > 1$. Then $$\|f\|^{\rm (av)}_{\mathcal{B}, \Omega, \mu}
\le \|f\|_{\mathcal{B}, \Omega, \mu} + \ln\left(\frac72\, \mu(\Omega)\right)\, \|f\|_{L_1(\Omega, \mu)} .$$
\[direction\] Let $\mu$ be a $\sigma$-finite Borel measure on $\mathbb{R}^2$ such that $\mu(\{x\})= 0,\;\forall x\in\mathbb{R}^2$. Let $$\label{sim}
\Sigma := \left\{\theta \in [0, \pi)\;:\;\exists\;l_{\theta}\mbox{ such that }\mu(l_{\theta}) > 0\right\},$$ where $l_{\theta}$ is a line in $\mathbb{R}^2$ in the direction of the vector $(\cos\theta, \sin\theta)$. Then $\Sigma$ is at most countable.
Let $$\Sigma_N := \left\{\theta \in [0, \pi)\;:\;\exists\; \l_{\theta} \mbox{ such that } \mu(l_{\theta}\cap B(0, N)) > 0\right\},$$ where $B(0, N)$ is the ball of radius $N\in\mathbb{N}$ centred at $0$. Then $$\Sigma = \underset{N\in\mathbb{N}}\cup \Sigma_N.$$It is now enough to show that $\Sigma_N$ is at most countable for $\forall N\in\mathbb{N}$. Suppose that $\Sigma_N$ is uncountable. Then there exists a $\delta > 0$ such that $$\Sigma_{N,\delta} := \left\{\theta \in [0, \pi)\;:\;\exists\; \l_{\theta} \mbox{ such that }\mu(l_{\theta}\cap B(0, N)) > \delta\right\}$$ is infinite. Otherwise, $\Sigma_N = \underset{n\in\mathbb{N}}\cup \Sigma_{N, \frac{1}{n}}$ would have been finite or countable. Now take distinct $\theta_1,..., \theta_k,... \in\Sigma_{N, \delta}$. Then $$\mu\left(l_{\theta_k}\cap B(0, N)\right) > \delta,\;\;\;\forall k\in\mathbb{N}\,.$$ Since $l_{\theta_j}\cap l_{\theta_k} ,\;j\neq k$ contains at most one point, then $$\mu\left(\underset{j \neq k}\cup (l_{\theta_j}\cap l_{\theta_k})\right) = 0.$$ Let $$\tilde{l}_{\theta_k}:= l_{\theta_k}\backslash\underset{j \neq k}\cup (l_{\theta_j}\cap l_{\theta_k})\,.$$ Then $\tilde{l}_{\theta_j}\cap\tilde{l}_{\theta_k} = \emptyset,\;j \neq k$ and $\tilde{l}_{\theta_k}\cap B(0, N) \subset B(0, N)$. So $$\sum_{k\in\mathbb{N}} \mu\left( \tilde{l}_{\theta_k}\cap B(0, N)\right) =
\mu\left(\underset{k\in\mathbb{N}}\cup (\tilde{l}_{\theta_k}\cap B(0, N))\right) \le \mu\left(B(0, N)\right) < \infty\,.$$ But $$\mu\left(\tilde{l}_{\theta_k}\cap B(0, N)\right) = \mu\left(l_{\theta_k}\cap B(0, N)\right) \ge \delta ,$$ which implies $$\sum_{k\in\mathbb{N}} \mu\left( \tilde{l}_{\theta_k}\cap B(0, N)\right) \geq \underset{k\in\mathbb{N}}\sum\delta = \infty\,.$$ This contradiction means that $\Sigma_N$ is at most countable for each $N\in\mathbb{N}$. Hence $\Sigma$ is at most countable.
\[cor-direct\] There exists $\theta_0 \in [0, \pi/2)$ such that $\theta_0 \notin \Sigma$ and $\theta_0 + \frac{\pi}{2} \notin \Sigma$.
The set $$\Sigma - \frac{\pi}{2} := \left\{ \theta - \frac{\pi}{2} \;: \theta\in \Sigma\right\}$$ is at most countable. This implies that there exists $$\theta_0 \in [0, \pi/2)\setminus\left(\Sigma \cup (\Sigma - \frac{\pi}{2})\right) .$$ Thus $\theta_0, \theta_0 + \frac{\pi}{2}\notin \Sigma$.
Let $Q$ be an arbitrary unit square with its sides in the directions determined by $\theta_0$ and $\theta_0 + \frac{\pi}{2}$ in Corollary \[cor-direct\]. For a given $x\in\overline{Q}$ and $t > 0$, let $Q_x(t)$ be the closed square centred at $x$ with sides of length $t$ parallel to those of $Q$.
\[measlemma2\] Suppose that $\Psi$ satisfies the $\Delta_2$-condition (see ). Then for every $f \in L_{\Psi}(Q, \mu)$, the function $t \longmapsto \mathcal{J}(t) :=\|f\|^{\textrm{(\textrm{av})}}_{\Psi, Q_x(t), \mu}$ is continuous and $\mathcal{J}(0+) = 0$.
Let $t > t_0 > 0$. Take any measurable function $g$ on $Q_x(t)$ such that $$\int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))$$ and consider $h_0 := \rho g$, where $\rho = \frac{\mu(Q_x(t_0))}{\mu(Q_x(t))} \le 1$. Then $$\begin{aligned}
\int_{Q_x(t_0)}\Phi(|h_0|)\,d\mu = \int_{Q_x(t_0)}\Phi(|\rho g|)\,d\mu \le \int_{Q_x(t)}\Phi(|\rho g|)\,d\mu\\ = \rho\int_{Q_x(t)}\Phi(| g|)\,d\mu \le \rho\mu(Q_x(t)) = \mu(Q_x(t_0)).\end{aligned}$$ Hence $$\begin{aligned}
0 &\le& \|f\|^{\textrm{(av)}}_{\Psi, Q_x(t), \mu} - \|f\|^{\textrm{(av)}}_{\Psi, Q_x(t_0), \mu}\\&=& \textrm{sup}\left\{ \left|\int_{Q_x(t)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g|)\,d\mu \le \mu(Q_x(t))\right\}\\ &-& \textrm{sup}\left\{ \left|\int_{Q_x(t_0)}fh\;d\mu\right| : \int_{Q_x(t_0)}\Phi(|h|)\,d\mu \le \mu(Q_x(t_0))\right\}\\&\le& \textrm{sup}\left\{ \left|\int_{Q_x(t)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g|)\,d\mu \le \mu(Q_x(t))\right\}\\
&-& \textrm{sup}\left\{\rho \left|\int_{Q_x(t_0)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g|)\,d\mu \le \mu(Q_x(t))\right\}\\&\le& \textrm{sup}\left\{\left|\int_{Q_x(t)}fg\;d\mu\right|- \rho\left|\int_{Q_x(t_0)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))\right\}\\&\le&\textrm{sup}\left\{\left|\int_{Q_x(t)\setminus Q_x(t_0)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))\right\} \\&+& (1 - \rho)\textrm{sup}\left\{\left|\int_{Q_x(t_0)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))\right\}.\end{aligned}$$
For every interval $I\subseteq Q$ parallel to the sides of $Q$, $\mu(I) = 0$. Then $\mu \left(Q_x(t)\setminus Q_x(t_0)\right) \longrightarrow \mu(\partial Q_x(t_0)) = 0$ as $t \longrightarrow t_0$.
Using the Hölder inequality (see ), we get $$\begin{aligned}
&&\sup\left\{\left|\int_{Q_x(t)\setminus Q_x(t_0)}fg\;d\mu\right| : \ \int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))\right\} \\
&&\le \sup_{\int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))} \|f\|_{\left(\Psi, Q_x(t)\setminus Q_x(t_0),\mu\right)}
\|g\|_{\Phi,Q_x(t)\setminus Q_x(t_0), \mu} \\
&&\le \|f\|_{\left(\Psi,Q_x(t)\setminus Q_x(t_0),\mu\right)} 2 \max\{1, \mu(Q_x(t))\}\end{aligned}$$(see and ). Since $\Psi$ satisfies the $\Delta_2$ condition, it follows from [@KR Theorems 9.4 and 10.3] that $$\lim_{t \longrightarrow t_0} \|f\|_{\left(\Psi,Q_x(t)\setminus Q_x(t_0),\mu\right)} = 0.$$ Further, $$\rho = \frac{\mu(Q_x(t_0))}{\mu(Q_x(t))} = 1 - \frac{\mu\left(Q_x(t)\setminus Q_x(t_0)\right)}{\mu(Q_x(t))}\longrightarrow 1 \;\;\textrm{as} \;\;
t \longrightarrow t_0.$$ Hence $$(1 - \rho) \sup\left\{\left|\int_{Q_x(t_0)}fg\;d\mu\right| : \int_{Q_x(t)}\Phi(|g(x)|)\,d\mu \le \mu(Q_x(t))\right\} \longrightarrow 0$$ as $t \longrightarrow t_0$. The case $t_0 > t > 0$ is proved similarly.
Finally, the equality $\mathcal{J}(0+) = 0$ follows from [@KR Theorems 9.4 and 10.3].
We will use the following pair of mutually complementary $N$-functions $$\label{thepair}
\mathcal{A}(s) = e^{|s|} - 1 - |s| , \ \ \ \mathcal{B}(s) = (1 + |s|) \ln(1 + |s|) - |s| , \ \ \ s \in \mathbb{R} .$$
Let $\mu$ be a positive Radon measure on $\mathbb{R}^2$. We say the measure $\mu$ is Ahlfors regular of dimension $\alpha \in (0, 2]$ if there exist positive constants $c_0$ and $c_1$ such that $$\label{Ahlfors}
c_0r^{\alpha} \le \mu(B(x, r)) \le c_1r^{\alpha}\;$$ for all $0< r \le \mathrm{diam(supp}\,\mu)$ and all $x\in$ $\mathrm{supp}\, \mu$, where $B(x, r)$ is a ball of radius $r$ centred at $x$ and the constants $c_0$ and $c_1$ are independent of the balls.
If the measure $\mu$ is $\alpha$-dimensional Ahlfors regular, then it is equivalent to the $\alpha$-dimensional Hausdorff measure (see, e.g., [@Dav Lemma 1.2] ). If $\mathrm{supp}\,\mu$ is unbounded, is satisfied for all $ r > 0$. For more details and examples of unbounded Ahlfors regular sets, see for example [@Dav; @HUT; @STR].
Suppose that $\mu$ is the usual one-dimensional Lebesgue measure on a horizontal or a vertical line. Then holds with $\alpha = 1$. This implies $\mu(I) \neq 0$ for every nonempty subinterval $I$ of that line. Hence the need of Lemma \[direction\] and Corollary \[cor-direct\] for the validity of Lemma \[measlemma2\] in this case.
Throughout the paper, we consider integrals and Orlicz norms with respect to $\mu$ over closed rather than open sets. This is because the $\mu$ measure of the boundary of a set may well be positive.
The main result {#mainresult}
===============
Let $\mathcal{H}$ be a Hilbert space and let $\mathbf{q}$ be a Hermitian form with a domain $\mbox{Dom}\, (\mathbf{q}) \subseteq \mathcal{H}$. Set $$\label{hermitian}
N_- (\mathbf{q}) := \sup\left\{\dim \mathcal{L}\, | \ \mathbf{q}[u] < 0, \,
\forall u \in \mathcal{L}\setminus\{0\}\right\} ,$$ where $\mathcal{L}$ denotes a linear subspace of $\mbox{Dom}\, (\mathbf{q})$. The number $N_- (\mathbf{q})$ is called the Morse index of $\mathbf{q}$. If $\mathbf{q}$ is the quadratic form of a self-adjoint operator $A$ with no essential spectrum in $(-\infty, 0)$, then by the variational principle, $N_- (\mathbf{q})$ is the number of negative eigenvalues of $A$ repeated according to their multiplicity (see, e.g., [@BerShu S1.3] or [@BirSol Theorem 10.2.3]).
Assume without loss of generality that $0\in \mbox{supp}\,\mu$ and $\mathrm{diam(supp}\,\mu) > 1$. Let $$J_n = [e^{2^{n - 1}}, e^{2^n}],\;\;n > 0\;\;\;J_0 := [e^{-1}, e],\;\;\;J_n = [e^{-2^{|n|}}, e^{-2^{|n|-1}}],\;\;n < 0,$$ and $$\label{meaeqn4}
G_n := \int_{|x| \in J_n}|\ln|x||V(x)\,d\mu(x), \;\;\; n\neq 0,\;\;\;\; G_0 := \int_{|x| \in J_0}V(x)d\mu(x).$$
If $\mbox{supp}\, \mu$ is bounded, there exists $m \in \mathbb{N}$ such that $$\left(2\frac{c_1}{c_0}\right)^{\frac{m -1}{\alpha}} < \mathrm{diam(supp}\,\mu) \le \left(2\frac{c_1}{c_0}\right)^{\frac{m}{\alpha}} .$$ Then there exists $\eta$ such that $$\label{etam}
\left(2\frac{c_1}{c_0}\right)^{-\frac{1}{\alpha}} < \eta \le 1 \ \ \mbox{ and } \ \
\mathrm{diam(supp}\,\mu) = \eta \left(2\frac{c_1}{c_0}\right)^{\frac{m}{\alpha}} .$$ If $\mbox{supp}\, \mu$ is unbounded, we just take $\eta =1$. Then we set $$\label{anc0c1}
Q_n := \left\{x\in\mathbb{R}^2 \;:\; \eta\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}} \le |x| \le
\eta\left(2\frac{c_1}{c_0}\right)^{\frac{n }{\alpha}}\right\}, \; n\in\mathbb{Z}$$ and $$\label{Dn}
\mathcal{D}_n := \|V\|^{(\textrm{av})}_{\mathcal{B}, Q_n, \mu}$$ (see ).
Define the operator by its quadratic form $$\begin{aligned}
&& \mathcal{E}_{V\mu,\mathbb{R}^2}[w] := \int_{\mathbb{R}^2}|\nabla w (x)|^2\,dx - \int_{\mathbb{R}^2}V(x)|w(x)|^2\,d\mu(x)\,,\\
&& \mathrm{Dom}(\mathcal{E}_{V\mu, \mathbb{R}^2}) = W^1_2(\mathbb{R}^2)\cap L^2(\mathbb{R}^2, Vd\mu).\end{aligned}$$ Let $N_-(\mathcal{E}_{V\mu, \mathbb{R}^2})$ denote the number of negative eigenvalues of counted according to their multiplicities, i.e. the Morse index of $\mathcal{E}_{V\mu, \mathbb{R}^2}$ defined by . Then we have the following result.
\[mainthm\] Let $\mu$ be a positive Radon measure on $\mathbb{R}^2$ that is Ahlfors regular and $V\ge 0$. Then there exist constants $A > 0$ and $c > 0$ such that $$\label{maineqn}
N_-(\mathcal{E}_{V\mu, \mathbb{R}^2}) \le 1 + 4 \sum_{G_n > 1/4} \sqrt{G_n} + A\sum_{\mathcal{D}_n > c} \mathcal{D}_n\,.$$
\[maincor\] Under the conditions of the above theorem, there exists a constant $B > 0$ such that $$\label{maineqncor}
N_-(\mathcal{E}_{V\mu, \mathbb{R}^2}) \le 1 + B\left(\int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, d\mu(x) + \|V\|_{\mathcal{B}, \mathbb{R}^2,\, \mu}\right) .$$
The proofs of the Theorem and the Corollary are given in sections \[proof\] and \[corproof\] respectively.
The Birman-Laptev-Solomyak method {#variational}
=================================
Our description of the Birman-Solomyak method of estimating $N_- (\mathcal{E}_V)$ follows [@BL; @Eugene; @Sol; @Sol2].
Let $(r, \theta)$ denote the polar coordinates in $\mathbb{R}^2$, $r\in\mathbb{R}_+,\;\theta\in[-\pi, \pi]$ and $$\label{radial}
w_{\mathcal{R}}(r) := \frac{1}{2\pi}\int_{-\pi}^{\pi}w(r, \theta)d\theta,\;\;\;w_{\mathcal{N}}(r, \theta) := w(r,\theta) - w_{\mathcal{R}}(r),$$ where $w\in C(\mathbb{R}^2\setminus\{0\})$. Then $$\label{fN0}
\int_{-\pi}^\pi w_{\mathcal{N}}(r, \theta)\, d\theta = 0 , \ \ \ \forall r > 0 ,$$ and it is easy to see that $$\int_{\mathbb{R}^2} w_{\mathcal{R}} v_{\mathcal{N}}\, dy = 0 , \ \ \ \forall
w, v \in C_0^{\infty}\left(\mathbb{R}^2\setminus\{0\}\right) .$$ Hence $w \mapsto Pw := w_{\mathcal{R}}$ extends to an orthogonal projection $P : L^2\left(\mathbb{R}^2\right) \to
L^2\left(\mathbb{R}^2\right)$.
Using the representation of the gradient in polar coordinates one gets $$\begin{aligned}
&& \int_{\mathbb{R}^2} \nabla w_{\mathcal{R}} \nabla v_{\mathcal{N}}\, dy =
\int_{\mathbb{R}^2} \left(\frac{\partial w_{\mathcal{R}}}{\partial r}
\frac{\partial v_{\mathcal{N}}}{\partial r} + \frac1{r^2}
\frac{\partial w_{\mathcal{R}}}{\partial \theta}
\frac{\partial v_{\mathcal{N}}}{\partial \theta}\right)\, dy \\
&& = \int_{\mathbb{R}^2} \frac{\partial w_{\mathcal{R}}}{\partial r}
\frac{\partial v_{\mathcal{N}}}{\partial r}\, dy =
\int_{\mathbb{R}^2} \left(\frac{\partial w}{\partial r}\right)_{\mathcal{R}}
\left(\frac{\partial v}{\partial r}\right)_{\mathcal{N}}\, dy = 0 , \ \ \
\forall w, v \in C^\infty_0\left(\mathbb{R}^2\setminus\{0\}\right) .\end{aligned}$$ Hence $P : W^1_2\left(\mathbb{R}^2\right) \to W^1_2\left(\mathbb{R}^2\right)$ is also an orthogonal projection.
Since $$\begin{aligned}
&& \int_{\mathbb{R}^2} |\nabla w|^2\, dx = \int_{\mathbb{R}^2} |\nabla w_{\mathcal{R}}|^2\, dx +
\int_{\mathbb{R}^2} |\nabla w_{\mathcal{N}}|^2\, dx , \\
&& \int_{\mathbb{R}^2} V |w|^2\, d\mu(x) \le 2 \int_{\mathbb{R}^2} V|w_{\mathcal{R}}|^2\, d\mu(x) +
2\int_{\mathbb{R}^2} V |w_{\mathcal{N}}|^2\, d\mu(x) ,\end{aligned}$$ we have $$\label{meaeqn1}
N_-(\mathcal{E}_{V\mu, \mathbb{R}^2}) \le N_-(\mathcal{E}_{\mathcal{R},2V\mu}) + N_-(\mathcal{E}_{\mathcal{N},2V\mu})$$ where $\mathcal{E}_{\mathcal{R},2V\mu}$ and $\mathcal{E}_{\mathcal{N}, 2V\mu}$ are the restrictions of the form $\mathcal{E}_{2V\mu, \mathbb{R}^2}$ to $PW^1_2(\mathbb{R}^2)$ and $(I -P)W^1_2(\mathbb{R}^2)$ respectively. Therefore to estimate $N_-\left(\mathcal{E}_{V\mu,\mathbb{R}^2}\right)$, it is sufficient to find estimates for $N_-\left(\mathcal{E}_{\mathcal{R},2V\mu}\right)$ and $N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu}\right)$.\
On the space $PW^1_2(\mathbb{R}^2)$, a simple exponential change of variables reduces the problem to a one-dimensional Schrödinger operator, which provides an estimate for $N_-\left(\mathcal{E}_{\mathcal{R},2V\mu}\right)$ in terms of weighted $L^1$ norms of $V$ (see , ). Theorem \[measthm3\] shows that this estimate is optimal in a sense (see also ).
On the space $(I - P)W^1_2(\mathbb{R}^2)$, one gets an estimate for $N_-\left(\mathcal{E}_{\mathcal{N},2V\mu}\right)$ in terms of Orlicz norms of $V$ (see and ). The variational principle (see, e.g., [@KS Lemma 3.2]) implies that $$\label{variat}
N_-\left(\mathcal{E}_{\mathcal{N},2V\mu}\right) \le \sum_{n\in\mathbb{Z}}N_-\left(\mathcal{E}_{\mathcal{N},2V\mu, Q_n}\right),$$ where $Q_n$ are the annuli defined in , $$\begin{aligned}
&& \mathcal{E}_{\mathcal{N}, 2V\mu, Q_n}[w] := \int_{Q_n}|\nabla w(x)|^2\,dx - 2\int_{Q_n}V(x)|w(x)|^2\,d\mu(x),\\
&& \mathrm{Dom}\;\left(\mathcal{E}_{\mathcal{N},2V\mu, Q_n}\right) = \left\{w\in (I - P)W^1_2(Q_n)\cap L^2\left(Q_n, Vd\mu\right)\right\}.\end{aligned}$$
The main reason for introducing the space $(I - P)W^1_2(\mathbb{R}^2)$ is that $$\label{Omegan0}
\int_{Q_n} w(x)\,dx = 0, \;\;\;\forall w\in (I - P)W^1_2(Q_n)$$ (cf. ), which allows one to use the Poincaré inequality and ensures that not all terms in the right-hand side of are necessarily greater or equal to 1.
The Ahlfors condition allows one to obtain estimates for $N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_n}\right)$ from those for $N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_1}\right)$ by scaling $x \longmapsto x\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}}$. So it is sufficient to find an estimate for $N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_1}\right)$.
Proof of Theorem \[mainthm\] {#proof}
============================
We need to find an estimate for the right-hand side of . We start with the first term. Let $I$ be an arbitrary interval in $\mathbb{R}_+$. Define a measure on $\mathbb{R}_+$ by $$\label{1dmeas}
\nu(I) := \int_{|x|\in I}V(x)\,d\mu(x).$$ Then (see ) $$\int_{\mathbb{R}^2}|w_{\mathcal{R}}(x)|^2 V(x)\,d\mu(x) = \int_{\mathbb{R}_+}|w_R(r)|^2d\nu(r).$$
Let $w \in PW^1_2(\mathbb{R}^2)$, $r = e^t$, $v(t) := w(x) = w_{\mathcal{R}}(r)$ (see ). Then $$\int_{\mathbb{R}^2}|\nabla w(x)|^2dx = 2\pi\int_{\mathbb{R}}|v'(t)|^2dt$$ and $$\begin{aligned}
\int_{\mathbb{R}^2}V(x) |w(x)|^2d\mu(x) &=& \int_{\mathbb{R}_+}|w_{\mathcal{R}}(r)|^2d\nu(r)= \int_{\mathbb{R}}|w_{\mathcal{R}}(e^t)|^2d\nu(e^t)\\&=&\int_{\mathbb{R}}|v(t)|^2\,d\nu(e^t).\end{aligned}$$ Let $$\label{meaeqn2}
\mathcal{G}_n := \frac{1}{2\pi}\int_{\mathbf{ I}_n}|t|\,d\nu(e^t), \;\;\; n\neq 0,\;\;\;\; \mathcal{G}_0 := \frac{1}{2\pi}\int_{\mathbf{ I}_0}d\nu(e^t) ,$$ where $$\mathbf{ I}_n := [2^{n - 1}, 2^n], \ n > 0 , \ \mathbf{ I}_0 := [-1, 1] , \ \
\mathbf{ I}_n := [-2^{|n|}, -2^{|n| - 1}], \ n < 0.$$ Then $$\label{meaeqn3}
N_-(\mathcal{E}_{\mathcal{R}, 2\nu}) \le 1 + 7.61 \sum_{\mathcal{G}_n > 0.046} \sqrt{\mathcal{G}_n}\,,$$ where $$\begin{aligned}
&& \mathcal{E}_{\mathcal{R}, 2\nu}[v] := \int_{\mathbb{R}}|v'(t)|^2\,dt - \int_{\mathbb{R}}|v(t)|^2\,d\nu(e^t),\\
&& \mathrm{Dom}(\mathcal{E}_{\mathcal{R}, 2\nu}) = W^1_2(\mathbb{R})\cap L^2(\mathbb{R}, d\nu)\end{aligned}$$ (see [@KS]). It follows from , and that $G_n = 2\pi\mathcal{G}_n$ and thus implies $$\label{meaeqn5}
N_-(\mathcal{E}_{\mathcal{R}, 2V\mu}) \le 1 + 4 \sum_{G_n > 1/4} \sqrt{G_n}.$$
Now, it remains to find an estimate for the second term in the right-hand side of (see ). We begin by stating some auxiliary results.\
Let $\varphi$ be a nonnegative increasing function on $[0, +\infty)$ such that $t\varphi(t^{-1})$ decreases and tends to zero as $t \longrightarrow\infty$. Further, suppose $$\label{maz2}
\int_u^{+\infty}t\sigma(t) dt \le cu\sigma(u),$$ for all $u > 0$, where $$\label{sigma}
\sigma(v) := v\varphi\left(\frac{1}{v}\right)$$ and $c$ is a positive constant.
\[measthm2\][[@Maz Theorem 11.8]]{} Let $\Psi$ and $\Phi$ be mutually complementary N-functions and let $\mu$ be a positive Radon measure on $\mathbb{R}^2$. Let $\varphi$ be the inverse function of $t \mapsto t\Phi^{-1}(t^{-1})$ and suppose it satisfies the above conditions. Then the best, possibly infinite, constant $A_1$ in $$\label{maz1}
\|w^2\|_{\Psi, \mathbb{R}^2, \mu} \le A_1\|w\|^2_{W^1_2(\mathbb{R}^2)} , \ \ \ \forall w\in W^1_2(\mathbb{R}^2)\cap C(\mathbb{R}^2)$$ is equivalent to $$\label{maz3}
B_1 = \sup\left\{|\log r| \mu(B(x, r))\Phi^{-1}\left(\frac{1}{\mu(B(x, r))}\right) : \ x \in \mathbb{R}^2,\, 0 < r < \frac{1}{2}\right\} ,$$ where $B(x, r)$ is a ball of radius $r$ centred at $x$.
Let $G\subset\mathbb{R}^2$ be a bounded set with Lipschitz boundary. Then there exists a bounded linear operator $$\label{opT}
T_G: W^1_2(G) \longrightarrow W^1_2(\mathbb{R}^2)$$ such that $$\begin{aligned}
&& (T_Gw)|_G = w, \;\;\;\forall w \in W^1_2(G), \\
&& T_Gw \in W^1_2(\mathbb{R}^2)\cap C(\mathbb{R}^2), \;\;\;\forall w \in W^1_2(G)\cap C\left(\overline{G}\right)\end{aligned}$$ (see [@Stein Ch.VI, Section 3]).
\[meascor\][(cf.[@Maz Corollary 11.8/2])]{} Consider the complementary N-functions $\mathcal{B}(t)= (1 + t)\ln(1 + t) - t$ and $\mathcal{A}(t)= e^t - 1 - t$. Let $G\subset\mathbb{R}^2$ be a bounded set with Lipschitz boundary. If a positive Radon measure $\mu$ on $\overline{G}$ satisfies the following estimate for some $\alpha > 0$ $$\label{ball}
\mu(B(x, r)) \le r^{\alpha}\;,\;\;\forall x\in\overline{G} \;\;\; \textrm{and}\;\;\; \forall r \in \left(0, \frac12\right)\,,$$ then the inequality $$\|w^2\|_{\mathcal{A}, \overline{G}, \mu} \le A_1 \|T_G\|^2 \|w\|^2_{W^1_2(G)} , \ \ \ \forall w \in W^1_2(G)\cap C(\overline{G})$$ holds with a constant $A_1$ (see ) depending only on $\alpha$.
First let us check that the conditions of Theorem \[measthm2\] are satisfied. Let $\varrho(t) := t\mathcal{B}^{-1}\left(\frac{1}{t}\right)$ and $\frac{1}{t} = \mathcal{B}(s)$. Then $\varrho(t) = \frac{s}{\mathcal{B}(s)}$. Since $\frac{d}{ds}\left(\frac{\mathcal{B}(s)}{s}\right) = -\frac{1}{s^2}\ln(1 + s) + \frac{1}{s}> 0$ for $s > 0$, the fraction $\frac{s}{\mathcal{B}(s)}$ is a decreasing function of $s$. It is also clear that $\frac{s}{\mathcal{B}(s)} \longrightarrow 0$ as $s \longrightarrow\infty$. Hence $\varrho(t)$ is an increasing function of $t$ and $\varrho(t) \longrightarrow 0$ as $t \longrightarrow 0+$. Further, $$\label{*}
\varrho(t) = t\mathcal{B}^{-1}\left(\frac{1}{t}\right) = \sqrt{2t}\left(1 + o(1)\right)\;\;\textrm{as}\;\;t\longrightarrow \infty$$ and $$\label{**}
\varrho(t) = t\mathcal{B}^{-1}\left(\frac{1}{t}\right) = \frac{1}{\ln\frac{1}{t}}\left(1 + o(1)\right)\;\;\textrm{as}\;\;t\longrightarrow 0$$ (see and in Appendix).
Let $\varphi(\tau) := \varrho^{-1}(\tau)$. Then $\varphi$ is an increasing function. Let $x = \varrho^{-1}\left(\frac{1}{t}\right)$. Then $x$ is a decreasing function of $t$, and $t = \frac{1}{\varrho(x)}$. Hence $$t\varphi(t^{-1}) = t \varrho^{-1}\left(\frac{1}{t}\right) = \frac{x}{\varrho(x)} = \frac{1}{\mathcal{B}^{-1}\left(\frac{1}{x}\right)}$$ is a decreasing function of $t$.
For small values of $\tau$, $$\label{***}
\varphi(\tau)= \tau e^{-\frac{1}{\tau}}e^{O(1)}$$ (see , ). Hence $$t\varphi(t^{-1}) = e^{-t}e^{O(1)} \longrightarrow 0 \ \mbox{ as } \ t \longrightarrow \infty$$ and (see ) $$\begin{aligned}
\int_u^{+\infty}t \sigma(t)\,dt = \int_u^{+\infty}t^2\varphi\left(\frac{1}{t}\right)\,dt = \int_u^{+\infty} te^{-t}e^{O(1)}\,dt \\
\le e^{O(1)} \int_u^{+\infty} te^{-t}\,dt = e^{O(1)} (u +1)e^{-u} \le 2 e^{O(1)} u e^{-u} \le \\
\le e^{O(1)} u^2\varphi\left(\frac{1}{u}\right)
= e^{O(1)} u\sigma(u)\;\;\;\textrm{as}\;\;\;u\longrightarrow +\infty
\end{aligned}$$ (see ).
For large values of $\tau$, $$\varphi(\tau) = \frac{\tau^2}{2}(1 + o(1))$$ (see ). Hence $$t\sigma(t)= t^2\varphi\left(\frac{1}{t}\right) = \frac{1}{2}\left(1 + o(1)\right)\;\;\;\textrm{as}\;\;\;t \longrightarrow 0+ ,$$ $$u\sigma (u) \;\longrightarrow \frac{1}{2} \ \mbox{ and } \
\int_u^{+\infty} t \sigma(t)\,dt \longrightarrow\;\textrm{constant} \;\;\;\textrm{as}\;\;\; u\longrightarrow 0+ .$$ Thus $\varphi(\tau)$ satisfies condition for all values of $u$.
Extend $\mu$ to $\mathbb{R}^2$ by $\mu(E) = 0$ for $E = \mathbb{R}^2\setminus \overline{G}$. It is easy to see that then holds for every $x \in \mathbb{R}^2$, and one has the following estimate for the constant $B_1$ in $$\begin{aligned}
B_1 &=& \sup\left\{|\ln r|\mu(B(x, r))\mathcal{B}^{-1}\left(\frac{1}{\mu(B(x, r))}\right) | \ 0 < r < \frac{1}{2}\right\} \\
&=& \sup_{0 < r < \frac{1}{2}}\frac{|\ln r|}{|\ln \mu(B(x, r))|}\left( 1 + o(1)\right) \le \textrm{const}\sup\frac{|\ln r|}{|\ln r^{\alpha}|}
= \frac{\textrm{const}}{\alpha}
\end{aligned}$$ (see and ). Thus one can take $A_1 \sim \frac{1}{\alpha}$ in . It follows from Theorem \[measthm2\] that $$%
\|w^2\|_{\mathcal{A}, \overline{G}, \mu} = \|(T_Gw)^2\|_{\mathcal{A}, \mathbb{R}^2, \mu} \le A_1\|T_Gw\|^2_{W^1_2(\mathbb{R}^2)} \le
A_1 \|T_G\|^2 \|w\|^2_{W^1_2(G)}$$ for all $w \in W^1_2(G)\cap C\left(\overline{G}\right)$.
We will use the following notation: $$\label{ave}
w_E := \frac{1}{|E|}\int_E w(x)\,dx\,,$$ where $E\subset\mathbb{R}^2$ is a set of a finite Lebesgue measure $|E|$.
\[measlemma3\*\] Let $G\subset\mathbb{R}^2$ be a bounded set with Lipschitz boundary and $\mu$ be a positive Radon measure satisfying . Then there exists a constant $A_2(G) > 0$ such that for any $V\in L_{\mathcal{B}}(\overline{G}, \mu), \;V\geq 0$, $$\label{maz6}
\int_{\overline{G}} V|w(x)|^2d\mu(x) \le A_2(G) \|V\|_{\mathcal{B}, \overline{G}, \mu}\int_{G}|\nabla w |^2dx$$ for all $w\in W^1_2(G)\cap C(\overline{G})$ with $w_G = 0$. One can take $$\label{AG}
A_2(G) = A_1 \|T_G\|^2 \left(1 + C_G\right),$$ where $A_1$ is the constant from Lemma \[meascor\] and $C_G$ is the optimal constant in the Poincaré inequality for $G$. In particular, in the case when $G = Q$ is a unit square with sides chosen in any direction, one can take $$\label{A2pi}
A_2 = A_2(Q)= A_1 \|T_Q\|^2 \left(1 + \pi^{-2}\right),$$ which depends only on $\alpha$.
The proof of , follows from the Hölder inequality for Orlicz spaces (see ), Lemma \[meascor\], and the Poincaré inequality (see, e.g., [@DL2 Ch. IV, §7, Sect. 2, Proposition 2]). Formula follows from the fact that the best constant in the Poincaré inequality equals $1/\lambda_2$, where $\lambda_2$ is the smallest positive eigenvalue of the Neumann Laplacian (see [@DL2 Ch. IV, §7, Sect. 2, Corollary 3]) and that the latter equals $\pi^2$ for the unit square $Q$ (see, e.g., [@DL3 Ch. VIII, §2, Sect. 8, (2.398)]).
\[measlemma3\] Suppose $\mu$ satisfies . Let $\Omega$ be a square centred in the support of $\mu$ with sides chosen in any direction. Then there exists a square $\Omega_0 \subseteq \Omega$ with the same centre such that for any $V\in L_{\mathcal{B}}\left(\overline{\Omega}, \mu\right)$, $V \geq 0$ the following estimate hiolds $$\label{maz8}
\int_{\overline{\Omega}}V(y)|w(y)|^2d\mu(y)
\le A_2 \frac{c_1}{c_0}4^{\alpha} \|V\|^{(\textrm{av})}_{ \mathcal{B},\overline{\Omega}, \mu}\int_{\Omega}|\nabla w(y)|^2 dy$$ for all $w\in W^1_2(\Omega)\cap C\left(\overline{\Omega}\right)$ with $w_{\Omega_0} = 0$ (see ). Here, $A_2$ is the same constant as in .
Let $R$ be the side length of $\Omega$. It is sufficient to prove in the case $\frac{R}{2} \le \mbox{diam}(\mbox{supp}\, \mu)$. Indeed, if $\frac{R}{2} > \mbox{diam}(\mbox{supp}\, \mu)$, then there exists a square $\Omega_1$ with the same centre as $\Omega$ and with the side length $R_1$ such that $R_1 < R$, $\frac{R_1}{2} \le \mbox{diam}(\mbox{supp}\, \mu)$, and $\overline{\Omega_1}\cap \mbox{supp}\, \mu = \overline{\Omega}\cap \mbox{supp}\, \mu$. Then would follow from a similar estimate for $\Omega_1$, since $$\int_{\overline{\Omega}}V(y)|w(y)|^2d\mu(y) = \int_{\overline{\Omega_1}}V(y)|w(y)|^2d\mu(y) \ \mbox{ and } \
\|V\|^{(\textrm{av})}_{ \mathcal{B},\overline{\Omega_1}, \mu} = \|V\|^{(\textrm{av})}_{ \mathcal{B},\overline{\Omega}, \mu} .$$ Below, we show that in the case $\frac{R}{2} \le \mbox{diam}(\mbox{supp}\, \mu)$, holds with $\Omega_0 = \Omega$.
There exist an orthogonal matrix $U \in \mathbb{R}^{2\times 2}$ and a vector $x_0 \in \mathbb{R}^2$ such that $\Omega = \xi(Q)$, where $\xi$ is the similarity transformation $\xi(y) = RUy + x_0$, $y \in \mathbb{R}^2$. Let $\tilde{V} := V\circ \xi$ and $\tilde{\mu} := \mu\circ \xi$. Take any $x \in \overline{Q}\cap \mbox{supp}\,\tilde{\mu}$, i.e. any $x \in \overline{Q}$ such that $\xi(x) \in \mbox{supp}\,\mu$. Since $\xi(B(x,r)) = B(\xi(x), Rr)$ for any $r > 0$, implies $$\label{Rr1}
c_0 (Rr)^{\alpha} \le \tilde{\mu}(B(x, r)) = \mu\left(\xi(B(x, r))\right) = \mu\left(B(\xi(x), Rr)\right) \le c_1 (Rr)^{\alpha}$$ for any positive $r \le \frac{1}{R}\, \mbox{diam}(\mbox{supp}\, \mu)$. It is clear that the latter restriction is not needed for the upper estimate in , since $\mu\left(B(\xi(x), Rr)\right)$ does not change as $r$ increases beyond $\frac{1}{R}\, \mbox{diam}(\mbox{supp}\, \mu)$. If $x \in \overline{Q}\setminus \mbox{supp}\,\tilde{\mu}$, then, obviously, $$\tilde{\mu}(B(x, r)) = 0, \ \ \ \forall r < \mbox{dist} \left(x, \mbox{supp}\,\tilde{\mu}\right).$$ If $r \ge \mbox{dist} \left(x, \mbox{supp}\,\tilde{\mu}\right)$, then there exists $x_1 \in \mbox{supp}\,\tilde{\mu}$ such that $|x- x_1| \le r$. Hence $B(x, r) \subset B(x_1, 2r)$, and it follows from that $$\tilde{\mu}(B(x, r)) \le \tilde{\mu}(B(x_1, 2r)) \le c_1 (2R)^\alpha r^\alpha .$$ Let $$c := \frac{1}{c_1 (2R)^{\alpha}}\, .$$ Then Lemma \[measlemma3\*\] applies to the measure $c\tilde{\mu}$. Using and the equality $$\int_Q|\nabla (w \circ\xi)(x)|^2 dx = \int_{\Omega}|\nabla w(y)|^2 dy ,$$ we get $$\begin{aligned}
\label{disk}
&&\int_{\overline{\Omega}}V(y)|w(y)|^2d\mu(y) = \frac{1}{c}\int_{\overline{Q}}V(\xi(x))|w(\xi(x))|^2d(c\mu(\xi(x)))\nonumber\\
&&= \frac{1}{c}\int_{\overline{Q}}\tilde{V}(x)|(w\circ\xi)(x)|^2d(c\tilde{\mu}(x))\nonumber\\
&&\le \frac{1}{c} A_2 \|\tilde{V}\|_{\mathcal{B}, \overline{Q}, c\tilde{\mu}}\int_Q|\nabla (w \circ\xi)(x)|^2 dx\nonumber \\
&&\le \frac{1}{c} A_2\, \frac{c}{\min\{1, c\tilde{\mu}\left(\overline{Q}\right)\}}\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{\Omega}, \mu}
\int_{\Omega}|\nabla w(y)|^2 dy .\end{aligned}$$ But $$\begin{aligned}
\label{disk1}
\frac{1}{\min\{1, c\tilde{\mu}\left(\overline{Q}\right)\}} &=& \max \left\{1, \frac{1}{c\tilde{\mu}\left(\overline{Q}\right)}\right\}
= \max \left\{1, \frac{c_1 (2R)^{\alpha}}{\mu\left(\overline{\Omega}\right)}\right\}\nonumber\\
&\le& \max\left\{ 1, \frac{c_1 (2R)^{\alpha}}{c_0\left(\frac{R}{2}\right)^{\alpha}}\right\}
= \frac{c_1}{c_0}4^{\alpha}.\end{aligned}$$ In the inequality above, we have used and the fact $\Omega$ contains a disk of radius $\frac{R}{2}$ centred in the support of $\mu$. Now, follows from and .
[Estimate may fail if $\Omega$ is not centred in the support of $\mu$ (see [@MK Example 3.2.11]).]{}
Let $G\subset\mathbb{R}^2$ be a bounded set with Lipschitz boundary such that $\mu\left(\overline{G}\right) > 0$. Let $G_0$ be the smallest closed square containing $G$ with sides chosen in the directions $\theta_0$ and $\theta_0 + \frac{\pi}{2}$ from Corollary \[cor-direct\]. Since $\mu\left(\overline{G}\right) > 0$, there exist $x \in \mbox{supp}\, \mu$ such that $x \in \overline{G} \subseteq G_0$. Let $G_1$ be the closed square centred at $x$ with sides chosen in the same directions as for $G_0$ and the side length twice that of $G_0$. Then $G_1 \supset G_0$. Finally, Let $G^*$ be the closed square with the same centre and the same directions of sides as $G_0$, and with the side length $3$ times that of $G_0$. Then $$\label{Gs}
\overline{G} \subseteq G_0 \subset G_1 \subset G^* .$$ Since $G_1$ is centred in $\mbox{supp}\, \mu$, Lemma \[measlemma3\] can be applied to it. On the other hand, an advantage of $G^*$ is that it does not depend on the choice of $x \in \mbox{supp}\, \mu$ and is uniquely defined by $G$ once the direction $\theta_0$ has been chosen. Hence one can define the following quantity $$\kappa_0(G) := \frac{\mu(G^*)}{\mu\left(\overline{G}\right)}\,.$$
Further, let $$\begin{aligned}
V_{*}(x) := \left\{\begin{array}{l}
V(x), \ \;\;\mbox{ if } x\in \overline{G}, \\ \\
0, \ \;\;\mbox{ if } x\notin \overline{G}.
\end{array}\right.\end{aligned}$$ Then $$\label{exteqn}
\|V_{*}\|^{(av)}_{\mathcal{B}, G_1, \mu} \le \|V_{*}\|^{(av)}_{\mathcal{B}, G^*, \mu}
= \|V\|^{(av), \kappa_0(G)}_{\mathcal{B}, \overline{G}, \mu} \le \kappa_0(G) \|V\|^{(av)}_{\mathcal{B}, \overline{G}, \mu}$$ (see Lemma \[lemma8\]).
Using the Poincaré inequality (see, e.g., [@DL2 Ch. IV, §7, Sect. 2, Proposition 2]), one gets the following estimate for operator $$\begin{aligned}
\label{ext1}
&& \|T_Gw\|^2_{W^1_2(G_1)} \le \|T_Gw\|^2_{W^1_2(G^*)} \le
\|T_Gw\|^2_{W^1_2(\mathbb{R}^2)} \nonumber \\
&& \le \|T_G\|^2\|w\|^2_{W^1_2(G)} \le \|T_G\|^2 (1 + C_G)\int_G|\nabla w(x)|^2\,dx\end{aligned}$$ for all $w\in W^1_2(G)$ with $w_G = 0$.
\[measlemma4\] Let $\mu$ be a positive Radon measure on $\mathbb{R}^2$ that is Ahlfors $\alpha$–regular and let $G\subset\mathbb{R}^2$ be a bounded set with Lipschitz boundary such that $\mu\left(\overline{G}\right) > 0$. Choose and fix a direction satisfying Corollary \[cor-direct\]. Further, let $Q_x(r)$ be the square with sides of length $r > 0$ in the chosen direction centred at $x\in \textrm{supp}\,\mu \cap \overline{G}$. Then for any $V\in L_{\mathcal{B}}(\overline{G}, \mu), \; V\geq 0$ and any $n\in \mathbb{N}$ there exists a finite cover of $\textrm{supp}\,\mu\cap \overline{G}$ by squares $Q_{x_k}(r_{x_k}), r_{x_k} > 0, k = 1, 2, ..., n_0$, such that $n_0\le n$ and $$\label{maz9}
\int_{\overline{G}} V(x)|w(x)|^2d\mu(x) \le A_3n^{-1}\|V\|^{(av)}_{\mathcal{B}, \overline{G}, \mu}\int_G|\nabla w(x)|^2\,dx$$ for all $w\in W^1_2(G)\cap C(\overline{G})$ with $(T_G w)_{Q_{x_k}(r_{x_k})} = 0, k = 1,..., n_0$ and $w_G = 0$, where $$\label{A3def}
A_3 = C_\alpha \frac{c_1}{c_0} \|T_G\|^2(1 + C_G) \kappa_0(G)^2$$ and the constant $C_\alpha$ depends only on $\alpha$.
Let $N\in\mathbb{N}$ be a bound (see, e.g., [@FM Theorem 2.7]) in the Besicovitch covering Lemma (see, e.g., [@Guz Ch. 1 Theorem 1.1]). If $n \le \kappa_0(G) N$, take $n_0 = 1$ and let $Q_{x_1}(r_{x_1})$ be the square $\Omega_0$ from Lemma \[measlemma3\] with $\Omega = G_1$. Then it follows from , , and that for all $w\in W^1_2(G)\cap C(\overline{G})$ with $(T_G w)_{Q_{x_1}(r_{x_1})} = 0$ and $w_G = 0$, $$\begin{aligned}
\label{B3}
&&\int_{\overline{G}} V(x)|w(x)|^2\,d\mu(x) = \int_{G_1} V_*(x)|T_G w(x)|^2\,d\mu(x) \nonumber \\
&& \le A_2 \frac{c_1}{c_0}4^{\alpha}\|V_*\|^{(\textrm{av})}_{\mathcal{B}, G_1, \mu}\int_{G_1}|\nabla (T_G w)(x)|^2\,dx \nonumber\\
&& \le A_2 \frac{c_1}{c_0}4^{\alpha}\kappa_0(G) N n^{-1}\|V_*\|^{(\textrm{av})}_{\mathcal{B}, G^*, \mu}
\int_{G^*}|\nabla (T_G w)(x)|^2\,dx \nonumber \\
&&\le A_2 \frac{c_1}{c_0}4^{\alpha} \kappa_0(G) N n^{-1}\kappa_0(G) \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\|T_G\|^2(1 + C_G)
\int_{G}|\nabla w(x)|^2\,dx \nonumber \\
&& = B_2n^{-1}\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\int_{G}|\nabla w(x)|^2\,dx\,,\end{aligned}$$ where $B_2 := A_2 \frac{c_1}{c_0}4^{\alpha}\|T_G\|^2(1 + C_G) \kappa_0(G)^2 N$.
Now assume that $n > \kappa_0(G) N$. Lemma \[measlemma2\] implies that for any $x\in \textrm{supp}\,\mu\cap\overline{G}$, there is a closed square $Q_x(r_x)$ centred at $x$ such that $$\label{cts}
\|V_{*}\|^{(\textrm{av})}_{\mathcal{B}, Q_x(r_x), \mu} = \kappa_0(G) N n^{-1}\|V\|^{(av)}_{\mathcal{B}, \overline{G}, \mu}.$$ Since $\kappa_0(G) N n^{-1} < 1$, it is not difficult to see that $Q_x(r_x) \subseteq G^*$. Consider the covering $\Xi = \{Q_x(r_x)\}$ of $\textrm{supp}\,\mu\cap\overline{G}$. According to the Besicovitch covering Lemma, $\Xi$ has a countable or a finite subcover $\Xi'$ that can be split into $N$ subsets $\Xi'_j,\,j = 1, ..., N$ in such a way that the closed squares in each subset are pairwise disjoint. Applying Lemma \[lemma7\] and , one gets $$\begin{aligned}
\kappa_0(G) Nn^{-1}\|V\|^{(av)}_{\mathcal{B}, \overline{G}, \mu} \textrm{card}\,\Xi'_j &=& \underset{Q_{x}(r_{x})\in\Xi'_j}\sum \|V_{*}\|^{(\textrm{av})}_{\mathcal{B}, Q_{x}(r_{x}), \mu}\le \|V_{*}\|^{(\textrm{av})}_{\mathcal{B}, G^{*}, \mu}\\
&\le& \kappa_0(G) \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\,.\end{aligned}$$ Hence $\textrm{card}\,\Xi'_j \le nN^{-1}$ and $$n_0 := \textrm{card}\,\Xi' = \sum_{j= 1}^N \textrm{card}\,\Xi'_j \le n.$$ Again, using , and , one gets for all $w\in W^1_2(G)\cap C(\overline{G})$ with $(T_G w)_{Q_{x_k}(r_{x_k})} = 0, k = 1,..., n_0$ and $w_G = 0$, $$\begin{aligned}
&& \int_{\overline{G}} V(x)|w(x)|^2d\mu(x) = \int_{\textrm{supp}\,\mu \cap \overline{G}} V(x)|w(x)|^2d\mu(x) \\
&& \le \sum_{k = 1}^{n_0}\int_{Q_{x_k}(r_{x_k})} V_{*}(x)|(T_G w)(x)|^2\,d\mu(x)\\
&& \le A_2 \frac{c_1}{c_0}4^{\alpha}\sum_{k = 1}^{n_0} \|V_{*}\|^{(\textrm{av})}_{\mathcal{B}, Q_{x_k}(r_{x_k}), \mu}
\int_{Q_{x_k}(r_{x_k})}|\nabla (T_G w)(x)|^2dx\\
&& = A_2 \frac{c_1}{c_0}4^{\alpha} \kappa_0(G) N n^{-1}\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}
\sum_{k = 1}^{n_0}\int_{Q_{x_k}(r_{x_k})}|\nabla (T_G w)(x)|^2\,dx\\
&& = A_2 \frac{c_1}{c_0}4^{\alpha} \kappa_0(G) N n^{-1} \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}
\sum_{j = 1}^{N}\underset{Q_{x_k}(r_{x_k})\in\Xi'_j}\sum\int_{Q_{x_k}(r_{x_k})}|\nabla (T_G w)(x)|^2\,dx\\
&& \le A_2 \frac{c_1}{c_0}4^{\alpha} \kappa_0(G) N n^{-1}\|V\|^{(\textrm{av})}_{\mathcal{B}, G, \mu}
\sum_{j = 1}^{N}\int_{G^*}|\nabla (T_G w)(x)|^2\,dx\\
&& \le A_2 \frac{c_1}{c_0}4^{\alpha} \kappa_0(G) N^2 n^{-1}\|T_G\|^2(1 + C_G)\|V\|^{(\textrm{av})}_{\mathcal{B}, G, \mu}
\int_{G}|\nabla w(x)|^2\,dx \\
&& = C_1n^{-1}\|V\|^{(\textrm{av})}_{\mathcal{B}, G, \mu}\int_{G}|\nabla w(x)|^2\,dx ,\end{aligned}$$ where $C_1 := A_2 \frac{c_1}{c_0}4^{\alpha} \|T_G\|^2(1 + C_G) \kappa_0(G) N^2$. It is now left to take $$\label{A3}
A_3 := \max\left\{B_2, C_1\right\} = A_2 4^{\alpha} N \|T_G\|^2(1 + C_G) \frac{c_1}{c_0} \kappa_0(G)
\max\left\{\kappa_0(G), N\right\}.$$
\[G\] Let $\mu$ and $G$ be as in Lemma \[measlemma4\]. Then $$\label{A4ineq}
\int_{\overline{G}} V(x)|w(x)|^2d\mu(x) \le A_4 \|V\|^{(av)}_{\mathcal{B}, \overline{G}, \mu}\int_G|\nabla w(x)|^2\,dx$$ for all $w\in W^1_2(G)\cap C(\overline{G})$ with $w_G = 0$, where $$\begin{aligned}
\label{A4}
A_4 = 2\|T_G\|^2(1 + C_G) \left(A_2 \frac{c_1}{c_0}4^{\alpha} + \frac{\mathcal{B}^{-1}(1)}{|G|}\right) \kappa_0(G) .\end{aligned}$$
It follows from that $$\begin{aligned}
\left|\left(T_G w\right)_{G_1} \right|^2 = \left|\frac{1}{|G_1|}\int_{G_1} (T_G w)(x)\, dx\right|^2 \le
\frac{1}{|G_1|} \|T_Gw\|^2_{L_2(G_1)} \\
\le \frac{1}{|G|} \|T_G\|^2 (1 + C_G)\int_G|\nabla w(x)|^2\,dx .\end{aligned}$$ Using Lemma \[L1Orl\], one gets, similarly to , $$\begin{aligned}
&&\int_{\overline{G}} V(x)|w(x)|^2\,d\mu(x) = \int_{G_1} V_*(x)|T_G w(x)|^2\,d\mu(x) \\
&& \le 2 \int_{G_1} V_*(x)|T_G w(x) - \left(T_G w\right)_{G_1}|^2\,d\mu(x) \\
&& \ \ \ + 2 \int_{G_1} V_*(x) \left|\left(T_G w\right)_{G_1}\right|^2\,d\mu(x) \\
&& \le 2A_2 \frac{c_1}{c_0}4^{\alpha}\|V_*\|^{(\textrm{av})}_{\mathcal{B}, G_1, \mu}\int_{G_1}|\nabla (T_G w)(x)|^2\,dx \\
&& \ \ \ +2 \mathcal{B}^{-1}(1) \|V_*\|^{(\textrm{av})}_{\mathcal{B}, G_1, \mu} \frac{1}{|G|} \|T_G\|^2 (1 + C_G)\int_G|\nabla w(x)|^2\,dx \\
&& \le A_4 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\int_{G}|\nabla w(x)|^2\,dx\,,\end{aligned}$$ where $A_4$ is given by .
[If $\mu$ satisfies , then the measure $\frac1{c_1} \mu$ satisfies . Applying Lemma \[measlemma3\*\] to $\frac1{c_1} \mu$ and using (with $c = \frac{1}{c_1}$, $\Omega_1 = \Omega_2 = \overline{G}$, and $\xi(x) \equiv x$) one gets a version of with the following constant $$\label{A4'}
A'_4 = \frac{A_1 \|T_G\|^2 \left(1 + C_G\right)}{\min\left\{1, \frac1{c_1}\,\mu\left(\overline{G}\right)\right\}}$$ in place of $A_4$. The terms in and in that depend on the measure $\mu$ are $\frac{c_1}{c_0}$ and $\kappa_0(G)$. The latter can often be estimated above by a quantity that depends only on $\frac{c_1}{c_0}$ and $\alpha$ (see Examples \[exsq\] and \[exan\] below). On the other hand, contains the term $\frac1{c_1}\,\mu\left(\overline{G}\right)$. Although would also work for us (see ), we prefer to use as it matches better than . ]{}
\[exsq\] [Let $\Omega$ be a square centred in the support of $\mu$ with sides of length $R$ chosen in any direction. Then the side length of $\Omega^*$ does not exceed $3\sqrt{2}\, R$, and $$\mu(\Omega^*) \le c_1 \left(3\sqrt{2}\, R\right)^\alpha .$$ If $\frac{R}{2} \le \mbox{diam}(\mbox{supp}\, \mu)$, then $$\mu\left(\overline{\Omega}\right) \ge c_0\left(\frac{R}{2}\right)^{\alpha} \ \mbox{ and } \
\kappa_0(\Omega) = \frac{\mu(\Omega^*)}{\mu\left(\overline{G}\right)} \le \frac{c_1}{c_0}\left(6\sqrt{2}\right)^{\alpha}.$$ If $\frac{R}{2} > \mbox{diam}(\mbox{supp}\, \mu)$, then $\mu\left(\overline{\Omega}\right) = \mu(\Omega^*)$ and $\kappa_0(\Omega) = 1$.]{}
\[exan\] [Let $G$ be a circular annulus centred at a point $x$ in the support of $\mu$ with the radii $r$ and $R$ such that $$\frac{R}{r} \ge \left(2\frac{c_1}{c_0}\right)^{\frac{1}{\alpha}} \ \mbox{ and } \ R \le \mbox{diam}(\mbox{supp}\, \mu) .$$ Then the side length of the square $G^*$ equals $6R$, and $$\begin{aligned}
&& \mu\left(\overline{G}\right) = \mu\left(\overline{B(x, R)}\right) - \mu\left(B(x, r)\right) \ge c_0 R^\alpha - c_1 r^\alpha \\
&&\ge c_0 R^\alpha - c_1 \frac12 \frac{c_0}{c_1} R^ \alpha= \frac{c_0}{2} R^\alpha ,\\
&& \mu(G^*) \le c_1 (6R)^\alpha .\end{aligned}$$ Hence, $$\label{kappaan}
\kappa_0(G) \le \frac{c_1 (6R)^\alpha}{\frac{c_0}{2} R^\alpha} = 2\,\frac{c_1}{c_0}\, 6^\alpha .$$ Note also that $$\label{A4'an}
\frac{1}{\min\left\{1, \frac1{c_1}\,\mu\left(\overline{G}\right)\right\}} \le \frac{1}{\min\left\{1, \frac{c_0}{2c_1}\,R^\alpha\right\}}
= \max\left\{1, 2\, \frac{c_1}{c_0}\, R^{-\alpha}\right\}\, .$$ ]{}
As above, let $\mu$ be a positive Radon measure on $\mathbb{R}^2$ that is Ahlfors $\alpha$–regular and let $G\subset\mathbb{R}^2$ be a bounded set with Lipschitz boundary such that $\mu\left(\overline{G}\right) > 0$. Let $$\begin{aligned}
\label{qformG}
&& \mathcal{E}_{2V\mu, G}[w] : = \int_{G} |\nabla w(x)|^2 dx - 2\int_{\overline{G}} V(x) |w(x)|^2 d\mu(x) , \\
&& \mbox{Dom}\, (\mathcal{E}_{2V\mu, G}) =
\left\{w \in W^1_2\left(G\right)\cap L^2\left(\overline{G}, Vd\mu\right) | \ w_G = 0\right\} . \nonumber\end{aligned}$$
\[measlemma5\][(cf. [@Eugene Lemma 7.7])]{} $$\label{meain1}
N_- (\mathcal{E}_{2V\mu, G}) \le A_5
\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu} + 2 , \ \ \ \forall V \ge 0 ,$$ where $A_5 := 2A_3$ and $A_3$ is the constant in Lemma \[measlemma4\].
Let $n = \left[A_5\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\right] + 1$ in Lemma \[measlemma4\], where $[a]$ denotes the largest integer not greater than $a$. Take any linear subspace $\mathcal{L} \subset \mbox{Dom}\, (\mathcal{E}_{2V\mu, G})$ such that $$\dim \mathcal{L} > \left[A_5 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\right] + 2 .$$ Since $n_0 \le n$, there exists $w \in \mathcal{L}\setminus\{0\}$ such that $w_{Q_{x_k}(r_{x_k})} = 0$, $k = 1, \dots, n_0$ and $w_{G} = 0$. Then $$\begin{aligned}
\mathcal{E}_{2V\mu, G}[w] &=& \int_{G} |\nabla w( x)|^2dx - 2\int_{\overline{G}} V( x) |w( x)|^2 d\mu(x) \\
&\ge& \int_{G} |\nabla w(x)|^2 dx -
\frac{A_5 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}}{\left[A_5 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\right] + 1}\,
\int_{G} |\nabla w(x)|^2 dx \\
&\ge& \int_{G} |\nabla w(x)|^2 dx - \int_{G} |\nabla w(x)|^2 dx = 0 .
\end{aligned}$$ Hence $$N_- (\mathcal{E}_{2V\mu, G}) \le \left[A_5
\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu}\right] + 2 \le
A_5 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu} + 2.$$
\[measlemma5\*\] $$\label{meain2}
N_- (\mathcal{E}_{2V\mu, G}) \le A_6 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu} , \ \ \ \forall V\geq 0,$$ where $A_6 := 2A_3 + 4A_4$, and $A_3$, $A_4$ are the constants in and respectively.
By , $$2\int_{\overline{G}}V(x)|w(x)|^2d\mu(x) \le 2A_4\|V\|^{\textrm{(av)}}_{\mathcal{B}, \overline{G}, \mu}\int_{G}|\nabla w(x)|^2dx$$ for all $w\in W^1_2(G)\cap C(\overline{G})$ with $w_{G} = 0$.
If $\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu} \le \frac{1}{2A_4}$, then $N_- (\mathcal{E}_{2V\mu, G}) = 0$. If $\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu} > \frac{1}{2A_4}$, then Lemma \[measlemma5\] implies $$N_- (\mathcal{E}_{2V\mu, G}) \le A_5 \|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu} + 2
\le A_6\|V\|^{(\textrm{av})}_{\mathcal{B}, \overline{G}, \mu},$$ where $A_6 = A_5 + 4A_4 = 2A_3 + 4A_4$.
Assume that $0 \in \mbox{supp}\, \mu$. Let $\mathbb{Z}_\mu := \mathbb{Z}$ if $\mbox{supp}\, \mu$ is unbounded and $\mathbb{Z}_\mu := \mathbb{Z}\cap (-\infty, m]$ if $\mbox{supp}\, \mu$ is bounded (see ).
\[measlemma6\] There exists a constant $A_8 > 0$ such that $$\label{meaeqn6}
N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_n}\right) \le A_8 \|V\|^{(\textrm{av})}_{\mathcal{B}, Q_n ,\mu}, \ \ \ \forall V\ge 0, \
\forall n \in \mathbb{Z}_\mu$$ (see and ).
We start with the case $n =1$. It follows from Lemma \[measlemma5\*\] and Example \[exan\] that $$\label{fol}
N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_1}\right) \le A_8 \|V\|^{(\textrm{av})}_{\mathcal{B}, Q_1, \mu}\;,\;\;\;\forall V\ge 0,$$ with $$\begin{aligned}
A_8 &=& 2 C_\alpha \frac{c_1}{c_0} \|T_{Q_1}\|^2(1 + C_{Q_1}) \left(2\,\frac{c_1}{c_0}\, 6^\alpha\right)^2 \\
&& + 8\|T_{Q_1}\|^2(1 + C_{Q_1}) \left(A_2 \frac{c_1}{c_0}4^{\alpha} + \frac{\mathcal{B}^{-1}(1)}{|G_1|}\right) 2\,\frac{c_1}{c_0}\, 6^\alpha \\
&=& 8 C_\alpha 6^{2\alpha} \left(\frac{c_1}{c_0}\right)^3 \|T_{Q_1}\|^2(1 + C_{Q_1}) \\
&& + 16 \|T_{Q_1}\|^2(1 + C_{Q_1}) \left(A_2 \frac{c_1}{c_0}4^{\alpha} + \frac{\mathcal{B}^{-1}(1)}{|G_1|}\right) \frac{c_1}{c_0}\, 6^\alpha .\end{aligned}$$ As far as the dependence on the measure $\mu$ is concerned, $A_8$ depends only on the ratio $\frac{c_1}{c_0}$.\
Let $\xi : Q_1 \longrightarrow Q_n$ by given by $\xi(x) := x\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}}$. Let $\tilde{V} := V \circ \xi,\, \tilde{\mu} := \mu \circ \xi$ and $\tilde{w} := w \circ \xi$. Since $\xi(B(x,r)) = B\left(\xi(x), \left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}}r\right)$ for any $r > 0$, $\tilde{\mu}$ satisfies the following analogue of (cf. ) $$\tilde{c}_0r^{\alpha} \le \tilde{\mu}(B(x, r)) \le \tilde{c}_1r^{\alpha}$$ for all $0 < r \le$ diam(supp $\tilde{\mu}$), where $\tilde{c}_0 := c_0\left(2\frac{c_1}{c_0}\right)^{n-1}$, $\tilde{c}_1 := c_1\left(2\frac{c_1}{c_0}\right)^{n - 1}$, and $\frac{\tilde{c}_1}{\tilde{c}_0} = \frac{c_1}{c_0}$. Now, $$\begin{aligned}
&&\int_{Q_n}|\nabla w(y)|^2 dy - 2\int_{Q_n}V(y)|w(y)|^2d\mu(y)\\&& = \int_{Q_1}|\nabla\tilde{w}(x)|^2dx - 2
\int_{Q_1}\tilde{V}(x)|\tilde{w}(x)|^2d\tilde{\mu}(x).\end{aligned}$$ It follows from that $$\begin{aligned}
N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_n}\right) = N_-\left(\mathcal{E}_{\mathcal{N}, 2\tilde{V}\tilde{\mu}, Q_1}\right) \le
A_8 \|\tilde{V}\|^{(\textrm{av})}_{\mathcal{B}, Q_1, \tilde{\mu}}\;,\;\;\;\forall \tilde{V}\ge 0.\end{aligned}$$ It follows from with $c = 1$ that $\|\tilde{V}\|^{(\textrm{av})}_{\mathcal{B}, Q_1, \tilde{\mu}} = \|V\|^{(\textrm{av})}_{\mathcal{B}, Q_n, \mu}$. Thus $$N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu, Q_n}\right) \le A_8 \|V\|^{(\textrm{av})}_{\mathcal{B}, Q_n, \mu} , \;\;\;\;\;\forall V\ge 0.$$ Hence the scaling $x \longmapsto x\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}}$ allows one to reduce the case of any $n\in\mathbb{Z}_\mu$ to the case $n =1$.
We are now in position to derive an estimate for the second term in the right-hand side of from the variational principle (see, e.g., [@KS Lemma 3.2]). Note that $\mbox{supp}\, \mu\setminus\{0\} \subseteq \cup_{n \in \mathbb{Z}_\mu} Q_n$ and $\mu(\{0\}) = 0$, and that implies $$w|_{Q_n} \in \mathrm{Dom}(\mathcal{E}_{V\mu,Q_n}) , \ \ \ \forall w \in \mathrm{Dom}(\mathcal{E}_{\mathcal{N}, 2V\mu}) .$$ Hence, the above Lemma implies, for any $c < \frac{1}{A_8}$, $$\label{meaeqn6*}
N_-\left(\mathcal{E}_{\mathcal{N}, 2V\mu}\right) \le A_8\sum_{\mathcal{D}_n > c} \mathcal{D}_n\;,\;\;\;\forall V\ge 0$$ (see ). Thus Theorem \[mainthm\] follows from , and .
Proof of Corollary \[maincor\] {#corproof}
==============================
It is easy to see that $$\label{Gns}
\sum_{G_n > 1/4} \sqrt{G_n} \le \sum_{G_n > 1/4} 2 G_n \le 2 \sum_{n \in \mathbb{Z}} G_n .$$
Let $\mathbf{\Omega}_{-1}$ be the closed disc $\overline{B\left(0, e^{-1}\right)}$ and $\beta \in (0, \alpha)$. Then using , , and Fubini’s theorem one gets $$\begin{aligned}
&& \sum_{n < 0} G_n \le 2 \int_{|x| \le 1/e} V(x) |\ln|x||\, d\mu(x)
\le 2 \|V\|_{\mathcal{B}, \mathbf{\Omega}_{-1}, \mu}
\left\|\ln|\cdot|\right\|_{(\mathcal{A}, \mathbf{\Omega}_{-1}, \mu)} , \\
&& \int_{\mathbf{\Omega}_{-1}} \mathcal{A}\left(\beta\left|\ln|x|\right|\right)\, d\mu(x) \le
\int_{|x| \le 1/e} e^{\ln \frac{1}{|x|^\beta}}\, d\mu(x) \le \int_{|x| \le 1} \frac{1}{|x|^\beta}\, d\mu(x) \\
&& = \int_{|x| \le 1} \left(\beta\int_{|x|}^1 r^{-\beta -1}\, dr + 1\right)\, d\mu(x) \\
&& = \beta \int_0^1 r^{-\beta -1} \int_{|x| \le r} d\mu(x) dr + \int_{|x| \le 1} 1\, d\mu(x) \\
&& = \beta \int_0^1 r^{-\beta -1} \mu(B(0, r))\, dr + \mu(B(0, 1)) \le \beta \int_0^1 r^{-\beta -1} c_1 r^\alpha dr + c_1 \\
&& = c_1 \left(\frac{\beta}{\alpha - \beta} + 1\right) = c_1\, \frac{\alpha}{\alpha - \beta} =: A_9\end{aligned}$$ (We have $\sum_{n < 0} G_n \le 2 \int_{|x| \le 1/e} \cdots$ rather than $\sum_{n < 0} G_n = \int_{|x| \le 1/e} \cdots$ in the first inequality above because $G_n$ are integrals over domains with intersections that may have positive measure $\mu$ (see ): $$\mu\left(\left\{x \in \mathbb{R}^2 | \ |x| \in J_{n - 1}\right\}\cap \left\{x \in \mathbb{R}^2 | \ |x| \in J_n \right\}\right) =
\mu\left(\left\{x \in \mathbb{R}^2 | \ |x| = e^{-2^{|n|}}\right\}\right)$$ may be positive. A similar situation occurs in and in the proof of Lemma \[reversetr\] below.) Hence $$\left\|\ln|\cdot|\right\|_{(\mathcal{A}, \mathbf{\Omega}_{-1}, \mu)} \le \frac{1}{\beta}\, \max\{1, A_9\} =: A_{10}$$ (see ) and $$\label{A0B0}
\sum_{n < 0} G_n \le 2 A_{10} \|V\|_{\mathcal{B}, \mathbf{\Omega}_{-1}, \mu}
\le 2 A_{10} \|V\|_{\mathcal{B}, \mathbb{R}^2, \mu} .$$ Further, $$\begin{aligned}
\label{G0}
G_0 &=& \int_{e^{-1} \le |x| \le e} V(x)\, d\mu(x) \nonumber \\
&\le& \frac{1}{\ln\left(1 + e^{-1}\right)} \int_{e^{-1} \le |x| \le e} V(x) \ln(1 + |x|)\, d\mu(x) \nonumber \\
&\le& \frac{1}{\ln\left(1 + e^{-1}\right)} \int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, d\mu(x)\end{aligned}$$ and $$\label{logcomp}
\sum_{n > 0} G_n \le 2 \int_{|x| \ge e} V(x) \ln|x|\, d\mu(x) \le
2 \int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, d\mu(x) .$$ It follows from – that $$\label{Gnsest}
\sum_{G_n > 1/4} \sqrt{G_n} \le A_{11} \left(\int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, d\mu(x) + \|V\|_{\mathcal{B}, \mathbb{R}^2, \mu}\right) ,$$ where $$A_{11} = 2\max\left\{2A_{10}, \ \frac{1}{\ln\left(1 + e^{-1}\right)} + 2\right\} .$$
Let $\mathbf{\Omega}_0$ be the closed unit disc $\overline{B\left(0, 1\right)}$. It follows from Lemma \[lemma7\] and Corollary \[avequiv\] that $$\begin{aligned}
\label{Dns0}
&& \sum_{n \le 0} \mathcal{D}_n = \sum_{k \le 0} \mathcal{D}_{2k} + \sum_{k \le 0} \mathcal{D}_{2k - 1} \le
2 \|V\|^{(\textrm{av})}_{\mathcal{B}, \mathbf{\Omega}_0, \mu} \nonumber \\
&& \le 2 \max\left\{1, \mu\left(\mathbf{\Omega}_0\right)\right\} \|V\|_{\mathcal{B}, \mathbf{\Omega}_0, \mu}
\le 2 \max\left\{1, \mu\left(\mathbf{\Omega}_0\right)\right\} \|V\|_{\mathcal{B}, \mathbb{R}^2, \mu} .\end{aligned}$$
We need the following lemma to estimate $\sum_{n \ge 1} \mathcal{D}_n$.
[(cf. [@Eugene Lemma 8.1])]{}\[reversetr\] There exists $A_{12} > 0$ such that $$\sum_{n = 1}^\infty \|V\|_{\mathcal{B}, Q_n, \mu} \le A_{12}
\left(\|V\|_{\mathcal{B}, \mathbb{R}^2\setminus B(0, 1), \mu} +
\int_{|x| \ge 1} V(x) \ln(2 + \ln |x|)\, d\mu(x)\right)$$ for any $V \ge 0$.
Suppose first that $\|V\|_{(\mathcal{B}, \mathbb{R}^2\setminus B(0, 1), \mu)} = 1$ and let $$\alpha_n := \int_{Q_n} \mathcal{B}(V(x))\, d\mu(x) , \ \ \
\kappa_n := \|V\|_{(\mathcal{B}, Q_n, \mu)} , \ \ \ n \in \mathbb{N} .$$ Then $$\begin{aligned}
&&\kappa_n \le \|V\|_{(\mathcal{B}, \mathbb{R}^2\setminus B(0, 1), \mu)} = 1 , \\
&&\sum_{n = 1}^\infty \alpha_n = \sum_{n = 1}^\infty \int_{Q_n} \mathcal{B}(V(x))\, d\mu(x) \le
2\int_{\mathbb{R}^2\setminus B(0, 1)} \mathcal{B}(V(x))\, d\mu(x) = 2\end{aligned}$$ and it follows from Lemma \[elem\] that $$\begin{aligned}
1 = \int_{Q_n} \mathcal{B}\left(\frac{V(x)}{\kappa_n}\right) d\mu(x) \le
\int_{Q_n} \left(\frac{V(x)}{\kappa_n} +
2\frac{V(x)}{\kappa_n} \ln_+\frac{V(x)}{\kappa_n}\right) d\mu(x) \\
\le \frac1{\kappa_n} \int_{Q_n} \left(V(x) + 2 V(x) \ln_+V(x)\right) d\mu(x) +
\frac2{\kappa_n}\, \ln\frac1{\kappa_n}\, \|V\|_{L_1(Q_n, \mu)} \\
\le \frac4{\kappa_n}\, \alpha_n + \frac1{\kappa_n}\left(1 + 2 \ln\frac1{\kappa_n}\right)
\|V\|_{L_1(Q_n, \mu)} .\end{aligned}$$ Hence $$\kappa_n \le 4\alpha_n + \left(1 + 2 \ln\frac1{\kappa_n}\right)
\|V\|_{L_1(Q_n, \mu)}$$ and $$\begin{aligned}
&& \sum_{n = 1}^\infty \|V\|_{\mathcal{B}, Q_n, \mu} \le 2 \sum_{n = 1}^\infty \kappa_n =
2 \sum_{\kappa_n \le 1/n^2} \kappa_n + 2 \sum_{\kappa_n > 1/n^2} \kappa_n \\
&& \le 2 \sum_{n = 1}^\infty \frac1{n^2} + 8 \sum_{n = 1}^\infty \alpha_n +
2 \sum_{n = 1}^\infty (1 + 4 \ln n)\|V\|_{L_1(Q_n, \mu)} \\
&& \le \frac{\pi^2}{3} + 16 + 2 \sum_{n = 1}^\infty (1 + 4 \ln n)
\int_{\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}} \le |x| \le \left(2\frac{c_1}{c_0}\right)^{\frac{n}{\alpha}}} V(x)\, d\mu(x) \\
&& \le \frac{\pi^2}{3} + 16 + A_{13} \sum_{n = 1}^\infty
\int_{\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}} \le |x| \le \left(2\frac{c_1}{c_0}\right)^{\frac{n}{\alpha}}}
V(x) \ln(2 + \ln |x|)\, d\mu(x) \\
&& \le \frac{\pi^2}{3} + 16 + 2 A_{13} \int_{|x| \ge 1} V(x) \ln(2 + \ln |x|)\, d\mu(x) \\
&& \le A_{12} \left(\|V\|_{\mathcal{B}, \mathbb{R}^2\setminus B(0, 1), \mu} +
\int_{|x| \ge 1} V(x) \ln(2 + \ln |x|)\, d\mu(x)\right)\end{aligned}$$ (see ). The case of a general $V$ is reduced to $\|V\|_{(\mathcal{B}, \mathbb{R}^2\setminus B(0, 1), \mu)} = 1$ by the scaling $V \mapsto t V$, $t > 0$.
Using Lemmta \[avequivB\] and \[reversetr\] (see also Corollary \[avequiv\]), one gets $$\begin{aligned}
\label{Solimpl}
&& \sum_{n \ge 1} \mathcal{D}_n = \sum_{n \ge 1} \|V\|^{(\textrm{av})}_{\mathcal{B}, Q_n, \mu} \\
&& \le \sum_{n = 1}^\infty \|V\|_{\mathcal{B}, Q_n, \mu} +
\sum_{n = 1}^\infty \max\left\{0,\ln\left(\frac72\, \mu(Q_n)\right)\right\}\,
\int_{Q_n} V(x)\, d\mu(x) \\
&& \le \sum_{n = 1}^\infty \|V\|_{\mathcal{B}, Q_n, \mu} +
\sum_{n = 1}^\infty \max\left\{0,\ln\left(\frac72\, \left(2\frac{c_1}{c_0}\right)^n\right)\right\}\,
\int_{Q_n} V(x)\, d\mu(x) \\
&& \le \sum_{n = 1}^\infty \|V\|_{\mathcal{B}, Q_n, \mu} + A_{14}
\sum_{n = 1}^\infty n \int_{\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}} \le |x| \le \left(2\frac{c_1}{c_0}\right)^{\frac{n}{\alpha}}}
V(x)\, d\mu(x) \\
&& \le \sum_{n = 1}^\infty \|V\|_{\mathcal{B}, Q_n, \mu} + A_{15}
\sum_{n = 1}^\infty \int_{\left(2\frac{c_1}{c_0}\right)^{\frac{n -1}{\alpha}} \le |x| \le \left(2\frac{c_1}{c_0}\right)^{\frac{n}{\alpha}}}
V(x) \ln(1 + |x|)\, d\mu(x) \\
&& \le A_{16} \Big(\|V\|_{\mathcal{B}, \mathbb{R}^2\setminus B(0, 1), \mu} +
\int_{|x| \ge 1} V(x) \ln(2 + \ln |x|)\, d\mu(x) \\
&& \ \ \ \ \ + \int_{|x| \ge 1}
V(x) \ln(1 + |x|)\, d\mu(x)\Big) \\
&& \le A_{17} \left(\|V\|_{\mathcal{B}, \mathbb{R}^2, \mu}
+ \int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, d\mu(x)\right) , \ \ \ \forall V \ge 0 .\end{aligned}$$ Hence it follows from that $$\label{Dnsall}
\sum_{n \in \mathbb{Z}} \mathcal{D}_n \le A_{18} \left(\|V\|_{\mathcal{B}, \mathbb{R}^2, \mu}
+ \int_{\mathbb{R}^2} V(x) \ln(1 + |x|)\, d\mu(x)\right) .$$ Estimate now follows from Theorem \[mainthm\] and , .
Conluding remarks {#remark}
=================
For a sequence of numbers $(a_n)_{n \in \mathbb{Z}}$, let $$\left\|(a_n)_{n \in \mathbb{Z}}\right\|_{1,\infty} := \sup_{s > 0}\left(s \;\textrm{card}\{n\;:\;|a_n| > s\}\right) .$$ It is easy to see that $$\left\|(a_n)_{n \in \mathbb{Z}}\right\|_{1,\infty} \le \left\|(a_n)_{n \in \mathbb{Z}}\right\|_1 = \sum_{n \in \mathbb{Z}} |a_n| .$$ Also, $$\label{sqrtweak}
\sum_{|a_n| > c} \sqrt{|a_n|} \le \frac{2}{\sqrt{c}}\, \left\|(a_n)_{n \in \mathbb{Z}}\right\|_{1,\infty}$$ and $$\label{sqrtweak1}
\sum_{\gamma |a_n| > c} \sqrt{\gamma |a_n|} = O(\gamma) \ \mbox{ as } \ \gamma \longrightarrow + \infty \ \ \ \Longleftrightarrow \ \ \
\left\|(a_n)_{n \in \mathbb{Z}}\right\|_{1,\infty} < \infty$$ (see [@Eugene (49), (77), (78)]).
\[measthm3\] Let $V \ge 0$. If $N_-(\mathcal{E}_{\gamma V\mu, \mathbb{R}^2}) = O(\gamma)$ as $\gamma \longrightarrow + \infty$, then $\left\|(G_n)_{n \in \mathbb{Z}}\right\|_{1,\infty} < \infty$.
This follows by replacing the Lebesgue measure with $\mu$ in the proofs of [@Eugene Theorems 9.1 and 9.2].
The above theorem and show that the term $\sum_{G_n > 1/4} \sqrt{G_n}$ in is optimal in a sense. Although the same cannot be said about the term $\sum_{\mathcal{D}_n > c} \mathcal{D}_n$, the following theorem shows that it is optimal in the class of Orlicz norms. More precisely, no estimate of the type $$\label{type}
N_-(\mathcal{E}_{V\mu, \mathbb{R}^2}) \le \textrm{const} + \int_{\mathbb{R}^2}V(x)W(x)\,d\mu(x) + \textrm{const}\|V\|_{\Psi, \mathbb{R}^2, \mu}$$ can hold with a norm $\|V\|_{\Psi, \mathbb{R}^2, \mu}$ weaker than $\|V\|_{\mathcal{B}, \mathbb{R}^2, \mu}$ provided the weight function $W$ is bounded in a neighbourhood of at least one point in the support of $\mu$.
\[notype\][(cf. [@Eugene Theorem 9.4]) Let $W \ge 0$ be bounded in a neighbourhood of at least point in the support of $\mu$ and let $\Psi$ be an N-function such that $$\underset{s \longrightarrow \infty}\lim\frac{\Psi(s)}{\mathcal{B}(s)} = 0.$$ Then there exists a compactly supported $V\ge 0$ such that $$\int_{\mathbb{R}^2}V(x)W(x)\,d\mu(x) + \|V\|_{\Psi, \mathbb{R}^2, \mu} < \infty$$ and $N_-(\mathcal{E}_{V\mu, \mathbb{R}^2}) = \infty$. ]{}
Shifting the independent variable if necessary, we can assume that $0 \in$ supp $\mu$ and $W$ is bounded in a neighborhood of $0$. Let $r_0 > 0$ be such that $W$ is bounded in the open ball $B(0, r_0)$.
Let $$\beta(s) := \sup_{t \ge s}\, \frac{\Psi(t)}{\mathcal{B}(t)}\, .$$ Then $\beta$ is a non-increasing function, $\beta(s) \to 0$ as $s \to \infty$, and $\Psi(s) \le \beta(s) \mathcal{B}(s)$. Since $\Psi$ is an $N$-function, $\Psi(s)/s \to \infty$ as $s \to \infty$ (see section \[App\]). Hence there exists $s_0 \ge e^{\frac{1}{\alpha}} > 1$ such that $\Psi(s) \ge s$ and $\beta(s) \le 1$ for $s \ge s^{\alpha}_0$. Choose $\rho_k \in (0, 1/s_0)$ in such a way that $$\sum_{k = 1}^\infty \beta\left(\frac1{\rho^{\alpha}_k}\right) < \infty .$$ It follows from that $\forall r > 0$, the disk $B(0, r)$ contains points of the support of $\mu$ different from $0$. Let $x^{(1)}\in \mbox{supp}\,\mu \setminus\{0\}$ be such that $$|x^{(1)}| < \min\left\{\frac{2}{3}\, r_0, 2\rho_1\right\} .$$ One can choose $x^{(k)},\,k\in\mathbb{N}$ inductively as follows: suppose $x^{(1)}, ... , x^{(k)} \in$ supp$\,\mu \setminus\{0\}$ have been chosen. Take $x^{(k + 1)} \in$ supp $\mu\setminus\{0\}$ such that $$|x^{(k+1)}| < \min\left\{\frac{1}{3}|x^{(k)}|, 2\rho_{k+1}\right\}.$$ Since $|x^{(k+1)}| < \frac{1}{3}|x^{(k)}|$, it is easy to see that the open disks $B(x^{(k)}, \frac{1}{2}|x^{(k)}|)$, $k \in \mathbb{N}$ lie in $B(0, r_0)$ and are pairwise disjoint. Let $r_k := \frac{1}{2}|x^{(k)}|$. Then $r_k < \rho_k,\,k\in\mathbb{N}$. For a constant $A_{19} > 0$ to be specified later, let $$\begin{aligned}
& t_k := \frac{A_{19}}{\ln\frac1{r_k}}\, r_k^{-2\alpha} & \\ \\
& V(x) := \left\{\begin{array}{ll}
t_k , & x \in B\left(x^{(k)}, r_k^2\right) , \ k \in \mathbb{N}, \\
0 , & \mbox{otherwise.}
\end{array}\right. &\end{aligned}$$ Since the function $r \mapsto r^{\alpha}\ln\frac1{r}$ has maximum equal to $\frac{1}{\alpha e} $, one can choose $A_{19} > 0$ such that $A_{19}\alpha e > 1$ and $$t_k = \frac{A_{19}}{\ln\frac1{r_k}}\, r_k^{-2\alpha} = \frac{A_{19}}{r_k^{\alpha} \ln\frac1{r_k}}\, r_k^{-\alpha} > \frac1{r^{\alpha}_k}
> \frac1{\rho^{\alpha}_k} > s_0^\alpha \ge e .$$ Then $$\begin{aligned}
&& \int_{\mathbb{R}^2}\Psi(V(x))\,d\mu(x) = \sum_{k= 1}^{\infty}\Psi(t_k)\mu\left(B(x^{(k)}, r^2_k)\right) \le
\sum_{k= 1}^{\infty}\Psi(t_k)c_1 r^{2\alpha}_{k}\\
&& \le c_1\sum_{k =1}^{\infty}r^{2\alpha}_k\beta(t_k)\mathcal{B}(t_k)
\le c_1\sum_{k =1}^{\infty}r^{2\alpha}_k\beta(t_k)(1 + t_k)\ln(1 + t_k) \\
&& < 4c_1\sum_{k =1}^{\infty}r^{2\alpha}_k\beta(t_k)t_k\ln t_k
= 4c_1\sum_{k =1}^{\infty}\beta(t_k)\frac{A_{19}}{\ln\frac{1}{r_k}}\ln\frac{A_{19}}{r^{2\alpha}_k\ln\frac{1}{r_k}} \\
&& \le 4c_1A_{19}\sum_{k =1}^{\infty}\beta\left(\frac{1}{r^{\alpha}_k}\right)\frac{1}{\ln\frac{1}{r_k}}\ln\frac{A_{19} \alpha}{r_k^{2\alpha}}\\
&& \le \textrm{const} \sum_{k = 1}^{\infty} \beta\left(\frac{1}{r^{\alpha}_k}\right) \le
\textrm{const} \sum_{k = 1}^{\infty} \beta\left(\frac{1}{\rho^{\alpha}_k}\right) < \infty.\end{aligned}$$ Thus $\|V\|_{\Psi, \mathbb{R}^2, \mu} < \infty$ (see and ). Since $t_k > \frac{1}{r^{\alpha}_k} > s^{\alpha}_0$, one has $t_k \le \Psi(t_k)$ and $$\int_{\mathbb{R}^2}V(x)\,d\mu(x) \le \int_{\mathbb{R}^2}\Psi(V(x))\,d\mu(x) < \infty.$$ Since $W$ is bounded in $B(0, r_0)$, $$\int_{\mathbb{R}^2}V(x)W(x)\,d\mu(x) < \infty\,.$$ Let $$w_k(x) := \left\{\begin{array}{cl}
1 , & \ |x - x^{(k)}| \le r_k^2 , \\ \\
\frac{\ln (r_k/|x - x^{(k)}|)}{\ln(1/r_k)} , & \ r_k^2 < |x - x^{(k)}| \le r_k , \\ \\
0 , & \ |x - x^{(k)}| > r_k
\end{array}\right.$$ (cf. [@Grig]). Then $$\int_{\mathbb{R}^2} |\nabla w_k(x)|^2\, dx = \frac{2\pi}{\ln(1/r_k)}\,.$$ Further, $$\begin{aligned}
\int_{\mathbb{R}^2} V(x) |w_k(x)|^2\, d\mu(x) &\ge& \int_{B\left(x^{(k)}, r_k^2\right)} V(x)\, d\mu(x) =
t_k \mu\left(B\left(x^{(k)}, r_k^2\right)\right)\\ &\ge& t_k c_0 r^{2\alpha}_k = c_0\frac{A_{19}}{\ln \frac{1}{r_k}}\, .\end{aligned}$$ Hence for any $A_{19} > \frac{2\pi}{c_0}$, $$\mathcal{E}_{V\mu, \mathbb{R}^2}[w_k] < 0 , \ \ \ \forall k \in \mathbb{N}$$ and $N_- (\mathcal{E}_{V\mu, \mathbb{R}^2}) = \infty$.
Appendix: Proofs of , and {#APII}
===========================
Let $\mathcal{B}(s) = (1 + s)\ln(1 + s) - s = \frac{1}{t}$, then $s = \mathcal{B}^{-1}\left(\frac{1}{t}\right)$. For small values of $s$ (large values of $t$), using $$\ln(1 + s) = s - \frac{s^2}{2} + \frac{s^3}{3} + O\left(s^4\right),$$ we have $$(1 + s)\ln(1 + s) - s = \frac{s^2}{2} + O\left(s^3\right) = \frac{1}{t}.$$ One can write this in the form $$\begin{aligned}
\frac{s^2}{2} + s^2g(s) &=& \frac{1}{t},\;\;\;\;g(0)= 0,\\
\frac{s^2}{2}\left(1 + 2g(s)\right) &=& \frac{1}{t},\\
s\left(1 + h(s)\right) &=& \sqrt{\frac{2}{t}},\;\;\;\;h(0) = 0,\end{aligned}$$ where $g$ and $h$ are $C^{\infty}$ smooth functions in a neighbourhood of $0$. Let $f(s) = s\left(1 + h(s)\right)$. Then $f(0)= 0, f'(0) = 1$ and $(f^{-1})'(0) = 1$, which means that both $f$ and $f^{-1}$ are invertible in a neighbourhood of $0$, and $$s = f^{-1}\left(\sqrt{\frac{2}{t}}\right) = \sqrt{\frac{2}{t}} + O\left(\frac{1}{t}\right).$$ Thus $$\mathcal{B}^{-1}\left(\frac{1}{t}\right) = \sqrt{\frac{2}{t}}\left(1 + o(1)\right) \ \mbox{ as } \ t\longrightarrow\infty$$ and $$\label{larget}
t\mathcal{B}^{-1}\left(\frac{1}{t}\right) = \sqrt{2t}\left( 1 + o(1)\right) \ \mbox{ as } \ t\longrightarrow\infty.$$ For large values of $s$ (small values of $t$), let $\rho = 1 + s$ and $r = \frac{1}{t}$, then $$\rho\ln\rho - \rho + 1 = r.$$ Let $\rho = e^z$, then $$\label{asymp}
ze^z - r - e^z + 1 = 0.$$ This implies $$\begin{aligned}
(z -1)e^z &=& r -1, \\ (z -1 )e^{z -1} &=& \frac{r - 1}{e}.\end{aligned}$$ Let $w := z - 1 \,\,\,\,\,\, v:= \frac{r - 1}{e}$. Then $$\label{asymp1}
we^w = v.$$ The solution of is given by $$w = \ln v - \ln\ln v + \frac{\ln\ln v}{\ln v} + O\left(\left(\frac{\ln\ln v}{\ln v}\right)^2\right)$$ (see (2.4.10) and the formula following (2.4.3) in [@DEB]). So $$\begin{aligned}
z &=& 1 + \ln\frac{r - 1}{e} - \ln\ln\frac{r - 1}{e} + \frac{\ln\ln \frac{r- 1}{e}}{\ln \frac{r-1}{e}}\\
&& + \, O\left(\left(\frac{\ln\ln \frac{r- 1}{e}}{\ln \frac{r-1}{e}}\right)^2\right).\end{aligned}$$ Since $$\begin{aligned}
&&\ln (r-1) = \ln r + O\left(\frac{1}{r}\right),\\&&\ln\left(\ln (r -1) - 1\right) = \ln\ln r + O\left(\frac{1}{\ln r}\right),\end{aligned}$$ we get $$\begin{aligned}
z &=& \ln r - \ln\ln r + \frac{\ln\ln r}{\ln r} + O\left(\frac{1}{\ln r}\right)\\ &=&\ln\frac{1}{t} - \ln\ln\frac{1}{t} + \frac{\ln\ln\frac{1}{t}}{\ln\frac{1}{t}} + O\left(\frac{1}{\ln\frac{1}{t}}\right).\end{aligned}$$ This implies $$\rho = e^z = \frac{1}{t\ln\frac{1}{t}}\left( 1 + \frac{\ln\ln \frac{1}{t}}{\ln\frac{1}{t}} + O\left(\frac{1}{\ln\frac{1}{t}}\right)\right).$$ Hence $$t\mathcal{B}^{-1}\left(\frac{1}{t}\right) = \frac{1}{\ln\frac{1}{t}}\left(1 + \frac{\ln\ln \frac{1}{t}}{\ln\frac{1}{t}} + O\left(\frac{1}{\ln\frac{1}{t}}\right)\right)$$ implying $$\label{smallt}
t\mathcal{B}^{-1}\left(\frac{1}{t}\right) = \frac{1}{\ln\frac{1}{t}}\left( 1 + o(1)\right) \ \mbox{ as } \ t\longrightarrow 0.$$ Let $$\label{taut}
\tau := t\mathcal{B}^{-1}\left(\frac{1}{t}\right) .$$ Then $$\label{t1}
\ln\frac{1}{t} = \frac{1 + o(1)}{\tau}.$$ From $$\tau = \frac{1}{\ln\frac{1}{t}}\left(1 + \frac{\ln\ln \frac{1}{t}}{\ln\frac{1}{t}} + O\left(\frac{1}{\ln\frac{1}{t}}\right)\right) ,$$ we get $$\label{t2}
\ln\frac{1}{t} = \frac{1 + \frac{\ln\ln \frac{1}{t}}{\ln\frac{1}{t}} + O\left(\frac{1}{\ln\frac{1}{t}}\right)}{\tau}.$$ Now implies $$\begin{aligned}
\ln\frac{1}{t} &=& \frac{1 + \frac{\ln\frac{1 + o(1)}{\tau}}{1 + o(1)}\tau + O\left( \frac{\tau}{1 + o(1)}\right)}{\tau} \\&=& \frac{1 + (1 + o(1))\tau \ln \frac{1}{\tau} + O(\tau)}{\tau}\,.\end{aligned}$$ Substituting this into , one gets $$\begin{aligned}
\ln\frac{1}{t} &=& \frac{1 + \frac{\ln\frac{1 + (1 + o(1))\tau\ln\frac{1}{\tau} + O(\tau)}{\tau}}{1 + (1 + o(1))\tau\ln \frac{1}{\tau} + O(\tau)}\tau + O(\tau)}{\tau}\\ &=& \frac{1}{\tau} - \ln\tau + O(1).\end{aligned}$$ Hence $$\label{large}
t = \tau e^{-\frac{1}{\tau}}e^{O(1)} \ \mbox{ as } \ \tau \longrightarrow 0.$$
Acknowlegments {#acknowlegments .unnumbered}
==============
The first author is grateful to the Commonwealth Scholarship Commission in the UK, grant UGCA-2013-138, for the funding when he was a PhD student at King’s College London.
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[^1]: Department of Mathematics, Mbarara University of Science and Technology, Uganda. E-mail: [email protected]
[^2]: Department of Mathematics, King’s College London, WC2R 2LS, Strand, London, UK. E-mail: [email protected]
|
---
abstract: 'In this paper we give an explicit formula for the HOMFLY polynomial of a rational link (in particular, knot) in terms of a special continued fraction for the rational number that defines the given link.'
author:
- 'S.Duzhin[^1], M.Shkolnikov'
title: A formula for the HOMFLY polynomial of rational links
---
Rational links {#rat_links}
==============
Rational (or 2-bridge) knots and links constitute an important class of links for which many problems of knot theory can be completely solved and provide examples often leading to general theorems about arbitrary knots and links. For the basics on rational (2-bridge) knots and links we refer the reader to [@Lik] and [@KM]. As regards the definition, we follow [@Lik], while the majority of properties that we need, are to be found in a more detailed exposition of [@KM]. In particular, by *equivalence* of (oriented) links $L=K_1\cup K_2$ and $L'=K_1'\cup K_2'$ we understand a smooth isotopy of ${\mathbb{R}}^3$ which takes the union $K_1\cup K_2$ into the union $K_1\cup K_2$, possibly interchanging the components of the link.[^2]
Let $p$ and $q$ be mutually prime integers, $p>0$, $|{p\over q}|\leq 1$, and we have a continued fraction $${p \over q}=
\cfrac{1}{
b_1+\cfrac{1}{
b_2+\cfrac{1}{
\dots+\cfrac{1}{
b_{n-1}+\cfrac{1}{b_n}}}}}\ ,$$ where $b_i$ are nonzero integers (positive or negative). Below, we will use shorthand notation\[short\_cf\] $[b_1,b_2,\dots,b_n]$ for the continued fraction with denominators $b_1,b_2,\dots,b_n$. A theorem of Schubert (see, for instance, [@Lik; @KM]) says that the (isotopy type of the) resulting unoriented link does not depend on the choice of the continued fraction for the given number $p/q$.
The case $p=q=1$ is exceptional: it corresponds to the trivial knot which is the only rational, but not 2-bridge knot. On some occasions, it will be helpful to allow the numbers $b_i$ also take values $0$ and $\infty$ subject to the rules $1/0=\infty$, $1/\infty=0$, $\infty+x=\infty$.
Consider a braid on four strands corresponding to the word $A^{b_1}B^{b_2}A^{b_3}\dots$, where $A$ and $B$ are fragments depicted in Figure \[elements\] and concatenated from left to right.
$$\begin{array}{ccccccc}
\includegraphics[height=15mm]{Ap}
&\qquad&
\includegraphics[height=15mm]{Am}
&\qquad&
\includegraphics[height=15mm]{Bp}
&\qquad&
\includegraphics[height=15mm]{Bm} \\
A &\qquad& A^{-1} &\qquad& B &\qquad& B^{-1}
\end{array}$$
Then take the closure of this braid depending on the parity of $n$ (see Fig. \[closure\]).
We will call (non-oriented) diagrams obtained in this way *natural diagrams of rational links* and denote them by $D[b_1,b_2,\dots,b_n]$. We shall denote the link represented by this diagram as $L({p\over q})$. For odd denominators $L({p\over q})$ turns out to be a knot, while for even denominators it is a two-component link. Such knots and links are called *2-bridge* or *rational*.
**Example.** We have, among others, the following two continued fractions for the rational number 4/7 (we use shorthand notation, see page ): $${4\over 7} = [1,1,3] = [2,-4].$$ These fractions correspond to the natural link diagrams shown in Figure \[nat\_diag\].
![Two natural diagrams of the table knot $5_2$[]{data-label="nat_diag"}](k5_2a.eps "fig:"){height="14mm"} ![Two natural diagrams of the table knot $5_2$[]{data-label="nat_diag"}](k5_2b.eps "fig:"){height="14mm"}
Orientations
============
Note that, if a natural diagram represents a two-component link, then the two vertical leftmost fragments belong to different components. If they are oriented in the same direction, as shown in Figure \[pos\_orient\], then we call the diagram *positive* and denote it by $D^+[b_1,b_2,\dots,b_n]$.
If the orientation of one of the components is reversed, then we call it *negative* and denote by $D^-[b_1,b_2,\dots,b_n]$. It does not matter which component of the link is reversed, because the change of orientation of both components yields the same link, see [@KM]. As we will see later (Lemmas \[plusminus\] and \[dual\]), the choice between the corresponding links does not depend on a particular continued fraction expansion of the number $p/q$. This makes the notations $L^+(p/q)$ and $L^-(p/q)$ well-defined.
Let $p'=p-q$, if $p>0$, and $p'=p+q$, if $p<0$. According to [@KM], we have: $L^-({p\over q})=L^+({p'\over q})$, therefore, in principle, it is sufficient to study only the totality of all positive rational links. In the case of knots (when $q$ is odd), the two oppositely oriented knots are isotopic, and we have $L({p\over q})=L({p'\over q})$\[chnge\_knot\] (again, see [@KM]). Therefore, it is sufficient to study only the knots with an even numerator (cf. Lemma \[even\_prod\] below).
Another important operation on links is the reflection in space; it corresponds to the change of sign of the corresponding rational number: $p/q\mapsto-p/q$, see [@KM].
The two symmetry operations on rational links generate a group ${\mathbb{Z}}_2\times{\mathbb{Z}}_2$; they are transparently exemplified by the examples $p/q=1/4, -1/4, 3/4, -3/4$, which correspond to the four versions of the so called *Solomon knot* (although it is actually a two-component link):
$$\begin{aligned}
L^+(1/4)\leftrightarrow D^+[4]&=\raisebox{-7mm}{\includegraphics[height=14mm]{sol1_4}}\\[2mm]
L(-1/4)\leftrightarrow D^+[-4]&=\raisebox{-7mm}{\includegraphics[height=14mm]{sol-1_4}}\\[2mm]
L(3/4)\leftrightarrow D^+[1,3]&=\raisebox{-7mm}{\includegraphics[height=14mm]{sol3_4}}\\[2mm]
L(-3/4)\leftrightarrow D^+[-1,-3]&=\raisebox{-7mm}{\includegraphics[height=14mm]{sol-3_4}}\end{aligned}$$
By dragging the lower strand of the diagram for $L(3/4)$ upwards we get the diagram for $L(-1/4)$ with the opposite orientation of the upper strand. The same is true for the pair $L(-3/4)$ and $L(1/4)$.
HOMFLY polynomial {#HOMFLY}
=================
In 2004–2005 Japanese mathematicians S.Fukuhara [@Fuk] and Y.Mizuma [@Miz] found independently different explicit formulae for the simplest invariant polynomial of 2-bridge links: the Conway (Alexander) polynomial. The aim of the present paper is to establish a formula for a more general HOMFLY polynomial $P$ in terms of the number $p/q$ that defines the rational link.
The HOMFLY polynomial [@Lik; @PS; @CDM] is a Laurent polynomial in two variables $a$ and $z$ uniquely defined by the following relations (we use the normalization of [@Atlas] and [@CDM]; other authors may write the same polynomial in different pairs of variables, for example, Lickorish [@Lik] uses $l=\sqrt{-1}a$ and $m=-\sqrt{-1}z$):
$$\label{skein}
P(\bigcirc)=1,\ aP(L_+)-a^{-1}P(L_-)=zP(L_0),$$
where $L_+$, $L_-$ and $L_0$ are links that differ inside a certain ball as shown in Figure \[homfly\_fig\].
As we mentioned in the previous section, in the case of rational knots, the change of orientation gives the same (isotopic) knot, while for links it is important to distinguish between the two essentially different orientations (this number is two, not four, because the change of orientation on both components gives the same rational link).
There is a simple formula relating the HOMFLY polynomials of a knot (link) with that of its mirror reflection ($a\mapsto-a^{-1}$, $z\mapsto z$), so in principle it is enough to study only the knots (links) described by positive fractions.
HOMFLY polynomials of some links are given below in Figure \[homfly\_torus\] and Table \[homflytab\].
Reduction formula {#reduc}
=================
Consider a family of links $L_n$ for $n$ even, which coincide everywhere but in a certain ball, where they look as shown in Fig. \[counter\].a, \[counter\].b and \[counter\].c. Moreover, we define the link $L_\infty$ by Fig. \[counter\].d. That is, we consider a family of links with a distinguished block where the strands are counter-directed. A formula similar to what we are going to prove, can also be established for co-directed strands, but for our purposes the following Proposition is sufficient. It expresses the value $P(L_n)$ through $P(L_0)$ and $P(L_\infty)$.
a) $n>0$\
b) $n<0$\
c) $n=0$\
d) $n=\infty$\
\[counter\]
\[P\_of\_Ln\] $$P(L_n)=a^nP(L_0)+z{{1-a^n}\over{a-a^{-1}}}P(L_\infty)\ .$$
Proceed by induction on $n$.
1\) For $n=0$ the assertion is trivially true.
2\) Suppose it is true for $n-2$. The skein relation (\[skein\]) shows that $zP(L_\infty)=aP(L_{n-2})-a^{-1}P(L_n)$. Substituting here the assumed formula for $P(L_{n-2})$, we can express $P(L_n)$ as follows: $$\begin{aligned}
P(L_n)&=a^2P(L_{n-2})-zaP(L_\infty)\\
&=a^2\big(a^{n-2}P(L_0)+z{1-a^{n-2}\over a-a^{-1}}P(L_\infty)\big)-zaP(L_\infty)\\
&=a^nP(L_0)+z\big(a^2{1-a^{n-2}\over a-a^{-1}}-a\big)P(L_\infty)\\
&=a^nP(L_0)+z{1-a^n\over a-a^{-1}}P(L_\infty)\ .\end{aligned}$$ The positive branch of induction is thus proved.
3\) Suppose the assertion holds for a certain value of $n$. Prove it for the value $n-2$. To do so, it is enough to reverse the argument in the previous item. This completes the proof of the proposition.
For even values of $n$ the fraction $(1-a^n)/(a-a^{-1})$ is actually a Laurent polynomial, namely, $-a-a^3-\dots-a^{n-1}$, if $n>0$, and $a^{-1}+a^{-3}+\dots+a^{n+1}$, if $n<0$.
Let $T_{2,n}$ be the torus link with counter-directed strands (shown in the picture on the right). Then $\displaystyle{P(T_{2,n})=z^{-1}a^n(a-a^{-1})+z{{1-a^n}\over{a-a^{-1}}}}$.
{width="\textwidth"}
Notice that $n$ is even. Consider the family of links $L_m=T_{2,m}$, where $m$ is an arbitrary even number. Outside of the grey ellipse all the links of this family are the same, and inside it they look as shown on Fig \[even\_fragm\]. Therefore, we fall under the assumptions of Lemma \[P\_of\_Ln\], and it only remains to note that $P(L_0)=z^{-1}(a-a^{-1})$ and $P(L_{\infty})=1$.
Particular cases of this Corollary for $n=0,\ \pm 2,\ \pm 4$ give the well-known values of the HOMFLY polynomial for the two unlinked circles, the Hopf link and the two (out of the total four) versions of the oriented “Solomon knot”, see Figure \[homfly\_torus\].
$z^{-1}(a^3-a)-za$\
$z^{-1}(a-a^{-1})$\
$z^{-1}(a^{-1}-a^{-3})+za^{-1}$\
$z^{-1}(a^5-a^3)-z(a+a^3)$\
$z^{-1}(a^{-3}-a^{-5})+z(a^{-1}+a^{-3})$\
Canonical orientation of rational links {#canon}
=======================================
\[even\_prod\] Suppose the numbers $p$ and $q$ are mutually prime and $|{p\over q}|<1$. The number ${p\over q}$ has a continued fraction expansion with non-zero even denominators if and only if the product $pq$ is even, and if such an expansion exists, it is unique.
1\) Necessity: if $p/q=[b_1,b_2,\dots,b_n]$ with all $b_i$’s even, then $pq$ is even. We shall prove that by induction on the length $n$ of the continued fraction. The induction base is evident. Now, $${p\over q}=[b_1,b_2,\dots,b_n]={1\over b_1+[b_2,\dots,b_n]}
={1\over b_1+p'/q'}={q'\over b_1q'+p'}\ .$$ By the induction assumption, one (and only one!) of $p'$ or $q'$ is even. Since $b_1$ is even, it follows that either the numerator or the denominator of the last fraction is even, so their product is even and, since the numbers $p'$ and $q'$ are mutually prime and ${p'}<{q'}$, this fraction is irreducible and smaller than 1 by absolute value.
2\) Sufficiency: if the product $pq$ is even, then the irreducible fraction $p/q$ allows for a continued fraction with even denominators.
If $q=\pm2$, then the expansion clearly exists. We proceed by induction on $|q|$. Among the numbers $[{q\over p}]$ and $[{q\over p}]+1$ one is even, call it $b$. The number $b-{q\over p}$ can be written as an irreducible fraction ${p'\over q'}$. Note that $b$ cannot be 0, because $|q/p|>1$. Then $|{p'\over q'}|<1$ and ${p\over q}={1\over{b+{p'\over q'}}}$, where we have $|q'|<|q|$. Similarly to the argument in the previous section we infer that the product $p'q'$ is even. By the induction assumption ${p'\over q'}$ has a continued fraction expansion with even denominators. This completes the proof of sufficiency.
We proceed to the proof of uniqueness, using induction on the length of the continued fraction. For $p=1$ the assertion is trivial. Suppose that $$[b_1,b_2,\dots,b_n]=[c_1,c_2,\dots,c_n]$$ where all the numbers $b_i$ and $c_i$ are even, and several last terms of the sequence $c_i$ may be $\infty$ (which means that this sequence is actually shorter than the first one). Then $$b_1+[b_2,\dots,b_n]=c_1+[c_2,\dots,c_n].$$ Therefore, $$|b_1-c_1|=\big|[b_2,\dots,b_n]-[c_2,\dots,c_n]\big| < 2$$ But the number $|b_1-c_1|$ is even, hence $b_1=c_1$.
The lemma is proved.
The continued fraction expansion with even denominators and the corresponding natural diagram will be referred to as the *canonical expansion* of a rational number and the *canonical diagram* of a rational link (defined up to a rotation, see Lemma \[dual\]).
The parity of the denominator of a rational number is always opposite to the parity of the length of its even (canonical) continued fraction expansion. That is, for knots the canonical expression is of even length, while for links it is of odd length.
Now we are in a position to define a canonical *oriented* rational link.
Let $${p\over q}=[b_1,b_2,\dots,b_n]
=\cfrac{1}{
b_1+\cfrac{1}{
b_2+\cfrac{1}{
\dots+\cfrac{1}{
b_{n-1}+\cfrac{1}{b_n}}}}},$$ where $q$ and all $b_i$ are even. The diagram $D[b_1,\dots,b_n]$ taken with the positive orientation, denoted by $D^+[b_1,\dots,b_n]$, will be referred to as the *canonical diagram* of the oriented link $L^+(p/q)$.
We will use the canonical diagrams for the proof of the main theorem. However, for this theorem to make sense, we must check that the oriented link $L^+(p/q)$ does not depend on a particular choice of the continued fraction for the rational number $p/q$ and, especially, that it does not change when $p/q$ is changed by $\bar{p}/q$ where $p\bar{p}\equiv 1 \mod
2q$. We will prove these facts immediately.
\[plusminus\] Suppose that $p/q=[b_1,\dots,b_n]=[c_1,\dots,c_m]$ where $b_i$ and $c_i$ are non-zero integers. Then the natural diagram $D^+[b_1,\dots,b_n]$ and $D^+[c_1,\dots,c_m]$ are oriented isotopic.
We will prove that every natural diagram $D^+[b_1,\dots,b_n]$ is oriented isotopic to the canonical (even) natural diagram. To do so, we follow the induction argument used in Lemma \[even\_prod\] (sufficiency part). In fact, the induction step used there consists of one of the two transformations on the sequence $\{b_1,b_2,\dots,b_n\}$:
\(1) $[S,s,t,T]\longmapsto[S,s+1,-1,1-t,T]$, if $t>0$,
\(2) $[S,s,t,T]\longmapsto[S,s-1,1,-1-t,T]$, if $t<0$,
where $s,t$ are arbitrary integers and $S,T$ are arbitrary sequences.
The algorithm is to find the first from the left occurrence of an odd number and apply one of these rules. If $T=\emptyset$ and $t=\pm1$, then we use the rule $[S,s,\pm1]\mapsto[S,s\pm1]$ instead. Note that the situation when all numbers $b_i$, $1\leq i\leq n-1$ are even, while $b_n$ is odd, is impossible, because it corresponds to a knot rather than to a two-component link.
The proof of Lemma \[even\_prod\] assures that, in this process, the denominator of the rational fraction monotonically decreases, and thus the algorithm is finite.
Each step of the algorithm, when depicted on natural diagrams, shows that during this process the equivalence of oriented links is preserved (even with numbering of components).
The previous lemma justifies the notation $L^+(p/q)$.
\[dual\] If $p\bar{p}\equiv 1 \mod 2q$, then the links $L^+(p/q)$ and $L^+(\bar{p}/q)$ are oriented isotopic.
Making the rotation of the canonical diagram $D^+[b_1,b_2,\dots,b_n]$ around a vertical axis, we obtain the canonical diagram $D^+[b_n,b_{n-1},\dots,b_1]$, and it is easy to show (by induction on $n$) that these two continued fractions have the same denominators, and their numerators are related as indicated in the statement of the lemma. (Remind that, for links, the number $n$ is odd). We see that the two corresponding links are isotopic with the orientation of both components changed. But the total change of orientation is a link equivalence (see [@KM]).
In a canonical diagram of a rational link, due to the fact that all blocks are of even length, the strands are everywhere counter-directed. Therefore, Lemma \[P\_of\_Ln\] can be applied recursively: $$\label{iterP}
\begin{aligned}
P(D^+[b_1,\dots,b_n])=&\,a^{{\varepsilon}_nb_n} P(D^+[b_1,\dots,b_{n-1},0])
+z{{1-a^{{\varepsilon}_nb_n}}\over{a-a^{-1}}} P(D^+[b_1,\dots,b_{n-1},\infty])\\
=&\,a^{{\varepsilon}_nb_n} P(D^+[b_1,\dots,b_{n-2}])
+z{{1-a^{{\varepsilon}_nb_n}}\over{a-a^{-1}}} P(D^+[b_1,\dots,b_{n-1}]),
\end{aligned}$$ because the following two pairs of diagrams are equivalent as links: $D^+[b_1,\dots,b_{n-1},0]=D^+[b_1,\dots,b_{n-2}]$ and $D^+[b_1,\dots,b_{n-1},\infty]=D^+[b_1,\dots,b_{n-1}]$. The sign ${\varepsilon}_n=(-1)^{n-1}$ comes from our convention of counting the number of twists in the first and the second layers of a natural diagram (see Figure \[elements\] — the powers of $A$ and $B$ correspond to odd and even values of $n$, respectively).
For a given sequence $[b_1,\dots,b_n]$ denote $x_n=P(D^+[b_1,\dots,b_n])$. Then Equation \[iterP\] can be rewritten as $$\label{iterPx}
x_n = z{{1-a^{(-1)^{n-1}b_n}}\over{a-a^{-1}}}x_{n-1}
+ a^{(-1)^{n-1}b_n}x_{n-2}$$ which makes sense when $n>2$. Drawing the diagrams and applying skein relation (\[skein\]) for the cases $n=2$ and $n=1$, we can see that Equation (\[iterPx\]) still holds for these values, if we set $x_0=1$ and $x_{-1}=z^{-1}(a-a^{-1})$.
Main theorem
============
Our aim is to find a closed form formula for $x_n$ in terms of $a$ and $z$. To do this, it is convenient to first consider a more general situation.
\[recursion\] Let $r_n$ and $l_n$ be elements of a certain commutative ring $R$. Define recurrently the sequence $x_n$, $n\geq-1$, of polynomials from $R[z]$ by the relation $$x_n=zl_nx_{n-1} + r_nx_{n-2},\ n\geq 1,$$ where $x_{-1}$ and $x_{0}$ are fixed elements of $R$. Let $C$ be the set of all integer sequences $c=\{c_1,c_2,\dots,c_l\}$ where $c_1>c_2>\dots>c_l$, $c_1=n$, $c_i-c_{i+1}=1\text{ or }2$, $c_l=0\text{ or }-1$, and only one of the numbers 0 and $-1$ is present in the sequence $c$ (that is, if $c_l=-1$, then $c_{l-1}\not=0$). Then $x_n$ can be expressed as the following polynomial in $z$ with coefficients depending on the elements $l_i$, $r_i$ and the initial conditions $x_0$, $x_{-1}$: $$x_n=\sum_{c\in C} z^{k(c)} x_{c_l}
\prod_{i\in{\lambda}(c)}l_{c_i} \prod_{i\in{\rho}(c)}r_{c_i},$$ where ${\lambda}(c)=\{i\mid c_i-c_{i+1}=1\}$, ${\rho}(c)=\{i\mid c_i-c_{i+1}=2\}$ and $k(c)=|{\lambda}(c)|=\#\{i\mid c_i-c_{i+1}=1\}$.
Essentially, the written formula describes the computational tree for the calculation of $x_n$. Note that the recurrence is of depth 2, that is, the element $x_n$ is expressed through $x_{n-1}$ and $x_{n-2}$. Therefore, the computational tree is best represented as a layered tree where each layer matches the $l_i$’s and $r_i$’s with the same $i$. We draw the $l$-edges (of length 1) to the left and the $r$-edges (of length 2) to the right. The exponent of $z$ for each directed path from the vertex at level $n$ to a vertex at levels 0 or $-1$ in this tree corresponds to the number of left-hand edges. Any path in such a tree is uniquely determined by a sequence of levels $c$ with the above listed properties. In the picture, you can see an example of such a tree for $n=5$.
{width="\textwidth"}
To obtain the formula for the HOMFLY polynomial of an arbitrary rational link $L^\pm(p/q)$, we combine Lemma \[recursion\] with formula \[iterPx\]. For the sake of unification, we first make some preparations:
- If $q$ is odd (that is, we deal with a knot) and $p$ is odd, too, then we change $p$ to $p'=p-q$, if $p>0$, or to $p'=p+q$, if $p<0$. Then $L(p/q)=L(p'/q)$, the numerator of the fraction becomes even, hence Lemma \[even\_prod\] applies and formula (\[iterPx\]) is valid.
- If $q$ is even and the link is negative, then we use the property $L^-(p/q)=L^+(p'/q)$, where $p'$ is computed by the same rule as above. Below, we will simply write $L(p/q)$ instead of $L^+(p/q)$.
Now the main result reads:
Suppose that $p$ is even and $q$ is odd or $p$ is odd and $q$ is even. Let $[b_1,b_2,\dots,b_n]$ be the canonical continued fraction for the number $p/q$ (all numbers $b_i$ are even, positive or negative, see Lemma \[even\_prod\]). Then $$\label{main}
P(L(p/q))=
\sum_{c\in C} z^{k(c)} x_{c_l}
\prod_{i\in{\lambda}(c)}{1-a^{(-1)^{c_i-1}b_{c_i}}\over a-a^{-1}}
\prod_{i\in{\rho}(c)}a^{(-1)^{c_i-1}b_{c_i}},$$ where
- $C$ is the set of all integer sequences $c=\{c_1,c_2,\dots,c_l\}$ with $c_1>c_2>\dots>c_l$, $c_i-c_{i+1}=1\text{ or }2$, $c_1=n$, $c_l=0\text{ or }-1$, and only one of the numbers $0$ and $-1$ is present in the sequence $c$ (that is, if $c_l=-1$, then $c_{l-1}\not=0$),
- ${\lambda}(c)=\{i\mid c_i-c_{i+1}=1\}$,
- ${\rho}(c)=\{i\mid c_i-c_{i+1}=2\}$,
- $k(c)=|{\lambda}(c)|=\#\{i\mid c_i-c_{i+1}=1\}$,
- $x_0=1$ and $x_{-1}=z^{-1}(a-a^{-1})$.
The proof was actually given above.
**Example.** The canonical expansion of the fraction $4/7$ is $[2,-4]$. We have $n=2$, and there are three possibilities for the sequence $c$:
\(1) $c=\{2,1,0\}$, then ${\lambda}(c)=\{2,1\}$, $k(c)=2$, ${\rho}(c)=\emptyset$, $c_l=0$,
\(2) $c=\{2,1,-1\}$, then ${\lambda}(c)=\{2\}$, $k(c)=1$, ${\rho}(c)=\{1\}$, $c_l=-1$,
\(2) $c=\{2,0\}$, then ${\lambda}(c)=\emptyset$, $k(c)=0$, ${\rho}(c)=\{2\}$, $c_l=0$,
Then formula (\[main\]) gives:
$$\begin{aligned}
P(L(4/7))&=z^2\cdot{1-a^2\over a-a^{-1}}\cdot{1-a^4\over a-a^{-1}}
+z\cdot z^{-1}(a-a^{-1})\cdot{1-a^4\over a-a^{-1}}\cdot a^2 +a^4\\
&=z^2(a^2+a^4)+(a^2+a^4-a^6)\end{aligned}$$
In formula \[main\] one can, in principle, collect the terms with equal powers of $z$. The formulation of this result is rather involved, and we need first to introduce necessary notations.
Let $\alpha=p/q$ be a nonzero irreducible rational number between $-1$ and $1$. We denote by $n=\nu(\alpha)$ the length of the canonical continuous fraction for $\alpha$, and by $\alpha'$, the number $\alpha+1$, if $\alpha<0$, and $\alpha-1$, if $\alpha>0$. Now, let $$\rho_k(\alpha)=
\sum_{\substack{{C\subseteq{{\overline{1,n}}},\ C\cap(C-1)=\O\ } \\
{|C|=(n-k)/2}}}\prod_{m\in C}a^{(-1)^{m+1}b_m}
\prod_{\substack{{m\in{\overline{1,n}}}
\\ {m\notin C\cup(C-1)}}}(1-a^{(-1)^{m+1}b_m})$$ where $\overline{1,n}=\{1,2,\dots,n\}$ and $C-1$ is understood as the set of all numbers $c-1$, where $c\in C$.
Then we have:
Let $q$ be odd, that is, $L(\alpha)$ is a knot. Then:
1\) If $p$ is even, then $$P(L(\alpha))=\sum_{\substack{{{0\leq k\leq \nu(\alpha)}} \\
{k\equiv 0 \mod 2}}}z^{k}(a-a^{-1})^{-k}\rho_{k}(\alpha)$$
2\) If $p$ is odd, then $$P(L(\alpha))=\sum_{\substack{{{0\leq k\leq \nu(\alpha')}} \\
{k\equiv 0 \mod 2}}}z^{k}(a-a^{-1})^{-k}\rho_{k}(\alpha')$$
Let $q$ be even, that is, $L(\alpha)$ is a two-component link. Then:
3\) If the two components are counterdirected, then $$P(L^{+}(\alpha))=\sum_{\substack{{{-1\leq k\leq \nu(\alpha)}} \\
{k\equiv 1 \mod 2}}}z^{k}(a-a^{-1})^{-k}\rho_{k}(\alpha)$$
4\) If the two components are codirected, then $$P(L^{-}(\alpha))=\sum_{\substack{{{-1\leq k\leq \nu({\alpha'})}} \\
{k\equiv 1 \mod 2}}}z^{k}(a-a^{-1})^{-k}\rho_{k}({\alpha'})$$
The theorem can be proved by first collecting the terms with equal powers of $z$ in the statement of Lemma \[recursion\] and then using induction on $\nu(\alpha)$; we do not give the details here. Although Theorem 2 is in a sense more explicit than Theorem 1, it is less practical; in particular, the formula of Theorem 1 is better suited for programming purposes.
Computer calculations
=====================
The formula for $P(L(p/q))$ can be easily programmed. The source code of the program, written by the second author and tested by the first one, as well as the resulting table of HOMFLY polynomials for rational links with denominators not exceeding 1000, are presented online at [@Calc]. Below, we give a short excerpt of that big table which is enough to know the polynomials of all rational knots and links with denominators no greater than 9, if one uses the following rules (see [@KM]):
1. $P(L^+(-p/q))$ is obtained from $P(L^+(p/q))$ by the substitution $a\mapsto -a^{-1}$.
2. The knots $L(p_1/q)$ and $L(p_2/q)$ are equivalent, if $p_1p_2\equiv 1\mod q$.
3. The links $L^+(p_1/q)$ and $L^+(p_2/q)$ are oriented equivalent, if $p_1p_2\equiv 1\mod 2q$.
In Table \[homflytab\], the first column (R) gives the notation of the rational link (knot) as $L(p/q)$ (in the case of links, this means $L^+(p/q)$), the second column (T) contains the standard notation of that link (knot) from Thistlethwaite (Rolfsen) tables (see [@Atlas]; the bar over a symbol means mirror reflection, the star is for the change of orientation of one component), and the third column (H) is for the values of the HOMFLY polynomial. Note that we list HOMFLY polynomials for both orientations of each rational link, while the famous Knot Atlas [@Atlas] shows them for only one orientation of two-component links.
$$\begin{array}{c|c|l}
R & T & H \\
\hline
L({1\over2}) & L_2a_1 & z^{-1}(a^3-a)-za\rule{0pt}{15pt} \\[1mm]
L({2\over3}) & 3_1 & (2a^2-a^4)+z^2a^2 \\[1mm]
L({1\over4}) & L_4a_1 & z^{-1}(-a^3+a^5) + z(-a-a^3)\\[1mm]
L({3\over4}) & L_4a_1^* & z^{-1}(-a^3+a^5)+z(-3a^3+a^5)-z^3a^3\\[1mm]
L({2\over5}) & 4_1 & (a^{-2}-1+a^2)-z^2 \\[1mm]
L({4\over5}) & 5_1 & (3a^4-2a^6)+z^2(4a^4-a^6)+z^4a^4 \\[1mm]
L({1\over6}) & L_6a_3 & z^{-1}(-a^5+a^7)+z(-a-a^3-a^5) \\[1mm]
L({5\over6}) & L_6a_3^* & z^{-1}(a^7-a^5)+z(3a^7-6a^5)+z^3(a^7-5a^5)-z^5a^5\\[1mm]
L({2\over7}) & 5_2 & (a^2+a^4-a^6)+z^2(a^2+a^4) \\[1mm]
L({6\over7}) & 7_1 & (4a^6-3a^8)+z^2(10a^6-4a^8)+z^4(6a^6-a^8)+z^6a^6\\[1mm]
L({1\over8}) & L_8a_{14}^* & z^{-1}(-a^7+a^9)+z(-a-a^3-a^5-a^7)\\[1mm]
L({3\over8}) & \overline{L_5a_1} & z^{-1}(-a^{-1}+a)+z(a^{-3}-2a^{-1}+a)-z^3a^{-1}\\[1mm]
L({7\over8}) & L_8a_{14} & z^{-1}(a^9-a^7)+z(6a^9-10a^7)+z^3(5a^9-15a^7)
+z^5(a^9-7a^7)-z^7a^7\\[1mm]
L({2\over9}) & 6_1 & (a^{-2}-a^2+a^4)+z^2(-1-a^2)\\[1mm]
L({8\over9}) & 9_1 & (5a^8-4a^{10})+z^2(20a^8-10a^{10})+z^4(21a^8-6a^{10})
+z^6(8a^8-a^{10})+z^8a^8
\end{array}$$
Concluding remarks
==================
1\. As the Conway polynomial is a reduction of the HOMFLY polynomial, Theorem 1 gives a formula for the Conway polynomial of rational links by the substitutions $a=1$, $z=t$ (the fraction $(1-a^n)/(a-a^{-1})$ is first transformed to a Laurent polynomial and becomes equal to $-n/2$).
2\. Since the Jones polynomial is a reduction of the HOMFLY polynomial, Theorem 1 leads to a formula for the Jones polynomial of rational links by the substitutions $a=t^{-1}$, $z=t^{1/2}-t^{-1/2}$.
3\. The famous open problem whether a knot must be trivial if its Jones polynomial is 1, has a simple positive solution for rational knots. Indeed, the value $|J(-1)|$ is equal to the determinant of the knot, and the determinant of a rational knot is its denominator (see [@KM]).
4\. *Open problem*. Can one generalize formula \[main\] to *all* links? The results of L.Traldi [@Tr] show that it can be generalized to at least some non-rational links, although his formula is less explicit than ours.
Acknowledgements
================
The authors are grateful to M.Karev who read the manuscript and indicated several inaccuracies. We also thank S.Chmutov for pointing out the relation of our investigations with papers [@Jaeg] and [@Tr], and L.Traldi for valuable comments on his paper [@Tr].
[9]{}
, web resource `http://katlas.math.toronto.edu/` maintained by Scott Morrison and Dror Bar-Natan.
, web document, online at `http://www.pdmi.ras.ru/~arnsem/dataprog/`.
S.Chmutov, S.Duzhin and J.Mostovoy, [*Introduction to Vassiliev knot invariants*]{}, draft, 20.07.2010, online at `http://www.pdmi.ras.ru/~duzhin/papers/cdbook/`.
Shinji Fukuhara, [*Explicit formulae for two-bridge knot polynomials.*]{} J. Aust. Math. Soc. [**78**]{} (2005), p. 149–166.
F.Jaeger, [*Tutte polynomials and Link polynomials.*]{} Proceedings of the Amer Math Soc, v. 103, no. 2 (1988), p. 647–654.
W.B.R.Lickorish, [*An introduction to knot theory*]{}, Springer-Verlag New York, Inc. (1997).
Yoko Mizuma, [*Conway polynomials of two-bridge knots.*]{} Kobe J. Math. [**21**]{} (2004), p. 51–60.
Hitoshi Murakami, [*Email to S.Duzhin*]{}, November 2010.
K.Murasugi, [*Knot Theory and Its Applications*]{}, Birkhäuser, 1996.
V.Prasolov, A.Sossinsky, [*Knots, links, braids and 3-manifolds*]{}. Translations of Mathematical Monographs [**154**]{}, Amer. Math. Soc., Providence, RI, 1997.
L.Traldi, [*A Dichromatic Polynomial for Weighted Graphs and Link Polynomials.*]{} Proceedings of the Amer Math Soc, v. 106, no. 1 (1989), p. 279–286.
[^1]: Supported by grants RFBR 08-01-00379 and NSh-8462.2010.1.
[^2]: In fact [@HM], all rational links are *interchangeable*, that is, there is an isotopy of a rational link onto itself that interchanges the components.
|
---
abstract: 'We design a perfect zero-knowledge proof system for recognition if two permutation groups are conjugate. It follows, answering a question posed by O. G. Ganyushkin, that this recognition problem is not NP-complete unless the polynomial-time hierarchy collapses.'
author:
- |
Oleg Verbitsky\
[Department of Algebra]{}\
[Faculty of Mechanics & Mathematics]{}\
[Kyiv National University]{}\
[Volodymyrska 60]{}\
[01033 Kyiv, Ukraine]{}
title: |
\
\
Zero-Knowledge Proofs of the Conjugacy\
for Permutation Groups
---
= 45
\[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Proposition]{} \[theorem\][Remark]{}
\[theorem\][Question]{}
Introduction
============
Let $S_m$ be a symmetric group of order $m$. We suppose that an element of $S_m$, a permutation of the set $\{1,2,\ldots,m\}$, is encoded by a binary string of length $l=\lceil\log_2m!\rceil$, $m(\log_2m-O(1))\le l\le m\log_2m$. Given $v\in S_m$, $y\in S_m$, and $Y\subseteq S_m$, we denote $y^v=v^{-1}yv$ and $Y^v={\left\{ \hspace{0.5mm} y^v :
\hspace{0.5mm} y\in Y \right\}}$. Two subgroups $G$ and $H$ of $S_m$ are [*similar*]{} if their actions on $\{1,2,\ldots,m\}$ are isomorphic or, equivalently, if $G=H^v$ for some $v\in S_m$. If $X\subseteq S_m$, let ${\langle X \rangle}$ denote the group generated by elements of $X$.
We address the following algorithmic problem.
[[Similitude of Permutation Groups]{}]{}\
[*Given:*]{} $A_0, A_1\subseteq S_m$.\
[*Recognize if:*]{} $A_0$ and $A_1$ are similar.
Note that the [[Equality of Permutation Groups]{}]{} problem, that is, recognition if ${\langle A_0 \rangle}={\langle A_1 \rangle}$ reduces to recognition, given $X\subseteq S_m$ and $y\in S_m$, if $y\in {\langle X \rangle}$. Since the latter problem is known to be solvable in time bounded by a polynomial of the input length [@Sim; @FHL], so is [[Equality of Permutation Groups]{}]{}. As a consequence, [[Similitude of Permutation Groups]{}]{} belongs to NP, the class of decision problems whose yes-instances have polynomial-time verifiable certificates. The similitude of ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ is certified by a permutation $v$ such that ${\langle A_1 \rangle}={\langle A_0^v \rangle}$.
Another problem, [[Isomorphism of Permutation Groups]{}]{}, is to recognize if ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are isomorphic. This problem also belongs to NP (E. Luks, see [@Bab:dm Corollary 4.11]). Furthermore, it is announced [@BKL] that [[Isomorphism of Permutation Groups]{}]{} belongs to the complexity class coAM (see Section \[s:prel\] for the definition). By [@BHZ] this implies that [[Isomorphism of Permutation Groups]{}]{} is not NP-complete unless the polynomial-time hierarchy collapses to its second level (for the background on computational complexity theory the reader is referred to [@GJo])
O. G. Ganyushkin [@Gan] posed a question if a similar non-completeness result can be obtained for [[Similitude of Permutation Groups]{}]{}. In this paper we answer this question in affirmative. We actually prove a stronger result of independent interest, namely, that [[Similitude of Permutation Groups]{}]{} has a perfect zero-knowledge interactive proof system. It follows by [@AHa] that [[Similitude of Permutation Groups]{}]{} belongs to coAM and is therefore not NP-complete unless the polynomial-time hierarchy collapses.
Informally speaking, a zero-knowledge proof system for a recognition problem of a language $L$ is a protocol for two parties, the prover and the verifier, that allows the prover to convince the verifier that a given input belongs to $L$, with high confidence but without communicating the verifier any information (the rigorous definitions are in Section \[s:prel\]). Our zero-knowledge proof system for [[Similitude of Permutation Groups]{}]{} uses the underlying ideas of the zero-knowledge proof systems designed in [@GMR] for the [Quadratic Residuosity]{} and in [@GMW] for the [Graph Isomorphism]{} problem. In particular, instead of direct proving something about the input groups ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$, the prover prefers to deal with their conjugates ${\langle A_0 \rangle}^w$ and ${\langle A_1 \rangle}^w$ via a random permutation $w$. The crucial point is that these random groups are indistinguishable by the verifier because they are identically distributed, provided ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are similar. However, we here encounter a complication: the verifier may actually be able to distinguish between ${\langle A_0 \rangle}^w$ and ${\langle A_1 \rangle}^w$ based on particular representations of these groups by their generators. Overcoming this complication, which does not arise in [@GMR; @GMW], is a novel ingredient of our proof system.
Our result holds true even for a more general problem of recognizing if ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are conjugated via an element of the group generated by a given set $U\subseteq S_m$. We furthermore observe that a similar perfect zero-knowledge proof system works also for the [[Element Conjugacy]{}]{} problem of recognizing, given $a_0,a_1\in S_m$ and $U\subseteq S_m$, if $a_1=a_0^v$ for some $v\in{\langle U \rangle}$. A version of this problem where $a_0,a_1\in{\langle U \rangle}$ was proved to be in coAM in [@Bab:dm Corollary 12.3 (i)]. Note that the proof system developed in [@Bab:dm] uses different techniques and is not zero-knowledge.
Preliminaries {#s:prel}
=============
Every decision problem under consideration can be represented through a suitable encoding as a recognition problem for a language $L$ over the binary alphabet. We denote the [*length*]{} of a binary word $w$ by $|w|$.
An [*interactive proof system*]{} ${\{ V,P \}}$, further on abbreviated as IPS, consists of two probabilistic Turing machines, a polynomial-time [*verifier*]{} $V$ and a computationally unlimited [*prover*]{} $P$. The input tape is common for the verifier and the prover. The verifier and the prover also share a communication tape which allows message exchange between them. The system works as follows. First both the machines $V$ and $P$ are given an input $w$ and each of them is given an individual random string, $r_V$ for $V$ and $r_P$ for $P$. Then $P$ and $V$ alternatingly write messages to one another in the communication tape. $V$ computes its $i$-th message $a_i$ to $P$ based on the input $w$, the random string $r_V$, and all previous messages from $P$ to $V$. $P$ computes its $i$-th message $b_i$ to $V$ based on the input $w$, the random string $r_P$, and all previous messages from $V$ to $P$. After a number of message exchanges $V$ terminates interaction and computes an output based on $w$, $r_V$, and all $b_i$. The output is denoted by ${\{ V,P \}}(w)$. Note that, for a fixed $w$, ${\{ V,P \}}(w)$ is a random variable depending on both random strings $r_V$ and $r_P$.
Let $\epsilon(n)$ be a function of a natural argument taking on positive real values. We say that ${\{ V,P \}}$ is an [*IPS for a language $L$ with error $\epsilon(n)$*]{} if the following two conditions are fulfilled.
[*Completeness.*]{} If $w\in L$, then ${\{ V,P \}}(w)=1$ with probability at least $1-\epsilon(|w|)$.\
[*Soundness.*]{} If $w\notin L$, then, for an arbitrary interacting probabilistic Turing machine $P^*$, ${\{ V,P^* \}}(w)=1$ with probability at most $\epsilon(|w|)$.
We will call any prover $P^*$ interacting with $P$ on input $w\notin L$ [*cheating*]{}. If in the completeness condition we have ${\{ V,P \}}(w)=1$ with probability 1, we say that ${\{ V,P \}}$ has [*one-sided error*]{} $\epsilon(n)$.
An IPS is [*public-coin*]{} if the concatenation $a_1\ldots a_k$ of the verifier’s messages is a prefix of his random string $r_V$. A [*round*]{} is sending one message from the verifier to the prover or from the prover to the verifier. The class AM consists of those languages having IPSs with error $1/3$ and with number of rounds bounded by a constant for all inputs. A language $L$ belongs to the class coAM iff its complement $\{0,1\}^*\setminus L$ belongs to AM.
\[prop:gsi\] Every IPS for a language $L$ can be converted into a public-coin IPS for $L$ with the same error at cost of increasing the number of rounds in 2.
Given an IPS ${\{ V,P \}}$ and an input $w$, let ${\mbox{view}}_{V,P}(w)=(r'_V,a_1,b_1,\ldots,a_k,b_k)$ where $r'_V$ is a part of $r_V$ scanned by $V$ during work on $w$ and $a_1,b_1,\ldots,a_k,b_k$ are all messages from $P$ to $V$ and from $V$ to $P$ ($a_1$ may be empty if the first message is sent by $P$). Note that the verifier’s messages $a_1,\ldots,a_k$ could be excluded because they are efficiently computable from the other components. For a fixed $w$, ${\mbox{view}}_{V,P}(w)$ is a random variable depending on $r_V$ and $r_P$.
An IPS ${\{ V,P \}}$ is [*perfect zero-knowledge on $L$*]{} if for every interacting polynomial-time probabilistic Turing machine $V^*$ there is a probabilistic Turing machine $M_{V^*}$, called a [*simulator*]{}, that on every input $w\in L$ runs in expected polynomial time and produces output $M_{V^*}(w)$ which, if considered as a random variable depending on a random string of $M_{V^*}$, is distributed identically with ${\mbox{view}}_{V^*,P}(w)$. This notion formalizes the claim that the verifier gets no information during interaction with the prover: everything that the verifier gets he can get without the prover by running the simulator. According to the definition, the verifier learns nothing even if he deviates from the original program and follows an arbitrary probabilistic polynomial-time program $V^*$. We will call the verifier $V$ [*honest*]{} and all other verifiers $V^*$ [*cheating*]{}. If, for all $V^*$, $M_{V^*}$ is implemented by the same simulator $M$ running $V^*$ as a subroutine, we say that ${\{ V,P \}}$ is [*black-box simulation*]{} zero-knowledge.
We call $\epsilon(n)$ [*negligible*]{} if $\epsilon(n)<n^{-c}$ for every $c$ and all $n$ starting from some $n_0(c)$. The class of languages $L$ having IPSs that are perfect zero-knowledge on $L$ and have negligible error is denoted by PZK.
\[prop:aha\] $\mbox{PZK}\subseteq\mbox{coAM}$.
The [*$k(n)$-fold sequential composition*]{} of an IPS ${\{ V,P \}}$ is the IPS ${\{ V',P' \}}$ in which $V'$ and $P'$ on input $w$ execute the programs of $V$ and $P$ sequentially $k(|w|)$ times, each time with independent choice of random strings $r_V$ and $r_P$. At the end of interaction $V'$ outputs 1 iff ${\{ V,P \}}(w)=1$ in all $k(|w|)$ executions. The initial system ${\{ V,P \}}$ is called [*atomic*]{}.
\[prop:seqrep\]
1. If ${\{ V',P' \}}$ is the $k(n)$-fold sequential composition of ${\{ V,P \}}$, then $$\max_{P^*}{ {\bf P} \left[ {\{ V',P^* \}}(w)=1 \right] }=
{\left( \max_{P^*}{ {\bf P} \left[ {\{ V,P^* \}}(w)=1 \right] } \right)}^{k(|w|)}.$$ Consequently, if ${\{ V,P \}}$ is an IPS for a language $L$ with one-sided constant error $\epsilon$, then ${\{ V',P' \}}$ is an IPS for $L$ with one-sided error $\epsilon^{k(n)}$.
2. (Goldreich-Oren [@GOr], see also [@Gol Lemma 6.19]) If in addition ${\{ V,P \}}$ is black-box simulation perfect zero-knowledge on $L$, then ${\{ V',P' \}}$ is perfect zero-knowledge on $L$.
In the [*$k(n)$-fold parallel composition*]{} ${\{ V'',P'' \}}$ of ${\{ V,P \}}$, the program of ${\{ V,P \}}$ is executed $k(|w|)$ times in parallel, that is, in each round all $k(|w|)$ versions of a message are sent from one machine to another at once as a long single message. In every parallel execution $V''$ and $P''$ use independent copies of $r_V$ and $r_P$. At the end of interaction $V''$ outputs 1 iff ${\{ V,P \}}(w)=1$ in all $k(|w|)$ executions.
\[prop:parrep\] If ${\{ V'',P'' \}}$ is the $k(n)$-fold parallel composition of ${\{ V,P \}}$, then $$\max_{P^*}{ {\bf P} \left[ {\{ V'',P^* \}}(w)=1 \right] }=
{\left( \max_{P^*}{ {\bf P} \left[ {\{ V,P^* \}}(w)=1 \right] } \right)}^{k(|w|)}.$$
Group Conjugacy
===============
We consider the following extension of [[Similitude of Permutation Groups]{}]{}.
[[Group Conjugacy]{}]{}\
[*Given:*]{} $A_0,A_1,U\subseteq S_m$.\
[*Recognize if:*]{} ${\langle A_1 \rangle}={\langle A_0 \rangle}^v$ for some $v\in{\langle U \rangle}$.
\[thm:grconj\] [[Group Conjugacy]{}]{} is in PZK.
Designing a perfect zero-knowledge interactive proof system for [[Group Conjugacy]{}]{}, we will make use of the following facts due to Sims [@Sim; @FHL].
1. There is a polynomial-time algorithm that, given $X\subseteq S_m$ and $y\in S_m$, recognizes if $y\in{\langle X \rangle}$. As a consequence, there is a polynomial-time algorithm that, given $X\subseteq S_m$ and $Y\subseteq S_m$, recognizes if ${\langle X \rangle}={\langle Y \rangle}$.
2. There is a probabilistic polynomial-time algorithm that, given $X\subseteq S_m$, outputs a random element of ${\langle X \rangle}$. Here and further on, by a [*random element*]{} of a finite set $Z$ we mean a random variable uniformly distributed over $Z$.
Given $A\subseteq S_m$ and a number $k$, define $$G(A,k)={\left\{ \hspace{0.5mm} (x_1,\ldots,x_k) :
\hspace{0.5mm} x_i\in S_m, {\langle x_1,\ldots,x_k \rangle}={\langle A \rangle} \right\}}.$$
In the sequel, the length of the binary encoding of an input $A_0,A_1,U\subseteq S_m$ will be denoted by $n$. We set $k=4m$. On input $(A_0,A_1,U)$, the IPS we design is the $n$-fold sequential repetition of the following 3-round system. We will say that the verifier $V$ [*accepts*]{} if ${\{ V,P \}}(A_0,A_1,U)=1$ and [*rejects*]{} otherwise.
If $(A_0,A_1,U)$ is yes-instance of [[Group Conjugacy]{}]{}, $P$ finds an element $v\in{\langle U \rangle}$ such that ${\langle A_1 \rangle}={\langle A_0 \rangle}^v$.
[*1st round.*]{}
$P$ generates a random element $u\in{\langle U \rangle}$, computes $A=A_1^u$, chooses a random element $(a_1,\ldots,a_k)$ in $G(A,k)$, and sends $(a_1,\ldots,a_k)$ to $V$. $V$ checks if all $a_i\in S_m$ and, if not (this is possible in the case of a cheating prover), halts and rejects.
[*2nd round.*]{}
$V$ chooses a random bit $\beta\in\{0,1\}$ and sends it to $P$.
[*3rd round.*]{}
[*Case $\beta=1$.*]{} $P$ sends $V$ the permutation $w=u$. $V$ checks if $w\in{\langle U \rangle}$ and if ${\langle a_1,\ldots,a_k \rangle}={\langle A_1^w \rangle}$.
[*Case $\beta\ne 1$*]{} (this includes the possibility of a message $\beta\notin\{0,1\}$ produced by a cheating verifier). $P$ computes $w=vu$ and sends $w$ to $V$. $V$ checks if $w\in{\langle U \rangle}$ and if ${\langle a_1,\ldots,a_k \rangle}={\langle A_0^w \rangle}$.
$V$ halts and accepts if the conditions are checked successfully and rejects otherwise.
We now need to prove that this system is indeed an IPS for [[Group Conjugacy]{}]{} and, moreover, that it is perfect zero-knowledge.
[*Completeness. *]{}
To show that the prover is able to follow the above protocol, we have to check that $G(A,k)\ne\emptyset$ for $k=4m$. The latter is true by the fact that every subgroup of $S_m$ can be generated by at most $m-1$ elements [@Jer]. If ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are conjugate via an element of ${\langle U \rangle}$ and the prover and the verifier follow the protocol, then ${\langle a_1,\ldots,a_k \rangle}={\langle A \rangle}={\langle A_1^u \rangle}={\langle A_0^{vu} \rangle}$. Therefore, the verifier accepts with probability 1 both in the atomic and the composed systems.
[*Soundness. *]{}
Assume that ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are not conjugate via an element of ${\langle U \rangle}$ and consider an arbitrary cheating prover $P^*$. Observe that if both ${\langle a_1,\ldots,a_k \rangle}={\langle A_1^u \rangle}$ and ${\langle a_1,\ldots,a_k \rangle}={\langle A_0^w \rangle}$ with $u,w\in{\langle U \rangle}$, then ${\langle A_1 \rangle}={\langle A_0 \rangle}^{wu^{-1}}$. It follows that $V$ rejects for at least one value of $\beta$ and, therefore, in the atomic system $V$ accepts with probability at most 1/2. By Proposition \[prop:seqrep\] (1), in the composed system $V$ accepts with probability at most $2^{-n}$.
[*Zero-knowledge. *]{}
We will need the following fact.
\[lem:gen\] Let $G$ be a subgroup of $S_m$ and $a_1,\ldots,a_k$ be random independent elements of $G$.
1. If $k=4m$, then ${\langle a_1,\ldots,a_k \rangle}=G$ with probability more than 1/2.
2. If $k=8m$, then ${\langle a_1,\ldots,a_k \rangle}=G$ with probability more than $1-2^{-m}$.
We will estimate from above the probability that ${\langle a_1,\ldots,a_k \rangle}\ne G$. This inequality is equivalent with the condition that all ${\langle a_1 \rangle}$, ${\langle a_1,a_2 \rangle}$, …, ${\langle a_1,\ldots,a_k \rangle}$ are proper subgroups of $G$. Assume that this condition is true. Since every subgroup chain in $S_m$ has length less than $2m$ [@Bab:chain; @CST], less than $2m-1$ inclusions among ${\langle a_1 \rangle}\subseteq{\langle a_1,a_2 \rangle}\subseteq\cdots\subseteq
{\langle a_1,\ldots,a_k \rangle}$ are proper. In other words, less than $2m-1$ of the events $a_2\notin{\langle a_1 \rangle}$, $a_3\notin{\langle a_1,a_2 \rangle}$, …, $a_k\notin{\langle a_1,\ldots,a_{k-1} \rangle}$ occur. Equivalently, there occur more than $k-2m$ of the events $a_2\in{\langle a_1 \rangle}$, $a_3\in{\langle a_1,a_2 \rangle}$, …, $a_k\in{\langle a_1,\ldots,a_{k-1} \rangle}$.
Let $p=|H|/|G|$ be the maximum density of a proper subgroup $H$ of $G$. Given $a_1,\ldots,a_i\in G$, define $E(a_1,\ldots,a_i)$ to be an arbitrary subset of $G$ fixed so that
(i)
: $E(a_1,\ldots,a_i)$ has density $p$ in $G$, and
(ii)
: $E(a_1,\ldots,a_i)$ contains ${\langle a_1,\ldots,a_i \rangle}$ if the latter is a proper subgroup of $G$.
If ${\langle a_1,\ldots,a_k \rangle}\ne G$, there must occur more than $k-2m$ of the events $$\label{eq:events}
a_2\in E(a_1),\ a_3\in E(a_1,a_2),\ldots,a_k\in E(a_1,\ldots,a_{k-1}).$$ It suffices to show that the probability of so many occurrences in [(\[eq:events\])]{} is small enough. Set $X_i(a_1,\ldots,a_k)$ to be equal to 1 if $a_{i+1}\in E(a_1,\ldots,a_i)$ and to 0 otherwise. In these terms, we have to estimate the probability that $$\label{eq:bernul}
\sum_{i=1}^{k-1} X_i > k-2m.$$ It is easy to calculate that an arbitrary set of $l$ events in [(\[eq:events\])]{} occurs with probability $p^l$. Hence the events [(\[eq:events\])]{} as well as the random variables $X_1,\ldots,X_{k-1}$ are mutually independent, and $X_1,\ldots,X_{k-1}$ are successive Bernoulli trails with success probability $p$.
If $k=4m$, the inequality [(\[eq:bernul\])]{} implies that strictly more than a half of all the trails are successful. Since $p\le 1/2$, this happens with probability less than 1/2 and the item 1 of the lemma follows.
If $k=8m$, the inequality [(\[eq:bernul\])]{} implies $$\frac1{k-1}\sum_{i=1}^{k-1}X_i>p+\epsilon$$ with deviation $\epsilon=1/4$ from the mean value $p={ {\bf E} \left[ \frac1{k-1}\sum_{i=1}^{k-1}X_i \right] }$. By the Chernoff bound [@ASp Theorem A.4], this happens with probability less than $\exp{\left( -2\epsilon^2(k-1) \right)}\allowbreak =\exp(-m+\frac18)<2^{-m}$. This proves the item 2 of the lemma.
By Proposition \[prop:seqrep\] (2) it suffices to show that the atomic system is black-box simulation perfect zero-knowledge. We describe a probabilistic simulator $M$ that uses the program of $V^*$ as a subroutine and, for each $V^*$, runs in expected polynomial time. Assume that the running time of $V^*$ is bounded by a polynomial $q$ in the input size. On input $(A_0,A_1,U)$ of length $n$, $M$ will run the program of $V^*$ on the same input with random string $r$, where $r$ is the prefix of $M$’s random string of length $q(n)$. In all other cases of randomization, $M$ will use the remaining part of its random string.
Having received an input $(A_0,A_1,U)$, the simulator $M$ chooses a random element $w\in{\langle U \rangle}$ and a random bit $\alpha\in\{0,1\}$. Then $M$ randomly and independently chooses elements $a_1,\ldots,a_k$ in ${\langle A_\alpha^w \rangle}$ and checks if $$\label{eq:succ}
{\langle a_1,\ldots,a_k \rangle}={\langle A_\alpha^w \rangle}.$$ If [(\[eq:succ\])]{} is not true, $M$ repeats the choice of $a_1,\ldots,a_k$ again and again until [(\[eq:succ\])]{} is fulfilled. By Lemma \[lem:gen\] (1), $M$ succeeds in at most 2 attempts on average. The resulting sequence $(a_1,\ldots,a_k)$ is uniformly distributed on $G(A_\alpha^w,k)$. Then $M$ computes $\beta=V^*(A_0,A_1,U,r,a_1,\ldots,a_k)$, the message that $V^*$ sends $P$ in the 2-nd round after receiving $P$’s message $a_1,\ldots,a_k$. If $\beta$ and $\alpha$ are simultaneously equal to or different from 1, $M$ halts and outputs $(r',a_1,\ldots,a_k,\beta,w)$, where $r'$ is the prefix of $r$ that $V^*$ actually uses after reading the input $(A_0,A_1,U)$ and the prover’s message $a_1,\ldots,a_k$. If exactly one of $\beta$ and $\alpha$ is equal to 1, then $M$ restarts the same program from the very beginning with another independent choice of $w$, $\alpha$, and $a_1,\ldots,a_k$. Notice that it might happen that in unsuccessful attempts $V^*$ used a prefix of $r$ longer than $r'$.
We first check that, for each $V^*$, the simulator $M$ terminates in expected polynomial time whenever $A_0$ and $A_1$ are conjugated via an element of ${\langle U \rangle}$. Since $V^*$ is polynomial-time, one attempt to pass the body of $M$’s program takes time bounded by a polynomial of $n$. Observe that $\alpha$ and $(r,a_1,\ldots,a_k)$ are independent. Really, independently of whether $\alpha=0$ or $\alpha=1$, $r$ is a random string of length $q(n)$ and $(a_1,\ldots,a_k)$ is a random element of $G(A,k)$, where $A$ itself is a random element of the orbit ${\left\{ \hspace{0.5mm} A_0^w :
\hspace{0.5mm} w\in{\langle U \rangle} \right\}}={\left\{ \hspace{0.5mm} A_1^w :
\hspace{0.5mm} w\in{\langle U \rangle} \right\}}$ under the conjugating action of ${\langle U \rangle}$ on subsets of $S_m$. It follows that $\alpha$ and $\beta$ are independent and therefore an execution of the body of $M$’s program is successful with probability 1/2. We conclude that on average $M$’s program is executed twice and this takes expected polynomial time.
We finally need to check that, whenever $A_0$ and $A_1$ are conjugated via an element of ${\langle U \rangle}$, for each $V^*$ the output $M(A_0,A_1,U)$ is distributed identically with ${\mbox{view}}_{V^*,P}(A_0,A_1,U)$. Notice that both the random variables depend on $V^*$’s random string $r$. It therefore suffices to show that the distributions are identical when conditioned on an arbitrary fixed $r$. Denote these conditional distributions by $D_M(A_0,A_1,U,r)$ and $D_{V^*,P}(A_0,A_1,U,r)$. We will show that both $D_M(A_0,A_1,U,r)$ and $D_{V^*,P}(A_0,A_1,U,r)$ are uniform on the set $$\begin{aligned}
S=\Bigl\{ \hspace{0.5mm} (a_1,\ldots,a_k,\beta,w)\ : \
w\in{\langle U \rangle},&&\hspace{-0.8em} \beta=V^*(A_0,A_1,U,r,a_1,\ldots,a_k),\\
&& (a_1,\ldots,a_k)\in G(A^w_{\delta(\beta)},k)\Bigr\},\end{aligned}$$ where $\delta(\beta)$ is equal to 1 if $\beta=1$ and to 0 otherwise.
Let $v\in{\langle U \rangle}$, such that ${\langle A_1 \rangle}={\langle A_0 \rangle}^v$, be chosen by the prover $P$ on input $(A_0,A_1,U)$. Given $x_1,\ldots,x_k\in G(A_1,k)$ and $u\in{\langle U \rangle}$, define $\phi(x_1,\ldots,x_k,u)=(a_1,\ldots,a_k,\beta,w)$ by $a_i=x_i^u$ for all $i\le k$, $\beta=V^*(A_0,A_1,U,r,a_1,\ldots,a_k)$, and $w=v^{1-\delta(\beta)}u$. As easily seen, $\phi(x_1,\ldots,x_k,u)\in S$.
The map $\phi{:G(A_1,k)\times{\langle U \rangle} \rightarrow S}$ is one-to-one.
Define $\psi(a_1,\ldots,a_k,\beta,w)=(x_1,\ldots,x_k,u)$ by $u=v^{\delta(\beta)-1}w$ and $x_i=a_i^{u^{-1}}$ for all $i\le k$. It is not hard to check that the map $\psi$ is the inverse of $\phi$.
Observe now that if $(x_1,\ldots,x_k,u)$ is chosen at random uniformly in $G(A_1,k)\times{\langle U \rangle}$, then $\phi(x_1,\ldots,x_k,u)$ has distribution $D_{V^*,P}(A_0,A_1,U,r)$. By Claim we conclude that $D_{V^*,P}(A_0,A_1,U,r)$ is uniform on $S$.
As a yet another consequence of Claim, observe that if a random tuple $(a_1,\ldots,a_k,\allowbreak \beta,w)$ is uniformly distributed on $S$, then its prefix $(a_1,\ldots,a_k)$ is a random element of $G(A,k)$, where $A$ is a random element of the orbit ${\left\{ \hspace{0.5mm} A_0^w :
\hspace{0.5mm} w\in{\langle U \rangle} \right\}}={\left\{ \hspace{0.5mm} A_1^w :
\hspace{0.5mm} w\in{\langle U \rangle} \right\}}$ under the conjugating action of ${\langle U \rangle}$ on subsets of $S_m$. This suggests the following way of generating a random element of $S$. Choose uniformly at random $\alpha\in\{0,1\}$, $w\in{\langle U \rangle}$, $(a_1,\ldots,a_k)\in G(A_\alpha^w,k)$ and, if $$\label{eq:guess}
\delta{\left( V^*(A_0,A_1,U,r,a_1,\ldots,a_k) \right)}=\alpha,$$ output $(a_1,\ldots,a_k,V^*(A_0,A_1,U,r,a_1,\ldots,a_k),w)$; otherwise repeat the same procedure once again independently. Under the condition that [(\[eq:guess\])]{} is fulfilled for the first time in the $i$-th repetition, the output is uniformly distributed on $S$. Notice now that this sampling procedure coincides with the description of $D_M(A_0,A_1,U,r)$. It follows that $D_M(A_0,A_1,U,r)$ is uniform on $S$. The proof of the perfect zero-knowledge property of our proof system for [[Group Conjugacy]{}]{} is complete.
The following corollary immediately follows from Theorem \[thm:grconj\] by Proposition \[prop:aha\] and the result of [@BHZ].
\[cor:grconj\] [[Group Conjugacy]{}]{} is in coAM and is therefore not NP-complete unless the polynomial-time hierarchy collapses.
We also give an alternative proof of this corollary that consists in direct designing a two-round IPS ${\{ V,P \}}$ with error 1/4 for the complement of [[Group Conjugacy]{}]{} and applying Proposition \[prop:gsi\]. More precisely, we deal with the [Group Non-Conjugacy]{} problem of recognizing, given $A_0,A_1,U\subseteq S_m$, if there is no $v\in{\langle U \rangle}$ such that ${\langle A_1 \rangle}={\langle A_0 \rangle}^v$.
Set $k=8m$. The below IPS is composed twice in parallel.
[*1st round.*]{}
$V$ chooses a random bit $\alpha\in\{0,1\}$, a random element $u\in{\langle U \rangle}$, and a sequence of random independent elements $a_1,\ldots,a_k\in{\langle A_\alpha^u \rangle}$. Then $V$ sends $(a_1,\ldots,a_k)$ to $P$.
[*2nd round.*]{}
$P$ determines $\beta$ such that ${\langle a_1,\ldots,a_k \rangle}$ and ${\langle A_\beta \rangle}$ are conjugate via an element of ${\langle U \rangle}$ and sends $\beta$ to $V$.
$V$ accepts if $\beta=\alpha$ and rejects otherwise.
[*Completeness. *]{}
By Lemma \[lem:gen\] (2), ${\langle a_1,\ldots,a_k \rangle}={\langle A_\alpha^u \rangle}$ with probability at least $1-2^{-m}$. If this happens and if ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are not conjugated via ${\langle U \rangle}$, the group ${\langle a_1,\ldots,a_k \rangle}$ is conjugated via ${\langle U \rangle}$ with precisely one of ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$. In this case $P$ is able to determine $\alpha$ correctly. Therefore $V$ accepts with probability at least $1-2^{-m}$ in the atomic system and with probability at least $1-2^{-m+1}$ in the composed system.
[*Soundness. *]{}
If ${\langle A_0 \rangle}$ and ${\langle A_1 \rangle}$ are conjugated via ${\langle U \rangle}$, then for both values $\alpha=0$ and $\alpha=1$, the vector $(a_1,\ldots,a_k)$ has the same distribution, namely, it is a random element of $A^k$, where $A$ is a random element of the orbit ${\left\{ \hspace{0.5mm} A_0^w :
\hspace{0.5mm} w\in{\langle U \rangle} \right\}}={\left\{ \hspace{0.5mm} A_1^w :
\hspace{0.5mm} w\in{\langle U \rangle} \right\}}$ under the conjugating action of ${\langle U \rangle}$ on subsets of $S_m$. It follows that, irrespective of his program, $P$ guesses the true value of $\alpha$ with probability 1/2. With the same probability $V$ accepts in the atomic system. By Proposition \[prop:parrep\], in the composed system $V$ accepts with probability 1/4.
Note that ${\{ V,P \}}$ is perfect zero-knowledge only for the honest verifier but may reveal a non-trivial information for a cheating verifier.
Element Conjugacy
=================
This section is devoted to the following problem. [$a_1=a_0^v$ for some $v\in{\langle U \rangle}$]{} L. Babai [@Bab:dm] considers a version of this problem with $a_0,a_1\in{\langle U \rangle}$ and proves that it belongs to coAM. His result holds true not only for permutation groups but also for arbitrary finite groups with efficiently performable group operations, in particular, for matrix groups over finite fields. It is easy to see that Theorem \[thm:grconj\] carries over to [[Element Conjugacy]{}]{}.
\[thm:elconj\] [[Element Conjugacy]{}]{} is in PZK.
The proof system designed in the preceding section for [[Group Conjugacy]{}]{} applies to [[Element Conjugacy]{}]{} as well. Moreover, the proof system for [[Element Conjugacy]{}]{} is considerably simpler. In place of groups ${\langle A_0^u \rangle}$ and ${\langle A_1^u \rangle}$ we now deal with single elements $a_0^u$ and $a_1^u$ and there is no complication with representation of ${\langle A_0^u \rangle}$ and ${\langle A_1^u \rangle}$ by generating sets.
We now notice relations of [[Element Conjugacy]{}]{} with the following problem considered by E. Luks [@Luk] (see also [@Bab:handbook Section 6.5]). Given $x\in S_m$, let $C(x)$ denote the centralizer of $x$ in $S_m$. [$x,y\in S_m$, $U\subseteq S_m$]{}[$C(x)\cap{\langle U \rangle} y\ne\emptyset$]{}
Since, given a permutation $x$, one can efficiently find a list of generators for $C(x)$, this is a particular case of the [Coset Intersection]{} problem of recognizing, given $A,B\subseteq S_m$ and $s,t\in S_m$, if the cosets ${\langle A \rangle}s$ and ${\langle B \rangle}t$ intersect.
\[prop:red\] [[Element Conjugacy]{}]{} and [[Centralizer and Coset Intersection]{}]{} are equivalent with respect to the polynomial-time many-one reducibility.
We first reduce [[Element Conjugacy]{}]{} to [[Centralizer and Coset Intersection]{}]{}. Given permutations $a_0$ and $a_1$, it is easy to recognize if they are conjugate in $S_m$ and, if so, to find an $s$ such that $a_1=a_0^s$. The set of all $z\in S_m$ such that $a_1=a_0^z$ is the coset $C(a_0)s$. It follows that ${\langle U \rangle}$ contains $v$ such that $a_1=a_0^v$ iff $C(a_0)$ and ${\langle U \rangle}s^{-1}$ intersect.
A reduction from [[Centralizer and Coset Intersection]{}]{} to [[Element Conjugacy]{}]{} is based on the fact that $C(x)$ and ${\langle U \rangle}y$ intersect iff $x$ and $yxy^{-1}$ are conjugated via an element of ${\langle U \rangle}$.
Note that, while the reduction we described from [[Element Conjugacy]{}]{} to [[Centralizer and Coset Intersection]{}]{} works only for permutation groups, the reduction in the other direction works equally well for arbitrary finite groups with efficiently performable group operations, in particular, for matrix groups over finite fields.
We now have three different ways to prove that [[Element Conjugacy]{}]{} is in coAM and is therefore not NP-complete unless the polynomial-time hierarchy collapses. First, this fact follows from Theorem \[thm:elconj\] by Proposition \[prop:aha\]. Second, one can use Proposition \[prop:red\] and the result of [@Bab:dm Corollary 12.2 (d)] that [Coset Intersection]{} is in coAM. Finally, one can design a constant-round IPS for the complement of [[Element Conjugacy]{}]{} as it is done in the preceding section for the complement of [[Group Conjugacy]{}]{}.
We conclude with two questions.
Is there any reduction between [[Group Conjugacy]{}]{} and [Coset Intersection]{}? We are not able to prove an analog of Proposition \[prop:red\] for groups because, given $A_0,A_1\subseteq S_m$ such that ${\langle A_1 \rangle}={\langle A_0 \rangle}^v$ for some $v\in S_m$, we cannot efficiently find any $v$ with this property (otherwise we could efficiently recognize the [[Similitude of Permutation Groups]{}]{}).
Does [[Element Conjugacy]{}]{} reduce to [[Group Conjugacy]{}]{}? Whereas Corollary \[cor:grconj\] gives us an evidence that [[Group Conjugacy]{}]{} is not NP-complete, we have no formal evidence supporting our feeling that [[Group Conjugacy]{}]{} is not solvable efficiently. A reduction from [[Element Conjugacy]{}]{} could be considered such an evidence as [[Element Conjugacy]{}]{} is not expected to be solvable in polynomial time [@Bab:kyoto page 1483].
Note that the conjugacy of permutations $a_0$ and $a_1$ via an element of a group ${\langle U \rangle}$ does not reduce to the conjugacy of the cyclic groups ${\langle a_0 \rangle}$ and ${\langle a_1 \rangle}$ via ${\langle U \rangle}$ because ${\langle a_0 \rangle}$ and ${\langle a_1 \rangle}$ can be conjugated by conjugation of another pair of their generators, while such a new conjugation may be not necessary via ${\langle U \rangle}$. For example, despite the groups ${\langle (123) \rangle}$ and ${\langle (456) \rangle}$ are conjugated via ${\langle (14)(26)(35) \rangle}$, the permutations $(123)$ and $(456)$ are not.
### Acknowledgement {#acknowledgement .unnumbered}
I appreciate useful discussions with O. G. Ganyushkin.
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---
abstract: 'We prove new cases of the inverse Galois problem by considering the residual Galois representations arising from a fixed newform. Specific choices of weight $3$ newforms will show that there are Galois extensions of ${{\mathbb Q}}$ with Galois group ${\operatorname{PSL}}_2({{\mathbb F}}_p)$ for all primes $p$ and ${\operatorname{PSL}}_2({{\mathbb F}}_{p^3})$ for all odd primes $p \equiv \pm 2, \pm 3, \pm 4, \pm 6 \pmod{13}$.'
address: 'Department of Mathematics, Cornell University, Ithaca, NY 14853, USA'
author:
- David Zywina
title: Modular forms and some cases of the inverse Galois problem
---
Introduction
============
The asks whether every finite group is isomorphic to the Galois group of some extension of ${{\mathbb Q}}$. There has been much work on using modular forms to realize explicit simple groups of the form ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^r})$ as Galois groups of extensions of ${{\mathbb Q}}$, cf. [@MR0419358], [@MR1352266], [@MR1800679], [@MR1879665], [@MR2512358]. For example, [@MR1800679]\*[§3.2]{} shows that ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^2})$ occurs as a Galois group of an extension of ${{\mathbb Q}}$ for all primes $\ell$ in a explicit set of density $1-1/2^{10}$ (and for primes $\ell \leq 5000000$). Also it is shown in [@MR1800679] that ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^4})$ occurs as a Galois group of an extension of ${{\mathbb Q}}$ when $\ell \equiv 2,3 \pmod{5}$ or $\ell\equiv \pm 3,\pm 5,\pm 6,\pm 7 \pmod{17}$.
The goal of this paper is to try to realize more groups of the form ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^r})$ for *odd* $r$. We will achieve this by working with newforms of odd weight; the papers mentioned above focus on even weight modular forms (usual weight $2$). We will give background and describe the general situation in §\[SS:general result\]. In §\[SS:weight 3 level 27\] and §\[SS:weight 3 level 160\], we will use specific newforms of weight $3$ to realize many groups of the form ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^r})$ with $r$ equal to $1$ and $3$, respectively.
Throughout the paper, we fix an algebraic closure ${{\overline{\mathbb Q}}}$ of ${{\mathbb Q}}$ and define the group $G:= \operatorname{Gal}({{\overline{\mathbb Q}}}/{{\mathbb Q}})$. For a ring $R$, we let ${\operatorname{PSL}}_2(R)$ and ${\operatorname{PGL}}_2(R)$ be the quotient of ${\operatorname{SL}}_2(R)$ and ${\operatorname{GL}}_2(R)$, respectively, by its subgroup of scalar matrices (in particular, this notation may disagree with the $R$-points of the corresponding group scheme ${\operatorname{PSL}}_2$ or ${\operatorname{PGL}}_2$).
General results {#SS:general result}
---------------
Fix a non-CM newform $f(\tau)=\sum_{n=1}^\infty a_n q^n$ of weight $k>1$ on $\Gamma_1(N)$, where the $a_n$ are complex numbers and $q=e^{2\pi i \tau}$ with $\tau$ a variable of the complex upper-half plane. Let $\varepsilon\colon ({{\mathbb Z}}/N{{\mathbb Z}})^\times \to \CC^\times$ be the nebentypus of $f$.
Let $E$ be the subfield of $\CC$ generated by the coefficients $a_n$; it is also generated by the coefficients $a_p$ with primes $p\nmid N$. The field $E$ is a number field and all the $a_n$ are known to lie in its ring of integers ${{\mathcal O}}$. The image of $\varepsilon$ lies in $E^\times$. Let $K$ be the subfield of $E$ generated by the algebraic integers $r_p:=a_p^2/\varepsilon(p)$ for primes $p\nmid N$; denote its ring of integer by $R$.
Take any non-zero prime ideal $\Lambda$ of ${{\mathcal O}}$ and denote by $\ell=\ell(\Lambda)$ the rational prime lying under $\Lambda$. Let $E_\Lambda$ and ${{\mathcal O}}_\Lambda$ be the completions of $E$ and ${{\mathcal O}}$, respectively, at $\Lambda$. From Deligne [@Deligne71-179], we know that there is a continuous representation $$\rho_\Lambda \colon G \to {\operatorname{GL}}_2({{\mathcal O}}_\Lambda)$$ such that for each prime $p\nmid N\ell$, the representation $\rho_\Lambda$ is unramified at $p$ and satisfies $$\label{E:trace and det}
\operatorname{tr}(\rho_\Lambda(\operatorname{Frob}_p)) = a_p \quad \text{ and } \quad \det(\rho_\Lambda(\operatorname{Frob}_p))= \varepsilon(p) p^{k-1}.$$ The representation $\rho_\Lambda$ is uniquely determined by the conditions (\[E:trace and det\]) up to conjugation by an element of ${\operatorname{GL}}_2(E_\Lambda)$. By composing $\rho_\Lambda$ with the natural projection arising from the reduction map ${{\mathcal O}}_\Lambda \to {{\mathbb F}}_\Lambda:={{\mathcal O}}/\Lambda$, we obtain a representation $$\bbar\rho_\Lambda \colon G \to {\operatorname{GL}}_2({{\mathbb F}}_\Lambda).$$ Composing $\bbar\rho_\Lambda$ with the natural quotient map ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)\to {\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$, we obtain a homomorphism $$\bbar\rho_\Lambda^{{\operatorname{proj}}}\colon G \to {\operatorname{PGL}}_2({{\mathbb F}}_\Lambda).$$ Define the field ${{\mathbb F}}_\lambda:=R/\lambda$, where $\lambda:= \Lambda\cap R$. There are natural injective homomorphisms ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda) \hookrightarrow {\operatorname{PGL}}_2({{\mathbb F}}_\lambda) \hookrightarrow {\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ and ${\operatorname{PSL}}_2({{\mathbb F}}_\Lambda) \hookrightarrow {\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ that we shall view as inclusions.
The main task of this paper is to describe the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ for all $\Lambda$ outside of some *explicit* set. The following theorem of Ribet gives two possibilities for $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ for all but finitely many $\Lambda$.
\[T:Ribet\] There is a finite set $S$ of non-zero prime ideals of $R$ such that if $\Lambda$ is a non-zero prime ideal of ${{\mathcal O}}$ with $\lambda:=R\cap \Lambda \notin S$, then the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to either ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$.
As noted in §3 of [@MR2806684], this is an easy consequence of [@MR819838].
We will give a proof of Theorem \[T:Ribet\] in §\[S:effective Ribet\] that allows one to compute such a set $S$. There are several related results in the literature; for example, Billerey and Dieulefait [@MR3188630] give an version of Theorem \[T:Ribet\] when the nebentypus $\varepsilon$ is trivial.
We now explain how to distinguish the two possibilities from Theorem \[T:Ribet\]. Let $L\subseteq \CC$ be the extension of $K$ generated by the square roots of the values $r_p=a_p^2/\varepsilon(p)$ with $p\nmid N$; it is a finite extension of $K$ (moreover, it is contained in a finite cyclotomic extension of $E$).
\[T:PSL2 vs GL2\] Let $\Lambda$ be a non-zero prime ideal of ${{\mathcal O}}$ such that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate to ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$, where $\lambda=\Lambda\cap R$. After conjugating $\bbar\rho_\Lambda$, we may assume that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G) \subseteq {\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$. Let $\ell$ be the rational prime lying under $\Lambda$.
\[T:PSL2 vs GL2 i\] If $k$ is odd, then $\bbar\rho_{\Lambda}^{{\operatorname{proj}}}(G) = {\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ if and only if $\lambda$ splits completely in $L$.
\[T:PSL2 vs GL2 ii\] If $k$ is even and $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is even, then $\bbar\rho_{\Lambda}^{{\operatorname{proj}}}(G) = {\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ if and only if $\lambda$ splits completely in $L$.
\[T:PSL2 vs GL2 iii\] If $k$ is even, $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is odd, and $\ell\nmid N$, then $\bbar\rho_{\Lambda}^{{\operatorname{proj}}}(G) = {\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$.
From Theorem \[T:PSL2 vs GL2\], we see that it is more challenging to produce Galois extensions of ${{\mathbb Q}}$ with Galois group ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^r})$ with odd $r$ if we focus solely on newforms with $k$ even. However, it is still possible to obtain such groups in the excluded case $\ell | N$.
Parts (\[T:PSL2 vs GL2 ii\]) and (\[T:PSL2 vs GL2 iii\]) of Theorem \[T:PSL2 vs GL2\] are included for completeness, see [@MR1879665]\*[Proposition 1.5]{} for an equivalent version in the case $k=2$ due to Dieulefait. Surprisingly, there has been very little attention in the literature given to the case where $k$ is odd (commenting on a preprint of this work, Dieulefait has shared several explicit examples worked out with Tsaknias and Vila). In §\[SS:weight 3 level 27\] and \[SS:weight 3 level 160\], we give examples with $k=3$ and $L=K$ (so $\lambda$ splits in $L$ for any $\lambda$).
An example realizing the groups $\text{PSL}_2({{\mathbb F}}_\ell)$ {#SS:weight 3 level 27}
------------------------------------------------------------------
We now give an example that realizes the simple groups ${\operatorname{PSL}}_2({{\mathbb F}}_\ell)$ as Galois groups of an extension of ${{\mathbb Q}}$ for all primes $\ell\geq 7$. Let $f=\sum_{n=1}^\infty a_n q^n$ be a non-CM newform of weight $3$, level $N=27$ and nebentypus $\varepsilon(a)=\big(\frac{-3}{a}\big)$. We can choose $f$ so that[^1] $$\begin{aligned}
f =q &+ 3iq^2 - 5q^4 - 3iq^5 + 5q^7 - 3iq^8 + 9q^{10} - 15iq^{11} - 10q^{13} + \cdots;\end{aligned}$$ the other possibility for $f$ is its complex conjugate $\sum_n \bbar{a}_n q^n$.
The subfield $E$ of $\CC$ generated by the coefficients $a_n$ is ${{\mathbb Q}}(i)$. Take any prime $p\neq 3$. We will see that $\bbar{a}_p = \varepsilon(p)^{-1} a_p$. Therefore, $a_p$ or $i a_p$ belongs to ${{\mathbb Z}}$ when $\varepsilon(p)$ is $1$ or $-1$, respectively, and hence $r_p=a_p^2/\varepsilon(p)$ is a square in ${{\mathbb Z}}$. Therefore, $L=K={{\mathbb Q}}$.
In §\[SS:3 27\], we shall verify that Theorem \[T:Ribet\] holds with $S=\{2,3,5\}$. Take any prime $\ell \geq 7$ and prime $\Lambda\subseteq{{\mathbb Z}}[i]$ dividing $\ell$. Theorem \[T:PSL2 vs GL2\] with $L=K={{\mathbb Q}}$ implies that $\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is isomorphic to ${\operatorname{PSL}}_2({{\mathbb F}}_\ell)$. The following theorem is now an immediate consequence (it is easy to prove directly for the group ${\operatorname{PSL}}_2({{\mathbb F}}_5)\cong A_5$).
\[T:example Serre\] For each prime $\ell\geq 5$, there is a Galois extension $K/{{\mathbb Q}}$ such that $\operatorname{Gal}(K/{{\mathbb Q}})$ is isomorphic to the simple group ${\operatorname{PSL}}_2({{\mathbb F}}_\ell)$.
In §5.5 of [@MR885783], J-P. Serre describes the image of $\bbar\rho_{(7)}$ and proves that it gives rise to a ${\operatorname{PSL}}_2({{\mathbb F}}_7)$-extension of ${{\mathbb Q}}$, however, he does not consider the image modulo other primes. Note that Serre was actually giving an example of his conjecture, so he started with the ${\operatorname{PSL}}_2({{\mathbb F}}_7)$-extension and then found the newform $f$.
Theorem \[T:example Serre\] was first proved by the author in [@Zywina-PSL2] by considering the Galois action on the second $\ell$-adic étale cohomology of a specific surface. One can show that the Galois extensions of [@Zywina-PSL2] could also be constructed by first starting with an appropriate newform of weight $3$ and level $32$.
Another example {#SS:weight 3 level 160}
---------------
We now give an example with $K\neq {{\mathbb Q}}$. Additional details will be provided in §\[SS:3 160\]. Let $f=\sum_n a_n q^n$ be a non-CM newform of weight $3$, level $N=160$ and nebentypus $\varepsilon(a)=\big(\frac{-5}{a}\big)$.
Take $E$, $K$, $L$, $R$ and ${{\mathcal O}}$ as in §\[SS:general result\]. We will see in §\[SS:3 160\] that $E=K(i)$ and that $K$ is the unique cubic field in ${{\mathbb Q}}(\zeta_{13})$. We will also observe that $L=K$.
Take any [odd]{} prime $\ell$ congruent to $\pm 2$, $\pm 3$, $\pm 4$ or $\pm 6$ modulo $13$. Let $\Lambda$ be any prime ideal of ${{\mathcal O}}$ dividing $\ell$ and set $\lambda= \Lambda \cap R$. The assumption on $\ell$ modulo $13$ implies that $\lambda= \ell R$ and that ${{\mathbb F}}_\lambda \cong {{\mathbb F}}_{\ell^3}$. In §\[SS:3 160\], we shall compute a set $S$ as in Theorem \[T:Ribet\] which does not contain $\lambda$. Theorem \[T:PSL2 vs GL2\] with $L=K$ implies that $\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is isomorphic to ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)\cong {\operatorname{PSL}}_2({{\mathbb F}}_{\ell^3})$. The following is an immediate consequence.
If $\ell$ is an odd prime congruent to $\pm 2$, $\pm 3$, $\pm 4$ or $\pm 6$ modulo $13$, then the simple group ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^3})$ occurs as the Galois group of an extension of ${{\mathbb Q}}$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
Thanks to Henri Darmon for pushing the author to find the modular interpretation of the Galois representations in [@Zywina-PSL2]. Thanks also to Ravi Ramakrishna and Luis Dieulefait for their comments and corrections. Computations were performed with `Magma` [@Magma].
The fields $K$ and $L$ {#S:inner twists}
======================
Take a newform $f$ with notation and assumptions as in §\[SS:general result\].
The field $K$ {#SS:inner twists}
-------------
Let $\Gamma$ be the set of automorphisms $\gamma$ of the field $E$ for which there is a primitive Dirichlet character $\chi_\gamma$ that satisfies $$\label{E:ap twist}
\gamma(a_p) = \chi_\gamma(p) a_p$$ for all primes $p\nmid N$. The set of primes $p$ with $a_p\neq 0$ has density $1$ since $f$ is non-CM, so the image of $\chi_\gamma$ lies in $E^\times$ and the character $\chi_\gamma$ is uniquely determined from $\gamma$.
Define $M$ to be $N$ or $4N$ if $N$ is odd or even, respectively. The conductor of $\chi_\gamma$ divides $M$, cf. [@MR617867]\*[Remark 1.6]{}. Moreover, there is a quadratic Dirichlet character $\alpha$ with conductor dividing $M$ and an integer $i$ such that $\chi_\gamma$ is the primitive character coming from $\alpha \varepsilon^i$, cf. [@MR617867]\*[Lemma 1.5(i)]{}.
For each prime $p\nmid N$, we have $\bbar{a}_p = \varepsilon(p)^{-1} a_p$, cf. [@MR0453647]\*[p. 21]{}, so complex conjugation induces an automorphism $\gamma$ of $E$ and $\chi_\gamma$ is the primitive character coming from $\varepsilon$. In particular, $\Gamma \neq 1$ if $\varepsilon$ is non-trivial.
More generally, we could have instead considered an embedding $\gamma\colon E \to \CC$ and a Dirichlet character $\chi_\gamma$ such that (\[E:ap twist\]) holds for all sufficiently large primes $p$. This gives the same twists, since $\gamma(E)=E$ and the character $\chi_\gamma$ is unramified at primes $p\nmid N$, cf. [@MR617867]\*[Remark 1.3]{}.
The set $\Gamma$ is in fact an abelian subgroup of $\operatorname{Aut}(E)$, cf. [@MR617867]\*[Lemma 1.5(ii)]{}. Denote by $E^{\Gamma}$ the fixed field of $E$ by $\Gamma$.
\[L:fields agree\]
\[L:fields agree i\] We have $K=E^\Gamma$ and hence $\operatorname{Gal}(E/K)=\Gamma$.
\[L:fields agree ii\] There is a prime $p\nmid N$ such that $K={{\mathbb Q}}(r_p)$.
Take any $p\nmid N$. For each $\gamma\in \Gamma$, we have $$\gamma(r_p) = \gamma(a_p^2)/\gamma(\varepsilon(p)) = \chi_\gamma(p)^2 a_p^2/\gamma(\varepsilon(p)) = a_p^2/\varepsilon(p) = r_p,$$ where we have used that $\chi_\gamma(p)^2 = \gamma(\varepsilon(p))/\varepsilon(p)$, cf. [@MR617867]\*[proof of Lemma 1.5(ii)]{}. This shows that $r_p$ belong in $E^\Gamma$ and hence $K\subseteq E^\Gamma$ since $p\nmid N$ was arbitrary. To complete the proof of the lemma, it thus suffices to show that $E^\Gamma={{\mathbb Q}}(r_p)$ for some prime $p\nmid N$.
For $\gamma\in \Gamma$, let $\widetilde\chi_\gamma\colon G \to \CC^\times$ be the continuous character such that $\widetilde\chi_\gamma(\operatorname{Frob}_p)=\chi_\gamma(p)$ for all $p\nmid N$. Define the group $H = \bigcap_{\gamma\in \Gamma} \ker \widetilde\chi_\gamma$; it is an open normal subgroup of $G$ with $G/H$ is abelian. Let ${{\mathcal K}}$ be the subfield of ${{\overline{\mathbb Q}}}$ fixed by $H$; it is a finite abelian extension of ${{\mathbb Q}}$.
Fix a prime $\ell$ and a prime ideal $\Lambda | \ell$ of ${{\mathcal O}}$. In the proof of Theorem 3.1 of [@MR819838], Ribet proved that $E^\Gamma={{\mathbb Q}}(a_v^2)$ for a positive density set of finite place $v\nmid N\ell$ of ${{\mathcal K}}$, where $a_v:=\operatorname{tr}(\rho_\Lambda(\operatorname{Frob}_v))$. There is thus a finite place $v\nmid N\ell$ of ${{\mathcal K}}$ of degree $1$ such that $E^\Gamma={{\mathbb Q}}(a_v^2)$. We have $a_v=a_p$, where $p$ is the rational prime that $v$ divides, so $E^\Gamma={{\mathbb Q}}(a_p^2)$. Since $v$ has degree $1$ and ${{\mathcal K}}/{{\mathbb Q}}$ is abelian, the prime $p$ must split completely in ${{\mathcal K}}$ and hence $\chi_\gamma(p)=1$ for all $\gamma\in \Gamma$; in particular, $\varepsilon(p)=1$. Therefore, $E^\Gamma={{\mathbb Q}}(r_p)$.
The field $L$
-------------
Recall that we defined $L$ to be the extension of $K$ in $\CC$ obtained by adjoining the square root of $r_p = a_p^2/\varepsilon(p)$ for all $p\nmid N$. The following allows one to find a finite set of generators for the extension $L/K$ and gives a way to check the criterion of Theorem \[T:PSL2 vs GL2\].
\[L:L description\]
\[L:L description i\] Choose primes $p_1,\ldots, p_m \nmid N$ that generate the group $({{\mathbb Z}}/M{{\mathbb Z}})^\times$ and satisfy $r_{p_i}\neq 0$ for all $1\leq i\leq m$. Then $L=K(\sqrt{r_{p_1}},\ldots,\sqrt{r_{p_m}})$.
\[L:L description ii\] Take any non-zero prime ideal $\lambda$ of $R$ that does not divide $2$. Let $p_1,\ldots, p_m$ be primes as in (\[L:L description i\]). Then the following are equivalent:
$
\lambda$ splits completely in $L$,
for all $p\nmid N$, $r_p$ is a square in $K_\lambda$,
for all $1\leq i \leq m$, $r_{p_i}$ is a square in $K_\lambda$.
Take any prime $p\nmid N$. To prove part (\[L:L description i\]), it suffices to show that $\sqrt{r_p}$ belongs to the field $L':=K(\sqrt{r_{p_1}},\ldots,\sqrt{r_{p_m}})$. This is obvious if $r_p=0$, so assume that $r_p\neq 0$. Since the $p_i$ generate $({{\mathbb Z}}/M{{\mathbb Z}})^\times$ by assumption, there are integers $e_i\geq 0$ such that $p\equiv p_1^{e_1} \cdots p_m^{e_m} \pmod{M}$. Take any $\gamma\in \Gamma$. Using that the conductor of $\chi_\gamma$ divides $M$ and (\[E:ap twist\]), we have $$\gamma\Big( \frac{a_p}{{\prod}_i a_{p_i}^{e_i}} \Big) =\frac{\chi_\gamma(p)}{\chi_\gamma({\prod}_i p_i^{e_i})} \cdot \frac{a_p}{{\prod}_i a_{p_i}^{e_i}} = \frac{\chi_\gamma(p)}{\chi_\gamma(p)}\cdot \frac{a_p}{{\prod}_i a_{p_i}^{e_i}} = \frac{a_p}{{\prod}_i a_{p_i}^{e_i}},$$ Since $E^\Gamma =K$ by Lemma \[L:fields agree\](\[L:fields agree i\]), the value $a_p/{\prod}_i a_{p_i}^{e_i}$ belongs to $K$; it is non-zero since $r_p\neq 0$ and $r_{p_i}\neq 0$. We have $\varepsilon(p) = \prod_i \varepsilon(p_i)^{e_i}$ since the conductor of $\varepsilon$ divides $M$. Therefore, $$\frac{r_p}{{\prod}_i r_{p_i}^{e_i}} = \frac{a_p^2}{{\prod}_i (a_{p_i}^2)^{e_i}} = \bigg(\frac{a_p}{{\prod}_i a_{p_i}^{e_i}}\bigg)^2 \in (K^\times)^2.$$ This shows that $\sqrt{r_p}$ is contained in $L'$ as desired. This proves (\[L:L description i\]); part (\[L:L description ii\]) is an easy consequence of (\[L:L description i\]).
Finding primes $p_i$ as in Lemma \[L:L description\](\[L:L description i\]) is straightforward since $r_p\neq 0$ for all $p$ outside a set of density $0$ (and the primes representing each class $a\in ({{\mathbb Z}}/M{{\mathbb Z}})^\times$ have positive density). Lemma \[L:L description\](\[L:L description ii\]) gives a straightforward way to check if $\lambda$ splits completely in $L$. Let $e_i$ be the $\lambda$-adic valuation of $r_{p_i}$ and let $\pi$ be a uniformizer of $K_\lambda$; then $r_{p_i}$ is a square in $K_\lambda$ if and only if $e$ is even and the image of $r_{p_i}/\pi^{e_i}$ in ${{\mathbb F}}_\lambda$ is a square.
Proof of Theorem \[T:PSL2 vs GL2\]
==================================
We may assume that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$. For any $n\geq 1$, the group ${\operatorname{GL}}_2({{\mathbb F}}_{2^n})$ is generated by ${\operatorname{SL}}_2({{\mathbb F}}_{2^n})$ and its scalar matrices, so ${\operatorname{PSL}}_2({{\mathbb F}}_{2^n})={\operatorname{PGL}}_2({{\mathbb F}}_{2^n})$. The theorem is thus trivial when $\ell=2$, so we may assume that $\ell$ is odd.
Take any $\alpha\in {\operatorname{PGL}}_2({{\mathbb F}}_\lambda)\subseteq {\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ and choose any matrix $A\in {\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$ whose image in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ is $\alpha$. The value $\operatorname{tr}(A)^2/\det(A)$ does not depend on the choice of $A$ and lies in ${{\mathbb F}}_\lambda$ (since we can choose $A$ in ${\operatorname{GL}}_2({{\mathbb F}}_\lambda)$); by abuse of notation, we denote this common value by $\operatorname{tr}(\alpha)^2/\det(\alpha)$.
\[L:PSL2 p criterion\] Suppose that $p\nmid N\ell$ is a prime for which $r_p \not\equiv 0\pmod{\lambda}$. Then $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p)$ is contained in ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ if and only if the image of $a_p^2/(\varepsilon(p) p^{k-1})=r_p/p^{k-1}$ in ${{\mathbb F}}_\lambda^\times$ is a square.
Define $A:= \bbar\rho_{\Lambda}(\operatorname{Frob}_p)$ and $\alpha:=\bbar\rho_\Lambda(\operatorname{Frob}_p)$; the image of $A$ in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ is $\alpha$. The value $\xi_p:=\operatorname{tr}(\alpha)^2/\det(\alpha) = \operatorname{tr}(A)^2/\det(A)$ agrees with the image of $a_p^2/(\varepsilon(p)p^{k-1})= r_p/p^{k-1}$ in ${{\mathbb F}}_\Lambda$. Since $r_p \in R$ is non-zero modulo $\lambda$ by assumption, the value $\xi_p$ lies in ${{\mathbb F}}_\lambda^\times$. Fix a matrix $A_0 \in {\operatorname{GL}}_2({{\mathbb F}}_\lambda)$ whose image in ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$ is $\alpha$; we have $\xi_p = \operatorname{tr}(A_0)^2/\det(A_0)$. Since $\xi_p \neq 0$, we find that $\xi_p$ and $\det(A_0)$ lies in the same coset in ${{\mathbb F}}_\lambda^\times/({{\mathbb F}}_\lambda^\times)^2$.
The determinant gives rise to a homomorphism $d\colon {\operatorname{PGL}}_2({{\mathbb F}}_\lambda) \to {{\mathbb F}}_\lambda^\times/( {{\mathbb F}}_\lambda^\times)^2$ whose kernel is ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$. Define the character $$\xi \colon G \xrightarrow{\bbar\rho_\Lambda^{{\operatorname{proj}}}} {\operatorname{PGL}}_2({{\mathbb F}}_\lambda) \xrightarrow{d} {{\mathbb F}}_\lambda^\times/({{\mathbb F}}_\lambda^\times)^2.$$ We have $\xi(\operatorname{Frob}_p) = \det(A_0) \cdot ( {{\mathbb F}}_\lambda^\times)^2 = \xi_p \cdot ( {{\mathbb F}}_\lambda^\times)^2$. So $\xi(\operatorname{Frob}_p)=1$, equivalently $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p) \in {\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$, if and only if $\xi_p \in {{\mathbb F}}_\lambda^\times$ is a square.
Let $M$ be the integer from §\[SS:inner twists\].
\[L:rp progressions\] For each $a\in ({{\mathbb Z}}/M\ell{{\mathbb Z}})^\times$, there is a prime $p\equiv a \pmod{M\ell}$ such that $r_p \not\equiv 0 \pmod{\lambda}$.
Set $H= \bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$; it is ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$ by assumption. Let $H'$ be the commutator subgroup of $H$. We claim that for each coset $\kappa$ of $H'$ in $H$, there exists an $\alpha\in \kappa$ with $\operatorname{tr}(\alpha)^2/\det(\alpha)\neq 0$. If $H'={\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$, then the claim is easy; note that for any $t\in {{\mathbb F}}_\lambda$ and $d\in {{\mathbb F}}_\lambda^\times$, there is a matrix in ${\operatorname{GL}}_2({{\mathbb F}}_\lambda)$ with trace $t$ and determinant $d$. When $\#{{\mathbb F}}_\lambda\neq 3$, the group ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ is non-abelian and simple, so $H'={\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$. When $\#{{\mathbb F}}_\lambda=3$ and $H={\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$, we have $H'={\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$. It thus suffices to prove the claim in the case where ${{\mathbb F}}_\lambda={{\mathbb F}}_3$ and $H={\operatorname{PSL}}_2({{\mathbb F}}_3)$. In this case, $H'$ is the unique subgroup of $H$ of index $3$ and the cosets of $H/H'$ are represented by $\left(\begin{smallmatrix} 1 & b \\ 0 & 1 \end{smallmatrix}\right)$ with $b\in {{\mathbb F}}_3$. The claim is now immediate in this remaining case.
Let $\chi \colon G\twoheadrightarrow ({{\mathbb Z}}/M\ell{{\mathbb Z}})^\times$ be the cyclotomic character that satisfies $\chi(\operatorname{Frob}_p) \equiv p \pmod{M\ell}$ for all $p\nmid M\ell$. The set $\bbar\rho_\Lambda(\chi^{-1}(a))$ is thus the union of cosets of $H'$ in $H$. By the claim above, there exists an $\alpha\in \bbar\rho_\Lambda^{{\operatorname{proj}}}(\chi^{-1}(a))$ with $\operatorname{tr}(\alpha)^2/\det(\alpha)\neq 0$. By the Chebotarev density theorem, there is a prime $p\nmid M\ell$ satisfying $p\equiv a \pmod{M\ell}$ and $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p)= \alpha$. The lemma follows since $r_p/p^{k-1}$ modulo $\lambda$ agrees with $\operatorname{tr}(\alpha)^2/\det(\alpha)\neq 0$.
**Case 1:** *Assume that $k$ is odd or $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is even.*
First suppose that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)={\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$. By Lemma \[L:rp progressions\], there are primes $p_1,\ldots, p_m \nmid N\ell$ that generate the group $({{\mathbb Z}}/M{{\mathbb Z}})^\times$ and satisfy $r_{p_i}\not\equiv 0 \pmod{\lambda}$ for all $1\leq i\leq m$. By Lemma \[L:PSL2 p criterion\] and the assumption $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)={\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$, the image of $r_{p_i}/{p_i}^{k-1}$ in ${{\mathbb F}}_\lambda$ is a non-zero square for all $1\leq i \leq m$. For each $1\leq i \leq m$, the assumption that $k$ is odd or $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is even implies that $p_i^{k-1}$ is a square in ${{\mathbb F}}_\lambda$ and hence the image of $r_{p_i}$ in ${{\mathbb F}}_\lambda$ is a non-zero square. Since $\lambda \nmid 2$, we deduce that each $r_{p_i}$ is a square in $K_\lambda$. By Lemma \[L:L description\](\[L:L description ii\]), the prime $\lambda$ splits completely in $L$.
Now suppose that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)={\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$. There exists an element $\alpha \in {\operatorname{PGL}}_2({{\mathbb F}}_\lambda) - {\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ with $\operatorname{tr}(\alpha)^2/\det(\alpha) \neq 0$. By the Chebotarev density theorem, there is a prime $p\nmid N\ell$ such that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p) = \alpha$. We have $r_p \equiv \operatorname{tr}(\alpha)^2/\det(\alpha) \not\equiv 0\pmod{\lambda}$. Since $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p) \notin {\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$, Lemma \[L:PSL2 p criterion\] implies that the image of $r_p/p^{k-1}$ in ${{\mathbb F}}_\lambda$ is not a square. Since $k$ is odd or $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is even, the image of $r_p$ in ${{\mathbb F}}_\lambda$ is not a square. Therefore, $r_p$ is not a square in $K_\lambda$. By Lemma \[L:L description\](\[L:L description ii\]), we deduce that $\lambda$ does not split completely in $L$.\
**Case 2:** *Assume that $k$ is even, $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is odd, and $\ell\nmid N$.*
Since $\ell\nmid N$, there is an integer $a\in {{\mathbb Z}}$ such that $a\equiv 1 \pmod{M}$ and $a$ is not a square modulo $\ell$. By Lemma \[L:rp progressions\], there is a prime $p\equiv a \pmod{M\ell}$ such that $r_p\not\equiv 0 \pmod{\lambda}$.
We claim that $a_p\in R$ and $\varepsilon(p)=1$. With notation as in §\[SS:inner twists\], take any $\gamma\in \Gamma$. Since the conductor of $\chi_\gamma$ divides $M$ and $p\equiv 1 \pmod{M}$, we have $\gamma(a_p)=\chi_\gamma(p) a_p =a_p$. Since $\gamma\in \Gamma$ was arbitrary, we have $a_p \in K$ by Lemma \[L:fields agree\]. Therefore, $a_p\in R$ since it is an algebraic integer. We have $\varepsilon(p)=1$ since $p\equiv 1\pmod{N}$.
Since $a_p\in R$ and $r_p\not\equiv 0 \pmod{\lambda}$, the image of $a_p^2$ in ${{\mathbb F}}_\lambda$ is a non-zero square. Since $k$ is even, $p^k$ is a square in ${{\mathbb F}}_\lambda$. Since $p$ is not a square modulo $\ell$ and $[{{\mathbb F}}_\lambda:{{\mathbb F}}_\ell]$ is odd, the prime $p$ is not a square in ${{\mathbb F}}_\lambda$. So the image of $$a_p^2/(\varepsilon(p)p^{k-1}) = p\cdot a_p^2/p^k$$ in ${{\mathbb F}}_\lambda$ is not a square. Lemma \[L:PSL2 p criterion\] implies that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p) \notin {\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$. Therefore, $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)={\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$.
An effective version of Theorem \[T:Ribet\] {#S:effective Ribet}
===========================================
Take a newform $f$ with notation and assumptions as in §\[SS:general result\]. Let $\lambda$ be a non-zero prime ideal of $R$ and let $\ell$ be the prime lying under $\lambda$. Let $k_\lambda$ be the subfield of ${{\mathbb F}}_\lambda$ generated by the image of $r_p$ modulo $\lambda$ with primes $p\nmid N\ell$. Take any prime ideal $\Lambda$ of ${{\mathcal O}}$ that divides $\lambda$.
In this section, we describe how to compute an explicit finite set $S$ of prime ideals of $R$ as in Theorem \[T:Ribet\]. First some simple definitions:
- Let ${{\mathbb F}}$ be an extension of ${{\mathbb F}}_\Lambda$ of degree $\gcd(2,\ell)$.
- Let $e_0=0$ if $\ell\geq k-1$ and $\ell\nmid N$, and $e_0=\ell-2$ otherwise.
- Let $e_1=0$ if $N$ is odd, and $e_1=1$ otherwise.
- Let $e_2=0$ if $\ell\geq 2k$, and $e_2=1$ otherwise.
- Define ${{\mathcal M}}=4^{e_1} \ell^{e_2}\prod_{p|N} p$.
We will prove the following in §\[S:proof theorem criterion\].
\[T:criteria\] Suppose that all the following conditions hold:
\[P:criteria a\] For every integer $0\leq j \leq e_0$ and character $\chi\colon ({{\mathbb Z}}/N{{\mathbb Z}})^\times \to {{\mathbb F}}^\times$, there is a prime $p\nmid N\ell$ such that $\chi(p)p^j \in {{\mathbb F}}$ is not a root of the polynomial $x^2 - a_px + \varepsilon(p)p^{k-1} \in {{\mathbb F}}_\Lambda[x]$.
\[P:criteria b\] For every non-trivial character $\chi \colon ({{\mathbb Z}}/{{\mathcal M}}{{\mathbb Z}})^\times \to \{\pm 1\}$, there is a prime $p\nmid N\ell$ such that $\chi(p)=-1$ and $r_p \not\equiv 0 \pmod{\lambda}$.
\[P:criteria c\] If $\#k_\lambda \notin \{4,5\}$, then at least one of the following hold:
- $\ell > 5k-4$ and $\ell \nmid N$,
- $\ell \equiv 0, \pm 1 \pmod{5}$ and $\#k_\lambda\neq\ell$,
- $\ell \equiv \pm 2\pmod{5}$ and $\#k_\lambda\neq \ell^2$,
- there is a prime $p\nmid N\ell$ such that the image of $a_{p}^2/(\varepsilon(p) p^{k-1})$ in ${{\mathbb F}}_\lambda$ is not equal to $0$, $1$ and $4$, and is not a root of $x^2-3x+1$.
\[P:criteria d\] If $\#k_\lambda \notin \{3,5,7\}$, then at least one of the following hold:
- $\ell > 4k-3$ and $\ell \nmid N$,
- $\#k_\lambda\neq \ell$,
- there is a prime $p\nmid N\ell$ such that the image of $a_{p}^2/(\varepsilon(p) p^{k-1})$ in ${{\mathbb F}}_\lambda$ is not equal to $0$, $1$, $2$ and $4$.
\[P:criteria e\] If $\#k_\lambda \in \{5,7\}$, then for every non-trivial character $\chi \colon ({{\mathbb Z}}/ 4^{e_1}\ell N{{\mathbb Z}})^\times\to \{\pm 1\}$ there is a prime $p\nmid N\ell$ such that $\chi(p)=1$ and $a_{p}^2/(\varepsilon(p) p^{k-1})\equiv 2 \pmod{\lambda}$.
Then the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2(k_\lambda)$ or ${\operatorname{PGL}}_2(k_\lambda)$.
Note that the above conditions simplify greatly if one also assumes that $\ell\nmid N$ and $\ell>5k-4$.
Though we will not prove it, Theorem \[T:criteria\] has been stated so that all the conditions (\[P:criteria a\])–(\[P:criteria e\]) hold if and only if $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate to ${\operatorname{PSL}}_2(k_\lambda)$ or ${\operatorname{PGL}}_2(k_\lambda)$. In particular, after considering enough primes $p$, one will obtain the minimal set $S$ of Theorem \[T:Ribet\] (one could use an effective version of Chebotarev density to make this a legitimate algorithm).
Let us now describe how to compute a set of exceptional primes as in Theorem \[T:Ribet\]. Define $M=N$ if $N$ is odd and $M=4N$ otherwise. Set ${{\mathcal M}}':=4^{e_1} \prod_{p|N} p$. We first choose some primes:
- Let $q_1,\ldots, q_n$ be primes congruent to $1$ modulo $N$.
- Let $p_1,\ldots, p_m \nmid N$ be primes with $r_{p_i}\neq 0$ such that for every non-trivial character $\chi \colon ({{\mathbb Z}}/{{\mathcal M}}' {{\mathbb Z}})^\times \to \{\pm 1\}$, we have $\chi(p_i)=-1$ for some $1\leq i \leq m$.
- Let $p_0 \nmid N$ be a prime such that ${{\mathbb Q}}(r_{p_0})=K$.
That such primes $p_1,\ldots, p_m$ exist is clear since the set of primes $p$ with $r_p\neq 0$ has density $1$. That such a prime $q$ exists follows from Lemma \[L:fields agree\] (the set of such $q$ actually has positive density). Define the ring $R':={{\mathbb Z}}[a_{q}^2/\varepsilon(q)]$; it is an order in $R$.
Define $S$ to be the set of non-zero primes $\lambda$ of $R$, dividing a rational prime $\ell$, that satisfy one of the following conditions:
- $\ell \leq 5k-4$ or $\ell\leq 7$,
- $\ell | N$,
- for all $1\leq i \leq n$, we have $\ell=q_i$ or $r_{q_i} \equiv (1+q_i^{k-1})^2 \pmod{\lambda}$,
- for some $1\leq i \leq m$, we have $\ell=p_i$ or $r_{p_i} \equiv 0 \pmod{\lambda}$,
- $\ell=q$ or $\ell$ divides $[R:R']$.
Note that the set $S$ is *finite* (the only part that is not immediate is that $r_{q_i} - (1+q_i^{k-1})^2\neq 0$; this follows from Deligne’s bound $|r_{q_i}|=|a_{q_i}| \leq 2q_i^{(k-1)/2}$ and $k>1$). The following is our effective version of Theorem \[T:Ribet\].
\[T:effective version\] Take any non-zero prime ideal $\lambda \notin S$ of $R$ and let $\Lambda$ be any prime of ${{\mathcal O}}$ dividing $\lambda$. Then the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to either ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$.
Let $\ell$ be the rational prime lying under $\lambda$. We shall verify the conditions of Theorem \[T:criteria\].
We first show that condition (\[P:criteria a\]) of Theorem \[T:criteria\] holds. Take any integer $0\leq j \leq e_0$ and character $\chi\colon ({{\mathbb Z}}/N{{\mathbb Z}})^\times \to {{\mathbb F}}^\times={{\mathbb F}}_\Lambda^\times$. We have $\ell>5k-4>k-1$ and $\ell \nmid N$ since $\lambda \notin S$, so $e_0=0$ and hence $j=0$. Take any $1\leq i \leq n$. Since $q_i\equiv 1\pmod{N}$ and $j=0$, we have $\chi(q_i)q_i^j=1$ and $\varepsilon(q_i)=1$. Since $\lambda \notin S$, we also have $q_i \nmid N \ell$ ($q_i\nmid N$ is immediate from the congruence imposed on $q_i$). If $\chi(q_i)q_i^j=1$ was a root of $x^2-a_{q_i}x+\varepsilon(q_i)q_i^{k-1}$ in ${{\mathbb F}}_\Lambda[x]$, then we would have $a_{q_i} \equiv 1 + q_i^{k-1} \pmod{\Lambda}$; squaring and using that $\varepsilon(q_i)=1$, we deduce that $r_{q_i} \equiv (1+q_i^{k-1})^2 \pmod{\lambda}$. Since $\lambda\notin S$, we have $r_{q_i} \not \equiv (1+q_i^{k-1})^2 \pmod{\lambda}$ for some $1\leq i \leq n$ and hence $\chi(q_i)q_i^j$ is not a root of $x^2-a_{q_i}x+\varepsilon(q_i)q_i^{k-1}$.
We now show that condition (\[P:criteria b\]) of Theorem \[T:criteria\] holds. We have $e_2=0$ since $\lambda\notin S$, and hence ${{\mathcal M}}'={{\mathcal M}}$. Take any non-trivial character $\chi \colon ({{\mathbb Z}}/{{\mathcal M}}{{\mathbb Z}})^\times \to \{\pm 1\}$. By our choice of primes $p_1,\ldots,p_m$, we have $\chi(p_i)=-1$ for some $1\leq i \leq m$. The prime $p_i$ does not divide $N\ell$ (that $p_i\neq \ell$ follows since $\lambda\notin S$). Since $\lambda\notin S$, we have $r_{p_i}\not\equiv 0 \pmod{\lambda}$.
Since $\lambda\notin S$, the prime $\ell\nmid N$ is greater that $7$, $4k-3$ and $5k-4$. Conditions (\[P:criteria c\]), (\[P:criteria d\]) and (\[P:criteria e\]) of Theorem \[T:criteria\] all hold.
Theorem \[T:criteria\] now implies that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to either ${\operatorname{PSL}}_2(k_\lambda)$ or ${\operatorname{PGL}}_2(k_\lambda)$. It remains to prove that $k_\lambda={{\mathbb F}}_\lambda$. We have $q\neq \ell$ since $\lambda \notin S$. The image of the reduction map $R' \to {{\mathbb F}}_\lambda$ thus lies in $k_\lambda$. We have $\ell\nmid [R:R']$ since $\lambda\notin S$, so the map $R'\to {{\mathbb F}}_\lambda$ is surjective. Therefore, $k_\lambda={{\mathbb F}}_\lambda$.
Proof of Theorem \[T:criteria\] {#S:proof theorem criterion}
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Some group theory
-----------------
Fix a prime $\ell$ and an integer $r\geq 1$.
A of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$ is a subgroup conjugate to the subgroup of upper triangular matrices.
A of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$ is a subgroup conjugate to the subgroup of diagonal matrices. A of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$ is a subgroup that is cyclic of order $(\ell^r)^2-1$. Fix a Cartan subgroup ${{\mathcal C}}$ of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$. Let ${{\mathcal N}}$ be the normalizer of ${{\mathcal C}}$ in ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$. One can show that $[{{\mathcal N}}:{{\mathcal C}}] = 2$ and that $\operatorname{tr}(g)=0$ and $g^2$ is scalar for all $g\in {{\mathcal N}}-{{\mathcal C}}$.
\[L:subgroups\] Fix a prime $\ell$ and an integer $r\geq 1$. Let $G$ be a subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$ and let $\bbar{G}$ be its image in ${\operatorname{PGL}}_2({{\mathbb F}}_{\ell^r})$. Then at least one of the following hold:
1. $G$ is contained in a Borel subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$,
2. $G$ is contained in the normalizer of a Cartan subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$,
3. $\bbar{G}$ is isomorphic to $\mathfrak{A}_4$,
4. $\bbar{G}$ is isomorphic to $\mathfrak{S}_4$,
5. $\bbar{G}$ is isomorphic to $\mathfrak{A}_5$,
6. $\bbar G$ is conjugate to ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell^s})$ or ${\operatorname{PGL}}_2({{\mathbb F}}_{\ell^s})$ for some integer $s$ dividing $r$.
This can be deduced directly from a theorem of Dickson, cf. [@MR0224703]\*[Satz 8.27]{}, which will give the finite subgroups of ${\operatorname{PSL}}_2({\overline{{{\mathbb F}}}}_\ell) = {\operatorname{PGL}}_2({\overline{{{\mathbb F}}}}_\ell)$. The finite subgroups of ${\operatorname{PGL}}_2({{\mathbb F}}_{\ell^r})$ have been worked out in [@Xander].
\[L:easy image PGL\] Fix a prime $\ell$ and an integer $r\geq1$. Take a matrix $A\in {\operatorname{GL}}_2({{\mathbb F}}_{\ell^r})$ and let $m$ be its order in ${\operatorname{PGL}}_2({{\mathbb F}}_{\ell^r})$.
Suppose that $\ell \nmid m$. If $m$ is $1$, $2$, $3$ or $4$, then $\operatorname{tr}(A)^2/\det(A)$ is $4$, $0$, $1$ or $2$, respectively. If $m=5$, then $\operatorname{tr}(A)^2/\det(A)$ is a root of $x^2-3x+1$.
If $\ell | m$, then $\operatorname{tr}(A)^2/\det(A) = 4$.
The quantity $\operatorname{tr}(A)^2/\det(A)$ does not change if we replace $A$ by a scalar multiple or by a conjugate in ${\operatorname{GL}}_2({\overline{{{\mathbb F}}}}_\ell)$. If $\ell\nmid m$, then we may thus assume that $A=\left(\begin{smallmatrix} \zeta & 0 \\ 0 & 1 \end{smallmatrix}\right)$ where $\zeta\in {\overline{{{\mathbb F}}}}_\ell$ has order $m$. We have $\operatorname{tr}(A)^2/\det(A) = \zeta+\zeta^{-1} + 2$, which is $4$, $0$, $1$ or $2$ when $m$ is $1$, $2$, $3$ or $4$, respectively. If $m=5$, then $ \zeta+\zeta^{-1} + 2$ is a root of $x^2-3x+1$. If $\ell| m$, then after conjugating and scaling, we may assume that $A=\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ and hence $\operatorname{tr}(A)^2/\det(A) =4$.
Image of inertia at $\ell$
--------------------------
Fix an inertia subgroup ${{\mathcal I}}_\ell$ of $G=\operatorname{Gal}({{\overline{\mathbb Q}}}/{{\mathbb Q}})$ for the prime $\ell$; it is uniquely defined up to conjugacy. The following gives important information concerning the representation $\bbar\rho_\Lambda|_{{{\mathcal I}}_\ell}$ for large $\ell$. Let $\chi_\ell \colon G\twoheadrightarrow {{\mathbb F}}_\ell^\times$ be the character such that for each prime $p\nmid \ell$, $\chi_\ell$ is unramified at $p$ and $\chi_\ell(\operatorname{Frob}_p)\equiv p \pmod{\ell}$.
\[L:inertia\] Fix a prime $\ell \geq k-1$ for which $\ell\nmid 2N$. Let $\Lambda$ be a prime ideal of ${{\mathcal O}}$ dividing $\ell$ and set $\lambda=\Lambda\cap R$.
\[L:inertia i\] Suppose that $r_\ell \not\equiv 0 \pmod{\lambda}$. After conjugating $\bbar\rho_\Lambda$ by a matrix in ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$, we have $$\bbar\rho_\Lambda|_{{{\mathcal I}}_\ell} = \left(\begin{matrix} \chi_\ell^{k-1}|_{{{\mathcal I}}_\ell} & * \\ 0 & 1 \end{matrix}\right)$$ In particular, $\bbar\rho_\Lambda^{{\operatorname{proj}}}({{\mathcal I}}_\ell)$ contains a cyclic group of order $(\ell-1)/\gcd(\ell-1,k-1)$.
\[L:inertia ii\] Suppose that $r_\ell \equiv 0 \pmod{\lambda}$. Then $\bbar\rho_\Lambda|_{{{\mathcal I}}_\ell}$ is absolutely irreducible and $\bbar\rho_\Lambda({{\mathcal I}}_\ell)$ is cyclic. Furthermore, the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}({{\mathcal I}}_\ell)$ is cyclic of order $(\ell+1)/\gcd(\ell+1,k-1)$.
Parts (\[L:inertia i\]) and (\[L:inertia ii\]) follow from Theorems 2.5 and Theorem 2.6, respectively, of [@MR1176206]; they are theorems of Deligne and Fontaine, respectively. We have used that $r_\ell=a_\ell^2/\varepsilon(\ell) \in R$ is congruent to $0$ modulo $\lambda$ if and only if $a_\ell\in {{\mathcal O}}$ is congruent to $0$ modulo $\Lambda$.
Borel case {#SS:Borel case 0}
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Suppose that $\bbar\rho_\Lambda(G)$ is a reducible subgroup of ${\operatorname{GL}}_2({{\mathbb F}})$. There are thus characters $\psi_1,\psi_2 \colon G \to {{\mathbb F}}^\times$ such that after conjugating the ${{\mathbb F}}$-representation $G \xrightarrow{\bbar\rho_\Lambda} {\operatorname{GL}}_2({{\mathbb F}}_\Lambda) \subseteq {\operatorname{GL}}_2({{\mathbb F}})$, we have $$\bbar\rho_\Lambda = \left(\begin{matrix} \psi_1 & * \\ 0 & \psi_2 \end{matrix}\right).$$ The characters $\psi_1$ and $\psi_2$ are unramified at each prime $p\nmid N\ell$ since $\bbar\rho_\Lambda$ is unramified at such primes.
\[L:unramified LCFT\] For each $i\in \{1,2\}$, there is a unique integer $0\leq m_i < \ell-1$ such that $\psi_i\chi_\ell^{-m_i} \colon G \to {{\mathbb F}}^\times$ is unramified at all primes $p\nmid N$. If $\ell \geq k-1$ and $\ell\nmid N$, then $m_1$ or $m_2$ is $0$.
The existence and uniqueness of $m_i$ is an easy consequence of class field theory for ${{\mathbb Q}}_\ell$. A choice of embedding ${{\overline{\mathbb Q}}}\subseteq {{\overline{\mathbb Q}}}_\ell$ induces an injective homomorphism $G_{{{\mathbb Q}}_\ell}:=\operatorname{Gal}({{\overline{\mathbb Q}}}_\ell/{{\mathbb Q}}_\ell) \hookrightarrow G$. Let ${{\mathbb Q}}_\ell^{{\operatorname{ab}}}$ be the maximal abelian extension of ${{\mathbb Q}}_\ell$ in ${{\overline{\mathbb Q}}}_\ell$. Restricting $\psi_i$ to $G_{{{\mathbb Q}}_\ell}$, we obtain a representation $\psi_i\colon G_{{{\mathbb Q}}_\ell}^{{\operatorname{ab}}}:=\operatorname{Gal}({{\mathbb Q}}_\ell^{{\operatorname{ab}}}/{{\mathbb Q}}_\ell) \to {{\mathbb F}}^\times$. By local class field, the inertia subgroup ${{\mathcal I}}$ of $G_{{{\mathbb Q}}_\ell}^{{\operatorname{ab}}}$ is isomorphic to ${{\mathbb Z}}_\ell^\times$. Since $\ell$ does not divide the cardinality of ${{\mathbb F}}^\times$, we find that $\psi_i|_{{{\mathcal I}}}$ factors through a group isomorphic to ${{\mathbb F}}_\ell^\times$. The character $\psi_i|_{{{\mathcal I}}}$ must agree with a power of $\chi_\ell|_{{{\mathcal I}}}$ since $\chi_\ell\colon G_{{{\mathbb Q}}_\ell} \to {{\mathbb F}}_\ell^\times$ satisfies $\chi_\ell({{\mathcal I}})={{\mathbb F}}_\ell^\times$ and ${{\mathbb F}}_\ell^\times$ is cyclic.
The second part of the lemma follows immediately from Lemma \[L:inertia\].
Take any $i\in \{1,2\}$. By Lemma \[L:unramified LCFT\], there is a unique $0\leq m_i < \ell-1$ such that the character $$\tilde{\psi}_i:=\psi_i\chi_\ell^{-m_i}\colon G\to {{\mathbb F}}^\times$$ is unramified at $\ell$ and at all primes $p\nmid N$. There is a character $\chi_i \colon ({{\mathbb Z}}/N_i{{\mathbb Z}})^\times \to {{\mathbb F}}^\times$ with $N_i\geq 1$ dividing some power of $N$ and $\ell\nmid N_i$ such that $\tilde\psi_i(\operatorname{Frob}_p) = \chi_i(p)$ for all $p\nmid N\ell$. We may assume that $\chi_i$ is taken so that $N_i$ is minimal.
\[L:Ni divides N\] The integer $N_i$ divides $N$.
We first recall the notion of an Artin conductor. Consider a representation $\rho\colon G\to \operatorname{Aut}_{{{\mathbb F}}}(V)$, where $V$ is a finite dimensional ${{\mathbb F}}$-vector space. Take any prime $p\neq \ell$. A choice of embedding ${{\overline{\mathbb Q}}}\subseteq {{\overline{\mathbb Q}}}_p$ induces an injective homomorphism $\operatorname{Gal}({{\overline{\mathbb Q}}}_p/{{\mathbb Q}}_p) \hookrightarrow G$. Choose any finite Galois extension $L/{{\mathbb Q}}_p$ for which $\rho(\operatorname{Gal}({{\overline{\mathbb Q}}}_p/L))=\{I\}$. For each $i\geq 0$, let $H_i$ be the $i$-th ramification subgroup of $\operatorname{Gal}(L/{{\mathbb Q}}_p)$ with respect to the lower numbering. Define the integer $$f_p(\rho)= \sum_{i\geq 0} [H_0:H_i]^{-1}\cdot \dim_{{{\mathbb F}}} V/V^{H_i}.$$ The of $\rho$ is the integer $N(\rho):=\prod_{p\neq \ell} p^{f_p(\rho)}$.
Using that the character $\tilde\psi_i\colon G \to {{\mathbb F}}^\times$ is unramified at $\ell$, one can verify that $N(\tilde\psi_i)=N_i$. Consider our representation $\bbar\rho_\Lambda \colon G \to {\operatorname{GL}}_2({{\mathbb F}})$. For a fixed prime $p\neq \ell$, take $L$ and $H_i$ as above. The semisimplification of $\bbar\rho_\Lambda$ is $V_1\oplus V_2$, where $V_i$ is the one dimensional representation given by $\psi_i$. We have $f_p(\psi_1)+f_p(\psi_2) \leq f_p(\bbar\rho_\Lambda )$ since $\dim_{{{\mathbb F}}} V^{H_i} \leq \dim_{{{\mathbb F}}} V_1^{H_i} + \dim_{{{\mathbb F}}} V_2^{H_i}$. By using this for all $p\neq \ell$, we deduce that $N(\psi_1) N(\psi_2)=N_1 N_2$ divides $N(\bbar\rho_\Lambda)$. The lemma follows since $N(\bbar\rho_\Lambda)$ divides $N$, cf. [@MR987567]\*[Prop. 0.1]{}.
Fix an $i\in \{1,2\}$; if $\ell \geq k-1$ and $\ell\nmid N$, then we may suppose that $m_i=0$ by Lemma \[L:unramified LCFT\]. Since the conductor of $\chi_i$ divides $N$ by Lemma \[L:Ni divides N\], assumption (\[P:criteria a\]) implies that there is a prime $p\nmid N\ell$ for which $\chi_i(p) p^{m_i} \in {{\mathbb F}}$ is not a root of $x^2-a_p x + \varepsilon(p)p^{k-1} \in {{\mathbb F}}[x]$. However, this is a contradiction since $$\chi_i(p) p^{m_i} = \tilde\psi_i(\operatorname{Frob}_p) \chi_\ell(\operatorname{Frob}_p)^{m_i} = \psi_i(\operatorname{Frob}_p)$$ is a root of $x^2-a_p x + \varepsilon(p)p^{k-1}$.
Therefore, the ${{\mathbb F}}$-representation $\bbar\rho_\Lambda$ is irreducible. In particular, $\bbar\rho_\Lambda(G)$ is not contained in a Borel subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$.
Cartan case
-----------
\[L:rule out non-split Cartan\] The group $\bbar\rho_\Lambda(G)$ is not contained in a Cartan subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$.
Suppose that $\bbar\rho_\Lambda(G)$ is contained in a Cartan subgroup ${{\mathcal C}}$ of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$. If $\ell=2$, then ${{\mathcal C}}$ is reducible as a subgroup of ${\operatorname{GL}}_2({{\mathbb F}})$ since ${{\mathbb F}}/{{\mathbb F}}_\Lambda$ is a quadratic extension. However, we saw in §\[SS:Borel case 0\] that $\bbar\rho_\Lambda(G) \subseteq {{\mathcal C}}$ is an irreducible subgroup of ${\operatorname{GL}}_2({{\mathbb F}})$. Therefore, $\ell$ is odd. If ${{\mathcal C}}$ is split, then $\bbar\rho_\Lambda(G)$ is a reducible subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$. This was ruled out in §\[SS:Borel case 0\], so ${{\mathcal C}}$ must be a non-split Cartan subgroup with $\ell$ odd.
Recall that the representation $\bbar\rho_\Lambda$ is *odd*, i.e., if $c\in G$ is an element corresponding to complex conjugation under some embedding ${{\overline{\mathbb Q}}}\hookrightarrow \CC$, then $\det(\bbar\rho_\Lambda(c))=-1$. Therefore, $\bbar\rho_\Lambda(c)$ has order $2$ and determinant $-1\neq 1$ (this last inequality uses that $\ell$ is odd). A non-split Cartan subgroup ${{\mathcal C}}$ of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$ is cyclic and hence $-I$ is the unique element of ${{\mathcal C}}$ of order $2$. Since $\det(-I)= 1$, we find that $\bbar\rho_\Lambda(c)$ does not belong to ${{\mathcal C}}$; this gives the desired contradiction.
Normalizer of a Cartan case {#SS:Cartan case}
---------------------------
Suppose that $\bbar\rho_\Lambda(G)$ is contained in the normalizer ${{\mathcal N}}$ of a Cartan subgroup ${{\mathcal C}}$ of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$. The group ${{\mathcal C}}$ has index $2$ in ${{\mathcal N}}$, so we obtain a character $$\beta_\Lambda\colon G \xrightarrow{\bbar\rho_\Lambda} {{\mathcal N}}\to {{\mathcal N}}/{{\mathcal C}}\cong \{\pm 1\}.$$ The character $\beta_\Lambda$ is non-trivial since $\bbar\rho_\Lambda(G) \not\subseteq {{\mathcal C}}$ by Lemma \[L:rule out non-split Cartan\].
\[L:Cartan character\] The character $\beta_\Lambda$ is unramified at all primes $p\nmid N\ell$. If $\ell \geq 2k$ and $\ell \nmid N$, then the character $\beta_\Lambda$ is also unramified at $\ell$.
The character $\beta_\Lambda$ is unramified at each prime $p\nmid N\ell$ since $\bbar\rho_\Lambda$ is unramified at such primes. Now suppose that $\ell \geq 2k$ and $\ell\nmid N$. We have $\ell>2$, so $\ell \nmid |{{\mathcal N}}|$ and hence Lemma \[L:inertia\] implies that $\bbar\rho_\Lambda({{\mathcal I}}_\ell)$ is cyclic. Moreover, Lemma \[L:inertia\] implies that $\bbar\rho_\Lambda^{{\operatorname{proj}}}({{\mathcal I}}_\ell)$ is cyclic of order $d\geq (\ell-1)/(k-1)$. Our assumption $\ell\geq 2k$ ensures that $d>2$.
Now take a generator $g$ of $\bbar\rho_\Lambda({{\mathcal I}}_\ell)$. Suppose that $\beta_\Lambda$ is ramified at $\ell$ and hence $g$ belongs to ${{\mathcal N}}-{{\mathcal C}}$. The condition $g\in {{\mathcal N}}-{{\mathcal C}}$ implies that $g^2$ is a scalar matrix and hence $\bbar\rho_\Lambda^{{\operatorname{proj}}}({{\mathcal I}}_\ell)$ is a group of order $1$ or $2$. This contradicts $d>2$, so $\beta_\Lambda$ is unramified at $\ell$.
Let $\chi$ be the primitive Dirichlet character that satisfies $\beta_\Lambda(\operatorname{Frob}_p)=\chi(p)$ for all primes $p\nmid N\ell$. Since $\beta_\Lambda$ is a quadratic character, Lemma \[L:Cartan character\] implies that the conductor of $\chi$ divides ${{\mathcal M}}$. The character $\chi$ is non-trivial since $\beta_\Lambda$ is non-trivial. Assumption (\[P:criteria b\]) implies that there is a prime $p\nmid N\ell$ satisfying $\chi(p)=-1$ and $r_p \not\equiv 0 \pmod{\lambda}$. We thus have $g\in{{\mathcal N}}-{{\mathcal C}}$ and $\operatorname{tr}(g)\neq 0$, where $g:=\bbar\rho_{\Lambda}(\operatorname{Frob}_p) \in {{\mathcal N}}$. However, this contradicts that $\operatorname{tr}(A)=0$ for all $A\in {{\mathcal N}}-{{\mathcal C}}$.
Therefore, the image of $\bbar\rho_\Lambda$ does not lie in the normalizer of a Cartan subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$.
$\mathfrak{A}_5$ case {#SS:A5 case}
---------------------
Assume that $\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is isomorphic to $\mathfrak{A}_5$ with $\#k_\lambda \notin \{4,5\}$.
The image of $r_p/p^{k-1}= a_p^2/(\varepsilon(p)p^{k-1})$ in ${{\mathbb F}}_\lambda$ is equal to $\operatorname{tr}(A)^2/\det(A)$ with $A=\bbar\rho_\Lambda(\operatorname{Frob}_p)$. Every element of $\mathfrak{A}_5$ has order $1$, $2$, $3$ or $5$, so Lemma \[L:easy image PGL\] implies that the image of $r_p/p^{k-1}$ in ${{\mathbb F}}_\lambda$ is $0$, $1$, $4$ or is a root of $x^2-3x+1$ for all $p\nmid N\ell$. If $\lambda | 5$, then Lemma \[L:easy image PGL\] implies that $k_\lambda={{\mathbb F}}_5$ which is excluded by our assumption on $k_\lambda$. So $\lambda \nmid 5$ and Lemma \[L:easy image PGL\] ensures that $k_\lambda$ is the splitting field of $x^2-3x+1$ over ${{\mathbb F}}_\ell$. So $k_\lambda$ is ${{\mathbb F}}_\ell$ if $\ell \equiv \pm 1 \pmod{5}$ and ${{\mathbb F}}_{\ell^2}$ if $\ell \equiv \pm 2 \pmod{5}$.
From assumption (\[P:criteria c\]), we find that $\ell > 5k-4$ and $\ell \nmid N$. By Lemma \[L:inertia\], the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ contains an element of order at least $(\ell-1)/(k-1) > ((5k-4)-1)/(k-1) = 5$. This is a contradiction since $\mathfrak{A}_5$ has no elements with order greater than $5$.
$\mathfrak{A}_4$ and $\mathfrak{S}_4$ cases {#SS:S4 case}
-------------------------------------------
Suppose that $\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is isomorphic to $\mathfrak{A}_4$ or $\mathfrak{S}_4$ with $\#k_\lambda \neq 3$.
First suppose that $\#k_\lambda \notin \{5,7\}$. The image of $r_p/p^{k-1}= a_p^2/(\varepsilon(p)p^{k-1})$ in ${{\mathbb F}}_\lambda$ is equal to $\operatorname{tr}(A)^2/\det(A)$ with $A=\bbar\rho_\Lambda(\operatorname{Frob}_p)$. Since every element of $\mathfrak{S}_4$ has order at most $4$, Lemma \[L:easy image PGL\] implies that $r_p/p^{k-1}$ is congruent to $0$, $1$, $2$ or $4$ modulo $\lambda$ for all primes $p\nmid N\ell$. In particular, $k_\lambda={{\mathbb F}}_\ell$. By assumption (\[P:criteria d\]), we must have $\ell>4k-3$ and $\ell\nmid N$. By Lemma \[L:inertia\], the group $\bbar\rho_\Lambda(G)$ contains an element of order at least $(\ell-1)/(k-1) > ((4k-3)-1)/(k-1) = 4$. This is a contradiction since $\mathfrak{S}_4$ has no elements with order greater than $4$.
Now suppose that $\#k_\lambda\in \{5,7\}$. By assumption (\[P:criteria e\]), with any $\chi$, there is a prime $p\nmid N\ell$ such that $a_p^2/(\varepsilon(p)p^{k-1}) \equiv 2 \pmod{\lambda}$. The element $g:=\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p)$ has order $1$, $2$, $3$ or $4$. By Lemma \[L:easy image PGL\], we deduce that $g$ has order $4$. Since $\mathfrak{A}_4$ has no elements of order $4$, we deduce that $H:=\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is isomorphic to $\mathfrak{S}_4$. Let $H'$ be the unique index $2$ subgroup of $H$; it is isomorphic to $\mathfrak{A}_4$. Define the character $$\beta \colon G \xrightarrow{\bbar\rho_\Lambda^{{\operatorname{proj}}}} H \to H/H' \cong \{\pm 1\}.$$ The quadratic character $\beta$ corresponds to a Dirichlet character $\chi$ whose conductor divides $4^e \ell N$. By assumption (\[P:criteria e\]), there is a prime $p\nmid N\ell$ such that $\chi(p)=1$ and $a_p^2/(\varepsilon(p)p^{k-1})\equiv 2 \pmod{\lambda}$. Since $\beta(\operatorname{Frob}_p)=\chi(p)=1$, we have $\bbar\rho_\Lambda^{{\operatorname{proj}}}(\operatorname{Frob}_p) \in H'$. Since $H'\cong \mathfrak{A}_4$, Lemma \[L:easy image PGL\] implies that the image of $a_p^2/(\varepsilon(p)p^{k-1})$ in ${{\mathbb F}}_\lambda$ is $0$, $1$ or $4$. This contradicts $a_p^2/(\varepsilon(p)p^{k-1})\equiv 2 \pmod{\lambda}$.
Therefore, the image of $\bbar\rho_\Lambda^{{\operatorname{proj}}}$ is not isomorphic to either of the groups $\mathfrak{A}_4$ or $\mathfrak{S}_4$.
End of proof
------------
In §\[SS:Borel case 0\], we saw that $\bbar\rho_\Lambda(G)$ is not contained in a Borel subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$. In §\[SS:Cartan case\], we saw that $\bbar\rho_\Lambda(G)$ is not contained in the normalizer of a Cartan subgroup of ${\operatorname{GL}}_2({{\mathbb F}}_\Lambda)$.
In §\[SS:A5 case\], we showed that if $\#k_\lambda\notin \{4,5\}$, then $\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is not isomorphic to $\mathfrak{A}_5$. We want to exclude the cases $\#k_\lambda \in \{4,5\}$ since ${\operatorname{PSL}}_2({{\mathbb F}}_4)$ and ${\operatorname{PSL}}_2({{\mathbb F}}_5)$ are both isomorphic to $\mathfrak{A}_5$.
In §\[SS:S4 case\], we showed that if $\#k_\lambda\neq 3$, then $\bbar\rho^{{\operatorname{proj}}}_\Lambda(G)$ is not isomorphic to $\mathfrak{A}_4$ and not isomorphic to $\mathfrak{S}_4$. We want to exclude the case $\#k_\lambda=3$ since ${\operatorname{PSL}}_2({{\mathbb F}}_3)$ and ${\operatorname{PGL}}_2({{\mathbb F}}_3)$ are isomorphic to $\mathfrak{A}_4$ and $\mathfrak{S}_4$, respectively.
By Lemma \[L:subgroups\], the group $\bbar\rho_\Lambda^{{{\operatorname{proj}}}}(G)$ must be conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2({{\mathbb F}}')$ or ${\operatorname{PGL}}_2({{\mathbb F}}')$, where ${{\mathbb F}}'$ is a subfield of ${{\mathbb F}}_\Lambda$. One can then show that ${{\mathbb F}}'$ is the subfield of ${{\mathbb F}}_\Lambda$ generated by the set $\{ \operatorname{tr}(A)^2/\det(A) : A \in \bbar\rho_\Lambda(G)\}$. By the Chebotarev density theorem, the field ${{\mathbb F}}'$ is the subfield of ${{\mathbb F}}_\Lambda$ generated by the images of $a_p^2/(\varepsilon(p) p^{k-1})=r_p/p^{k-1}$ with $p\nmid N\ell$. Therefore, ${{\mathbb F}}'=k_\lambda$ and hence $\bbar\rho_\Lambda^{{{\operatorname{proj}}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2(k_\lambda)$ or ${\operatorname{PGL}}_2(k_\lambda)$.
Examples
========
Example from §\[SS:weight 3 level 27\] {#SS:3 27}
--------------------------------------
Let $f$ be the newform from §\[SS:weight 3 level 27\]. We have $E={{\mathbb Q}}(i)$. We know that $\Gamma\neq 1$ since $\varepsilon$ is non-trivial. Therefore, $\Gamma=\operatorname{Gal}({{\mathbb Q}}(i)/{{\mathbb Q}})$ and $K=E^{\Gamma}$ equals ${{\mathbb Q}}$. So $\Gamma$ is generated by complex conjugation and we have $\bbar{a}_p = \varepsilon(p)^{-1} a_p$ for $p\nmid N$. As noted in §\[SS:weight 3 level 27\], this implies that $r_p$ is a square in ${{\mathbb Z}}$ for all $p\nmid N$ and hence $L$ equals $K={{\mathbb Q}}$. Fix a prime $\ell=\lambda$ and a prime ideal $\Lambda | \ell$ of ${{\mathcal O}}={{\mathbb Z}}[i]$.
Set $q_1=109$ and $q_2=379$; they are primes that are congruent to $1$ modulo $27$. Set $p_1=5$, we have $\chi(p_1)=-1$, where $\chi$ is the unique non-trivial character $({{\mathbb Z}}/3{{\mathbb Z}})^\times \to \{\pm 1\}$. Set $q=5$; the field ${{\mathbb Q}}(r_q)$ equals $K={{\mathbb Q}}$ and hence ${{\mathbb Z}}[r_{q}]={{\mathbb Z}}$.
One can verify that $a_{109}=164$, $a_{379}=704$ and $a_5=-3i$, so $r_{109}=164^2$, $r_{379}=704^2$ and $r_5=3^2$. We have $$\label{E:disc fact}
r_{109} - (1+109^2)^2 = -2^2\cdot 3^3\cdot 7\cdot 19\cdot 31\cdot 317 \quad \text{ and }\quad
r_{379} - (1+379^2)^2 = -2^2 \cdot 3^3\cdot 2647 \cdot 72173.$$ So if $\ell\geq 11$, then there is an $i\in \{1,2\}$ such that $\ell \neq q_i$ and $r_{q_i}\not\equiv (1+q_i^2)^2\pmod{\ell}$.
Let $S$ be the set from §\[S:effective Ribet\] with the above choice of $q_1$, $q_2$, $p_1$ and $q$. We find that $S = \{2,3,5,7,11\}$. Theorem \[T:effective version\] implies that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2({{\mathbb F}}_\ell)$ when $\ell>11$.\
Now take $\ell \in \{7,11\}$. Choose a prime ideal $\Lambda$ of ${{\mathcal O}}$ dividing $\ell$. We have $e_0=e_1=e_2=0$ and ${{\mathcal M}}=3$. The subfield $k_\ell$ generated over ${{\mathbb F}}_\ell$ by the images of $r_p$ modulo $\ell$ with $p\nmid N\ell$ is of course ${{\mathbb F}}_\ell$ (since the $r_p$ are rational integers). We now verify the conditions of Theorem \[T:criteria\].
We first check condition (\[P:criteria a\]). Suppose there is a character $\chi\colon ({{\mathbb Z}}/27{{\mathbb Z}})^\times \to {{\mathbb F}}_\ell^\times$ such that $\chi(q_2)$ is a root of $x^2-a_{q_2} x + \varepsilon(q_2) q_2^2$ modulo $\ell$. Since $q_2\equiv 1\pmod{27}$ and $a_{q_2}=704$, we find that $1$ is a root of $x^2-a_{q_2} x + q_2^2 \in {{\mathbb F}}_\ell[x]$. Therefore, $a_{q_2} \equiv 1+q_2^2 \pmod{\ell}$ and hence $r_{q_2}^2 = a_{q_2}^2 \equiv (1+ q_2^2)^2 \pmod{\ell}$. Since $\ell \in\{7,11\}$, this contradicts (\[E:disc fact\]). This proves that condition (\[P:criteria a\]) holds.
We now check condition (\[P:criteria b\]). Let $\chi\colon ({{\mathbb Z}}/3{{\mathbb Z}})^\times \to \{\pm 1\}$ be the non-trivial character. We have $\chi(5)=-1$ and $r_5 = 9 \not\equiv 0 \pmod{\ell}$. Therefore, (\[P:criteria b\]) holds.
We now check condition (\[P:criteria c\]). If $\ell=7$, we have $\ell \equiv 2 \pmod{5}$ and $\#k_\ell = \ell \neq \ell^2$, so condition (\[P:criteria c\]) holds. Take $\ell=11$. We have $a_5^2/(\varepsilon(5) 5^2) = 9/5^2 \equiv 3 \pmod{11}$, which verifies (\[P:criteria c\]).
Condition (\[P:criteria d\]) holds since $\#k_\ell = 5$ if $\ell = 7$, and $\ell>4k-3=9$ and $\ell\nmid N$ if $\ell=11$.
Finally we explain why condition (\[P:criteria e\]) holds when $\ell=7$. Let $\chi\colon ({{\mathbb Z}}/7\cdot 27 {{\mathbb Z}})^\times \to\{\pm 1\}$ be any non-trivial character. A quick computation shows that there is a prime $p\in \{13,37,41\}$ such that $\chi(p)=1$ and that $a_p^2/(\varepsilon(p) p^2) \equiv 2 \pmod{7}$.
Theorem \[T:criteria\] implies that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2({{\mathbb F}}_\ell)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\ell)$. Since $L=K$, the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ isomorphic to ${\operatorname{PSL}}_2({{\mathbb F}}_{\ell})$ by Theorem \[T:PSL2 vs GL2\](\[T:PSL2 vs GL2 i\]).
Example from §\[SS:weight 3 level 160\] {#SS:3 160}
---------------------------------------
Let $f$ be a newform as in §\[SS:weight 3 level 160\]; we have $k=3$ and $N=160$. The `Magma` code below verifies that $f$ is uniquely determined up to replacing by a quadratic twist and then a Galois conjugate. So the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$, up to isomorphism, does not depend on the choice of $f$.
eps:=\[c: c in Elements(DirichletGroup(160)) | Order(c) eq 2 and Conductor(c) eq 20\]\[1\]; M:=ModularSymbols(eps,3); F:=NewformDecomposition(NewSubspace(CuspidalSubspace(M))); assert \#F eq 2; \_,chi:=IsTwist(F\[1\],F\[2\],5); assert Order(chi) eq 2;
Define $b = \zeta_{13}^1 + \zeta_{13}^5+\zeta_{13}^8 +\zeta_{13}^{12}$, where $\zeta_{13}$ is a primitive $13$-th root of unity in ${{\overline{\mathbb Q}}}$ (note that $\{1,5,8,12\}$ is the unique index $3$ subgroup of ${{\mathbb F}}_{13}^\times$). The characteristic polynomial of $b$ is $x^3 + x^2 - 4x + 1$ and hence ${{\mathbb Q}}(b)$ is the unique cubic extension of ${{\mathbb Q}}$ in ${{\mathbb Q}}(\zeta_{13})$. The code below shows that the coefficient field $E$ is equal to ${{\mathbb Q}}(b,i)$ (it is a degree $6$ extension of ${{\mathbb Q}}$ that contains roots of $x^3 + x^2 - 4x + 1$ and $x^2+1$).
f:=qEigenform(F\[1\],2001); a:=\[Coefficient(f,n): n in \[1..2000\]\]; E:=AbsoluteField(Parent(a\[1\])); Pol<x>:=PolynomialRing(E); assert Degree(E) eq 6 and HasRoot(x\^3+x\^2-4\*x+1) and HasRoot(x\^2+1);
Fix notation as in §\[SS:inner twists\]. We have $\Gamma\neq 1$ since $\varepsilon$ is non-trivial. The character $\chi_\gamma^2$ is trivial for $\gamma \in \Gamma$ (since $\chi_\gamma$ is always a quadratic character times some power of $\varepsilon$). Therefore, $\Gamma$ is a $2$-group. The field $K=E^\Gamma$ is thus ${{\mathbb Q}}(b)$ which is the unique cubic extension of ${{\mathbb Q}}$ in $E$. Therefore, $r_p = a_p^2/\varepsilon(p)$ lies in $K={{\mathbb Q}}(b)$ for all $p\nmid N$.
The code below verifies that $r_3$, $r_7$ and $r_{11}$ are squares in $K$ that do not lie in ${{\mathbb Q}}$ (and in particular, are non-zero). Since $3$, $7$ and $11$ generate the group $({{\mathbb Z}}/40{{\mathbb Z}})^\times$, we deduce from Lemma \[L:L description\] that the field $L=K(\{\sqrt{r_p}:p\nmid N\})$ is equal to $K$.
\_,b:=HasRoot(x\^3+x\^2-4\*x+1); K:=sub<E|b>; r3:=K!(a\[3\]\^2/eps(3)); r7:=K!(a\[7\]\^2/eps(7)); r11:=K!(a\[11\]\^2/eps(11)); assert IsSquare(r3) and IsSquare(r7) and IsSquare(r11); assert r3 notin Rationals() and r7 notin Rationals() and r11 notin Rationals();
Let $N_{K/{{\mathbb Q}}}\colon K\to {{\mathbb Q}}$ be the norm map. The following code verifies that $N_{K/{{\mathbb Q}}}(r_3) = 2^6$, $N_{K/{{\mathbb Q}}}(r_7) = 2^6$, $N_{K/{{\mathbb Q}}}(r_{11}) = 2^{12}5^4$, $N_{K/{{\mathbb Q}}}(r_{13})=2^{12} 13^2$, $N_{K/{{\mathbb Q}}}(r_{17}) = 2^{18}5^2$, and that $$\label{E:end gcd}
\gcd\Big( 641\cdot N_{K/{{\mathbb Q}}}(r_{641}-(1+642^2)^2), \, 1061\cdot N_{K/{{\mathbb Q}}}(r_{1061}-(1+1061^2)^2)\Big) = 2^{12}.\\$$
r13:=K!(a\[13\]\^2/eps(13)); r17:=K!(a\[17\]\^2/eps(17)); assert Norm(r3) eq 2\^6 and Norm(r7) eq 2\^6 and Norm(r11) eq 2\^12\*5\^4; assert Norm(r13) eq 2\^12\*13\^2 and Norm(r17) eq 2\^18\*5\^2; r641:=K!(a\[641\]\^2/eps(641)); r1061:=K!(a\[1061\]\^2/eps(1061)); n1:=Integers()!Norm(r641-(1+641\^2)\^2); n2:=Integers()!Norm(r1061-(1+1061\^2)\^2); assert GCD(641\*n1,1061\*n2) eq 2\^12;
Set $q_1=641$ and $q_2=1061$; they are primes congruent to $1$ modulo $160$. Let $\lambda$ be a prime ideal of $R$ dividing a rational prime $\ell>3$. From (\[E:end gcd\]), we find that $\ell\neq q_i$ and $r_{q_i} \not\equiv(1+q_i^2)^2 \pmod{\lambda}$ for some $i\in \{1,2\}$ (otherwise $\lambda$ would divide $2$).
Set $p_1=3$, $p_2= 7$ and $p_3=11$. For each non-trivial quadratic characters $\chi\colon ({{\mathbb Z}}/40 {{\mathbb Z}})^\times \to \{\pm 1\}$, we have $\chi(p_i)=-1$ for some prime $i\in\{1,2,3\}$ (since $3$, $7$ and $11$ generate the group $({{\mathbb Z}}/40{{\mathbb Z}})^\times$). From the computed values of $N_{K/{{\mathbb Q}}}(r_p)$, we find that $r_{p_i} \not\equiv 0 \pmod{\lambda}$ for all $i\in \{1,2,3\}$ and all non-zero prime ideals $\lambda \nmid N$ of $R$.
Set $q=3$. We have noted that $r_{q}\in K-{{\mathbb Q}}$, so $K={{\mathbb Q}}(r_{q})$. The index of the order ${{\mathbb Z}}[r_{q}]$ in $R$ is a power of $2$ since $N_{K/{{\mathbb Q}}}(q)$ is a power of $2$.
Let $S$ be the set from §\[S:effective Ribet\] with the above choice of $q_1$, $q_2$, $p_1$, $p_2$, $p_3$ and $q$. The above computations show that $S$ consists of the prime ideals $\lambda$ of $R$ that divide a prime $\ell\leq 11$.\
Now let $\ell$ be an odd prime congruent to $\pm 2$, $\pm 3$, $\pm 4$ or $\pm 6$ modulo $13$. Since $K$ is the unique cubic extension in ${{\mathbb Q}}(\zeta_{13})$, we find that the ideal $\lambda:= \ell R$ is prime in $R$ and that ${{\mathbb F}}_\lambda\cong{{\mathbb F}}_{\ell^3}$. The above computations show that $\lambda \notin S$ when $\ell \notin \{3,7,11\}$. Theorem \[T:effective version\] implies that if $\ell\notin \{3,7,11\}$, then $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$, where $\Lambda$ is a prime ideal of ${{\mathcal O}}$ dividing $\lambda$. So if $\ell \notin \{3,7,11\}$, the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ isomorphic to ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)\cong {\operatorname{PSL}}_2({{\mathbb F}}_{\ell^3})$ by Theorem \[T:PSL2 vs GL2\](\[T:PSL2 vs GL2 i\]) and the equality $L=K$.\
Now take $\lambda = \ell R$ with $\ell \in \{3,7,11\}$; it is a prime ideal. Choose a prime ideal $\Lambda$ of ${{\mathcal O}}$ dividing $\lambda$. We noted above that ${{\mathbb Z}}[r_3]$ is an order in $R$ with index a power of $2$; the same argument shows that this also holds for the order ${{\mathbb Z}}[r_7]$. Therefore, the field $k_\lambda$ generated over ${{\mathbb F}}_\ell$ by the images of $r_p$ modulo $\lambda$ with $p\nmid N\ell$ is equal to ${{\mathbb F}}_\lambda$. Since $\#{{\mathbb F}}_\lambda =\ell^3$, we find that conditions (\[P:criteria c\]), (\[P:criteria d\]) and (\[P:criteria e\]) of Theorem \[T:criteria\] hold.
We now show that condition (\[P:criteria a\]) of Theorem \[T:criteria\] holds for our fixed $\Lambda$. We have $e_0=0$, so take any character $\chi\colon ({{\mathbb Z}}/N{{\mathbb Z}})^\times \to {{\mathbb F}}_\Lambda^\times$. We claim that $\chi(q_i)\in {{\mathbb F}}_\Lambda$ is not a root of $x^2-a_{q_i} x + \varepsilon(q_i) q_i^2$ for some $i\in \{1,2\}$. Since $q_i\equiv 1\pmod{N}$, the claim is equivalent to showing that $a_{q_i} \not\equiv 1 + q_i^2 \pmod{\Lambda}$ for some $i\in \{1,2\}$. So we need to prove that $r_{q_i} \equiv (1 + q_i^2)^2 \pmod{\lambda}$ for some $i\in \{1,2\}$; this is clear since otherwise $\ell$ divides the quantity (\[E:end gcd\]). This completes our verification of (\[P:criteria a\]).
We now show that condition (\[P:criteria b\]) of Theorem \[T:criteria\] holds. We have $r_p \not\equiv 0 \pmod{\lambda}$ for all primes $p\in \{3,7,11,13,17\}$; this is a consequence of $N_{K/{{\mathbb Q}}}(r_p)\not\equiv 0 \pmod{\ell}$. We have ${{\mathcal M}}= 120$ if $\ell=3$ and ${{\mathcal M}}=40$ otherwise. Condition (\[P:criteria b\]) holds since $({{\mathbb Z}}/{{\mathcal M}}{{\mathbb Z}})^\times$ is generated by the primes $p\in \{3,7,11,13,17\}$ for which $p\nmid {{\mathcal M}}\ell$.
Theorem \[T:criteria\] implies that $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ is conjugate in ${\operatorname{PGL}}_2({{\mathbb F}}_\Lambda)$ to ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)$ or ${\operatorname{PGL}}_2({{\mathbb F}}_\lambda)$. Since $L=K$, the group $\bbar\rho_\Lambda^{{\operatorname{proj}}}(G)$ isomorphic to ${\operatorname{PSL}}_2({{\mathbb F}}_\lambda)\cong {\operatorname{PSL}}_2({{\mathbb F}}_{\ell^3})$ by Theorem \[T:PSL2 vs GL2\](\[T:PSL2 vs GL2 i\]).
[^1]: More explicitly, take $f=\tfrac{i}{2} g \theta_0 - \tfrac{1+i}{2} g \theta_1 + \tfrac{3}{2} g \theta_2$, where $g:= q \prod_{n \geq 1} (1-q^{3n})^2 (1-q^{9n})^2$ and $\theta_j:= \sum_{x,y \in {{\mathbb Z}}} q^{3^j(x^2+xy+y^2)}$, cf. [@MR885783]\*[p. 228]{}.
|
---
author:
- 'T. Karalidi[^1]'
- 'D. M. Stam'
- 'D. Guirado'
date: 'Received 18 March 2013 / Accepted 22 May 2013'
title: |
Flux and polarization signals of\
spatially inhomogeneous gaseous exoplanets
---
Introduction
============
Since the discovery of the first exoplanet orbiting a main sequence star by @mayorqueloz95, more than 850 exoplanets have been detected up to today. The refinement of the detection methods and the instrumentation, such as the highly successful space missions CoRoT (COnvection, ROtation & planetary Transits) [@baglin06; @magali10] and Kepler [@koch98], and ground-based telescope instruments like HARPS (High Accuracy Radial Velocity Planet Searcher) [@pepe04] have led to an almost exponential increase of the number of detected planets per year.
The next step of exoplanet research is the characterization of detected exoplanets: what is the composition and structure of their atmospheres, and, for rocky exoplanets, their surface? In the near future, instruments like SPHERE (Spectro–Polarimetric High–Contrast Exoplanet Research) [@dohlen08; @roelfsema11] on the VLT (Very Large Telescope) and GPI (Gemini Planet Imager) [@macintosh08] on the Gemini North telescope, and more in the future, EPICS (Exoplanet Imaging Camera and Spectrograph) [@kasper10] on the E-ELT, will characterize gaseous exoplanets in relatively wide orbits around their stars, and possibly super-Earths around nearby stars, using combinations of spectroscopy and broadband polarimetry.
In exoplanet research, polarimetry helps to detect a planet because the direct stellar light is usually unpolarized [see @kemp87], while the starlight that has been reflected by a planet will usually be (linearly) polarized because it has been scattered by atmospheric particles and/or it has been reflected by the planetary surface (if present). Polarimetry helps to confirm the detection of an exoplanet, because background sources will usually be unpolarized, or have a direction of polarization that excludes a relation with the star. A first detection of the polarization signal of an exoplanet was claimed by @berdyugina08 [@berdyugina11]. The power of polarization in characterizing planetary atmospheres and surfaces has been demonstrated through observations of Solar System planets (including Earth itself)[see, for example @hansenhovenier74; @hansentravis74; @mishchenko90; @tomasko09]. For gaseous and terrestrial–type exoplanets numerical calculations have clearly shown the added information on planetary characteristics that can be derived from polarimetry [e. g. @stam03; @stamhovenier04; @saar03; @seager00; @stam08; @karalidi11].
The numerical studies mentioned above pertain to planetary model atmospheres that are vertically inhomogeneous, but horizontally homogeneous. Partly, this has to do with the computational effort: fully including all orders of scattering and polarization in radiative transfer calculations requires orders more computing time than the scalar radiative transfer calculations that are commonly used to model reflected fluxes (note that ignoring polarization in radiative transfer calculations introduces errors in calculated fluxes, see e.g. @stam05 and references therein).
In this paper, we present numerically calculated total flux and polarization signals of unpolarized incident starlight that is reflected by gaseous exoplanets that are both vertically and horizontally inhomogeneous. For the horizontal inhomogeneities, we use banded structures similar to the belts and zones that circle Jupiter and Saturn, cyclonic spots such as the long lived Great Red Spot on Jupiter and the Great Dark Spot that Voyager–2 observed on Neptune, and polar hazes such as those covering the north and south poles of Jupiter and Saturn. In particular the latter hazes are known to strongly polarize the incident sunlight, as observed at small phase angles from Earth [@schmid11], and at intermediate phase angles from spacecraft observations [see e.g. @west83; @smith84].
For exoplanets, observations will yield disk–integrated flux and polarization signals. As we will explore in this paper, local horizontal inhomogeneities might reveal themselves through temporal variations in flux and/or polarization signals when planets rotate with respect to the observer, or because the flux and/or polarization signals of the horizontally inhomogeneous planets deviate from those of horizontally homogeneous planets. The dependence of a total flux signal on the composition and structure of a planetary atmosphere is different than that of a polarization signal. Therefore, a horizontally inhomogeneous planet’s total flux and polarization phase functions could be similar to those of different horizontally homogeneous planets. Such differences could be used to detect spatial features like belts and zones, cyclonic spots, and/or polar hazes.
This paper is organized as follows. In Sect. \[section2\], we describe polarized light, our radiative transfer algorithm, and the model planetary atmospheres. In Sect. \[sec:ss\_props\], we present the single scattering properties of the cloud and haze particles in the model atmospheres. Section \[sec:gaseousplanets\] shows the calculated flux and polarization signals of different types of spatially inhomogeneous model planets: with zones and belts (Sect. \[sec:zonesbelts\]), cyclonic spots (Sect. \[sec:cyclonicspots\]), and polar hazes (Sect. \[sec:polarhazes\]). Section \[sect\_summary\], finally, contains a summary and our conclusions.
Description of the numerical simulations {#section2}
========================================
Definitions of flux and polarization
------------------------------------
Starlight that has been reflected by a planet can be described by a flux vector $\pi\vec{F}$, as follows $$\label{eq:first}
\pi\vec{F}= \pi\left[\begin{array}{c}
F \\ Q \\ U \\ V \end{array}\right],$$ where parameter $\pi F$ is the total flux, parameters $\pi Q$ and $\pi
U$ describe the linearly polarized flux and parameter $\pi V$ the circularly polarized flux [see e.g. @hansentravis74; @hovenier04]. Although not explicitly shown in Eq. \[eq:first\], all four parameters depend on the wavelength $\lambda$, and their dimensions are W m$^{-2}$m$^{-1}$. Parameters $\pi Q$ and $\pi U$ are defined with respect to a reference plane, for which we chose the planetary scattering plane, i.e. the plane through the centers of the star, the planet and the observer. Parameter $\pi V$ of starlight that is reflected by a planet is generally small [@hansentravis74] and we will ignore it in our simulations. This can be done without introducing significant errors in our calculated fluxes $\pi F$, $\pi Q$ and $\pi U$ [@stam05]. The degree of (linear) polarization $P$ of flux vector $\pi \vec{F}$ is defined as follows $$P=\frac{\sqrt{Q^{2}+U^{2}}}{F},
\label{eq:poldef}$$ which is independent of the choice of reference plane.
Unless stated otherwise, we assume that the starlight that is incident on a model planet is unpolarized [@kemp87] and that the model planets are mirror-symmetric with respect to the planetary scattering plane. In that case, $\pi U$ equals zero, and we can use an alternative definition for the degree of polarization, namely $$P_\mathrm{s}= - \frac{Q}{F},
\label{eq:polsign}$$ with the subscript [*s*]{} referring to [*signed*]{}. For $P_\mathrm{s} > 0$ ($< 0$), the reflected light is polarized perpendicular (parallel) to the reference plane.
We will present calculated fluxes that are normalized such that at a planetary phase angle $\alpha$ equal to 0$^\circ$ (i.e. seen from the middle of the planet, the angle between the star and the observer equals 0$^\circ$), the total reflected flux $\pi F$ equals the planet’s geometric albedo $A_\mathrm{G}$ [see e.g. @stamhovenier04]. We will indicate the hence normalized total flux by $\pi F_\mathrm{n}$ and the associated linearly polarized fluxes by $\pi Q_\mathrm{n}$ and $\pi U_\mathrm{n}$. The normalized fluxes that we present in this paper can straightforwardly be scaled to absolute fluxes of a particular planetary system by multiplying them with $r^2/d^2$, where $r$ is the spherical planet’s radius and $d$ the distance between the planet and the observer, and with the stellar flux that is incident on the planet. In our calculations, we furthermore assume that the distance between the star and the planet is large enough to assume that the incident starlight is uni-directional. Since the degree of polarization $P$ (or $P_\mathrm{s}$) is a relative measure, it doesn’t require any scaling.
Our calculations cover phase angles $\alpha$ from 0$^\circ$ to 180$^\circ$. Of course, the range of phase angles an exoplanet exhibits as it orbits its star, depends on the orbital inclination angle. Given an orbital inclination angle $i$ (in degrees), one can observe the exoplanet at phase angles ranging from $90^\circ - i$ to $90^\circ + i$. Thus, an exoplanet in a face–on orbit ($i=0^\circ$) would always be observed at a phase angle equal to 90$^\circ$, while the phase angles of an exoplanet in an edge–on orbit ($i=90^\circ$) range from 0$^\circ$ to 180$^\circ$, the complete range that is shown in this paper. Note that the actual range of phase angles an exoplanet can be observed at will depend strongly on the observational technique that is used, and e.g. on the angular distance between a star and its planet.
Our radiative transfer code
---------------------------
Our radiative transfer code to calculate the total and polarized fluxes that are reflected by model planets is based on the efficient adding–doubling algorithm described by @dehaan87. It fully includes single and multiple scattering and polarization, and assumes that locally, the planetary atmosphere is plane–parallel. We will use a version of the code that applies to horizontally homogeneous planets [@stamhovenier04; @stam06; @stam08], and a (more computing-time-consuming) version that applies to horizontally inhomogeneous planets [@karalidi12b]. In the latter code, a model planet is divided into pixels that are small enough to be considered horizontally homogeneous. Reflected stellar fluxes are then calculated for all pixels that are both illuminated and visible to the observer and then summed up to acquire the disk–integrated total and polarized reflected fluxes. Since the adding–doubling code uses the local meridian plane (which contains both the local zenith direction and the propagation direction of the reflected light) as the reference plane, we have to rotate locally calculated flux vectors to the planetary scattering plane before summing them up. Following @karalidi12b, we divide our model planets into pixels of $2^\circ\times2^\circ$ (latitude $\times$ longitude).
Our model planets {#sect:modelplanets}
-----------------
The atmospheres of our model planets consist of homogeneous, plane–parallel layers that contain gases and, optionally, clouds or hazes. Here, we use the term ’haze’ for optically thin layers of submicron–sized particles, while ’clouds’ are thicker and composed of larger particles. The model atmospheres are bounded below by black surfaces, i.e. no light is entering the atmospheres from below. The ambient atmospheric temperature and pressure profiles are representative for midlatitudes on Jupiter [see @stamhovenier04]. Given the temperatures and pressures across an atmospheric layer, and the wavelength $\lambda$, the gaseous scattering optical thickness of each atmospheric layer is calculated according to @stam99, using a depolarization factor that is representative for hydrogen–gas, namely 0.02 [see @hansentravis74]. At $\lambda=0.55$ $\mu$m, the total gaseous scattering optical thickness of our model atmosphere is 5.41. We ignore absorption by methane, and choose wavelengths in the continuum for our calculations. In particular, when broad band filters are being used for the observations, the contribution of reflected flux from continuum wavelengths will contribute most to the measured signal.
The physical properties of the clouds and hazes across a planet like Jupiter vary in time [for an overview, see e.g. @west04]. Here, we use a simple atmosphere model that suffices to show the effects of clouds and hazes on the flux and polarization signals of Jupiter–like exoplanets. Our model atmospheres have an optically thick tropospheric cloud layer that is composed of ammonia ice particles (their properties are presented in Sect. \[sec:ss\_props\]). The bottom of this cloud layer is at a pressure of 1.0 bar. We vary the top of the cloud between 0.1 and 0.5 bars. The cloud top pressure of 0.1 bars is representative for the so–called zonal bands on Jupiter. In the zones, the clouds typically rise up higher into the atmosphere than in the adjacent belts where the cloud top pressures can be up to a few hundred mbar higher [see @ingersoll04]. We set the optical thickness of the clouds in the zones at 21 (at 0.75 $\mu$m) and in the belts it varies from 20 to 6.02 as the cloud top pressure varies from 0.1 bar to 0.5 bar. On Jupiter, the clouds are overlaid by a stratospheric, photochemically produced haze layer. The haze layers over in particular both polar regions, provide strong polarization signals indicating that they consist of small aggregated particles [@west91]. To avoid introducing too many variables, we only use haze layers over the polar regions of our model planets.
We will present results for horizontally homogeneous model planets and for model planets with bands of clouds divided into zones and belts that run parallel to the equator, which lies in the planet’s equatorial plane. Our banded model planets are mirror–symmetric: measured from the equator in either the northern or the southern direction, we chose the latitudes that bound the belts and zones as follows: 0$^\circ$–8$^\circ$ (zone), 8$^\circ$–24$^\circ$ (belt), 24$^\circ$–40$^\circ$ (zone), 40$^\circ$–60$^\circ$ (belt), 60$^\circ$–90$^\circ$ (zone). These latitudes correspond roughly to the most prominent cloud bands of Jupiter [see e.g. @depaterlissauer02]. The northern and southern polar hazes extend upward, respectively downward, from a latitude of 60$^\circ$. Vertically these hazes extend between $\sim$0.0075 bar and $\sim$0.0056 bar, and we give them an optical thickness of 0.2 at 0.55 $\mu$m. The single scattering albedo of the haze particles is 0.995 at 0.55 $\mu$m.
Single scattering properties of the cloud and haze particles {#sec:ss_props}
============================================================
The tropospheric cloud particles {#sec:nh3ice}
--------------------------------
Thermodynamic models of the jovian atmosphere indicate that the upper tropospheric cloud layers should consist of ammonia ice particles [see for example @sato79; @simonmiller01; @depaterlissauer02]. Galileo NIMS and Cassini CIRS data, however, indicated that spectrally identifiable ammonia ice clouds cover only very small regions on the planet [see, respectively @baines02; @wong04]. As put forward by e.g. @atreya05, this apparent contradiction could be explained if the ammonia ice particles are coated by in particlar hydrocarbon haze particles settling from the stratosphere. Thus, only the highest and freshest ammonia ice clouds would show identifiable spectral features. @atreya05 also mention that the strength of the spectral features would depend on the sizes and shapes of the ice crystals. In this paper, we assume that the upper tropospheric clouds in our model atmospheres are indeed composed of ammonia ice particles, without modelling specific spectral features.
![Single scattering $F$ and $P_\mathrm{s}$ of our ammonia (NH$_3$) ice cloud particles and the polar haze particles as functions of the planetary phase angle $\alpha$ at three different wavelengths. The curves for the NH$_3$ ice particles are: black, solid line ($\lambda=0.55~\mu$m), red, dotted line (0.75 $\mu$m), green, dashed–dotted line (0.95 $\mu$m). The curves for the polar haze particles are: grey, dashed–triple–dotted line (0.55 $\mu$m), orange, long–dashed line (0.75 $\mu$m), magenta, dashed line (0.95 $\mu$m). The blue, dashed lines are the Rayleigh scattering curves at 0.55 $\mu$m.[]{data-label="fig:ss_ice"}](karalidi1a.ps "fig:"){width="85mm"} ![Single scattering $F$ and $P_\mathrm{s}$ of our ammonia (NH$_3$) ice cloud particles and the polar haze particles as functions of the planetary phase angle $\alpha$ at three different wavelengths. The curves for the NH$_3$ ice particles are: black, solid line ($\lambda=0.55~\mu$m), red, dotted line (0.75 $\mu$m), green, dashed–dotted line (0.95 $\mu$m). The curves for the polar haze particles are: grey, dashed–triple–dotted line (0.55 $\mu$m), orange, long–dashed line (0.75 $\mu$m), magenta, dashed line (0.95 $\mu$m). The blue, dashed lines are the Rayleigh scattering curves at 0.55 $\mu$m.[]{data-label="fig:ss_ice"}](karalidi1b.ps "fig:"){width="85mm"}
Our ammonia ice particles are assumed to be spherical with a refractive index of $n=1.48~+~0.01i$ (assumed to be constant across the spectral region of our interest) (as adopted from @romanescu10 and @gibson05, for the real and imaginary part respectively) and with their sizes described by a standard size distribution [see @hansentravis74] with and effective radius $r_\mathrm{eff}$ of 0.5$\mu$m, and an effective variance $v_\mathrm{eff}$ of 0.1 [@stamhovenier04]. We calculate the single scattering properties of the ammonia ice particles using Mie theory as described by @derooijvdstap87.
Figure \[fig:ss\_ice\] shows the flux and degree of linear polarization $P_\mathrm{s}$ of unpolarized incident light with $\lambda$=0.55 $\mu$m, 0.75 $\mu$m, and 0.95 $\mu$m, respectively, that is singly scattered by the ice particles as functions of the planetary phase angle $\alpha$. Note that $\alpha= 180^\circ - \Theta$, with $\Theta$ the conventional single scattering angle, defined as $\Theta=0^\circ$ for forward scattered light. All scattered fluxes have been normalized such that their average over all scattering directions equals one [see Eq. 2.5 of @hansentravis74]. At $\alpha=0^\circ$ (180$^\circ$) the light is scattered in the backward (forward) direction. For comparison, we have added the curves for (Rayleigh) scattering by gas molecules at $\lambda=0.55~\mu$m (these curves are fairly wavelength independent). As can be seen in the figure, our spherical ice particles are moderately forward scattering and the scattered fluxes show a prominent feature (a local minimum) around $\alpha=12^\circ$ at $\lambda=0.55~\mu$m, around $20^\circ$ at 0.75 $\mu$m, and (much less pronounced) around $25^\circ$ at 0.95 $\mu$m.
The degree of polarization $P_\mathrm{s}$ of the light that is singly scattered by our ammonia ice particles is negative across almost the whole phase angle range. The light is thus polarized parallel to the scattering plane, which contains both the incident and the scattered beams. The local minima in scattered fluxes have associated local minima in $P_\mathrm{s}$ (at slightly shifted values of $\alpha$).
We should note here that the vast majority of giant planets imaged so far, are too hot to contain ammonia ice particles in their atmospheres [see e.g. @bonnefoy13; @konopacky13]. These planets though, are so far away from their parent star and so hot that the contribution of the reflected starlight to the disk–integrated signal is very small in comparison to the thermal radiation of the planets. Even though the thermal radiation can also be polarized (through scattering by the cloud particles, see @dekok11) these planets are out of the scope of this paper.
Polar haze particles {#sec:hazeparts}
--------------------
We model Jupiter’s polar haze particles as randomly oriented aggregates of equally sized spheres. To generate the aggregates (needed for calculating the single scattering properties of these particles), we use a cluster–cluster aggregation (CCA) method that starts with the formation of particle–cluster aggregates (PCA) by sequentially adding spheres from random directions to an existing cluster, as shown in the upper part of Fig. \[fig:particles1\]. Next, we combine several PCA–particles, as shown in the lower part of Fig. \[fig:particles1\]. For both PCA and CCA, the coagulation process finishes when the maximum distance between any pair of monomers of the aggregate exceeds a certain limit (in Fig. \[fig:particles1\]: $d_\mathrm{c}$ for PCA and $d_\mathrm{p}$ for CCA). With the later assumption, we limit the size of the generated particles, which is needed due to the computational limitations of the numerical methods to calculate the single scattering properties of the particles (see below). We use CCA–particles rather than PCA–particles in the model atmospheres because they can yield the high polarization values that have been observed at the poles of Jupiter [see e.g. @schmid11]. Because PCA–particles are more compact, light is on average scattered more within each particle, which decreases the degree of polarization of the scattered light.
![Sketch of the two aggregation–mechanisms to build the polar haze particles. With PCA, identical monomers are sticked together until the maximum distance between two monomers is larger than a given limiting distance $d_\mathrm{c}$. With CCA, several PCA–particles are combined until the maximum distance between two monomers of the whole particle is larger than $d_\mathrm{p}$.[]{data-label="fig:particles1"}](karalidi2.ps){width="85mm"}
![A picture of a model aggregate haze particle that we used for our single light scattering calculations.[]{data-label="fig:particles2"}](karalidi3.ps){width="85mm"}
Figure \[fig:particles2\] shows a model aggregate haze particle that we generated and for which we calculated the single scattering matrices and other optical properties. The particle consists of 94 spherical monomers. The radius of each monomer is approximately 0.035 $\mu$m, and the volume-equivalent-sphere radius of the whole particle 0.16 $\mu$m. Calculations of the single scattering matrix and other optical properties of these particles were performed using the T-matrix theory combined with the superposition theorem [@mackowski11], at $\lambda=0.55$, 0.75 and 0.95 $\mu$m, and adopting a refractive index of $1.5+i0.001$, which corresponds to that of benzene [suggested to exist on the jovian poles by e.g. @friedson02]. In Fig. \[fig:ss\_ice\] we show the flux and polarization of unpolarized incident light that is singly scattered by the haze particles at the three different wavelengths, together with the Rayleigh curves at $\lambda=0.55~\mu$m. The single scattering albedo of the particles is 0.995 at 0.55 $\mu$m.
From comparing the different lines in Fig. \[fig:ss\_ice\], it is clear that the haze particles are more forward scattering than the ammonia ice particles, and that their scattered flux shows less angular features. The degree of linear polarization of the light scattered by the haze particles is very different from that of the cloud particles: it is positive at almost all phase angles (hence the light is polarized perpendicular to the scattering plane) and it reaches values larger than 0.7 near $\alpha=90^\circ$. The main reason that the polarization phase function of the haze particles differs strongly from that of the cloud particles while their flux phase functions are quite similar, is that the latter depends mostly on the size of the whole particle, while the polarization phase function depends more on the size of the smallest scattering particles, which have radii of about 0.035 $\mu$m, in the case of the aggregate particles.
The maximum single scattering polarization of our aggregate particles is slightly higher than that derived by @west91. This is most likely due to the properties of our haze particles: our monomers are smaller than those used by @west91, which have radii near 0.06 $\mu$m, sometimes mixed with monomers with radii of 0.03 $\mu$m. In addition, the particles in @west91 were generated using the diffusion-limited aggregation (DLA) method, in which monomers follow random paths towards the aggregate, and which yields more compact particles than those produced by our CCA–method [see @meakin83].
Reflected flux and polarization signals of the model planets {#sec:gaseousplanets}
============================================================
In this section, we present fluxes and degrees of linear polarization for three different types of spatial inhomogeneities that occur on gaseous planets in the Solar System: zones and belts (Sect. \[sec:zonesbelts\]), cyclonic spots (Sect. \[sec:cyclonicspots\]), and polar hazes (Sect. \[sec:polarhazes\]). We will compare the flux and polarization signals of the spatially inhomogeneous planets with those of horizontally homogeneous planets to investigate whether or not such spatial inhomogeneities would be detectable.
Zones and belts {#sec:zonesbelts}
---------------
The model atmospheres in this section contain only clouds, no hazes. Figures \[fig:top02\_550nm\]–\[fig:top02\_950nm\] show the flux $\pi F_\mathrm{n}$ and the degree of linear polarization $P_\mathrm{s}$ as functions of $\alpha$ at $\lambda=0.55~\mu$m (Fig. \[fig:top02\_550nm\]), 0.75 $\mu$m (Fig. \[fig:top02\_750nm\]), and 0.95 $\mu$m (Fig. \[fig:top02\_950nm\]), for horizontally homogeneous planets with the bottom of the cloud layer at 1.0 bar, and the top at 0.1, 0.2, 0.3, 0.4, or 0.5 bar. Also shown in these figures, are $\pi F_\mathrm{n}$ and $P_\mathrm{s}$ for horizontally inhomogeneous model planets each with a cloud top pressure of 0.1 bar in the zones and with cloud top pressures ranging from 0.2 bar (Fig. \[fig:top02\_550nm\], \[fig:top02\_750nm\] and \[fig:top02\_950nm\]) to 0.5 bar (Fig. \[fig:top03\_550nm\]) in the belts. The latitudinal borders of the zones and belts have been described in Sect. \[sect:modelplanets\].
![Total flux $\pi F_\mathrm{n}$ and degree of polarization $P_\mathrm{s}$ of starlight with $\lambda=0.55$ $\mu$m that is reflected by a cloud covered model planet. The bottom of the clouds is at 1.0 bar, while the cloud top pressure varies. For the horizontally homogeneous planets, the cloud top pressures are as follows: 0.1 bar (red, dotted line), 0.2 bar (green, dashed line), 0.3 bar (blue, dashed–dotted line), 0.4 bar (grey, dashed–triple-dotted line), 0.5 bar (magenta, long–dashed line). The spatially inhomogeneous planet (black, solid line) has a cloud top pressure of 0.1 bar in the zones, and 0.2 bar in the belts.[]{data-label="fig:top02_550nm"}](karalidi4a.ps "fig:"){width="85mm"} ![Total flux $\pi F_\mathrm{n}$ and degree of polarization $P_\mathrm{s}$ of starlight with $\lambda=0.55$ $\mu$m that is reflected by a cloud covered model planet. The bottom of the clouds is at 1.0 bar, while the cloud top pressure varies. For the horizontally homogeneous planets, the cloud top pressures are as follows: 0.1 bar (red, dotted line), 0.2 bar (green, dashed line), 0.3 bar (blue, dashed–dotted line), 0.4 bar (grey, dashed–triple-dotted line), 0.5 bar (magenta, long–dashed line). The spatially inhomogeneous planet (black, solid line) has a cloud top pressure of 0.1 bar in the zones, and 0.2 bar in the belts.[]{data-label="fig:top02_550nm"}](karalidi4b.ps "fig:"){width="85mm"}
![Total flux $\pi F_\mathrm{n}$ and degree of polarization $P_\mathrm{s}$ of starlight with $\lambda=0.55$ $\mu$m that is reflected by a spatially inhomogeneous model planet. The cloud top pressure is 0.1 bar in the zones and 0.2 bar (black, solid line), 0.3 bar (red, dashed line), 0.4 bar (blue, dashed–dotted line) or 0.5 bar (grey, dashed–triple–dotted line) in the belts. []{data-label="fig:top03_550nm"}](karalidi5a.ps "fig:"){width="85mm"} ![Total flux $\pi F_\mathrm{n}$ and degree of polarization $P_\mathrm{s}$ of starlight with $\lambda=0.55$ $\mu$m that is reflected by a spatially inhomogeneous model planet. The cloud top pressure is 0.1 bar in the zones and 0.2 bar (black, solid line), 0.3 bar (red, dashed line), 0.4 bar (blue, dashed–dotted line) or 0.5 bar (grey, dashed–triple–dotted line) in the belts. []{data-label="fig:top03_550nm"}](karalidi5b.ps "fig:"){width="85mm"}
For each model planet and each wavelength, total flux $\pi F_\mathrm{n}$ at $\alpha=0^\circ$ equals the planet’s geometric albedo $A_\mathrm{G}$. With increasing wavelength, $A_\mathrm{G}$ decreases slightly, because of the decreasing cloud optical thickness with $\lambda$, and the decreasing single scattering phase function in the backscattering direction (see Fig. \[fig:ss\_ice\]). With increasing $\alpha$, $\pi F_\mathrm{n}$ decreases smoothly for all model atmospheres. The angular feature around $\alpha= 12^\circ$ for the horizontally homogeneous planets with the highest cloud layers, can be retraced to the single scattering phase function (Fig. \[fig:ss\_ice\]). The strength of the feature in the planetary phase functions decreases with $\lambda$, just like that in the single scattering phase functions. The decrease of the feature with increasing cloud top pressure is due to the increasing thickness of the gas layer overlying the clouds. With increasing $\lambda$, the difference between the total fluxes reflected by the model atmospheres decreases, mostly because of the decrease of Rayleigh scattering above the clouds with $\lambda$.
Interestingly, $\pi F_\mathrm{n}$ is insensitive to the cloud top pressure around $\alpha=125^\circ$ at $\lambda=0.55$ $\mu$m (Fig. \[fig:top02\_550nm\]). With increasing $\lambda$, the phase angle where this insensitivity occurs decreases: from about 110$^\circ$ at $\lambda=0.75~\mu$m (Fig. \[fig:top02\_750nm\]), to about 90$^\circ$ at $\lambda=0.95~\mu$m (Fig. \[fig:top02\_950nm\]). Thus precisely across the phase angle range where exoplanets are most likely to be directly detected because they are furthest from their star, reflected fluxes do not give access to the cloud top altitudes.
![Same as in Fig. \[fig:top02\_550nm\], except for $\lambda=$0.75 $\mu$m.[]{data-label="fig:top02_750nm"}](karalidi6a.ps "fig:"){width="85mm"} ![Same as in Fig. \[fig:top02\_550nm\], except for $\lambda=$0.75 $\mu$m.[]{data-label="fig:top02_750nm"}](karalidi6b.ps "fig:"){width="85mm"}
![Same as in Fig. \[fig:top02\_550nm\], except for $\lambda=$0.95 $\mu$m.[]{data-label="fig:top02_950nm"}](karalidi7a.ps "fig:"){width="85mm"} ![Same as in Fig. \[fig:top02\_550nm\], except for $\lambda=$0.95 $\mu$m.[]{data-label="fig:top02_950nm"}](karalidi7b.ps "fig:"){width="85mm"}
The degree of linear polarization, $P_\mathrm{s}$, shows the typical bell-shape around approximately $\alpha=90^\circ$, that is due to Rayleigh scattering of light by gas molecules (Fig. \[fig:top02\_550nm\]). With increasing cloud top altitude, hence decreasing Rayleigh scattering optical thickness above the clouds, the features of the single scattering phase function of the cloud particles become more prominent. This is especially obvious at the longer wavelengths, i.e. at $0.75~\mu$m and 0.95 $\mu$m, where the Rayleigh scattering optical thickness above the clouds is smaller by factors of about $(0.55/0.75)^4$ and $(0.55/0.95)^4$, respectively (Figs. \[fig:top02\_750nm\] and \[fig:top02\_950nm\]). In particular, the negative polarized feature below $\alpha=30^\circ$, that is due to light singly scattered by the cloud particles (see Fig. \[fig:ss\_ice\]) becomes more prominent.
Figure \[fig:top02\_550nm\] clearly shows that, unlike the reflected flux, $P_\mathrm{s}$ is sensitive to cloud top altitudes across planetary phase angles that are important for direct detections. The reason is that $P_\mathrm{s}$ is very sensitive to the Rayleigh scattering optical thickness above the clouds, as has been known for a long time from observations of Solar System planets, such as the ground-based observations of Venus [@hansenhovenier74], and remote-sensing observations of the Earth by instruments such as POLDER on low–orbit satellites [@knibbe00]. As expected, with increasing $\lambda$, the sensitivity of $P_\mathrm{s}$ to the cloud top altitude decreases (see Figs. \[fig:top02\_750nm\] and \[fig:top02\_950nm\]).
Figures \[fig:top02\_550nm\]–\[fig:top02\_950nm\] also show $\pi F_\mathrm{n}$ and $P_\mathrm{s}$ of horizontally inhomogeneous planets with zones and belts. In all figures, the cloud top pressure of the zones is 0.1 bar while that at the top of the belts varies from 0.2 bar (Figs. \[fig:top02\_550nm\], \[fig:top02\_750nm\], and \[fig:top02\_950nm\]) to 0.5 bar (Fig. \[fig:top03\_550nm\]). The shapes of the flux and polarization phase functions of these horizontally inhomogeneous planets are very similar to those of the horizontally homogeneous planets: one could easily find a horizontally homogeneous model planet with a cloud top pressure between 0.1 and 0.3 bar that would fit the curves pertaining to the horizontally inhomogeneous planets. The cloud top pressure that would provide the best fit would be slightly different when fitting the flux or the polarization curves. For example, fitting the flux reflected by an inhomogeneous planet with cloud top pressures in the belts at 0.4 bar (blue dashed–dotted line of Fig. \[fig:top03\_550nm\]) would require a homogeneous planet with its cloud top pressure at 0.2 bar, while fitting the polarization would require a cloud top pressure of about 0.18 bar. Such small differences would most likely disappear in the measurement errors. With increasing $\lambda$, the effects of the cloud top pressure decrease, in particular in $\pi F_\mathrm{n}$. Covering a broad spectral region would thus not help in narrowing the cloud pressures down.
Cyclonic spots {#sec:cyclonicspots}
--------------
Other spatial features on giant planets in our Solar System are (anti–)cyclonic storms that show up as oval–shaped spots. Famous examples are Jupiter’s Great Red Spot (GRS) that appears to have been around for several hundreds of years and Neptune’s Great Dark Spot (GDS) that was discovered in 1989 by Voyager–2, but that seems to have disappeared [@hammel95]. Recent spots on Uranus were presented by @hammel09 and @sromovsky12. To study the effect of localized spots on reflected flux and polarization signals of exoplanets, we use a Jupiter–like model atmosphere with a spot of NH$_3$ ice clouds extending between 0.75 and 0.13 bar. The clouds in the spot have an optical thickness of 36 at $\lambda=0.5~\mu$m [@simonmiller01]. We model the spot as a square of 26$^\circ$ in longitude by 22$^\circ$ in latitude with an optical thickness of $\sim$30 at 0.55 $\mu$m. Additionally, our model planet has zones and belts spatially distributed across the planet as described before, extending between 0.56 to 0.18 bar in the zones, and from 1.0 to 0.5 bar in the belts. The cloud optical thickness in the belts is 6.02, and in the zones 21.0 at $\lambda=0.75~\mu$m.
![$\pi F_\mathrm{n}$ and $P_\mathrm{s}$ as functions of the planet’s rotation angle (in degrees), for a Jupiter–like model planet with a spot on its equator. Curves are shown for spots covering 0.3% (blue, dashed–dotted line), 1% (red, dashed line), 2% (green, dotted line), or 4% (black, solid line) of the planetary disk. The planetary phase angle $\alpha$ is 90$^\circ$ and $\lambda=0.55~\mu$m. Calculations have been done at rotation angle steps of 20$^\circ$.[]{data-label="fig:jup3"}](karalidi8a.ps "fig:"){width="85mm"} ![$\pi F_\mathrm{n}$ and $P_\mathrm{s}$ as functions of the planet’s rotation angle (in degrees), for a Jupiter–like model planet with a spot on its equator. Curves are shown for spots covering 0.3% (blue, dashed–dotted line), 1% (red, dashed line), 2% (green, dotted line), or 4% (black, solid line) of the planetary disk. The planetary phase angle $\alpha$ is 90$^\circ$ and $\lambda=0.55~\mu$m. Calculations have been done at rotation angle steps of 20$^\circ$.[]{data-label="fig:jup3"}](karalidi8b.ps "fig:"){width="85mm"}
Figure \[fig:jup3\] shows reflected fluxes $\pi F_\mathrm{n}$ and degree of polarization $P_\mathrm{s}$ at $\lambda=0.55~\mu$m, as functions of the planet’s rotation angle. The planet’s phase angle is 90$^\circ$. The spot is on the planet’s equator (which coincides with the planetary scattering plane). Recall that both Jupiter and Saturn have rotation periods on the order of 10 hours, while Uranus and Neptune rotate in about 17 and 16 hours, respectively. With a 10-hour rotation period and $\alpha=90^\circ$, a small spot would cross the illuminated and visible part of the disk in about 2.5 hours. Curves are shown for different sizes of the spot: covering at maximum 0.3 %, 1%, 2%, or 4% of the planet’s disk, respectively. For comparison: the GRS covers about 6 % of Jupiter’s disk. Note that the calculations for Fig. \[fig:jup3\] have been done per 20$^\circ$ rotation of the planet, due to computational restrictions.
As can be seen in Fig. \[fig:jup3\], the reflected fluxes $\pi F_\mathrm{n}$ hardly change upon the passage of the spot across the illuminated and visible part of the planetary disk: even for the largest spot located at the equator, the maximum change in $\pi F_\mathrm{n}$ is a few percent. In $P_\mathrm{s}$, the transiting spots also leave a change of at most a few percent (absolute, since $P_\mathrm{s}$ is a relative measure itself). For spots located at higher latitudes of the planet, the effects are even smaller. With increasing wavelength, the sensitivity of both $\pi F_\mathrm{n}$ and $P_\mathrm{s}$ to the cloud top altitude decreases. At longer wavelengths, the effects of a spot would thus be smaller than shown in Fig. \[fig:jup3\].
Polar hazes {#sec:polarhazes}
-----------
The poles of Jupiter and Saturn are covered by stratospheric hazes. In particular, when seen at phase angles around 90$^\circ$, Jupiter’s polar hazes yield strongly polarized signals. This high polarization can be explained by haze particles that consist of aggregates of particles that are small compared to the wavelength, and that polarize the incident sunlight as Rayleigh scatterers [@west91], with a high degree of polarization at scattering angles around 90$^\circ$. We are interested in whether strongly polarized polar hazes will leave a trace in the disk-integrated polarization signal of a planet.
![Reflected flux $\pi F_\mathrm{n}$ at $\lambda=0.55~\mu$m for every pixel on the disk of a Jupiter–like model planet containing clouds in zones and belts, and polar stratospheric hazes at latitudes above (below) 60$^\circ$ (-$60^\circ$). The phase angles are: 0$^\circ$ (top) and 90$^\circ$ (below).[]{data-label="fig:flx_a90"}](karalidi9a.ps "fig:"){width="85mm"} ![Reflected flux $\pi F_\mathrm{n}$ at $\lambda=0.55~\mu$m for every pixel on the disk of a Jupiter–like model planet containing clouds in zones and belts, and polar stratospheric hazes at latitudes above (below) 60$^\circ$ (-$60^\circ$). The phase angles are: 0$^\circ$ (top) and 90$^\circ$ (below).[]{data-label="fig:flx_a90"}](karalidi9b.ps "fig:"){width="85mm"}
We use a Jupiter–like model planet with clouds in belts and zones as in Sect. \[sec:zonesbelts\]. The cloud top pressure of the belts is 0.3 bar. Starting at latitudes of $60^\circ$, the north and south poles of each model planet are covered by polar haze particles as described in Sect. \[sec:hazeparts\]. The optical thickness of the haze is 0.2 at $\lambda=$0.55, 0.75, and 0.95 $\mu$m.
In Figs. \[fig:flx\_a90\] and \[fig:pol\_a90\], we show, respectively $\pi F_\mathrm{n}$ and $P_S$ at $\lambda=0.55$ $\mu$m of starlight that is locally reflected by pixels on the visible and illuminated part of the planetary disk for $\alpha=0^\circ$ and $90^\circ$. In Fig. \[fig:disk\_ints\], we have plotted the disk–integrated reflected flux $\pi F_\mathrm{n}$ and degree of polarization $P_\mathrm{s}$ as functions of $\alpha$, for the Jupiter–like model planet with and without polar hazes, and for $\lambda=0.55$, 0.75, and 0.95 $\mu$m.
The reflected flux across the planetary disk (Fig. \[fig:flx\_a90\]) shows clear differences between the zones and the belts, especially around the center of the planetary disk (for both values of $\alpha$). The reflected fluxes due to the polar hazes do not stand out against those due to the clouds in the belts and the zones. The reflected flux pattern across the visible and illuminated part of the planetary disk for $\alpha=90^\circ$ is very similar to that for $\alpha=0^\circ$, with the brightest regions in the center, and the darkest at the limb and, for $\alpha=90^\circ$, at the terminator. Integrated across the planetary disk (Fig. \[fig:disk\_ints\]), there is very little difference between $\pi F_\mathrm{n}$ of the planets with and without haze. From the disk–integrated reflected fluxes it would thus be impossible to derive the presence of the polar hazes. Note that our simulations pertain to broadband fluxes. Observations across gaseous absorption bands, such as those of methane, could provide more information about the presence of hazes, because the latter would decrease the depths of the bands [see e.g. @stamhovenier04]. Of course, in order to use absorption band depths to derive altitudes of hazes and/or clouds, independent information about the mixing ratios of the absorbing gases would be essential.
The degree of polarization across the planetary disk (Fig. \[fig:pol\_a90\]), also shows clear differences between the zones and the belts, but mostly near the limb and, for $\alpha=90^\circ$, the terminator of the planet. For $\alpha=0^\circ$, the observed light has been scattered in the backward direction, and the degree of polarization is close to zero across the central region of the disk. Towards the limb, the degree of polarization increases to reach values as high as 0.15 towards the northern and southern limbs, and then it decreases again. The relatively high polarization values are due to significantly polarized second order scattered light (the singly scattered light contributes virtually no polarized light), while the low values at the limb are due to light that has been singly scattered in the backscattering direction (cf. Fig. \[fig:ss\_ice\]).
![Same as in Fig. \[fig:flx\_a90\] except for $P_\mathrm{s}$. Note the different color scales.[]{data-label="fig:pol_a90"}](karalidi10a.ps "fig:"){width="85mm"} ![Same as in Fig. \[fig:flx\_a90\] except for $P_\mathrm{s}$. Note the different color scales.[]{data-label="fig:pol_a90"}](karalidi10b.ps "fig:"){width="85mm"}
![Disk–integrated $\pi F_\mathrm{n}$ and $P_\mathrm{s}$ as functions of phase angle $\alpha$ for the Jupiter–like model planet with polar hazes as used in Figs. \[fig:flx\_a90\] and \[fig:pol\_a90\] at: $\lambda= 0.55$ $\mu$m (black, solid lines), 0.75 $\mu$m (red, dotted lines), and 0.95 $\mu$m (green, dashed lines). Also shown are the lines for model planets with the clouds but without the hazes: $\lambda= 0.55$ $\mu$m (blue, dashed–dotted lines), 0.75 $\mu$m (grey, dashed–triple–dotted lines), and 0.95 $\mu$m (magenta, long–dashed lines).[]{data-label="fig:disk_ints"}](karalidi11a.ps "fig:"){width="85mm"} ![Disk–integrated $\pi F_\mathrm{n}$ and $P_\mathrm{s}$ as functions of phase angle $\alpha$ for the Jupiter–like model planet with polar hazes as used in Figs. \[fig:flx\_a90\] and \[fig:pol\_a90\] at: $\lambda= 0.55$ $\mu$m (black, solid lines), 0.75 $\mu$m (red, dotted lines), and 0.95 $\mu$m (green, dashed lines). Also shown are the lines for model planets with the clouds but without the hazes: $\lambda= 0.55$ $\mu$m (blue, dashed–dotted lines), 0.75 $\mu$m (grey, dashed–triple–dotted lines), and 0.95 $\mu$m (magenta, long–dashed lines).[]{data-label="fig:disk_ints"}](karalidi11b.ps "fig:"){width="85mm"}
For $\alpha=90^\circ$, the scattering angle of the singly scattered light is 90$^\circ$ (cf. Fig. \[fig:ss\_ice\]). The degree of polarization is a few percent at the center of the illuminated and visible part of the planetary disk, where the contribution of multiple scattered light is significant, and increases towards the terminator and the limb. The higher values (up to 0.90 towards the northern and southern limbs, and up to 0.50 towards the western limb and terminator) are due to singly scattered light. We have repeated our calculations for $\lambda= 0.75$ (not shown), and found that for $\alpha=0^\circ$ the polar hazes produce a fractional polarization signal $Q/F$ of $\sim 6$%, which is similar to what was observed in that wavelength range by @schmid11. The degree of polarization due to the polar hazes will depend strongly on the single scattering polarization phase function of the particles and hence on the wavelength. @kolokolova12 present calculated polarization phase functions of large fluffy and compact aggregates with application to polarization observations of comets, and clearly show the increase of the maximum degree of polarization around $\alpha=90^\circ$ ($\Theta=90^\circ$) with increasing $\lambda$ for particles built using the CCA–method.
While the polar hazes strongly influence the degree of polarization of local regions on the model planet, integrated across the planetary disk (Fig. \[fig:disk\_ints\]), they have a small ($\lambda=0.55~\mu$m) to negligible ($\lambda=0.95~\mu$m) influence. Like with the reflected fluxes (see above), the polar hazes could have a stronger effect across gaseous absorption bands, where the observed reflected starlight originates from higher altitudes in the planetary atmosphere.
Summary and conclusions {#sect_summary}
=======================
In view of upcoming instruments for the direct detections of fluxes and polarization signals of starlight that is reflected by orbiting exoplanets, like SPHERE [@dohlen08; @roelfsema11] on the VLT, GPI [@macintosh08] on the Gemini North telescope, and, further in the future, EPICS [@kasper10] on the E-ELT, we have presented numerically calculated disk–integrated, broadband flux and polarization signals of horizontally inhomogeneous gaseous exoplanets in order to investigate whether or not spatial inhomogeneities such as those found on gas giants in the Solar System could be identified on gaseous exoplanets. The spatial inhomogeneities that we have modeled are: cloud top altitudes belts and zones due to varying cloud top altitudes (Sect. \[sec:zonesbelts\]), cyclonic spots (Sect. \[sec:cyclonicspots\]), and polar hazes (Sect. \[sec:polarhazes\]).
We have calculated the total flux and polarization signals of the exoplanets due to the mentioned spatial inhomogeneities using an adding–doubling radiative transfer code that fully includes all orders of scattering and linear and circular polarization. Here, we ignore the circular polarization because it is very small and neglecting it does not introduce significant errors in the calculated total and linearly polarized fluxes [@stam05]. Circular polarization will also not be measured by the above mentioned instruments. We only consider exoplanets in wide orbits, such that they can be spatially resolved from their parent star, and ignore diffracted light from the parent star and starlight that is scattered by e.g. exozodiacal dust in the planetary system (we thus assume that the observations can be corrected for these two types of background signals).
A version of the adding–doubling radiative transfer code that handles vertically inhomogeneous, but horizontally homogeneous exoplanets has been described by @stamhovenier04. In our version of the code [see @karalidi12b; @karalidi12c], we divide a horizontally inhomogeneous planet into horizontally homogeneous pixels. For each pixel, we calculate the reflected total and polarized fluxes and add up these local fluxes (including rotations between local and planetary reference planes for the polarized fluxes) to obtain the disk–integrated total and polarized fluxes at a given planetary phase angle. We compare signals calculated for horizontally inhomogeneous planets with those of horizontally homogeneous planets to investigate the appearance of the spatial inhomogeneities in the disk–integrated signals.
Cyclonic spots on a planet will rotate in and out of the view of the observer depending on their daily rotation periods (because we consider exoplanets in wide orbits, they are unlikely to be tidally locked to their star), and will thus give rise to time–varying total and polarized fluxes. Our model exoplanets with zones and belts, and those with the polar hazes are symmetric with respect to their axis of rotation. The total flux and polarization signals of these planets will thus depend on the planetary phase angle, but not on daily rotations of a planet. For these model exoplanets, the presence of horizontal inhomogeneities could presumably be derived from different effects the inhomogeneities have on the total flux and on the degree of linear polarization of the reflected starlight.
Our Jupiter–like model exoplanets have thick cloud layers consisting of spherical ammonia ice particles. In Sect. \[sec:zonesbelts\], we model belts and zones by choosing different pressure levels for the tops of the clouds, while the base of the clouds is fixed at 1 bar. The pressure at the top of the zonal clouds is fixed at 0.1 bar, while the pressure at the top of the belts is varied between 0.2 bar and 0.5 bar. Comparing the total and polarized fluxes of the horizontally inhomogeneous planets with those of horizontally homogeneous cloudy planets, it is clear that both the reflected total fluxes and the polarization are sensitive to the cloud top altitude. The polarization appears to be more useful to derive cloud top altitudes than the reflected flux, because the polarization is most sensitive to the cloud top altitude at planetary phase angles around 90$^\circ$, which are favorable for direct imaging of exoplanets. At these phase angles, the sensitivity of the total flux to the cloud top altitude is small, and it decreases with increasing wavelength, because of the decreasing Rayleigh scattering optical thickness above the clouds.
The shape of the polarization phase function for a planet with belts and zones is very similar to that of a horizontally homogeneous planet with a cloud top pressure between that of the zones and the belts (the precise value depends on the latitudes covered by the belts and zones and by the cloud top pressures across the belts and the zones). Combining polarization phase functions at different broadband wavelengths will not help to reveal the presence of horizontal inhomogeneities. Combining total flux phase functions with polarization phase functions will also fail to reveal the presence of inhomogeneities because of the lack of sensitivity of the total flux phase functions to the cloud top pressures, especially at the phase angles that are favorable for polarimetry (by lack of independent information about the planet radius, the total flux phase functions will also have error bars that preclude deriving meaningful cloud top pressures from them).
We model a cyclonic spot as a localized cloudy region with a lower cloud top pressure than the surrounding clouds: in the spot, the cloud top pressure is 0.13 bar, while outside the spot, the cloud top pressure is 0.5 bar at the edges and 0.18 bar at the central latitudes. Our results for model planets with a cyclonic spot on the planet’s equator that covers about 4% of the planetary disk show that the change in the reflected total flux as the spot moves in and out of view of the observer is less than a percent at a planetary phase angle of 90$^\circ$. The change in the degree of polarization is a few percent (in absolute sense). The temporal changes in the total flux and the degree of polarization decrease if the spot is located further away from the planetary scattering plane. If a planet has more than one spot, a temporal analysis of flux and polarization time series might reveal the locations and sizes of the most prominent spots.
Polar hazes that very likely consist of aggregates of small monomers cover the poles of Jupiter and Saturn, yielding a locally relatively high degree of polarization [@west91]. We modeled polar hazes on Jupiter–like planets using fluffy aggregates covering latitudes north and southwards of, respectively, 60$^\circ$ and -60$^\circ$. In the locally reflected total fluxes (i.e. spatially resolved across the planet), the polar hazes do not leave a significant trace, and, not surprisingly, neither do they in the disk-integrated total fluxes. In the degree of polarization of the locally reflected light, the polar hazes do show up. At a phase angle of 0$^\circ$, light that has been scattered twice by the haze particles has a relatively high degree of polarization (compared to the light that has been singly scattered in the backward direction by the gas molecules or the haze particles, and that is virtually unpolarized). At a phase angle of 90$^\circ$, light that has been singly scattered by the haze particles has a high degree of polarization, because the monomers that form the aggregated haze particles scatter like Rayleigh scatterers.
When integrated over the planetary disk, however, the polar hazes change the degree of linear polarization by only a few percent when compared to a planet without polar hazes at $\lambda=0.55$ $\mu$m, and less at longer wavelengths. The shape of the polarization phase function of the planet with polar hazes is similar to that without hazes.
Finally, we note that the vast majority of giant planets discovered so far are very hot [see e.g. @knutson12; @demooij13] and their atmospheric chemistry and temperature–pressure profiles vary considerably from those of the giant planets of our solar system [see e.g. @moses11; @huitson12; @madhusudhan12b and references there in]. While these close-in planets orbit too close to their parent star to be directly detectable and are thus out of the scope of this paper, they have taught us that giant planets can exhibit a large variety of properties. Even the exoplanets that orbit a few AU from their parent star, and that could thus be directly detectable, might be shown in the near future to vary considerably in their properties.
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[^1]: Present address: Steward Observatory, 933 N Cherry Ave, Tucson, AZ 85721, USA
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---
abstract: 'When testing multiple hypothesis in a survey –e.g. many different source locations, template waveforms, and so on– the final result consists in a set of confidence intervals, each one at a desired confidence level. But the probability that at least one of these intervals does not cover the true value increases with the number of trials. With a sufficiently large array of confidence intervals, one can be sure that at least one is missing the true value. In particular, the probability of false claim of detection becomes not negligible. In order to compensate for this, one should increase the confidence level, at the price of a reduced detection power. False discovery rate control[@Benjamini95] is a relatively new statistical procedure that bounds the number of mistakes made when performing multiple hypothesis tests. We shall review this method, discussing exercise applications to the field of gravitational wave surveys.'
address: 'Dipartimento di Fisica, Università di Trento and INFN, Gruppo collegato di Trento, Sezione di Padova, via Sommarive, 14, 38050, Povo, TN, Italy'
author:
- L Baggio and G A Prodi
title: 'False discovery rate: setting the probability of false claim of detection'
---
Introduction
============
The motivation for controlling the false discovery rate (FDR) –i.e. the fraction of false alarms in a collection of candidate detections– jumped to our attention as we were involved in data analysis for the IGEC [@IGEC-PRD], the network of resonant detectors that searched for coincident burst gravitational wave (GW) signals in the years 1997-2000. Even if the detectors involved in IGEC were rather similar, there were obvious configurations (special choice of detector pairs, three-fold instead of double coincidence) or cuts of the data (higher or lower threshold on event amplitude) characterized by lower background counts, or higher duty time. We did not have *a priori* a good reason to prefer one configuration or cut more than others, as we do not know a priori the intensity of the signal, hence the efficiency. Therefore, we decided at the beginning a fairly long list of interesting choices, in order to perform many analyses in parallel, and eventually to quote the results for each trial.
The results were expressed as confidence intervals on the expectation value for the number of counts in coincidence due to GW. When unveiling the final results, one of the confidence intervals at 90% coverage was not including the null hypothesis (i.e. zero counts). Of course this can be somewhat expected by chance when the number of trials is very high. It was possible to compute accurately that with 30% probability we had a chance that at least one of the tests falsely rejected the null hypothesis.
Total
-- ------------- -- ---------
**$m_0$-B** $m_0$
$m_1$
**$m$-R** **$m$**
: \[tab:notation\] Quick-reference notation chart for the variables used in . $m$ is the total number of performed tests (trial factor), $m_0$ and $m_1$ the real number of underlying off-source and on-source tests. The number of *actually* positive tests is R, given by S true positives and B spurious claims. An ideal experiment would neither treat background as signal (type I error) nor do the reverse (type II errors).
The probability of at least one false claim in a set of trials is known as *family-wise error rate* (FWER). It is not difficult to devise a method to control this quantity *before* going to the results: we just have to increase the confidence in the single trial (say 99%, or 99.99% coverage) in order to keep the FWER much lower than one. The drawback is that the resulting confidence interval would be much larger, and consequently the power of the search would fall dramatically. This is a consequence of the request that *not even in a single case* the null hypothesis is rejected when it is true.
A very reasonable compromise was suggested by Benjamini and Hochberg[@Benjamini95]. They remark that in many practical cases, when having one or more false claim is not by itself unacceptable, we could just be happy if –on average– *most* of the claims were real. In other words, they propose to bound FDR instead of FWER.
There are many topics in GW search which would benefit from this kind of procedure. For instance:
- all sky surveys: many source directions and polarizations are tried in parallel;
- template banks;
- eyes-wide-open searches: many alternative analysis pipelines, with different amplitude thresholds, signal duration, and so on are applied on the same data.
- periodic updates of results: every new science run is a chance for a “discovery" (“Maybe next one is the good one");
- Many graphical representations or aggregations of the data (“With a slight change in the binning, the ‘signal’ shows up better")
This work means not to be a complete review of the state-of-art techniques about FDR control, but hopefully it will be a stimulus for whoever is involved in multiple-test data analysis issues.
In the following sections, we shall use the notation reported in .
Description of the method {#sec:description}
=========================
Preliminary remarks
-------------------
In order to decide whether the results of a measurement are compatible with being generated by noise only (*null hypothesis*, H0) or instead they contain a signal (*alternative hypothesis*, H1) the textbook procedure is to set up a test statistic $t$ from the measures themselves. If $F_0(t)$ is the distribution of $t$ when the H0 holds, then the $p$-value of $t$ is defined as $p = F_0(t) = \Pr(t_0>t|\forall t_0)$. By construction $p$ is uniformly distributed between 0 and 1:
$$\label{eq:background}
\Pr(p<p_0|0 \le p_0 \le 1)=p_0$$
It is of paramount importance that the distribution $F_0$ is known. It is always wise to check a priori models with a goodness-of-fit test, when there are enough off-source data available. This is not always the case, but often there are surrogate procedures (e.g. data permutation) which give fresh independent samples of the background process, removing at the same time the effect of real signals, if any are present in the data. For instance, in the case of IGEC, the *resampling procedure* consisted in adding a delay to the time reference of one of the detectors in the network, such that the coincident signal is lost, while the background expectation value of coincidence counts is approximately unchanged. In case the data are not compliant with the model, at worse resampled data may allow to estimate $F_0$ by empirical fit.
As for H1, it is usually unknown, but for our purposes it is sufficient assuming that the signal can be distinguished from the noise, i.e. $\Pr(p<p_0|0 \le p_0 \le 1) \ne p_0$. The sketch in (*top left*) illustrates the concept.
For a single hypothesis test, the condition “reject null if $p<\alpha$" leads to false positives with probability $\alpha$. In case of multiple tests, we deal with a set ${\bf p} \equiv \{p_1, p_2, \dots p_m\}$ of $p$-levels, which need not to derive from the same test statistics, nor they should refer to same tested null hypothesis. $m$ is called the trial factor. We select discoveries using a threshold $T({\bf p})$: “reject null if $p_j<T({\bf p})$".
Controlling Type I errors (B)
-----------------------------
The *uncorrected testing* would just use the same threshold for each test: $T({\bf p})= \alpha$. The probability that at least one rejection is wrong grows as $P(B>0) = 1- (1- \alpha)^m \approx m\alpha$.
Therefore, as in the IGEC case, false discovery is guaranteed for $m$ large enough.
The other extreme solution, usually referred to as the *Bonferroni procedure* [@Bonferroni], controls the FWER in the most stringent manner, by requiring that $P(B>0) \le \alpha$. This is achieved by the choice $T({\bf p})= \alpha/m$ While this approach makes mistakes rare, the cost is low efficiency ($S\approx0$).
Controlling false discovery rate (B/R) {#ssec:fdr}
--------------------------------------
In order to trade-off between B=0 and S=0, the FDR control focuses on the ratio of false discoveries to the total number of claims:
$$\label{eq:fdr}
% MathType!MTEF!2!1!+-
% feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l
% bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R
% Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
% caGaaeqabaaaamaaaOqaaiaadAeacaWGebGaamOuaiabggMi6oaace
% aabaqbaeqabiGaaaqaamaalyaabaGaamOqaaqaaiaadkfaaaGaeyyy
% IO7aaSGbaeaacaWGcbaabaGaaiikaiaadkeacqGHRaWkcaWGtbGaai
% ykaaaaaeaacaqGPbGaaeOzaiaabccacaWGsbGaeyOpa4JaaGimaaqa
% aiaaicdaaeaacaqGPbGaaeOzaiaabccacaWGsbGaeyypa0JaaGimaa
% aaaiaawUhaaaaa!4A10!
FDR \equiv \left\{ {\begin{array}{*{20}c}
{{B \mathord{\left/
{\vphantom {B R}} \right.
\kern-\nulldelimiterspace} R} \equiv {B \mathord{\left/
{\vphantom {B {(B + S)}}} \right.
\kern-\nulldelimiterspace} {(B + S)}}} & {{\rm if }R > 0} \\
0 & {{\rm if }R = 0} \\
\end{array}} \right.$$
This can be done with a proper choice of $T({\bf p})$. The original procedure suggested by Benjamini and Hochberg (BH) is extremely simple, involving only trivial algebraic operations. It consists in the following steps.
- sort the $p$-values in ascending order: $\{p_1, p_2, \dots p_m | i < j \Rightarrow p_i \le p_j \}$;
- choose your desired FDR $q$ (in case no signal source is actually present during the observation, then the procedure is equivalent to the Bonferroni procedure with $\alpha=q$);
- define $c(m)=1$ if $p$-values are independent or positively correlated; otherwise $c(m) = \sum\nolimits_{j = 1}^m {{1 \mathord{\left/ {\vphantom {1 j}} \right. \kern-\nulldelimiterspace} j}} $;
- determine the threshold $T({\bf p})=p_j$ by finding the index $j$ such that $p_k>k(q/m) /c(m)$ when $k>j$ (see for a visual representation of this condition).
The above procedure with $c(m)=1$ was shown [@Benjamini95] to control the expectation value[^1] of FDR at least at level $q$ in the case when all $m$ tests are independent. However, it was later proved to control FDR when tests are positively correlated [@Benjamini01] (for instance, multivariate normal data where the covariance matrix has all positive elements). The alternative definition of $c(m)$ given above controls FDR in the most general case [@Benjamini01], but at the cost of reduced efficiency.
There is a nice back-of-the-envelope plausibility argument which can be found in [@Miller01] for the simple case when signals are easily separable (e.g. signals with high signal-to-noise ratios). In this case we expect their $p$-level to be very low, and correspondingly in the cumulative histogram of $p$-levels we shall see a step with height $S$ near $p \approx 0$, see (*bottom right*). We see also that there is only one intersection point for the BH procedure, such that $$\label{eq:miller}
T({\bf p})/R = q/m$$ On the other hand, the threshold $T({\bf p})$ can be expressed on average by $B/m_0$ (this is a special case of ). Substituting this value in we obtain $$B/R = q m_0/m \le q$$ For a rigorous proof see [@Benjamini01].
 The probability density function of $p$-values when data come from a mixed model can be thought as the sum of a uniform distribution (background) and a biased one (signal). (*b*) The Benjamini-Hochberg procedure (BH) consists in plotting the cumulative histogram of the $p$-values of the $m$ trials (*continuos line*) and looking for intersections with a line drawn from the origin and with slope equal to $m \cdot c(m)/q$ (*dashed line*). The null hypothesis is rejected for all data with $p$-value between 0 and the abscissa of the highest intersection point.\
(*c*) Sketch of histogram of $p$-values and (*d*) corresponding cumulative histogram, in a case of easily separable signals. The BH procedure applied to this case can be easily shown to control FDR (see for details).](fig1.eps "fig:"){width="8cm"}  The probability density function of $p$-values when data come from a mixed model can be thought as the sum of a uniform distribution (background) and a biased one (signal). (*b*) The Benjamini-Hochberg procedure (BH) consists in plotting the cumulative histogram of the $p$-values of the $m$ trials (*continuos line*) and looking for intersections with a line drawn from the origin and with slope equal to $m \cdot c(m)/q$ (*dashed line*). The null hypothesis is rejected for all data with $p$-value between 0 and the abscissa of the highest intersection point.\
(*c*) Sketch of histogram of $p$-values and (*d*) corresponding cumulative histogram, in a case of easily separable signals. The BH procedure applied to this case can be easily shown to control FDR (see for details).](fig2.eps "fig:"){width="7cm"}
Numerical test of the method {#sec:example}
============================
We now demonstrate this procedure with a simple example. Suppose we are given the results of 50 counting experiments, labeled by the index $i$. Their background is modeled as a Poisson random variable, with the same[^2] known expectation value $N_b$ for all $i$.
We consider two possible cases: in the first one, we draw 50 independent measures, in the other case we generate correlation by summing neighbor bins (i.e., if $n_c^i$ represent independent counts in the $i$-th bin, then the 50 correlated counts ${n'}_c^i$ are defined as ${n'}_c^i=n_c^i+n_c^{i-1}$, where $n_c^{0}\equiv n_c^{50}$). We investigated different background levels (from $N_b=0.01$ to $N_b=50$) and different number of detected signals ($N_s=0-6$), assuming –for sake of simplicity– that each bin can have either one or zero counts due to true signals.
In order to decide the presence of a signal we use the one-tail Poisson probability for the expected number of counts in each bin. In the results of a Monte Carlo simulation are shown. For each configuration (differing by average background and extent of true signals) we compute the average number of claims $R$, i.e. the number of bins for which the null hypothesis is rejected. We present the results for Bonferroni and BH tests, both tuned to bound the FWER at 1% when no signal is present.
Both procedures are working as expected, controlling the FWER and the FDR respectively at the desired level. For high background values they give as expected similar results. On the other side the efficiency of the Bonferroni procedure falls to zero for $N_b>0.01$, while the BH procedure is still effective, up to $N_b=0.05$ in this example.
In we can visualize how the BH procedure manage to grasp the signals promptly, as the background level lowers (see also ).
[ccccccccccccc]{} $N_s$ & **0.01** & **0.02** & **0.05** & **0.1** & **0.2** & **0.5** & **1** & **5** & **10** & **50**\
**0** & 0.005 &- &- & $10^{-4}$ & 3$\cdot10^{-4}$ & 0.003 & 0.007 & 0.008 & 0.003 & 0.004\
& 0.005 & 2$\cdot10^{-4}$ &- & $10^{-4}$ & 3$\cdot10^{-4}$ & 0.003 & 0.007 & 0.008 & 0.003 & 0.004\
**1** & 1.005 & 4$\cdot10^{-4}$ & 0.001 & 0.002 & 0.005 & 0.013 & 0.028 & 0.012 & 0.004 & 0.005\
& 1.010 & 0.019 & 0.001 & 0.002 & 0.005 & 0.013 & 0.028 & 0.012 & 0.004 & 0.005\
**2** & 2.004 & 5$\cdot10^{-4}$ & 0.002 & 0.004 & 0.009 & 0.021 & 0.047 & 0.016 & 0.005 & 0.005\
& 2.010 & 2.019 & 0.002 & 0.004 & 0.009 & 0.021 & 0.047 & 0.016 & 0.005 & 0.005\
**3** & 3.005 & 0.001 & 0.003 & 0.006 & 0.013 & 0.032 & 0.069 & 0.021 & 0.006 & 0.005\
& 3.010 & 3.019 & 0.004 & 0.006 & 0.013 & 0.032 & 0.069 & 0.021 & 0.006 & 0.005\
**4** & 4.005 & 0.001 & 0.004 & 0.008 & 0.017 & 0.043 & 0.086 & 0.027 & 0.007 & 0.006\
& 4.010 & 4.018 & 0.125 & 0.008 & 0.017 & 0.043 & 0.086 & 0.027 & 0.007 & 0.006\
**5** & 5.004 & 0.002 & 0.005 & 0.009 & 0.020 & 0.053 & 0.106 & 0.029 & 0.008 & 0.006\
& 5.009 & 5.018 & 5.046 & 0.009 & 0.020 & 0.053 & 0.106 & 0.029 & 0.008 & 0.006\
**6** & 6.004 & 0.002 & 0.006 & 0.013 & 0.024 & 0.061 & 0.124 & 0.034 & 0.010 & 0.007\
& 6.009 & 6.017 & 6.043 & 0.013 & 0.024 & 0.061 & 0.124 & 0.034 & 0.010 & 0.008\
[ccccccccccccc]{} $N_s$ & **0.01** & **0.02** & **0.05** & **0.1** & **0.2** & **0.5** & **1** & **5** & **10** & **50**\
**0** & 0.006 &- &- & $10^{-4}$ & 2$\cdot10^{-4}$ & 0.003 & 0.008 & 0.008 & 0.003 & 0.004\
& 0.010 & 0.010 &- & $10^{-4}$ & 2$\cdot10^{-4}$ & 0.003 & 0.009 & 0.008 & 0.003 & 0.004\
**1** & 1.005 & 3$\cdot10^{-4}$ & 0.001 & 0.002 & 0.005 & 0.013 & 0.030 & 0.012 & 0.004 & 0.005\
& 1.010 & 0.029 & 0.001 & 0.002 & 0.005 & 0.013 & 0.031 & 0.012 & 0.004 & 0.005\
**2** & 2.005 & 7$\cdot10^{-4}$ & 0.002 & 0.004 & 0.008 & 0.023 & 0.046 & 0.017 & 0.005 & 0.005\
& 2.009 & 2.018 & 0.006 & 0.004 & 0.008 & 0.023 & 0.047 & 0.017 & 0.005 & 0.006\
**3** & 3.004 & 0.001 & 0.003 & 0.005 & 0.012 & 0.032 & 0.067 & 0.022 & 0.006 & 0.005\
& 3.009 & 3.019 & 0.060 & 0.005 & 0.012 & 0.032 & 0.068 & 0.022 & 0.006 & 0.006\
**4** & 4.005 & 0.002 & 0.004 & 0.008 & 0.017 & 0.043 & 0.084 & 0.025 & 0.007 & 0.005\
& 4.009 & 4.017 & 0.143 & 0.008 & 0.017 & 0.043 & 0.085 & 0.025 & 0.007 & 0.006\
**5** & 5.004 & 0.002 & 0.005 & 0.010 & 0.019 & 0.051 & 0.107 & 0.029 & 0.008 & 0.007\
& 5.009 & 5.018 & 5.047 & 0.010 & 0.019 & 0.051 & 0.108 & 0.029 & 0.008 & 0.007\
**6** & 6.004 & 0.003 & 0.007 & 0.011 & 0.024 & 0.061 & 0.127 & 0.035 & 0.010 & 0.007\
& 6.009 & 6.016 & 6.044 & 0.013 & 0.024 & 0.061 & 0.127 & 0.035 & 0.010 & 0.007\
![\[fig:example\]A few samples from the Monte Carlo used to produce are displayed in detail. They refer to $N_s=5$, and the background is (*a*) $N_b=50$ (*b*) $N_b=0.5$ (*c*) $N_b=0.01$. In the plots above the cumulative histogram of the $p$-values is compared with the threshold given by the Bonferroni () and the BH () procedures.](fig3.eps){width="10.9cm"}
Conclusions
===========
When multiple tests are tried for the same data set, controlling FDR seems in general a wiser idea than just limiting type-I errors. Robust but simple procedures exist which (conservatively) control FDR in positively correlated tests, and also in the more general case (but at the cost of reduced efficiency).
This idea is relatively new in the astrophysics community, and we are not aware of any application in the GW community. Its application should be encouraged. Notice however that BH procedure is not the only one, and more complex –but approximate– strategies have been investigated (see for instance [@Storey02; @Yekutieli99]).
Acknowledgments
===============
We are indebted to James T. Linnemann (MSU) for introducing us to FDR.
Baggio acknowledges the hospitality of ICRR and the grant of Tokyo University.
References {#references .unnumbered}
==========
[1]{} Benjamini Y and Hochberg Y 1995 *J. Roy. Statist. Soc. Ser. B* **57** 289-300 Benjamini Y and Yekutieli D 2001 *Ann. Statist.* **29** 1165-88 Astone P 2003 [*Phys. Rev. D*]{} **68** 022001 Hochberg Y and Tamhane A C 1987 *Multiple Comparison Procedures* (New York: Wiley) Miller C J 2001 *A.J.* **122** 3492-505 Yekutieli D and Benjamini Y 1999 *J. Statist. Plann. Inference* **82** 171-96 Storey J D 2002 *J. Roy. Statist. Soc. Ser. B* **64** 479-98
[^1]: Of course, the quantity FDR is a random variable, as well as the $p$-values.
[^2]: To avoid degeneracy due to the discreteness of the test statistic (many results collapsing at the same $p$-values), we actually spread the background of the experiments in a range $\pm1\%$ around $N_b$).
|
---
abstract: |
In this presentation are discussed some problems, relevant with application of information technologies in nano-scale systems and devices. Some methods already developed in quantum information technologies may be very useful here. Here are considered two illustrative models: representation of data by quantum bits and transfer of signals in quantum wires.\
[**keywords–quantum; information; nanotechnologies**]{}
author:
- 'Alexander Yu. Vlasov'
title: '**Information Nano-Technologies: Transition from Classical to Quantum**'
---
Introduction {#Sec:Intro}
============
Let us recollect well known Feynman’s talks, relevant to presented theme. The first one, is the Caltech lecture “There’s plenty of room at the bottom” in 1959 [@plenty] often is considered between the origins of nanotecnnologies. The second one, is the keynote speech “Simulating physics with computers” [@feysim] in 1981 at the conference PhysComp’81 “Physics and Computations” about physical background of computing and information technologies. Main part of this talk was devoted to quantum processes.
In this speech was established some ideas, essential for the development of quantum computations and communication, but not only that. The simulation — is detailed modeling of a physical process. For quantum systems it is the especial challenge, because the formulations of the quantum theory is often similar with a “black box” [@ngnr; @vlnl] description.
A positive result of the research of [*physics of computations*]{} was understanding of principle possibility of information processing by devices with elements of the atomic size. Sometimes it was even necessary to critically revisit some widespread ideas. For example, elements in such a scale often may be more adopted for [*reversible*]{} operations, but most gates in standard computer design are [*irreversible*]{}.
Charles Bennett suggested a model of a reversible Turing machine and even denoted a similarity of such a model with DNA and RNA [@ben73]. The reversible Turing machine has direct generalization on quantum systems and it was demonstrated in few works of Paul Benioff, including the presentation on already mentioned PhysComp’81 [@ben82]. Feynman’s representation is more close to the modern description of computers by gates and circuits, but uses specific attributes of quantum mechanics [@feysim].
In Section \[Sec:qubits\] is reminded an abstract quantum analogue of classical bit. In Section \[Sec:quprop\] are briefly discussed distribution of signals in nanosystems and relevant quantum effects. In Section \[Sec:perf\] is revisited so-called perfect state transfer. Section \[Sec:Weyl\] is devoted to Weyl commutation relations.
Quantum bits {#Sec:qubits}
============
There is widespread notation $|0\rangle$ and $|1\rangle$ for two basic states of a quantum system, which often is called [*quantum bit*]{} or [*“qubit”*]{}. Feynman had used for manipulations with a qubit expressions with formal operators of [*annihilation*]{} and [*creation*]{}: ${\bf a}|1\rangle = |0\rangle$, ${\bf a}^*|0\rangle = |1\rangle$, ${\bf a}\,{\bf a}^* + {\bf a}^* {\bf a} = {\bf 1}$, ${\bf a}^2 =({\bf a}^*)^2 = {\bf 0}$.
Let us consider a set with eight qubits. Basic states of such a “quantum byte” may be described as strings of zeros and units: $|00000000\rangle$, $|00000001\rangle$, $\ldots$, $|11111111\rangle$. It can be simply estimated, there are $2^8 = 256$ basic states or $2^N$ for a system with $N$ particles. It is in agreement rather with the principles of quantum mechanics, than with the classical case. In “computer notation” it is clear enough even without more pedantic consideration of tensor product of linear spaces describing a state of the quantum system.
An illustrative classical picture still exists for one qubit and any state may be represented by direction of some “arrow,” like two basic states: “spin up” and “spin down”. For a classical case description of such a system also demands two parameters, [*e.g.*]{}, the Euler angles. But this visual correspondence disappears in a case with few systems, because in the classical world for description of $N$ “arrows” it would be necessary to use only $2N$ parameters instead of $2^N$.
Of course, classical bits may be represented in the classical model, as a discrete set with $2^N$ elements inside of a space with $2N$ [*continuous*]{} parameters. The quantum model with $2^N$ parameters also includes this set (Figure \[fig\_chain\]a), and here each element [*directly*]{} conforms to a continuous parameter.
a) ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:")
b) ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spinxket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin0ket.eps "fig:") ![Quantum spin chains[]{data-label="fig_chain"}](spin1ket.eps "fig:")
The quantum model corresponds to a classical one, if only states of [*separate*]{} qubits may differ from two fixed options of usual bit (Figure \[fig\_chain\]b). The difference $2^N\!\! - 2N$ is an approximate estimation of [*“non-classicality”*]{} and it grows very fast with the number of systems $N$.
*This consideration of complexity has relevance with presented theme, because the nano-scale domain describes aggregates with more than one quantum system, but it is still not big enough to use statistical laws. The quantum theory of information provides the convenient language for description of systems with not very big amount of elements due to appropriate level of abstraction.*
For example, the same model may be applied to different quantum systems. A spin one-half system was used in the visual picture above, but the [*qubit*]{} is a model for many other systems with two states, like photons or quantum dots. Multi-qubit systems like “quantum byte” also may be associated not only with spin chain (Figure \[fig\_chain\]), but with quantum dots arrays (Figure \[fig\_ddots\]) and other implementations [@qcroad].
![Quantum (double) dots[]{data-label="fig_ddots"}](ddotsn.eps)
Quantum signal propagation in nanosystems {#Sec:quprop}
=========================================
Let us discuss now application of some ideas to next generations of nanotechnologies. Such devices are still in a state of development and it may be reasonable to pay attention to processes in biosystems. Recent time active research is carried out with respect to descendants of most ancient “nanodevices” existing on the Earth about three milliards years or so.
It is the light-harvesting complex of some microorganisms. The importance of quantum effects for this case is already almost impossible to deny. The significant contribution for understanding here is due to works with participation of experts in quantum theory of information [@jo08; @ll08; @pl09; @wh09].
Let us consider a problem of the effective transfer of absorbed photon energy to different elements of a nanosystem. In the biological systems mentioned above the effectiveness may be about 99%. It is astonishing with taking into consideration of quantum uncertainty, because it apparently should hinder the optimal transfer.
Yet biophysical processes in such systems formally look as not relevant with information transfer, a set of problems and methods applied there are very similar with the statement of a question about an effective transmission of signals in a nanodevice with taking into account of quantum effects. [*E.g.*]{}, in paper [@ll08] is suggested a transfer model based on a quantum analogue of the random walk. In the classical case a chaotic motion may be considered as a quite effective way of a transport in complicated compact systems.
Uncertainty of positions and trajectories in quantum mechanics needs for the special consideration. There is well known approach with the suggestion about state localization due to the interaction with the environment. It is one possible explanation of the transition from quantum to classical world [@zur03]. Related ideas about decoherence assisted transport may be more or less directly used in some models about effective energy transfer in the light-harvesting complexes [@ll08; @pl09].
Some classical models are tested very well for the macroscopic level and it is not clear at that scale they are still work for nanosystems. It is reasonable to check a possibility of description of the effective transfer without appealing to semi-empiric regularities acting on the boundary between quantum and classical world [@zur03].
Indeed, the possibility of the “perfect” transfer of an excitation in “purely” quantum approach was also found recently [@ek04]. Similar methods was only briefly mentioned in the relation with the biophysical systems discussed above [@wh09].
In the works [@vl07; @vl08] were also considered some aspects of this approach, appropriate to the present discussion. It can be said, the model of perfect transfer is an analogue of shift register: $10000000$ $\rightarrow$ $01000000$ $\rightarrow$ $\ldots$ $\rightarrow$ $00000001$. Here unit corresponds to the excited state. Coefficients describing strength of interactions between adjacent nodes of the chain may be chosen in such a way, to ensure localization of excitation only for two ends of chain and perfect transfer from first to last node [@ek04].
Let us recollect some essential ideas. The quantum information science is most often related with quantum systems with [*finite*]{} number of (basic) states and it was quite clear from examples with qubits above. In more general case the term [*qudit*]{} is often used for a quantum system with $d$ states, [*e.g.*]{}, a particle with spin $s$ corresponds to $d=2s+1$. Qubit is the particular case with $d=2$ and $s=1/2$.
Other model of qudit is some particle in a lattice with $d$ locations. Two simple examples with $d$ nodes are [*ring*]{} (Figure \[fig\_latt\]a) [@FLP3 Chap. 15-4] and a [*line*]{} (Figure \[fig\_latt\]b) [@FLP3 Chap. 15-5]. If we consider a [*single*]{} electron in such a circular or linear system, the wave numbers $k_j$ of stationary states may be expressed as $k_j b = 2\pi j / d $ and $k_j b = \pi j / (d + 1) $ respectively [@FLP3 Chap. 15]. The energies in both cases are $$E_j = E_0 - 2A \cos (k_j b).
\label{Ek}$$ Here $b$ is distance between atoms and $A$ is amplitude of transition.
a)![Models with lattices[]{data-label="fig_latt"}](ring.eps "fig:") b)![Models with lattices[]{data-label="fig_latt"}](line.eps "fig:")
Understanding perfect state transfer {#Sec:perf}
====================================
Such chains may be used for quantum communications [@bo03], but a nonlinear dispersion law like Eq. (\[Ek\]) may be considered as a certain obstacle for the good transmission. Already mentioned earlier [*perfect*]{} scheme of transport has [*varying*]{} amplitudes of transition [@ek04] $$A_j = A \sqrt{j\,(d-j)}.
\label{Aj}$$
Such a model may appear more natural, if to consider a simpler equivalent [@vl07]. Indeed, in the continuous case the ideal transmission of a signal might be obtained with the linear law of dispersion $E(k) \propto k$. For the quantum case with the discrete lattice there is similar approach.
Let us denote a state with occupation of only $l$’th node as $|l\rangle$, [*cf*]{} [@FLP3 Figure 13-1]. A spin chain also may be used [@ek04] instead of the lattice with $d$ states. Yet, the chain has $2^d$ basic states (Figure \[fig\_chain\]a), only $d$-dimensional subspace is used [@ek04] and it illustrates rather standard correspondence between such lattices [@FLP3 Chap. 13] and spin waves in chains with exchange interaction [@FLP3 Chap. 15].
If to consider a [*ring*]{} (Figure \[fig\_latt\]a) with $d$ nodes, the ideal scheme of transfer could be described via [*cyclic shift*]{} operator $${\bf U} : |l\rangle \mapsto |\,l + 1 \mod d\rangle,
\quad l = 0,\ldots,d-1.
\label{shift}$$
Eigenvectors and eigenvalues of the operator may be simply found $${\bf U} |\kappa_j\rangle = \zeta_j |\kappa_j\rangle, \quad
\zeta_j = \exp\bigl(\frac{2\pi i}{d}\, j\bigr),
\label{Uk}$$ $$|\kappa_j\rangle =
\frac{1}{\sqrt{d}} \sum_{l=0}^{d-1} \exp\bigl(\frac{2\pi i}{d}\, l j\bigr)\, |l\rangle
\label{mom}$$ and produce “momentum” basis with $d$ states.
Similar states were already mentioned above in relation with a “molecular” ring, Figure \[fig\_latt\]. These states had the fixed wave number $k$. A simplest analogy of a continuous model with linear dispersion may be provided by Hamiltonian with eigenvectors $|\kappa_j\rangle$ described by Eq. (\[mom\]) and equidistant eigenvalues, [*i.e.*]{}, $${\bf H}\, |\kappa_j\rangle = \hbar\theta j \, |\kappa_j\rangle
\label{Hkj}$$ with unessential constant $\theta$.
Evolution of a system due to such Hamiltonian in the same basis may be expressed via a diagonal matrix, [*i.e.*]{}, $$|\kappa_j\rangle \mapsto \exp(-i \theta j t) |\kappa_j\rangle.
\label{jt}$$ It is convenient to choose time step $$\Delta t = \frac{2 \pi}{\theta\, d},
\label{Delt}$$ because the matrix of evolution of the quantum system for such a time period coincides with introduced earlier Eq. (\[shift\]) matrix of cyclic shift $${\bf U} = \exp(\frac{{\bf H} \Delta t}{i\hbar}),
\label{Uexp}$$ [*i.e.*]{}, it is expressed in basis $|\kappa_j\rangle$ as $${\bf U} : |\kappa_j\rangle \mapsto
\exp\bigl(\frac{2\pi i}{d}\, j\bigr)\, |\kappa_j\rangle.
\tag{\ref{Uk}$^\prime$}$$ So, on the one hand, ${\bf U}$ is a discrete analogue of operator with linear dispersion, on the other one, it ensures perfect transmission Eq. (\[shift\]) of local state along chain.
The advantage of such approach is very clear law of evolution Eq. (\[shift\]), but in the basis $|l\rangle$ there is no simple expression like Eq. (\[Hkj\]) for Hamiltonian used in Eq. (\[Uexp\]). The Hamiltonian in such basis may be found [@vl07 Eq. (10)], but it has nonzero values of transition amplitudes for any two sites, unlike initial models with nonzero elements only for adjacent locations [@FLP3].
It is useful to look for Hamiltonians with the same equidistant spectra, but nonzero values of transition only for $j \pm 1$. Analogues of angular momentum components operators like ${\bf J}_x$ or ${\bf J}_y$ have necessary properties and in [@ek04] was used such a formal Hamiltonian with only nonzero elements $\grave{H}_{j,j\pm 1}$ corresponding to Eq. (\[Aj\]) and proportional to ${\bf J}_x$ for some fictitious particle with spin $s=(d - 1)/2$ $$\setlength{\arraycolsep}{1pt}
\grave{\bf H} =
\vartheta\hbar\!\!
\begin{pmatrix}
0 \ \quad \sqrt{d-1} \ \quad 0 &\cdots\! & 0 \ \ \qquad 0 \\
\sqrt{d-1} \quad 0 \ \sqrt{2 (d-2)}& \cdots\! & 0 \ \ \qquad 0\\
0 \ \ \sqrt{2 (d-2)} \ \ 0 & \cdots\! & 0 \ \ \qquad 0 \\
\dotfill\ddots &\ddots &\ddots\dotfill \\
0 \ \ \qquad 0 \ \ \qquad 0 \, & \cdots\! & \ \ 0 \ \ \sqrt{d-1} \\
0 \ \ \qquad 0 \ \ \qquad 0 \, & \cdots\! & \sqrt{d-1} \ \ 0 \ \ %
\end{pmatrix}.
\label{Hj}$$
Evolution of a system with such a Hamiltonian is described by operator $${\bf R}(t) = \exp(\frac{\grave{\bf H} t}{i \hbar})
= \exp(\frac{\vartheta{\bf J}_x t}{i\hbar}).
\label{Rx}$$ It coincides with a revolution generated in $d$-dimensional representation of rotation group by ${\bf J}_x$ and familiar from theory of angular momentum [@FLP3 Chapt. 18]. Here is discussed a linear chain, but due to such representation there is an analogy with a ring — it can be considered formally as a chain with reflection on the boundaries.
There is a subtle problem, because instead of one-way transmission the state is rather circulating between two ends of the lattice with period $\pi/\vartheta$. It may be resolved by controlled state transfer, like [*quantum bots (qubots)*]{} discussed in [@vl07] or more cumbersome schemes.
The consideration above illustrates application of general methods for description of information transfer in the different types of [*quantum wires*]{}. From the one hand it takes into account quite specific properties of quantum systems, from the other one it draws a parallel with the traditional information science.
Weyl commutation relations {#Sec:Weyl}
==========================
Let us return to the question about problems with quantum uncertainty. The operator of shift ${\bf U}$ Eq. (\[shift\]) is an important attribute of quantum mechanics and used in so-called Weyl commutation relations [@WeylGQM].
For the continuous case such a formalism has the direct correspondence with the Heisenberg commutation relations [@WeylGQM; @QFT]. Indeed in some cases it is necessary to use the Heisenberg uncertainty relation with proper care due to subtleties with definition of domains of operators [@QFT; @PerQT], [*e.g.*]{}, for a ring with the periodic coordinate $0 \le q < 2 \pi$ the value $\Delta q$ is always finite, but $\Delta p = 0$ for eigenfunctions of ${\bf p}$, [*i.e.*]{}, $$u_k(q) = (2 \pi)^{-1/2} \exp(i k q),
\label{ukq}$$ and so $\Delta p \,\Delta q = 0$ [@QFT; @PerQT]. Here $|u_k(q)|^2$ is constant and $\Delta q = \pi/\sqrt{3}$ — it coincides with the standard deviation of the random variable $0 \le q < 2\pi$ with the uniform distribution.
More details may be found in [@PerQT Chapt. 4-3] (with notation $\theta$ and $p_\theta$ for ${\bf q}$ and ${\bf p}$ respectively). It should be mentioned, that operators ${\bf q}$ and ${\bf p}$ above were not “usual” coordinate and momentum for a particle [*on a line*]{} with uncertainty relation $\Delta p \,\Delta q \ge \hbar/2$. For a line $\Delta q$ is not limited by some fixed value and for $\Delta p \to 0$ we would have “unrestricted” plane waves instead of Eq. (\[ukq\]), [*i.e.*]{}, $\Delta q \to \infty$. Here the limit is an uniform distribution on an infinite line $-\infty < q < \infty$, instead of a bounded ring.
Qubit and qudit are said to be “discrete quantum variables” widely used in the quantum information science. For such systems the problem with proper definition of coordinate, momentum operators and analogues of uncertainty relations could look even worst, than for continuous ring, but [*Weyl quantization*]{} [@WeylGQM] may help to resolve that. The idea is to consider exponents of coordinate and momentum operators [@WeylGQM; @QFT] $${\bf U}(a) = \exp(i a {\bf p}), \quad
{\bf V}(b) = \exp(i b {\bf q}).
\label{UV}$$
In the continuous case the actions of such operators on a wave function $\psi(x)$ are represented as $${\bf U}(a) \psi(x) = \psi(x+\hbar a),\quad
{\bf V}(b) \psi(x) = e^{i b x} \psi(x),
\label{UVpsi}$$ and direct consequence of Heisenberg commutation relation $$[{\bf q},{\bf p}] \equiv
{\bf q} {\bf p} - {\bf p} {\bf q} = i \hbar
\label{xpcom}$$ is Weyl commutation relation [@WeylGQM; @QFT] $${\bf U}(a) {\bf V}(b) = \exp(i \hbar a b) {\bf V}(b) {\bf U}(a).
\label{UVcom}$$
Such a scheme has some advantage because [*Weyl pair*]{} ${\bf U}$, ${\bf V}$ with appropriate properties may be formally written even if ${\bf p}$ and ${\bf q}$ are not (well) defined. It may be even used for discrete quantum variables [@WeylGQM] and it was already reproduced above in example with operator of shift [**U**]{} Eq. (\[shift\]). The second operator in this case is $${\bf V} : |l\rangle \mapsto
\exp\bigl(\frac{2\pi i}{d}\, l\bigr)\, |l\rangle.
\label{clock}$$ So, ${\bf U}$ and ${\bf V}$ are $d{\times}d$ matrixes with commutation relation $${\bf U} {\bf V} = \exp\bigl(\frac{2\pi i}{d}\bigr)\, {\bf V} {\bf U}.
\label{UVVU}$$
These matrixes together with relation Eq. (\[UVVU\]) were introduced by Weyl [@WeylGQM]. Really, the “shift” ${\bf U}$ and “clock” ${\bf V}$ matrixes were considered even earlier in few works of J. J. Sylvester around 1882–1884. In quantum information science they are also known as “generalized Pauli matrixes” with an alternative notation ${\bf X}$ and ${\bf Z}$ [@high].
Other examples may be found elsewhere [@low]. Let us only consider less formally questions about uncertainty for discrete quantum variables. Famous Stern-Gerlach experiment demonstrates only two possible projections on some axis for spin one-half. For spin $s$ there are $2s+1$ projections. It is just obvious statement about [*quantization of angular momentum*]{}.
A belief about inevitable problems with quantum transport due to uncertainties of trajectories related with lack of examples with similar effects for some spatial properties of quantum systems. It is more common to expect quantization for energy levels, angular momentum, [*etc.*]{} Yet, in quantum information science were quite natural formal models with discrete spatial variables, [*e.g.*]{}, quantum cellular automata, quantum lattice gases [@RW95; @Mey96], [*etc.*]{}
Conclusion
==========
The quantum information technologies may be useful for construction of difficult nano-technology devices, because they are providing universal and compact way of understanding different processes with “systems of quantum systems.” It is an analogy with the application of usual information technologies for description in symbolic form of classical processes and objects.
In the paper were recollected few simple models: the representation of data by qubits and signal transfer in small quantum systems. These models may be quite familiar in area of quantum computing, but it should be emphasized, that main purpose of present work — is [*not*]{} theory of quantum algorithms adapted for cryptography. Even usual electronic computers were initially constructed for code-breaking and plain calculations, but nowadays they work as well in absolutely different areas.
It should be mentioned also, that this presentation is [*not*]{} concentrated on restricted question, how nanotechnologies could help to build a quantum computer to crack some ciphers. It is rather analyzed, how “quantum-computer-type of thinking” may help to understand and control nano-scale systems and devices.
0.55cm
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|
\#1[ ]{} \#1\#2 §[[**S**]{}]{}
[Global Demons in Field Theory :]{}\
\
Dimitri KUSNEZOV[^1]\
\
[*Yale University, New Haven, CT 06511-8167*]{}\
John SLOAN[^2]\
[*July 1992*]{}
[1. Introduction]{}
One traditional dynamical approach to simulating ensemble averages has been molecular dynamics (MD) algorithms. In the simplest of these, micro-canonical simulations, conjugate momenta are introduced for each degree of freedom in the ensemble, and the resulting system is time-evolved according to Hamilton’s equations of motion. The reversibility of Hamiltonian evolution then ensures detailed balance, i.e. that the simulation is a Markov process. If the system is sufficiently large, and the interactions sufficiently complex, one usually imposes the quasi-ergodic hypothesis, and hopes that the simulation will explore the desired ensemble. Unfortunately, it is known that Hamiltonian evolution conserves energy, and is therefore not ergodic. In fact, microcanonical simulations introduce an explicit factor of $\delta(H-E)$ into the measure of the ensemble being simulated. In order to use an MD algorithm to obtain the correct ensemble, some additional method must be introduced to integrate over the different energy surfaces $E$.
One method of dealing with this difficulty is embodied in the hybrid molecular dynamics (HMD) and hybrid Monte Carlo (HMC) algorithms[@hmca]–[@hmcc]. In these algorithms, the micro-canonical equations of motion are integrated along a ‘trajectory’ for a time $T$, after which the momenta are touched with a heat bath, changing the energy of the system. As with any numerical integration of Hamilton’s equations, finite step size errors will build up along the micro-canonical trajectories, leading to systematic errors in the ensemble generated by HMD. Although these errors can be controlled by making the step size sufficiently small, this can become costly. The HMC algorithm is designed to correct for these $dt$ errors. It does this by treating the configuration at the end of a micro-canonical trajectory as a proposal for a global update of the system, which is then accepted or rejected according to a Metropolis hit. If the equations of motion are reversible, this sequence of configurations is then a Markov chain which, given ergodicity, is guaranteed to produce the correct ensemble. The HMC and HMD algorithms are currently widely used in lattice gauge theory simulations, especially in systems involving dynamical fermions. Typically, HMC is used in theories where the action can be expressed as the volume integral of a local function, while HMD is used when it cannot.
Although these algorithms are generally quite robust, they do have one weakness, related to critical slowing down. Associated with any observable $O$ is an autocorrelation ‘time’ $\tau_O$, the time scale required for the simulation to produce a statistically independent measurement of $O$. This autocorrelation time will generally depend on the correlation length of the system as a power law[@adk], $$\tau_O =A\xi^z.\label{eq:texp}$$ where $z$ is the dynamical critical exponent, and $A$ and $z$ will depend on $O$. Critical slowing down occurs whenever $z>0$. $\tau_O$ represents the typical amount of simulation time it takes for a local change in $O$ to propagate across a correlated cluster. For multi-scale algorithms, such as cluster algorithms, one can hope to obtain $z=0$, since these algorithms are designed to change an entire correlated cluster simultaneously. Unfortunately, these algorithms are not easily generalized from one model to another, and have not yet been implemented for most lattice gauge theory models. At the other extreme, it is dangerously easy to obtain $z=2$ with an algorithm which involves only local updates. This just corresponds to diffusive transport of fluctuations through the correlated cluster. In principle, one should be able to do no better than $z=1$ with a local algorithm, since the local algorithm will restrict fluctuations to a finite propagation speed[@rutgers]. One can obtain $z=1$[@adk; @ktc] in HMC and HMD, but to do so requires a [*correlation length dependant tuning of the trajectory length.*]{} If the trajectories are too short, then the frequent randomization of the momenta causes the motion of the system through phase space to be diffusive (resulting in a $z$ of $2$), while if the trajectories are too long the quality of statistics goes down because energy conservation correlates the measurements along each trajectory. In addition, the optimum trajectory length is likely to differ for different observables, forcing an inefficient trajectory length for some of them. (This problem is especially severe for HMC, where only one measurement is allowed per trajectory. Running with trajectories twice as long as is necessary is therefore equivalent to a factor of two slow-down in the code.)
We would like to contrast these hybrid algorithms based on micro-canonical evolution to a purely dynamical approach we term [*global demons*]{}. Like the hybrid algorithms, global demons are an easily implemented, very general approach to simulating ensemble averages. Unlike the hybrid algorithms, the global demon formulation has nothing to do with Hamiltonian dynamics. For example, it can be defined completely in terms of coordinates alone if so desired: the presence or absence of a symplectic structure is irrelevant. This should be contrasted to the MD based algorithms, where a partition function whose action depends only on coordinates is usually augmented to include fictitious conjugate momenta in order to define a Hamiltonian or Poisson structure. Another way to say this is that while evolution using Hamiltonian dynamics generates a microcanonical ensemble, evolution in the global demon system generates a canonical ensemble. The global demon equations of motion are deterministic and time reversal invariant, and are designed to evolve through the physically accessible regions of configuration space (it is [*not*]{} a phase space) such that the trajectory fills configuration space with a density reproducing the correct ensemble. Consider, as an illustration, 1000 points in a phase space $(q,p)$ which lie equally spaced on a unit circle, as shown in the left column of Fig. 1, at $t=0$. The points are connected in order to see how neighboring points behave. Under microcanonical evolution of the 1-d harmonic oscillator equations of motion ($t=1$ is the natural time scale of the dynamics), $$\dot{q} = p,\qquad \dot{p} = -q \; ,$$ this circle will be preserved, and the points will rotate, preserving the figure at all later times. In contrast to Hamilton’s dynamics, global demon dynamics will result in a rapid dissemination of neighboring points through the space. The time evolution of the circle for a 1-d harmonic oscillator Hamiltonian $H=(p^2+q^2)/2$ is shown at time $t=10$ and $t=20$, in our canonical dynamics. What is striking is the speed at which neighboring points on the circle evolve to opposite sides of the the space. Even at $t=20$, one can see that the phase space density is nearing the desired ensemble $\exp(-\beta H)$. In the right column of Fig. 1, we have the same situation for an $SU(2)$ Hamiltonian $H=J_z^2/2$, whose phase space, parameterized by $(J_x,J_y,J_z)$, is the unit sphere. Here an initial condition of a circle at $J_z=0.5$ is evolved in a similar manner. Again, the rapid divergence of neighboring points is striking. Since global demons generate a deterministic chaotic dynamics, the danger of diffusive motion through phase space present in the hybrid and other stochastic algorithms is absent here. The one drawback to the global demons approach is that we have been unable to determine a way of making it exact, i.e. removing the $dt$ errors in the ensemble arising from finite step size.
In this article, we investigate an application of the global demon algorithm to a lattice field theory. We are interested in understanding the critical properties of the dynamics near phase transitions, and how tuning the dynamics can improve convergence. We also want to examine the correctness of measurements in this inexact dynamics, as well as ways to make it exact without recourse to stochastic techniques. We choose the 2D XY model, since it has been well studied in the past in a variety of algorithms, and has an infinite order phase transition around which we can investigate the manifestations of critical slowing down. In section 2, we present the global demon dynamics for unconstrained systems. The implementation of this dynamics for the XY model in section 3 explores the behavior of simulations as various parameters of the algorithm are tuned. We find that the algorithm is quite robust, obtaining good results with little tuning. In addition, we compare the critical behavior of the algorithm to several implementations of HMC. We conclude in section 4. Finally, in an appendix, we present a scheme which, in principle, should remove the finite $dt$ errors from the algorithm dynamically. Although our present implementation of the global demon algorithm is not exact, we have chosen to compare our results to HMC, rather than HMD. The main reason for this choice is that we are treating the HMC results as a control, and would like them to be as free of systematic errors as possible. Since the HMC and HMD algorithms are so similar, however, it is quite likely that qualitative conclusions about the critical behavior of HMC will also be correct for HMD.
[2. Global Demon Dynamics]{}
Let us consider a system characterized by an action $S(x)$ and coordinates $x=(x_1,...,x_n)$. The ensemble averages of this system will have the generic form $$\VEV{\co}= \frac{1}{ Z}\int\;{\cal D}\mu(x)\;e^{-\beta S(x)}\co,\label{eq:ens}$$ where $$Z = \int\; {\cal D}\mu(x)\; e^{-\beta S(x)},$$ is the normalization, and the measure ${\cal D}\mu(x)$ might include constraints (for example, symmetries associated with a Lie algebra). In this article, we are only concerned with the situation when the measure is trivial, ${\cal D}x$. If the variables $x=(q,p)$ include canonically conjugate coordinates and momenta, $S(x)$ can be taken as a Hamiltonian: $H(p,q)=S(x)$. Otherwise, in what is more or less standard practice, conjugate momenta are introduced and added to $S(x)$ to make the exponent in Eq. (\[eq:ens\]) resemble a Hamiltonian $$H(p,x) = \frac{1}{ 2}\sum_{i=1}^n p_i^2 + S(x),$$ and the measure is modified to ${\cal D}x{\cal D}p\exp(-\beta H)$, with appropriate normalization. Molecular dynamics is now easily implemented, leading to the equation of motion $$\dot{x_i} = p_i\, ,\qquad \dot{p_i} = -\partder{S(x)}{x_i},\qquad
(i=1,...,n).\label{eq:mcd}$$ This dynamics, combined with momentum refreshes and global metropolis hits, produce the HMD and HMC algorithms.
Let us now pass to the canonical dynamics for such a system[@cmda]–[@cmdc]. In contrast to the microcanonical dynamics, the energy and the symplectic structure are no longer preserved. Rather, the measure itself is preserved directly by the dynamics. We can define such dynamics in many ways. For instance, instead of Eq. (\[eq:mcd\]), we could take
$$\dot{x}_i = - \kappa\, \frac{\beta}{ n}\, \frac{dG(w)}{ dw}\;
F_{i}(x)\, ,\qquad (i=1,...,n).$$
Here $G(w)$ and $\fax$ are arbitrary functions of a global demon variable $w$ and coordinates $x$, respectively, and $\ka$ is a coupling constant. The number of global demon interactions (the right hand side of (7a)) is unrestricted. In this example we have used 1, while 2 or 3 are usually sufficient, regardless of $n$. This type of treatment can be viewed as a deterministic version of Parisi and Wu’s stochastic quantization[@pwu; @cmdb]. An alternative formulation of the $x$ dynamics which includes a relic of the underlying Hamiltonian $H(p,x)$ is $$\begin{aligned}
\dot{x}_i&= &p_i - \frac{\kappa_1\beta}{ n}\; \frac{dG_1}{dw_1}\; F_{1i}(x),\\
\dot{p}_i &= &-\partder{S(x)}{x_i} - \frac{\kappa_2\beta}{ n}\;
\frac{dG_2}{dw_2}\; F_{2i}(p).\end{aligned}$$
An important observation here is that while we can retain a Hamiltonian sub-structure to the dynamics, it is not responsible for the ergodicity in the full configuration space, and can be retained or altogether removed. This will have some effect on the convergence, since the Hamiltonian forces can provide additional decorrelation. In non-equilibrium simulations, it is more convenient to use (7b-c) since there is a closer link to the thermodynamics of $H(p,x)$[@cmdd]. Eqs. (7b-c) also have a microcanonical limit when $\kappa_\alpha=0$. Since we are going to compare the global demon approach to HMC, we retain the momenta to have a greater parity between the two algorithms.
With the introduction of the global demons $w_\alpha$, a larger configuration space $\{\phi\}$ must be defined, where $\phi=(x_1,...,p_n,w_1,...w_m)$ . In $\phi-$space we can define a new action $f$, which is determined by the equations of motion (7) in the following way: $$f(x,p,w) = S(x) + \frac{1}{ 2}\sum_{i=1}^n p_i^2 + \sum_{\alpha=1}^m
G_\alpha(w_\alpha),\qquad\quad \rho_f=e^{-\beta f}.\label{eq:meas}$$ Unlike $H(p,x)$ in molecular dynamics, $f$ is [*not*]{} preserved by the equations of motion. While the definition of $f$ is not unique (in the sense that the measure for the variables $w$ is arbitrary), it is natural as well as convenient in determining the dynamics of the global demons, by providing a particular solution to the continuity equation below. Eq. (\[eq:meas\]) defines the $\phi-$space measure as ${\cal D}x{\cal D}p{\cal D}w\exp(-\beta
f)=\rho_f\; {\cal D}\phi$. The global demon dynamics can then be determined by requiring that $\rho_f$ be a stationary solution of a generalized Liouville (continuity) equation in configuration space: $$0 = \partder{\rho_f}{t} + \sum_{i=1}^{n+m}
\partder{(\dot{\phi}_i\rho_f)}{\phi_i}.\label{eq:prob}$$ This is equivalent to requiring that the master equation, enforcing conservation of probability under evolution of the ensemble, be satisfied.
The equations of motion for the demons are now found by requiring that they, combined with the generalized dynamics of (7), satisfy Eq. (\[eq:prob\]). A direct substitution of $\rho_f$ and $\dot{\phi}_i$ (Eqs. (7b-c) ) into Eq. (\[eq:prob\]) allows one to solve for $\dot{w}$: $$\begin{aligned}
\dot{w}_1 &= &\frac{\kappa_1}{ n} \left( \beta F_{1i}
\partder{S}{x_i} - \partder{F_{1i}}{x_i}\right),\label{eq:grd}\\
\dot{w}_2 &=& \frac{\kappa_2}{ n}
\left( \beta F_{2i} p_i - \partder{F_{2i}}{p_i}\right).\nonumber\end{aligned}$$ If we had chosen to neglect the momenta and used (7a), the form of Eq. (\[eq:grd\]) would be unchanged. Eqs. (7) and (\[eq:grd\]) define a dynamics which by construction preserves the measure Eq. (\[eq:meas\]). (It is worth noting that while we have taken an exponential form for the density, in general we can take an arbitrary function $\rho$ and still use this same procedure.)
Microcanonical dynamics preserve the phase space volume exactly, since the divergence of the equations of motion, $$\partder{\dot{q}_i}{q_i} + \partder{\dot{p}_i}{p_i},$$ trivially vanishes by Hamilton’s equations of motion. The global demon equations of motion (7), (\[eq:grd\]), on the other hand, do allow for fluctuations in the $\phi$-space volume, which can be quite large. Writing these equations as $\dot{\phi}_i = {\cal F}_i(\phi)$, the divergence is explicitly $$\partder{\dot{\phi}_i}{\phi_i} =
-\frac{\beta}{ n}\left(\kappa_1\frac{dG_1}{ dw_1}
\partder{F_{1i}}{x_i} + \kappa_2
\frac{dG_2}{dw_2}\partder{F_{2i}}{p_i}\right).\label{eq:volf}$$ This local ‘breathing’ of $\phi-$space is controlled by the arbitrary functions $G$ and $F$. Although this behavior is not microcanonical, there is nevertheless an invariant quantity, called the pseudoenergy ${\cal E}$, which is preserved: $${\cal E} = f(x,p,w) + \frac{1}{\beta}\int_0^t dt'\;
\partder{\dot{\phi}_i}{\phi_i} .\label{eq:pse}$$ One can check directly that $\dot{\cal E}=0$.
There is clearly some freedom in defining the dynamics: the functions $G$ and $F_i$ and the coupling strength $\ka$. The only restriction on $G(w)$ is that the measure Eq. (\[eq:meas\]) is normalizable; in general the auxiliary variables $w$ can have any desired measure. In practice, highly non-linear functions are impractical since they will require small integration time steps. For these reasons, it is convenient to take $G=w^2/2$ or $G=w^4/4$. A necessary condition for $F_{i}$ is for it to be at least linear in its argument, the minimal requirement for the existence of the fluctuations in the volume (\[eq:volf\]). The precise relation to the fluctuations in a volume $V$, or equivalently, the instantaneous $\phi-$space compressibility, can be found using the divergence theorem $$\frac{dV}{ dt} =-\frac{\beta}{ n}\int_V {\cal D}\phi\;
\left(\kappa_1\frac{dG_1}{ dw_1} \partder{F_{1i}}{x_i} + \kappa_2
\frac{dG_2}{dw_2}\partder{F_{2i}}{p_i}\right).$$ In this paper, we do not explore the effect of different choices of $G_i$, $F_i$. Such studies have been done on smaller systems[@cmda]–[@cmdc].
Finally, we observe that the equations of motion (7), (\[eq:grd\]) will have no stable fixed points[@orbit]. This is the case since the sum of the Lyapunov exponents is related to the average rate of change of total volume of $\phi-$space. By the Liouville equation, this will necessarily vanish[@billa]: $$\sum_{i=1}^{n+m}\lambda_i = \VEV{\partder{\dot{\phi}_i}{\phi_i}}=0.$$
[3. Implementation for the 2D XY Model]{}
The 2D XY model consists of spins located on the sites of a two dimensional square lattice, which are free to rotate in the plane. The action is given by $$V(\theta) = -\sum_{<ij>} Re U_i U_j^\dagger
= -\sum_{<ij>} \cos{(\theta_i-\theta_j)},$$ where the sum is over nearest neighbors, and the $U_j\equiv e^{i\theta_j}$ are elements of $U(1)$ located at each lattice site $j$. In two dimensions, this model exhibits a Kosterliz-Thouless phase transition near $\beta\sim
1$[@kta]–[@kte]. Above the phase transition, the dynamics is dominated by dissociated vortex-antivortex pairs. These pairs become tightly bound below the phase transition, where the dynamics is dominated by spin waves. The K-T phase transition is infinite order, characterized by an exponentially diverging correlation length ($\xi$): $$\xi = a_\xi\exp( b_\xi (T-T_c)^{-\nu}).$$ Numerical simulations indicate similar critical behavior for finite lattices[@ktd; @kte]. Near the critical temperature, the system experiences an exponential increase in the correlation length $\xi$, which can lead to critical slowing down in simulations by virtue of Eq. (\[eq:texp\]).
[3.1 Equations of Motion]{}
The implementation of global demons to the XY model is straight forward. Following the equations of motion (7b-c) and (\[eq:grd\]), we have $$\begin{aligned}
\dot{\theta}_i & = & p_i - \frac{\kappa_2\beta}{ n}w_2\sin^3\theta_i
,\qquad (i=1,...,n),\label{eq:eoma}\\
\dot{p}_i & = &-\partder{V(\theta)}{\theta_i} -
\frac{\kappa_1\beta}{ n}w_1^3 p_i.\nonumber\end{aligned}$$ $F_i(\theta) = \sin^3\theta_i$ is chosen to respect the periodicity in $\theta$, while $G_i(p)=p_i$ has no such restriction. This choice was motivated only by simplicity, and in general, we could take more complicated interactions, and include additional global demons. The corresponding equations for the global demons are then $$\begin{aligned}
\dot{w}_1 & = &\kappa_1\left[ \frac{\beta}{ n}\sum_i p_i^2 -
1\right],\label{eq:eomb}\\
\dot{w}_2 & = &\frac{\kappa_2}{ n}\left[
\beta\sum_i\partder{V(\theta_i)}{\theta_i}\sin^3\theta_i
- 3\sum_i \sin^3\theta_i\cos\theta_i\right].\nonumber\end{aligned}$$ We used leapfrog integration, which included a Taylor expansion so that the $O(dt^2)$ errors in a time step cancel. The pairs $q$, $w_1$ and $p$, $w_2$ were updated in alternate steps. $w_1$ was taken with $q$ since they both involve momenta, and $w_2$ with $p$ since they both involve coordinates. Our general studies of the systematics of the model under tuning of parameters were performed on a $16^2$ lattice, while the studies of the critical exponents were done on a $64^2$ lattice, to allow longer correlation lengths. The particular choice of functions $F$ leads to a small non-ergodicity for this particular system: the momentum zero mode cannot change sign. (We have corrected for this by occasionally (every 64 trajectories) refreshing the momenta, using the same procedure as in HMD. In general, this is probably a good idea, to ensure that the evolution is ergodic. This particular non-ergodicity could also have been corrected by a small modification of the equations of motion.) We have verified that the equations of motion are correct to $O(dt^3)$ on a time step, leading to $O(dt^2)$ systematic errors in observables, by computing the $dt$ behavior of several observables in Fig. 2, demonstrating the quadratic behavior of the systematic error.
[3.2 Hybrid Monte Carlo]{}
We have used the critical properties of HMC as a benchmark for comparison of our global demon approach, studying three of its variations[@ktc]. The equations of motion are the same as those used for global demons, except with $\kappa_i=0$. By modifying the length of the HMC trajectory between momentum refreshes, we modify the decorrelation time[@hmcb]. The first variation, denoted HMC-1, has trajectories of length 1, where the highest frequency of the free theory is $(2\pi)^{-1}$. While this ‘standard’ choice is easy to implement, it suffers from severe critical slowing down, with $z=2$ in Eq. (\[eq:texp\]). The critical behavior should be improved by choosing the trajectory length proportional to the spatial correlation length $\xi$ of the system[@adk]. The two variations we consider are denoted HMC-S, for $T=\xi$, and HMC-L, for $T=2\pi\xi$. Again we point out that, in order to make this choice, we require the very information which we are attempting to measure. We were fortunate to have previous results for $\xi$ available to us[@kte], but in general this is not likely to be the case.
The integration time step was kept fixed at $dt=.1$ along trajectories of length $T$. This value of $dt$ was chosen so that the acceptance rate of the global Metropolis hit in the HMC algorithm was approximately $80\%$. For HMC, it is necessary to use random trajectory lengths for optimum relaxation[@hmcb]: we chose $T$ uniformly distributed on the interval $(.5\VEV{T}, 1.5\VEV{T}]$, with $\VEV{T} = 1$ for HMC-1, and $\VEV{T}\approx\xi, 2\pi\xi$ for HMC-S,L. We also made several runs of HMC-L using exponentially distributed random trajectory lengths. We found that this does not lead to any improvement over the runs with uniformly distributed trajectories, and may even have resulted in slightly noisier measurements. One interesting difference between HMC and global demons is that, for global demons, $T$ simply denotes time between measurements along a single trajectory - the evolution of the simulation is completely unaffected by the choice of $T$. At the end of a HMC trajectory we performed a global metropolis hit, after which we performed measurements and refreshed the momenta by choosing new, gaussian distributed $p_i$.
[3.3 Coupling Strength Dependence]{}
In the micro-canonical algorithms HMC and HMD, the rate at which the simulation covers phase space in the non-ergodic directions (i.e. changes energy) is controlled by the time between momentum refreshes. If the trajectories are too long, the system changes total energy very slowly, leading to autocorrelations on timescales proportional to the trajectory length. If the trajectories are too short, on the other hand, the motion between energy shells is rapid, but motion in the micro-canonical directions is diffusive, leading to a dynamical critical exponent of $2$. This means that, for large correlation lengths, the efficiency of the algorithm can vary as a power of $T/T_{opt}$, where $T_{opt}$ is the optimum trajectory length. As shown in Refs. [@hmcb; @adk], $T_{opt}$ should be proportional to the correlation length of the system, which is not known [*a priori*]{}. Thus it is necessary to perform a sensitive tuning which depends on a parameter measured in the simulation.
In the global demon algorithm, on the other hand, the parameter which controls energy (or action) non-conservation is just the coupling $\kappa$ of the demons to the system. In the limit $\kappa \rightarrow 0$, the demons decouple and ergodicity is lost. If $\kappa$ becomes too large, the equations of motion will suffer the characteristic instabilities of discretized dynamics. In contrast to the HMC and HMD algorithms, however, we can make an [*a priori*]{} choice of $\kappa$ which works quite well at all values of the correlation length. To do this, consider the change in total action, $\Delta S =
S(\phi_2)-S(\phi_1)$, in a single time step, $$\Delta S \simeq \Delta t\;
\partder{S}{\phi_i}\dot{\phi}_i\quad ,\label{eq:nab}$$ where $S$ now refers to the total global demon action $f$ in Eq. (\[eq:meas\]). Using equations (7)-(\[eq:grd\]) and (\[eq:nab\]), we see that $<|dS/dt|>$ is proportional to $\kappa$, with a constant of proportionality of order one. Thus, the change in $S$ along a trajetory of length $T$ should be $\Delta S \simeq \kappa T$. To set the scale of $\Delta
S$, we can compute its expectation value if two consecutive measurements are totally decorrelated: $$\sigma = \sqrt{\langle\Delta S^2\rangle}
= \sqrt{2(\langle S(\phi)^2\rangle -\langle
S(\phi)\rangle^2 )} = \frac{1}{\beta}\sqrt{2nC_s}\quad ,$$ where $C_s$ is the specific heat of the system. In order for the action to decorrelate between measurements (with $T=1$), we conclude that the optimal choice of $\kappa$ is likely to be $\kappa \approx {\cal O}(\sqrt{n})$. Note that this philosophy of forcing large fluctuations in energy along a trajectory is inherently different from HMC, where (in order to avoid prohibitively low acceptance rates) $\Delta S$ is ${\cal O}(1)$.
To investigate the behavior of the global demons algorithm under tuning, we ran a series of simulations on a $16^2$ lattice. (Except where otherwise indicated, we used an integration time step of $dt=0.1$ and measuring at intervals $T=1=10dt$; we will call this time between measurements a trajectory length, even though no momentum refresh is performed). Along the trajectory, we determine the square of the change in action between measurements, $\Delta S^2
= [S(t)-S(t-T)]^2$, which is then averaged along the entire trajectory to obtain $\sqrt{\VEV{\Delta S^2}}$. The result gives a guide as to how fast the trajectory can diffuse through configuration space. In Fig. 3, we plot the quantity $\overline{\Delta S}$, which we call the diffusiveness, defined by $$\overline{\Delta S} = \frac{ \sqrt{\VEV{\Delta S^2}}}{ \sigma},$$ as a function of coupling strength $\kappa$, for simulations at representative values of $\beta$ both above and below the phase transition. In the limit $\kappa_i=0$, the equations of motion (\[eq:eoma\])-(\[eq:eomb\]) are microcanonical and $S$ is preserved, as indicated in the figure. For small couplings, the microcanonical component of the dynamics is only slightly perturbed by the canonical component, and the ergodicity is weak. The convergence times here are quite large[@cho]. For $\kappa_i\sim\sqrt{2n}\sim 23$, the value of $\overline{\Delta S}$ can be seen to saturate near unity, the value expected when two consecutive measurements are uncorrelated; here the steps are quite large through configuration space. For larger values of the coupling, the change in action remains saturated. Convergence is also generally slower for larger $\kappa$, since the additional decorrelation produced by the microcanonical component to the dynamics is reduced. The reduction of $\overline{\Delta S}$ in Fig. 3 for $\beta\sim 1$ can be attributed to critical slowing down. However, while the correlation lengths become quite large, the dip in $\overline{\Delta S}$ is not so noticeable. [*In this respect, critical slowing down does not seem to hinder the dynamics, nor does it require any special tuning of $\kappa$.*]{}
In Fig. 4, the $\beta$ dependence of $\overline{\Delta S}$ is plotted for simulations at fixed coupling strengths. By selecting the $\kappa = 32$ curve, for instance, we see that we can study both the low and high temperature properties of the $XY$ model, as well as the phase transition, without modifying $\kappa$. While there is a small dip in the curves near $\beta\sim
1$, critical slowing down does not seem to strongly effect this measure of the dynamics as one approaches the phase transition from either side. An important result is that the couplings $\kappa$ are essentially independent of $\beta$ as well as the details of the physics of the model under study.
It should be emphasized that the runs with $\kappa\ll\sqrt{2n}$ converge slower as $\ka$ decreases, and ultimately do not converge for the microcanonical limit $\kappa=0$. We have checked the convergence of the dynamics to the proper ensemble by measuring and subsequently histogramming $\wa$ and $p_i$ along the trajectory and comparing them to their exact analytic distributions, finding that convergence is best above $\kappa\sim \sqrt{2n}$. We can also examine the $\kappa$ dependence of measurements at fixed $\beta$, illustrated in Fig. 5 for $\beta=1/1.1$. What we see is that the measurements for coupling strengths roughly 5-8 times saturation do not exhibit any systematic deviation as $\kappa$ increases. For much larger $\kappa$ (at fixed $dt$), there will be the characteristic instabilities associated with difference equations. However, the value $\kappa=\sqrt{2n}$ clearly is not near this instability limit, and we can safely use it. In the low $\kappa$ limit, we are close to micro-canonical dynamics, and $S$ begins to become approximately conserved. This slow diffusion results in long time correlations and poor statistics. This figure is typical of other temperatures, above and below the phase transition.
[3.4 Choice of Trajectory Length (i.e. Measurement Frequency)]{}
An indication of how rapidly measurements decorrelate is shown in Fig. 6. There we plot the diffusiveness $\overline {\Delta S}$ as a function of trajectory length $T$, for $\kappa_1=\kappa_2=16$ and $\beta=1$. We observe a saturation in the trajectory length near $T=2$. Because measurements do not effect the time evolution of the global demon trajectory (they are not associated with any Metropolis hit or momentum refresh), the choice of frequency of measurements is governed by the relative costs of the time evolution and measurement routines. Because our measurement algorithm was relatively inexpensive, we chose to use $T=1$.
[3.5 Observables]{}
The observables we measured include the energy $E$, lattice magnetization $M$, topological charge $Q$, defined by $$\begin{aligned}
E & =& -\frac{1}{ n}\sum_{<ij>}Re U_iU_j^\dagger,\nonumber\\
M & =& \frac{1}{ n} \sum_i U_i,\\
Q & =& \frac{1}{ n} \sum_p q_p.\nonumber\end{aligned}$$ The sum in $Q$ indicates the sum over all plaquettes of the number of positive topological charges occupying that plaquette. (For an exact definition of the topological charge, see e.g.[@hands].) Corresponding to these observables, we can define the specific heat and susceptibilities: $$\begin{aligned}
C_v &=& \beta^2 n (\VEV{E^2}-\VEV{E}^2),\nonumber\\
\chi_Q&=& n (\VEV{Q^2}-\VEV{Q}^2),\\
\chi_M&=& n (\VEV{ (ReM)^2} + \VEV{(ImM)^2}).\nonumber\end{aligned}$$ In both our global demon and HMC runs, we started with about $20000/T$ trajectories for thermalization, followed by $160,000/T$ trajectories of data, where $T$ is the trajectory length. Statistical errors in observables were obtained by binning measurements in bins of size $2^n$. The errors quoted in our tables use the smallest bin larger than $8\tau_M$, where $\tau_M$ is the integrated autocorrelation time of the total magnetization. The errors in the susceptibilities were obtained from the errors in the corresponding observables by assuming gaussian fluctuations on a timescale $\tau_{\cal O}$, where ${\cal O}$ is the appropriate observable. A selection of our observables are indicated in Table 1. We find that the global demon results usually agree with the HMC results within a few $\sigma$, which indicates that the systematic errors are not large. They could, of course, be further reduced by extrapolating to $dt=0.$
[3.6 Autocorrelation Functions and Decorrelation Times]{}
Because we have a dynamical algorithm, the trajectory has a memory, which will be reflected in the auto-correlation functions. This can be analyzed by examining auto-correlation functions of the observables $M$, $E$, $Q$, and $S$. When the couplings $\kappa$ are small, the dynamics is near the microcanonical limit, and decorrelation is very poor. Typical auto-correlation functions for $\ka_1=\ka_2=1$ are shown in Fig. 7 (dots) for $\beta=1/1.1$. Here we define $$\delta {\cal O}(t) = {\cal O}(t)-\VEV{\cal O},$$ as the fluctuation from the mean. As the couplings are increased near their optimum values, the ringing disappears, and decorrelation times become better defined quantities, indicated by the solid curve with $\kappa_1=\kappa_2=16$. The corresponding HMC autocorrelation functions are shown as well. Parenthetically, this type of ringing can also occur in HMC simulations if one uses a constant trajectory length. The comparison to our HMC runs at this $\beta$ is shown in Fig. 8.
A comparison of auto-correlation functions for the total lattice magnetization $M$ for global demons to the various implementations of HMC are shown in Fig. 9 for a selection of temperatures. It is clear that HMC does not significantly out perform global demons in terms of decorrelation, no matter how ‘optimal’ the trajectory length. The points in these curves indicate the actual number of data points. Hence while the number of global demon measurements is given by $t$, the optimal HMC runs have between one and two orders of magnitude smaller sampling rate in order to have similar decorrelation behavior.
[3.7 Critical Exponents]{}
The integrated autocorrelation times $\tau$ are defined for a given quantity ${\cal O}$ as $$\tau_{\cal O} = T\left[\frac{1}{2} + \sum_{t=1}^\infty
\frac{\VEV{\delta{\cal O}(0)\delta{\cal O}(t)} }
{\VEV{\delta{\cal O}(0)\delta{\cal O}(0)} }\right].$$ Note that we are measuring $\tau$ in units of total time evolved rather than number of trajectories. In Fig. 10 we present a ln-ln plot of the decorrelation times $\tau$ vs. the correlation length $\xi$ for several observables. The fit parameters are indicated in Table 2, and are only indicative of the critical behavior, since they will depend strongly on systematic effects. What is seen is that in almost every case, the global demon prefactor and critical exponent are smaller than the HMC results. The integrated autocorrelation times are tabulated in Table 3. In Ref. [@adk], it was argued that the critical exponent is unity when $T$ is proportional to the lowest frequency mode in the system for a free field theory. The results in Tables 2–3 seem to be good evidence that their results are qualitatively correct in an interacting field theory as well.
The critical behavior of global demon dynamics will also depend on the coupling strengths. In Fig. 11 we plot $\tau$ as a function of $\kappa$ for simulations at several values of $\beta$, on a $16^2$ lattice. As $\kappa$ increases, the system tend to decorrelate faster, again generally saturating above $\kappa\sim\sqrt{2n}$. In Fig. 12, the $\beta$ dependence of simulations at $\kappa=1,4,16,64$ are indicated. The $\kappa=1$ runs have the highest decorrelation times as expected, but we also observe that the high temperature phase is rather insensitive to the value of the coupling. Although convergence of the trajectory to the correct ensemble will always depend strongly on the coupling strength, the decorrelation times of both weakly ergodic and strongly ergodic trajectories are very similar. The effects of critical slowing down are particularly noticeable in $\tau_Q$ and $\tau_E$ near $\beta\lsim 1$. The peak in the $\tau_E$ is closely related to the dip in $\overline{\Delta S}$ in Fig. 4. The reason is that the diffusiveness measures the maximum rate at which the total energy $S$ can change, so if $\overline{\Delta S}\ll 1$, the potential energy $E$ will change slowly.
One might conclude from Figs. 11–12 that the coupling strength dependence is not too important, and that $\kappa\sim 2$ is roughly equivalent to $128$. Clearly, while the decorrelation times are indicative of the dynamics, they do not provide the complete picture of the situation. For example, information such as the ringing in the autocorrelation functions (see Fig. 7) average out, and are not strongly reflected in the value of $\tau$. We also see that simulations with very similar decorrelation times can have disparate values of the diffusiveness $\overline{\Delta S}$. But in all these guides, the tuning is consistently optimal for $\kappa\sim\sqrt{2n}$.
[4. Conclusions]{}
We have studied the global demon dynamical approach to simulating lattice regularized field theories. This method breaks away from the conventional Hamiltonian wisdom, defining a deterministically chaotic, time reversal invariant ‘canonical’ dynamics which rapidly fills configuration space with the desired ensemble. We have taken a particularly simple implementation of global demons using two coupling functions and examined its critical behavior, comparing to HMC using various trajectory lengths.
We have found that the algorithm is very stable under tuning of the various parameters. In particular, once $\kappa$ is large enough, the quality of the results seem to be independent of $\kappa$ until the simulation becomes unstable. It appears that $\kappa\approx\sqrt{2n}$ is a good rule of thumb for which the simulation will perform well in all regimes. In addition we found that the systematic errors were small (a few standard deviations), and that the critical slowing down properties of the algorithm were competitive with or better than the best implementations of HMC. The main fault with the algorithm is that it is not exact, i.e. there is a systematic error associated with the numerical integration. This problem is addressed in the appendix.
One advantage of this approach is that there is no barrier in principle to obtaining $z<1$ (note our estimated critical exponents for $S$, $E$ and $Q$ in Table 2). In contrast, local algorithms such as HMC and HMD are limited by $z\sim 1$[@rutgers]. In addition, the dynamical nature of the algorithm has allowed extensions to non-equilibrium situations[@cmdd]. One possible improvement which we have not examined is Fourier acceleration. This technique improves $z$ to $0$ for HMC in free field theory, and may help our approach.
The numerical computations in this work were performed on the Cray [Y-MP8/864]{} at the Ohio Supercomputer Center. We thank Aurel Bulgac, Robert Edwards, Rajan Gupta, Bill Hoover, Tony Kennedy, Greg Kilcup, Klaus Pinn, Junko Shigemitsu and Beth Thacker for useful conversations. This work was supported under DOE grants DE-AC02-ER01545 and DE-FG02-91ER40608.
[Appendix: Exact Dynamics?]{}
One major weakness of the global demon algorithm presented in this paper is the systematic error associated with finite step size. In HMC, this error is eliminated by performing a global metropolis hit before every measurement. Unfortunately, this technique has no hope of working for an algorithm with global demons, since the trajectories do not conserve energy. One might hope to use the pseudo-energy ${\cal E}$ in this capacity, since it is conserved. However, the memory term in Eq. (\[eq:pse\]) precludes this. For Hamiltonian systems, one can implement symplectic integrators[@yosh] to render the dynamics exact, or one can introduce a global Metropolis hit as in HMC. For our non-Hamiltonian ‘canonical’ dynamics, this procedure is not so clear. Although we have been unable to satisfactorily solve this problem, we mention here one approach which we have tried. We present this method because, although it is presently numerically impractical, it appears to be correct in principle. In addition, it should be applicable to any dynamical simulation which does not conserve energy, and is quite different from the traditional method of using metropolis hits to ensure exactness.
Consider the exact (i.e. $dt=0)$ equations of motion, $\dot{\phi} = F(\phi)$, which, by construction, preserve the measure $\rho(\phi)=\exp(-\beta f(\phi))$. When we discretize time, $\rho$ is no longer preserved exactly. Let us assume that there exists some measure, $\tilde \rho\not= \rho$, which is preserved by the discretized equations of motion $\phi_{n+1} = M(\phi_n)$, where $M$ is the time evolution operator. We can now define a correction factor $\alpha$ such that, up to normalization, $$\rho(\phi) = \alpha(\phi) \tilde \rho(\phi).$$ If we know $\alpha$, we can obtain the exact expectation values of observables through convolution: $$\VEV{\cal O} = \frac{ \int {\cal O}\rho d\phi }{ \int \rho d\phi }
= \frac{\int ({\cal O}\alpha ) {\tilde \rho} d\phi}{
\int \alpha {\tilde \rho} d\phi} = \frac{\VEV{ {\cal
O}\alpha}'}{\VEV{\alpha}'},$$ where the prime indicates evaluation in the $\tilde \rho$ ensemble. The discretized Liouville (continuity) equation tells us that after one integration time step, we preserve the measure $\tilde \rho$: $$\tilde \rho_{n+1}\; d\phi_{n+1} = \tilde \rho_n\; d\phi_n.$$ We can now use $(A.1)$ and $(A.3)$ to solve for the correction factor $\alpha$, through which we can measure exact observables: $$\alpha_{n+1} = \alpha_n \frac{d\phi_{n+1}}{ d\phi_n}
\frac{\rho_{n+1}}{ \rho_n}
= \alpha_n \; {\rm det}
\left(\partder{M}{\phi}\right)\;\frac{\rho_{n+1}}{ \rho_n},$$ where det$(\partial M/\partial\phi)$ is just the Jacobian of the map $\phi_n
\rightarrow \phi_{n+1}$. Thus, by choosing $\alpha_0=1$ at the initial point of the trajectory, we can compute $\alpha_n$ at all subsequent points along the trajectory.
Note that $\alpha$ should play a role very similar to the acceptance rate in HMC. When $dt$ is small, the distribution $\tilde \rho$ is very close to $\rho$, so $\alpha \approx 1$ and all of the configurations will be of approximately equal statistical weight. This corresponds to high acceptance rates in HMC. When $dt$ is large, on the other hand, $\alpha$ will be large in some regions and small in others, meaning that the simulation spends significant amounts of time in regions of low statistical importance. This is similar to the rejection of many proposed configurations when the acceptance is low. Also note that, although in general it is quite difficult, for our leapfrog algorithm the calculation of det$(\partial M/\partial\phi)$ is straight forward and only ${\cal O}(volume)$. The correction factor can also be written as $$\begin{aligned}
\alpha_n &=& \alpha_o\exp\left\{ \sum_{i=0}^{n-1} {\ell
n}\left[\partder{q_{i+1}}{q_i}\partder{p_{i+1}}{p_i}\right] - \beta H_i +
\beta H_0\right\}\\
&=& \alpha_o\exp\left\{-\beta\left[ {\cal E}_n - {\cal E}_o + O(\Delta
t^2)\right]\right\},\nonumber\end{aligned}$$ where ${\cal E}_n$ is the pseudo-energy (\[eq:pse\]) evaluated at $\phi_n$.
We have implemented this algorithm for small harmonic and anharmonic oscillator systems. In all runs, we found that, along the trajectory, the magnitude of $\alpha$ would occasionally rapidly decrease several orders of magnitude and never recover. We believe that this is due to an instability in the equations of motion: if the demons become too large at fixed $\kappa$ and $dt$, they begin to grow in an unbounded manner. It is quite likely that this means that a non-zero $\tilde \rho$ does not exist. We have tried decreasing $dt$ and various improvements in the equations of motion, but all of these fixes only decrease the frequency of drops - they do not eliminate them.
In the scheme described above, it is not obvious how to perform a momentum refresh. This is because the absolute magnitude of $\alpha$ is not known, only its ratio to the previous alpha along the trajectory. We will now show (again, assuming the existence of $\tilde\rho$) that the correct procedure is to reset $\alpha$ to $1$ after a gaussian momentum refresh. Assume, given $\tilde\rho$ and a refreshing scheme, that the probability of a trajectory beginning at $\phi$ is $P(\phi)$. In addition, let $\bar\alpha(\phi)$ be the correctly normalized function $\alpha$ discussed above and $\alpha(\phi | \phi_0) = \bar\alpha(\phi)/\bar\alpha(\phi_0)$ be the value of alpha obtained using ($A.4$) along the trajectory from $\phi_0$ to $\phi$ (note that $\alpha(\phi_0 | \phi_0) = 1$). Then $$\VEV{\cal O}
= \frac{ \int d\phi_0 P(\phi_0) \int d\phi {\cal O}\tilde\rho(\phi)
\alpha(\phi | \phi_0)}
{ \int d\phi_0 P(\phi_0) \int d\phi \tilde\rho(\phi)
\alpha(\phi | \phi_0)}
= \frac{N\VEV{ {\cal O}\bar\alpha}'}{ N\VEV{\bar\alpha}'}$$ where $N = \int d\phi_0(P(\phi_0)/\bar\alpha(\phi_0))$. We attempted to test this scheme on the systems discussed above, but we found that the statistical errors in the exact demon simulation tended to be larger than the systematic errors in a similar inexact simulation. Thus, it is presently more efficient to eliminate $dt$ errors by extrapolation techniques than by using the exact algorithm.
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[TABLE 1. Comparison of observables between algorithms. For each $\beta$, the four rows correspond to global demons, HMC-1, HMC-S, HMC-L, respectively. Measurements were performed on a $64^2$ lattice, with 160K/T statistics.]{}
[l@llllll]{}
------------------------------------------------------------------------
$\beta$ & $E$ & $C_v$ & $Q\times 100$ & $\chi_Q\times 100$ & $\mid M\mid$ & $\chi_M$\
\[2mm\]\
0.70 & 0.8287(1)& 0.760(3)& 6.241(2) & 4.49(2)& 0.0484(2)& 12.2(1)\
& 0.8299(1)& 0.765(5)& 6.235(2) & 4.50(3)& 0.0487(3)& 12.4(2)\
& 0.8297(2)& 0.750(7)& 6.233(2) & 4.40(3)& 0.0493(3)& 12.7(1)\
& 0.8300(3)&0.686(14)& 6.231(5) & 4.36(7)& 0.0490(3)& 12.5(1)\
\
\
0.78 & 0.9573(1)& 0.992(4)& 4.725(2) & 3.92(2)& 0.0683(3)& 24.3(2)\
& 0.9577(2)& 0.994(8)& 4.730(3) & 3.93(3)& 0.0683(6)& 24.3(4)\
& 0.9573(3)&1.010(12)& 4.731(3) & 3.94(4)& 0.0674(4)& 23.6(3)\
& 0.9582(5)&1.061(30)& 4.722(6) & 4.04(9)& 0.0683(5)& 24.3(3)\
\
\
0.82 & 1.0243(1)& 1.138(5)& 3.981(2) & 3.62(2)& 0.0848(5)& 37.2(4)\
& 1.0238(2)& 1.13(1) & 3.995(3) & 3.61(3)& 0.0828(9)& 35.8(8)\
& 1.0244(3)& 1.12(2) & 3.987(3) & 3.61(4)& 0.0828(7)& 35.7(6)\
& 1.0234(6)& 1.09(4) & 3.996(7) & 3.53(10)& 0.0837(7)& 36.4(6)\
\
\
1/1.1 & 1.1757(2)& 1.396(7)& 2.431(2) & 2.73(2)& 0.162(1)& 133(2)\
& 1.1750(3)& 1.40(2) & 2.448(4) & 2.76(4)& 0.151(3)& 116(4)\
& 1.1756(5)& 1.44(4) & 2.443(5) & 2.80(5)& 0.161(2)& 132(3)\
&1.1777(10)& 1.38(6) & 2.418(10)& 2.67(11)& 0.166(2)& 140(3)\
\
\
1/1.04 & 1.2632(3)& 1.512(9)& 1.638(3) & 2.14(2)& 0.295(3)& 409(7)\
& 1.2618(5)& 1.50(3) & 1.659(5) & 2.14(4)& 0.276(7)& 370(20)\
& 1.2618(7)& 1.50(6) & 1.657(6) & 2.14(6)& 0.281(4)& 374(10)\
&1.2630(13)& 1.76(17)& 1.647(12)& 2.33(15)& 0.291(5)& 401(11)\
\
TABLE 2. Comparison of estimated critical exponents $z$ and prefactor $A$ for global demons, HMC-1, HMC-S and HMC-L, where $\tau = A \xi^z$. Measurements were performed on a $64^2$ lattice, with 160K/T statistics.
[l@ll]{}
------------------------------------------------------------------------
& $z$ & $A$\
\[2mm\]\
\
\
Global Demons & 1.3& 2.4\
HMC-1 & 2.0 & 5.7\
HMC-S & 1.3 & 6.9\
HMC-L & 1.05 & 5.8\
\
\
\
\
Global Demons & 0.8& 1.0\
HMC-1 & 1.5 & 1.6\
HMC-S & 1.0 & 1.6\
HMC-L & 1.0 & 4.3\
\
\
\
\
Global Demons & 0.5& 0.8\
HMC-1 & 1.0 & 1.9\
HMC-S & 1.4 & 2.1\
HMC-L & 1.3 & 14\
\
\
\
\
Global Demons & 0.8& 0.8\
HMC-1 & 1.0 & 1.4\
HMC-S & 1.3 & 1.6\
HMC-L & 1.3 & 8.2\
\
[TABLE 3. Auto-correlation times for total magnetization $M$, a single spin $S$, topological charge $Q$ and internal energy $E$, and the magnetization correlation length $\xi$. Measurements were performed on a $64^2$ lattice, with 160K/T statistics.]{}
[l@ccccc]{}
------------------------------------------------------------------------
$\beta$ & $\tau_M$ & $\tau_S$ & $\tau_E$ & $\tau_Q$ & $\xi$\
\[2mm\]\
\
\
0.70 & 6.5 & 1.86 & 1.14 & 1.53 &2.2\
0.78 & 11.8 & 2.6 & 1.38 & 1.97 &3.4\
0.82 & 16.8 & 3.2 & 1.57 & 2.5 &4.3\
1/1.1 & 47. & 5.9 & 2.2 & 4.5 &9.4\
1/1.07 & 62. & 7.8 & 2.4 & 5.5 &12.8\
1/1.04 & 106. & 10.3 & 3.1 & 8.8 &18.4\
1.0 & 124. & 22. & 2.8 & 9.5 &31.\
\
\
\
\
0.70 & 27. & 5.1 & 4.1 & 3.2 &2.2\
0.78 & 62. & 9.2 & 5.9 & 4.8 &3.3\
0.82 & 106. & 12.1 & 7.9 & 6.2 &4.3\
1/1.1 & 380 & 36. & 15.7 & 13.5 &8.9\
1/1.04 & 1620 & 99. & 30. & 26. &17.1\
\
\
\
\
0.70 & 17.9 & 3.7 & 6.2 & 4.0 & 2.2\
0.78 & 32. & 5.5 & 12.6 & 8.0 & 3.3\
0.82 & 44. & 7.4 & 18. & 10.5 & 4.3\
1/1.1 & 126. & 14. & 50. & 31. & 9.3\
1/1.04 & 240 & 32. &132. & 59. & 17.5\
\
\
\
\
0.70 & 12.7 & 9.3 & 36. & 22. & 2.2\
0.78 & 21. & 15.5 & 66. & 38. & 3.4\
0.82 & 29. & 20. & 136. & 61. & 4.3\
1/1.1 & 64. & 40. & 144. & 132. & 9.5\
1/1.04 & 114. & 84. & 760 & 350 & 17.7\
\
[FIGURES ]{}
1. [Time evolution of $10^3$ connected points evolving according to ‘canonical’ rather than microcanonical dynamics for (a) the 1-d harmonic oscillator $H=(p^2+q^2)/2$ (left column; $t=0$ points lie on the unit circle), and (b) the $SU(2)$ Hamiltonian $H=J_z^2/2$ (right column; $t=0$ points lie on the circle $J_z=0.5$), both at $\beta=1$. The phase space in (a) is $(q,p)$ and in (b) it is the sphere parameterized by $(J_x,J_y,J_z)$. The rapid spreading of neighboring points is characteristic of global demon dynamics. The characteristic time scale is $t\sim 1$.]{}
2. [Finite time step extrapolations, demonstrating the global $O(dt^2)$ leapfrog error for (a) potential energy, (b) topological charge and (c) magnetization.]{}
3. [Diffusiveness of a global demon trajectory versus the coupling strength, for selected values of $\beta$. $\Delta S$ is measured at intervals of $T=1$. The ‘optimal’ value of $\difu=\sigma$ is indicated by the dashed line. As can be seen, the dynamics is not strongly affected by critical slowing down. Saturation occurs when $\kappa_\alpha\sim O(\sqrt{2n}) \sim 23$. The optimal coupling can be seen to be independent of both $\beta$ and the phase transition.]{}
4. [Diffusiveness of the global demons versus $\beta$. The dip at $\beta\sim 1$ is a result of the Kosterlitz-Thouless phase transition and critical slowing down. The dynamics is not strongly effected by the transition, so no additional tuning is required, and $\kappa$ can be taken as fixed for all $\beta$.]{}
5. [Measurement dependence on the coupling $\kappa$ at $\beta=1/1.1$.]{}
6. [Diffusiveness of a global demon trajectory a function of the trajectory length $T$, at $\beta = 1$, $\kappa_1=\kappa_2=16$. Saturation can be seen, indicating an optimal trajectory length of $T\sim O(1)$.]{}
7. [Autocorrelation functions for potential energy, topological charge, magnetization and spin, at $\beta=1/1.1$ with $\kappa_1=\kappa_2=1$ (dots) and $\kappa_1=\kappa_2=16$ (solid). As can be seen, the ringing vanishes as the coupling increases.]{}
8. [Autocorrelation functions for potential energy, topological charge, magnetization and spin, at $\beta=1/1.1$ for global demons with $\kappa_1=\kappa_2=16$ (solid), HMC-1 (dots), HMC-S (crosses) and HMC-L (boxes). For global demons and HMC-1, measurements are made every $t=1$, while for HMC-S,L, the boxes and crosses indicate the actual number of data points.]{}
9. [Lattice magnetization auto-correlation function at selected temperatures for global demons (solid), HMC-1 (dots), HMC-S (dashes) and HMC-L (boxes). The time axis for $\beta=0.7$ has been scaled by a factor of 0.1 to magnify the short time behavior.]{}
10. [The behavior of $\tau=a\xi^z$ is plotted near the phase transition on a $64^2$ lattice for (a) total magnetization, (b) the spin at the origin, (c) topological charge and (d) potential energy. In each figure, we indicate the results for HMC (boxes), HMC$-s$ (diamonds), HMC$-l$ (crosses) and global demons (circles). Critical exponents can be extracted from a linear fit. The results of the numerical fits are given in Table 2. Results are on a $64^2$ lattice with $160K/T$ statistics.]{}
11. [Decorrelation times versus coupling strength for $\beta =
0.5$ (squares), $0.7$ (diagonal crosses), $1/1.1$ (diamonds), $2.0$ (vertical crosses) and $4.0$ (circles) for (a) total magnetization, (b) a single spin, (c) topological charge and (d) potential energy.]{}
12. [Decorrelation times versus $\beta$ for couplings $\kappa_1=\kappa_2=1$ (crosses), $4$ (diamonds), $16$ (squares) and $64$ (circles) for (a) total magnetization, (b) a single spin, (c) topological charge and (d) potential energy. The rise at $\beta\sim 1$ is critical slowing down, and is especially evident for the energy and topological charge.]{}
[^1]: Bitnet: DIMITRI%NST@YALEVMS.
[^2]: Bitnet: SLOAN@OHSTPY.
|
---
abstract: 'We study a substitute for the matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the “singular case” $\alpha\beta=q^N\gamma\delta$, when the standard form of the matrix product ansatz of [@derrida1993exact] does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, $\varphi_1$, is defined on the entire algebra, and determines stationary probabilities for large systems on $L\geq N+1$ sites. The other functional, $\varphi_0$, is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on $L< N+1$ sites. Functional $\varphi_0$ vanishes on non-constant Askey-Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey-Wilson polynomials.'
address:
- |
Włodzimierz Bryc\
Department of Mathematical Sciences\
University of Cincinnati\
2815 Commons Way\
Cincinnati, OH, 45221-0025, USA.
- |
Marcin Świeca\
Department of Mathematical Sciences\
University of Cincinnati\
2815 Commons Way\
Cincinnati, OH, 45221-0025, USA. and Faculty of Mathematics and Information Science\
Warsaw University of Technology\
pl. Politechniki 1 00-661\
Warszawa, Poland
author:
- Włodzimierz Bryc
- Marcin Świeca
title: On matrix product ansatz for Asymmetric Simple Exclusion Process with open boundary in the singular case
---
This is the expanded version of the paper. It includes additional material that is typeset differently from the main body of the paper.
Introduction and main results
=============================
The Asymmetric Simple Exclusion Process (ASEP) with open boundary on sites $\{1,\dots,L\}$ is a continuous time Markov chain with state space $\{0,1\}^L$. Informally, particles may arrive at the left boundary at rate $\alpha>0$ and leave at rate $\gamma\geq 0$. A particle may move to the right at rate $1$ or to the left at rate $q<1$. It may leave at the right boundary at rate $\beta>0$ or a new particle may arrive there at rate $\delta\geq0$, see Fig. \[Fig1\]. More formal description of the evolution is given as Kolmogorov’s equations below.
(.5,1) circle \[radius=0.2\]; (1.5,1) circle \[radius=0.2\]; (2.5,1) circle \[radius=0.2\]; (5,1) circle \[radius=0.2\]; (6,1) circle \[radius=0.2\];
(7,1) circle \[radius=0.2\]; (9.5,1) circle \[radius=0.2\]; (10.5,1) circle \[radius=0.2\]; (-1,2.3) to \[out=-20,in=135\] (.5,1.5); at (-.2,2) [$\alpha$]{}; (10.5,1.5) to \[out=45,in=200\] (12,2.3); at (11.2,2) [$\beta$]{}; at (8.25,1) [$\cdots$]{}; at (3.75,1) [$\cdots$]{}; at (6.5,1.8) [$1$]{}; (6.1,1.5) to \[out=45,in=135\] (7,1.5); at (5.5,1.8) [$q$]{}; (5,1.5) to \[out=45,in=135\] (5.9,1.5); at (10,1.8) [$1$]{}; (9.6,1.5) to \[out=45,in=135\] (10.4,1.5); at (9,1.8) [$q$]{}; (8.4,1.5) to \[out=45,in=135\] (9.4,1.5); (-1,-.3) to \[out=0,in=-135\] (.5,0.6); at (-.2,0) [$\gamma$]{}; at (0.5,0) [$1$]{}; at (1.5,0) [$2$]{}; at (2.5,0) [$3$]{}; at (3.75,0) [$\cdots$]{}; at (6,0) [ $k$ ]{}; at (8.25,0) [$\cdots$]{}; at (10.5,0) [$L$]{}; at (9.5,0) [$L-1$]{}; (10.6,.7) to \[out=-45,in=180\] (12,-.3); at (11.2,0) [$\delta$]{};
We are interested in the steady state of the ASEP, so we focus on the stationary distribution of the Markov chain. The standard method relies on Kolmogorov’s prospective equations. Denoting by $P_t(\tau_1,\dots,\tau_L)$ the probability that Markov chain is in configuration $(\tau_1,\dots,\tau_L)\in
\{0,1\}^L$ at time t, we have $$\begin{gathered}
\label{DiffEq}
\frac{d}{dt}P_t(\tau_1,\dots,\tau_L)= \delta_{\tau_1=1}\left[\alpha P_t(0,\tau_2,\dots,\tau_L)-\gamma P_t(1,\tau_2,\dots,\tau_L)\right]\\
+\delta_{\tau_1=0}\left[\gamma P_t(1,\tau_2,\dots,\tau_L)-\alpha P_t(0,\tau_2,\dots,\tau_L)\right]\\
+\sum_{k=1}^{L-1} \delta_{\tau_k=1,\tau_{k+1}=0}\left[qP_t(\tau_1,\dots,\tau_{k-1},0,1,\tau_{k+2},\dots,\tau_L)- P_t(\tau_1,\dots,\tau_{k-1},1,0,\tau_{k+2},\dots,\tau_L)
\right] \\
+\sum_{k=1}^{L-1} \delta_{\tau_k=0,\tau_{k+1}=1}\left[P_t(\tau_1,\dots,\tau_{k-1},1,0,\tau_{k+2},\dots,\tau_L)- q P_t(\tau_1,\dots,\tau_{k-1},0,1,\tau_{k+2},\dots,\tau_L)
\right]\\
+\delta_{\tau_L=0}\left[\beta P_t(\tau_1,\dots,\tau_{L-1},1)-\delta P_t(\tau_1,\dots,\tau_{L-1},0)\right] \\
+\delta_{\tau_L=1}\left[\delta P_t(\tau_1,\dots,\tau_{L-1},0)-\beta P_t(\tau_1,\dots,\tau_{L-1},1)\right].\end{gathered}$$ The stationary distribution $P(\tau_1,\dots,\tau_L)$ of this Markov chain satisfies $\frac{d}{dt}P_t(\tau_1,\dots,\tau_L)=0$ so it solves the system of linear equations on the right hand side of . An ingenious method of determining the stationary probabilities for all $L$ was introduced by Derrida, Evans, Hakim and Pasquier in [@derrida1993exact], who consider infinite matrices and vectors that satisfy relations $$\begin{aligned}
{\mathbf{D}}{\mathbf{E}}-q{\mathbf{E}}{\mathbf{D}}&=&{\mathbf{D}}+{\mathbf{E}},\label{q-comm-Derrida}\\
\langle W|(\alpha {\mathbf{E}}-\gamma {\mathbf{D}})&=&\langle W| ,\label{W}\\
(\beta {\mathbf{D}}-\delta {\mathbf{E}})|V\rangle&=&|V\rangle. \label{V}\end{aligned}$$ The stationary probabilities are then computed as $$\label{MatrixSolution}
P(\tau_1,\dots,\tau_L)= \frac{\langle W|\prod_{j=1}^L (\tau_j{\mathbf{D}}+(1-\tau_j){\mathbf{E}})|V\rangle }{\langle W|({\mathbf{D}}+ {\mathbf{E}})^L|V\rangle}.$$
It has been noted in the literature that the above approach may fail: @essler1996representations [page 3384] point out that matrix representation runs into problems when $\alpha\beta=\gamma\delta$, and they point out the importance of a more general condition that $\alpha\beta-q^n\gamma\delta\ne0$ for $n=0,1,\dots$. We will call this a non-singular case.
The singular case when $\alpha\beta=q^N\gamma\delta$, is discussed by @Mallick-Sandow-1997 [Appendix A] in the context of finite matrix representations. Of course, this is a singular case for the matrix product ansatz, not for the actual Markov chain. To avoid singularity, [@lazarescu2013matrix] presents a perturbative generalization of the matrix product ansatz, which was used in [@gorissen2012exact] to derive exact current statistics for all values of parameters. Continuity of the ASEP with respect to its parameters is also used to derive recursion for stationary probabilities in [@Liggett-1975 proof of Theorem 2.3].
Solution for the singular case
------------------------------
Our goal is to analyze the singular case $\alpha\beta=q^N\gamma\delta$ directly. We consider an abstract noncommutative algebra ${{\mathcal M}}$ with identity ${\mathbf{I}}$ and two generators ${\mathbf{D}},{\mathbf{E}}$ that satisfy one relation $$\label{q-Com}
{\mathbf{D}}{\mathbf{E}}-q{\mathbf{E}}{\mathbf{D}}={\mathbf{D}}+{\mathbf{E}}.$$ The algebra consists of linear combinations of monomials ${\mathbf{X}}={\mathbf{D}}^{n_1}{\mathbf{E}}^{m_1}\dots {\mathbf{D}}^{n_k}{\mathbf{E}}^{m_k}$. It turns out that monomials in normal order, ${\mathbf{E}}^m{\mathbf{D}}^n$, form a basis for ${{\mathcal M}}$ as a vector space. We introduce increasing subspaces ${{\mathcal M}}_k$ of ${{\mathcal M}}$ that are spanned by the monomials in normal order of degree at most $k$, i.e., ${{\mathcal M}}_k$ is the span of $\{{\mathbf{E}}^m{\mathbf{D}}^n: m+n\leq k\}$. The abstract version of the matrix product ansatz for the singular case uses a pair of linear functionals $\varphi_0:{{\mathcal M}}_N\to{\mathds{C}}$ and $\varphi_1:{{\mathcal M}}\to{\mathds{C}}$.
\[T-0\] Suppose $\alpha,\beta, \gamma,\delta>0$ satisfy $\alpha \beta=q^N \gamma\delta$ for some $N=0,1,\dots$. Then there exists a pair of linear functionals $\varphi_0:{{\mathcal M}}_N\to{\mathds{C}}$ and $\varphi_1:{{\mathcal M}}\to{\mathds{C}}$ such that stationary probabilities for the ASEP are $$\label{phi2P}
P(\tau_1,\dots,\tau_L)= \frac{\varphi\left[\prod_{j=1}^L (\tau_j{\mathbf{D}}+(1-\tau_j){\mathbf{E}})\right]}{\varphi\left[({\mathbf{D}}+ {\mathbf{E}})^L\right]},$$ where $\varphi=\varphi_0$ if $1\leq L< N+1$ and $\varphi=\varphi_1$ if $L\geq N+1$. Furthermore, if $L=N+1$ then the stationary distribution is the product of Bernoulli measures $$P(\tau_1,\dots,\tau_{N+1}) = \prod_{j=1}^{N+1} p_j^{\tau_j} q_j^{1-\tau_j}$$ with $p_j=\frac{\alpha}{\alpha+\gamma q^{j-1}}$ and $q_j=1-p_j$.
If $\alpha,\beta>0$, $\gamma,\delta\geq 0$ are such that $\alpha \beta\ne q^n \gamma\delta$ for all $n=0,1,\dots$, then $\varphi_0$ is defined on ${{\mathcal M}}_\infty={{\mathcal M}}$, and holds with $\varphi=\varphi_0$ for all $L$.
We remark that part of the conclusion of the theorem is the assertion that the denominators in are non-zero for all $L$. Proposition \[P-positivity\] below determines their signs, which according to Remark \[Rem-signs\] may vary also in the non-singular case. The signs determine the direction of the [*current*]{} $J$ through the bond between adjacent sites, which is defined as $J=\Pr(\tau_k=1,\tau_{k+1}=0)-q\Pr(\tau_k=0,\tau_{k+1}=1)$. When $L\ne N+1$, we have $J=\varphi[({\mathbf{E}}+{\mathbf{D}})^{L-1}]/\varphi[({\mathbf{E}}+{\mathbf{D}})^{L}]$, so the current is negative for $2\leq L\leq N$, and positive for $L> N+1$. As noted in @Aneva-2009-Integrability [Section 3], the current vanishes for $L=N+1$ due to the detailed balance condition satisfied by the product measure.
The proof of Theorem \[T-0\] is given in Section \[sect:ProofT1\] and consist of recursive construction of the pair of functionals. In the construction, the left and right eigenvectors in and are replaced by the left and right invariance requirements: $$\label{W+} \varphi\left[(\alpha {\mathbf{E}}-\gamma{\mathbf{D}}){\mathbf{A}}\right]=\varphi[{\mathbf{A}}],$$ $$\label{V+} \varphi\left[{\mathbf{A}}(\beta {\mathbf{D}}-\delta{\mathbf{E}})\right]=\varphi[{\mathbf{A}}],$$ for all ${\mathbf{A}}\in{{\mathcal M}}$ when $\varphi=\varphi_1$ and for all ${\mathbf{A}}\in{{\mathcal M}}_{N-1}$ if $\varphi=\varphi_0$. By an adaptation of the argument from [@derrida1993exact], functionals that satisfy and give stationary probabilities, see Theorem \[T2\] for the precise statement. Similar modification of and in the matrix formulation appears in [@corteel2011tableaux Theorem 5.2].
In the singular case functional $\varphi_0$ is defined on $N(N+1)/2$-dimensional space ${{\mathcal M}}_N$. However, ${{\mathcal M}}_N$ is not an algebra, so this is different from the finite dimensional representations of the matrix algebra which were studied by @essler1996representations and @Mallick-Sandow-1997. In Appendix \[Sec:MatrixModel\] we present a “matrix model" for all $\alpha,\beta,\gamma,\delta$ with $0<q<1$ that was inspired by @Mallick-Sandow-1997. The model reproduces their finite matrix model when the parameters are chosen like in their paper, but exhibits lack of associativity for general parameters.
Relation to Askey-Wilson polynomials
------------------------------------
Ref. [@uchiyama2004asymmetric] shows that the stationary distribution of the open ASEP is intimately related to the Askey–Wilson polynomials. Here we extend this relation to cover also the singular case, when the Askey-Wilson polynomials do not have the Jacobi matrix, see discussion below.
In the context of ASEP, the Askey-Wilson polynomials depend on parameter $q$, and on four real parameters $a,b,c,d$ which are related to parameters of ASEP by the equations $$\label{AW-parameters}
\alpha=\frac{1- q}{(1+c)(1+d)}, \beta=\frac{1-q}{(1+a)(1+b)}, \; \gamma=-\frac{(1-q)cd}{(1+c)(1+d)}, \delta=-
\frac{ab(1-q)}{(1+a)(1+b)},$$ see [@Bryc-Wesolowski-2015-asep], [@essler1996representations (74)], [@uchiyama2004asymmetric], and (in a somewhat different parametrization) [@Mallick-Sandow-1997]. Since $\alpha,\beta>0$ and $\gamma,\delta\geq 0$, when solving the resulting quadratic equations without loss of generality we can choose $a,c>0$, and then $b,d\in(-1,0]$. The explicit expressions are $a=\kappa_+(\beta,\delta),b=\kappa_-(\beta,\delta), c=\kappa_+(\alpha,\gamma),
d=\kappa_-(\alpha,\gamma) $, where $$\label{kappa}
\kappa_{\pm}(u,v)=\frac{1-q-u+v\pm\sqrt{(1-q-u+v)^2+4u v}}{2u}$$ and appear in many reference, including [@essler1996representations (74)], [@uchiyama2004asymmetric], [@Bryc-Wesolowski-2015-asep], and in [@Mallick-Sandow-1997]. In this parametrization, the singularity condition becomes $abcd q^N=1$.
Recall the $q$-hypergeometric function notation $${_{r+1}\phi_r}\left(\begin{matrix}
a_1,\dots,a_{r+1}\\ b_1,\dots,b_r
\end{matrix}\middle|q;z\right) = \sum_{k=0}^\infty \frac{(a_1, a_2, \dots,a_{r+1};q)_k}{(q,b_1, b_2, \dots,b_r;q)_k}z^k.$$ Here we use the usual Pochhammer notation: $(a_1, a_2, \dots,a_r;q)_n=(a_1;q)_n(a_2;q)_n \dots(a_r;q)_n$ and $(a;q)_{n+1}=(1-a q^{n })(a;q)_{n}$ with $(a;q)_0=1$. Later, we will also need the $q$-numbers $[n]_q=1+q+\dots+q^{n-1}$ with the convention $[0]_q=0$, $q$-factorials $[n]_q!=[1]_q\dots [n]_q=(1-q)^n(q;q)_n$ with the convention $[0]_q!=1$ and the $q$-binomial coefficients $${\begin{bmatrix}n\\#2
\end{bmatrix}_q
}=\frac{[n]_q!}{[k]_q![n-k]_q!}.$$
We define the $n$-th Askey-Wilson polynomial using the $_4\phi_3$-hypergeometric function, which in the second expression we write more explicitly for all $x$ rather than for $x=\cos \psi$. $$\begin{gathered}
\label{AW}
p_n(x;a,b,c,d|q)=a^{-n}(ab, ac, ad;q)_n{_4\varphi_3}\left(\begin{matrix}
q^{-n},q^{n-1}abcd,a e^{i\psi},a e^{-i\psi} \\
ab, ac, ad
\end{matrix}\middle|q;q\right)
\\ =
a^{-n}(ab, ac, ad;q)_n\sum_{k=0}^n q^k \frac{(q^{-n},abcd q^{n-1};q)_k}{(q,ab,ac,ad;q)_k}\prod_{j=0}^{k-1}(1+a^2 q^{2j}-2a x q^j).\end{gathered}$$ Although this is not obvious from , it is known that $p_n(x;a,b,c,d|q)$ is invariant under permutations of parameters $a,b,c,d$, and that the polynomial is well defined for all $a,b,c,d\in {\mathds{C}}$. However, in the singular case the degree of the polynomial varies with $n$ somewhat unexpectedly. It is easy to see from that if $abcd q^N=1 $, then for $0\leq k\leq N+1$ the degree of polynomial $p_k(x;a,b,c,d) $ is $\min\{k,N+1-k\}$. In particular, the degrees may decrease and hence there is no three step recursion, or a Jacobi matrix.
Indeed, $p_n(x;a,b,c,d|q)=a^{-n}(ab, ac, ad;q)_n Q_n(x)$ with $$Q_n(\cos \psi)=_4\!\!\varphi_3\left(\begin{matrix}
q^{-n},q^{n-N-1},a e^{i\psi},a e^{-i\psi} \\
ab, ac, ad
\end{matrix}\middle|q;q\right),$$ and $Q_n(x)=Q_{N+1-n}(x)$.
The relation of $\varphi_0$ to Askey-Wilson polynomials is more conveniently expressed using a different pair of generators of algebra ${{\mathcal M}}$. Instead of ${\mathbf{E}},{\mathbf{D}}$, we consider elements ${\mbox{\fontfamily{phv}\selectfont d}}$ and ${\mbox{\fontfamily{phv}\selectfont e}}$ given by $$\label{DE2de}
{\mathbf{D}}=\theta^2{\mathbf{I}}+\theta {\mbox{\fontfamily{phv}\selectfont d}},\; {\mathbf{E}}=\theta^2{\mathbf{I}}+ \theta{\mbox{\fontfamily{phv}\selectfont e}}, \;\theta=1/\sqrt{1-q}.$$ (Similar transformation was used by several authors, including [@uchiyama2004asymmetric] and [@Bryc-Wesolowski-2015-asep].)
In this notation, ${{\mathcal M}}$ is then an algebra with identity with two generators ${\mbox{\fontfamily{phv}\selectfont d}},{\mbox{\fontfamily{phv}\selectfont e}}$ that satisfy relation $$\label{de-comute}
{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}-q{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}={\mathbf{I}}.$$ Recall that $\varphi_0$ is defined on ${{\mathcal M}}_N$ in the singular case, and on all of ${{\mathcal M}}$ in the non-singular case, which we can include in the statement of Theorem \[T-0\] and in the conclusion below by setting $N=\infty$. The action of $\varphi_0$ on Askey-Wilson polynomials can now be described as follows.
\[T3\] With ${\mbox{\fontfamily{phv}\selectfont x}}=
\frac{1}{2\theta}\left({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}\right)$, for $1\leq k< N+1$ we have $$\varphi_0\left[p_k({\mbox{\fontfamily{phv}\selectfont x}}\;;a ,b ,c,d\;|q)\right]=0.$$
More generally, for any non-zero $t\in{\mathds{C}}$ let $$\label{ed2x} {\mbox{\fontfamily{phv}\selectfont x}}_t=
\frac{1}{2\theta}\left(\tfrac1t{\mbox{\fontfamily{phv}\selectfont e}}+t{\mbox{\fontfamily{phv}\selectfont d}}\right).$$ Then $$\label{Phi0-mean} \varphi_0\left[p_k({\mbox{\fontfamily{phv}\selectfont x}}_t\;;at ,bt ,\tfrac{c }{t}, \tfrac{d}{t}\;|q)\right]=0 \mbox{ for } 1\leq k< N+1.$$
The proof of Theorem \[T3\] appears in Section \[sect:ProofT3\] and is fairly involved. It relies on evaluation of $\varphi_0$ on the family of continuous $q$-Hermite polynomials, on explicit formula for the connection coefficients between the $q$-Hermite polynomials and the Askey-Wilson polynomials which we did not find in the literature, and to complete the proof we need some non-obvious $q$-hypergeometric identities. In Appendix \[Sect:TASEP\] we discuss action of $\varphi_0$ and $\varphi_1$ on the Askey-Wilson polynomials in the much simpler case of Totaly Asymmetric Exclusion process where $q=0$.
Relation to orthogonality functional for the Askey Wilson polynomials
---------------------------------------------------------------------
In the non-singular case when $q^n abcd\ne 1$ for all $n=0,1,\dots$, the Askey-Wilson polynomials $\{p_n\}_{n=0,1,\dots}$ are of increasing degrees and satisfy the three step recursion @Askey-Wilson-85 [(1.24)]. According to Theorem \[T-0\] functional $\varphi_0$ is then defined on all of ${{\mathcal M}}$ and determines stationary probabilities for all $L\geq 0$. Theorem \[T3\] implies that $\varphi_0$ is an orthogonality functional for the Askey-Wilson polynomials, which encodes the relation between ASEP and Askey-Wilson polynomials that was discovered by Uchyama, Sasamoto and Wadati [@uchiyama2004asymmetric]. In particular, corresponds to [@uchiyama2004asymmetric formula (6.2)] with $\xi=t$.
Orthogonality can be seen as follows. Theorem \[T3\] says that $\varphi_0\left[p_n({\mbox{\fontfamily{phv}\selectfont x}}\;;a ,b ,c,d\;|q)\right]=0$ for all $n\geq 1$, and it is easy to check, see e.g. [@chihara2011introduction Proof of Favard’s theorem], that the latter property together with the three-step recursion for the Askey-Wilson polynomials implies orthogonality: $$\varphi_0\left[p_m({\mbox{\fontfamily{phv}\selectfont x}}\;;a ,b ,c,d |q)p_n({\mbox{\fontfamily{phv}\selectfont x}}\;;a ,b ,c,d\;|q)\right]=0$$ for all $m\ne n$. Note that this orthogonality relation holds without the additional conditions on $a,b,c,d$ that appear when orthogonality of polynomials $\{p_n\}$ is considered on the real line [@Askey-Wilson-85 Theorem 2.4], or on a complex curve [@Askey-Wilson-85 Theorem 2.3]. Since $\varphi_0\left[p_n({\mbox{\fontfamily{phv}\selectfont x}}\;;a ,b ,c,d\;|q)\right]\ne 0$ only for $n=0$, linearization formulas [@foupouagnigni2013connection] give the value of $$\begin{gathered}
\varphi_0\left[p_n^2({\mbox{\fontfamily{phv}\selectfont x}}\;;a,b,c,d\;|q)\right] = \frac { (ab,ac,ad;q)^2_n }{ a^{2n} }
\sum _ {L = 0}^{2n } \frac { q^
L \left( a b ,
a c ,
a d ;q \right)_L }
{\left(a b c d ;q \right)_L}
\sum _ {j = \max (0,
L - n)}^{\min (n,
L )} \frac {q^{j (j - L
)} \left(q^{-n},a b c d q^{n - 1};q\right)_j }
{ (q, ab, ac, ad;q)_j (q;q)_{ L -j}}\\
\times \sum _ {k =
0}^{\min (j, j - L + n)} \frac {q^
k \left(q^{-j},
a^2 q^{L + r};q\right)_k \left(q^{-n}, a b c d q^{n - 1} ;q\right)_{
k + L -j}} { (q)_k (a b, a c, a d;q)_{ k + L -j}},\end{gathered}$$ which may fail to be positive when $abcd>1$.
Somewhat more generally, in the notation of [@foupouagnigni2013connection] we have $$\varphi_0\left[p_m ({\mbox{\fontfamily{phv}\selectfont x}}\;;a,b,c,d\;|q)p_n ({\mbox{\fontfamily{phv}\selectfont x}}\;;a,b,c,d\;|q)\right]=L_0(m,n),$$ where $$\begin{gathered}
L_r(m,n)= \frac { q^{\frac {1} {2} r (r + 1)}(ab,ac,ad;q)_m (ab,ac,ad;q)_n }{(-1)^
r a^{m + n - r} \left(a b c d q^{r - 1} ;q\right)_r}
\sum _ {L = 0}^{m + n +
r}{\begin{bmatrix}L+r\\#2
\end{bmatrix}_q
} \frac { q^
L \left( a b q^r,
a c q^r,
a d q^r ;q\right)_L }
{\left(a b c d q^{2 r} ;q \right)_L}
\\
\times \sum _ {j = \max (0,
L - m + r)}^{\min (n,
L +
r)} \frac {q^{j (j - L -
r)} \left(q^{-n},a b c d q^{n - 1};q\right)_j }
{ (q, ab, ac, ad;q)_j (q;q)_{ L +
r-j}}\\
\times \sum _ {k =
0}^{\min (j, j - L + m - r)} \frac {q^
k \left(q^{-j},
a^2 q^{L + r};q\right)_k \left(q^{-m}, a b c d q^{m - 1} ;q\right)_{
k + L + r -j}} { (q)_k (a b, a c, a d;q)_{ k + L +
r-j}}.\end{gathered}$$ Numerical experiments suggest that $L_0(m,n)=0$ if $p_n,p_m$ have different degrees which, if true, would strengthen the conclusion of Theorem \[T3\] to the assertion of full orthogonality.
Proof of Theorem \[T-0\] {#sect:ProofT1}
========================
We begin with two observations from the literature which seem not to be widely known. The first observation is that the proof of Derrida, Evans, Hakim and Pasquier in [@derrida1993exact] is non-recursive, so it implies that an invariant functional on the finite-dimensional subspace ${{\mathcal M}}_L$ determines stationary probabilities for ASEP of size $L$.
\[T2\] Fix $L\in{\mathds{N}}$. Suppose that $\varphi$ is a linear functional on ${{\mathcal M}}_L $ such that $\varphi({\mathbf{E}}+{\mathbf{D}})^L\ne 0$. If invariance equations and hold for all ${\mathbf{A}}\in{{\mathcal M}}_{L-1}$, then the stationary probabilities for the ASEP of length $L$ are $$\label{MatrixSolution+}
P(\tau_1,\dots,\tau_L)= \frac{\varphi\left[\prod_{j=1}^L (\tau_j{\mathbf{D}}+(1-\tau_j){\mathbf{E}})\right]}{\varphi\left[({\mathbf{D}}+ {\mathbf{E}})^L\right]}.$$
The argument here is the same as the proof in [@derrida1993exact Section 11.1] for the matrix version, see also [@sandow1994partially Section III]. The important aspect of that proof is that it works with fixed $L$, i.e., that we do not need to use a recurrence that lowers the value of $L$ as in [@derrida1992exact formula (8)] or in [@Liggett-1975 Theorem 3.2]. We reproduce their argument for completeness and clarity.
For $L=1$ it is easily seen that the stationary distribution is $P(1)=\frac{\alpha+\delta}{\alpha+\beta+\gamma+\delta}$ with $P(0)=1-P(1)$. On the other hand, equations and give $\alpha \varphi[{\mathbf{E}}]-\gamma \varphi[{\mathbf{D}}]=\varphi[{\mathbf{I}}]$ and $\beta \varphi[{\mathbf{D}}]-\delta \varphi[{\mathbf{E}}]=\varphi[{\mathbf{I}}]$. The solution is: $$\varphi[{\mathbf{E}}]=
\begin{cases}
\frac{\beta+\gamma}{\alpha \beta -\gamma\delta} \varphi[{\mathbf{I}}] & \mbox{ if } \alpha\beta \ne \gamma\delta \\
\frac{\gamma}{\alpha+\gamma} & \mbox{ if } \alpha\beta = \gamma\delta
\end{cases}, \quad \;
\varphi({\mathbf{D}})=
\begin{cases}
\frac{\alpha+\delta}{\alpha \beta -\gamma\delta}\varphi({\mathbf{I}}) & \mbox{ if } \alpha\beta \ne \gamma\delta \\
\frac{\alpha}{\alpha+\gamma} & \mbox{ if } \alpha\beta = \gamma\delta
\end{cases},$$ where we note that $\varphi[{\mathbf{I}}]=0$ when $\alpha\beta=\gamma\delta$ and in this case we also used the normalization $\varphi[{\mathbf{E}}+{\mathbf{D}}]=1$ to determine the values. In both cases, a calculation shows that $$\frac{\varphi[{\mathbf{D}}]}{\varphi[{\mathbf{E}}]+\varphi[{\mathbf{D}}]}=\frac{\alpha+\delta}{\alpha+\beta+\gamma+\delta}$$ giving the correct value of $P(1)$.
Suppose that $L\geq 2$. Denote by $p(\tau_1,\dots,\tau_L)= \varphi\left[\prod_{j=1}^L (\tau_j{\mathbf{D}}+(1-\tau_j){\mathbf{E}})\right]$ the un-normalized probabilities. Since by assumption the denominator in is non-zero, it is enough to verify that the right hand side of vanishes on $p(\tau_1,\dots,\tau_L)$. That is, we want to show that $$\begin{gathered}
\label{MS2}
(\delta_{\tau_1=1}-\delta_{\tau_1=0})\left[\alpha p(0,\tau_2,\dots,\tau_L)-\gamma p(1,\tau_2,\dots,\tau_L)\right]\\
+\sum_{k=1}^{L-1}
(\delta_{\tau_k=0,\tau_{k+1}=1}-\delta_{\tau_k=1,\tau_{k+1}=0})\left[ p(\tau_1,\dots,\tau_{k-1},1,0,\tau_{k+2},\dots,\tau_L)-qp(\tau_1,\dots,\tau_{k-1},0,1,\tau_{k+2},\dots,\tau_L)
\right] \\
+(\delta_{\tau_L=0}-\delta_{\tau_L=1})\left[\beta p(\tau_1,\dots,\tau_{L-1},1)-\delta p(\tau_1,\dots,\tau_{L-1},0)\right] =0.\end{gathered}$$ Denote $$\label{XY}
{\mathbf{X}}_k=\prod_{j=1}^{k} (\tau_j{\mathbf{D}}+(1-\tau_j){\mathbf{E}}) \mbox{ and } {\mathbf{Y}}_k=\prod_{j=k}^L (\tau_j{\mathbf{D}}+(1-\tau_j){\mathbf{E}})$$ with the usual convention that empty products are ${\mathbf{I}}$. Relation implies that $$\begin{gathered}
p(\tau_1,\dots,\tau_{k-1},1,0,\tau_{k+2},\dots,\tau_L)-qp(\tau_1,\dots,\tau_{k-1},0,1,\tau_{k+2},\dots,\tau_L) \\=
\varphi[{\mathbf{X}}_{k-1}({\mathbf{D}}{\mathbf{E}}-q{\mathbf{E}}{\mathbf{D}}){\mathbf{Y}}_{k+2}]=\varphi[{\mathbf{X}}_{k-1}({\mathbf{D}}+{\mathbf{E}}){\mathbf{Y}}_{k+2}].\end{gathered}$$ Noting that $$\delta_{\tau_k=0,\tau_{k+1}=1}-\delta_{\tau_k=1,\tau_{k+1}=0}=(1-\tau_k)\tau_{k+1}-\tau_k(1-\tau_{k+1})=\tau_{k+1}-\tau_k,$$ the sum in becomes $$\sum_{k=1}^{L-1} (\tau_{k+1}-\tau_k)\varphi[{\mathbf{X}}_{k}({\mathbf{D}}+{\mathbf{E}}){\mathbf{Y}}_{k+2}].$$ Since $\tau_k,\tau_{k+1}\in\{0,1\}$, the difference $\tau_{k+1}-\tau_k$ can take only three values $0,\pm 1$. Considering all four possible cases, we get $$\begin{gathered}
(\tau_{k+1}-\tau_k)\varphi[{\mathbf{X}}_{k-1}({\mathbf{D}}+{\mathbf{E}}){\mathbf{Y}}_{k+2}]\\=(\tau_{k+1}-\tau_k)\left(\varphi[{\mathbf{X}}_{k-1}(\tau_{k+1}{\mathbf{D}}+(1-\tau_{k+1}){\mathbf{E}}){\mathbf{Y}}_{k+2}] +
\varphi[{\mathbf{X}}_{k-1}(\tau_{k}{\mathbf{D}}+(1-\tau_{k}){\mathbf{E}}){\mathbf{Y}}_{k+2}]\right) \\
=(\tau_{k+1}-\tau_k)\left(\varphi[{\mathbf{X}}_{k} {\mathbf{Y}}_{k+2}]+\varphi[{\mathbf{X}}_{k-1} {\mathbf{Y}}_{k+1}]\right)
\\
={\varepsilon}_k \varphi[{\mathbf{X}}_{k-1} {\mathbf{Y}}_{k+1}]-{\varepsilon}_{k+1}\varphi[{\mathbf{X}}_{k} {\mathbf{Y}}_{k+2}],\end{gathered}$$ where ${\varepsilon}_k=\delta_{\tau_k=1}-\delta_{\tau_k=0}=\pm1$. (For the last equality we need to notice that $ {\mathbf{X}}_{k-1}
{\mathbf{Y}}_{k+1}={\mathbf{X}}_{k} {\mathbf{Y}}_{k+2}$ when $\tau_k=\tau_{k+1}$.)
Thus $$\sum_{k=1}^{L-1} (\tau_{k+1}-\tau_k)\varphi[{\mathbf{X}}_{k}({\mathbf{D}}+{\mathbf{E}}){\mathbf{Y}}_{k+2}]=\sum_{k=1}^{L-1}({\varepsilon}_k \varphi[{\mathbf{X}}_{k-1} {\mathbf{Y}}_{k+1}]-{\varepsilon}_{k+1}\varphi[{\mathbf{X}}_{k} {\mathbf{Y}}_{k+2}])={\varepsilon}_1\varphi[{\mathbf{Y}}_2]-{\varepsilon}_L\varphi[{\mathbf{X}}_{L-1}].$$ By invariance we have $$\left[\alpha p(0,\tau_2,\dots,\tau_L)-\gamma p(1,\tau_2,\dots,\tau_L)\right] =\varphi[(\alpha{\mathbf{E}}-\gamma{\mathbf{D}}){\mathbf{Y}}_2]=\varphi[{\mathbf{Y}}_2]$$ $$\left[\beta p(\tau_1,\dots,\tau_{L-1},1)-\delta p(\tau_1,\dots,\tau_{L-1},0)\right]=\varphi[{\mathbf{X}}_{L-1}(\beta{\mathbf{D}}-\delta{\mathbf{E}})]=\varphi[{\mathbf{X}}_{L-1}].$$ So the left hand side of becomes $$-{\varepsilon}_1 \varphi[{\mathbf{Y}}_2]+{\varepsilon}_1\varphi[{\mathbf{Y}}_2]-{\varepsilon}_L\varphi[{\mathbf{X}}_{L-1}]+{\varepsilon}_L\varphi[{\mathbf{X}}_{L-1}]=0$$ proving .
The second observation is that stationary distribution for ASEP of length $L=N+1$ is given as an explicit product of Bernoulli measures. This has been noted in [@enaud2004large Section 5.2], see also [@Enaud2005 Section 4.6.2] and [@Aneva-2009-Integrability Section 3]. The proof consists of verification of detailed balance equations so that individual terms on the right hand side of vanish.
\[P1\] Suppose $\alpha\beta=q^N\gamma\delta$. If $L=N+1$ then the stationary distribution of the ASEP is the product of Bernoulli measures $$P(\tau_1,\dots,\tau_L)= \prod_{j=1}^L p_j^{\tau_j} q_j^{1-\tau_j}$$ with $p_j=\frac{\alpha}{\alpha+\gamma q^{j-1}}$ and $q_j=1-p_j$.
The stationary distribution for $L=1$ is $p_1=\frac{\alpha+\delta}{\alpha+\beta+\gamma+\delta}$. When $\alpha\beta=\gamma\delta$ this answer matches $p_1=\frac{\alpha}{\alpha+\gamma }$.
For $L\geq 2$ we can use . Inserting the product measure into the right hand side of , we get: $$\alpha P_t(0,\tau_2,\dots,\tau_L)-\gamma P_t(1,\tau_2,\dots,\tau_L)=\alpha\frac{\gamma}{\alpha+\gamma}\prod_{k>1}p_{k}^{\tau_k}q_k^{1-\tau_k}-\gamma \frac{\alpha}{\alpha+\gamma} \prod_{k>1}p_{k}^{\tau_k}q_k^{1-\tau_k}=0,$$ $$\begin{gathered}
\left[P_t(\tau_1,\dots,\tau_{k-1},1,0,\tau_{k+2}\dots)- q P_t(\tau_1,\dots,\tau_{k-1},0,1,\tau_{k+2}\dots)
\right] \\ =\prod_{i\leq k-1}p_{i}^{\tau_i}q_i^{1-\tau_i}\frac{\alpha}{\alpha+q^{k-1}\gamma}\frac{\gamma q^k}{\alpha+q^{k}\gamma}\prod_{j\geq k+2}p_{j}^{\tau_j}q_j^{1-\tau_j}
\\
-q\prod_{i\leq k-1}p_{i}^{\tau_i}q_i^{1-\tau_i}\frac{\gamma q^{k-1}}{\alpha+q^{k-1}\gamma}\frac{\alpha}{\alpha+q^{k}\gamma}\prod_{j\geq k+2}p_{j}^{\tau_j}q_j^{1-\tau_j}
=0.\end{gathered}$$ Finally, $$\begin{gathered}
\left[\beta P_t(\tau_1,\dots,\tau_{L-1},1)-\delta P_t(\tau_1,\dots,\tau_{L-1},0)\right]
=
\prod_{i\leq L-1}p_{i}^{\tau_i}q_i^{1-\tau_i}\left(\beta\frac{ \alpha}{\alpha+q^{L-1} \gamma} -\delta \frac{ \gamma q^{L-1}}{\alpha+q^{L-1} \gamma}\right)
=0,\end{gathered}$$ as $L=N+1$ and $\alpha\beta=q^N\gamma\delta$. This shows that the right hand side of is zero, i.e. the product measure is stationary.
Construction of the pair of invariant functionals
-------------------------------------------------
The construction starts with choosing a convenient basis for ${{\mathcal M}}$, consisting of monomials in normal order, with all factors ${\mbox{\fontfamily{phv}\selectfont e}}$ occurring before ${\mbox{\fontfamily{phv}\selectfont d}}$. Such monomials appear in many references, see e.g. @frisch1970parastochastics [pg 368], @bozejko97qGaussian [page 137], [@Mallick-Sandow-1997 page 4524].
Monomials in normal order $\{{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n: m,n=0,1,\dots\}$ are a basis of ${{\mathcal M}}$ considered as a vector space. In this basis ${{\mathcal M}}_k$ is the span of $\{{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n: m+n\leq k\}$.
It is easy to check by induction that $q$-commutation relation gives explicit expressions for “swaps" that recursively convert all monomials into linear combinations of monomials in normal order. We have $$\label{de^m}
{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n=q^m{\mbox{\fontfamily{phv}\selectfont e}}^m {\mbox{\fontfamily{phv}\selectfont d}}^{n+1}+[m]_q{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n.$$
Indeed, ${\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m =q^m{\mbox{\fontfamily{phv}\selectfont e}}^m {\mbox{\fontfamily{phv}\selectfont d}}+[m]_q{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}$ holds for $m=0,1$. For the induction step we use and get ${\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^{m+1} =q^m{\mbox{\fontfamily{phv}\selectfont e}}^m {\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}+[m]_q{\mbox{\fontfamily{phv}\selectfont e}}^{m}=q^m{\mbox{\fontfamily{phv}\selectfont e}}^m (q{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}+{\mathbf{I}}) +[m]_q{\mbox{\fontfamily{phv}\selectfont e}}^{m}=q^{m+1}{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}+(q^m+[m]_q){\mbox{\fontfamily{phv}\selectfont e}}^m=q^{m+1}{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}+[m+1]_q{\mbox{\fontfamily{phv}\selectfont e}}^m$. To get the general case of we just right-multiply the formula ${\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m =q^m{\mbox{\fontfamily{phv}\selectfont e}}^m {\mbox{\fontfamily{phv}\selectfont d}}+[m]_q{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}$ by ${\mbox{\fontfamily{phv}\selectfont d}}^n$.
Similarly, we get $$\label{d^ne}
{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n{\mbox{\fontfamily{phv}\selectfont e}}=q^n{\mbox{\fontfamily{phv}\selectfont e}}^{m+1} {\mbox{\fontfamily{phv}\selectfont d}}^{n}+[n]_q{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}.$$
As before, we only need to prove ${\mbox{\fontfamily{phv}\selectfont d}}^n{\mbox{\fontfamily{phv}\selectfont e}}=q^n{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}+[n]_q{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}$. The induction step is ${\mbox{\fontfamily{phv}\selectfont d}}^{n+1}{\mbox{\fontfamily{phv}\selectfont e}}={\mbox{\fontfamily{phv}\selectfont d}}^n({\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}})={\mbox{\fontfamily{phv}\selectfont d}}^n (q{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}+{\mathbf{I}})= q^{n+1}{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}+(q[n]_q+1){\mbox{\fontfamily{phv}\selectfont d}}^n= q^{n+1}{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}+[n+1]_q{\mbox{\fontfamily{phv}\selectfont d}}^n$.
(Formulas and holds also for $m=0$ or $n=0$ after omitting the term with $[0]_q=0$.)
The formulas imply that any monomial is a linear combination of monomials in normal order: $$\label{Ordered0}
{\mbox{\fontfamily{phv}\selectfont d}}^{n_1}{\mbox{\fontfamily{phv}\selectfont e}}^{m_1}\dots {\mbox{\fontfamily{phv}\selectfont d}}^{n_k}{\mbox{\fontfamily{phv}\selectfont e}}^{m_k}=q^I{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n+\sum_{i+j\leq m+n-1}a_{i,j}{\mbox{\fontfamily{phv}\selectfont e}}^i{\mbox{\fontfamily{phv}\selectfont d}}^j,$$ where $m=m_1+\dots m_k$, $n=n_1+\dots+n_k$ and $I=\sum_{i=1}^k\sum_{j=1}^i m_i n_j$ is the minimal number of inversions (length) of a permutation that maps ${\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n$ into ${\mbox{\fontfamily{phv}\selectfont d}}^{n_1}{\mbox{\fontfamily{phv}\selectfont e}}^{m_1}\dots {\mbox{\fontfamily{phv}\selectfont d}}^{n_k}{\mbox{\fontfamily{phv}\selectfont e}}^{m_k}$, see e.g. [@bjorner2006combinatorics]. Compare [@Mallick-Sandow-1997 Appendix A].
Formula shows that monomials in normal order span ${{\mathcal M}}$. To verify that they are linearly independent we consider a pair of linear mappings (endomorphism) ${\mbox{\fontfamily{phv}\selectfont D}_q}$ and ${\mbox{\fontfamily{phv}\selectfont Z}}$ acting on polynomials ${\mathds{C}}[z]$ which are the $q$-derivative and the multiplication mappings: $$({\mbox{\fontfamily{phv}\selectfont D}_q}p)(z)=\frac{p(z)-p(qz)}{(1-q)z},\; ({\mbox{\fontfamily{phv}\selectfont Z}}p)(z)=zp(z).$$ The mapping ${\mbox{\fontfamily{phv}\selectfont d}}\mapsto{\mbox{\fontfamily{phv}\selectfont D}_q}$ and ${\mbox{\fontfamily{phv}\selectfont e}}\mapsto{\mbox{\fontfamily{phv}\selectfont Z}}$ extends to homomorphism of algebra ${\mathds{C}}\langle{\mbox{\fontfamily{phv}\selectfont d}},{\mbox{\fontfamily{phv}\selectfont e}}\rangle$ of polynomials in noncommuting variables ${\mbox{\fontfamily{phv}\selectfont e}},{\mbox{\fontfamily{phv}\selectfont d}}$ to the algebra $End({\mathds{C}}[z])$. It is well known that ${\mbox{\fontfamily{phv}\selectfont D}_q}{\mbox{\fontfamily{phv}\selectfont Z}}-q{\mbox{\fontfamily{phv}\selectfont Z}}{\mbox{\fontfamily{phv}\selectfont D}_q}$ is the identity, so we get an induced homomorphism of algebras $${{\mathcal M}}={\mathds{C}}\langle{\mbox{\fontfamily{phv}\selectfont d}},{\mbox{\fontfamily{phv}\selectfont e}}\rangle/I\to End({\mathds{C}}[z]),$$ where $I$ is the two sided ideal generated by ${\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}-q{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}-{\mathbf{I}}$. Therefore, it is enough to prove linear independence of $\{{\mbox{\fontfamily{phv}\selectfont Z}}^m{\mbox{\fontfamily{phv}\selectfont D}_q}^n\}$. To prove the latter, consider a finite sum ${\mbox{\fontfamily{phv}\selectfont S}}=\sum_{m,n\geq 0} a_{m,n}{\mbox{\fontfamily{phv}\selectfont Z}}^m{\mbox{\fontfamily{phv}\selectfont D}_q}^n =0$ and suppose that some of the coefficients $a_{m,n}$ are non-zero. Let $n_*\geq 0$ be the smallest value of index $n$ among the non-zero coefficient $a_{m,n}$. We note that $${\mbox{\fontfamily{phv}\selectfont Z}}^m{\mbox{\fontfamily{phv}\selectfont D}_q}^n (z^{n_*})=\begin{cases}
0 & n>n_* \\
[n_*]_q! z^{m} &n=n_*
\end{cases}$$ Therefore, applying the sum ${\mbox{\fontfamily{phv}\selectfont S}}$ to the monomial $z^{n_*}\in{\mathds{C}}[z]$ we get $\sum_{m\in M} a_{m,n_*} [n_*]_q! z^{m} =0$, i.e., all $\{a_{m,n_*}: m\in M\}$ are zero, in contradiction to our choice of $n_*$. The contradiction shows that all coefficients must be zero, proving linear independence.
Using we remark that invariance conditions and with ${\mathbf{A}}\in{{\mathcal M}}_k$ can be written equivalently in our basis of monomials in normal order as $$\begin{aligned}
\label{W++} \alpha\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^n]-\gamma \varphi[{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]&=&\Delta(\gamma-\alpha)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n], \\
\label{V++} -\delta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n{\mbox{\fontfamily{phv}\selectfont e}}]+\beta \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}]&=&\Delta(\delta-\beta)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n],\end{aligned}$$ where $m+n\leq k$ and $\Delta(x)=(1/\theta+\theta x)$.
Recursive construction of the functionals
-----------------------------------------
We define linear functional $\varphi=\varphi_0$ or $\varphi=\varphi_1$ by assigning its values on all elements of the basis $\{{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n\}$ and then extending it to ${{\mathcal M}}_N$ or ${{\mathcal M}}$ by linearity. On the basis, we define $\varphi$ recursively, extending it from ${{\mathcal M}}_k$ to ${{\mathcal M}}_{k+1}$ in such a way that the invariance properties and hold.
### Initial values
We set $\varphi_0[{\mathbf{I}}]=1$. We set $$\label{phi-1-ini}
\varphi_1[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n]= \begin{cases}
0 & \mbox{if $m+n\leq N$}\\
\Pi^{-1}\alpha^n\gamma^m q^{m(m-1)/2} & \mbox{if $m+n= N+1$},
\end{cases}$$ where the normalizing constant $\Pi=\theta^{N+1}\prod_{j=1}^{N+1}(\alpha+q^{j-1}\gamma)$ is chosen so that $\varphi_1\left[({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}})^{N+1}\right]=1/\theta^{N+1}$.
Clearly, $\varphi_1\equiv 0$ on ${{\mathcal M}}_N$. We need to check that our initialization of $\varphi_1$ has the properties we need for the recursive construction: that invariance conditions hold for ${\mathbf{A}}\in{{\mathcal M}}_N$, and that $\varphi_1$ determines the stationary measure of ASEP with $L=N+1$.
\[L-phi12prod\] For monomials of degree $N+1$ we have $$\label{phi-prod}
\varphi_1[{\mathbf{D}}^{\tau_1}{\mathbf{E}}^{1-\tau_1}\dots {\mathbf{D}}^\tau_{N+1}{\mathbf{E}}^{1-\tau_{N+1}}]=\prod_{j=1}^{N+1}p_j^{\tau_j}q_j^{1-\tau_j},$$ where the weights $\{p_j\}$ come from stationary product measure in Proposition \[P1\]. Furthermore, and hold for ${\mathbf{A}}\in{{\mathcal M}}_N$.
Since $\varphi$ vanishes on polynomials of lower order, from it is easy to see that $$\varphi_1[{\mathbf{D}}^{\tau_1}{\mathbf{E}}^{1-\tau_1}\dots {\mathbf{D}}^\tau_{N+1}{\mathbf{E}}^{1-\tau_{N+1}}]=\theta^{N+1}
\varphi_1[{\mbox{\fontfamily{phv}\selectfont d}}^\tau_1{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_1}\dots {\mbox{\fontfamily{phv}\selectfont d}}^\tau_{N+1}{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_{N+1}}].$$ So we only need to show that $$\label{swap} \varphi_1[{\mbox{\fontfamily{phv}\selectfont d}}^{\tau_1}{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_1}\dots
{\mbox{\fontfamily{phv}\selectfont d}}^\tau_{N+1}{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_{N+1}}]=\prod_{j=1}^{N+1}p_j^{\tau_j}q_j^{1-\tau_j}/\theta^{N+1}.$$ It is easy to see that this formula holds true for $\varphi_1[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{N+1-m}]$. (In fact, this is how we defined $\varphi_1[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]$ when $m+n=N+1$.) All monomials of the form ${\mbox{\fontfamily{phv}\selectfont d}}^\tau_1{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_1}\dots
{\mbox{\fontfamily{phv}\selectfont d}}^\tau_{N+1}{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_{N+1}}$ can be obtained from monomials ${\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{N+1-m}$ in normal order by applying a finite number of adjacent transpositions, i.e. by swapping pairs of adjacent factors ${\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}$ or ${\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}$. (Adjacent transpositions are Coxeter generators for the permutation group, see e.g. [@bjorner2006combinatorics].) So to complete the proof we check that if the formula holds for some monomial, then it also holds after we swap the entries at adjacent locations $k,k+1$. Suppose that $$\theta^{N+1}\varphi_1[{\mathbf{X}}{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}{\mathbf{Y}}]=q_kp_{k+1}\Pi'=
\frac{\alpha \gamma q^{k-1}}{(\alpha+q^{k-1}\gamma)(\alpha+q^k\gamma)} \Pi',$$ with ${\mathbf{X}}={\mbox{\fontfamily{phv}\selectfont d}}^\tau_1{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_1}\dots {\mbox{\fontfamily{phv}\selectfont d}}^\tau_{k-1}{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_{k-1}}$, ${\mathbf{Y}}={\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_{k+2}}\dots
{\mbox{\fontfamily{phv}\selectfont d}}^\tau_{N+1}{\mbox{\fontfamily{phv}\selectfont e}}^{1-\tau_{N+1}}$ and $\Pi'=\prod_{ j\ne k,k+1} p_j^{\tau_j}q_j^{1-\tau_j}$. Multiplying this by $q$ and replacing $q{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}$ by $ {\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}-{\mathbf{I}}$, we get $$\theta^{N+1}\varphi_1[{\mathbf{X}}{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}{\mathbf{Y}}]=
\frac{\alpha \gamma q^{k}}{(\alpha+q^{k-1}\gamma)(\alpha+q^k\gamma)} \Pi'=p_kq_{k+1}\Pi',$$ as $\varphi$ vanishes on lower order monomials. So the swap preserves the expression on the right hand side of . The case when the factors at the $k,k+1$ locations are ${\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}$ is handles similarly.
To verify that and hold for ${\mathbf{A}}\in{{\mathcal M}}_N$ we show that and hold for $m+n\leq N$. Indeed, both sides are zero if $m+n\leq N-1$, and if $m+n=N$ then the right hand sides are still zero. The left hand side of is $$ \alpha^{n+1}\gamma^{m+1} \left(q^{m(m+1)/2}-q^m q^{m(m-1)/2}\right)/\Pi=0.$$ The left hand side of is $$\alpha^{n}\gamma^{m} q^{m(m-1)/2}\left(\alpha\beta-q^{n+m}\gamma\delta \right)/\Pi=0$$ by singularity assumption.
### Recursive step for $\varphi=\varphi_0$ or $\varphi_1$
Suppose $\varphi$ is defined on ${{\mathcal M}}_k$ and that invariance conditions hold for ${\mathbf{A}}\in{{\mathcal M}}_{k-1}$. If $m+n=k$ with $1\leq k\leq N-1$ (case of $\varphi_0$) or $k\geq N+1$ (case of $\varphi_1$). Define $$\begin{gathered}
\label{SolA} \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^n]=
\frac{\left(\beta\Delta(\gamma-\alpha)+\gamma\Delta(\delta-\beta)q^m\right)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]+\gamma\delta [n]_qq^m\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]+
\beta\gamma[m]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]}{(q^N-q^{m+n})\gamma\delta} ,\end{gathered}$$ $$\begin{gathered}
\label{SolB} \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}]
= \frac{\left(\alpha \Delta(\delta-\beta)+\delta \Delta(\gamma-\alpha)q^n\right)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]+
\alpha\delta[n]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]+\gamma\delta q^n[m]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]}{(q^N-q^{m+n})\gamma\delta},\end{gathered}$$ where $\Delta(x)=(1/\theta+\theta x)$ comes from and .
\[Rem:nonsingular\] If $\alpha\beta-q^n\gamma\delta \ne 0$ for all $n$, we define $\varphi_0$ on ${{\mathcal M}}$, replacing the above recursion with $$\begin{gathered}
\label{SolA0} \varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^n]=
\frac{\left(\beta\Delta(\gamma-\alpha)+\gamma\Delta(\delta-\beta)q^m\right)\varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]+\gamma\delta [n]_qq^m\varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]+
\beta\gamma[m]_q\varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]}{\alpha\beta-q^{m+n} \gamma\delta},\end{gathered}$$ $$\begin{gathered}
\label{SolB0} \varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}]= \frac{\left(\alpha \Delta(\delta-\beta)+\delta \Delta(\gamma-\alpha)q^n\right)\varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]+
\alpha\delta[n]_q\varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]+\gamma\delta q^n[m]_q\varphi_0[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]}{\alpha\beta-q^{m+n} \gamma\delta}.\end{gathered}$$
We need to make sure that this expression is well defined.
Fix $k\ne N$. Suppose $m'+n'=k+1$. Then $\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m'}{\mbox{\fontfamily{phv}\selectfont d}}^{n'}]$ is well defined: both formulas give the same answer when $(m',n')$ can be represented as $(m',n')=(m+1,n)$ and as $(m',n')=(m,n+1)$.
We proceed by contradiction. Suppose that $m,n$ is a pair of smallest degree $m+n$ where consistency fails. This means that and still hold for all pairs of lower degree but the solution with $m$ replaced by $m+1$ and $n$ replaced by $n-1$ does not match the solution in . We show that this cannot be true by verifying that $$\begin{gathered}
\label{consistency}
{\left(\beta\Delta(\gamma-\alpha)+\gamma\Delta(\delta-\beta)q^m\right)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]+\gamma\delta [n]_qq^m\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]+\beta\gamma[m]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]}
\\=\left(\alpha \Delta(\delta-\beta)+\delta \Delta(\gamma-\alpha)q^{n-1}\right)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]
\\+ \alpha\delta[n-1]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]+\gamma\delta q^{n-1}[m+1]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}].\end{gathered}$$ (Formally, the term with the factor $[n-1]_q$ should be omitted when $n=1$.) We shall verify this by appealing to and several times.
The difference between the left hand side and the right hand side of is $$\begin{gathered}
\Delta(\gamma-\alpha)\left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta q^{n-1}\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right)
+\Delta(\delta-\beta)\left(\gamma q^m\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]-\alpha\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right)\\+
\left(\gamma\delta [n]_qq^m\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]-\alpha\delta[n-1]_q
\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]\right)
\\+ \left(\beta\gamma[m_q]\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta\gamma q^{n-1}[m+1]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right).\end{gathered}$$ Since $q^m[n]_q=q^m[n-1]q+q^{m}q^{n-1}$ and $q^{n-1}[m+1]q=q^{n-1}[m_q]+ q^{m}q^{n-1}$, canceling the appropriate terms in the last two terms of the sum we rewrite the above as $$\begin{gathered}
\Delta(\gamma-\alpha)\left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta q^{n-1}\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right)
+\Delta(\delta-\beta)\left(\gamma q^m\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]-\alpha\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right)\\+
\delta[n-1]_q\left(\gamma q^m\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]-\alpha
\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]\right)+ \gamma[m_q]\left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta q^{n-1}\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right).\end{gathered}$$ We now use and . We get $$\begin{gathered}
\Delta(\gamma-\alpha)\left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}{\mbox{\fontfamily{phv}\selectfont e}}]\right)+\Delta(\gamma-\alpha) \delta[n-1]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]
\\
+\Delta(\delta-\beta)\left(\gamma \varphi[{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]-\alpha\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right)-\Delta(\delta-\beta)\gamma[m]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\\+
\delta[n-1]_q\left(\gamma\varphi[{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]-\alpha
\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]\right)-\gamma\delta[n-1]_q [m]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]
\\+ \gamma[m_q]\left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}{\mbox{\fontfamily{phv}\selectfont e}}]\right)+\gamma\delta[m_q][n-1]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}].\end{gathered}$$ After canceling $\gamma\delta[m_q][n-1]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]$ we re-group the sum into three terms. $$\begin{gathered}
\Delta(\gamma-\alpha)\left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}{\mbox{\fontfamily{phv}\selectfont e}}]\right) -
\Delta(\delta-\beta)\left(\alpha\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]-\gamma \varphi[{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}]\right)\\
+\delta[n-1]_q\left(\Delta(\gamma-\alpha) \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]
-
\left(\alpha
\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]-\gamma\varphi[{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-2}]\right)\right)
\\
\gamma[m]_q\left( \left(\beta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^n]-\delta \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}{\mbox{\fontfamily{phv}\selectfont e}}]\right)-\Delta(\delta-\beta)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}] \right)
.\end{gathered}$$ From and we see that each of the three terms is zero, proving .
Formulas and extend $\varphi$ from ${{\mathcal M}}_{k}$ to ${{\mathcal M}}_{k+1}$.
Invariance conditions and hold for ${\mathbf{A}}\in{{\mathcal M}}_{k}$.
We verify and with $m+n\leq k$. By inductive assumption and hold when $m+n< k$, so we only need to consider $m+n=k$.
Using “swap identities" and we rewrite these relations as $$\begin{aligned}
\label{W+++} \alpha\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^n]-q^m\gamma \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}]&=&\Delta(\gamma-\alpha)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n]+ \gamma[m]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n}]\\
\label{V+++} -q^n\delta\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^n]+\beta \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}]&=&\Delta(\delta-\beta)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n]+\delta [n]_q\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n-1}],\end{aligned}$$ with the solution given in and . By linearity this establishes invariance conditions for all ${\mathbf{A}}\in{{\mathcal M}}_k$.
Signs of $\varphi$ on monomials
-------------------------------
To verify that $\varphi[({\mathbf{E}}+{\mathbf{D}})^L]\ne 0$, we will need the following version of a formula discussed in [@Mallick-Sandow-1997 Appendix A].
If ${\mathbf{X}}= {\mathbf{E}}^{m_1}\dots {\mathbf{D}}^{n_k}{\mathbf{E}}^{m_k}{\mathbf{D}}^{n_k}$ is a monomial of degree $m+n$ with $m=m_1+\dots+m_k$, $n=n_1+ \dots+n_k$, then there exist non-negative integers $b_j,c_j$ and monomials ${\mathbf{Y}}_j,{\mathbf{Z}}_j $ of degree $m+n$ such that $$\label{ED-expansion}
{\mathbf{X}}{\mathbf{E}}=q^n {\mathbf{E}}{\mathbf{X}}+\sum_j b_j {\mathbf{Y}}_j \mbox{ and } {\mathbf{D}}{\mathbf{X}}=q^m {\mathbf{X}}{\mathbf{D}}+\sum_j c_j {\mathbf{Z}}_j.$$
Denote ${\mathbf{S}}={\mathbf{E}}+{\mathbf{D}}$. Suppose that formulas hold for ${\mathbf{X}}$ with $k\geq 0$ factors. Then for $n=n_{k+1}$ and $m=m_{0}$ by repeated applications of we get $$\label{ED-expansion1}
{\mathbf{D}}^n{\mathbf{E}}=q{\mathbf{D}}^{n-1}{\mathbf{E}}{\mathbf{D}}+{\mathbf{D}}^{n-1}{\mathbf{S}}=q^2{\mathbf{D}}^{n-2}{\mathbf{E}}{\mathbf{D}}^2+{\mathbf{D}}^{n-2}{\mathbf{S}}{\mathbf{D}}+{\mathbf{D}}^{n-1}{\mathbf{S}}=
\dots=q^n{\mathbf{E}}{\mathbf{D}}+\sum_{j=0}^{n-1} {\mathbf{D}}^{n-1-j}{\mathbf{S}}{\mathbf{D}}^{j}$$ and $$\label{ED-expansion2}
{\mathbf{D}}{\mathbf{E}}^m=q {\mathbf{E}}{\mathbf{D}}{\mathbf{E}}^{m-1}+{\mathbf{S}}{\mathbf{E}}^{m-1}=q^2 {\mathbf{E}}^2 {\mathbf{D}}{\mathbf{E}}^{m-2}+{\mathbf{E}}{\mathbf{S}}{\mathbf{E}}^{m-2}+{\mathbf{S}}{\mathbf{E}}^{m-1}=\dots=q^m{\mathbf{E}}^m{\mathbf{D}}+\sum_{j=0}^{m-1} {\mathbf{E}}^j{\mathbf{S}}{\mathbf{E}}^{m-1-j}.$$ Clearly, ${\mathbf{D}}^{n-1-j}{\mathbf{S}}{\mathbf{D}}^{j}={\mathbf{D}}^{n-j-1}{\mathbf{E}}{\mathbf{D}}^{j}+{\mathbf{D}}^{n}$ is the sum of monomials of degree $n$ and ${\mathbf{E}}^j{\mathbf{S}}{\mathbf{E}}^{m-1-j}={\mathbf{E}}^{m}+{\mathbf{E}}^j{\mathbf{D}}{\mathbf{E}}^{m-1-j}$ is the sum of monomials of degree $m$. We now multiply by ${\mathbf{X}}{\mathbf{E}}^{m_{k+1}}$ from the left and use the induction assumption. Similarly, we multiply by ${\mathbf{D}}^{n_0}{\mathbf{X}}$ from the right and use the induction assumption. This establishes by induction.
\[P-positivity\] If $\alpha\beta=q^N\gamma\delta$ then
1. $(-1)^L\varphi_0[({\mathbf{E}}+{\mathbf{D}})^L]>0$ for $L=0,\dots,N$
2. $\varphi_1[({\mathbf{E}}+{\mathbf{D}})^L]>0$ for $L\geq N+1$.
\[Rem-signs\] An inspection of our argument shows that in the non-singular case with $\alpha\beta \ne q^n \gamma\delta$ for all $n$, we have $\varphi_0[({\mathbf{E}}+{\mathbf{D}})^L]\ne0$ for all $L$. More precisely, define $M=\min\{n\geq 0: \alpha\beta>q^n \gamma\delta\}$, with $M=0$ when $\alpha\beta>\gamma\delta$. Then
1. $(-1)^L\varphi_0[({\mathbf{E}}+{\mathbf{D}})^L]>0$ for $0\leq L\leq M$
2. $(-1)^M\varphi_0[({\mathbf{E}}+{\mathbf{D}})^L]>0$ for $L\geq M+1$.
In particular, the current $J=\varphi_0[({\mathbf{E}}+{\mathbf{D}})^L]/\varphi_0[({\mathbf{E}}+{\mathbf{D}})^{L-1}]$ undergoes a phase transition as the system size grows: $J<0$ for $1\leq L\leq M$ and $J>0$ for $L\geq M+1$.
Both proofs are similar and consist of showing that for $\varphi=\varphi_0$ and for $\varphi=\varphi_1$ the value $\varphi[{\mathbf{X}}]$ on a monomial ${\mathbf{X}}={\mathbf{E}}^{m_1}{\mathbf{D}}^{n_1}\dots {\mathbf{E}}^{m_k}{\mathbf{D}}^{n_k}$ is real, and that for all monomials ${\mathbf{X}}$ of the same degree $L=m+n$ with $m=m_1+\dots m_k$, $n=n_1+\dots+n_k$, the sign of $\varphi[{\mathbf{X}}]$ is the same. We begin with the recursive proof for functional $\varphi=\varphi_0$ where the signs alternate with $L$. Then we will indicate how to modify the proof for $\varphi=\varphi_1$ where the signs are all positive.
For $L=0$ we have $(-1)^L\varphi[{\mathbf{X}}]=1>0$ by the initialization of $\varphi_0$. Suppose that $(-1)^L\varphi[{\mathbf{X}}]>0$ holds for all monomials ${\mathbf{X}}={\mathbf{E}}^{m_1}{\mathbf{D}}^{n_1}\dots {\mathbf{E}}^{m_k}{\mathbf{D}}^{n_k}$ with $m=m_1+\dots+m_k=m$, $n=n_1+\dots+n_k=n$ of degree $L=m+n <N$. A monomial ${\mathbf{Y}}$ of degree $L+1$ arises from a monomial ${\mathbf{X}}$ of degree $L$ in one of the following ways: ${\mathbf{Y}}={\mathbf{E}}{\mathbf{X}}$, ${\mathbf{Y}}={\mathbf{X}}{\mathbf{D}}$, ${\mathbf{Y}}={\mathbf{D}}{\mathbf{X}}$, or ${\mathbf{Y}}= {\mathbf{X}}{\mathbf{E}}$. Our goal is to show that in each of these cases $\varphi[{\mathbf{Y}}]$ is a real number of the opposite sign than $\varphi[{\mathbf{X}}]$.
Cases ${\mathbf{Y}}={\mathbf{E}}{\mathbf{X}}$ and ${\mathbf{Y}}={\mathbf{X}}{\mathbf{D}}$ are handled together, and are needed for the other two cases. From and applied with ${\mathbf{A}}={\mathbf{X}}={\mathbf{E}}^{m_1}{\mathbf{D}}^{n_1}\dots {\mathbf{E}}^{m_k}{\mathbf{D}}^{n_k}$ we get $$\alpha\varphi[{\mathbf{E}}{\mathbf{X}}]-\gamma\varphi[{\mathbf{D}}{\mathbf{X}}]=\varphi({\mathbf{X}}) \mbox{ and }
-\delta\varphi[{\mathbf{X}}{\mathbf{E}}]+\beta\varphi[{\mathbf{X}}{\mathbf{D}}]=\varphi({\mathbf{X}}).$$ Applying to ${\mathbf{D}}{\mathbf{X}}$ and to ${\mathbf{X}}{\mathbf{E}}$ we get $$\begin{aligned}
\alpha\varphi[{\mathbf{E}}{\mathbf{X}}]-q^{m}\gamma\varphi[ {\mathbf{X}}{\mathbf{D}}]=d_1 \\
-q^{n}\delta\varphi[{\mathbf{E}}{\mathbf{X}}]+\beta\varphi[{\mathbf{X}}{\mathbf{D}}]=d_2.\end{aligned}$$ where by inductive assumption $d_1 =\varphi({\mathbf{X}})+\gamma\sum_{j}c_j\varphi({\mathbf{Z}}_j)$ is the sum of non-zero real numbers of the same sign $(-1)^L$, and similarly $ d_2 $ is real and has the sign $(-1)^L$. The solution of this system is $$\label{sign-check}
\varphi[{\mathbf{E}}{\mathbf{X}}]=\frac{\left|\begin{matrix}
d_1 &-q^m \gamma \\
d_2 & \beta
\end{matrix}\right|}{\left|\begin{matrix}
\alpha & -q^m\gamma \\
-q^n \delta & \beta
\end{matrix}\right|} \mbox{ and } \varphi[ {\mathbf{X}}{\mathbf{D}}]=\frac{\left|\begin{matrix}
\alpha & d_1\gamma \\
-q^n \delta & d_2
\end{matrix}\right|}{\left|\begin{matrix}
\alpha & -q^m\gamma \\
-q^n \delta & \beta
\end{matrix}\right|}.$$ Since the numerators have sign $(-1)^L$ and the denominator $\alpha\beta-q^{L}\gamma\delta=\gamma\delta(q^N-q^L)<0$, this establishes the conclusion for all monomials ${\mathbf{Y}}={\mathbf{E}}^{m_1+1}{\mathbf{D}}^{n_1}\dots {\mathbf{E}}^{m_k}{\mathbf{D}}^{n_k}$ and ${\mathbf{Y}}={\mathbf{E}}^{m_1}{\mathbf{D}}^{n_1}\dots {\mathbf{E}}^{m_k}{\mathbf{D}}^{n_k+1}$ of degree $m+n+1=L+1$.
To handle the case ${\mathbf{Y}}={\mathbf{D}}{\mathbf{X}}$, we use already established information about the sign of monomial $\varphi({\mathbf{E}}{\mathbf{X}})$. Using , we see that the sign of $\gamma\varphi[{\mathbf{D}}{\mathbf{X}}]=\alpha \varphi[{\mathbf{E}}{\mathbf{X}}]-\varphi[{\mathbf{X}}]$ is $(-1)^{L+1}$, and similarly determines the sign of $\delta\varphi[{\mathbf{X}}{\mathbf{E}}]=\beta\varphi[{\mathbf{X}}{\mathbf{D}}]-\varphi[{\mathbf{X}}]$ as $(-1)^{L+1}$.
The proof for $\varphi=\varphi_1$ is similar, starting with formula which establishes positivity for $L=N+1$. We then use to prove that $\varphi_1[{\mathbf{E}}{\mathbf{X}}]>0$ and $\varphi_1[{\mathbf{X}}{\mathbf{D}}]>0$, noting that in the case of $\varphi_1$ we have $d_1,d_2>0$ and that the denominator $\alpha\beta-q^{L}\gamma\delta=\gamma\delta(q^N-q^L)>0$ as $L\geq N+1$. Finally, applying $\varphi_1$ to we see that $\varphi_1[{\mathbf{D}}{\mathbf{X}}]>0$ and $\varphi_1[{\mathbf{X}}{\mathbf{E}}]>0$.
Functional $\varphi_0$ satisfies invariance conditions and , and $\varphi_0\left[({\mathbf{E}}+{\mathbf{D}})^L\right]\ne 0$ for $L\leq N$ by Proposition \[P-positivity\]. Therefore, by Theorem \[T2\] we get for $L\leq N$. In the non-singular case, by Remark \[Rem:nonsingular\] functional $\varphi_0$ is defined on ${{\mathcal M}}$ and by Remark \[Rem-signs\] we have $\varphi_0[({\mathbf{E}}+{\mathbf{D}})^L]\ne 0$ for all $L$, so Theorem \[T2\] applies.
Functional $\varphi_1$ satisfies invariance conditions and by Lemma \[L-phi12prod\] and construction. Proposition \[P-positivity\] states that $\varphi_1\left[({\mathbf{E}}+{\mathbf{D}})^L\right]>0$ for $L\geq N+1$. Therefore, by Theorem \[T2\] we get for all $L\geq N+2$. Proposition \[P1\] gives the stationary distribution for $L=N+1$, and Lemma \[L-phi12prod\] shows that this case also arises from .
Proof of Theorem \[T3\] {#sect:ProofT3}
=======================
Denote $\varphi_{k,n}=\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^n]$, where $\varphi$ is either $\varphi_0$ or $\varphi_1$. (The latter is needed only for the second part of Theorem \[T-Hermit2\].) We first rewrite and using Askey-Wilson parameters . $$\begin{gathered}
\label{SolA1}
\varphi_{m+1,n}=\frac{1}{1-abcd q^{m+n}}\Big(
\theta\left(c+d- cd (a+b)q^m\right)\varphi_{m,n}- cd [m]_q\varphi_{m-1,n}+abcd q^m [n]_q\varphi_{m,n-1}
\Big),\end{gathered}$$ $$\begin{gathered}
\label{SolB1}
\varphi_{m,n+1}=\frac{1}{1-abcd q^{m+n}}\Big(\theta\left(a+b - ab(c+d)q^n\right)\varphi_{m,n}-ab [n]_q \varphi_{m,n-1}+abcd q^n [m]_q\varphi_{m-1,n}
\Big).\end{gathered}$$
In fact, it might be simpler to use to rewrite and and then solve the system of equations. Equations and simplify to $$\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m+1}{\mbox{\fontfamily{phv}\selectfont d}}^n]+cd\varphi[{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]=\theta (c+d)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]$$ $$ab \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^{m}{\mbox{\fontfamily{phv}\selectfont d}}^n{\mbox{\fontfamily{phv}\selectfont e}}]+ \varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^{n+1}]=\theta (a+b)\varphi[{\mbox{\fontfamily{phv}\selectfont e}}^m{\mbox{\fontfamily{phv}\selectfont d}}^n]$$
Our proof relies heavily on monic continuous $q$-Hermite polynomials which are defined by the three step recurrence $$\label{EQ-H}
x H_n(x)=H_{n+1}(x)+[n]_qH_{n-1}(x)$$ with initial values $H_0(x)=1$ and $H_{-1}(x)=0$. Somewhat more generally, for $t\in{\mathds{C}}$ we consider polynomials $H_n(x;t)$ defined by the three step recurrence $$\label{q-H(t) recursion}
x H_n(x;t)=H_{n+1}(x;t)+t[n]_qH_{n-1}(x;t)$$ with initial values $H_0(x;t)=1$ and $H_{-1}(x;t)=0$. These two families of polynomials are related by a simple formula $H_n(x;t^2)=t^{n }H_n(x/t)$.
These polynomials are convenient because when evaluated at ${\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}$ they have explicit expansion in the basis of monomials in normal order. The following version of [@bozejko97qGaussian Corollary 2.8] follows from .
\[L-BKS\] $$H_n(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}\;;t)=\sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
} t^k{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}.$$
Since $H_0(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}})={\mathbf{I}}$ and $H_1(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}})=t{\mbox{\fontfamily{phv}\selectfont e}}{\mbox{\fontfamily{phv}\selectfont d}}^0+{\mbox{\fontfamily{phv}\selectfont e}}^0{\mbox{\fontfamily{phv}\selectfont d}}$, we only need to verify that holds. We have $$(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}})\sum_{k=0}^n{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} t^k{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}-t [n]_q\sum_{k=0}^{n-1} {\begin{bmatrix}n-1\\#2
\end{bmatrix}_q
} t^k{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-k} = \sum_{k=0}^{n+1} {\begin{bmatrix}n+1\\#2
\end{bmatrix}_q
} t^k{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}$$ Using , the left hand side is $$\begin{gathered}
\sum_{k=0}^n{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} t^{k+1}{\mbox{\fontfamily{phv}\selectfont e}}^{k+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}+\sum_{k=0}^n{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} t^k{\mbox{\fontfamily{phv}\selectfont d}}{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}-t[n]_q\sum_{k=0}^{n-1}
{\begin{bmatrix}n-1\\#2
\end{bmatrix}_q
} t^k{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-1-k}
\\=
t^{n+1}{\mbox{\fontfamily{phv}\selectfont e}}^{n+1}+\sum_{k=0}^{n-1}{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} t^{k+1}{\mbox{\fontfamily{phv}\selectfont e}}^{k+1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}+\sum_{k=1}^nq^k{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} t^k
{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n+1-k}+{\mbox{\fontfamily{phv}\selectfont d}}^{N+1}
\\+
[n]_q\sum_{k=1}^n{\begin{bmatrix}n-1\\#2
\end{bmatrix}_q
} t^k {\mbox{\fontfamily{phv}\selectfont e}}^{k-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-k} -[n]_q\sum_{k=1}^{n} {\begin{bmatrix}n-1\\#2
\end{bmatrix}_q
} t^{k}{\mbox{\fontfamily{phv}\selectfont e}}^{k-1}{\mbox{\fontfamily{phv}\selectfont d}}^{n-k}
\\=
t^{n+1}{\mbox{\fontfamily{phv}\selectfont e}}^{n+1}+\sum_{k=1}^{n}\left({\begin{bmatrix}n\\#2
\end{bmatrix}_q
} +q^k{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} \right) t^k{\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n+1-k}+{\mbox{\fontfamily{phv}\selectfont d}}^{N+1}+0
= \sum_{k=0}^{n+1} {\begin{bmatrix}n+1\\#2
\end{bmatrix}_q
} t^k {\mbox{\fontfamily{phv}\selectfont e}}^k{\mbox{\fontfamily{phv}\selectfont d}}^{n-k} ,\end{gathered}$$ as $${\begin{bmatrix}n\\#2
\end{bmatrix}_q
}+q^k{\begin{bmatrix}n\\#2
\end{bmatrix}_q
}={\begin{bmatrix}n+1\\#2
\end{bmatrix}_q
}$$
We now introduce two numerical sequences: $$\label{EQ-G}
G_n(t):=\varphi_0\left[ H_n(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}\;;t)\right],$$ where $0\leq n<N+1$ (we include here non-singular case by allowing $N=\infty$), and $$\label{EQ-F}
F_n(t):=\varphi_1\left[H_{n+N}(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}\;;t)\right], \; n\geq 1.$$ It turns out that these sequences satisfy similar recursions.
\[T-Hermit2\] For $0\leq n <N$ we have $$\begin{gathered}
\label{Eq:MArcin2}
G_{n+1}(t)=\frac{\theta}{1-abcdq^n}\left((a+b)(1-t cd)G_n(qt)+(c+d)(t-q^n a b)G_n(t)\right) \\
-\theta^2\frac{1-q^n}{1-a b c d q^n}\left(ab(1-t cd)G_{n-1}(qt)+t cd(t-a b q^n)G_{n-1}(t)\right)
\mathcal{}\end{gathered}$$ with $G_0(t)=1$ and $G_{-1}(t)=0$.
For $n\geq 1$ we have $$\begin{gathered}
\label{Eq:MArcin1}
F_{n+1}(t)=\frac{\theta}{1- q^{n }}\left((a+b)(1-t cd)F_n(qt)+(c+d)(t-q^{n+N} a b)F_n(t)\right) \\
-\theta^2\frac{1-q^{n+N}}{1- q^{n }}\left(ab(1-t cd)F_{n-1}(qt)+t cd(t-a b q^{n+N})F_{n-1}(t)\right)\end{gathered}$$ with $$F_1(t)= \frac{1}{\theta^{N+1}}\prod_{j=0}^{N}\frac{1-cd t q^j}{1-cd q^j} =\frac{(tcd;q)_{N+1}}{\theta^{N+1}(cd;q)_{N+1}}$$ and $F_{0}(t)=0$.
Using the well known identity ${\begin{bmatrix}n+1\\#2
\end{bmatrix}_q
}=q^k{\begin{bmatrix}n\\#2
\end{bmatrix}_q
}+{\begin{bmatrix}n\\#2
\end{bmatrix}_q
}$ we write $$\begin{gathered}
G_{n+1}(t)=\sum_{k=0}^{n+1}{\begin{bmatrix}n+1\\#2
\end{bmatrix}_q
}\varphi_{k,n+1-k} t^k
=\varphi_{0,n+1}+\sum_{k=1}^{n}{\begin{bmatrix}n+1\\#2
\end{bmatrix}_q
}\varphi_{k,n+1-k} t^k+\varphi_{n+1,0}t^{n+1}
\\=\sum_{k=0}^{n}{\begin{bmatrix}n\\#2
\end{bmatrix}_q
} q^k\varphi_{k,n+1-k}+t \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}\varphi_{k+1,n-k} t^k = A+B \mbox{ (say)}.\end{gathered}$$ Applying to expression $A$ we get
$$\begin{gathered}
(1-abcd q^n) A= \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}q^k t^k \theta(a+b)\varphi_{k,n-k}
-\sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}q^kt^k \theta ab(c+d)q^{n-k}\varphi_{k,n-k}
\\
-ab \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}[n-k]_qq^k t^k \varphi_{k,n-1-k}+abcd \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}[k]_q q^k t^k q^{n-k}\varphi_{k-1,n-k}
\\= \theta (a+b)G_n(qt)-\theta ab(c+d)q^n G_n(t)-ab [n]_q G_{n-1}(qt)+abcd [n]_q q^n t G_{n-1}(t)\end{gathered}$$
Similarly applying to expression $B$ we get $$\begin{gathered}
(1-abcd q^n) B= t \theta (c+d)\sum_{k=0}^n{\begin{bmatrix}n\\#2
\end{bmatrix}_q
}t^k\varphi_{k,n-k}-cd (a+b)\theta t \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
} q^k t^k \varphi_{k,n-k}\\
-t cd \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
} [k]_q t^k \varphi_{k-1,n-k}+t abcd \sum_{k=0}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}[n-k]_q t^k q^k \varphi_{k,n-1-k}
\\= \theta t(c+d)G_n(t)-\theta t cd (a+b) G_n(qt)-t cd [n]_qG_{n-1}(t)+t abcd [n]_q G_{n-1}(qt).\end{gathered}$$ Since $[n]_q=\theta^2(1-q^n)$, we get .
To determine the initial $F_1(t)$ we apply Lemma \[L-BKS\] and formula which in parameters becomes $$\varphi_{k,N+1-k}=\frac{(-cd)^kq^{\frac{k(k-1)}{2}}}{\theta^{N+1}\prod_{j=0}^{N}(1-cdq^j)}=\frac{1}{\theta^{N+1}(cd;q)_{N+1}}(-cd)^kq^{\frac{k(k-1)}{2}}.$$ We have $$\begin{gathered}
F_1(t)=\varphi_1[H_{N+1}(t{\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}};t)]= \sum_{k=0}^{N+1}{\begin{bmatrix}N+1\\#2
\end{bmatrix}_q
} t^{ k}\varphi_{k,N+1-k}=
\frac{1}{\theta^{N+1}(cd;q)_{N+1}}\sum_{k=0}^{N+1}{\begin{bmatrix}N+1\\#2
\end{bmatrix}_q
} t^{ k} (-cd)^kq^{\frac{k(k-1)}{2}}
\\
=\frac{ \prod_{j=0}^N(1-cdtq^j)} {\theta^{N+1}\prod_{j=0}^N(1-cdq^j)}
,\end{gathered}$$ where we used Cauchy’s $q$-binomial formula . The remaining steps of the proof are similar to the proof of recursion and are omitted.
For completeness, we include the omitted steps in the expanded version of the paper.
$$\begin{gathered}
F_{n+1}(t)=\sum_{k=0}^{N+n+1}{N+n+1\brack k}_q\varphi_{k,N+n+1-k}t^k\\
=\varphi_{0,N+n+1}+\sum_{k=1}^{N+n}{N+n+1\brack k}_q\varphi_{k,N+n+1-k}t^k+\varphi_{N+n+1,0}t^{N+n+1}\\
=\varphi_{0,N+n+1}+\sum_{k=1}^{N+n}\left({N+n\brack k}_q q^k+{N+n\brack k-1}_q\right)\varphi_{k,N+n+1-k}t^k+\varphi_{N+n+1,0}t^{N+n+1}\\
=\sum_{k=0}^{N+n}{N+n\brack k}_q q^k \varphi_{k,N+n-k+1}t^k+t\sum_{k=0}^{N+n}{N+n\brack k}_q \varphi_{k+1,N+n-k}t^k =A'+B' \mbox{ (say)}.
\end{gathered}$$
Applying to expression $A'$ we get $$\begin{gathered}
(1-q^{n}) A'= \sum_{k=0}^{n+N} {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
}q^k t^k \theta(a+b)\varphi_{k,n+N-k}
-\sum_{k=0}^{n+N} {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
}q^kt^k \theta ab(c+d)q^{n+N-k}\varphi_{k,n+N-k}
\\
-ab \sum_{k=0}^{n+N} {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
}[n+N-k]_qq^k t^k \varphi_{k,n+N-1-k}+abcd \sum_{k=0}^{n+N} {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
}[k]_q q^k t^k q^{n+N-k}\varphi_{k-1,n+N-k}
\\= \theta (a+b)F
_n(qt)-\theta ab(c+d)q^{n+N} F_n(t)-ab [n+N]_q F_{n-1}(qt)+abcd [n+N]_q q^{n+N} t F_{n-1}(t)
\end{gathered}$$ Similarly applying to expression $B'$ we get $$\begin{gathered}
(1-q^{n}) B'= t \theta (c+d)\sum_{k=0}^{n+N}{\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
} t^k\varphi_{k,n+N-k}-cd (a+b)\theta t \sum_{k=0}^{n+N} {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
} q^k t^k \varphi_{k,n+N-k}\\
-t cd \sum_{k=0}^{n+N} {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
} [k]_q t^k \varphi_{k-1,n+N-k}+t abcd \sum_{k=0}^n {\begin{bmatrix}n+N\\#2
\end{bmatrix}_q
}[n+N-k]_q t^k q^k \varphi_{k,n+N-1-k}
\\= \theta t(c+d)F_n(t)-\theta t cd (a+b) F_n(qt)-t cd [n+N]_qF_{n-1}(t)+t abcd [n+N]_q F_{n-1}(qt).
\end{gathered}$$ Since $[n+N]_q=\theta^2(1-q^{n+N})$, we get .
We now want to express the $q$-Hermite polynomials as linear combinations of the Askey-Wilson polynomials. We will start with the following two explicit formulas for the connection coefficients, relating $q$-Hermite polynomials with Al-Salam-Chihara polynomials in the first step, and then with Askey-Wilson polynomials in the second step. (This topic is well studied, see e.g. [@foupouagnigni2013connection; @tcheutia2014connection] and the references therein, so both formulas should be known; but we were not able to locate them in the literature.)
\[HerToAWCon\] For $a,b\in{\mathds{C}}$, the connection coefficients in the expansion $$p_n(x;0,0,0,0|q)=\sum_{k=0}^{n}c_{n,k}p_k(x;a,b,0,0|q)\label{AW2H0}$$ are $$\label{c_nk}
c_{n,k}=\sum_{\ell=k}^n {\begin{bmatrix}n\\#2
\end{bmatrix}_q
}{\begin{bmatrix}\ell\\#2
\end{bmatrix}_q
}a^{n-\ell} b^{\ell-k}.$$ If $a\ne 0$, the connection coefficients in the expansion $$p_n(x;0,0,0,0|q)=\sum_{\ell=0}^ne_{n,\ell}(a,b,c,d)p_\ell(x;a,b,c,d|q)$$ are $$e_{n,\ell}(a,b,c,d)=\sum_{k=\ell}^nc_{n,k}{k \brack \ell}_q\frac{q^{\ell(\ell-k)}(abq^\ell;q)_{k-\ell}}{a^{k-\ell}(abcdq^{\ell-1};q)_\ell}{_3\phi_2}\left(\begin{matrix}
q^{\ell-k},\ acq^\ell,\ adq^\ell\\\ 0,\ abcdq^{2\ell}
\end{matrix}\middle|q;q\right).$$
Since holds trivially when $a=b=0$, by symmetry of $p_k(x;a,b,0,0|q)$ in parameters $a,b$, we can assume $a\ne 0$. From we see that $$\label{AW2H1} p_n(x;a,0,0,0|q)=\sum_{k=0}^n C_{n,k}p_k(x;a,b,0,0|q),$$ where $$C_{n,k}=\frac{q^{k(k-n)}}{a^{n-k}}{n \brack k}_q {_2\phi_1}\left(\begin{matrix}
q^{k-n},\ abq^k\\ 0
\end{matrix}\middle|q;q\right)=\frac{q^{k(k-n)}}{a^{n-k}}{n \brack k}_q(abq^k)^{n-k}={n \brack k}_qb^{n-k}$$ (we used formula .) In particular is valid also for $a=0$. Setting $a=0$ in , using symmetry again, and renaming $b$ as $a$ we get $$\label{AW2H2} p_n(x;0,0,0,0|q)=\sum_{k=0}^n {n \brack k}_qa^{n-k}p_k(x;a,0,0,0|q).$$ Combining with proves that $$\label{AW2H3} p_n(x;0,0,0,0|q)=\sum_{k=0}^n c_{n,k} p_k(x;a,b,0,0|q),$$ where $c_{n,k}$ is given by . This formula holds for all $a,b$.
Next we prove the second connection formula for $a\ne 0$. From follows that the coefficient $C'_{k,\ell}$ in the expansion $$\label{AW2H4} p_k(x;a,b,0,0|q)=\sum_{\ell=0}^n C'_{k,\ell} p_\ell(x;a,b,c,d|q)$$ is equal to $${k \brack \ell}_q\frac{q^{\ell(\ell-k)}(abq^\ell;q)_{k-\ell}}{a^{k-\ell}(abcdq^{\ell-1};q)_\ell}{_3\phi_2}\left(\begin{matrix}
q^{\ell-k},\ acq^\ell,\ adq^\ell\\\ 0,\ abcdq^{2\ell}
\end{matrix}\middle|q;q\right).$$ This ends the proof, since $e_{n,\ell}(a,b,c,d)=\sum_{k=\ell}^nc_{n,k}C'_{k,\ell}$.
Suppose that the degrees of polynomials $p_k$ are $k$ for $k=0,1,\dots,n$. (Recall that this fails for large $n$ if $q^Nabcd=1$ for some $N=0,1,\dots$.) Denote by $\{a_{n,k}(a,b,c,d)\}$ the coefficients in the expansion $$\label{H2P}
H_n(x)=\sum_{k=0}^n a_{n,k}(a,b,c,d)p_k\left(\frac{x}{2\theta};a,b,c,d|q\right),$$ where $H_n(x)=H_n(x;1)$ is given by .
We will need explicit formula for the coefficient $A_n(a,b,c,d):=a_{n,0}(a,b,c,d)$. Since $a_{n,k}(a,b,c,d)$ are invariant under permutations of $a,b,c,d$, without loss of generality we assume $a\ne 0$. This is enough for our purposes, as we have $a,c>0$ for the parameters arising from ASEP.
$$\label{Marcin-SolA}
A_n(a,b,c,d)=\theta^n\sum_{k=0}^nc_{n,k}\frac{(ab;q)_k}{a^k}{_3\phi_2}\left(\begin{matrix}
q^{-k},\ ac,\ ad\\\ 0,\ abcd
\end{matrix}\middle|q;q\right),$$
with $c_{n,k}$ given by .
By comparing the three step recursions, it is clear that $H_n(x)=\theta^n p_n(\frac{x}{2\theta};0,0,0,0|q)$. Hence, by Proposition \[HerToAWCon\], $A_n(a,b,c,d)=\theta^n e_{n,0}(a,b,c,d)$.
It turns out that $A_n(a,b,c,d)$ is related to the moment of the $n$-th $q$-Hermite polynomial introduced in .
\[L:A=G\] For $0\leq n<N$, $a,c>0$ and $t\ne 0$ we have $$t^n G_n(1/t^2)=A_n(at,bt,c/t,d/t).$$
For the proof, we need to rewrite both sides of this equation.
For the next lemma, we write $G_{n}(z)$ as $G_n(z;a,b,c,d)$ with explicitly written Askey-Wilson parameters. In this notation, Proposition \[L:A=G\] says $t^n G_n(1/t^2;a,b,c,d)=A_n(at,bt,c/t,d/t)$, which is the same as $t^n G_n(1/t^2;a/t,b/t,ct,dt)=A_n(a,b,c,d)$.
Expression $$\label{B-sub} B_n(a,b,c,d):=
(abcd;q)_n\frac{G_n(t^2; at, bt, c/t,d/t)}{\theta^n t^n}$$ does not depend on $t$ and satisfies the following recursion for $0\leq n <N$: $$\begin{gathered}
\label{B-rec}
B_{n+1}( a,b,c,d)= (a +b)(1-cd)q^{n/2}B_n(
a/\sqrt{q},b/\sqrt{q},c\sqrt{q},d\sqrt{q})+(c+d)(1-q^n ab) B_n(
a,b,c,d)\\
-(1-q^n)(1-abcd q^{n-1}) \left(ab (1-cd)q^{(n-1)/2}B_{n-1}(
a/\sqrt{q},b/\sqrt{q},c\sqrt{q},d\sqrt{q}) +cd (1-ab q^n)B_{n-1}(a,b,c,d)\right)
\end{gathered}$$ with the initial value $B_0(a,b,c,d)=1$, and $B_{-1}(a,b,c,d)=0$.
Denote by $\widetilde G_n(t^2;a,b,c,d)$ the right hand side of . Inserting this expression into we get recursion $$\begin{gathered}
\label{tildeG}
\widetilde G_{n+1}(t^2;a,b,c,d)=(a +b)(1-cd)\widetilde G_n(q t^2;a,b,c,d)+(c+d)(1-q^n ab) \widetilde G_n(
t^2;a,b,c,d)\\
-(1-q^n)(1-abcd q^{n-1}) \left(ab (1-cd) \widetilde G_{n-1}(q t^2;a,b,c,d)+cd (1-ab q^n)\widetilde G_{n-1}( t^2;a,b,c,d)\right)
\end{gathered}$$ with the coefficients that do not depend on $t$. Since the initial condition $\widetilde G_{-1}=0$ and $\widetilde G_{0}=1$ does not depend on $t$, therefore the solution of the recursion does not depend on $t$. We check this by induction, assuming that this assertion holds for $\widetilde G_0,\dots,\widetilde G_n$. Denoting $\tilde t=t\sqrt{q}$ we have $$\begin{gathered}
\widetilde G_n(q t^2;a,b,c,d)= (abcd;q)_n\frac{G_n(q t^2; at, bt, c/t,d/t)}{\theta^n t^n} = \\
=q^{n/2} (abcd;q)_n\frac{G_n(\tilde t ^2; \frac{a}{\sqrt{q}}\tilde t, \frac{b}{\sqrt{q}}\tilde t, \sqrt{q} c/\tilde t,\sqrt{q}d/\tilde
t)}{\theta^n \tilde t^n }=q^{n/2} B_n( a/\sqrt{q},b/\sqrt{q},c\sqrt{q},d\sqrt{q}).\end{gathered}$$ Thus shows that $\widetilde G_{n+1}(t^2;a,b,c,d)$ does not depend on $t$, and recursion follows.
Next we rewrite the right hand side of the equation in Proposition \[L:A=G\]. Denote $$\widetilde A_n(a,b,c,d)=
(abcd;q)_n A_n(a,b,c,d)/\theta^n = (abcd;q)_n \sum_{k=0}^nc_{n,k}\frac{(ab;q)_k}{a^k}{_3\phi_2}\left(\begin{matrix}
q^{-k},\ ac,\ ad\\\ 0,\ abcd
\end{matrix}\middle|q;q\right).$$ We rewrite this as $$\widetilde A_n(a,b,c,d)=(abcd;q)_n\sum_{k=0}^n (ab;q)_k c_{n,k}\beta_k$$ with $$\beta_k(a,b,c,d)=\frac{1}{a^k}{_3\phi_2}\left(\begin{matrix}
q^{-k},\ a d,\ a c\\0 ,\ a b c d
\end{matrix}\middle|q;q\right)=\frac{1}{a^k}\sum_{j=0}^k \frac{(q^{-k}, ad, ac;q)_j}{(q, abcd;q)_j}q^j.$$ In order to prove Proposition \[L:A=G\] it is enough to show that $\widetilde A_n(a,b,c,d)=B_n(
a,b,c,d)$. Since both expressions are $1$ when $n=0$, we only need to verify that $\widetilde A_n(a,b,c,d)$ satisfies recursion . To accomplish this goal, we need auxiliary recursions for the coefficients $c_{n,k}$ and $\beta_k$.
\[lemmarek1\] With the usual convention that $c_{n,k}=0$ if $k>n$ or $k<0$, for all $n\geq 0$ and all $k$, we have $$\label{lemarek1A}
c_{n+1,k}=c_{n,k-1}+q^k(a+b)c_{n,k}-q^k(1-q^n)ab\cdot c_{n-1,k}.$$ Furthermore, for $n\geq 1$ and $ 0\leq k\leq n$ we have $$\label{lemarek1B}
(1-q^{k+1})c_{n,k+1}=(1-q^n)c_{n-1,k}.$$
Let $h_{n}(x)=p_n(x;0,0,0,0|q)$ and $Q_n(x)=p_n(x;a,b,0,0|q)$. Then is $$h_n(x)=\sum_{k=0}^n c_{n,k}Q_k(x), \; n\geq 0.$$ Comparing the three step recursions $$2xh_n(x)=h_{n+1}(x)+(1-q^n)h_{n-1}(x)$$ and $$2xQ_n(x)=Q_{n+1}(x)+q^n(a+b)Q_n(x)+(1-q^n)(1-q^{n-1}ab)Q_{n-1}(x),$$ we get $$\label{c***}
c_{n+1,k}=c_{n,k-1}+q^k(a+b)c_{n,k}+(1-q^{k+1})(1-q^kab)c_{n,k+1}-(1-q^n)c_{n-1,k}.$$ Indeed, expanding both sides of $2xh_n(x)=h_{n+1}(x)+(1-q^n)h_{n-1}(x)$ and applying the second recurrence to the expansion on left hand side we get $$\sum_{k=0}^n c_{n,k}\left(Q_{k+1}(x)+q^k(a+b)Q_k(x)+(1-q^k)(1-q^{k-1}ab)Q_{k-1}(x)
\right)
= \sum_{k=0}^{n+1}c_{n+1,k}Q_k(x)+(1-q^n)\sum_{k=0}^{n-1}c_{n-1,k}Q_k(x).$$ The formula follows by comparing the coefficients at $Q_k(x)$.
Since $c_{n,k}=c_{n,k}(a,b)$ is a homogeneous polynomial of degree $n-k$ in variables $a$ and $b$, we can separate the components of recursion into the pair of recursions. The terms of degree $n-k-1$ give . The terms of degree $n+1-k$ give $c_{n+1,k}=c_{n,k-1}+q^k(a+b)c_{n,k}-(1-q^{k+1})q^k a b \cdot c_{n,k+1}$, which gives after using .
\[corrolaryocnk\] $$(ab;q)_kc_{n+1,k}-(1-q^nab)(ab;q)_{k-1}c_{n,k-1}=(a+b)\left(\frac{ab}{q};q\right)_kq^kc_{n,k}-(1-q^n)ab \left(\frac{ab}{q};q\right)_kq^kc_{n-1,k}.$$
It’s enough to prove that $$(1-abq^{k-1})c_{n+1,k}=(a+b)\left(1-\frac{ab}{q}\right)q^k c_{n,k}+(1-q^nab)c_{n,k-1}-(1-q^n)ab\left(1-\frac{ab}{q}\right)q^kc_{n-1,k}.$$ Since $c_{n,k}$ is a homogeneous polynomial of degree $n-k$ in variables $a$ and $b$ this is equivalent to a pair of identities $$c_{n+1,k}=q^k(a+b)c_{n,k}+c_{n,k-1}-q^k(1-q^n)ab\cdot c_{n-1,k},\label{corrolaryocnkeq1}$$ which is , and $$-abq^{k-1}c_{n+1,k}=-ab(a+b)q^{k-1}c_{n,k}-q^nab\cdot c_{n,k-1}+(1-q^n)a^2b^2q^{k-1}c_{n-1,k}.\label{corrolaryocnkeq2}$$ To prove it is enough to verify that $$q^kc_{n+1,k}=q^k(a+b)c_{n,k}+q^{n+1}c_{n,k-1}-q^k(1-q^n)ab\cdot c_{n-1,k}.$$ To do this, we subtract this expression from and use .
We get $(1-q^k)c_{n+1,k}= (1-q^{n+1})c_{n,k-1} $.
We also need the following recursion which was discovered by Mathematica package [qZeil]{} [@paule1997mathematica], but for which we have a standard proof.
\[lammarek2\] For $0\leq n <N$, $a\ne 0$ and $b,c,d\in{\mathds{C}}$ we have $$\label{qZeil} (1-a b c d q^{n}) \beta_{n+1}(a,b,c,d)= \left(c +d - c d (a+b) q^{n}\right) \beta_{n}(a,b,c,d)-
c d \left(1-q^{n}\right) \beta_{n-1}(a,b,c,d).$$ The initial condition for this recursion is $\beta_{0}=1, \beta_{-1}=0$.
For $a\ne 0$, consider the Al-Salam–Chihara polynomials $$\label{Q-AlSalam}
\widetilde Q_n(x;a,b)=\frac{a^n}{(ab;q)_n}p_n(x;a,b,0,0|q)={_3}\phi_2\left(\begin{matrix}
q^{-n},a e^{i\psi},a e^{-i\psi} \\
0,\ ab
\end{matrix}\middle|q;q\right),$$ where $x=\cos\psi$. The three step recursion for polynomials $\widetilde Q_n(x)$ is $$\label{ThreeStepKoek}
2x\widetilde Q_n(x;a,b)=a^{-1}(1-abq^n)\widetilde Q_{n+1}(x;a,b)
+(a+b)q^n\widetilde Q_n(x;a,b)+a(1-q^n)\widetilde Q_{n-1}(x;a,b)$$ with $\widetilde Q_0(x;a,b)=1$ and $\widetilde Q_{-1}(x;a,b)=0$. (Compare e.g. [@koekoek1998askey (3.8.4)], who normalize the polynomials differently.) For $c,d>0$ let $x_*=\frac12\left(\sqrt{\tfrac{c}{d}}+\sqrt{\tfrac{d}{c}}\right)$. It is easy to see that $$\widetilde Q_n\left(x_*;a\sqrt{cd},b\sqrt{cd}\right)={_3}\phi_2\left(\begin{matrix}
q^{-n},\ ac,\ ad \\
0,\ abcd
\end{matrix}\middle|q;q\right)=a^n\beta_n(a,b,c,d).$$ Indeed, to extend polynomial $\widetilde Q_n(x)$ from $x=\cos \psi\in[-1,1]$ to $x>1$ we replace $e^{\pm i\psi}$ in by $x\pm \sqrt{x^2-1}$. These expressions evaluate to $\sqrt{c/d}$ and $\sqrt{d/c}$ at $x=\frac12\left(\sqrt{\tfrac{c}{d}}+\sqrt{\tfrac{d}{c}}\right)$.
Recursion implies that $$\begin{gathered}
\left(\sqrt{\tfrac{c}{d}}+\sqrt{\tfrac{d}{c}}\right)a^n\beta_n=
\frac{1}{a\sqrt{cd}}(1-abcdq^n)a^{n+1}\beta_{n+1}
+\left(a\sqrt{cd}+b\sqrt{cd}\right)q^na^n\beta_n
+a\sqrt{cd}(1-q^n)a^{n-1}\beta_{n-1}.\end{gathered}$$ This implies for $a\ne 0$ and $c,d>0$. We now use the fact that $\beta_{n}(a,b,c,d)$ is a rational function of $a,b,c,d$, with the denominator that has factors $a^k$ and $1-abcd q^k$, $0\leq k\leq n<N$. Thus recursion extends to all $a,b,c,d$ within the domain of $\beta_{n}(a,b,c,d)$.
We will show that $\widetilde A_n(a,b,c,d)=(abcd;q)_n\sum_{k=0}^n (ab;q)_k c_{n,k}\beta_k$ satisfies recursion .
We first note that $c_{n,k}(a/\sqrt{q},b/\sqrt{q})=q^{(k-n)/2}c_{n,k}(a,b)$ and $\beta_k(a/\sqrt{q},b/\sqrt{q},c\sqrt{q},d\sqrt{q})=q^{k/2}\beta_k(a,b,c,d)$. We therefore want to show that $$\begin{aligned}
\frac{\widetilde A_{n+1}(a,b,c,d)}{(abcd;q)_n}&=&(a+b)(1-cd)\sum_{k=0}^{n}\left(\frac{ab}{q};q\right)_kq^kc_{n,k}\beta_k\\
&&+(c+d)(1-q^nab)\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k}\beta_k\\
&&-(1-q^n)ab(1-cd)\sum_{k=0}^{n-1}\left(\frac{ab}{q};q\right)_kq^kc_{n-1,k}\beta_k\\
&&-(1-q^n)cd(1-abq^{n})\sum_{k=0}^{n}\left(ab;q\right)_kc_{n-1,k}\beta_k.\end{aligned}$$ We will be working with the right hand side of this equation. The sum of the first and the third term is equal to $$(1-cd)\sum_{k=0}^n\left[(a+b)\left(\frac{ab}{q};q\right)_kq^kc_{n,k}-(1-q^n)ab\left(\frac{ab}{q};q\right)_kq^k c_{n-1,k}\right]\beta_k.$$ By Corollary \[corrolaryocnk\] this is equal $$(1-cd)\sum_{k=0}^{n}\left(ab;q\right)_kc_{n+1,k}\beta_k-(1-cd)(1-abq^n)\sum_{k=0}^{n}\left(ab;q\right)_{k-1}c_{n,k-1} \beta_k=$$ $$=(1-abcdq^n)\sum_{k=0}^{n}\left(ab;q\right)_kc_{n+1,k}\beta_k-cd(1-abq^n)\sum_{k=0}^{n}\left(ab;q\right)_kc_{n+1,k}\beta_k$$ $$-(1-cd)(1-abq^n)\sum_{k=0}^{n}\left(ab;q\right)_{k-1}c_{n,k-1}\beta_k,$$ since $(1-cd)=(1-abcdq^n)-cd(1-abq^n)$.\
It follows that what we want to show is $$\frac{\widetilde A_{n+1}(a,b,c,d)}{(abcd;q)_n}=(1-abcdq^n)\sum_{k=0}^{n}\left(ab;q\right)_kc_{n+1,k}\beta_k+(1-abq^n) S,$$ where $$S=S_1-S_2-S_3-S_4=\overbrace{(c+d)\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k}\beta_k}^{S_1}-\overbrace{(1-q^n)cd\sum_{k=0}^{n}\left(ab;q\right)_kc_{n-1,k}\beta_k}^{S_2}$$ $$-\overbrace{cd\sum_{k=0}^{n}\left(ab;q\right)_kc_{n+1,k}\beta_k}^{S_3}-\overbrace{(1-cd)\sum_{k=0}^{n}\left(ab;q\right)_{k-1}c_{n,k-1}\beta_k}^{S_4}.$$ We will finish the proof by showing that $S$ is equal to $(1-abcdq^n)\left(ab;q\right)_{n}c_{n+1,n+1}\beta_{n+1}$.
By Lemma \[lemmarek1\] $$S_3=cd\sum_{k=0}^{n}\left(ab;q\right)_k c_{n+1,k}\beta_k=S'_3+S''_3+S'''_3=$$ $$\overbrace{cd(a+b)\sum_{k=0}^{n}\left(ab;q\right)_kq^kc_{n,k}\beta_k}^{S'_3}+\overbrace{cd\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k-1}\beta_k}^{S''_3}\overbrace{-cd\sum_{k=0}^{n}\left(ab;q\right)_kq^k(1-q^n)ab\cdot c_{n-1,k}\beta_k}^{S'''_3}.$$ Since $cd\left(ab;q\right)_k=cd\left(ab;q\right)_{k-1}-abcdq^{k-1}\left(ab;q\right)_{k-1}=-(1-cd)\left(ab;q\right)_{k-1}+(1-abcdq^{k-1})\left(ab;q\right)_{k-1}$ we see that $$S''_3=\overbrace{-(1-cd)\sum_{k=0}^{n}\left(ab;q\right)_{k-1}c_{n,k-1}\beta_k}^{I_1}+\overbrace{\sum_{k=0}^{n}(1-abcdq^{k-1})\left(ab;q\right)_{k-1}c_{n,k-1}\beta_k}^{I_2}=I_1+I_2$$ Writing $-abq^k=(1-abq^k)-1$ we can rewrite $S'''_3$ as $$S'''_3=cd\sum_{k=0}^{n}\left(ab;q\right)_{k+1} \underbrace{(1-q^n)c_{n-1,k}}_{\textrm{Lemma}\ \ref{lemmarek1}}\beta_k-(1-q^n)cd\sum_{k=0}^{n}\left(ab;q\right)_{k} \beta_k$$ $$=\overbrace{cd\sum_{k=0}^{n}\left(ab;q\right)_{k+1} (1-q^{k+1})c_{n,k+1}\beta_k}^{J_1}\overbrace{-(1-q^n)cd\sum_{k=0}^{n}\left(ab;q\right)_{k}c_{n-1,k} \beta_k}^{J_2}=J_1+J_2.$$ Combining all the transformation together we obtain $$S=\left(S_1-S'_3-J_1\right)-\underbrace{(S_2+J_2)}_{=0}-\underbrace{(I_1+S_4)}_{=0}-I_2.$$ The first expression is equal $$S_1-S'_3-J_1=\overbrace{\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k}\left[cd-cd(a+b)q^k\right]\beta_k}^{S_1-S'_3}-\overbrace{cd\sum_{k=0}^{n}\left(ab;q\right)_{k}(1-q^{k})c_{n,k}\beta_{k-1}}^{J_1}$$ $$=\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k}\underbrace{\left\{\left[cd-cd(a+b)q^k\right]\beta_k-cd(1-q^k)\beta_{k-1}\right\}}_{\textrm{Lemma}\ \ref{lammarek2}}$$ $$=\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k}(1-abcdq^k)\beta_{k+1}.$$ Hence $$S=\overbrace{\sum_{k=0}^{n}\left(ab;q\right)_kc_{n,k}(1-abcdq^k)\beta_{k+1}}^{=S_1-S'_3-J_1}-\overbrace{\sum_{k=0}^{n}(1-abcdq^{k-1})\left(ab;q\right)_{k-1}c_{n,k-1}\beta_k}^{I_2}$$ $$=(ab;q)_nc_{n,n}(1-abcdq^n)\beta_{n+1}.$$ This ends the proof, as $c_{n+1,n+1}=c_{n,n}=1$.
The proof does not use explicitly singularity condition $q^Nabcd=1$, except for the constraints that it implies on the domain of $\varphi_0$ and on the degrees of the polynomials $\{p_k: k=1,\dots,N\}$.
For $k=1$ this is a calculation, which is also covered by the induction step. Suppose that $p_k$ is of degree $k$ and $$\varphi_0\left[p_k\left( {\mbox{\fontfamily{phv}\selectfont x}}_t;at,bt,c/t,d/t\middle|q\right)\right]=0$$ for $k=1,\dots, n$. Suppose that polynomial $p_{n+1}$ is of degree $n+1$. Then, recalling , we have $$\begin{gathered}
H_{n+1}({\mbox{\fontfamily{phv}\selectfont e}}/t^2+{\mbox{\fontfamily{phv}\selectfont d}}\;;1/t^2)= H_{n+1}(2\theta {\mbox{\fontfamily{phv}\selectfont x}}_t/t;1/t^2)= \frac{1}{t^{n+1}}H_{n+1}(2\theta {\mbox{\fontfamily{phv}\selectfont x}}_t)
\\= \frac{1}{t^{n+1}}
\sum_{k=0}^{n+1} a_{n+1,k}(at,bt,c/t,d/t)p_k\left({\mbox{\fontfamily{phv}\selectfont x}}_t;at,bt,c/t,d/t\middle|q\right)\end{gathered}$$ by . Since $p_0=1$, by inductive assumption we have $$\varphi_0\left[H_{n+1}({\mbox{\fontfamily{phv}\selectfont e}}/t^2+{\mbox{\fontfamily{phv}\selectfont d}}\;;1/t^2) \right] =\frac{1}{t^{n+1}}a_{n+1,0}(a t,bt,c/t,d/t)+\frac{1}{t^{n+1}}a_{n+1,n+1}\varphi_0\left[ p_{n+1}\left({\mbox{\fontfamily{phv}\selectfont x}}_t;at,bt,c/t,d/t\middle|q\right)\right].$$ This shows that $\varphi_0[p_{n+1}\left({\mbox{\fontfamily{phv}\selectfont x}}_t;at,bt,c/t,d/t\middle|q\right)]=0$, provided that $a_{n+1,n+1}\ne 0$, which holds true due to the assumption on the degree of $p_{n+1}$, and provided that $$a_{n+1,0}(a t,bt,c/t,d/t)=t^{n+1}G_{n+1}(1/t^2),$$ which holds true by Proposition \[L:A=G\].
Conclusions
===========
In this paper we construct a functional $\varphi_0$, or a pair of functionals $\varphi_0,\varphi_1$, on an abstract algebra that give stationary probabilities for an ASEP of length $L$ with arbitrary parameters. Formula for the probabilities is a substitute for the celebrated matrix product ansatz [@derrida1993exact]. Our approach avoids an associativity pitfall that may arise in matrix product models. In Appendix \[Sec:MatrixModel\] we exhibit an example of such a matrix model that satisfies the usual conditions yet cannot be used to compute stationary probabilities.
While verifying that our functionals give non-zero answers for un-normalized probabilities, we noted an interesting phenomenon of reversal of current as $L$ increases when $\alpha\beta<\gamma\delta$ and $0<q<1$ .
In the non-singular case, we showed that functional $\varphi_0$ may serve as an orthogonality functional for the Askey-Wilson polynomials with fairly general parameters. Part of this connection persists in the singular case $\alpha\beta=q^N\gamma\delta$ when the degrees of the first $N$ Askey-Wilson polynomials do not exceed $(N+1)/2$. In Appendix \[Sect:TASEP\] we found concise explicit formulas for the (formal) “Cauchy-Stieltjes” transforms of both functionals when $q=0$.
Acknowledgements {#acknowledgements .unnumbered}
----------------
The authors thank Peter Paule for sharing mathematica software packages [qZeil]{} and [qMultiSum]{} developed in Research Institute for Symbolic Computation at the University of Linz and Dr. Daniel Tcheutia for helpful comments on the early draft of the paper. Marcin Świeca’s research was partially supported by grant 2016/21/B/ST1/00005 of National Science Centre, Poland.
Auxiliary identities
====================
Here we collect $q$-hypergeometric formulas used in this paper. Cauchy’s $q$-binomial formula is $$\label{pochtopot}
(x;q)_n=\sum_{k=0}^n{n \brack k}_q(-1)^kq^{\frac{k(k-1)}{2}}x^k.$$ Heine’s summation formula [@gasper2004basic (1.5.3)] reads $${_2\phi_1}\left(\begin{matrix}
q^{-n},\ b\\ c
\end{matrix}\middle|q;q\right)=\frac{(c/b;q)_n}{(c;q)_n}b^n.\label{twophione}$$ We also need the connection coefficients of the Askey-Wilson polynomials.
If $a_4\neq 0$ then $$p_n(x;b_1,b_2,b_3,a_4|q)=\sum_{k=0}^nc_{n,k}p_k(x;a_1,a_2,a_3,a_4|q),$$ where $$\begin{gathered}
\label{conectiocoefficient}
c_{n,k}=(b_1b_2b_3a_4;q)_k\frac{q^{k(k-n)}(q;q)_n(b_1a_4q^k,b_2a_4q^k,b_3a_4q^k;q)_{n-k}}{a_4^{n-k}(q;q)_{n-k}(q,a_1a_2a_3a_4q^{k-1};q)_k}
\\ \times{_5\phi_4}\left(\begin{matrix}
q^{k-n},\ b_1b_2b_3a_4q^{n+k-1},\ a_1a_4q^k,\ a_2a_4q^k,\ a_3a_4q^k\\b_1a_4q^k, \ \ b_2a_4q^k,\ \ b_3a_4q^k,\ \ a_1a_2a_3a_4q^{2k}
\end{matrix}\middle|q;q\right).\end{gathered}$$
Totally asymmetric case {#Sect:TASEP}
=======================
Our recursions simplify when $q=0$, i.e., the case of Totaly Asymmetric Exclusion Process. Then the conclusion of Theorem \[T3\] can be derived more directly, and there is also additional information about $\varphi_1$ in the singular case $abcd=1$.
For $q=0$, [@Askey-Wilson-85] relate Askey-Wilson polynomials $p_n$ to the Chebyshev polynomials $U_n$ of second kind. Denote by $s_j(a,b,c,d)$ the $j$-th symmetric function, i.e. $s_1=a+b+c+d$, $s_2=ab+ac+ad+bc+bd+cd$, $s_3=abc+abd+acd+bcd$, $s_4=abcd$. Then with $U_{-1}=0$ we have $$\begin{aligned}
p_0&=&U_0 \\
p_1&=&(1-s_4) U_1+(s_3-s_1)U_0
\\
p_2&=&U_2-s_1U_1+(s_2-s_4)U_0 \\
p_n&=&U_n-s_1U_{n-1}+s_2U_{n-2}-s_3U_{n-3}+s_4U_{n-4} \mbox{ for } n\geq 3.\end{aligned}$$ Recall that $G_n(1)=\varphi_0[H_n({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}})]=\varphi_0[U_n({\mbox{\fontfamily{phv}\selectfont x}})]$. So in the non-singular case the conclusion of Theorem \[T3\] follows from the following relations between $G_n(1)$. $$\begin{aligned}
(1 - s_4) G_1(1) + (s_3 - s_1) G_0(1)&=&0 \label{Eqnt-1}\\
G_2(1)-
s_1 G_1(1)+ (s_2 - s_4) G_0(1) &=&0 \label{Eqnt-2}\\
G_n(1) - s_1 G_{n-1}(1) +
s_2 G_{n-2}(1) - s_3 G_{n-3}(1) +
s_4 G_{n-4}(1)&=&0, \quad \; n\geq 3 \label{Eqnt-n}.\end{aligned}$$ These relations can be established by analyzing explicit solutions of recursion . We first determine the initial (irregular) solutions $$G_1(t)=\frac{(c+d) (t-a b)+(a+b) (1-c d t)}{1-a b c d}$$ and $$G_2(t)=t (c+d)G_1(t) +\frac{(a+b) (1-c d t) (a+b- a b (c+d))}{1-a b c d} -a b (
1-cd t)-c d t^2$$ which we use with $t=1$ to verify and . Next, we use with $t=0$ and $n\geq 1$ to determine $\alpha_n=G_n(0)$ from the recursion of order $2$, $$\label{G0}
\alpha_{n+1}(0)=(a+b)\alpha_n -ab \alpha_{n-1} .$$ Since in our setting arising from ASEP parameters $b\leq 0<a$ are not equal, the general solution is $$\alpha_n=C_1 a^n+C_2 b^n.$$ The constants $C_1, C_2 $ are determined from the initial values of $G_0(0)=1$ and $G_1(0)=\frac{a+b-a b (c+d)}{1-a b c d}$. We get $$\alpha_n=
\frac{ (
1-b c) (1-b d)}{(a-b) (1-a b c d)}a^{n+1}+ \frac{(1-a c) (1-a d)}{(b-a) (1-a b c d)} b^{n+1}.$$ Next we solve the recursion for $z_n=G_n(1)$. This is now a non-homogeneous recursion $$z_{n+1} =(1-cd)((a+b)\alpha_n-ab \alpha_{n-1})+(c+d)z_n -cd z_{n-1},$$ which we simplify using into $$z_{n+1} =(1-cd)\alpha_{n+1}+(c+d)z_n-cd z_{n-1}.$$ Since $d\leq 0<c$, the general solution of this recursion is $$G_n(1)=z_n= B_1 a^{n+3}+B_2 b^{n+3}+K_1 c^{n+3}+K_2 d^{n+3}, \; n\geq 0$$ where $$B_1=\frac{(1-b c) (1-b d) (1-c d)}{(a-b) (a-c) (a-d) (1-a b c d)}, \; \quad B_2=\frac{(1-a c) (1-a d) (1-c d)}{(b-a) (b-c) (b-d) (1-a b c d)}$$ come from the undetermined coefficients method and $$K_1=\frac{(1-a b) (1-a d) (1-b d) }{(c-a)
(c-b) (c-d) (1-a b c d)} , \; \quad K_2=\frac{ (1-a b) (1-a c) (1-b c)
}{(d-a) (d-b) (d-c) (1-a b c d)}$$ come from matching the initial values. It turns out that the explicit values of the constants are only needed for verification of the initial equations, as equation holds for any linear combination of $a^n,b^n,c^n,d^n$.
Proceeding in similar way we can also derive a version of Theorem \[T3\] that relates functional $\varphi_1$ to Askey-Wilson polynomials. We have $$F_0(t) =0, \quad
F_1(t)=\frac{1-cdt}{1-cd}.$$ The recursion for $\alpha_n=F_n(0)$ is , so using the above initial values we get the solution $$F_n(0)=\frac{a^{n}-b^{n}}{(a-b)(1-cd)}, \quad n\geq0.$$ The recursion for $F_n(1)$ is $$F_{n+1}(1)=(c+d)F_n(1)-c d F_{n-1}(1)+\frac{a^n-b^n}{a-b}, \quad n\geq 1.$$ Here the constants are simpler and a calculation gives $$\label{Fq=0}
F_{n}(1)= \frac{a^{n+2}}{(a-b)(a-c)(a-d)}+ \frac{b^{n+2}}{(b-a)(b-c)(b-d)}+\frac{c^{n+2}}{(c-a)(c-b)(c-d)}+\frac{d^{n+2}}{(d-a)(d-b)(d-c)},\; n\geq0 .$$ Noting that in the singular case $p_1$ is a constant, we have $\varphi_1[p_n({\mbox{\fontfamily{phv}\selectfont x}})]=0$ for all $n=0,1,\dots$.
To avoid the irregularity with $p_1$ in the singular case, we can also consider the following family of polynomials: $$\begin{aligned}
q_0(x)&=&U_0(x)\\
q_1(x)&=&U_1(x)+(s_3-s_1)U_0 (x)\\
q_2(x)&=& U_2(x)-s_1 U_1(x)+(s_2-s_4) U_0(x) \\
q_n(x)&=& U_n(x)-s_1U_{n-1}(x)+s_2U_{n-2}(x)-s_3U_{n-3}(x)+s_4U_{n-4}(x), \quad n\geq 3.\end{aligned}$$ Since $2xU_n=U_{n+1}+U_{n-2}$, polynomials $q_n$ satisfy the following finite perturbation of the constant three step recursion: $$\begin{aligned}
2x q_0&=&q_1+(s_1-s_3)q_0 \\
2x q_1&=&q_2+s_3 q_1+(s_4-s_2+s_3(s_1-s_3))q_0\\
2x q_2&=&q_3+q_1 \\
2x q_n&=&q_{n+1}+q_{n-1}, \quad n\geq 2.\end{aligned}$$ As previously, implies that $\varphi_1[q_1({\mbox{\fontfamily{phv}\selectfont x}})]=1$ and $\varphi_1[q_n({\mbox{\fontfamily{phv}\selectfont x}})]=0$ for $n\geq 2$. Since $x^kq_n$ is a linear combination of $g_{n-k},g_{n-k+1},\dots,g_{n+k}$ this implies that $$\varphi_1[q_k({\mbox{\fontfamily{phv}\selectfont x}})q_n({\mbox{\fontfamily{phv}\selectfont x}})]=0 \mbox{ for } |n-k|\geq 2.$$
Motivated by the generating function $\sum_{n=0}^\infty H_n(x)z^n=1/(1+z^2-xz)$ lets denote by $\varphi[(1+z^2 -({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}) z)^{-1}]$ the power series $\sum_{n=0}^\infty \varphi[H_n({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}})]z^n$. We can now summarize the above formulas more concisely.
If $abcd\ne 1$ then for $|z|$ small enough $$\varphi_0[(1+z^2 -({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}) z)^{-1}]=1+z^2 abcd +
\frac{ z abcd(a+b+c+d-(1/a+1/b+1/c+1/d)) }{(1-az)(1-bz)(1-cz)(1-dz)}.$$ If $abcd= 1$ then for $|z|$ small enough $$\varphi_1[(1+z^2 -({\mbox{\fontfamily{phv}\selectfont e}}+{\mbox{\fontfamily{phv}\selectfont d}}) z)^{-1}]=\frac{z}{(1-a z) (1-b z) (1-c z) (1-d z)}.$$
The first expression matches the formula from [@Szpojankowki2010 Theorem 4.1] who computed the integral of $1/(1+z^2-xz)$ with respect to the Askey-Wilson measure with $q=0$ under the assumptions which in our setting boil down to $ac \leq 1$ and $abcd<1$.
A matrix model {#Sec:MatrixModel}
==============
According to @Mallick-Sandow-1997 stationary probabilities for ASEP with large $L $ can be computed from a finite matrix model when the parameters satisfy condition $q^{m}ac=1$ for some $m\geq 0$. Here we present a version of this model, together with a caution about a subtle issue that may affect some infinite matrix models.
Recalling that in we chose $a>0$, for $q> 0$ we consider two infinite matrices $${\mathbf{E}}=\theta^2 \begin{bmatrix}
1+\frac{1}{a} & 0 & 0& & \dots &0& \dots\\
1 & 1+ \frac{1}{ a q} & 0& & \dots & \\
0& 1 &\ddots & & & \\
\vdots & &\ddots &&\ddots& \vdots \\
\\ 0 & 0&\dots&0& 1& 1+\frac{1}{a q^{n-1}} \\ \\
\vdots &&&& & \ddots&\ddots
\end{bmatrix} \; {\mathbf{D}}=\theta^2 \begin{bmatrix}
1+a & 0 & 0 & \dots & \\
0 & 1+a q &0& \dots & \\
0&0 &\ddots & &
\\ 0& 0 &\dots & 1+a q^{n-1} \\ \\
\vdots &\vdots&&& \ddots
\end{bmatrix}.$$ It is straightforward to verify that identity is satisfied. Conditions and become recursions for the components of the vectors $$\langle W|=[w_1,w_2,\dots] \mbox{ and } |V\rangle =[v_1,v_2,\dots]^T.$$ In parametrization , conditions and become and , and the resulting recursions are $$\frac{1}{a q^{k-1}} w_k+w_{k+1}=(c+d)w_k-a cd q^{k-1}w_k,$$ $$ab \left(v_{k-1}+\frac{1}{a q^{k-1}}v_k\right)=(a+b)v_k-a q^{k-1} v_k.$$
Conditions and are $(1-q)\langle W|({\mathbf{E}}+c d
{\mathbf{D}})=(1+c)(1+d)\langle W|$ and $(1-q)(ab{\mathbf{E}}+{\mathbf{D}})|V\rangle=(1+a)(1+b)|V\rangle$.
To derive and , we insert into the above equations, and simplify the expressions.
To derive the recursions as written above, we compute $${\mbox{\fontfamily{phv}\selectfont d}}=\theta\begin{bmatrix}
a & 0 & 0& & \dots & & \\
0 & a q & 0& & \dots & \\
0& 0 &a q^2 & & & \\
\vdots & &\ddots &&\ddots& \\
\\ 0 & 0&\dots&0& & a q^{k-1} \\ \\
\vdots &&&& \ddots&\ddots
\end{bmatrix}, \quad
{\mbox{\fontfamily{phv}\selectfont e}}=\theta\begin{bmatrix}
\frac{1}{a} & 0 & 0& & \dots & & \\
1 & \frac{1}{a q} & 0& & \dots & \\
0& 1 &\frac{1}{a q^2} & & & \\
\vdots & &\ddots &&\ddots& \\
\\ 0 & 0&\dots&0& 1& \frac{1}{a q^{k-1}} \\ \\
\vdots &&&& \ddots&\ddots
\end{bmatrix}.$$
With $w_1=v_1=1$, the solutions are explicit $$\label{w}
w_{n}= \prod_{k=1}^{n-1}\left(c+d -acd q^{k-1}-\frac{1}{a q^{k-1}}\right)= \frac{(a c,ad;q)_{n-1}}{(-a)^{n-1}q^{(n-1)(n-2)/2}},$$ $$\label{v}
v_n= \frac{ a^{n-1}b^{n-1}}{\prod_{k=1}^{n-1}\left(a (1-q^k)+b(1-1/q^k)\right)}
= \frac{(-a)^{n-1}q^{n(n-1)/2}}{(q, qa/b;q)_{n-1}}.$$
We remark that since $a>0$ and $b\leq 0$ the second expression for $v_n$ is well defined only if $b<0$, i.e,. when $\delta>0$, see . When $b=0$, from the first expression we get $V=[1,0,0,\dots]^T$, and the formulas we discuss below are not valid.
We therefore get explicit formula $$\label{WV}
\langle W |{\mathbf{I}}|V\rangle = \sum_{k=1}^{\infty} v_kw_k = \sum_{k=1}^{\infty} q^{k-1}\frac{(ac,ad;q)_{k-1}} {(q,a q/b;q)_{k-1}} ={_2\phi_1}\left(\begin{matrix}
ac,\ a d \\\ q a /b
\end{matrix}\middle|q;q\right),$$ valid for $0<q<1$. Somewhat more generally, since ${\mbox{\fontfamily{phv}\selectfont d}}$ introduced in is a diagonal matrix with the sequence $\{\theta a q^{k-1}\}$ on the diagonal, we get $$\label{Wd^LV}
\langle W |{\mbox{\fontfamily{phv}\selectfont d}}^L|V\rangle =a^L\theta^L {_2\phi_1}\left(\begin{matrix}
ac,\ a d \\\ q a /b
\end{matrix}\middle|q;q^{L+1}\right).$$ (We will use this formula for $L=0,1$ in Section \[Sec:Warn\].)
We now consider the case when parameters $a,c$ are such that $ac q^m=1$ for some integer $m\geq0$. In this case the infinite series terminate as formula gives $w_n=0$ for all $n\geq m+2$. Since each monomial ${\mathbf{X}}$ is a lower-triangular matrix, in this case components $v_k$ with $k\geq m+2$ do not enter the calculation of $\langle W|{\mathbf{X}}|V\rangle$, so we can truncate ${\mbox{\fontfamily{phv}\selectfont e}}, {\mbox{\fontfamily{phv}\selectfont d}}, {\mathbf{I}}$ to their $m+1$ by $m+1$ upper left corners, recovering a version of the finite matrix model from @Mallick-Sandow-1997.
Using one can show that $${_2\phi_1}\left(\begin{matrix}
q^{-m},\ a d \\\ q a /b
\end{matrix}\middle|q;q\right)=
\frac{(bdq^{-m};q)_m}{(bc;q)_m}.$$
Applying transformation we rewrite ${_2\phi_1}\left(\begin{matrix}
q^{-m},\ a d \\\ q a /b
\end{matrix}\middle|q;q\right)$ as $$\begin{gathered}
(ad)^m \frac{(q/(bd);q)_m}{(a q/b;q)_m}
=q^{-m^2} \frac{(q-bd)(q^2-bd)\dots(q^{m}-bd)}{(1-bc)(q^{-1}-bc)\dots (q^{1-m}-bc)}
\\=q^{-m^2} \frac{q^{m(m+1)/2}(1-bd/q)(1-bd/q^2)\dots(1-bd/q^m)}{q^{-m(m-1)/2}(1-bc)(1-q bc)\dots (1-q^{m-1}bc)}
=\frac{(bdq^{-m};q)_m}{(bc;q)_m}.\end{gathered}$$
Thus, in agreement with findings in @Mallick-Sandow-1997, $$\langle W |{\mathbf{I}}|V\rangle=\frac{(bdq^{-m};q)_m}{(bc;q)_m}$$ vanishes if and only if $bd\in\{q,q^2,\dots,q^{m}\}$, i.e., in the singular case when $q^Nabcd=1$ for some $N=0,\dots,m-1$. One would expect that in this case the matrix model should be related to functional $\varphi_1$ by a simple renormalization but we have not verified the details.
In the non-singular case (but still with $q^m ac=1$) the answer is straightforward. Due to shared recursion and initialization at ${\mathbf{I}}$, it is clear that functional $\varphi_0$ is indeed related to the matrix model by $$\label{phi0Mallick}
\langle W | {\mathbf{X}}|V\rangle= \frac{(bdq^{-m};q)_m}{(bc;q)_m} \varphi_0[{\mathbf{X}}].$$
A natural question then arises how the functionals $\varphi_0$, or $ \varphi_1$, are related to this matrix model for more general parameters $a,b,c,d$. The surprising answer is that there is no such relation, as we explain next.
A caution about matrix models {#Sec:Warn}
-----------------------------
It is known, [@keremedis1988associativity; @bossaller2019associativity], but perhaps this is not appreciated enough, that multiplication of infinite matrices may fail to be associative for other reasons than divergence. And precisely this difficulty afflicts the above matrix model when $acq^n\ne1$ for all $n$. To see the source of the difficulty, we rewrite and as $$\label{WeV} \langle W| {\mbox{\fontfamily{phv}\selectfont e}}=\theta (c+d)\langle W|- c d \langle W|{\mbox{\fontfamily{phv}\selectfont d}},$$ $$\label{WdV} ab {\mbox{\fontfamily{phv}\selectfont e}}|V\rangle=\theta(a+b)|V\rangle - {\mbox{\fontfamily{phv}\selectfont d}}|V\rangle.$$ To indicate more clearly the order of matrix multiplications, lets denote vector $\langle W| {\mbox{\fontfamily{phv}\selectfont e}}$ by $\langle \tilde W|$ and vector ${\mbox{\fontfamily{phv}\selectfont e}}|V\rangle$ by $|\tilde V\rangle$. Using and , we should be able to compute product $\langle W|{\mbox{\fontfamily{phv}\selectfont e}}|V\rangle$ of three matrices either as $\langle \tilde W|V\rangle$, or as $\langle W |\tilde V\rangle$. From the first calculation we get $$\langle \tilde W|V\rangle =\theta (c+d) {_2\phi_1}\left(\begin{matrix}
q^{-m},\ a d \\\ q a /b
\end{matrix}\middle|q;q\right) - a c d \theta {_2\phi_1}\left(\begin{matrix}
q^{-m},\ a d \\\ q a /b
\end{matrix}\middle|q;q^2\right)$$ where we used with $L=0$ and $L=1$ on the right hand side. The second calculation gives a different answer $$ab \langle W |\tilde V\rangle =\theta (a+b) {_2\phi_1}\left(\begin{matrix}
ac,\ a d \\\ q a /b
\end{matrix}\middle|q;q\right) - a \theta {_2\phi_1}\left(\begin{matrix}
ac,\ a d \\\ q a /b
\end{matrix}\middle|q;q^2\right).$$ In fact, we have $$\langle\tilde W|V\rangle=\theta \sum_{k=1}^\infty \left(\frac{1}{a q^{k-1}} w_k+w_{k+1}\right)v_k$$ $$\langle W|\tilde V\rangle=\theta \sum_{k=1}^\infty w_k\left(v_{k-1}+\frac{1}{a q^{k-1}} v_k\right) \mbox{ with $v_{-1}=0$}.$$ So from and we get $$\langle \tilde W | V\rangle-\langle W|\tilde V\rangle= \lim_{n\to\infty}\sum_{k=1}^n(w_{k+1}v_k-w_kv_{k-1})=
\lim_{n\to\infty} w_{n+1}v_n =-\frac{\theta}{a}\frac{ (ac,ad;q)_\infty}{ (q,qa/b;q)_\infty}.$$ This shows that in general multiplication of matrices $\langle W|$, ${\mbox{\fontfamily{phv}\selectfont e}}$ and $|V\rangle$ is not associative. Since $d\leq 0$, the two answers match only when $q^mac=1$ for some $m$, i.e. in the terminating case. This is precisely the case considered by [@Mallick-Sandow-1997], and of course multiplication of finite dimensional matrices is associative.
This established the following hypergeometric function identity $$\label{FasleId}
a(1-abcd){_2\phi_1}\left(\begin{matrix}
ac,\ a d \\\ q a /b
\end{matrix}\middle|q;q^2\right) =(a+b-ab(c+d)){_2\phi_1}\left(\begin{matrix}
ac,\ a d \\\ q a /b
\end{matrix}\middle|q;q\right)+b \frac{ (ac,ad;q)_\infty}{ (q,qa/b;q)_\infty}.$$
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|
---
author:
- 'N. Chornay'
- 'N. A. Walton'
bibliography:
- 'bib.bib'
title: 'Searching for central stars of planetary nebulae in *Gaia* DR2'
---
[Accurate distance measurements are fundamental to the study of Planetary Nebulae (PNe) but have long been elusive. The most accurate and model-independent distance measurements for galactic PNe come from the trigonometric parallaxes of their central stars, which were only available for a few tens of objects prior to the *Gaia* mission.]{} [Accurate identification of PN central stars in the *Gaia* source catalogues is a critical prerequisite for leveraging the unprecedented scope and precision of the trigonometric parallaxes measured by *Gaia*. Our aim is to build a complete sample of PN central star detections with minimal contamination.]{} [We develop and apply an automated technique based on the likelihood ratio method to match candidate central stars in *Gaia* Data Release 2 (DR2) to known PNe in the HASH PN catalogue, taking into account the BP–RP colours of the *Gaia* sources as well as their positional offsets from the nebula centres. These parameter distributions for both true central stars and background sources are inferred directly from the data.]{} [We present a catalogue of over 1000 *Gaia* sources that our method has automatically identified as likely PN central stars. We demonstrate how the best matches enable us to trace nebula and central star evolution and to validate existing statistical distance scales, and discuss the prospects for further refinement of the matching based on additional data. We also compare the accuracy of our catalogue to that of previous works.]{}
Introduction
============
Planetary Nebulae (PNe) are an end stage of life for low and intermediate mass stars, a relatively short step on their evolutionary path after they depart from the tip of the Asymptotic Giant Branch (AGB) [@herwigAGB]. The star sheds its outer layers, growing brighter and hotter before ultimately cooling into a white dwarf. Ultraviolet light from the star ionises this rapidly expanding shell of gas, which reaches typical sizes on the order of lightyears over the tens of thousands of years during which it is visible.
PNe are important in galactic evolution for their enrichment of the interstellar medium (ISM) with heavier elements [@johnsonnucleosynthesis; @karakas_lattanzio_2014_nucleosynthesis], joining other mechanisms of stellar mass loss such as supernovae. Their brightness and resulting visibility over large distances make PNe valuable chemical probes of not only the Milky Way but also nearby galaxies [@kwitterwhitepaper]. In addition, the Planetary Nebula Luminosity Function (PNLF) forms a useful rung in the cosmic distance ladder [@ciardullo2012pnlf].
The number of PNe present in our galaxy at any given time is small relative to the stellar population on account of their short lifespans. However the set of 3500 or so confirmed and likely PNe that have been discovered only represents a fraction of those expected to be visible if all stars in a certain mass range go through the PN phase [@demarcobinaries2006], with this inconsistency leaving open questions as to whether there are further requirements for PN formation, namely binary interactions [@jonesboffinnaturebinaries2017]. Understanding of PNe is limited in part by difficulties in constraining their distances [@smith2015]. Accurate distances are critical for meaningful astrophysical characterisation of PNe, from measuring physical sizes of individual PNe and the absolute magnitudes of their central stars to determining their lifetimes and formation rates.
The rapid evolution of PN central stars (CSPNe) generally prevents the application of usual methods of distance determination such as isochrone fitting. Thus a variety of distance measurement techniques have been developed, which fall into two broad categories (following @frewsurfacebrightness2016, henceforth FPB16).
Primary techniques measure the distances to individual PNe with varying degrees of accuracy and assumptions. Most involve modelling – either of the nebula’s expansion [@expansion2018], of the environment (e.g. extinction distances, cluster or bulge membership, or location in external galaxies), or of the CSPN itself. The most direct primary distances measurements come from trigonometric parallaxes of CSPNe, but until recently these have only been available for few nearby objects with measurements from USNO [@usno2007] and the HST [@hst2009].
Secondary, or statistical distance scales, rely on finding a broadly applicable relationship that provides a means of estimating a physical parameter of the PN such as its physical size, given a distance-independent measurement, such as nebula surface brightness. The distance can then be determined from the relation of the physical parameter to a measured one, for example through comparison of physical to angular size, in a manner analogous to the distance modulus for stars with known absolute magnitudes. The determination of the relationships underlying such secondary methods require a calibrating set of objects whose distances are known independently.
Most PN distance estimates rely on secondary methods, but these methods are only as good as the quality and purity of the distances used to calibrate them. Incorrect distances or polluting objects can inflate errors well beyond the uncertainties stemming from measurement errors and intrinsic scatter. Thus, improved primary distances to a set of PNe provides a twofold benefit, as it betters not only the distances to that set of objects but also, through improved calibration of statistical distance scales, to the population as a whole.
Good primary distance measurements are rare. In their statistical distance scale calibration, FPB16 deemed only around 300 galactic and extragalactic PNe to have sufficiently reliable primary distances. The galactic selection they chose represented only around 5% of confirmed galactic PNe, and their more relatively accurate primary distances were for extragalactic objects.
However, the situation is now changing with the recently launched *Gaia* mission [@gaiamission], which is conducting astrometric measurements – positions, parallaxes, and proper motions – of over a billion stars in the Milky Way, including many CSPNe. @stanghellini2017 found a small number of CSPNe with parallax measurements in *Gaia* DR1 [@gaiadr1], while @kimeswenger2018 found a larger sample in the most recent *Gaia* data release, DR2 [@gaiadr2], using a manual matching technique and a limited input catalogue. Most recently, @gapn and @stanghellinipreprint searched *Gaia* DR2 using more complete input catalogues, but relied on a position-based matching approach that creates a risk of contamination. Fully exploiting the data from *Gaia* for the study of PNe requires a CSPN sample that is both complete and pure, which is time-consuming and difficult to achieve manually. As astronomical datasets become larger and are updated more frequently, automated techniques become increasingly useful, offering not only improved speed and consistency but also adaptability, making it easier to incorporate new data as it becomes available. In the case of *Gaia*, future data releases will have more detections with improved photometry and parallaxes, so an automated technique will allow these to quickly be taken advantage of.
Our work aims to provide a more complete sample of CSPNe in *Gaia* DR2 and to lay the groundwork for future data releases, through an automated matching process that we have developed that takes into account both relative position and colour information of *Gaia* sources. In the remainder of this work we present the technique, the resulting catalogue of CSPNe in *Gaia* DR2, a comparison of this catalogue to previous works, and finally some initial applications that use the subsets of this catalogue with the most accurate *Gaia* parallaxes for astrophysical characterisation and distance scale evaluation.
Methods
=======
Our starting point is the Hong Kong/AAO/Strasbourg H$\alpha$ (HASH) PN catalogue[^1] from @hashpn. This catalogue represents the most complete catalogue of PNe available, containing at the time of writing[^2] around 2500 spectroscopically confirmed PNe (following the criteria in @pndiagnostics) and 1000 possible and likely PNe, as well as objects that are commonly confused with PNe, such as HII regions, symbiotic stars, and reflection nebulae. HASH also collects together additional information about individual nebulae such as fluxes, angular sizes, and spectra; however it lacks structured positional data on known CSPNe.
PN catalogues have historically suffered from positional inaccuracies. Some uncertainty is intrinsic to PNe as extended objects: they have varied morphologies, and the full extents of their nebulae may not be visible, so different assessments of the nebula position are possible. Moreover in the cases where there is a known or apparent central star, some catalogues adopt this star’s position as that of the PN, though the nebula and star positions can have significant offsets.
In MASH, the precursor to HASH, @mashpn based their positions on the geometric centre of the visible nebula, and claimed uncertainties on the order of 1$"$ to 2$"$. They found notable disagreements between their measured positions for known PNe and those in existing catalogues. This can in part be due to catalogue inhomogenaity noted above. However outright misidentification can also occur, particularly for compact PNe: the HASH authors found positions in the online SIMBAD database that were simply incorrect. We see in Sect. \[comparison\] that such errors are still present.
HASH promises a homogeneous set of PN coordinates, primarily based on centroiding narrowband H$\alpha$ imagery of the PNe. This along with its completeness motivates our choice of it as an input catalogue. A disadvantage is that the coordinates contained in the catalogue may not correspond to the central star coordinates even for PNe with known central stars. Information about CSPNe is scattered across the literature and often the coordinates themselves are not identified [@weidmannCSPNe].
In addition to astrometric parameters (positions, parallaxes and proper motions), the *Gaia* satellite measures fluxes in three bands: the wider *G* band covering visible wavelengths and extending partway into the near infrared (330 to 1050nm), and narrower bands *G*$_{BP}$ and *G*$_{RP}$, covering the blue and red halves of the spectrum respectively [@gaiaphotometry]. For well-behaved sources the *BP* and *RP* fluxes essentially add to produce the total flux in *G*, with this degeneracy giving a single measurement of colour as the magnitude difference across any two passbands (e.g. BP – RP).
The second *Gaia* data release, DR2, contains around 1.7 billion sources, with about 1.3 billion sources having full sets of astrometric parameters (as opposed to positions only) and a similar number (1.4 billion) having full photometry from which colour information can be derived (rather than magnitude in *G* only; this is possible because the photometry is performed by different instruments, and the blue and red photometers (BP and RP) are less sensitive and more susceptible to contamination from nearby sources).
Not all CSPNe will appear as sources in *Gaia* DR2, for a variety of possible reasons: being too faint, having insufficiently many detections, or being obscured by foreground stars, dust, or the nebula itself. Indeed the fraction of PNe with secure CSPN identifications is small: @parkercspnpercentage noted it as 25%.
CSPNe that are detected may not be the closest sources to the centre of the visible nebula, especially if the full extent of the nebula is not apparent, or in high density regions such as the galactic centre and plane. PN progenitors are hot (blue) stars, but they may not appear blue in the *Gaia* BP – RP colourspace, due to reddening effects or, in the case of binary CSPNe, the presence of a main sequence or giant companion whose light dominates. With binary systems we are still interested in the detected companion as it provides an equally useful parallax measurement.
For many if not most PNe, our expectation is that the true central star, if it is visible, will be closest to the centre of the nebula. However, many CSPNe will not be detected by *Gaia*, and in those cases the closest stars to the nebula centre will be field stars. Our goal is as much to avoid these imposters as it is to recover true CSPNe, as their inclusion skews any further analysis based on their properties. Thus, a matching approach is required that considers more than just taking the nearest neighbour in the *Gaia* DR2 catalogue for each PN in HASH.
Catalogue Matching
------------------
We treat the search for CSPNe as a catalogue matching problem, one of finding correspondences between known PNe and sources in *Gaia* DR2. The problem of catalogue matching arises often in astronomy, usually in the context of matching objects detected at different wavelengths. It has been well studied, and has a common solution, the likelihood ratio method (@sutherlandsaunders1992, henceforth SS92), which provides a principled statistical approach towards determining the reliability of candidate matches. We briefly describe the method here for reference, following SS92.
Suppose we have a sparse primary catalogue and a dense secondary catalogue, and wish to match objects between them. Given a pair of candidate counterparts in two different catalogues, the idea of the likelihood ratio method is to compare two competing hypotheses: the objects are actually the same (a genuine match), or merely coincidental. In the simplest version, if the positions in the catalogues are offset from each other by an angular separation $r$, the likelihood ratio is the ratio of the probability of finding true counterparts with measured positions separated by $r$ to the probability of finding chance objects with that separation. That is, the likelihood ratio is a ratio of two probability densities. If we assume that Gaussian positional uncertainties are present only in the second catalogue, with standard deviation $\sigma$, the distribution of separations follows a Rayleigh distribution with parameter $\sigma$. Likewise, assuming a constant background density $\rho$, the density of spurious objects at a radial separation $r$ from the primary source position simply increases linearly with $r$ (considering a ring of increasing size). This gives a likelihood ratio[^3] $$\label{eq:simpleLR}
L = \frac{\textup{Rayleigh}(r; \sigma)}{2\pi r \rho}.$$ The likelihood ratio can also incorporate additional properties such as colour and magnitude, and prior probability of finding a match. If we consider the colour $c$ in addition to separation $r$, and furthermore assume that they are independent, we get $$\label{eq:LR}
L = \overbrace{Q}^{\text{prior}} \underbrace{\frac{P(c|\textup{genuine})}{P(c|\textup{chance})}}_{\text{colour term}} \overbrace{\frac{P(r|\textup{genuine})}{P(r|\textup{chance})}}^{\text{separation term}}$$ where $Q$ is the prior probability of there being a match for the object in the secondary catalogue.
While the likelihood ratio is valid for individual sources in isolation, we should consider the likelihood ratios for all candidate matches together for given primary catalogue object. This is done through the reliability, which for the $i$th candidate is $$\label{eq:reliablity}
R_i = \frac{L_i}{\sum_j L_j + (1 - Q)}.$$ Reliability serves as the probability of a match being correct, with the nice properties that the sum of reliabilities of all candidate matches for a given object is at most 1, and the expectation of that sum is the identification rate $Q$.
Likelihood Ratio Method for CSPNe
---------------------------------
In our approach we take HASH as the primary, or leading catalogue, and the far denser *Gaia* DR2 catalogue as the secondary catalogue. We consider the BP – RP colour of *Gaia* sources in addition to their positional offsets, motivated by the expectation of PN progenitor stars being hotter and thus having bluer colours.
We derive a prior distribution of CSPN colours from the data in a manner similar to the idea of a “self-constructed priors” described in @nwaycrossmatching and elsewhere, forming priors over relevant parameters from the empirical distribution of those parameters seen for matches that have high confidence from position alone. As we do not have positional uncertainties for the PNe themselves, we take the idea one step further, using an approach similar to co-training in semi-supervised learning [@cotraining] to derive these from the data as well.
The approach is iterative, involving first determining an approximate colour prior from nearest neighbouring sources, using the high confidence results from that to derive the positional uncertainties, and finally refining the initial colour prior based on the secure matches from position alone (\[fig:flowchart\]). Essentially we consider the colour and separation terms in Eq. \[eq:LR\] separately and alternate between them.
![Outline of the steps used in the matching process.[]{data-label="fig:flowchart"}](figs/flowchart.pdf){width="\hsize"}
### Nearest Neighbour Selection
We select all *Gaia* sources within 60$"$ of the roughly 2500 confirmed PNe in HASH (`PNstat = T`). We take the closest *Gaia* source to each PN location (applying a generous separation cutoff of half the PN radius plus 2$"$), and compare the empirical distribution of BP – RP colours of these nearest neighbour sources with the other (non-nearest neighbour) sources (Fig. \[fig:colourhistogram\]).[^4]
While the caveats we mentioned previously apply, we do expect that most true central stars will be the nearest sources to the PN centres, and that a significant fraction of nearest neighbours will indeed be true central stars. Thus the nearest neighbour and non-nearest neighbour colour distributions should approximate $P(c|\textup{genuine})$ and $P(c|\textup{chance})$, with some contamination in both directions. The effect of the contamination is to push the ratio of these densities towards 1, but the structure should still be preserved.
We use this initial colour density ratio to select the subset of *Gaia* sources within our radius cutoff (not necessarily nearest neighbours) that have high colour density ratios above a threshold.[^5] These sources are confident matches based on their colour alone, and are used in the next step to determine the positional uncertainties.
![Histogram of BP – RP colours for three different sets of *Gaia* sources covering both iterations of the colour density ratio computation. The grey histogram shows the distribution of colours of background sources, which does not change visibly between iterations. The initial colour distribution derived from nearest neighbouring sources is indicated by the red dashed line. Already there is a clear overdensity of blue BP – RP colours. The colour distribution of the final selection based on separation is indicated in blue. The lower panel shows the density ratio in colour space, also for both iterations, with the final density ratio indicated by the black line, and the initial ratio derived nearest versus non-nearest neighbours shown by the dashed red line. All densities are for sources with a well behaved BP/RP excess factor; in practice the densities and ratio function are 2-D, in order to take excess factor into account as well.[]{data-label="fig:colourhistogram"}](figs/bp_rp_hist_grey.png){width="\hsize"}
### Positional Uncertainty and Background Density Estimation {#positionaluncertainty}
The angular separations from the sources selected by colour to their PN centres (upper half of Fig. \[fig:separationhistogram\]) range from fractions of an arcsecond to tens of arcseconds, and the concentration of very nearby sources combined with the long tail is not well described by a single Rayleigh distribution as in Eq. \[eq:simpleLR\]. This is not unexpected as there are many possible sources of disagreement between the PN position and that of its central star:
- catalogue inhomogeneity, in particular whether the catalogue position was based on nebula or central star
- for positions based on central star positions, inherent uncertainty in that measurement
- for positions not based on central stars, uncertainties as to the true location of the nebula centre, and also the possibility of offset due to relative motion between the central star and the nebula
- effects of proper motion (different measurement times)
- *Gaia* measurements errors (negligible relative to to other sources of uncertainty, so we do not include the *Gaia* errors explicitly)
Thus to estimate the distribution of PN centre separations for true CSPNe ($P(r|\textup{genuine})$), we fit a mixture of Rayleigh distributions, with one distribution per PN – *Gaia* source pair in our colour selection. Each individual distribution is fit to the maximum likelihood parameter for the angular separation between the *Gaia* source and the PN centre. This construction approach simplifies the estimate and ensures the the separation density ratio is smooth, strictly decreasing, and behaves well near zero.
Some sources of uncertainty are dependent on the nebula size. Thus we re-weight the mixture to reflect offsets to CSPNe for PNe of similar sizes. For example, for a PN with a radius of 60$"$ the mixture will be be dominated by PN – *Gaia* source pairs where the radius of the PN is between around 30$"$ and 120$"$.
The other component of the term in the angular separation likelihood ratio is the density of background sources, $rho$. We estimate this locally for each PN by counting the *Gaia* sources found within the 1’ search window. We choose this approach over other methods such as taking the separation to the $n$th nearest object for its simplicity. The density will be the same for all candidate CSPNe for a given PN, so it does not affect the relative ranking, only the confidence.
![Histogram of the separations of the “high-confidence” sources (selected by colour) from their PN centres, along with, for comparison, a Rayleigh distribution with a similar mode in red, and a uniform density of background sources in grey. The lower panel shows separation density ratio resulting from the derived mixture of Rayleigh distributions compared to that from the single Rayleigh distribution in the upper panel. In practice the mixture is re-weighted depending on the radius of the PN.[]{data-label="fig:separationhistogram"}](figs/separation_hist_grey.png){width="\hsize"}
### Colour Prior Refinement
The estimate of the positional uncertainties can be used to refine the estimate of the colour density ratio. Now, instead of splitting *Gaia* sources into nearest and non-nearest neighbours, we use those sources that are highly likely to be CSPNe from their positions alone as matches, and those that are unlikely to be CSPNe based on their positions as non-matches, leaving sources out for which the position by itself is inconclusive.[^6] This removes many of the contaminants from the previous estimation based on nearest neighbour, showing a stronger preference for blue colours and a decreased score assigned to redder *Gaia* sources – in essence, increasing the contrast in the colour density ratio function.
In principle we could alternate back and forth between updating the distances and colour distributions, but the updated colour prior does not significantly change which sources meet the threshold used for the selection at the end of Sect. \[positionaluncertainty\]. Thus further iteration is not necessary.
### Final Steps
The final piece of the likelihood ratio function is $Q$, the identification rate. This scales all likelihood ratios, but does not change the ranking. We choose a value for $Q$ of 0.5, which we will verify later.
We calculate the likelihood ratios for all *Gaia* sources within each 60$"$ search window, though we enforce an additional separation cutoff of half the radius of the PN plus 2$"$. We do this for all confirmed and possible or likely PNe in HASH, though only the confirmed PNe were used in deriving the priors.
Following SS92, we compute the reliability of candidate sources for each PN, using that as our scoring metric.
Matching Results {#results}
================
The reliability distribution of the highest ranked candidate for each PN is strongly bimodal (Fig. \[fig:results\], upper left), meaning that for most PNe our method has either selected a single *Gaia* source as the best central star candidate with high confidence or rejected all nearby sources. The mean sum of reliabilities for the 2480 confirmed PNe in HASH is 0.53, consistent with our chosen value of $Q$.[^7] We focus the remainder of our analysis on these confirmed PNe, as they are most relevant for scientific applications.
![The histogram on the upper left shows the reliabilities of highest ranked candidate central stars for each PN. Overplotted is the mirrored cumulative distribution function (CDF) of that distribution, with the cutoffs and counts for best and potential matches highlighted. The two scatter plots show the distribution of the matches in colour / separation space and in galactic coordinates, with blue circles being likely matches, grey circles being possible matches, and red circles being rejected sources. Larger circles correspond to PN with larger angular sizes.[]{data-label="fig:results"}](figs/results_all.png){width="\hsize"}
Based on the shape of the reliability distribution, we choose 0.8 as our threshold for likely matches and 0.2 as our threshold for possible matches. Applying these thresholds, we find 1086 likely matches and 381 possible matches, representing 44% and 15% respectively of the total number of confirmed PNe.
The highest confidence matches are *Gaia* sources that are both blue and within fractions of an arcsecond of the HASH position. However either of these criteria alone can be sufficient - our method also finds more distant blue sources and accepts red sources that are very central (Fig. \[fig:results\], upper right).
The greatest matching success rates are for extended PNe away from the galactic centre and away from the disc (lower half of Fig. \[fig:results\]), where the PNe tend to be nearer, the density of background objects lower, and the visible light from stars less reddened by dust. Most of the uncertain matches are towards the galactic centre; cursory inspection of these shows that many are missing colours in *Gaia* and that their positional offsets are too large to accept the candidates based on position alone. These could benefit from the incorporation of additional photometry or spectroscopy, or, for those that do have colours, reddening estimates. It is interesting to observe that few PNe have multiple plausible candidate CSPNe; the choice is generally between a single best candidate and the conclusion that the CSPN as not been detected by *Gaia* at all.
Comparison with Previous Works {#comparison}
------------------------------
We compare our matching results to the published catalogues from @kimeswenger2018 (henceforth KB18) and @gapn (henceforth GS19), two previous works on central stars of PNe in *Gaia* DR2 that used different input catalogues and relied on different cross matching approaches. At the time of writing the catalogue of @stanghellinipreprint is not available, but the paper suggests that the cross matching is position based and thus may suffer from the same limitations of that method as GS19.
KB18 relied on visual and literature searches, using a smaller input catalogue of 728 PNe with radio distances from [@sh10] (henceforth SH10). They found matches for 382 out of the 728 objects in the input catalogue. All of the objects in SH10 (and KB18) are in HASH as well, though 4% of them are listed as non-PNe (Symbiotic Stars, HII regions, etc.) and 1% have unconfirmed PN status. Thus we focus our comparison on the 95% of PNe that are confirmed.
![Reliabilities of our best candidate central star matches for confirmed PNe in HASH compared to the reliabilities of the matches for those same PNe published by KB18 (top) and GS19 (bottom). Points along the diagonal of the scatter plots represent PNe for which our method found the same highest-ranked match as the previous work, while off-diagonal points indicate ranking disagreements. Lower points indicate that our method assigned a low reliability to the matches found in previous works, whether or not it found the same best match. These low reliability candidates are excluded from our analysis and our published best matches catalogue. The histograms reflect the total counts of reliabilities of matches from previous works, with the top histogram for KB18 also including reliabilities of best matches for all PNe in SH10, including those for which KB18 did not find matches. The red hatching indicates notable disagreements; that is those for which our method found a different match that also met the reliability threshold for inclusion in our catalogue (our reliability > 0.2).[]{data-label="fig:kb18reliability"}](figs/comparisons.png){width="\hsize"}
We find good agreement overall (upper half of Fig. \[fig:kb18reliability\]) for 90% of the KB18 matches (that is, we find the same best match and assign reliability > 0.2). The remaining 10% of objects are evenly split between those where we found no good match at all (e.g. Fig. \[fig:kb18comparison\]) and those where we scored another candidate higher (e.g. Fig. \[fig:kb18comparison\]). Most cases where we scored another candidate higher seem likely to be due to differences in input catalogue positions, with the positions in HASH generally being better centred than those in SIMBAD. There are a few cases for nebulae with particularly large angular sizes in which the SIMBAD position is that of the central star, and has a significant offset from the position in HASH. These sources are sometimes matched by our method but with low reliabilties on account of the angular separation. This reflects a tradeoff between purity and completeness with our method; these blue sources at high positional separation (those in the upper left corner of the upper right scatter plot of Fig. \[fig:results\]) merit further review.
We do believe that any potentially incorrect associations in KB18 are unlikely to substantially change the results of their distance comparisons, because of restrictions on colour and the outlier cuts that they used in their regressions.
Of the remaining PNe in SH10 not matched by KB18, we find good matches (reliability > 0.8) for 38% of them, slightly lower than our overall rate of 44%. These new matches from SH10 that are not in KB18 tend to lack secure colour information, either missing BP–RP colours altogether or having a high BP/RP excess factor due to nebular contamination or crowded fields.[^8] Some are also for compact PNe where the central star is likely not visible through the nebula.[^9] We have spot-checked several of these in imaging data and our matching appears good, though confirmatory colour measurements will require improved *Gaia* photometry (which should come in EDR3 for some of the sources lacking colour), data from other surveys such as VPHAS+ [@vphassurvey], or followup spectroscopy.
GS19 drew their input catalogue from a variety of sources including confirmed PNe in HASH, ending up with 2554 objects, a similar number to our input catalogue derived solely from HASH. They used a positional cross-match limited to 5$"$ and cross-checks with SIMBAD to whittle this down to a list of 1571 objects with central star matches, before manually selecting the final subset of 211 of those matches with the best parallaxes as their “golden” sample. Only this sample was published. As with KB18, we focus on the subset listed as confirmed PNe in HASH, leaving us with 178 objects, which are plotted in the lower half of Fig. \[fig:kb18reliability\].
Three very large (700“ to 1900” diameter) and asymmetric objects in GS19 have confirmed central stars outside our 60" search radius; these are absent from our catalogue but included in GS19.[^10]
While our method agreed with GS19 for much of the remaining 175 confirmed PNe in their catalogue, it also rejected just over 20% of their matched *Gaia* sources (assigning reliabilty < 0.2) and found another 10% of them to be uncertain (0.2 < reliability < 0.8). One third of the rejected sources have other matches that are scored significantly higher (e.g. Fig. \[fig:kb18comparison\]), while for the remaining two thirds our method rejected all options (e.g. Fig. \[fig:kb18comparison\]). The rejected sources from GS19 tend to have redder colours, larger angular separations, and larger parallaxes, suggesting that they are more likely to be nearby field stars. Such contamination would, for example, bias an estimate of the PN population by overestimating the number of nearby PNe.
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Our matching has its own limitations. It does not draw on literature other than the HASH PN catalogue, and it does not take into account other features that could be relevant, e.g. the magnitude of the candidate. It is important to treat the matches that we provide probabilistically and appropriately in context. If a candidate source has a precise parallax that strongly disagrees with other reliable distance measures or leads to an implausible physical nebula size, that adds weight to the source in fact being coincidental, especially in the absence of strong evidence from other non-positional features such as colour.[^11]
Catalogue
---------
Our full best matches catalogue (Table B.1) is available online at the CDS, containing the highest reliability matches from *Gaia* DR2 for all true, likely, and possible PNe in HASH, for which the reliability is at least 0.2 (our threshold for a possible match). The catalogue contains, for each PN, the PN G identifier and name, the *Gaia* DR2 source ID for the single best match for that PN, and the reliability of that match determined by our algorithm. The reliability should be used as a filter to limit analysis to high confidence matches (e.g. reliability > 0.8), with the particular threshold being dependent on the application. In addition the table contains selected columns from the *Gaia* catalogue and from HASH, denoting the position, colour, magnitude, and parallax of the best matches as well as the given PN position and radius. These columns are particularly relevant to the matching and to the science results presented in this work.
Applications
============
The subset of matched *Gaia* sources with parallaxes offers a significant increase in the number of primary galactic PN distance measurements, even with additional restrictions on parallax uncertainties or other quality indicators (Fig. \[fig:parallaxerrors\]). We present some indicative results using these distances to characterise PN physical properties and revisit the statistical distance scale of FPB16.
Physical Parameters
-------------------
Accurate distance measurements enable us to transform angular PN radii to physical radii. Combined with kinematical assumptions physical radii can determine the age of the nebula. Moreover we can also determine the luminosity of the central star, which is also related to its age and thereby its position on the evolutionary track between AGB and White Dwarf stages.
![Histogram of PN physical radii derived from *Gaia* parallaxes of matched CSPNe with various relative parallax error cutoffs. For comparison with Fig. \[fig:hrdiagram\], circle sizes used to denote physical radii are shown in the lower panel.[]{data-label="fig:radii"}](figs/radius_hist.png){width="\hsize"}
The distribution of physical radii is shown in Fig. \[fig:radii\] for PNe whose matched central stars have relative parallax error better than 20%. With these errors parallax inversion produces relatively well-behaved distance estimations [@bailorjonesparallax], which we deem acceptable for the indicative results that we present, particularly since we are not making overall population characterisations that would be biased by this sort of selection.
For central stars in *Gaia* with both full astrometric solutions and photometry, we can combine the *Gaia* G band magnitude and the distanceF estimate from the parallax to estimate the absolute magnitude, and plot this against BP – RP colour in an observational Hertzsprung–Russell diagram (HRD), following @gaiahrdiagram (Fig. \[fig:hrdiagram\]). Even without correction for reddening and extinction, most of our matches occupy an otherwise sparsely populated region of the HRD, bluer than the main sequence and giant branch but also brighter than white dwarfs.
![PN central stars plotted on an observational HR diagram, with the circular markers scaled according to the physical radii of the PNe as in Fig. \[fig:radii\]. Filled circles indicate objects with the lowest uncertainties. Individual PNe referenced in the text are coloured red rather than blue and accompanied by the PN name. Red lines represent CSPN tracks from @bertalomicspntracks for solar metallicity and various initial masses, with the green portions of the line denoting time since leaving the AGB of between 1000 and 20000 years, indicative of the sorts of timescales during which a PN could be visible. The peak temperatures of these tracks, through which the stars evolve relatively quickly, are located at an absolute *Gaia* magnitude around 5 (see text for details). In the background, the grey points are the other sources that were loaded in the 60$"$ search windows, with $\sigma_\omega/\omega < 10\%$. They trace out the main sequence (MS) and giant branch. The beginning of the AGB is also labeled, with its position taken from @gaiahrdiagram. White dwarfs are shown separately, as they are too rare to appear otherwise, with the grey contours in the lower left representing the 10, 30, and 50% density contours of the observed high confidence white dwarf candidates from @gaiawhitedwarfcandidates, where the same quality cuts have been made as for the background points.[]{data-label="fig:hrdiagram"}](figs/hr_diagram_new_tracks.png){width="\hsize"}
### Theoretical Tracks
Comparison to theoretical tracks requires mapping between the physical stellar parameters of effective temperature $T_\textup{eff}$ and bolometric luminosity $L$ and the observed *Gaia* BP – RP colours and G magnitude.
The goal of the *Gaia* astrophysical parameters inference system (Apsis) is to perform the mapping starting from *Gaia* observations, deriving the mapping based on machine learning techniques [@gaiaapsisprelaunch]. It will ultimately use the *Gaia* photospectrometry and account for reddening and extinction as well. In *Gaia* DR2, temperatures are available for less than 10% of sources, and luminosities for less than half of those. Moreover, because of the limited temperature range of the training data, the $T_\textup{eff}$ values that the model does produce do not go above 10000 K [@gaiadr2apsis], making them unhelpful for the much higher temperature range expected for CSPNe.
Instead we perform the mapping in the opposite direction, transforming physical parameters into observables. Such transformations were provided pre-launch for main sequence and giant stars by @jordigaiaphotometry, and for white dwarfs in a followup paper by @gaiawdphotometry. Though the latter transformations cover higher surface gravities and extend the range of effective temperatures, the two together still miss most of the CSPN evolutionary tracks we wish to cover. Fortunately, the transformation for higher temperature objects such as CSPNe is largely independent of metallicity and surface gravity. We used the revised BP and RP passbands from @gaiaphotometry and assumed blackbody spectral energy distributions (SEDs) to generate expected BP - RP colours. We found that these were bluer than expected from the pre-launch papers and speculate that this is due to higher sensitivity than expected at the lowest wavelengths. The bluer tracks are a better fit for the observed data. For transforming luminosity into absolute *Gaia* G magnitude we adopt the bolometric corrections from @gaiawdphotometry for a surface gravity (log *g*) of 7, fitting a cubic spline to extrapolate to the highest temperature regime. The G passband more closely matches the nominal, pre-launch passband, so we do not expect these calculations to change significantly.
Using these transformations, we plot a selection of tracks from @bertalomicspntracks for solar metallicity (Z$_0$ = 0.01, versus 0.0134 for the sun) and a range of initial masses. Bolometric corrections change the shape of the tracks from those in the temperature versus luminosity space, with higher temperature objects at the same luminosity having more of their flux at ultraviolet wavelengths outside the *Gaia* G band. Thus peak temperature occurs at a G absolute magnitude of around 5, with higher temperatures appearing fainter.
The theoretical tracks are relatively near each other in the BP – RP colour space, as the *Gaia* BP – RP colours are not highly sensitive in this temperature range. Between this and the degeneracy between temperature and reddening, we are not able to constrain initial masses and ages from the *Gaia* DR2 photometry alone (as was the case for white dwarfs in @gaiawhitedwarfcandidates). Such determinations require additional photometry or spectroscopy to better constrain and disentangle reddening and temperature. The *Gaia* estimated distances combined with dust maps may prove useful in this regard.
### Discussion
![Reddening values E(B-V) and their given uncertainties taken from FPB16’s statistical distance compilation plotted against *Gaia* BP – RP colours for all matches with reliability > 0.8 (not limited by parallax uncertainties). Known and suspected binary systems taken from the compilation of David Jones are highlighted as black squares. Objects lying below the trend (objects appearing red in *Gaia* with low reddening) could be binary systems or have significant reddening internal to the nebula (not considered by FPB16), or could have dubious identification. Relevant individual objects mentioned in the text are shown in red.[]{data-label="fig:reddening"}](figs/reddening_annotated.png){width="\hsize"}
We find that most of our matched central star colours and absolute magnitudes are well explained by the theoretical tracks plus reddening effects, and that the physical sizes of the nebulae are consistent with the evolutionary direction of their central stars (in that younger and therefore brighter central stars have less evolved nebulae).
Several of the CSPNe are inconsistent with the evolutionary tracks; that is those with relatively red BP – RP colours whose de-reddened projection onto the theoretical tracks is a poor fit. We focus on those with that are for resolved nebulae (so that the *Gaia* detection is of the central star rather than the nebula itself) and have low photometric excess factors (indicating well behaved photometry; colour uncertainty from high excess factors likely dominates any flux uncertainties).
One explanation is that these are binary systems where the light from actual progenitor of the PN is dominated by a main sequence companion. A few examples that we checked are LoTr 5 (PN G339.9+88.4) and BE Uma / LTNF 1 (PN G144.8+65.8), which have the largest absolute latitudes of the red sample, meaning that their colours are less likely to be reddened. Both of these are in fact known binary systems (@joneslotr5ngc1514binary and @fergusonbeumbinary respectively), as is NGC 1514 (PN G165.5-15.2) in the first reference. WeBo 1 (PN G135.6+01.0), the reddest star in the sample with a large BP–RP value of 1.9, is also a binary [@webo1binary], while the second reddest star, the central star of PMR 1 (PN G272.8+01.0) with BP–RP equal to 1.7, is noted in the literature to simply be heavily reddened [@pmr1wolfrayet]. These are highlighted in Fig. \[fig:hrdiagram\] and Fig. \[fig:reddening\] along with the other individual PNe mentioned in this section.
One star that does appear inconsistent with the nebula size versus absolute magnitude trend is NGC 2438 (PN G231.8+04.1). Its parallax measurement places it around 422pc away, with less than 10% error. However this distance is inconsistent with other distance determinations for the PN: the statistical scale of @frewsurfacebrightness2016 has it at 1.54kpc $\pm$ 0.44kpc, consistent with an even further away estimate from central star modelling that was used as part of that paper’s calibration. We believe that the parallax errors in this case may not be well behaved, supported by the source having high astrometric excess noise (and renormalised unit weight error (RUWE) = 2.39). Assuming NGC 2438 to be further away removes tensions, leading to a larger physical nebula size and a brighter central star.
The HRD additionally suggests the possibility of further refinement of the matching itself based on parallaxes, with the derived absolute magnitudes disambiguating between possibly reddened CSPNe and main sequence stars, and the derived distances allowing for the calculation of a physical radius which can then be checked for compatibility with other knowledge about the PN. Stars that do lie on the main sequence and have plausible parallaxes merit further investigation as possible mismatches, binaries, or reddened single central stars.
Statistical Distance Scales
---------------------------
The parallaxes from *Gaia* also offer an opportunity to evaluate and ultimately refine statistical distance scales. We focus on the H$\alpha$ surface brightness to physical radius relation from FPB16.
The idea underlying the distance scale of FPB16 is that the measured H$\alpha$ surface brightness of a PN is independent of its distance (ignoring the effects of extinction, the ratio of the apparent brightness of an extended object to its apparent area is constant), and can be related to the PN’s physical radius, with that relation derived from a set of calibrating objects with known distances. Using the calibrated relationship to estimate the physical radius of a PN and comparing that to its angular radius then provides an estimate of its distance.
![Relative parallax errors $\sigma_\omega/\omega$ for the best matches (reliability > 0.8) subsample of confirmed PNe, along with the cumulative counts below various reliability thresholds for positive parallaxes (in black). The bins at either end represent the counts or matches with $\sigma_\omega/\omega$ falling outside of the range (-0.5, 1.5). Within the sample, those parallaxes meeting more stringent criteria (reliability > 0.98, $\sigma_\omega$ < 0.2 mas, RUWE < 2, `visibility_periods_used` > 8) are indicated by the darker shaded area of the histogram. This subset is used for the @frewsurfacebrightness2016 distance comparisons as it improves error behaviour without the biases introduced by cuts on $\sigma_\omega/\omega$.[]{data-label="fig:parallaxerrors"}](figs/sigma_hist.png){width="\hsize"}
The calibrating set contained just over 300 objects that were considered to have sufficiently reliable distance estimates based on a variety of primary techniques. Half of these objects were extragalactic; these tended to have the most relatively accurate distance estimates, but exhibit a slightly different trend to that of galactic objects. A more extensive set of primary distances such as those from the parallaxes of the matched CSPNe in this work allow calibrating the scale based on a more representative galactic sample: as shown in Fig. \[fig:parallaxerrors\] the parallaxes with errors < 20% is already the same size as the galactic calibrating set used in FPB16, and future data releases will only improve on this.
The caveats present in using parallaxes to estimate distances are well known [@xlurigaiaparallaxes]; in particular naive parallax inversion does not produce a statistically sound distance estimate for any reasonable choice of prior, and any attempt to limit an analysis to parallaxes with relative error $\sigma_\omega/\omega$ below some threshold (as was done in the previous section), or even to positive parallaxes (as inverting negative parallaxes is unphysical) introduces biases. We can avoid these caveats by staying in the space of parallaxes, where the errors are well behaved.
### Distance Ratios
The notion of distance ratios described in @smith2015 avoids parallax inversions and the problems that come with them. The idea is to “anchor” a given statistical distance scale using a set of parallax measurements by taking the product over objects of measured parallaxes $\omega'$ and estimated statistical distances $d'_s$ to form a distance ratio $$\label{eq:distance_ratio}
R_S = \omega' d'_s$$ the expectation of which is the true, underlying distance ratio for that distance scale. This holds regardless of the true distribution of distances and only requires that the errors have zero means. The associated uncertainty is $$\label{eq:distance_ratio_error}
\sigma^2_R = d^2_S\sigma^2_\omega + \omega^2\sigma^2_S + \sigma^2_\omega\sigma^2_S$$ where $d_S$ is the true distance $d$ multiplied by the distance ratio $R_S$ and $\sigma_S$ is the standard error on the statistical distance.
The distance ratio can be used to measure both errors in the intercept of the statistical relation (through deviations in the mean distance ratio away from 1) and in its slope (through correlations between the distance ratio and the estimated physical radius or statistical distance). It can also serve as a measurement of the intrinsic scatter of the relationship, though Smith noted that this can be biased.
Smith used the technique to evaluate FPB16, using for comparison the trigonometric parallaxes from the HST and USNO as well as a set of spectroscopic parallaxes that covered a greater distance. He found good agreement, with a mean distance ratio of $1.01\pm0.08$, though the comparison set unavoidably overlapped with the calibrating set of FPB16. We perform a similar analysis here using the *Gaia* parallaxes, which are completely independent from those calibrating distances.
### Methods {#methods}
We consider the set of confirmed PNe for which FPB16 published statistical distances (1024 in total) and for which we have found reliable central star matches with high quality parallax measurements.
To limit the effect of poorly behaved parallaxes we apply quality cuts similar to those used in @gaiahrdiagram. We require a slightly higher number of observations than the threshold for inclusion into the Gaia data release, that is $\texttt{visibility\_periods\_used} > 8$. Additionally we set an upper limit on the renormalised unit weight error (RUWE) of 1.4 as recommended by @gaiaruwe. This is a goodness-of-fit statistic that indicates how well the *Gaia* astrometric solution matches that expected for a single star [@gaiaastrometry]. Finally, we apply a cut on the absolute uncertainty of the parallax itself, of $\sigma_\omega < 0.2$ mas. Critically this is different from a cut on *relative* parallax error in that it does not depend on the estimated parallax, so it avoids introducing truncation biases.[^12] Though these should still have the same expected distance ratio value, removing these highly uncertain parallaxes reduces the overall uncertainty the average distance ratio.
To avoid the effect of any incorrect matches we also apply a stricter reliability cut of 0.98, which keeps the vast majority of the matches.
The quality cuts leave us with 160 objects out of the 636 objects from FPB16 that we matched.
For many PNe, FPB16 provide multiple distance estimates - one based on a general trend, and one based on a subtrend for PNe that are classified as either optically thick or optically thin. The subtrend relationships have different slopes from each other and lower scatter. The calibrating set in FPB16 was chosen to represent a range of PN properties, and is balanced between optically thin and thick objects. If the subset that we compare is a different mixture, it will deviate from the mean trend even if the distances are correct. We consider this in the next section.
### Results
![Histograms of distance ratios $R_S$ and normalised distance ratios $R_S / \sigma_S$derived from comparison between *Gaia* parallaxes and statistical distances (using subtrends) from FPB16. Ratios are plotted for both the higher quality set of parallaxes (see text) and rejected parallaxes for comparison, in dark and light blue respectively. The plot on the left shows the raw distance ratios, with the mean value of 1.03 $\pm$ 0.06 for the best quality parallax set. On the right the distance ratios have been re-centered around $R_S=1$ and divided by their estimated uncertainties $\sigma_S$. Though the distribution of distance ratios is not expected to be Gaussian, a standard normal distribution is overplotted for comparison. Below is a scatter plot depicting the distance ratios of the best parallax subset against the physical radius derived from the statistical distance. Marker colours and shapes show morphological classifications taken from HASH. Trends in this plot (that is, a correlation between distance ratio and radius) would be indicative of a slope differing from that derived in FPB16. Filled markers have $R_S$ within $2.5\sigma_S$ of 1. Outliers are empty markers, with the two outliers specifically mentioned in the text highlighted. The correlation coefficients are 0.18 and 0.08 with and without the outliers respectively. The former is very weakly significant, while the latter is not. []{data-label="fig:distanceratios"}](figs/distance_ratios_all.png){width="\hsize"}
Using the mean trends gives a mean distance ratio of $1.15 \pm 0.07$ for the 160 objects that passed the quality cuts, while using the subtrends reduces this ratio to $1.03 \pm 0.06$ (the uncertainties are calculated via bootstrapping). We note that the matched PNe show a preference for optically thin PNe relative to the mixture of thick and thin PNe that formed the mean trend in FPB16, which could be due to optically thin PNe being more likely to have visible central stars. The subtrend in FPB16 for optically thin PNe has such PNe having lower surface brightnesses for the same physical radius, which translates to the mean trend overestimating the physical radii for these objects and thus overestimating their distances. This is consistent with the difference in mean distance ratios we see comparing the mean trend and subtrends.
The results using the subtrends are shown in Fig. \[fig:distanceratios\]. On average *Gaia* parallaxes are consistent with the FPB16 statistical distances. This is not surprising given the many extragalactic distances in the set of calibrating distances and the use of parallaxes in the calibration itself, which mean that the distance scale is unlikely to deviate from a true distances by the factors of 2 that older scales suffered from [@smith2015]. There is a slight suggestion of a dependency on physical radius but the uncertainties are too large to draw a meaningful conclusion. Grouping by morphology (lower half of Fig. \[fig:distanceratios\]), we find no significant deviations from a mean distance ratio of 1, with round PNe having the largest deviation at $1.15 \pm 0.12$.
We see a few notable outliers, objects for which $|(R_S - 1) / \sigma_S|$ is large. The mean trend statistical distances for both K 1-6 (PNG 107.0+21.3) (1.85 $\pm$ 0.53 kpc) and Abell 28 (PNG 158.8+37.1) (1.67 $\pm$ 0.48 kpc) appear to be significant overestimates relative to their central star parallaxes, which place both of them within 500 pc. The distances from the thin trend (1.45 $\pm$ 0.27 kpc and 1.29 $\pm$ 0.25 kpc respectively) are smaller and thus closer, but the significance of the disagreement is greater due to the smaller uncertainties in statistical distances for that population of objects. K 1-6 was studied in @k1-6frew2011, which noted tensions between different distance estimates for that nebula in terms of its surface brightness and the properties its binary central stars; they adopted a distance of 1kpc, halfway between FPB16’s statistical distances and that suggested by the parallax from *Gaia*. They also noted a range of possible distances based on the spectroscopic parallax of the binary central star companion, with the short end of those distances being consistent with the now observed trigonometric parallax. The *Gaia* parallax for the blue central star of Abell 28 places it in the population of “subluminous” PNe noted in Sect. 4.3.4 of FPB16, with Abell 28 then occupying a place in the surface brightness versus physical radius plane near that of RWT 152 (PNG 219.2+07.5) (the parallax of the central star of RWT 152 itself is consistent with both the primary and statistical measurements).
On the other end of the scale there are several objects for which the parallaxes indicate larger distances than the statistical ones. There is a suggestive excess of elliptical / bipolar objects in this set that would match the trend that FPB16 noted of bipolar objects having higher surface brightnesses, but even comparing the calibrating distances of those objects the *Gaia* parallaxes shows significant disagreement by up to a factor of 2, for example for Hen 2-11 (PN G259.1+00.9), whose parallax of 0.5 mas gives it a 2$\sigma$ distance range of 1.25 to 5 kpc from *Gaia*, outside of the relatively confident 730 pc estimate derived from modelling of its binary star by @hen211jonesboffin that was also used in the calibrations by FPB16. One possibility is that the parallaxes are themselves skewed by binarity, as in *Gaia* DR2 only single stars are modelled. Also, as the uncertainties in statistical distances are correlated with the statistical distances themselves, statistical distances that are underestimates also have underestimated uncertainties, which in turn means that the uncertainty in the distance ratio is underestimated. This effect was noted by @smith2015.
### Discussion
The fraction of outliers would increase significantly if we lowered the reliability threshold of our method and accepted nearest neighbour *Gaia* sources that we had not considered to be matches. The selection of such mismatches based on distance ratios is biased towards nearby objects, which tend to have lower parallax errors (on account of being brighter) and larger parallaxes that more tightly constrain their distances. Such mismatches will become more noticeable in future data releases as parallax uncertanties tighten, however, even mismatches that are individually consistent within errors will globally skew any calibration or evaluation, making it important to have a robust selection process to begin with. As with the HRD in the previous section, additional data, in this case distance priors based on statistical distances derived from nebula properties, can be used to further refine the matching by placing bounds on reasonable parallaxes.
Ultimately the *Gaia* parallaxes will offer a new opportunity to calibrate statistical distance scales such as that of FPB16 using galactic PNe and bring the uncertainties closer to the intrinsic scatter of the relationship. Trigonometric parallaxes provide the most direct means of measuring distances, but their properties mean that they require a proper prior on the underlying distances which must be accounted for at the level of the derived relationship rather than for individual distances such as those published in the catalogue by @bailerjonesdistances. Selection effects may be present as well as certain types of PNe may be more amenable to distance determination from central star parallaxes. Performing such a calibration is beyond the scope of this particular work, but we believe that the uniform matching performance of our automated technique will offer a good basis for such work in the future, in particular with the improved data in the forthcoming *Gaia* EDR3.
Conclusions
===========
We have used a novel application of the likelihood ratio method to automatically match central stars of planetary nebulae in the HASH PN catalogue with sources in *Gaia* DR2 based on their position and colours, with a particular focus on accuracy and consistency that contrasts with previous works. Our catalogue of matches includes confidence scores, and is the largest available for *Gaia* DR2 at the time of writing. We have described a few examples of how this catalogue and the new data offered by *Gaia* will enable future science, and discussed the importance of accurate matching in achieving these aims. We emphasize that the certainty of the matching itself should be considered holistically in any analysis.
There are opportunities for further refinement of our matches based on additional data. Photometry from other surveys could disambiguate where *Gaia* colours are lacking, though *Gaia* itself will improve significantly on this front in the future with the full BP/RP spectrophotometry (low resolution spectra). As noted in the previous section, the candidate central star sources with the best parallaxes can be further evaluated on their plausibility as central stars based on their positions in the HR diagram and whether the resulting distance is compatible with the angular size and surface brightness of the nebula itself. Equally, outliers in these parameter spaces can point to interesting sources and systems for followup and further study, such as binary central stars.
Our automated method makes it possible to easily and quickly update the catalogue based on future *Gaia* data releases and future PN discoveries. In particular, we will be able to leverage the improved completeness and more precise astrometric measurements in those future data releases to better understand the galactic PN population.
This research has made use of data from the European Space Agency (ESA) mission [*Gaia*]{},[^13] processed by the [*Gaia*]{} Data Processing and Analysis Consortium (DPAC).[^14] Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the [*Gaia*]{} Multilateral Agreement.
This research has also made use of the HASH PN database,[^15] of Astropy,[^16] a community-developed core Python package for Astronomy [@astropy:2013; @astropy:2018], and of “Aladin sky atlas” developed at CDS, Strasbourg Observatory, France.
Parts of this research were based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID 177.D-3023, as part of the VST Photometric H$\alpha$ Survey of the Southern Galactic Plane and Bulge (VPHAS+),[^17] as well as data obtained as part of the INT Photometric H$\alpha$\] Survey of the Northern Galactic Plane (IPHAS)[^18] carried out at the Isaac Newton Telescope (INT). The INT is operated on the island of La Palma by the Isaac Newton Group in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias. All IPHAS data are processed by the Cambridge Astronomical Survey Unit (CASU), at the Institute of Astronomy in Cambridge.
This research was supported through the Cancer Research UK grant A24042.
Implementation Details
======================
The probability densities functions (PDFs) used to calculate the likelihood ratio (Eq. \[eq:LR\]) are estimated from the data themselves. The particular methods and parameters were chosen with the overall aim of producing smooth density ratios with few extrema and to thereby avoid overfitting.
Colour Density Ratio Estimation
-------------------------------
Recall that our goal is to estimate the density in BP – RP colour space of true CSPNe and of non-CSPNe (background sources). We determine our estimates empirically by choosing representative examples of both kinds of sources based on their positions.
The BP and RP fluxes measured by *Gaia* can be contaminated by light from nearby sources (within a couple of arcseconds), particularly in densely populated or nebulous regions. For well-behaved sources with no contamination, it is expected that the total flux measured in the BP and RP passbands should approximately match that of the G band, which does not have the same possibility of contamination. Deviations from this relation are indicated in the catalogue by a large photometric excess factors, and @gaiaphotometry suggests using cuts based on this factor to select photometrically well-behaved sources for applications relying on colour information. Rather than ignoring the colours of these high excess factor sources completely with hard cuts, we incorporate the excess factor into our density estimation, treating the source colour space as two-dimensional.
We bin sources by excess excess factor (distance above the locus of well-behaved colours, that is $\texttt{phot\_bp\_rp\_excess\_factor}-1.3\times\texttt{bp\_rp}^2$, taken from @gaiaphotometry) with overlapping bins. We compute the density ratio within each bin as a function of BP – RP alone, and then smoothly interpolate to get the density ratio values for excess factors between bin centres (interpolating towards 1 for high excess, corresponding the colour density ratio of 1 for sources lacking colours). Thus while the density ratio function has a two-dimensional domain, colour densities are only ever one-dimensional in BP – RP. We thereby hope to treat excess factor as only a quality indicator.
We estimate the density ratios at each BP – RP value within a single bin non-parametrically using kernel density estimation with a Gaussian kernel. Because of the highly varying density, we use a balloon estimator, in which the kernel bandwidth (the standard deviation of the Gaussian in this case) is variable and is inversely proportional to the local density at the sample point. We estimate the local density from the distance to the $n$th nearest neighbour in BP – RP, so the kernel bandwidth is effectively proportional to distance to the $n$th nearest neighbour. The bandwidth is clipped to lie within a range of well-behaved values. To avoid artefacts from mismatched kernel widths, the same kernel width is used for both the numerator and denominator of the density ratio, with the kernel chosen based on the numerator density (the density of BP – RP colours for candidate central stars chosen based on distance or nearest neighbour), since there are fewer such sources.
Sources used to estimate the background colour density (either non-nearest neighbour sources in the first iteration or sources with low separation density ratios in the second iteration) are weighted by the inverse of the local spatial source density $\rho$. The idea of this is that each PN neighbourhood is given equal weighting in the denominator of the colour density ratio estimation (the colour density for background sources). Each PN neighbourhood is by default equally weighted in the estimate for genuine match colours (the numerator), since all neighbourhoods contribute (at most) a single candidate genuine match and are thus weighted equally in that calculation.
Separation Density Ratio Estimation
-----------------------------------
The set of sources used to estimate the separation density ratio is those with a colour likelihood ratio $> 20$ from the initial (nearest neighbour) colour density ratio estimation. We apply a cutoff on the separation $s$ to these sources, requiring that $s < r_{PN} + 2"$ where $r_{PN}$ is the PN radius in arcseconds, with the addition of $2"$ reflecting our expectation that the relative positional uncertainty is greater for smaller PNe. There are $n$ *Gaia* sources that met our cutoff, having separations $s_i,\ i=1\dots n$. These sources are associated with confirmed ($\texttt{PNstat} = \texttt{T}$ in HASH) PNe with radii < 600$"$ (including unresolved PNe with no size information in HASH, which we treat for the purposes of binning as having radii of 0.25$"$).
As noted in Sect. \[positionaluncertainty\] and Fig. \[fig:separationhistogram\], the distribution of separations $s$ does not match well with a single Rayleigh distribution, unsurprising given the multiple sources of positional uncertainty. However adopting a fully non-parametric approach does not work as well as it did for the colour density in the previous section.
The PDF of a Rayleigh distribution is $$\textup{Rayleigh}(r; \sigma) = \frac{r}{\sigma^2}e^{-r^2 / 2\sigma^2}$$ which has the convenient property that the $r$ term cancels with the $r$ in the PDF for a constant density of background sources, that is $2\pi r \rho$, giving a likelihood ratio that levels off at a finite value as the separation approaches 0. This reflects the fact that while finding a background source with a very small separation is highly unlikely, so is finding a true counterpart source.
To preserve these properties we form our distribution as by mixing $n$ Rayleigh distributions $$f(r) = \sum_{i=1}^n w_i\ \textup{Rayleigh}(r; \sigma_i)$$ with parameters $\sigma_i$ each corresponding to the maximum likelihood estimates (MLEs) from a single separation $s_i$, that is $\sigma_i=s_i/{\sqrt 2}$. This mixture captures the behaviour of the empirical distribution while ensuring that the resulting density ratio is smooth, strictly decreasing, and well behaved near 0.
Another advantage of this mixture approach is that the mixture can be reweighted to fit different PN sizes, reflecting the expected dependence in positional uncertainty on the size of the PN. Rather than identical weights $w_i=1/n$, we choose mixture weights for a PN with radius $r_{PN}$ as $$w_i \propto \exp\frac{(\log_2 r_{PN} - \log_2 r_{PN_i})^2}{2\sigma^2}$$ scaled so that $\sum_i{w_i}=1$. We consider log radii as the logarithm is scale invariant, and choose a standard deviation $\sigma=0.5$ so that most of the influence comes from PN with radii $r_{PN_i}$ within a factor of 2 of the given PN radius.
Justification of Nearest Neighbour Approximation
------------------------------------------------
We form our initial estimate of the colour density ratio by splitting our candidate set into nearest and non-nearest neighbours, and use the candidate points with the highest colour density ratio as a kind of initial training set for learning the positional uncertainties. This density estimation (and indeed the second iteration based on position) is contaminated in both directions, with many background sources in the nearest neighbour set (standing in for the CSPNe set) and some true CSPNe in the non-nearest neighbour set (standing in for the background distribution).
The effect of this contamination is to push the density ratio towards 1 (the density ratio becomes 1 in the limit where the two distributions contain the same proportions of true CSPNe and background sources). We can still learn useful and valid information from the colour provided that true CSPNe make up a larger proportion of the nearest neighbour set than they do of the non-nearest neighbour set, which we expect will be the case as the non-nearest neighbour set is so much larger to begin with.
Results Table
=============
[^1]: <https://hashpn.space>
[^2]: Version 4.6 of the HASH PN catalogue, downloaded on 16 June, 2019, was used to produce the published catalogue for this work.
[^3]: The version of the likelihood ratio in Eq. \[eq:simpleLR\] is related to the figure of merit used in the *Gaia* cross matches with external catalogues [@gaiacrossmatching2017; @gaiacrossmatching2019].
[^4]: We use kernel density estimation with Gaussian kernels, and consider the BP/RP excess factor in addition to the colour itself. Binning by excess factor allows us to still derive useful information from the colours of sources that are not within the locus of well-behaved sources suggested by @gaiaphotometry. Objects with no BP – RP colour are ignored in this selection (the presence or absence of colour in itself is considered to be uninformative, giving a colour density ratio of 1).
[^5]: For PNe with multiple nearby high colour density ratio sources, we take the closest.
[^6]: We calculate reliability based on separation only, using sources with reliability > 0.8 as our positive examples and those with reliability < 0.2 as negative ones.
[^7]: Otherwise we could iteratively update $Q$ and recalculate the reliabilities until they converge. Indeed, the 415 likely and 663 possible PNe have mean reliabilities of 0.45 and 0.34 respectively, indicating a lower success rates for these unconfirmed PNe and also that the chosen $Q$ value of 0.5 is thus inconsistent for them. The relationship between PN status and central star matching success is expected given that a clearly visible central star contributes towards confirming the PN status of a nebula.
[^8]: Examples are M 1-18 (PN G231.4+04.3) and Mz 1 (PN G322.4-02.6) with high excess factors, and Hen 1-6 (PN G065.2-05.6) and K 1-4 (PN G001.0+01.9) with no colour.
[^9]: An example is KFL 19 (PN G003.3-07.5).
[^10]: These are Sh 2-188 (PN G128.0-04.1), FP J1824-0319 (PN G026.9+04.4), and FP J0905-3033 (PN G255.8+10.9).
[^11]: An example of an implausible match is that of the “likely” PN Abell 19 (PN G200.7+08.4). The centrally located star is nearby at around 250 pc away, with small parallax uncertainties. The absolute *Gaia* G magnitude of 5.55 derived from this parallax and BP – RP colour of 1.04 place the star neatly on the main sequence. While the colour could be explained by reddening, significant reddening is unlikely given the star’s relatively close proximity. Moreover the resulting physical radius of 0.05 pc is more consistent with a very young PN, which would then have a much brighter central star. Thus we can conclude that this candidate is more likely to be a nearby field star.
[^12]: Since parallax uncertainties are magnitude dependent, this cut does favour brighter sources, which is anyway unavoidable as those are more likely to be detected to begin with.
[^13]: <https://www.cosmos.esa.int/gaia>
[^14]: <https://www.cosmos.esa.int/web/gaia/dpac/consortium>
[^15]: <http://hashpn.space>
[^16]: <http://www.astropy.org>
[^17]: [www.vphas.eu](www.vphas.eu)
[^18]: [www.iphas.org](www.iphas.org)
|
---
title: |
Supplementary information:\
Search for Axion-Like Particles produced in [$e^+e^-$]{}collisions at
---
Comparisons between data and simulated sample distributions of variables used in the selection and fit are given in Fig. \[fig:sup\_mgg\_650\]-\[fig:sup\_deltatheta\]. Fig. \[fig:bestfit\] shows the fit for $m_a=0.477$[$\text{GeV/c}^2$]{}that resulted in the largest significance observed. We provide an additional textfile with numerical results the observed 95% CL upper limit on the ALP cross section $\sigma_a$ (pb), and the observed 95% CL upper limit on the photon coupling [$g_{a\gamma\gamma}$]{}(GeV$^{-1}$) as a function of ALP mass $m_a$ (GeV).
|
---
abstract: 'This study examines statistical performance of tests for time-varying properties under misspecified conditional mean and variance. When we test for time-varying properties of the conditional mean in the case in which data have no time-varying mean but have time-varying variance, asymptotic tests have size distortions. This is improved by the use of a bootstrap method. Similarly, when we test for time-varying properties of the conditional variance in the case in which data have time-varying mean but no time-varying variance, asymptotic tests have large size distortions. This is not improved even by the use of bootstrap methods. We show that tests for time-varying properties of the conditional mean by the bootstrap are robust regardless of the time-varying variance model, whereas tests for time-varying properties of the conditional variance do not perform well in the presence of misspesified time-varying mean.'
author:
- |
Daiki Maki[^1]\
Doshisha University
- |
**Yasushi Ota** [^2]\
Okayama University of Science
title: 'Testing for time-varying properties under misspecified conditional mean and variance'
---
Introduction
============
Many economic and financial time series data have time-varying properties. Their properties are roughly classified into two types. One is a property about the conditional mean. Constant and/or autoregressive parameters change with time as time-varying properties for the mean. Another time-varying property is the conditional variance of the error term for a regression model. Variance of the error term, that is, volatility, is frequently modeled by the autoregressive conditional heteroskedasticity (ARCH). As introduced by Dahlhaus and Rao (2006), a property of the time-varying model for volatility is that the parameters of volatility models change with time. Amado and Teräsvirta (2013) and Kim and Kim (2016) recently provided a time-varying volatility model and its applications.
Linearity tests are usually used to investigate time-varying properties. When we test for time-varying properties of the conditional mean, we assume a homoskedastic variance or a correctly specified linear variance process for the error term of the condtional mean. When we test for time-varying properties of the conditional variance, we assume a correctly specified linear process for a mean. However, it is difficult to know which time-varying property for the mean or variance is present a priori. This challenge implies that researchers may erroneously test for time-varying properties of the mean, although volatility actually has time-varying properties. Similarly, researchers might erroneously test for time-varying properties of the variance, although the mean actually has time-varying properties. In fact, Pitarakis (2004) and Perron and Yamamoto (2019) showed that a test for a change in regression coefficients (variance) ignoring or misspecifying the presence of change in variance (regression coefficients) causes poor statistical properties. For the influence of ARCH on inference of misspecified models, Lumsdaine and Ng (1999) showed over-rejection of ARCH tests in the presence of misspecified conditional mean models. Van Dijk, Franses, and Lucas (1999) demonstrated size distortions of ARCH tests when a process has additive outliers. Balke and Kapetanios (2007) pointed out that supurious ARCH effects appear when nonlinearity of the mean is ignored. If we were to perform erronous tests and fail to obtain reliability from the derived results, the correct model construction and evaluation are difficult. Accordingly, it is important to clarify the influence of tests on the misspecified models.
This study examines the influence of time-varying tests on misspecified conditional mean and variance. In particular, we clarify the statistical performance of tests for time-varying variance when a process has time-varying mean with homoskedastic variance. We also investigate statistical performance of tests for time-varying mean when a process has time-varying variance with a linear mean model. Some studies, including Lumsdaine and Ng (1999), van Dijk, et al. (1999), and Balke and Kapetanios (2007), investigated problems of misspecified models in ARCH tests. However, previous studies have not clarified the influence of time-varying tests on misspecified conditional mean and variance. As mentioned above, it is important to clarify time-varying properties correctly. This study uses a logistic smooth transition function to model time-varying mean and variance. This model smoothes threshold or structural break models in time. The time-varying tests used in this study are based on the method introduced by Luukkonen, Saikkonen, and Teräsvirta (1988a). They depeloped linearity tests using a Taylor series approximation to overcome the identification problem pointed out by Davies (1977, 1987).
The simulation results in this study provide evidence that the asymptotic test for time-varying mean has size distortions when the conditional variance model is misspecified. However, the wild bootstrap method introduced by Liu (1988) improves the size distortions. In fact, Becker and Hurn (2009) provided evidence that the wild bootstrap improves size properties when the conditional mean is being tested. We can test for time-varying mean by using the wild bootstrap without depending on the form of volatility. When we test for time-varying variance in the presence of a misspecified conditional mean, asymptotic tests have large size distortions. The properties are not improved by the use of the bootstrap ARCH test proposed by Gel and Chen (2012) and the wild bootstrap. The results show that the wild bootstrap tests for time-varying mean are robust regardless of the misspecified conditional variance, whereas tests for time-varying variance do not perform well in the presence of misspecified conditional mean.
The remainder of this paper is organized as follows. Section 2 presents the tests for time-varying mean and variance. Section 3 provides statistical properties of the tests for time-varying mean and variance in the presence of misspecified conditional models. Finally, Section 4 summarizes and concludes.
Tests for time-varying mean and variance
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We assume the following regression model to test for time-varying mean$^1$. $$y_t=\alpha_0 + \beta_0 y_{t-1} + (\alpha_1 + \beta_1 y_{t-1}) F (t, \ \gamma, \ c)+ u_{t},$$ where $u_t \sim N(0, \sigma^2)$ and $F(t, \ \gamma, \ c)$ is a transition function to model time-varying properties of (1). (1) has a time-varying constant and slope for $y_{t-1}$ depending of the value of $F(t, \ \gamma, \ c)$. Lin and Teräsvirta, T. (1994) provided the asymptotic theory for time-varying regression models. (1) reduces to a linear autoregressive model for $F(t, \ \gamma, \ c)=0$ and/or $\alpha_1=0$ and $\beta_1= 0$. $F(t, \ \gamma, \ c)$ is denoted as $$F(t, \ \gamma, \ c)= (1+ \exp \{ - \gamma (t-c) \})^{-1}-\frac{1}{2},$$ where $\gamma$ is a parameter of smoothness and $c$ is a threshold parameter. We assume $\gamma>0$ and $c>0$. $F(t, \ \gamma, \ c)$ is bounded between $-1/2$ and $1/2$. $F(t, \ \gamma, \ c)$ moves toward $-1/2$ when $t < c$ and small $\gamma (t-c)$, and moves toward $1/2$ when $t > c$ and large $\gamma (t-c)$. $F(t, \ \gamma, \ c)$ takes 0 for $t=c$. $F(t, \ \gamma, \ c)$ with $\gamma=\infty$ equals a structural break model, because $F(t, \ \gamma, \ c)$ with $\gamma=\infty$ becomes the indicator function that takes the value -1/2 or 1/2. For example, Figures 1 and 2 demonstrate the values of $F(t, \ \gamma, \ c)$. While Figure 1 depicts the graph of $F(t, \ \gamma, \ c)$ with $T=200$, $\gamma=(0.01,0.1)$, and $c=T/2$, Figure 2 depicts it with $T=1,000$, $\gamma=(0.01,0.1)$, and $c=T/2$. Figure 1 shows that $F(t, \ \gamma, \ c)$ with $\gamma=0.01$ is almost linear, whereas $F(t, \ \gamma, \ c)$ with $\gamma=0.1$ has a smooth change around $T=100.$ For Figure 2, $F(t, \ \gamma, \ c)$ with $\gamma=0.01$ has a smooth change around $T=500$. By contrast, $F(t, \ \gamma, \ c)$ with $\gamma=0.1$ has a rapid chagnge around $T=500.$ The time-varying properties of (1) depend on the values of $t$, $\gamma$, and $c$.
The null and alternative hypotheses to test for time-varying mean in (1) are denoted as follows $$H_0: \gamma=0, \ \ \ \ \ H_1: \gamma>0.$$ (1) becomes a simple autoregressive (AR) model under $H_0$ and has a time-varying constant and slope under $H_1$ with $\alpha_1$ and $\beta_1$. Note that we cannot test for $\gamma=0$ directly, because the null hypothesis has an identification problem about $\alpha_1$ and $\beta_1$, which are identified only under the alternative hypothesis with $\gamma >0$. Davies (1977, 1987) discussed the identification problem whereby a nuisance parameter is present only under the alternative hypothesis. A solution for the identification problem is to use the Taylor series approximation introduced by Luukkonen, Saikkonen, and Teräsvirta (1988a).
When we use a Taylor series approximation around $\gamma=0$, (1) with $F(t, \gamma, c)$ is given by $$y_t=\phi_0 + \phi_1 y_{t-1} + \phi_2 t + \phi_3 t y_{t-1} + e_{t},$$ where $e_t$ is an error term for the new regression model. We can formulate the null and alternative hypotheses to test for time-varying mean in (4) as follows. $$H_0: \phi_2=\phi_3=0, \hspace{5ex} H_1: H_0 \hspace{1ex} \mbox{is not true}.$$ We rewrite (4) as $$y_t=\Phi^{\prime} Y_t +e_t,$$ where $\Phi=(\phi_0, \phi_1, \phi_2, \phi_3)^{\prime}$ and $Y_t=(1, y_{t-1}, t, t y_{t-1})^{\prime}$. The usual test for (5) uses the following Wald statisic. $$M_a=\frac{1}{\hat{\sigma}^2}\hat{\Phi}_2^{\prime} \Bigg [R \bigg (\sum_{t=1}^T Y_t Y_t^{\prime} \bigg)^{-1} R^{\prime} \Bigg]^{-1} \hat{\Phi_2},$$ where $\hat{\Phi_2}$ is the estimate of $\Phi_2=(\phi_2, \phi_3)^{\prime}$, $\hat{\sigma}^2$ is the estimate of the residual variance obrained from (4), and $R$ is the matrix such that $R\hat{\Phi}=\hat{\Phi}_2$. $M_a$ has an $F$ distribution with $(2, T-4)$ degrees of freedom under the null hypothesis.
Standard asymptotic tests for linearity often cause spurious nonlinearity when errors have heteroskedastic variance, including ARCH, generalized ARCH (GARCH), stochastic volatility, and structural breaks. A method to relieve the influence is to use heteroskedasticity consistent covariance matrix estimator (HCCME) introduced by White (1980). This is a popular method to test for linearity in the presence of heteroskedasticity, since test statistics based on the HCCME asymptotically have the same distribution as the original test statistics under the null hypothesis. However, linearity tests using HCCME do not perform well under the null hypothesis of linearity with ARCH and GARCH type errors particularly for finite samples. HCCME cannot improve the influence of heteroskedastic variance on inference of the conditional mean sufficiently. As pointed out by Pavlidis, Paya, and Peel (2013), the presence of ARCH errors complicate tests for lineality because ARCH errors are like nonlinear processes for conditional mean. Some studies, including Pavlidis, Paya, and Peel (2010) and Maki (2014), demonstrated this problem. An better method is the wild bootstrap proposed by Liu (1988). This can resample data with unknown heteroskedastic variance and yeild better statistical performance than the asymptotic and the HCCME tests can. The wild bootstrap test for time-varying mean takes the following procedure.\
Step 1. Estimate regression model (4) and compute test statistic (6).\
Step 2. Estimate the regression model under the null hypothesis with $\phi_2=\phi_3=0$ and obtain the parameter estimates $\hat{\phi}_0$ and $\hat{\phi}_1$ and residuals denoted as $\hat{e}_{0t}$.\
Step 3. Generate the bootstrapped sample as follows: $$y_t^{\ast}=\hat{\phi}_0+\hat{\phi}_1 y_{t-1} + e_{t}^{\ast},$$ where $e_{t}^{\ast}=\epsilon_{t} \hat{e}_{0t}$. We set $\epsilon_{t}$ to independent and identically distributed $N(0,1)$$^2$.\
Step 4. Estimate regression model (4) using the generated bootstrap sample (7) and compute test statistic (6) denoted as $M_{wb}$.\
Step 5. Repeat steps 3 and 4 many times.\
Step 6. Compute the bootstrap $p$-value as follows: $$P(M_{wb})=\frac{1}{M}\sum_{m=1}^M I(M_{wb} > M_a),$$ where $M$ is the number of bootstrap iterations and $I(\cdot)$ is an indicator function such that $I(\cdot)$ is 1 if $(\cdot)$ is true and 0 otherwise. Usually, more than 1,000 times are enough for the number of bootstrap iterations. Andrews and Buchinsky (2000) and Davidson and MacKinnon (2000) discussed the problem of the number of bootstrap iterations. The null hypothesis is rejected if (8) is smaller than a significant level.
We next explain the test for time-varying smooth transition variance. Consider the following data generating process (DGP). $$\begin{aligned}
y_t=\alpha_0 + \beta_0 y_{t-1} + u_{t}, \\
u_t=h_t \epsilon_t, \\
h_t^2=a_0+b_0 u_{t-1}^2+ (a_1+b_1 u_{t-1}^2)F(t, \ \gamma, \ c), \\
F(t, \ \gamma, \ c)= (1+ \exp \{ - \gamma (t-c) \})^{-1}-\frac{1}{2}. \end{aligned}$$ (9) has a linear AR model for mean and time-varying smooth transition ARCH process (11) for variance. While Hagerud (1996) proposed the smooth transition GARCH model with the transition variable $u_{t-1}$, (12) has the transition variable $t$. The transition function for time-varying ARCH is similar to that of time-varying mean$^{3}$. Since (11) also has an identification problem, we use a Taylor series approximation around $\gamma=0$ in order to test for time-varying smooth transition ARCH. (11) is rewritten as $$h_t^2=\rho_0+\rho_1 u_{t-1}^2+ \rho_2 t + \rho_3 t u_{t-1}^2 +\upsilon_t,$$ where $\upsilon_t$ is an error term, including white noise and the remainder term from the Taylor approximation. The null hypothesis and alternative hypothesis of test for time-varying ARCH are given by $$H_0: \rho_1=\rho_2=\rho_3=0, \ \ \ \ \ H_1: H_0 \ \mbox{is not true}.$$ (14) nests ARCH test$^{4}$. In order to test for (14), we first estimate (9) and obtain residuals. The test statistic is $$V_a=\frac{(SSR(h)_0-SSR(h)_1)/3}{SSR(h)_1/(T-4)},$$ where $SSR(h)_0$ is the sum of the squared residuals obtained from the estimation of (13) with $\rho_1=\rho_2=\rho_3=0$ and $SSR(h)_1$ is the sum of the squared residuals obtained from the estimation of (13). $V_a$ has an $F$ distribution with $(3, T-4)$ degrees of freedom under the null hypothesis$^{5}$.
Gel and Chen (2012) introduced a new bootstrap test for ARCH to improve the size and power of the asymptotic test. We use their approach to test for time-varying ARCH. The procedure is denoted as follows.\
Step 1. Estimate regression model (13) and compute test statistic (15).\
Step 2. Estimate the regression model under the null hypothesis with $\rho_1=\rho_2=\rho_3=0$ and obtain the parameter estimates $\hat{\rho}_0$ and residuals denoted as $\hat{\upsilon}_{0t}$.\
Step 3. Generate the bootstrapped sample as follows: $$h_t^{\ast}=\hat{\rho}_0+ \upsilon_{t}^{\ast},$$ where $\upsilon_{t}^{\ast}$ is randomly selected from $(\hat{\upsilon}_{01} \cdots \hat{\upsilon}_{0T})$.\
Step 4. Estimate regression model (13) using the generated bootstrap sample (16) and compute test statistic (15) denoted as $V_{b}$.\
Step 5. Repeat steps 3 and 4 many times.\
Step 6. Compute the bootstrap $p$-value as follows: $$P(V_{b})=\frac{1}{M}\sum_{m=1}^M I(V_{b} > V_a).$$ The null hypothesis is rejected if the $p$-value (17) is smaller than a significant level. We also use the wild bootstrap test for (14). This is almost smilar to the bootstrap test shown above. The main difference is Step 3. The wild bootstrap approach replace $\upsilon_{t}^{\ast}$ as $\tilde{\upsilon}_{t}=\hat{\upsilon}_{0t} \eta_t$, where $\eta_t \sim N(0,1)$. We denote the wild bootstrap test for time-varying ARCH as $V_{wb}$. The $p$-value is given by $$P(V_{wb})=\frac{1}{M}\sum_{m=1}^M I(V_{wb} > V_a).$$
Statistical properties of time-varying tests
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This section examines the statistical properties of the time-varying tests reviewed in section 2. We conduct Monte Carlo simulations to compare the size and power of the test statistics. The simulations are based on 10,000 replications, the nominal level at 5%, and sample sizes with $T=100$, $200$, $400$, and $1000$. We show only nominal power properties and do not present size-corrected power because the purpose of the paper is to compare the tests and usual applied practitioners do not use size-corrected tests. Bootstrap tests have 1,000 replications. In order to avoid the effect of initial conditions, data with $T+100$ are generated. The initial 100 samples are discarded and we use the data with sample size $T$. Tests compared in this section are denoted as $M_a$, $M_{wb}$, $V_a$, $V_{b}$, and $V_{wb}$.
First, as a benchmark, autoregressive processes with homoskedastic error are generated. $$y_t=\alpha_0 + \beta_0 y_{t-1} + u_{t}, \\$$ where $u_{t} \sim \mbox{i.i.d.}N(0,1)$. $\alpha_0$ is set to $\alpha_0=1$. The persistent parameter $\beta_0$ is set to $\beta_0=0.3$ and $0.9$ in order to investigate the influence of persistence of the process on time-varying tests. Since (19) has no time-varying properties in mean and variance, the rejection frequencies of each test demonstrate the empirical size for (19). Table 1 presents the rejection frequencies of each test. When the persistent parameter is $\beta_0=0.3$, $V_a$ has slight under-rejection for $T=100$, $200$, and $400$. Other tests have the empirical size near to 5% nominal level and reasonable size properties regardless of sample size. Although time-varying tests for mean, $M_a$ and $M_{wb}$, have over-rejection in small samples with $\beta_0=0.9$, the over-rejection decreases with the increased sample size. Unlike time-varying tests for mean, time-varying tests for variance do not have size distortions, except for $V_a$ with $T=100$, which has slight under-rejection.
Next, we consider the following DGP to examine the power properties of $M_a$ and $M_{wb}$ and the size properties of $V_a$, $V_b$, and $V_{wb}$ under the following time-varying mean model. $$\begin{aligned}
y_t=1 + 0.3 y_{t-1} + (\alpha_1 + \beta_1 y_{t-1}) F (\cdot)+ u_{t}, \\
F(\cdot)= (1+ \exp \{ - \gamma (t-c) \})^{-1}-\frac{1}{2}. \end{aligned}$$ Parameters for the time-varying mean are set to $(\alpha_1,\beta_1)=(0,0.3),$ $(0,0.6),$ $(0.5,0.3),$ and $(1,0.3)$, respectively. $F(\cdot)$ moves from $-1/2$ to $1/2$. For $(\alpha_1,\beta_1)=(1,0.3)$, (20) has time-varying properties from $y_t=0.5 + 0.15 y_{t-1} + u_{t}$ to $y_t=1.5 + 0.45 y_{t-1} + u_{t}$. We examine the effect of the magnitude of change for constant and AR parameters on performance of the tests. Smoothness parameter $\gamma$ in $F(\cdot)$ is set to $\gamma=(0.01,0.1)$. $\gamma=0.01$ means more smooth and slight change in $F(\cdot)$ than $\gamma=0.1$. In addition, the threshold parameter $c$ is set to $c=T/2^{6}$. The simulation results are shown in Table 2. The powers of $M_a$ and $M_{wb}$ are low for parameter sets $(\alpha_1,\beta_1,\gamma)=(0,0.3,0.01)$ with $T=100$ and $200$. They increase rapidly with the large sample size. We can observe that the magnitude of $\alpha_1$ and $\beta_1$ affects the powers of $M_a$ and $M_{wb}$. For example, the powers of $M_a$ and $M_{wb}$ for $(\alpha_1,\beta_1)=(0,0.3)$ with $T=200$ are 0.134 and 0.132, respectively, whereas those for $(\alpha_1,\beta_1)=(1,0.3)$ with $T=200$ are 0.694 and 0.703, respectively. When the smoothness parameter is $\gamma=0.1$, $M_a$ and $M_{wb}$ have sufficient power. $\gamma$ has clear impact on the performance of $M_a$ and $M_{wb}$. Generally, we observe that $M_a$ and $M_{wb}$ have ability to find time-varying mean properties in the presence of homoskedastic variance.
For $V_a$, $V_b$, and $V_{wb}$, the results in Table 2 show the empirical size properties of the tests, because (20) has homoskedastic variance. $V_a$ performs well in small samples. It has over-rejection for $\gamma=0.01$ and $T=1,000$. In particular, $V_a$ has a rejection frequency 0.325 and large size distortions when $(\alpha_1, \beta_1)=(0, 0.6)$ and $T=1,000.$ We find that the increase in the time-varying AR parameter causes high rejection frequencies for the time-varying variance test under the null hypothesis of homoskecastic variance. Although the change of a constant of (20) also increases the rejection frecencies, the effect is not larger than the increase in the AR parameter. The rejection frecencies of $V_a$ for $T=1,000$ with $(\alpha_1,\beta_1)=(0,0.3)$ and $(\alpha_1,\alpha_1)=(1,0.3)$ are 0.068 and 0.119, respectively. Similar properties are observed for $V_b$. The results imply that the bootstrap time-varying ARCH test does not perform well under the null hypothesis of homoskedastic variance with a misspecified conditional mean. By contrast, the size properties of $V_{wb}$ outperform those of $V_a$ and $V_b$. $V_{wb}$ does not have over-rejection even for $T=1,000$. However, $V_{wb}$ has size distortions for $\gamma=0.1$. Actually, the rejection frequencies of $V_{wb}$ for $T=1,000$ with $(\alpha_1,\beta_1)=(0,0.6)$ and $(\alpha_1,\alpha_1)=(1,0.3)$ are 0.253 and 0.113, respectively. $V_{wb}$ does not improve the size properties when the DGP has time-varying mean with homoskedastic error. $V_a$ and $V_b$ for $\gamma=0.1$ have larger size distortions than those for $\gamma=0.01$.
When we test for time-varying ARCH in the presence of a misspecified conditional mean, the tests cannot lead to reliable results, particularly for a large change in time-varying parameters with a large smoothness parameter $\gamma.$ As shown by Balke and Kapetanios (2007), neglected nonlinearity in the conditional mean cause spurious heteroskedasticity because the conditional variance includes extra nonlinearity and lags. This is more clear when $T$ increases$^{7}$. If the mean is misspeficied, the residual includes nonlinearity of the mean. Large $b_1$ and sample size lead to stronger nonlinearity. This is the reason for poor performance of the time-varying variance test. Therefore, time-varying ARCH tests in the presence of a misspecified conditional mean have size distortions.
Table 3 reports rejection frequencies of the tests for the AR process with ARCH error. The DGP is the following. $$\begin{aligned}
y_t=1 + 0.3 y_{t-1} + u_{t}, \\
u_t =h_t \epsilon_t, \\
h_t^2=a_0+b_0 u_{t-1}^2 \end{aligned}$$ We set a constant parameter for ARCH to $a_0=1$. Persistence parameter $b_0$ for ARCH is set to $b_0=0.3$, $0.6$, and $0.9$ in order to examine the effect of persistence of variance on the performance of the tests. $b_0<1$ is the necessary and sufficient condition for a weak stationarity of a semi-strong process. For (22), (23), and (24), the rejection frequencies of $M_a$ and $M_{wb}$ show the empirical size, because the process does not have time-varying mean. The performances of $V_a$, $V_b$, and $V_{wb}$ show frequencies of finding ARCH properties. $M_a$ overrejects the null hypothesis at 5% significance and its over-rejection increases when the persistence parameter and/or sample size increases. $M_a$ is sensitive to persistence in the presence of the ARCH effect. $M_{wb}$ has slight over-rejection only for $b_0=0.9$ with large samples. However, $M_{wb}$ clearly outperforms $M_a$ and has reasonable and acceptable rejection frequencies in most cases. This is due to the property that the wild bootstrap can deal with heteroskedasticity of unknown form when the conditional mean is correctly specified. We observe the difference of power properties among $V_a$, $V_b$, and $V_{wb}$. $V_a$ and $V_b$ have sufficient power to find the ARCH effect. Their abilities to detect the ARCH effect increases for large $b_0$ and $T$. The ability of $V_{wb}$ is lower than that of $V_a$ and $V_b$. For example, the powers of $V_a$, $V_b$, and $V_{wb}$ for $b_1=0.3$ and $T=400$ are 0.937, 0.937, and 0.260, respectively. While the size properties of $V_{wb}$ are superior to $V_{a}$ and $V_{b}$ under a misspecified conditional mean reported in Table 2, the powers of $V_{wb}$ are clearly inferior to $V_{a}$ and $V_{b}$. The higher powers of $V_{a}$ and $V_{b}$ are due to size distortions presented in Table 2. The comparison indicates that the use of these tests is not effective in the viewpoint of size and power because the size corrected tests are needed.
We finally investigate the statistical properties of the tests for the AR process with time-varying ARCH error. The DGP is given by $$\begin{aligned}
y_t=1 + 0.3 y_{t-1} + u_{t},\\
u_t = h_t \epsilon_t, \\
h_t^2=1+0.3 u_{t-1}^2 + (a_1 + b_1 u_{t-1}^2) F (\cdot), \\
F(t, \ \gamma, \ c)= (1+ \exp \{ - \gamma (t-\frac{T}{2}) \})^{-1}-\frac{1}{2}. \end{aligned}$$ Time-varying parameters $(a_1,b_1)$ are set to $(a_1,b_1)=(0,0.3),$ $(0,0.6),$ $(0.5,0.3),$ and $(1,0.3)$, respectively. We consider DGP with $\gamma=0.01$ and $0.1.$ These settings are similar to those in Table 2, in which the DGP has time-varying mean. The results are presented in Table 4. $M_{wb}$ does not have over-rejection for all cases and is close to the nominal size at 5%. This means that $M_{wb}$ is a reliable test for time-varying mean and does not lead to spurious time-varying mean. Although $M_a$ has size distortions under time-varying ARCH error, the rejection frequencies do not depend on the size of parameters $a_1$, $b_1$, and $\gamma$ too much. The rejection frequencies mainly depend on sample size and slowly increase when the sample size increases. The results are similar to those of Zhou (2013) and Boldea, Cornea-Madeira, and Hall (2019). They show that the wild bootstrap is asymptotically valid for a change in mean or regression coefficients in the presence of conditional and unconditional heteroskedasticity$^{8}$. The wild bootstrap can replicate unknown heteroskedasticity of the errors. The property brings valid tests for linearity of the mean.
The powers of $V_b$ are a little better than those of $V_a$. These tests have sufficient power to find time-varying ARCH effects. Furthermore, the increase in the smoothness parameter $\gamma$ brings higher power of time-varying ARCH tests. In particular, the effect is clear for $(a_1,b_1)=(1,0.3)$. When the sample size is $T=100$, the powers of $V_a$ and $V_b$ are 0.387 and 0.429 for $\gamma=0.01$, respectively, and 0.774 and 0.813 for $\gamma=0.1$, respectively. Note that the results of $V_{wb}$ are clearly different from those of $V_a$ and $V_b$. The powers of $V_{wb}$ are lower than those of $V_a$ and $V_b$ in all cases. For example, when the process has $(a_1,b_1)=(0.5,0.3)$ and $T=200$, the powers of $V_a$, $V_b$, and $V_{wb}$ are 0.724, 0.744, and 0.154, respectively. $V_a$ and $V_b$ cause large over-rejections under the null hypothesis with time-varying mean, whereas they have higher power under the alternative hypothesis. By contrast, $V_{wb}$ has better size properties than $V_a$ and $V_b$ do, whereas $V_{wb}$ has clearly lower power than $V_a$ and $V_b$ do. The Monte Carlo simulation results provide evidence that $M_{wb}$ tests yeild reliable results than do $M_a$ under ARCH and time-varying ARCH, and time-varying ARCH tests do not perform well in the presence of time-varying mean.
Summary and conclusion
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This study examined the statistical performance of time-varying tests under misspecified conditional mean and variance. Although time-varying properties are frequently observed in various economic and financial data, it is difficult to know which time-varying property for mean or variance is present a priori. Researchers may employ misspecified conditional models and obtain unreliable results. Therefore, it is important to clarify the statistical properties of time-varying tests in misspecified conditional mean or variance models. Monte Carlo simulation results reveal that asymptotic tests for time-varying mean have size distortions when the variance model is misspecified, whereas the wild bootstrap method improves the size distortions. Wild bootstrap tests for time-varying mean are robust regardless of the misspecified variance, and can lead to reliable results of tests for time-varying mean. However, bootstrap tests in addition to asymptotic tests for time-varying variance have size distortions in the presence of the misspecified conditional mean. Tests for time-varying variance do not perform well and not provide reliable results in the presence of misspecified conditional mean. Robust time-varying variance tests in the presence of the misspecified conditional mean remain for further study.
[Footnotes]{}\
1\. Although (1) can take a general model with $p$ lags, we consider only a single-lag model to simplify the investigation in this study. The same is true of variance.\
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2. Davidson and Flachaire (2008) discussed the method of the wild bootstrap, including other distributions for $\epsilon_{t}$.\
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3. (12) includes conditional and unconditional heteroskedasticity. While conditional heteroskedasticity has $b_0>0$ and/or $b_1>0$ with $F(\cdot) \neq 0$, unconditional heteroskedasticity has $a_0>0$ and/or $a_1>0$ with $F(\cdot) \neq 0$ in addition to $b_0=b_1=0$.\
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4. It is possible to test for the null hypothesis of $\rho_2=\rho_3=0$. This means that in the null hypothesis, there is an ARCH error and in the alternative hypothesis, there is a time-varying ARCH error. However, it is difficult to know that the variance process has ARCH properties a priori. When we have the null hypothesis $\rho_2=\rho_3=0$, we have to test for the hypothesis whether variance has ARCH error before testing for $\rho_2=\rho_3=0$. This takes a two-step approach and cannot test for time-varying ARCH directly. This may make comparisons among the tests difficult and lead to misleading results. Therefore, we adopt hypothesese in (14) to avoid these problems and simply test for time-varying ARCH.\
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5. The test statistic is also expressed by $T R^2$, where $R^2$ is the coefficient of the determinantion of (13). $T R^2$ asymptotically has $\chi^2(3)$ distribution. See Gel and Chen (2012)\
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6. We conduct Monte Carlo experiments under various other situations. For example, while tests have larger rejection frequencies when $c$ is smaller than $T/2$, they have smaller rejection frequencies when $c$ is larger than $T/2$. However, differences among the tests are similar to those of $c=T/2$. The results for other parameter sets $(\alpha_1,\beta_1, \gamma, \mbox{and} \ c)$ are available from the author on request.\
\
7. See Luukkonen et al. (1988b) for similar results.\
\
8. Zhang and Wu (2019) proposes a nonparametric test for a change in regression coefficients in the presence of time-varying variance.
[99]{}
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Testing for nonlinearity in mean in the presence of heteroskedasticity, *Economic Analysis & Policy* 39, 313-326.
Bootstrapping structural change tests, *Journal of Econometrics* (in press)
Combined Lagrange multiplier test for ARCH in vector autoregressive models, *Econometrics and Statistics* 1, 62-84.
Statistical inference for time-varying ARCH processes, *The Annals of Statistics* 34, 1075-1114.
Bootstrap tests: how many bootstraps? *Econometric Reviews* 19, 55-68.
The wild bootstrap, tamed at last, *Journal of Econometrics* 146, 162-169.
Hypothesis testing when a nuisance parameter is present only under the alternative, *Biometrika* 64, 247-254.
Hypothesis testing when a nuisance parameter is present only under the alternative, *Biometrika* 74, 33-43.
Robust Lagrange multiplier test for detecting ARCH/GARCH effect using permutation and bootstrap, *The Canadian Journal of Statistics* 40, 405-426.
A new non-linear GARCH model, EFI Economic Research Institute, Stockholm School of Economics.
Capital asset pricing model: A time-varying volatility approach, *Journal of Empirical Finance* 37, 268-281.
Testing the constancy of regression parameters against continuous structural change, *Journal of Econometrics* 62, 211-228.
Bootstrap procedure under some non-i.i.d. models, *Annals of Statistics* 16, 1696-1708.
Testing for ARCH in the presence of a possibly misspecified conditional mean, *Journal of Econometrics* 93, 257-279.
Testing linearity against smooth transition autoregressive models, *Biometrika* 70, 491-499.
Testing linearity in univariate time series models, *Scandinavian Journal of Statistics* 15, 161-175.
A comparison of linearity tests based on wild bootstrap, *Advances and Applications in Statistics* 4(3), 93-107.
Specifying smooth transition regression models in the presence of conditional heteroskedasticity of unknown form, *Studies in Nonlinear Dynamics and Econometrics* 14(3), 1-40.
Nonlinear causality tests and multivariate conditional heteroskedasticity: a simulation study, *Studies in Nonlinear Dynamics and Econometrics* 17(3), 297-312.
Pitfalls of two-step testing for changes in the error variance and coefficients of a linear regression model, *Econometrics* 7(2), 22, 1-11.
Least squares estimation and tests of breaks in mean and variance under misspecification, *Econometrics Journal* 7, 32-54.
Testing for ARCH in the presence of additive outliers, *Journal of Applied Econometrics* 14, 539-562.
Testing for smooth transition nonlinearity in the presence of outliers, *Journal of Business & Economic Statistics* 17, 217-235.
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A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity, *Econometrica* 14, 1261-1295.
Testing for structural changes in linear regressions with time-varying variance, *Communications in Statistics-Theory and Methods* (in press)
Heteroscedasticity and autocorrelation robust structural change detection, *Journal of the American Statistical Association* 108, 726-740.
Table 1: Rejection frequencies under AR with homoskedastic error
$M_a$ $M_{wb}$ $V_a$ $V_b$ $V_{wb}$
--------------- ------- ---------- ------- ------- ----------
$\beta_0=0.3$
$T=100$ 0.049 0.051 0.034 0.046 0.051
$T=200$ 0.048 0.049 0.041 0.045 0.054
$T=400$ 0.053 0.054 0.040 0.050 0.051
$T=1000$ 0.046 0.054 0.046 0.050 0.046
$\beta_0=0.9$
$T=100$ 0.106 0.104 0.035 0.047 0.048
$T=200$ 0.080 0.074 0.043 0.049 0.044
$T=400$ 0.054 0.061 0.045 0.049 0.051
$T=1000$ 0.056 0.057 0.045 0.050 0.045
Table 2: Rejection frequencies under time-varying AR with homoskedastic error
----------------------------- ------- ---------- ------- ------- ---------- ------- ---------- ------- ------- ----------
$M_a$ $M_{wb}$ $V_a$ $V_b$ $V_{wb}$ $M_a$ $M_{wb}$ $V_a$ $V_b$ $V_{wb}$
$\alpha_1=0, \beta_1=0.3$
$T=100$ 0.056 0.061 0.036 0.043 0.048 0.380 0.382 0.036 0.046 0.042
$T=200$ 0.134 0.132 0.040 0.048 0.049 0.789 0.794 0.036 0.051 0.035
$T=400$ 0.625 0.630 0.049 0.053 0.041 0.986 0.988 0.042 0.067 0.035
$T=1000$ 1 0.999 0.068 0.072 0.031 1 1 0.119 0.086 0.036
$\alpha_1=0, \beta_1=0.6$
$T=100$ 0.091 0.090 0.035 0.043 0.048 0.943 0.941 0.055 0.071 0.029
$T=200$ 0.428 0.424 0.041 0.053 0.045 1 0.999 0.126 0.133 0.040
$T=400$ 0.996 0.997 0.064 0.076 0.027 1 1 0.286 0.296 0.083
$T=1000$ 1 1 0.325 0.332 0.057 1 1 0.704 0.703 0.253
$\alpha_1=0.5, \beta_1=0.3$
$T=100$ 0.085 0.091 0.036 0.042 0.044 0.880 0.875 0.038 0.046 0.034
$T=200$ 0.357 0.365 0.040 0.047 0.048 0.998 0.999 0.049 0.054 0.045
$T=400$ 0.990 0.992 0.046 0.055 0.038 1 1 0.068 0.077 0.044
$T=1000$ 1 1 0.074 0.078 0.035 1 1 0.139 0.144 0.070
$\alpha_1=1, \beta_1=0.3$
$T=100$ 0.121 0.129 0.036 0.042 0.042 0.994 0.994 0.040 0.052 0.031
$T=200$ 0.694 0.703 0.036 0.043 0.044 1 1 0.070 0.080 0.044
$T=400$ 1 1 0.042 0.052 0.034 1 1 0.127 0.132 0.058
$T=1000$ 1 1 0.119 0.125 0.040 1 1 0.300 0.311 0.113
----------------------------- ------- ---------- ------- ------- ---------- ------- ---------- ------- ------- ----------
Table 3: Rejection frequencies under AR with ARCH error
$M_a$ $M_{wb}$ $V_a$ $V_b$ $V_{wb}$
----------- ------- ---------- ------- ------- ----------
$b_0=0.3$
$T=100$ 0.079 0.049 0.372 0.407 0.065
$T=200$ 0.086 0.052 0.681 0.705 0.120
$T=400$ 0.098 0.052 0.937 0.937 0.260
$T=1000$ 0.100 0.053 0.999 0.999 0.558
$b_0=0.6$
$T=100$ 0.107 0.049 0.686 0.719 0.184
$T=200$ 0.144 0.057 0.946 0.956 0.323
$T=400$ 0.191 0.051 0.998 0.999 0.498
$T=1000$ 0.231 0.055 1 1 0.674
$b_0=0.9$
$T=100$ 0.129 0.055 0.802 0.841 0.253
$T=200$ 0.201 0.057 0.976 0.985 0.366
$T=400$ 0.282 0.062 0.998 0.999 0.474
$T=1000$ 0.401 0.060 0.999 1 0.561
Table 4: Rejection frequencies under AR with time-varying ARCH error
-------------------- ------- ---------- ------- ------- ---------- ------- ---------- ------- ------- ----------
$M_a$ $M_{wb}$ $V_a$ $V_b$ $V_{wb}$ $M_a$ $M_{wb}$ $V_a$ $V_b$ $V_{wb}$
$a_1=0, b_1=0.3$
$T=100$ 0.079 0.049 0.381 0.407 0.071 0.072 0.049 0.418 0.454 0.098
$T=200$ 0.091 0.055 0.692 0.706 0.128 0.088 0.053 0.738 0.756 0.175
$T=400$ 0.101 0.051 0.946 0.950 0.276 0.105 0.054 0.962 0.963 0.329
$T=1000$ 0.116 0.056 0.999 0.999 0.576 0.117 0.050 0.999 1 0.597
$a_1=0, b_1=0.6$
$T=100$ 0.082 0.051 0.384 0.414 0.069 0.079 0.059 0.521 0.560 0.153
$T=200$ 0.091 0.055 0.712 0.733 0.144 0.095 0.059 0.863 0.875 0.298
$T=400$ 0.109 0.053 0.962 0.970 0.333 0.115 0.056 0.991 0.991 0.458
$T=1000$ 0.151 0.054 1 1 0.627 0.146 0.061 1 1 0.645
$a_1=0.5, b_1=0.3$
$T=100$ 0.077 0.054 0.381 0.414 0.071 0.072 0.055 0.563 0.600 0.202
$T=200$ 0.086 0.054 0.724 0.744 0.154 0.083 0.052 0.910 0.922 0.384
$T=400$ 0.102 0.056 0.976 0.981 0.385 0.096 0.050 0.998 0.998 0.570
$T=1000$ 0.125 0.055 1 1 0.725 0.118 0.052 1 1 0.746
$a_1=1, b_1=0.3$
$T=100$ 0.073 0.054 0.387 0.429 0.082 0.074 0.053 0.774 0.813 0.410
$T=200$ 0.091 0.055 0.752 0.790 0.196 0.081 0.057 0.992 0.995 0.595
$T=400$ 0.101 0.054 0.989 0.996 0.539 0.090 0.055 1 1 0.723
$T=1000$ 0.118 0.052 1 1 0.813 0.117 0.053 1 1 0.816
-------------------- ------- ---------- ------- ------- ---------- ------- ---------- ------- ------- ----------
{width="13cm" height="7cm"}\
{width="13cm" height="7cm"}\
Solid line has $\gamma=0.01$. Dashed line has $\gamma=0.1$.
[^1]: This research was supported by KAKENHI (Grant number: 16K03604). Address: Faculty of Commerce, Doshisha University, Karasuma-higashi-iru, Imadegawa-dori, Kamigyo-ku, Kyoto Japan 602-8580 (E-mail:[email protected])
[^2]: This research was supported by KAKENHI (Grant number: 18K03439). Faculty of Management, Okayama University of Science, 1-1 Ridaicyou, Okayama City, Okayama Japan 700-0005 (E-mail:[email protected])
|
---
abstract: '$\epsilon$ Ind A is one of the nearest sun-like stars, located only 3.6 pc away. It is known to host a binary brown dwarf companion, $\epsilon$ Ind Ba/Bb, at a large projected separation of 6.7$''$, but radial velocity measurements imply that an additional, yet unseen component is present in the system, much closer to $\epsilon$ Ind A. Previous direct imaging has excluded the presence of any stellar or high-mass brown dwarf companion at small separations, indicating that the unseen companion may be a low-mass brown dwarf or high-mass planet. We present the results of a deep high-contrast imaging search for the companion, using active angular differential imaging (aADI) at 4 $\mu$m, a particularly powerful technique for planet searches around nearby and relatively old stars. We also develop an additional PSF reference subtraction scheme based on locally optimized combination of images (LOCI) to further enhance the detection limits. No companion is seen in the images, although we are sensitive to significantly lower masses than previously achieved. Combining the imaging data with the known radial velocity trend, we constrain the properties of the companion to within approximately 5-20 $M_{\rm jup}$, 10-20 AU, and $i > 20^{\rm o}$, unless it is an exotic stellar remnant. The results also imply that the system is probably older than the frequently assumed age of $\sim$1 Gyr.'
author:
- |
M. Janson$^{1}$[^1], D. Apai$^{2}$, M. Zechmeister$^{3}$, W. Brandner$^{3}$, M. Kürster$^{3}$, M. Kasper$^{4}$, S. Reffert$^{5}$, M. Endl$^{6}$, D. Lafrenière$^{1}$, K. Gei[ß]{}ler$^{3}$, S. Hippler$^{3}$, Th. Henning$^{3}$\
\
$^{1}$Department of Astronomy, University of Toronto, 50 St George St, Toronto, M5S 3H4, Canada\
$^{2}$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA\
$^{3}$Max Planck Institute for Astronomy, Königstuhl 17, Heidelberg, D-69117, Germany\
$^{4}$ESO, Karl-Schwarzschild-Strasse 2, Garching bei München, D-85748, Germany\
$^{5}$Landessternwarte, Königstuhl 12, Heidelberg, D-69117, Germany\
$^{6}$McDonald Observatory, 1 University Station, Austin, TX 78712, USA
date: 'N/A'
title: 'Imaging search for the unseen companion to $\epsilon$ Ind A – Improving the detection limits with 4 $\mu$m observations [^2]'
---
\[firstpage\]
stars: low-mass, brown dwarfs – planetary systems
Introduction {#sec_intro}
============
Direct imaging of exoplanets is a field of research in rapid development. The past year has seen a number of interesting planet candidates imaged directly around stars. Most notable is arguably HR 8799, showing three planetary companions (see Marois et al. 2008) so far. These planets have been shown to exhibit Keplerian motion around the star, and have estimated masses in the range of 7-10 $M_{\rm jup}$ from theoretical models (Baraffe et a. 2003). The system is known to also host a debris disk (e.g. Moor et al. 2006; Rhee et al. 2007). Along with the fact that there are three almost equal-mass companions, all orbiting the star in what appears to be a co-planar manner, this implies that the objects most likely formed in a circumstellar disk, by a process distinct from the star or brown dwarf formation process. This can be contrasted with the case of the 2M1207 system (Chauvin et al. 2005), where the low primary-to-secondary mass ratio is more reminiscent of a brown dwarf binary system than a star-planet system. Two other intriguing planet candidates in systems with debris disks were reported around the same time as HR 8799 b/c/d: Fomalhaut b (Kalas et al. 2008) and Beta Pic b (Lagrange et al. 2009), though additional follow-up observations would be desirable to provide more information on these systems.
$\epsilon$ Ind A is a K4V-type southern sky star located at a distance of 3.6 pc, with a very high proper motion of 4.7$''$ yr$^{-1}$ (Perryman et al. 1997), see Table \[indtab0\]. Comoving on the sky at the same rate, at a separation of 6.7$'$, is $\epsilon$ Ind B, which was first detected by Scholz et al. (2003), and shortly thereafter resolved into the brown dwarf binary $\epsilon$ Ind Ba/Bb (McCaughrean et al. 2004) as it is known today. Being the most nearby binary brown dwarf, and with a physical separation small enough to determine its dynamical mass within a reasonable timeframe, $\epsilon$ Ind Ba/Bb will be a benchmark object for the physical understanding of brown dwarfs. However, in addition to the Ba/Bb components, the $\epsilon$ Ind system may provide an additional possibility to study an even lower-mass and cooler object. $\epsilon$ Ind A displays a linear radial velocity trend (Endl et al. 2002; Zechmeister et al., in prep.) which, unless due to some exotic stellar remnant, is indicative of a giant planet or very low-mass brown dwarf companion. If this object could be directly imaged, it would constitute yet another important benchmark object, given its probable low mass, low temperature, and the possibility to estimate both its luminosity as well as its dynamical mass within a reasonable timeframe. It would also be the closest planet or very low-mass brown dwarf companion directly detected outside of our Solar System.
[ll]{} Property & Value\
Right Ascension & 22 03 21.66\
Declination & -56 47 09.5\
Spectral Type & K4.5V\
Distance & $3.626 \pm 0.009$ pc\
Proper Motion & 4705 mas yr$^{-1}$\
Mass & 0.7 $M_{\rm sun}$\
Age & 1-5 Gyr\
Here, we will present our deep imaging campaign of $\epsilon$ Ind, using narrow-band 4$\mu$m imaging with two different high-contrast techniques. One technique is pure active angular differential imaging (aADI) as already implemented and demonstrated as a powerful technique for detecting close companions to bright nearby stars (Janson et al. 2008). In aADI (also known as roll subtraction), images are taken at two different instrument rotator angles, and one is subtracted from the other, thereby removing the bulk of the stellar PSF including static instrumental speckles. The other technique is a combination of PSF reference subtraction and aADI (PSFR+aADI), using the LOCI (locally optimized combination of images) algorithm developed by Lafreniere et al. (2007). We also compare the performance of aADI at 4$\mu$m with the same technique in the L’-band. The concept of using aADI with L’ was proven by Kasper et al. (2007; the usefulness of L’ for high-contrast imaging purposes was also independently demonstrated by Hinz et al. 2006). The concept of using NB4.05 to enhance the physical contrast was introduced in Janson et al. (2008).
The outline is as follows: In Sect. \[sec\_obs\], we summarize the observational parameters and ambient conditions of the observing runs. The two paths of data reduction employed are described in Sect. \[sec\_datared\]. This is followed by a presentation of the results and the associated analysis in Sect. \[sec\_results\], including a comparison between filters (Sect. \[sec\_filters\]), a comparison between techniques (Sect. \[sec\_methods\]), a discussion of what we can learn from the dynamical input in Sect. \[sec\_dynamics\], and the final detection limits and their interpretation in Sect. \[sec\_detlimits\]. Finally, we conclude in Sect. \[sec\_conclusions\].
Observations {#sec_obs}
============
The data presented here are based on two different sets of VLT/NACO observations of $\epsilon$ Ind. One set of observations consisted of deep imaging with aADI in the NB4.05 filter, executed in service mode and split into two equal observing blocks (henceforth observations A1 and A2), on 31 Oct 2008 and 2 Nov 2008. The other set of observations were taken in visitor mode on 3 Jul 2008, as part of a larger survey searching for planets around a volume-limited sample of nearby stars (Apai et al., in prep.). Those observations were less deep, but consisted of aADI imaging in both the NB4.05 and broad-band L’ filters (henceforth observations B1 and B2, respectively). For observations A1 and A2, the same strategy was used as in previous observations (see Janson et al. 2008): Jittering was applied to enable a good subtraction of the thermal background, and the aADI was performed at two different instrument rotator angles, using a differential angle of 33 degrees. For B1 and B2, a four-point large-throw dithering scheme was applied for the background subtraction purposes, and the differential angle used for aADI was 20 degrees. All observations were taken with the L27 objective, providing a pixel scale of 27 mas/pixel, and a field of view of 28$''$ by 28$"$. The weather conditions and observational parameters of each run are listed in Table \[indtab1\].
[lllll]{} & A1 & A2 & B1 & B2\
Date (2008) & 31 Oct & 2 Nov & 3 Jul & 3 Jul\
Filter & NB4.05 & NB4.05 & NB4.05 & L’\
Seeing$^a$ & 0.9$''$ & 0.7$''$ & 1.1$''$ & 1.0$''$\
Strehl$^b$ & 83% & 79% & 85% & 83%\
Humidity & 8% & 13% & 3% & 3%\
Coh. time$^a$ & 2.6 ms & 3.4 ms & 2.3 ms & 2.4 ms\
Frames & 19 & 19$^c$ & 30 & 33\
(per angle) & & & & \
DIT & 1.0 s & 1.0 s & 0.2 s & 0.2 s\
NDIT & 61 & 61 & 150 & 150\
Tot. time & 1159 s & 1159 s$^c$ & 900 s & 990 s\
(per angle) & & & & \
Values given by the atmospheric seeing monitor at a wavelength of 500 nm.
Strehl ratio given by the AO system, rescaled to the observing wavelengths.
Two frames were de-selected for 0$^{\rm o}$, hence the effective time is 1037 s for that case.
Data reduction {#sec_datared}
==============
The data reduction for A1 and A2 was done differently with respect to the two different techniques applied, hence separate descriptions are provided below. The reduction of B1 and B2 was only done with aADI.
Reduction for the purpose of aADI {#sec_aadired}
---------------------------------
Since observations A1 and A2 were taken in the same way as the observations of Janson et al. (2008), largely the same data reduction could be applied as in the case for aADI purposes: The most basic reduction steps (e.g. flat fielding, bad pixel removal, background subtraction) were provided by the ESO automatic pipeline. The subsequent steps were performed with our dedicated IDL pipeline: The images were shifted using bilinear interpolation to a common center determined through cross-correlation, and to an absolute center using center of gravity. Low-frequency filtering was applied by subtracting a smoothed counterpart of each image produced by convolution with a Gaussian kernel with 0.5$''$ FWHM (the FWHM of the stellar PSF was about 120 mas). All the images corresponding to different rotation angles were subtracted from each other, and the results from the two nights were coadded. As an alternative analysis, all images were also de-rotated back to a common angle and coadded. This procedure yields a single co-added signature of any companion, which provides a useful alternative way to look at the data in the background-limited regime, with respect to the aADI-subtracted data which instead produces two independent signatures of half the amplitude each.
For observations B1 and B2, the data reduction was performed with the IDL routines as described above, with the exception that the centering was determined based on a sub-frame of 200x200 pixels around the star instead of the whole frame. This was done in order to avoid influence from residual features from the sky subtraction and the different dithering scheme.
The residuals as function of separation from the star were calculated from the standard deviation of all pixels in an annulus corresponding to each separation step. The physical brightness contrast was derived from the 3$\sigma$ residuals by division of the peak value of the primary. Since the primary was saturated in the science images (the saturation radius was about 5-6 pixels), this had to be calibrated. For runs A1 and A2, this was done by introducing a neutral density filter into the optical path during acquisition, thus getting non-saturated images of the primary, allowing to determine a renormalized peak value. For run B1, the same non-saturated images as for A1 and A2 were used, which could be done since the Strehl ratio was stable and almost equal between the epochs. For B2, an image was taken of a fainter photometric standard star during the night.
Reduction for the purpose of PSFR+aADI {#sec_psfrred}
--------------------------------------
In applications involving LOCI, it is preferable to maximize the number of PSF representations of a target or reference star, hence for this case, reduction was done on individual frames, with the combination of frames only performed at the very end. The PSF star used was $\epsilon$ Eri, which is practically ideal for the purpose given the similar spectral type, brightness, and the fact that observations exist taken under almost identical circumstances. All individual target and reference star frames were manually subjected to flat fielding, dark subtraction and bad pixel correction using calibration frames provided by ESO. Low-frequency filtering was applied as described above. A master sky frame was then produced by taking the median of the frames, where the stellar image is randomly placed in each frame, thereby removing the star altogether. The individual frames revealed ring-like structures in the background that could be reproduced in the master sky frame. By subtracting the master sky frame from each individual frame, the pattern could be removed. The pattern was found to be constant during the extent of an observation, but variable between observations (e.g., the frames corresponding to $\epsilon$ Ind and $\epsilon$ Eri were different from each other), and is probably related to instrumental dust emitting at 4$\mu$m. The full background subtraction obtained in this way was found to be equally good as that delivered by the ESO ’jitter’ routine. It is interesting to note, that given the fact that the pattern is constant during an observation, it should be possible to calibrate it out of a generic observation by making a master sky observation directly before or after the target observation. Hence, for any observation dedicated to the detection of point sources, it should be possible to achieve the same degree of background subtraction with and without jittering. This is an important realization with respect to high-contrast imaging at these wavelengths using techniques such as passive ADI (pADI) or coronagraphy, where it is desirable to maintain the stellar primary at a fixed position on the detector, and simultaneously achieve the best possible background subtraction. In summary, there appears to be no conflict between these two requirements, as long as the master sky calibration step is performed during observations.
PSFR and aADI were performed separately, in sequence. For PSFR, every target and reference image was de-rotated such that the spider patterns were aligned. For each target frame, an optimized PSF reference frame was then produced from the full set of reference frames using the LOCI (Lafreniere et al. 2007) algorithm and subtracted from the target frame. The optimization was performed in 10 regions, five for the image range contaminated by the four spiders, covering different radial sections of the PSF, and five for the image range not contaminated by spiders, also covering different radial sections. The spider optimization areas were rectangular with a fixed width of 25 pixels, inner radii of 10, 20, 40, 70, and 120 pixels, and outer radii of 60, 70, 90, 150, and 200 pixels. The remaining areas were annuli excluding the spider regions, between inner radii of 20, 30 40, 50, and 60 pixels and outer radii of 50, 60, 70, 80, and 100 pixels. The subtractions were performed sequentially outwards with the subtraction zone defined from the inner radius of the optimization zone and outwards. Following this procedure, each of the target frames were re-rotated to their true parallactic angle. The aADI step was then performed through a second LOCI PSF construction, using all 33$^{\rm o}$ frames as PSF library for each 0$^{\rm o}$ frame, and vice versa. For this case, the optimization regions were simply five annuli between inner radii of 10, 20, 30, 50, and 70 pixels and outer radii of 40, 50, 60, 70 and 100 pixels. The optimization regions were chosen to provide a good balance between the two main criteria of LOCI: to maximize the efficiency of stellar PSF structure subtraction, and minimize subtraction of actual companions. The latter was tested by generating a series of runs where false companions had been introduced in the target frame – in total 3600 companions distributed between 10 and 100 pixels separation from the center of the star, and over all azimuthal angles. The partial subtractions in each case were used to construct a radial profile of conserved companion flux fractions. As expected, a significant flux loss occurs at 10 pixels, but decreases rapidly outwards. At 100 pixels, the fraction of restored companion flux approaches unity, as indeed expected, given that the LOCI optimization is not applied beyond 100 pixels for the vast majority of the image space.
Finally, all frames corresponding to each rotator angle were combined using 3$\sigma$-clipping. The radial profile of residuals was created in the same way as for aADI, but with the additional step that it was normalized by the radial profile of conserved companion flux fraction to provide an accurate measure of the actual achieved contrast.
Results and discussion {#sec_results}
======================
The output images from runs A1+A2 from each of the two reduction paths are shown in Fig. \[ei\_imgs\_mrg\]. No companion candidates were detected in the images. In the following, we discuss the implications of this result, and compare the methods used.
{width="15cm"}
Comparison between filters {#sec_filters}
--------------------------
Although the B1 and B2 images are less deep than A1+A2, the fact that they were obtained for the same target at about the same time, and with an almost identical observational setup, makes them ideal for comparing L’ and NB4.05 imaging for planet detection purposes around bright stars. A comparison was already made in Janson et al. (2008) between L’ aADI, NB4.05 aADI, and SDI+aADI (from Janson et al. 2007). While a fully relevant comparison could be made between SDI+aADI and NB4.05 aADI, where NB4.05 aADI was found to perform better under all circumstances, the comparison with L’ was preliminary, since no comparable data was available. Instead, the comparison between L’ and NB4.05 was based entirely on physical contrast given by the theoretical models, and the instrumental contrast was assumed to be the same. While this is relevant for a large part of the parameter space, there will in reality be differences in instrumental contrast due to differences in Strehl ratio, PSF diffraction, and thermal background between the filters. Using the B1 and B2 observations, we can now provide a comparison that takes all these issues into account.
The comparison was done by translating the brightness contrasts into mass detection limits using the spectral and photometric evolutionary models of Baraffe et al. (2003) and Burrows et al. (2003) for various ages. The method is described in detail in Janson et al. (2008). Note that the comparison is done for almost identical observing time, and with virtually no difference in overheads, i.e. the telescope time investment is also the same in both cases. As expected, the instrumental contrast is almost identical in the contrast-linited range, confirming the assumptions of the previous analysis, and thus the difference in the inner range is almost entirely set by the expected flux distribution of the companion. We show an example that demonstrates the favourable spectral range of NB4.05 in Fig. \[xmpl\_spec\_sm\], for 10$M_{\rm jup}$ and 15$M_{\rm jup}$ objects, at an age of 1 Gyr. The flux density is higher in NB4.05 than in both L’-band and M-band. For cooler objects, the bulk of the flux moves redward, hence M-band becomes better in terms of flux density, but the thermal background is also much worse in M-band. The improvement of NB4.05 over L’ increases further for cooler objects.
![Example of two model spectra from Burrows et al. (2003), and the corresponding flux densities in filters L’, NB4.05, and M. Upper lines: A 15$M_{\rm jup}$ object. Lower lines: A 10$M_{\rm jup}$ object. The age is 1 Gyr in both cases.[]{data-label="xmpl_spec_sm"}](mj_fig2.eps){width="8.0cm"}
We show the results of the detection limit comparison for 1 Gyr, 3 Gyr, and 5 Gyr in Fig. \[brg\_vs\_lband\]. It is seen that for all these ages, NB4.05 performs better in the contrast-limited inner part, and L’ performs better in the outer background-limited part, as expected. The crossover point for $\epsilon$ Ind A in our dataset is at about 4$''$. The position of the crossover point will vary as a function of stellar brightness and integration time. The brighter the star and the longer the integration time, the larger the parameter range where NB4.05 will be favourable, and vice versa. We conclude that NB4.05 is likely to be an excellent choice for very deep planet search imaging close to bright stars, although it should be noted that this depends on the validity of the theoretical models. A first test of the models could be provided by the HR 8799 system.
![Comparison between L’ (dashed line) and NB4.05 (solid line) imaging for high-contrast purposes, for 1 Gyr (top panel), 3 Gyr, (middle panel), and 5 Gyr (bottom panel). As expected, NB4.05 provides a better performance than L’ in the inner image range, and the opposite is true in the outer range. The comparison is based on sets B1 and B2, note that the A1+A2 detection limits are better.[]{data-label="brg_vs_lband"}](mj_fig3.eps){width="8.0cm"}
Comparison of aADI and PSFR+aADI {#sec_methods}
--------------------------------
As can be seen in the images (Fig. \[ei\_imgs\_mrg\]), the main difference between aADI and PSFR+aADI is that spiders are more efficiently removed in the latter case. However, the impact of this is largely cosmetic, as a comparable amount of flux is lost from the companion in the spider regions. This can be clearly seen in a comparison of the respective contrast curves for the two methods (see Fig. \[comp\_aadi\_loci\]), which have been normalized with respect to flux losses. PSFR+aADI slightly improves the performance at large separations, but provides no improvement at all for small separations. It should be noted that the results are based on a single PSF reference star (though with multiple representations) – it would be preferable to use multiple reference stars, and doing so might substantially improve the performance. In any case, we do not reach as promising results as those achieved with PSFR using LOCI on space-based HST data (see Lafreniere et al. 2009), where a significant improvement over aADI is readily seen. As we have demonstrated, 4 $\mu$m imaging provides a very high Strehl ratio, so if this was the limiting PSF stability factor at this level of contrast, we should have expected an improvement in the inner image range. Hence, the results imply that other PSF effects become dominant once the Strehl ratio is high enough, such as low-order aberrations arising in the telescope, and differences in PSF representation resulting from dithering. This in turn implies that a stable telescope configuration is the best way forward for improving the contrast in 4 $\mu$m imaging even further. There is an obvious and well-tested technique for achieving this, called passive ADI, in which the pupil is stabilized during observations, while the field is allowed to rotate (see Marois et al. 2006). Indeed, the LOCI algorithm was originally designed for this purpose (Lafreniere et al. 2007). In fact, we have a passive ADI sequence at 4 $\mu$m showing exquisite performance at small separations, but those data are taken with a different telescope and of a different target, so a rigorous comparison can not be made. The passive ADI data will be part of a separate publication.
![Comparison of contrast for aADI (solid line) and PSRF+aADI (dashed line). The performance is generally very similar.[]{data-label="comp_aadi_loci"}](mj_fig4.eps){width="8.0cm"}
Input from dynamics {#sec_dynamics}
-------------------
There exist extensive radial velocity measurements of $\epsilon$ Ind A, as well as some limited astrometric information, which can be used to constrain the properties of any sufficiently massive companion, as discussed in the following.
### The radial velocity linear trend {#sec_rvtrend}
The linear radial velocity trend of $\epsilon$ Ind A was first reported by Endl et al. (2002). The original dataset covered an observational baseline of about 5.2 years, taken with the ESO CES instrument in the period 1992-1998. Since then, HARPS data have been taken from 2003 to 2008 (Zechmeister et al., in prep.). The linear trend of the HARPS data is consistent with that of the aforementioned CES data, with a slope of 4.4 m s$^{-1}$ yr$^{-1}$. Hence, we adopt this slope over the 16 year total baseline. One contributor to the linear trend is secular acceleration. This is the apparent acceleration that an observer measures over time in projected motion (in this case along the line of site) of an object with constant velocity, due to the actual motion in 3D space. Using all the measured spatial coordinates and velocity components of $\epsilon$ Ind, the secular acceleration can be calculated to 1.8 m s$^{-1}$ yr$^{-1}$. This is quite large, due to the fast motion of $\epsilon$ Ind in the plane of the sky, but still leaves a 2.6 m s$^{-1}$ yr$^{-1}$ trend that must be due to actual acceleration.
Since $\epsilon$ Ind Ba/Bb is known to be physically bound to $\epsilon$ Ind A, it needs to be tested whether it could be responsible for the observed trend. We do this with the following order-of-magnitude estimate: The projected separation between A and Ba/Bb is about 1500 AU, hence for masses of 0.7$M_{\rm sun}$, 0.047$M_{\rm sun}$, and 0.028$M_{\rm sun}$ respectively (see McCaughrean et al. 2004), the orbital period of the A/B system is at least 66000 years for a circular Keplerian orbit. Such an orbit would lead to a radial velocity semi-amplitude for $\epsilon$ Ind A of 62 m s$^{-1}$, which in turn gives an average peak-to-peak acceleration of $4*10^{-3}$ m s$^{-1}$ yr$^{-1}$. Thus, the gravitational influence of $\epsilon$ Ind Ba/Bb is several orders of magnitude too small to make any significant contribution to the observed trend.
With $\epsilon$ Ind Ba/Bb out of the picture, we are left with closer, as of yet unseen companions. A previous H- and K-band imaging campaign (Gei[ß]{}ler et al. 2007) has excluded the presence of stellar and massive brown dwarf companions, down to 53 $M_{\rm jup}$ outside of a projected separation of 1.5 AU and 21 $M_{\rm jup}$ outside of 4.7 AU. This also excludes white dwarfs, since at ages up to several Gyrs, they are much brighter in H-and K-band than a 50$M_{\rm jup}$ object (see e.g. Holberg & Bergeron 2006 and Baraffe et al. 2003). Stellar objects outside of the field of view can be excluded, as they would be detectable with wide-field or all-sky surveys such as 2MASS (Skrutskie et al. 2006). While more exotic forms of stellar remnants (e.g. neutron stars) can perhaps not be categorically excluded, for the remainder of this paper we will assume that the observed acceleration is due to a low-mass brown dwarf or giant planet. The combined constraints from the imaging and the radial velocity trend are given in Sect. \[sec\_detlimits\].
### Astrometry {#sec_astrometry}
As will be seen in the following, the companion is expected to have a mass in the range of $\sim$5-20$M_{\rm jup}$, and an orbital semi-major axis in the range of $\sim$10-20 AU. At the distance of the $\epsilon$ Ind system, this corresponds to a strong astrometric amplitude signature imposed on the primary of about 15-60 mas. However, with an orbital period of a few decades, it would not be possible to detect orbital motion with, e.g., *Hipparcos* data alone. On the other hand, one might expect a systematic difference between the proper motion as measured by *Hipparcos* versus that measured in long-term ground based monitoring, such as from the Fifth Fundamental Catalog (FK5). This type of signature is referred to as $\Delta \mu$ binarity, see Wielen et al. (2001). For $\epsilon$ Ind, an approximate conversion between the FK5 and HIP systems implies that there is a difference between the *Hipparcos* and FK5 proper motions of $\Delta \mu _{\alpha} = -0.23 \pm 1.68$ mas yr$^{-1}$, and $\Delta \mu _{\delta} = -2.5 \pm 0.98$ mas yr$^{-1}$. This corresponds in total to a significance level of $F=2.54$, where the $F$ value is roughly to the same level of confidence as the equivalent $\sigma$-number for Gaussian statistics (Wielen et al. 2001). In other words, there is an indication of a companion in the data, but not at a very high level of significance. We can make an order-of-magnitude estimation of whether these numbers are consistent with the RV companion by assuming that the orbital motion is completely averaged out in the FK5 data, that the orbit is circular, and that a sufficiently small fraction of the orbit is covered by *Hipparcos* such that local curvature in the motion during that period is negligible. The limiting cases quoted above then yield astrometric motions of $\pi*15$ mas in 32 years and $\pi*60$ mas in 89 years respectively, i.e. 1.5 mas yr$^{-1}$ and 2.0 mas yr$^{-1}$, both of which are consistent with the given $\Delta \mu$ within the errors. Hence, the astrometry is indeed consistent with the RV trend, though we reiterate that the significance is rather limited for the astrometry.
Detection limits {#sec_detlimits}
----------------
Since A1+A2 are the deepest images, they provide the strongest detection limits, and therefore we concentrate on them in this section. The instrumental contrast for A1+A2 is determined at each separation as the maximum performance out of the aADI and PSFR+aADI contrasts at that separation. The corresponding mass limits for A1+A2, calculated in the same way as for B1 and B2 in section \[sec\_filters\], are shown in Fig. \[mlim\_rv\] for ages of 1, 3, and 5 Gyr. Several age determinations exist pointing to an age in the range of 1 Gyr for $\epsilon$ Ind (e.g. Lachaume et al. 1999; Barnes 2007). However, preliminary analysis of the astrometric masses of $\epsilon$ Ind Ba/Bb (Cardoso et al. 2008) implies that the components are probably under-luminous with respect to model predictions (Baraffe et al. 2003) at 1 Gyr, such that the $\epsilon$ Ind system has to be significantly older, perhaps up to 5 Gyr, if the models are accurate (which may not be the case, see e.g. Dupuy et al. 2009). On the other hand, such an old age would be incompatible with the observed spectra of $\epsilon$ Ind Ba and Bb according to the analysis of Kasper et al. (2009). It is with these uncertainties in mind that we consider the full range of 1 to 5 Gyr in our analysis.
![Detection limits at 1, 3, and 5 Gyr for $\epsilon$ Ind A. The solid line that increases outwards is the mass as function of semi-major axis corresponding to the 2.6 m s$^{-1}$ yr$^{-1}$ slope of the observed RV trend, and the dashed line is the reference for minimum projected separation at a typical inclination of 60$^{\rm o}$, both under the assumption of a circular orbit. The dotted vertical line is the minimum semi-major axis from the RV baseline at $q=2$.[]{data-label="mlim_rv"}](mj_fig5.eps){width="8.0cm"}
Also plotted is the mass as function of semi-major axis derived from the slope of the linear trend, under the assumption that the inclination is $60^{\rm o}$ (the mean inclination of randomly oriented orbits). The minimum possible semi-major axis is set by the minimum possible period, which in turn is some multiple $q$ of the observational baseline. The exact value of $q$ depends on the amount of curvature present in the trend, the determination of which would be an over-interpretation of the data at hand. As discussed in Janson et al. (2008), $q=1$ would be the most conservative limit possible to set, but it is unrealistic, since it would require a discrete change in velocity state. Here, we set $q=2$, which is still conservative, and more realistic.
The mass limits and RV trend shown in Fig. \[mlim\_rv\] provide a good illustration of the detectability of the dynamical companion under the assumption of a circular orbit. However, given the large eccentricity spread of the exoplanet population outside of 0.1 AU, it is necessary to perform more detailed simulations in order to constrain the possible physical and orbital parameters of the companion. The method for doing so is described in detail in Janson et al. (2008), and we follow it here for $q=2$. In brief, based on the empirical distribution of eccentricities for known exoplanets outside of 0.1 AU, we simulate all possible orbits and orbital phases and test whether they are consistent with the observed linear trend. The fraction of such orbits as function of semi-major axis is named $\phi$. Out of these allowed orbits, we test what fraction would lead to a detectable companion. This fraction as a function of semi-major axis is termed $\chi$. One addition has been made to this procedure with respect to what was presented in Janson et al. (2008): In the case of $\epsilon$ Eri, the plane of the disk, the rotational plane of the star, and the orbital plane of the planet candidate $\epsilon$ Eri b all gave a consistent orbital inclination of about 30$^{\rm o}$, hence this number was fixed in the simulations. In the case of $\epsilon$ Ind, we have no prior information of the inclination, hence it is treated as a free parameter in the simulations. This is done by performing the simulations over several different inclination angles and averaging the results. The input inclination angles are set to correspond to the actual probability of a given inclination occurring (i.e., accurately taking into account that the inclination is more likely to be edge-on than face-on).
![Detection probability in our images as function of semi-major axis for 1, 3, and 5 Gyr. Also plotted is $\phi$, the fraction of orbits at a given semi-major axis that are consistent with the linear RV trend, denoted with the subscript “All” to signify that it is independent of age, in contrast to $\chi$.[]{data-label="probfig_q2"}](mj_fig6.eps){width="8.0cm"}
The results of the simulation are shown in Fig. \[probfig\_q2\]. It can be seen that if the age is 1 Gyr, the probability of detecting the companion is always about 90% or higher for any semi-major axis, hence since no companion is detected, it is quite unlikely that the system is that young. On the other hand, if the age is 3 Gyr, or even 5 Gyr as discussed above, there is still a substantial parameter range in which the companion could hide. Given these results, in approximate numbers we can constrain the planet or brown dwarf mass to about 5-20 $M_{\rm jup}$ and its semi-major axis to about 10-20 AU. Also, the inclination must be larger than at least 20$^{\rm o}$, otherwise the projection effects could never bring the companion close enough to the star to hide it, and the actual mass would be sufficiently larger than the projected mass to make it brighter than the background at any reasonable age.
Conclusions {#sec_conclusions}
===========
We have attempted to image the indirectly discovered companion to $\epsilon$ Ind A, using imaging in the 4$\mu$m filter as well as the L’-band. As expected, 4$\mu$m imaging was found to be a preferable choice over L’-band in the inner, contrast-limited regime, whereas the opposite was found to be true in the outer, background-limited range. This conclusion is based on theoretical models that ultimately need to be confirmed through observations of known planets. The overlap occurs at a radius of 4$''$ in our images, a number that will depend on target brightness and integration time. Two PSF subtraction techniques were employed: regular active ADI as used previously, and a new combination of techniques, using PSF reference subtraction and aADI with the LOCI algorithm. While PSFR+aADI performs slightly better at large separations, the techniques are virtually indistinguishable for most of the contrast-limited regime. Using more than one PSF reference star may change this picture. In addition, the method of combining 4$\mu$m imaging and LOCI is also well suited for passive ADI, which has the potential to substantially enhance the performance even further.
In spite of the high sensitivities achieved in our images, we did not detect any potential companion candidate. Unless the known radial velocity companion to $\epsilon$ Ind A is a neutron star or even more exotic stellar remnant, the non-detection in all images implies that the system is probably older than 1 Gyr, possibly consistent with preliminary results presented by Cardoso et al. (2008). Furthermore, we can constrain the planet or brown dwarf mass to within approximately 5-20 $M_{\rm jup}$, the semi-major axis to $\sim$10-20 AU, and the inclination to $>$20$^{\rm o}$. An analysis based on astrometry from FK5 and *Hipparcos* is consistent with such a companion. Given the high significance of the RV trend, the fact that we can exclude all stellar, white dwarf and high-mass brown dwarf companions, and the fact that exotic stellar remnants are rare, it seems very plausible that $\epsilon$ Ind A is one of the nearest stars to host a massive giant planet or very low-mass object. Furthermore, it is likely that this companion would be detectable through further imaging with either the presently available facilities, or facilities that come online in the relatively near future. Hence, $\epsilon$ Ind is a high-profile target for the study of substellar objects, even aside from the fact that it hosts the nearest binary brown dwarf.
Finally, we note that no sophisticated coronagraph adapted for observations beyond 3$\mu$m presently exists on any of the 8m-class or larger AO-assisted telescopes (although simple coronagraphs do exist, e.g. a Lyot coronagraph for NACO). The potential coronagraphic performance is intimately connected to the adaptive optics performance, which leads to an interest in coronagraphs in the context of ’extreme AO’ facilities currently in development (e.g. Petit et al. 2008). However, given the fact that a demonstrated Strehl ratio in the range of 85% can be reached even with NACO at 4$\mu$m, an ’extreme AO’-type performance in this particular wavelength range is available already today. The development of a coronagraph for this wavelength range could therefore be another promising avenue to further increase the near-future capacity of detecting extrasolar planets through direct imaging.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors wish to thank Marten van Kerkwijk and Yanqin Wu for useful discussion. The study made use of the CDS and SAO/NASA ADS online services. M.J. is supported through the Reinhardt postdoctoral fellowship from the University of Toronto.
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\[lastpage\]
[^1]: Reinhardt fellow. E-mail: [email protected]
[^2]: Based on observations collected at the European Southern Observatory, Chile (ESO No. 082.D-0251 and 081.C-0430).
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abstract: 'Humans can experience fake body parts as theirs just by simple visuo-tactile synchronous stimulation. This body-illusion is accompanied by a drift in the perception of the real limb towards the fake limb, suggesting an update of body estimation resulting from stimulation. This work compares body limb drifting patterns of human participants, in a rubber hand illusion experiment, with the end-effector estimation displacement of a multisensory robotic arm enabled with predictive processing perception. Results show similar drifting patterns in both human and robot experiments, and they also suggest that the perceptual drift is due to prediction error fusion, rather than hypothesis selection. We present body inference through prediction error minimization as one single process that unites predictive coding and causal inference and that it is responsible for the effects in perception when we are subjected to intermodal sensory perturbations.'
author:
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bibliography:
- 'pl.bib'
- 'selfception.bib'
- 'ninaRHI.bib'
title: |
Drifting perceptual patterns suggest prediction errors fusion rather than hypothesis selection:\
replicating the rubber-hand illusion on a robot [^1] [^2]
---
Sensorimotor self, rubber-hand illusion, predictive coding, robotics
Introduction {#sec:intro}
============
Distinguishing between our own body and that of others is fundamental for our understanding of the self. By learning the relationship between sensory and motor information and integrating them into a common percept, we gradually develop predictors about our body and its interaction with the world [@lanillos2017enactive]. This body learning is assumed to be one of the major processes underlying embodiment. Body-ownership illusions, like the rubber hand illusion [@botvinick1998rubber], are the most widely used methodology to reveal information about the underlying mechanisms, helping us to understand how the sensorimotor self is computed. Empirical evidence has shown that embodiment is flexible, adaptable, driven by bottom-up and top-down modulations and sensitive to illusions.
We replicated the passive rubber hand illusion on a multisensory robot and compared it with human participants, therefore gaining insight into the perceptive contribution to self-computation. Enabling a robot with human-like self-perception [@lanillos2016yielding] is important for: i) improving the machine adaptability and providing safe human-robot interaction, and ii) testing computational models for embodiment and obtaining some clues about the real mechanisms. Although some computational models have already been proposed for body-ownership, agency and body-illusions, the majority of them are restricted to the conceptual level or simplistic simulation [@kilteni2015over]. Examining real robot data and using body illusions as a benchmark for testing the underlying mechanism enriches the evaluation considerably. To the best of our knowledge, this is the first study replicating the rubber hand illusion on an artificial agent and comparing it to human data.
We showed that when inferring the robot body location through prediction error minimization [@friston2005theory], the robot limb drifting patterns are similar to those observed in human participants. Human and robot data suggest that the perception of the real hand and the rubber hand location drifts to a common location between both hands. This supports the idea that, instead of selecting one of two hypotheses (common cause for stimulation vs. different causes) [@Samad.2015], visual and proprioceptive information sources are merged generating an effect similar to averaging both hypotheses [@Erro.2018].
The remainder of this paper is as follows: Sec. \[sec:method\] describes current rubber-hand illusion findings and its neural basis; Sec. \[sec:model\] defines the computational model designed for the robot; Sec. \[sec:setup\] describes the experimental setup for both human participants and the robot; Sec. \[sec:results\] presents the comparative analysis of the drifting patterns; Finally, Sec. \[sec:discussion\] discusses body estimation within the prediction error paradigm as the potential cause of the perceptual displacement.
Background {#sec:method}
==========
Rubber-hand illusion
--------------------
Botvinick and Cohen [@Botvinick.1998] demonstrated that humans can embody a rubber hand only by means of synchronous visuo-tactile stimulation of the rubber hand and their hidden real hand. This was measured using a questionnaire about the illusion, but also by proprioceptive localization of the participant’s real hand. After experiencing the illusion, the perception of their own hand’s position had drifted towards that of the rubber hand. Since then, multiple studies have replicated the illusion under different conditions (for a review, see [@Kilteni.2015]). Collectively, these studies show that top-down expectations about the physical appearance of a human hand, resulting from abstract internal body models, and bottom-up sensory information, especially spatiotemporal congruence of the stimulation and distance between the hands, influence embodiment of the fake hand [@Tsakiris.2005]. In [@AymerichFranch.2017], they were even able to induce body-ownership on a robotic arm. The common assumption is that the spatial representations of both hands are merged, as a result of minimizing the error between predicted sensory outcomes from seeing the stimulation of the rubber hand and the actual sensory outcome of feeling the stimulation of one’s own hand [@Tsakiris.2010]. Recently, [@Erro.2018] undermined this by showing that not only the perception of the real hand’s position is drifted towards the rubber hand (proprioceptive drift), but the one of the rubber hand is drifted towards the real hand as well (visual drift), i.e. to a common location between both hands. In [@Samad.2015], they proposed a Bayesian Causal Inference Model for this prediction error minimization, considering visual, tactile and proprioceptive information, weighted according to their precision. In combination with the prior probability of assuming a common cause or different causes for the conflicting multi-sensory information, the posterior probability of each hypothesis is computed. A common cause, i.e. ownership of the rubber hand, is assumed if the posterior probability exceeds a certain threshold. This binarity of the illusion, however, is at variance with findings of [@Tsakiris.2005], demonstrating a continuous proprioceptive drift of the stimulated hand. The proprioceptive drift was shown to increase exponentially during the first minute of stimulation and increasing further over the following four minutes (Fig. \[fig:overview:proprioceptiveDrift\]). Although the reported onset of the illusion ranged from 11 seconds [@Ehrsson.2004] to 23 seconds, with 90 percent of subjects experiencing it within the first minute of stimulation [@Kalckert.2017], the ongoing drifting suggests a continuous, rather than a discrete mechanism, being involved in embodiment.
The proposed computational models in the literature of body-ownership illusions need further verification from experimental data. Several studies showed reduced illusion scores for larger, as compared to smaller, distances between the real and the rubber hand [@Lloyd.2007; @Zopf.2010; @Preston.2013; @Pritchard.2016; @Ratcliffe.2017] (Fig. \[fig:overview:illusionDistance\]), though only few studies measured the proprioceptive drift in dependence on the distance between the two hands [@Zopf.2010], [@Preston.2013]. While [@Zopf.2010] found an increased proprioceptive drift for a larger distance, the relative amount of drift (i.e. corrected for distance) did not differ between the small and large distances. In [@Preston.2013], they replicated this result, as long as the fake hand was near the body. If the real hand was closer to body midline than the fake hand, increasing distance between both hands decreased the proprioceptive drift.
In the present study, we systematically varied the distance between both hands. The real hand, however stayed in the same position for all conditions and only the fake hand had a varying distance from the real hand in anatomically plausible positions. In [@Lloyd.2007], where the distance between both hands was varied by displacing the fake hand in relation to the real hand, the fake hand was also increasingly rotated with increasing distance. Rotational differences, nevertheless, may influence the illusion [@Makin.2008], probably confounding the results of [@Lloyd.2007]. In the current study, we systematically examined the influence of the distance between the rubber and the real hand on proprioceptive and visual drift. This provided the basis for validating the computational model proposed in Sec. \[sec:model\] and comparing the drift of the body estimation in different distances between the robot and humans.
Body illusions in the brain
---------------------------
A seminal contribution to possible neuronal mechanisms underlying the rubber hand illusion came from [@graziano2000coding], who discovered parietal neurons in the primate brain coding for the position of the real arm and a plosturally plausible fake monkey arm. Several fMRI studies looked into the neural correlates for body-ownership illusions in humans (see [@Makin.2008] for a review). Three areas were consistently found activated during the rubber hand illusion: posterior parietal cortex (including intra-parietal cortex and temporo-parietal junction), premotor cortex and lateral cerebellum. The cerebellum is assumed to compute the temporal relationship between visual and tactile signals, thus playing a role in the integration of visual, tactile and proprioceptive body-related signals [@Ehrsson.2005], [@Guterstam.2013]. The premotor and intra-parietal cortex are multisensory areas, also integrating visual, tactile and proprioceptive signals present during the rubber hand illusion [@Guterstam.2016]. In [@Makin.2008], they differentiated the role of posterior parietal cortex, being responsible for the recalibration of visual and tactile coordinate systems into a common reference frame, and the role of premotor cortex, being responsible for the integration of signals in this common, hand-centered reference frame. Although it is known that these areas participate in evoking the rubber hand illusion, little is known about the underlying computations [@Apps.2014]. In [@Zeller.2016], they used dynamic causal modeling during the rubber hand illusion to confirm to some extent that, during the illusion, visual information is weighted more than proprioceptive information - which would be congruent with predictive coding models. During the illusion the intrinsic connectivity of the somatosensory cortex was reduced, indicative of less somatosensory precision, while the extrinsic connectivity between the lateral occipital complex and premotor cortex was increased, indicative of more attention to visual input. Further functional evidence for the proposed computations is needed.
Computational model {#sec:model}
===================
![Rubber hand illusion modelled as a body estimation problem solved using prediction error minimization. Visual features of the rubber hand are incorporated when there is synchronous visuo-tactile stimulation, though it is constrained by the prior belief and the expected location of the hand according to the generative visual forward model and the estimated body configuration.[]{data-label="fig:model"}](Overview_small2.jpg){width="0.9\columnwidth"}
We formalized the rubber hand illusion as a body estimation problem under the predictive processing framework. The core idea behind this is that all features and sensory modalities are contributing to refine body estimation through the minimization of the errors between sensations and predictions [@lanillos2018adaptive]. During synchronous visuo-tactile stimulation, the most plausible body configuration is perturbed due to the merging of visual and proprioceptive information. This is coherent with the drift of both the real hand and the rubber hand as the participants are just pointing to the prediction of their hand according to the current body configuration. Figure \[fig:model\] shows how sensory modalities or features are contributing to the estimation of the participant´s limb.
We define $x$ as the latent space variable that expresses the body configuration. We model the problem as inferring the most plausible body configuration $\hat{x}$ given the sensation likelihood and the prior: $P(\hat{x}|s) = p(s|\hat{x}) p(\hat{x})$. We further define $s_p$, $s_v$, $s_{vt}$ as the proprioceptive, visual and visuo-tactile sensation respectively. Assuming independence of the different sources of information we get:
$$\begin{aligned}
P(\hat{x}|s) = p(s_p|x)p(s_v|x)p(s_{vt}|x) p(x)\end{aligned}$$
The perception or estimation of the body is then solved by learning an approximation of the forward model for each feature or modality $s =g(x)$ and minimizing a lower bound on the KL-divergence known as negative free energy F [@friston2005theory; @bogacz2015tutorial]. $$\begin{aligned}
\log P(\hat{x}|s) = -F = \sum_i \log p(s_i|x) + \log p(x)\end{aligned}$$ We obtain the estimated value of the latent variables through gradient descent minimization $\hat{x}= \frac{\partial F}{x}$:
$$\begin{aligned}
\hat{x} = \sum_i \underbrace{\frac{(s_i - g_i(\hat{x}))}{\sigma_{s_i}}}_{\text{error expected sensation}} \!\!\!\!\!\!g'_i(\hat{x}) - \underbrace{\frac{s_x- \mu_x}{\sigma_x}}_{\text{error prior}}\end{aligned}$$
We assume that all sensations / features follow a Gaussian distribution with (linear or non-linear) mean $g_i(x)$ and variance $\sigma_{s_i}$. The forward models learned should be differentiable with respect to the body configuration ($g'_i(\hat{x}) = \partial g_i(\hat{x})/\partial x$).
By rewriting the prediction error as $e = s-g(x)$ and defining $\mu_x$ as the prior belief about the body configuration, the dynamics of the body perception model are described by (see Appendix for derivation and [@lanillos2018adaptive] for the detailed algorithm): $$\begin{aligned}
&\dot{x} = -e_x + e_p + e_v g'_v(\hat{x}) + e_{vt} g'_{vt}(\hat{x})\nonumber\\
&\dot{e}_x = s_x - \mu_x - \sigma_x e_x \nonumber\\
&\dot{e}_p = s_p - \hat{x} - \sigma_p e_p\nonumber\\
&\dot{e}_v = s_v - g_v(\hat{x}) - \sigma_v e_v\nonumber\\
&\dot{e}_{vt} = s_{vt} - g_{vt}(\hat{x}) - \sigma_{vt} e_{vt}\nonumber\\
&\dot{\mu}_x = \mu_x + \lambda e_x\end{aligned}$$ where $\lambda$ is the learning ratio parameter that specifies how fast the prior of the body configuration $\mu_x$ is adjusted to the prediction error.
The visual forward function $g_v$ and its derivative are calculated using Gaussian process estimation (see Sec. \[sec:setup\]). The visuo-tactile generative function is computed by means of a hand-crafted likelihood, which uses the visual $o_v$ and tactile $s_t$ stimulation information (temporal $h_s$ and spatial $h_t$), and the expected position of the hand $g_v(\hat{x})$: $$\begin{aligned}
\label{eq:touch}
g_t(\hat{x}) = s_t \cdot h_s \cdot h_t = s_t a_1 e^{-b_1\sum(g_v(\hat{x}) - o_v)^2} \cdot a_2e^{-b_2 \delta^2}\end{aligned}$$ where $a_1,b_1,a_2,b_2$ are parameters that shape the likelihood of the spatial plausibility and have been tuned in accordance with the data acquired from human participants; $\delta$ is the level of synchrony of the events (e.g. timing difference between the visual and the tactile event); and $o_v$ is the other agent end-effector location in the visual field.
Experimental Set-up {#sec:setup}
===================
Participants selection
----------------------
20 volunteers (mean age: 25, 75 % female) took part in the experiment. They received 8 euros per hour in compensation. All participants were right-handed, had no disability of perceiving touch on their right hand, did not wear nail polish and did not have any special visual features (e.g. scars / tattoos) on their right hand. They had no neurological or psychological disorders, as indicated by self-report, and normal or corrected-to-normal vision. None of them had experienced the rubber hand illusion before. All participants gave informed consent prior to the experiment.
Humans experiment details
-------------------------
We performed the rubber-hand illusion experiment, focusing on the proprioceptive drift of the real and the visual drift of the rubber hand, as a function of the distance between both hands. The experiment, depicted in Fig. \[fig:setup\], comprised six conditions, each with a different distance between the real hand and the rubber hand. The participant’s real right hand was placed in a box, with the index finger 20cm away from the participant’s body midline. The rubber hand was placed with its index finger 15cm, 20cm, 25cm, 30cm, 35cm or 40cm away from the participant’s real right hand (5cm, 0cm, -5cm, -10cm, -15cm or -20cm away from the participant’s body midline respectively).
Participants sat in front of a wooden box, placing their hands near the outer sides of the box. They wore a rain cape covering their body and arms. In one of the arms, a rubber hand was placed such that it seemed coherent with the body. With their left hand they held a computer mouse. Each trial consisted of four phases: localization of the real hand, localization of the rubber hand, the stimulation phase and post-stimulation localization of both hands.
1. First, we covered the box with a wooden top and a blanket above it, so that no visual cues could be used. Participants had to indicate where they currently perceived the location of the index finger of their right hand, pointing with the mouse on a horizontal line presented on the screen. The line did cover the whole length of the box.
2. After fixating the rubber hand for one minute, we again covered the box and the same task was repeated for the rubber hand.
3. The box was remodeled, removing the cover and introducing a vertical board next to the participant’s right hand so that it was not visible to the participant (Fig. \[fig:setup:1\]). Then the experimenter began stimulating the rubber and the real hand synchronously with two similar paintbrushes, starting at the index finger, continuing to the little finger and then starting at the index finger again, with one brush stroke each about two seconds.
4. Subsequently, participants were again asked to indicate where they perceived the index finger of the real or the rubber hand, starting with the real or the rubber hand in randomized fashion. The box was covered during the localization.
5. At the end of each trial, participants were asked to answer ten questions related to the illusion adapted from [@Kammers.2009], presented randomized on the screen, using a continuous scale from -100 (indicating strong disagreement) to 100 (indicating strong agreement).
For the localization trials, a horizontal line was presented on the screen opposite to the box, with the screen having the same size as the box. The localization trials were repeated ten times to account for high variance. The proprioceptive drift and visual drift were calculated by subtracting the average of the first localization phase from the second localization phase for the real hand and the rubber hand separately. The illusion index was calculated by subtracting the average response to control statements S4-S10 from the average response to illusion statements S1-S3 [@Abdulkarim.2016]. Between all phases participants were blindfolded, so they did not observe the remodeling, which might potentially have impeded the illusion.
Robot experiment details
------------------------
We tested the model on the multisensory UR-5 arm of robot TOMM [@dean2017tomm], as depicted in Fig. \[fig:setup:2\]. The proprioceptive input data were three joint angles with added noise (shoulder$_1$, shoulder$_2$ and elbow - Fig. \[fig:data:a1\]). The visual input was an rgb camera mounted on the head of the robot, with $640\times480$ pixels definition. The tactile input was generated by multimodal skin cells distributed over the arm [@mittendorfer2011humanoid].
Learning $g(x)$ from visual and proprioceptive data
---------------------------------------------------
In order to learn the sensory forward model, we applied Gaussian process (GP) regression: $g_v(x) \sim \mathcal{GP}$. We programmed random trajectories in the joint space that resembled horizontal displacements of the arm. Figure \[fig:data:a1\] shows the extracted data: noisy joint angles and visual location of the end-effector, obtained by color segmentation. To learn the visual forward model $s_v = g_v(x)$, each sample was defined as the input joint angles sensor values $x= (x_1, x_2, x_3)$ and the output $s_v = (i,j)$ pixel coordinates. As an example, Figure \[fig:data:a2\] shows the learned visual forward model by GP regression with 46 samples (red dots). It describes the horizontal mean and variance (in pixels) with respect to two joints angles. The GP learning and its partial derivative computation with respect to $x$ is described in the Appendix \[sec:appendix:gp\].
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Extracting visuo-tactile data
-----------------------------
We used proximity sensing information (infrared sensors) from 117 different skin cells to discern when the arm was being touched. The sensor outputed a filtered signal $\in (0,1)$. From the other’s hand visual trajectory and the skin proximity activation, we computed the level of synchrony between the two patterns (Fig. \[fig:sync:a1\]). Timings for tactile stimuli $s_t$ were obtained by setting a threshold over the proximity value: prox $> 0.7 \rightarrow$ activation. Timings for the other’s trajectory events were obtained through the velocity components. Detected initial and end positions of the visual touching are depicted in Fig. \[fig:sync:a1\] (right, green circles).
Results {#sec:results}
=======
We compared the drifting data extracted from the rubber-hand illusion experiment in human participants and the robot. In order to obtain the robot results, we fixed in advance the model parameters for the learning and the body estimation stages. $g_v(x)\sim \mathcal{GP}$ learning hyperparameters: signal variance $\sigma_n = \exp(0.02)$ and kernel length scale $l=\exp(0.1)$. The integration step was $\Delta_t= 0.05$ ($20Hz$) and the error variances were $\sigma_x \in \mathcal{R}^3 = [1,1,1]$, $\sigma_p \in \mathcal{R}^3 = [1,1,1]$, $\sigma_{vt} \in \mathcal{Z}^2= [2,2]$. The adaptability rate of $\mu_x$ was $\lambda = 1$. The visual feature from the real hand $s_v$ was not used in the rubber hand illusion experiment as the arm was covered. Finally, the visuo-tactile function (Eq. \[eq:touch\]) parameters were: $b_1= \frac{\sigma_t}{d_{max}^2}$, where $\sigma_t=0.001$ and $d_{max}=0.0016$; $b_2 = 25$; and $a_1=a_2=1$. The robot drift was computed by subtracting the estimated end-effector position $g_v(\hat{x})$ and the ground truth location, and $\hat{x}$ was dynamically updated minimizing the prediction error using the proposed model.
Comparative analysis
--------------------
Figure \[fig:results\] shows the proprioceptive drift comparison. Fig. \[fig:results:1\] shows similar drifting patterns in both the robot and the human participants. A drift towards the fake hand emerges in both cases when the distance is small and then vanishes with longer distances. The prior information used for the tactile likelihood function parameters is modulated when the effect is taking place, as the error will start propagating when the gradient of the function is noticeable. Furthermore, the relative drift (Fig. \[fig:results:2\]) also showed that, for close distances, the amount of displacement is the same, and then it decreases until vanishing. The robot was tested on even closer distances than humans, since the human experimental setup was not equipped for distances beneath 15cm. The large increase in proprioceptive drift for 10cm distance between fake and real hand is an interesting prediction for human data, that could be tested in future work.
Human data analysis
-------------------
Data exceeding a range of two standard deviations around the mean was excluded from further analysis. T-tests were used to test the proprioceptive drift, the visual drift and the illusion score in each condition against zero. In the first three conditions the proprioceptive drift was significantly different from zero, while it was not in the other three conditions (Table \[tab:ttest\]). Employing Bonferroni-Correction only leaves a trend towards significance in the 20cm distance condition. However, the average of the first three conditions is still significantly different from zero $M: 13.72$, $SD: 14.84$, $p <.001$) while the average of the other conditions is not ($M 8.24:$, $SD: 19.79$, $p > .05$). The visual drift was only significantly different from zero in the 30cm distance condition ($M: 14.01$, $SD: 29.72$, $p < .01$). The illusion score was significantly different from zero in all conditions (all $p < .05$). Partial Pearson correlations between illusion score and proprioceptive drift, illusion score and visual drift and between proprioceptive drift and visual drift were not significant (all $p > .05$).
**Condition** **mean** **std** **df** **t-value** **p-value**
--------------- ---------- --------- -------- ------------- -------------
15cm 14.89mm 27.08mm 19 2.46 .024
20cm 12.79mm 18.99mm 17 2.86 .011
25cm 13.22mm 23.51mm 16 2.32 .034
30cm 4.39mm 15.52mm 16 1.17 .260
35cm 7.84mm 18.00mm 18 1.90 .074
40cm 5.45mm 31.34mm 17 0.74 .471
: Descriptive and inference statistics from proprioceptive drift data in each condition.
\[tab:ttest\]
Robot model analysis
--------------------
We analyzed the internal variables of the proposed model during the visuo-tactile stimulation and the induced end-effector estimation drift towards the fake arm. Figure \[fig:resultsrh:a2\] shows the robot camera view with the final end-effector estimation overlaid after 12 seconds. Depending on the different enabled modalities (proprioceptive, visuo-tactile and proprioceptive+visuo-tactile), body estimation evolved differentially, accordingly the prediction of the end-effector $g_v(x)$. Fig. \[fig:resultsrh:a3\] shows the evolution of the body configuration in term of joint angles and the corresponding prediction errors. We did initialize the robot belief in a wrong body configuration to further show the adaptability of the model. During the first five seconds, the system converged to the real body configuration. Afterwards, when perturbing with synchronous visuo-tactile stimulation, a bias appeared on the body joints. This implies a drift of the robot end-effector towards the location of the fake arm visual feature. Tactile perturbations are shown as prediction error bumps (yellow line). Fig. \[fig:resultsrh:a3\], top plot, also shows how smooth body configuration output $\mu_{x_{1:3}}$ is (blue line). The robot inferred the most plausible body joints angles given the sensory information, which in this case yielded a horizontal drift on the estimated end-effector location. A video of the evolution of the variables during the artificial rubber-hand illusion experiment can be found at <http://web.ics.ei.tum.de/~pablo/rubberICDL2018PL.mp4>.
Discussion: Body estimation as an explanation for the perceptual drift {#sec:discussion}
======================================================================
It has been shown that during the rubber hand illusion, the location of the real hand is perceived to be closer to the rubber hand than before. Similarly, the location of the rubber hand is mislocalized towards the real hand [@Erro.2018]. Our results from the robot and humans support the former finding: in our predictive coding scheme, the representation of both hands merged into a common location between both hands, due to inferring one’s body’s location from minimizing free energy. This body estimation generated a drift of the perceived location of the real hand towards the equilibrium location, which was visible in the data from both the humans and the robot. In comparative analysis, the patterns of the drift resembled each other, both in terms of absolute and relative values. The three closest tested distances showed a substantial proprioceptive drift. All other distances showed a smaller drift, approaching zero. For these, the distance between the fake and the real hand was probably too large for the fake hand to be fully embodied, supporting [@Samad.2015] simulation data exhibiting a reduced illusion probability for distances over 30cm. Although previous and the present research support predictive coding as a probable underlying mechanism of the rubber hand illusion, other accounts can not be ruled out by the present work.
Human illusion score data, however, did not mirror the proprioceptive shift pattern found. For all distances, illusion scores ranged between 20 and 35, which on our continuous scale up to 100 resembles illusion scores previously found from 1 on a discrete scale to 3, e.g. [@Lloyd.2007]. Given this, we can assume that we were able to induce the illusion in every condition. Illusion scores and the proprioceptive drift, additionally, were not correlated. This supports the current debate that body-ownership illusions and the drift are two different, but related, processes [@Abdulkarim.2016]. The proprioceptive drift is an unconscious process - in contrast to the illusion, which is consciously accessible to the subjects. Hence, it might be possible that the predictive coding formulation in its unconscious form can explain drifting patterns, while it is not as such sufficient to explain body-ownership illusions.
In contrast to [@Erro.2018], we did not find a conclusive visual drift in the human experiment. From participants’ personal communication, we know that many used visual landmarks to estimate the position of the rubber hand, but of the real hand also. Differences in this strategy for localization would not only account for the high variance we observed in the drift data, but also for the small magnitude of the proprioceptive drift - as compared to the values reported in other studies (e.g. [@Zopf.2010]). Beyond that, the method we used for localization is probably prone to high variance due to small mouse movements. Although we tried to account for that by repeating the localization ten times, more trials might be needed as performed in [@Samad.2015]. Arguably, however, the lack of visual drift in our study does not contradict the predictive coding scheme. Actually, some of our participants communicated that they experienced that the representation of both hands merged together. This is supported by the positive mean (14.16) of the actual control statement S10 “It felt as if the rubber hand and my own right hand lay closer to each other", which was even higher than the mean response (-4.95) to the actual illusion statement S3 “I felt as if the rubber hand were my hand". Further investigations, accounting for the variance in localization, are needed to support this conjecture.
The computational model presented here also generates predictions about the temporal dynamics of the rubber hand illusion. The constant accumulation of information resulting in an also accumulated drift of the body estimation (see \[fig:resultsrh:a2\]) is comparable to findings from [@Tsakiris.2005] (see \[fig:overview\]), who also found an accumulation of the drift over time in humans. To provide a finer temporal comparison, the dynamics of the human illusion should be further investigated.
Conclusion {#sec:conclusions}
==========
We implemented the rubber hand illusion experiment on a multisensory robot. The perception of the real hand’s position drifted towards the rubber hand, following a similar pattern in humans and the robot. We suggest that this proprioceptive drift resulted from a merging body estimation between both hands. This supports an account of the proprioceptive drift underlying body-ownership illusions in terms of the predictive coding scheme. Future work will address the mechanisms behind awareness of body-ownership.
Free energy gradient {#sec:appendix:partialF}
====================
$$p(\hat{x}|s_p,s_v) = p(s_p|\hat{x})p(s_v|\hat{x})p(s_{vt}|\hat{x}) p(\hat{x})$$
Applying logarithms we get the negative free energy formulation: $$F = \ln p(s_p|\hat{x}) + \ln p(s_v|\hat{x}) + \ln p(s_{vt}|\hat{x}) + \ln p(\hat{x})$$ Substituting the probability distributions by their functions $f(.;.)$, and under the Laplace approximation [@friston2008hierarchical] and assuming normally distributed noise, we can compute the negative free energy as: $$\begin{aligned}
\label{eq:freeenergyexample}
F =& \ln f(s_p; g_p(x), \sigma_p) + \ln f(s_p; g_p(x), \sigma_p) + \ln f(x; \mu_x, \sigma_x) \nonumber\\
=& - \frac{(\hat{x} - \mu_x)^2}{2\sigma_x} + \nonumber\\
& - \frac{(s_p - g_p(\hat{x}))^2}{2\sigma_p} -\frac{(s_v - g_v(\hat{x}))^2}{2\sigma_v} -\frac{(s_{vt} - g_{vt}(\hat{x}))^2}{2\sigma_{vt}}
\nonumber\\
& \quad + \frac{1}{2} \left[ -\ln \sigma_x - \ln \sigma_{s_p} - \ln \sigma_{s_v} - \ln \sigma_{s_{vt}}\right] + C. \end{aligned}$$ In order to find $\hat{x}$ in a gradient-descent scheme we minimize Eq. \[eq:freeenergyexample\] through the following differential equation: $$\begin{aligned}
\dot{x} =& - \frac{\hat{x}-\mu_x}{\sigma_x} + \nonumber\\
&+ \frac{s_p - g_p(\hat{x})}{\sigma_p} g_p'(\hat{x}) +\frac{s_v - g_v(\hat{x})}{\sigma_v} g_v'(\hat{x}) +\frac{s_{vt} - g_{vt}(\hat{x})}{\sigma_{vt}} g'_{vt}(\hat{x})\end{aligned}$$ In the case that $\hat{x}$ is equivalent to $g_{p}(x)$ like using the joint angles values directly as the body configuration, then the proprioceptive error can be rewritten as: $s_p - \hat{x}$ and the gradient becomes 1. Generalizing for $i$ sensors we finally have: $$\begin{aligned}
\frac{\partial F}{\partial \hat{x}} = -\frac{(\hat{x} -\mu_x)}{\sigma_x} + \sum_i\frac{\partial{g_i(\hat{x})^T}}{\partial{\hat{x}}} \frac{s_i-g_i(\hat{x})}{\sigma_i}\end{aligned}$$
GP regression {#sec:appendix:gp}
=============
Given sensor samples $\overline{s}$ from the robot in several body configurations $\overline{x}$ and the covariance function $k(x_i,x_j)$, the *training* is performed by computing the covariance matrix $K(X,X)$ on the collected data with noise $\sigma_n^2$: $$\label{eq:covariance}
k_{ij} = \sigma_n^2\mathbf{I} + k(x_i,x_j) \quad | \forall i,j \in \overline{x}$$
The *prediction* of the sensory outcome $s$ given $x$ is then computed as [@rasmussen2005GPM]: $$\label{eq:GPmean}
g(\hat{x}) = k(\hat{x},X) K(X,X)^{-1} \overline{s}= k(\hat{x},X) \boldsymbol{\alpha}$$ where $\boldsymbol{\alpha} = \text{choleski}(K)^T \backslash ( \text{choleski}(K) \backslash \overline{s})$.
Finally, in order to compute the gradient of the posterior $g(x)'$ we differentiate the kernel [@mchutchon2013differentiating], and obtain its prediction analogously as Eq. \[eq:GPmean\]: $$\begin{aligned}
g(\hat{x})' &= \frac{\partial k(\hat{x},X)}{\partial \hat{x}} K(X,X)^{-1} \overline{s} = \frac{\partial k(\hat{x},X)}{\partial \hat{x}} \boldsymbol{\alpha}\end{aligned}$$ Using the squared exponential kernel with the Mahalanobis distance covariance function, the derivative becomes: $$\begin{aligned}
\label{eq:gderivative}
g(\hat{x})' = -\Lambda^{-1} (\hat{x} - X)^T (k(\hat{x},X)^T \cdot \boldsymbol{\alpha})\end{aligned}$$ where $\Lambda$ is a matrix where the diagonal is populated with the length scale for each dimension ($\text{diag}(1/l^2)$) and $\cdot$ is element-wise multiplication.
[^1]: This work has been supported by SELFCEPTION project (www.selfception.eu) European Union Horizon 2020 Programme (MSCA-IF-2016) under grant agreement n. 741941 and the ENB Master Program in Neuro-Cognitive Psychology at Ludwig-Maximilians Universität. Video to this paper: http://web.ics.ei.tum.de/pablo/rubberICDL2018PL.mp4
[^2]: Accepted for publication at 2018 IEEE International Conference on Development and Learning and Epigenetic Robotics
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---
abstract: 'Controlling electromagnetic energy is essential for an efficient and sustainable society. A key requirement is concentrating magnetic energy in a desired volume of space in order to either extract the energy to produce work or store it. Metamaterials have opened new possibilities for controlling electromagnetic energy [@tjc_book; @zheludev]. Recently, a superconductor-ferromagnetic metamaterial that allows unprecedented concentration and amplification of magnetic energy, and also its transmission at distance through free space, has been devised theoretically [@concentrator]. Here we design and build an actual version of the superconductor-ferromagnetic metamaterial and experimentally confirm these properties. We show that also a ferromagnetic metamaterial, without superconducting parts, can achieve concentration and transmission of energy with only a slight decrease in the performance. Transmission of magnetic energy at a distance by magnetic metamaterials may provide new ways of enhancing wireless power transmission, where efficiency depends critically on the magnetic coupling strength between source and receiver.'
author:
- 'Jordi Prat-Camps, Carles Navau, and Alvaro Sanchez'
title: Experimental realization of magnetic energy concentration and transmission at a distance by metamaterials
---
Electromagnetic fields power our society. Achievements such as the ubiquitous availability of electric power and the globally connected world through internet have required an exquisite control of electromagnetic fields - how to create, transport and use them- and the development of continuously improving materials. An essential part of this effort is in the control and manipulation of static magnetic fields. They play a fundamental role, for instance, in the generators that provide the energy for our appliances and devices, or in the writing heads of computer memories for storing information.
Metamaterials recently joined in to open new possibilities in the control of electromagnetic fields [@pendry; @controlling; @review_TO; @zheludev]. Novel ways of concentrating energy at different scales and wavelengths have been achieved by metamaterials. Examples range from electromagnetic energy concentration by plasmonics at the nanoscale [@schuller; @aubry] to thermal energy concentration at larger scale [@NarayanaT; @Han12]. Metamaterials have also created new possibilities of controlling static magnetic fields. For example, magnetic cloaks have been proposed and experimentally realized using superconductors combined with magnetic materials [@wood; @magnus; @ourAPL; @antimagnet; @narayana; @gomory; @carpet_magnetic].
In this work we focus on concentration of magnetic energy. A particularly needed feature is to achieve large magnetic field values not only by close contact with the sources but also at a distant point separated by an air gap, as in wireless power transmission. Also, it is important that magnetic energy can be concentrated in an empty region, where an antenna o sensor can be placed. With these goals in mind, we theoretically designed in [@concentrator] an infinite cylindrical shell with extreme radial and angular permeability components $\mu_{\rho} \rightarrow \infty$ and $\mu_{\theta} \rightarrow 0$, respectively. We showed that such a shell would concentrate an external applied field in its interior. It increases the field magnitude by a factor that is simply the ratio of outer to inner shell radii, which can be tuned to achieve large values. The shell would also expel the field of an interior source to the exterior, increasing its magnitude. These two properties, fully analytically demonstrated, allowed to propose different combinations of shells to obtain unprecedented control of magnetic fields. For example, fields could be concentrated at points distant from the source through free space, which could be relevant to enhance wireless transmission of power.
The ideal concentrating shell required extreme magnetic properties, not directly found in natural materials. To solve this, we proposed magnetic metamaterials made of alternated superconductor (SC) and ferromagnetic (FM) pieces to approximate the required anisotropic permeabilities. These pieces could be wedges or rectangular prisms [@concentrator]. Theoretical calculations indicated that a good performance should be obtained by metamaterial shells made of alternated wedge-shaped FM and SC pieces and behavior would improve with the number of elements [@sust]. However, ideal materials with linear behaviour and with very large permeability for the FMs and zero permeability for the SCs were assumed. Also, only infinitely long geometries were considered in the calculations. Thus, an experimental validation with actual materials and realistic geometries is needed to confirm these ideas.
In this work we design and construct actual metamaterial shells to experimentally confirm the properties of magnetic concentration, expulsion, and concentration at a distance. We will find that all these properties are achieved when using existing (not mathematically ideal) SC and FM materials, when having shells with short lengths, and when discretizing the continuous ideal material into a set of rectangular prisms. As an important step towards the feasibility of implementing our ideas in the technology, we will also experimentally demonstrate that metamaterial shells composed of only FMs separated with air gaps (without the use of SCs and their associated cryogenics) show good properties as well. Finite-element numerical calculations will simulate and help to interpret the experimental results.
We build a metamaterial cylindrical shell made of 36 rectangular prisms of alternated SC and FM materials (see Fig. 1a and Appendix for details). The FM pieces are made of a commercial high-permeability metallic alloy (mu-metal) whilst the SC ones of a commercial coated conductor. The pieces are radially distributed and fixed in a non-magnetic plastic support specially designed and made by a 3-D printer (Fig. 1b,c).
We first experimentally study the magnetic concentration properties of our metamaterial shell. Two Helmholtz coils created a uniform external field ($B_a$) perpendicular to the shell axis. A Hall probe was used to measure the concentrated field ($B^{\rm IN}_z$) in the central point inside the shell (see the sketch of the experimental setup in Fig. 2a and Appendix). Measurements were first performed with both the metamaterial and the probe submerged in liquid nitrogen, below the critical temperature of the SC, $T_c$. The magnetic-field enhancement factor at the center for the ideal infinitely long metamaterial should be 4, corresponding the ratio between outer to inner shell radii [@concentrator]. Measurements below $T_c$ showed that field was increased by a factor $2.70$ with respect to applied field (green symbols and dashed fitting line in Fig. 2b). This number is in excellent agreement with the 3D numerical simulation of this system (green solid line and Fig. 2d), which considered the actual finite dimensions and assumed ideal magnetic properties of the materials ($\mu_{FM}\to \infty$ and $\mu_{SC}\to 0$) . Simulation also shows that, despite the finite shell length, magnetic field achieves a very homogenous concentration [@concentrator], which we experimentally confirmed.
We next study if similar concentration results can be produced with simplified versions of the metamaterial. Because using SCs requires cryogenics and this may limit the applicability of the device, we measured the concentration of magnetic field at temperatures above $T_c$, at which the SCs are deactivated. A concentrating factor of $2.23$ (red symbols and dashed fitting line in Fig. 2b) was still achieved under these conditions. The corresponding simulation with only 18 FM pieces (red solid line and Fig. 2e) perfectly matched these measurements. Simulations also illustrate the role played by the two different materials. The SC layers in the SC-FM metamaterial (Fig. 2d) prevent angular components of magnetic field in the shell volume, so that magnetic field is basically guided radially to the central hole by the FM pieces. In the only-FM metamaterial shell (Fig. 2e), this role played by the SC parts is now taken up by the air gaps. This works only approximately, as can be seen from the presence of some field in the inner parts of the FM-metamaterial shell volume (Fig. 2e).
We next experimentally study the expelling properties of our metamaterial shell by measuring the field at its exterior when a dipolar-like magnetic field source is present in the shell interior. A small coil was placed at the center of the shell with its axis on the $z$ direction (see Fig. 3a) and was fed with a constant current. The field $B_z$ was measured outside the shell along the centered line as a function of the distance $z$. Measurements below $T_c$ (green symbols in Fig. 3b) show the expelled field was increased by a factor of about 2.4 in all the positions with respect to the measurements of the bare coil (blue symbols). Above $T_c$, the shell also increased the field in all exterior points by a factor of about 2, confirming its good behavior even without SCs. These measurements agree well with the corresponding 3D simulations performed assuming ideal materials (solid lines in Fig. 3b and Fig. 3c-e). Plots of the field at median planes show how the metamaterial shell expels most of the field from its interior to the exterior (Fig. 3d) as already discussed for the concentration case. When using solely FM pieces (Fig. 3e) the field expulsion is reduced.
The combination of the expulsion and concentration properties can be used to transfer magnetic energy at a distance, as theoretically anticipated in [@concentrator]. To experimentally demonstrate this property we used the studied shell with a coil inside as a source and construct a second shell as a receiver (see Figs. 4a and b). The second shell is an only-FM metamaterial shell. It consists of 18 FM pieces, without superconducting parts. We separate both shells at a distance of 70mm from center to center, so there is an air gap of 10mm between them. Measurements are shown in Fig. 4c, for two cases. When $T<T_c$ (upper plot), the shell with the source behaves like a full SC-FM metamaterial and, even though the receiver shell is a only-FM metamaterial shell, the measured magnetic field inside the receiver is increased by an average factor of around 7.5 with respect to the field of the naked coil. When $T>T_c$ (lower plot), the average factor is still around 6. In both cases it is seen that not only the magnetic field is magnified in the inner region of the receiver shell but also the field gradient is enlarged, as predicted by the theory [@concentrator]. The main result here is that these experiments are a proof-of-principle of the theory for magnetic energy transmission with metamaterials. The particular obtained numbers can be further optimized. The field-increase factor of 7.5 or 6 and also the air gap distance, can be both easily turned into larger values by changing the geometry, like reducing the inner radii of both shells [@concentrator]. Moreover, modifications of the studied magnetic metamaterials can be explored for changing the magnetic field shape in the space around the shell as well.
When a second coil is placed in the receiver shell, the enhanced field transmission is equivalent to increasing the magnetic coupling between both coils, which may have applications in wireless transmission of power [@persp; @Kim13]. In wireless systems, including the recent non-radiative power transfer proposals [@kurs], the mutual inductive coupling between the source and the receive resonators is a key parameter to increase the efficiency. As discussed in [@DaHuang], because the distance between the source and receiver is so much smaller than the wavelength, the relevant field distribution is quasistatic and the inductive coupling relates predominantly to magnetic flux emanating from one coil captured by the second coil. This is precisely what we achieve with our metamaterial shells. Actually, magnetic metamaterials have been already proposed for increasing the magnetic coupling in wireless systems in the form of magnetic lenses. This strategy requires a negative permeability material [@choi; @urzhumov; @DaHuang; @mitsu], whereas our metamaterial is made of non-resonant materials with positive permeability.
Even though our ideas have been confirmed strictly in the static limit, there are good indications that for low frequencies (up to tenths of kilohertz at least) the materials will behave as required. Recently, a magnetic cloak made of similar FM and SC materials designed to operate at the dc regime was shown to maintain cloaking properties for low-frequency applied magnetic fields [@accloak]. For wireless power applications, further work should be done to study the performance of these magnetic metamaterials at higher frequencies, including eventual losses in the components.
To sum up, we have experimentally demonstrated the unique properties that superconductor-ferromagnetic metamaterials can offer in terms of magnetic energy concentration, expulsion, and transmission through free space. Although the original theory was derived for infinitely samples and assumed ideal materials, our results confirm that these properties can be achieved in finite geometries and using commercially available materials. Finite-element calculations precisely reproduce the measured data and thus can be used to obtain optimized designs. Even when the superconducting parts are removed, very good behaviour is obtained with only ferromagnetic metamaterials. This hints at possible application of the results to technology, in particular in increasing the magnetic coupling between distant circuits, an essential factor for enhancing wireless transmission of power.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Elena Bartolome and EUSS for support, Fedor Gomory and Jan Souc for helpful discussions, SuperPower for providing the coated conductors, and Spanish Consolider Project NANOSELECT (CSD2007-00041) and MAT2012- 35370 for financial support. JPC acknowledges a FPU grant form Spanish Government (AP2010-2556).
Corresponding author {#corresponding-author .unnumbered}
====================
Alvaro Sanchez ([email protected])
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![ The metamaterial shell was made of 18 pieces of FM material and 18 of SC material, [**a**]{}, which were radially and alternately displaced. To correctly position these pieces a non-magnetic support structure, [**b**]{}, was fabricated. This plastic structure was 3D-printed having 36 lodgings homogeneously separated in which the SC and FM pieces fit, [**c**]{}. The resulting metamaterial shell, [**d-e**]{}, had an approximate interior and exterior radii of $R_1=7.5$mm and $R_2=30$mm respectively, and a height of 30mm. []{data-label="fig1"}](Fig1.pdf){width="100.00000%"}
![ [**a**]{} Sketch of the experimental setup used to measure the magnetic field concentration. Two Helmhotlz coils (red) with a radius of 70mm were used to create a uniform field. The metamaterial shell was placed in the central region (SC pieces represented in orange and FM ones in dark gray) and the Hall probe (blue) measured the field in the central interior point (some pieces have been sketched translucent to show the precise measuring point). [**b**]{} The measured field inside the metamaterial shell is plotted as a function of the field applied by the coils $B_a$ (in symbols) and for two different working conditions; measurements for $T<T_c$ in green and for $T>T_c$ in red (horizontal and vertical error bars $\Delta B_a=0.04$mT and $\Delta B^{\rm IN}=0.002$mT respectively, have been omitted for clarity). The corresponding linear fits are presented in dashed lines (whose slopes are 2.70 and 2.23 for $T<T_c$ and $T>T_c$, respectively), together with the simulations results in solid lines (whose slopes are 2.83 and 2.28 for $T<T_c$ and $T>T_c$, respectively), showing a good agreement. The blue line shows the field created by the bare coils in the central point, and is relevant to show the improvement achieved by the metamaterial shell in both working conditions. The calculated $B_z$ field component is plotted in the planes IN and OUT, [**c**]{}, for the case of 18 SC alternated with 18 FM (i.e. $T<T_c$), [**d**]{}, and for the case of only 18 FM ($T>T_c$), [**e**]{}.[]{data-label="fig2"}](Fig2.pdf){width="100.00000%"}
{width="100.00000%"}
{width="100.00000%"}
APPENDIX {#appendix .unnumbered}
========
Metamaterial shell construction {#metamaterial-shell-construction .unnumbered}
-------------------------------
The concentrating metamaterial shell consisted of 18 FM pieces and 18 SC pieces with a rectangular shape of 22x30mm (See Fig. 1). The former were made of a commercial mu-metal foil with a thickness of $0.3$mm from Vacuumschmelze, which was cut to the appropriate size. The SC parts were made of type-II SC strip 12mm width (SuperPower SCS12050). Each part was made of two pieces of strip, cut with a length of 30mm and parallel fixed with adhesive tape with an approximate overlap of 2mm.
To ensure the appropriate placement and alignment of the SC and FM pieces, a non-magnetic plastic support was designed and built. The support had a hollowed cylindrical shape, with an external radius of 31mm, an interior one of 6.5mm and a height of 33mm and had 36 lodgings homogeneously distributed with a consecutive angular spacing of $10^{\circ}$. It was provided with openings in the lateral face and also in the interior walls to ensure the free circulation of the liquid nitrogen during the measurements in which it was required. The support was built with a 3D printer and was made of ABS thermoplastic polymer.
Concentration measurements and simulations {#concentration-measurements-and-simulations .unnumbered}
------------------------------------------
The Helmholtz coils used to generate the uniform applied field had a radius of 70mm and were aligned and separated a distance of 70mm. We used a Hall probe model HHP-NP from Arepoc to measure the magnetic fields. Measurements below $T_c$ were done by submerging the metamaterial shell and the Hall probe into liquid nitrogen. The measurements where done increasing the intensity in the coils from 0 to 0.6A and measuring the field in the shell hole. Measurements were repeated several times for each configuration.
Expulsion measurements and simulations {#expulsion-measurements-and-simulations .unnumbered}
--------------------------------------
The small coil used to generate the field had an approximate diameter of 10mm and a height of 10mm and was vertically centered in the interior hole of the metamaterial shell. The field was measured by placing the Hall probe at the same height outside the shell. Measurements at $T>T_c$ were performed by submerging the metamaterial shell, the coil and the Hall probe into liquid nitrogen. Measurements were done increasing the field in the coil from 0A to 1A, and measuring the corresponding field. The transmitted field was calculated as the slope of the resulting field-intensity plot, which allowed to separate it from the ambient fields and the field created by the remanent magnetization of the ferromagnets. This process was repeated mutiple times in each measuring position, and in all the cases the plots field-intensity showed a very linear behavior, with no sign of saturation. In the simulations, the coil was modeled as a uniformly magnetized cylinder with the same size of the real coil. The magnetization was tunned to match the calculated plot of the coil alone (blue solid line) with the corresponding experimental measurements (blue symbols). SC and FM pieces were assumed ideal.
Concentration at distance measurements and construction of a only-FM shell {#concentration-at-distance-measurements-and-construction-of-a-only-fm-shell .unnumbered}
--------------------------------------------------------------------------
For these measurements we built a second metamaterial shell consisting of 18 FM pieces (made of the same mu-metal foil) with a rectangular shape similar to that of the first shell. The pieces where homogeneously distributed with an angular spacing of $20^{\circ}$ and where fixed in a non-magnetic extruded polystyrene foam layer in which 18 lodgings where made. The resulting shell had an approximate interior and exterior radii of $R_1=7.5$mm and $R_2=30$ respectively, and a height of 30mm. The field was measured inside the second shell for different positions $z$ using the procedure previously described, and for temperatures above and below $T_c$.
Numerical simulations {#numerical-simulations .unnumbered}
---------------------
3D simulations where obtained by the AC/DC module of Comsol Multiphysics software assuming ideal materials ($\mu^{SC}=10^{-5}$ and $\mu^{FM}=10^{5}$). No free parameters were used for the calculations.
|
---
abstract: 'We study light transmission through a homeotropically oriented nematic liquid crystal cell and solve self-consistently a nonlinear equation for the nematic director coupled to Maxwell’s equations. We demonstrate that above a certain threshold of the input light intensity, the liquid-crystal cell changes abruptly its optical properties due to the light-induced Freedericksz transition, demonstrating multistable hysteresis-like dependencies in the transmission. We suggest that these properties can be employed for tunable all-optical switching photonic devices based on liquid crystals.'
author:
- 'Andrey E. Miroshnichenko, Igor Pinkevych, and Yuri S. Kivshar'
title: 'Light-induced multistability and Freedericksz transition in nematic liquid crystals'
---
Introduction
============
Liquid crystals (LCs) play an important role in the modern technologies being used for numerous applications in electronic imaging, display manufacturing, and optoelectronics [@lmbvgc94; @ick94]. A large variety of electro-optical effects that may occur in LCs can be employed for a design of photonic devices. For example, the property of LCs to change its orientational structure and the refractive index in the presence of a static electric field suggests one of the most attractive and practical schemes for tuning the photonic bandgap devices [@kbsj99; @yoshino]. Nonlinear optical properties of LCs and multistability of light transmission are of a great interest for the future applications of LCs in photonics [@fs97].
Light polarized perpendicular to the LC director changes its orientation provided the light intensity exceeds some threshold value [@aszvfknknnslc80]. This effect is widely known as *the light-induced Freedericksz transition* (LIFT), and its theory was developed more than two decades ago in a number of the pioneering papers [@byznvtysc81; @ick81; @sddsmayrs81]. In particular, Zeldovich [*et al.*]{} [@byznvtysc81] demonstrated that the light-induced Freedericksz transition can generally be treated as the second-order orientational transition, but in some types of LCs hysteresis-like dependencies and two thresholds can be observed, for the increasing and decreasing intensity of the input light. The results obtained later by Ong [@hlo83] confirmed that for the MBBA nematics the Freedericksz transition is of the second order and there is no hysteresis behavior, whereas for the PAA nematics the Freedericksz transition is of the first order and the hysteresis-like behavior with two distinct thresholds should be observed. Although these conclusions have been confirmed to some extent in later experiments [@hlo83], the theory developed earlier was based on the geometrical optics and by its nature is approximate. The similar approximation was used later [@rshnvtbyz83] for taking into account a backward wave in a LC film placed in a Fabry-Perot resonator, where it was shown that the threshold of the Freedericksz transition depends periodically on the LC cell thickness.
Nonlinear optical properties of a nematic LC film in a Fabry-Perot interferometer was studied by Khoo [*et al.*]{} [@ickjyhrnvcys83], who considered the propagation of light polarized under an acute angle to the LC director and observed experimentally bistability in the output light intensity caused by giant nonlinearity of the LC film. Cheung [*et al.*]{} [@mmcsddyrs83] observed experimentally the effects of multistability in a similar system, including oscillations of the output light intensity.
However, in spite of numerous theoretical studies and experimental observations, a self-consistent theory of the light-induced Freedericksz transition based on a systematic analysis of the coupled equations for the nematic director and electromagnetic field is still missing. Therefore, the purpose of this paper is twofold. First, we consider a general problem of the light transmission through a homeotropically-oriented nematic LC and analyze the specific conditions for the multistability and light-induced Freedericksz transition, for possible applications in all-optical switching photonic devices. Second, for the first time to our knowledge, we consider this problem self-consistently and solve numerically a coupled system of the stationary equations for the director and Maxwell’s equations. We present our results for two kinds of nematic liquid crystal, para-azoxyanisole (PAA) and Np-methoxybenzylidene-np-butylaniline (MBBA), which show quite dissimilar behavior of the nematic director at the Freedericksz transition in the previous theoretical studies [@hlo83], and also discuss light transmission and bistability thresholds as functions of the cell thickness.
The paper is organized as follows. Sections II and III present our basic equations and outline our numerical approach. Section IV summarizes our results for two kinds of nematic liquid crystal and discusses in detail both bistability and hysteresis-type behavior of the light transmission. Section V concludes the paper.
Basic equations
===============
We consider a nematic LC cell confined between two planes ($z=0$ and $z=L$) with the director initially oriented along the $z$ axis (see Fig. \[fig1\]). The LC cell interacts with a normally incident monochromatic electromagnetic wave described by the electric field ${\bf E}({\bf r}, t)$, $$\label{eq1} {\bf E}({\bf r},t)=\frac{1}{2}\left[{\bf E}({\bf
r})e^{-i\omega t}+ {\bf E}^\ast ({\bf r})e^{i\omega t}\right].$$
![\[fig1\] (colour online) Schematic representation of the problem. A LC cell is placed between two walls ($z=0,z=L$), the vector ${\bf n}$ describes the molecules orientation in the cell. ](fig1.eps){width="50mm"}
To derive the basic equations, we write the free energy of the LC cell in the presence of the electromagnetic wave as follows [@byznvtysc81] $$\label{eq2}
F= \int (f_{\rm el} +f_E) dV,$$ where $$f_{\rm el} =\frac{K_{11}}{2}(\nabla\cdot{\bf n})^2+\frac{K_{22}}{2}({\bf n}\cdot\nabla\times{\bf n})^2 +\frac{K_{33}}{2}[ {{\bf
n}\,\times \nabla\times{\bf n}}]^2,$$ $$f_E =-\frac{1}{8\pi}\varepsilon _{ik} E_i E_k ^\ast\;,\;\;\; \varepsilon
_{ik} =\varepsilon _\bot \delta _{ik} +\varepsilon _an_in_k.$$ Here $f_{\rm el} $ is the LC elastic energy density, $f_E $ is a contribution to the free energy density from the light field, ${\bf n}$ is the nematic director, $K_{ii}$ are the elastic constants, $\varepsilon_{ik} $ is the LC dielectric permittivity tensor, $\varepsilon _a =\varepsilon _\parallel -\varepsilon _\bot >0$ describes anisotropy of the LC dielectric susceptibility, where $\varepsilon_\parallel $ and $\varepsilon_\bot $ are the main components of the tensor $\varepsilon _{ik} $ parallel and perpendicular to the director, respectively.
We assume that outside the LC cell the electric field is directed along the $x$ axis (see Fig. \[fig1\]), which can cause the director reorientation in the $xz$ plane inside the LC cell. When the incident beam is broad, we can describe it as a plane wave, so that all functions inside the LC cell will depend only on the $z$-coordinate. Therefore, we can seek the spatial distribution of the nematic director in the form $$\label{eq3} {\bf n}({\bf r})= {\bf e}_x \,\sin \,\phi (z)+{\bf e}_z
\,\cos \,\phi (z),$$ where $\phi$ is the angle between the director and the $z$ axis (see Fig. \[fig1\]), ${\bf e}_x$ and ${\bf e}_z$ are the unit vectors in the Cartesian coordinate frame.
After minimizing the free energy (\[eq2\]) with respect to the director angle $\phi$, we obtain the stationary equation for the LC director orientation in the presence of the light field
$$\begin{aligned}
\label{eq4} (K_{11}\sin^2\phi +K_{33}\cos^2\phi )\frac{d^2\phi}{d
z^2}-(K_{33}-K_{11})\sin\phi\cos\phi \left(\frac{d \phi}{d
z}\right)^2
+\frac{\varepsilon_a\varepsilon_\parallel\varepsilon_\bot}{16\pi}
\frac{\sin2\phi}{(\varepsilon_\bot +\varepsilon_a
\cos^2\phi)^2}\left|{E_x}\right|^2=0\;,\end{aligned}$$
where we take into account that, as follows from Maxwell’s equations, the electric vector of the light field inside the LC cell has the longitudinal component $E_z(z)=-(\varepsilon_{zx}/\varepsilon_{zz})E_x (z)$.
From Maxwell’s equations, we obtain the scalar equation for the $x$-component of the electric field, $$\label{eq5} \frac{d^2E_x }{d z^2}+k^2\frac{\varepsilon_\bot
\varepsilon _\parallel}{\varepsilon_\bot +\varepsilon_a
\cos^2\phi}E_x =0,$$ where $k=2\pi \lambda /c$, and $\lambda $ is the wavelength of the incident light. The time-averaged $z$-component of the Poynting vector, $S_z =(c/8\pi)E_x H_y^\ast$, remains unchanged inside the LC cell [@byznvtysc81; @hlo83], and it can be used for characterizing different regimes of the nonlinear transmission.
Numerical approach
==================
We solve the system of coupled nonlinear equations (\[eq4\]) and (\[eq5\]) in a self-consistent manner together with the proper boundary conditions. For the director, we assume the strong anchoring at the cell boundaries, i.e. $$\label{eq6} \phi(0)=\phi(L)=0,$$ whereas for the electric field we consider the standard scattering conditions $$\label{eq7}
E_x (0)=E_{\rm in} +E_{\rm ref} ,\quad E_x (L)=E_{\rm out}.$$ Here $E_{\rm in}$, $E_{\rm ref}$, and $E_{\rm out}$ are the amplitudes of the incident, reflected, and outgoing waves, respectively. In all equations above we consider the magnetic susceptibility $\mu=+1$, and the refractive index outside the cell $n_s =1$, also taking into account that $H_y =(1/ik)(dE_x /dz)$.
The boundary conditions (\[eq7\]) imply that we consider two counter-propagating waves on the left side of the LC cell, incoming and reflecting, whereas only an outgoing wave appears on the right side. Therefore, in order to solve this nonlinear problem, first we fix the amplitude of the outgoing wave $E_{\rm out}$. It allows us to find the unique values of the incident $E_{\rm in}$ and reflected $E_{\rm ref}$ waves.
Equation for the director (\[eq4\]) is similar to a general-type equation for a nonlinear pendulum with the fixed boundary conditions (\[eq6\]). This means that we should look for its periodic solutions with the period $2L$. In fact, there exist many periodic solutions of Eq. (\[eq4\]). First of all, a trivial solution $\phi(z)=0$ corresponds to the undisturbed orientation distribution of the director and the absolute minimum of the free energy (\[eq2\]). The Freedericksz transition occurs when this trivial solution becomes unstable for larger values of the input light intensity, and the director angle $\phi(z)$ becomes nonzero. We find this solution numerically by using the well-known [*shooting method*]{} [@nr]. By fixing the amplitude of the outgoing wave $E_{\rm out}$ and taking $\phi(L)=0$ at the right boundary, we find the values of the derivative $(d\phi/dz)_{z=L}$ such that after integration we obtain a vanishing value of the director angle at the left boundary, $\phi(0)=0$. By analyzing the nonlinear equation (\[eq4\]) in a two-dimensional phase space, we can show that the corresponding solution lies just below the separatrix curve, and it has no node between the points $z=0$ and $z=L$. This observation allows us to reduce significantly the parameter region for the required values of the derivative $(d\phi/dz)_{z=L}$. From the obtained set of solutions we chose the solution that corresponds to the absolute minimum of the free energy (\[eq2\]).
![\[fig2\] (colour online) (a,b) Spatial distributions of the field amplitude $|E_x|$ in the cell of MBBA, [*before*]{} (dashed) and [*after*]{} (solid) the light-induced Freedericksz transition, $L=\lambda /n_0$, $\lambda =6328\,A$, $n_0 =1.544$. (b) Spatial distributions of the director deviation angle $\phi(z)$ in the cell of MBBA [*after*]{} the light-induced Freedericksz transition, for $L=\lambda /n_0 $ (solid), $L=100\mu m$ (dashed), are shown together with the function $\phi_0 \sin (\pi z/L)$ at $\phi_0 =
1.483$ (dash-dotted). ](fig2.eps){width="80mm"}
We also take into account the fact that a finite energy barrier can appear between the minima of the free energy which correspond to the trivial and nontrivial solutions for the director orientation angle $\phi(z)$. When the light intensity decreases adiabatically, the director does not return to its initial undisturbed position at the threshold value of the “up” Freedericksz transition, but it remains in a disturbed state which corresponds to a local minimum of the free energy; thus, the transition to the state $\phi(z)=0$ takes place only when this energy barrier disappears. This leads to a hysteresis-like dependence of the director and the different threshold values for the “up” and “down” transitions in the director orientation.
{width="160mm"}
Results and discussions
=======================
We solve the nonlinear transmission problem for two kinds of nematic liquid crystals, para-azoxyanisole (PAA) and Np-methoxybenzylidene-np-butylaniline (MBBA), which possess different signs of the parameter $B=(1-9\epsilon_{||}/(4\epsilon_{\bot})-(K_{33}-K_{11})/K_{33})/4$, which appears in the geometrical optics approximation [@byznvtysc81; @hlo83]. According to their approach , the sign of this parameter $B$ determines the order of the Freedericksz transition. For PAA $B<00$ and the Freedericksz transition should be of the first order, while for MBBA $B>0$ and there should be the second order transition.
We take the following physical parameters [@hlo83]: (a) for PAA, $K_{11}
=9.26\cdot 10^{-7}dyn$, $K_{33} =18\cdot 10^{-7}dyn$, $n_0 =1.595$, $n_e =1.995$, at $\lambda =4800\,A$, and (b) for MBBA, $K_{11}
=6.95\cdot 10^{-7}dyn$, $K_{33} =8.99\cdot 10^{-7}dyn$, $n_0
=1.544$, $n_e =1.758$, at $\lambda =6328\,A$; and consider two values for the cell thickness, $L=\lambda /n_0$ and $L=100\mu m$.
{width="160mm"}
Spatial distributions of the electric field amplitude $|E_x(z)|$ in the LC cell before and after the light-induced Freedericksz transition occurs is presented in Fig. \[fig2\](a) for the parameters of MBBA and the cell thickness $L=\lambda /n_0$. For the other value of the LC cell thickness ($L=100\mu m$), the spatial distribution of the electric field is similar, but the number of the oscillations of the electric field $|E_x|$ inside the LC cell increases due to a larger value of $L/\lambda$. For PAA, a very similar distribution of the electric field is found. Thus, we reveal an essentially inhomogeneous spatial distribution of the electric field inside the LC cell, and the functions $|E_x(z)|$ are different before and after the Freedericksz transition.
Spatial distribution of the director orientation angle $\phi(z)$ inside the LC cell after the Freedericksz transition is shown in Fig. \[fig2\](b) for the parameters of MBBA, for $L=\lambda
/n_0 $ and $L=100\mu m$, respectively. On the same plot, we show the function $\phi_0\sin(\pi z/L)$ at $\phi_0 =1.483$ for comparison. We notice that the position of the maximum of the director deviation angle can shift from the point $z=L/2$, as a consequence of an asymmetric distribution of the field $|E_x(z)|$ inside the LC cell. Spatial distribution of the director angle $\phi(z)$ in the PAA cell has the same character as that shown in Fig. \[fig2\](b) for MBBA.
In Fig. \[fig3\], we present our numerical results for a change of the maximum deformation angle $\phi_{\rm max}$ of the director as a function of the power density inside LC $S_z$ for increasing and decreasing light intensity, for both PAA and MBBA and two values of the cell thickness, $L=\lambda/n_0$ and $L=100\mu m$. For both kinds of LC, we observe a hysteresis-like dependence of the angle $\phi_{\rm max
}$ and two different thresholds of the light-induced director reorientation: $S_z^{\prime}$, for the increasing intensity, and ${S_z}^{\prime\prime}$, for the decreasing intensity. In both cases, these two thresholds correspond to the first-order transition. The results are similar for two values of the LC cell thickness, see Figs. \[fig3\](a-d). Thus, our results suggest that at the light-induced Freedericksz transition the cells of both kinds of LCs, PAA and MBBA, reveal hysteresis-like behavior with the respect to $S_z$.
Dependencies of the amplitude of the outgoing wave $|E_{\rm out}|$ on the amplitude of the incident wave $|E_{\rm in}|$ are shown in Figs. \[fig4\](a-d) for the parameters of both PAA and MBBA. Depending on the LC cell thickness $L$, the cell transmission is characterized by either hysteresis or multistability with respect to the incident wave amplitude. In the case of small thickness of the LC cell ($L=\lambda/n_0$) only the hysteresis-like transmission is observed; it is caused by the hysteresis behavior of the director reorientation between “up” and “down” thresholds, as presented in Figs. \[fig4\](a,c). However, for larger thickness ($L=100\mu
m$) we observe the transmission multistability, above the “up” threshold for increasing light intensity, and above the “down” threshold for decreasing light intensity \[see Figs. \[fig4\](b,d)\]. Multistability in our system is similar to that of a nonlinear resonator, and is it determined by the resonator properties of a finite thickness of the LC cell.
![\[fig5\] (colour online) Thresholds of the director reorientation for increasing (solid) and decreasing (dotted) light intensities vs. the cell thickness $L$: (a) PAA, (b) MBBA. ](fig5.eps){width="80mm"}
The thresholds of the director reorientation for increasing and decreasing light intensities are shown in Figs. \[fig5\](a,b), for PAA and MBBA, respectively, as functions of the normalized thickness of the LC cell. Similar to the results of the geometrical optics approximation [@byznvtysc81; @hlo83], the threshold values are proportional to $(1/L)^2$, but they increase approximately in two times due to an essentially inhomogeneous spatial distribution of the electric field inside the LC cell. A similar increase of the threshold value for an inhomogeneous distribution of the electric field in the LC cell was also mentioned by Lednei [*et al.*]{} [@lednei95]. In addition, for the “up” threshold we observe an additional periodic dependence of the threshold value on the cell thickness $L$, which is typical for resonators and is caused by an interference of two counter-propagating waves in the LC cell. This result agrees with the results obtained for LC in a Fabry-Perot resonator [@rshnvtbyz83]. The “up” threshold is determined by a competition between the electric field forces and elastic forces of the liquid crystal, and thus the interference distribution of the electric field in the LC cell is important. However, the “down” threshold is defined by the condition of the disappearance of a barrier between the local and absolute minima of the LC free energy [@byznvtysc81]. We suppose that difference of these mechanisms leads to the different type of $L$-dependencies for the “up” and “down” thresholds.
We should mention that our results differ qualitatively from the results of earlier studies [@byznvtysc81; @hlo83], where for MBBA both hysteresis and bistability were not predicted. In the simplest case of one traveling wave [@byznvtysc81; @hlo83], the conservation of the value of $S_z$ during the Freedericksz transition leads to the conservation of the electromagnetic field amplitudes at the boundaries of the LC cell. However, in the general case there always exists a reflected wave, so that we have $S_z =(c/8\pi)E_x(0)H_y^\ast (0)=(c/8\pi )(E_{\rm
in}+E_{\rm ref})(E_{\rm in} -E_{\rm ref})=(c/8\pi )(E_{\rm
in}^2-E_{\rm ref}^2)=S_{\rm in}-S_{\rm ref}$, where $S_{\rm in}$, $
S_{\rm ref}$ are the power densities of the incident and reflected waves, respectively. In such a case, the conservation of $S_z $ does not require the conservation of $S_{\rm in}$ and $S_{\rm ref}$, so that the amplitudes of the electromagnetic fields at $z=0$ can change at the Freedericksz transition, as is seen in Fig. \[fig2\]. Thus, the problem solved in this paper and that in Refs. [@byznvtysc81; @hlo83] corresponds to different boundary conditions. Therefore, we suggest that experimentally observed the second order Freedericksz transition for MBBA liquid crystal [@ick81; @csillag81] is caused by the weak reflection from the boundaries of LC cell. In that situation, the single wave approximation can be used and results obtained in Refs. [@byznvtysc81; @hlo83] become valid.
Conclusions
===========
We have analyzed the light transmission through homeotropically-oriented cell of a nematic liquid crystal, and studied multistability and light-induced Freedericksz transition. We have solved numerically the coupled stationary equations for the nematic director and electric field of the propagating electromagnetic wave, for two kinds of liquid crystals (PAA and MBBA). We have found that the liquid crystals of both kinds possess multistability and hysteresis behavior in the transmission characterized by two thresholds of the director reorientation, so that for the increasing and decreasing light intensities the Freedericksz transition is of the first order.
We have demonstrated that the resonator effects of the liquid-crystal cell associated with the light reflection from two boundaries are significant, and they are responsible, in particular, for the observed periodic dependence of the threshold values and multistability of the transmitted light as a function of the cell thickness. We expect that these features will become important for the study of periodic photonic structures with holes filled in liquid crystals [@aemipysk06] where multiple reflection effects and nonlinear light-induced Freedericksz transition should be taken into account for developing tunable all-optical switching devices based on the structure with liquid crystals.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the Australian Research Council. Yuri Kivshar thanks B.Ya. Zeldovich, M.A. Karpierz, and I.C. Khoo for useful discussions.
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|
---
address:
- '*Departamento de Física Teórica, Universidad de Zaragoza, 50009 Zaragoza, Spain.*'
- ' Departamento de Matemáticas e Informática, Universidad Pública de Navarra, 31006 Pamplona, Spain.'
author:
- 'M. Asorey$\,$ , D. Garcia-Alvarez'
- 'J.L. López'
date:
title: 'Non-analyticities in three-dimensional gauge theories'
---
epsf.tex newreferences.tex
plus 2pt minus 1pt
In 2+1 dimensions the number of degrees of freedom of massive and massless relativistic particles is the same. This peculiar behaviour permits a smooth transition from massless to massive regimes in the same theory without the need of extra fields. In gauge theories this transition can be simply achieved by the addition of a Chern-Simons term to the ordinary Yang-Mills action . For the same reasons there is no protection against the existence of radiative quantum corrections which either generate or suppress the topological mass.
The special characteristics of Chern-Simons term and its peculiar transformation law under large gauge transformations requires the quantization of its coupling constant $k\in \IZ$ when the gauge group is compact. The constraint arises in the covariant formalism as a consistency condition for the definition of the euclidean functional integral due to the special transformation properties of the Chern-Simons action under large gauge transformations . In the canonical formalism it appears as a necessary condition for the integration of Gauss law on the physical states . Both interpretations are based on non-infinitesimal symmetries and therefore the quantization condition can not be inferred from perturbative arguments. However, unexpectedly the perturbative contributions of quantum fluctuations do not seem to change the integer nature of the Chern-Simons coupling constant in most of standard renormalization schemes –. From a pure quantum field theory point of view this behavior is bizarre because in absence of perturbative symmetry constraints there must always exist regularization schemes where the effective values of the coupling constants of marginal local terms are arbitrary. Indeed, such regularization schemes exist but require a fine tuning of the leading ultraviolet behavior of parity even and parity odd terms of regulators .
The perturbative quantum corrections are not the only contributions of quantum fluctuations. There exist additional contributions to the effective gauge action which cannot be obtained in perturbation theory because they are not analytical on gauge fields. The presence of such non-analytic contributions in one loop approximation is more evident in the case of regularizations which do not preserve the integer value character of the effective Chern-Simons coupling. They appear as necessary to compensate the anomalous transformation law of Chern-Simons terms under large gauge transformations. The role of those terms is crucial to understand the finite temperature behaviour of gauge theories in $2+1$ dimensions –. They are similar to the well known non-analytic terms which appear in the $\eta$–invariant of the spectral asymmetry of the operator $\ast d^{\ }_A+d_A\ast$ induced by the changes of signs in the spectral flow .
The study of non-analytic terms of the effective action and their physical implications is the main goal of our analysis. The discontinuities associated to these terms yield singularities which in the case of Chern-Simons theory seem to be mere artifacts of perturbation theory. The origin of the singularities is the same that appears in ordinary gauge theories in presence of massless quarks in the fundamental representations. In this case the singularities do have a simple physical origin, the existence of zero-modes of Dirac operator.
The main result of the paper is the proof that this kind of non-analyticities are regularization dependent which provides a further support to the claim that different renormalization methods define in fact different physical theories. The perturbative corrections to the coupling constant can also be different and depend on the regularization method but those differences can be compensated in general by the choice of different renormalization schemes. However, the presence of different non-analytic contributions cannot be changed by the choice of appropriate renormalization schemes. In some way this provides a physical meaning to the non-perturbative constraint that requires the coupling of counterterms must be an integer value. The meaning of the restriction is that the analytic behaviour of the effective partition function cannot be changed by the renormalization scheme and provides a novel physical role to the choice of regularization method.
Parity anomaly and framing anomaly have a common origin in the existence of odd quantum effects. Because of their dependence on the regularization method it is possible, thus, to find out some regularization regimes where both anomalies are absent.
Finally, the regularization dependence of these phenomena is also responsible of the failure of simple attempts to define a Zamolodchikov’s c-function in terms of gravitational terms in order to generalize of Zamolodchikov c-theorem for three-dimensional theories.
In the limit of infinite topological mass the gauge theory reduces to a Chern-Simons topological theory governed by the action $${ k}\ S_{\rm CS}(A) = {{ k}\over 4\pi}
\int_{M}{\rm Tr}\left(A\land
dA +{2\over 3}A\land A\land A\right) \qquad k\in
{\field Z}$$ where the coupling constant $k$ must be an integer for compact groups to have a consistent quantization . Let us consider $SU(N)$ gauge field theories for simplicity.
The theory is superenormalizable also in this topological limit. In the Hamiltonian formalism divergences appear in the normalization of physical states and the hermitian product of the Hilbert space . The removal of these divergences generates a shift in the renormalized Chern-Simons coupling constant $k_{\rm R}= k+ N$. In the covariant formalism the propagator is very singular because of the large gauge symmetry of the theory originated by its topological character. In perturbation theory one way of improving the UV behaviour of the propagator without breaking gauge invariance is by introducing higher derivative regulating terms into the classical action, e.g. $$S_{\rm \Lambda}(A)= S_{\rm CS}(A)+ S^+_{\rm R}(A)$$ $$S_{\rm R}^+(A)={\lambda_+\over \Lambda}
\int_{M}{\rm Tr}\, F_{\mu\nu}(A) \left({\mathbb{ I}}+
{\bbm{\Delta}_A \over \Lambda^2}\right)^m F_{\mu\nu}(A),$$ where $\bbm{\Delta}_A=d^\ast_Ad^{\phantom \ast}_A+d^{\phantom \ast}_Ad^\ast_A $ is the covariant laplacian. For large enough values of the exponent $m$ there are not UV superficial divergences in diagrams with more than one loop. However, one loop divergences need of an extra Pauli-Villars regularization . The resulting one loop effective action has no divergences even after the removal of the ultraviolet regulator $\Lambda\to\infty$. The renormalized perturbative effective action is of the form with $k_{\rm R}= k+ N$. The first non-trivial contribution to ${\rm \Gamma}(A^R)$ arises from the four point function .
The Hamiltonian approach yields similar results, but this coincidence is not based on general symmetry principles. Thus, it should be possible the existence of a regularization where the renormalization of $k$ is not a simple shift of $k$ by N units. Indeed, there exist other gauge invariant regularizations, e.g. . $$S_{\rm \Lambda}(A)= S_{\rm CS}(A)+ S^-_{\rm R}(A)$$ $$S_{\rm R}^-(A)={\lambda_-\over \Lambda^2}
\int_{M}{\rm Tr}\, { \epsilon^{\alpha\sigma\mu}}\,
F_{\alpha\nu}(A) \left(\mathbb{I}+
{\bbm{\Delta}_A \over \Lambda^2}\right)^n
D^\sigma_A\left(\mathbb{I}+
{\bbm{\Delta}_A \over \Lambda^2}\right)^n \! F_{\mu\nu}(A)$$ which after removing one-loop divergences yield an effective action like , but without radiative contributions to the effective value of the coupling constant $k_{\rm R}= k$. Even more general regularizations can be conceived, e.g. $$S_{\rm \Lambda}(A)= S_{\rm CS}(A)+ S^+_{\rm R}(A)+
S^-_{\rm R}(A)$$ In that case the result depends on the relative weights $\lambda_->0$ and $\lambda_+>0$ of $S^+_{\rm R}$ and $S^-_{\rm R}$ $$k_{\rm R}= \begin{cases}
{
{k + N}\phantom{\Big[}}& {{\rm if}\ m > 2n+1/2}\cr% \displaystyle
{k + {2 N \over\pi}
\arctan{\lambda_+\over\lambda_- }}& {{\rm if}\ m = 2n+1/2}\cr
{{k} \phantom{\Big[}}& {{\rm if}\ m < 2n+1/2 }\cr
\end{cases}$$
In these very general regularization schemes the radiative corrections to the coupling constant present three different regimes which depend on the interplay between the ultraviolet behaviors of parity even terms $S^+_{\rm R}$ of the regularized action and the parity odd terms of $S^-_{\rm R}$.
In the first regime the leading ultraviolet terms are parity even. The effective coupling constant gets shifted by $N$ ( $k\to k+N$), due to one loop gluonic radiative corrections. The third regime is characterized by an ultraviolet behavior dominated by parity odd terms and the absence of radiative corrections to $k$. In the transition regime parity even and parity odd terms have the same ultraviolet behavior and the quantum corrections to $k$ can take any real value which depends on the relative coefficients of the leading terms of parity even and parity odd interactions.
The phenomenon can be pictorially understood by looking at the way the shift of $k_R$ is generated. In fact, $${ k_R}=k+ {2N\over\pi}
\int_{0}^{\infty}{d\Phi \over 1+\Phi^2}=k+{N\over \pi}\arctan \Phi(\infty)$$ and the behaviour of $$\Phi={\lambda_+ p(1+p^2)^m\over
1+\lambda_- p^2(1+p^2)^{2n}}$$ is dictated by the form of $S_{\rm \Lambda}(A)$
The actual value of the effective coupling constant can always be modified by a different choice of renormalization scheme because the Chern-Simons term is local and can be added as a conterterm. However, as pointed out in the previous section, the behaviour of Chern-Simons term under large gauge transformations requires that the bare coupling constant $k$ must be an integer number otherwise the quantum theory will be inconsistent, e.g. the functional integral will be ill defined. Such a constraint is a pure non-perturbative requirement, because large gauge transformations map small fields into large gauge fields and, therefore, they are genuine non-perturbative symmetries. In consequence, although in perturbation theory any local BRST invariant counterterm is valid, only counterterms which preserve the non-perturbative consistency condition can be added to the bare action. The condition imposes a very stringent constraint on counterterms which have to preserve the integer valued character of the bare coupling constant $k$. In particular, if the effective value of $k_R$ is not an integer, one cannot reduce the physical behavior of the system to the standard integer valued case by a consistent renormalization. Thus, the first and third regularization schemes are generic and equivalent from the physical point of view but the transition regime $m=2n+\ha$ can not be reduced any of the other two regimes by the choice of a different scheme renormalization. In fact, the regime defines a new different theory.
In the generic case there is a correspondence of Chern-Simons states and the primary fields of rational conformal field theories . In the transition regime the corresponding two-dimensional theory will be non-rational. In this sense, the transition regularization really defines a new type of theory,
The existence of different regimes in the regularization of Chern-Simons theory opens new possibilities for the analysis parity anomaly.
This insight is further supported by the existence of a straightforward connection between one loop corrections of Chern-Simons theory and the determinant of a massless fermion in the adjoint representation . Indeed, the second variation of Chern-Simons term and the corresponding ghost terms in a covariant Landau gauge yields an operator $\SS_A$ which is equivalent to the square of Dirac operator $(\Dsla^{^{ \rm ad}})^2$ for adjoint fermions. Therefore, $${\det}{ \Dsl_{\af A}^{^{ \rm ad}} }=
{\rm e}^{-{1\over 2}\Gamma^{[1]}{(A)}}.$$ The effect of the existence of different regularization regimes is more intriguing because gauge invariance seems to be broken in the transition regime. Indeed, three different regimes can be generated by the following regularization of Dirac operator $$\Dsl_{\sc A}^\Lambda=
{ \Dsl_{\sc A}}+ \lambda_+
{\Dsl_{\sc A}^2 \over \Lambda}
\left(\mathbb{I}+{\Dsl_{\sc A}^2\over
\Lambda^2}\right)^{m}
+\lambda_-{\Dsl_{\sc A}^3
\over \Lambda^2}\left(\mathbb{I}+{\Dsl_{\sc A}^2
\over \Lambda^2}\right)^{2n}$$ with $\lambda_\pm>0$ and the corresponding Pauli-Villars regulators. In that case the effective Chern-Simons coupling behaves in a similar way to the case of pure Chern-Simons theory.
$$k_{\rm R}= \begin{cases}
%\displaystyle
{{N}{\phantom{\Bigr[ \Bigl]}}} & {{\rm if}\ m > 2n+{1\over 2 }}\cr
% \displaystyle
{ {2 N \over \pi} \arctan{\lambda_+\over\lambda_- }}& {{\rm if}\ m = 2n+{1\over 2 }}\cr
%\displaystyle
{{0} {\phantom{\Bigr[ }}} &{{\rm if}\ m <
2n+ {1\over 2 }}\end{cases}$$ If the fermions are in the fundamental representation of $SU(N)$ the result is analogous $$k_{\rm R}= \begin{cases}
%\displaystyle
{{1\over 2}
\phantom{\Bigr[ \Bigl]}}
& {{\rm if}\ m > 2n+{1\over 2 }}\cr
%\displaystyle
{ {1 \over\pi} \arctan{\lambda_+\over\lambda_- }} &{{\rm if}\ m = 2n+{1\over 2 }}\cr
%\displaystyle
{{0}
{\phantom{\Bigr[ }}}
&{{\rm if}\ m <
2n+ {1\over 2 }}\end{cases}$$ Although the Pauli-Villars regularization method used here is completely gauge invariant also under large gauge transformation, gauge invariance under those transformations seems to be broken in the first two cases because the effective Chern-Simons term is not invariant. The puzzle is solved by noticing that the analytic perturbative radiative corrections do not exhaust all quantum corrections to the effective action. In fact, gauge invariance requires that the full radiative corrections must have a non-analytic conterpart which permit to recover full gauge invariance. Indeed, $${\rm \Gamma}^{\rm }(A^R)={\rm \Gamma}_R(A^R)+
i {\rm \Gamma}_I(A^R)\qquad{\rm with }\quad{\Gamma^{}_{I}(A)}={ k_{\rm R}}
\, S_{\rm CS}(A)+ h(A)$$ where $ h(A)$ has a non-analytic dependence on $A$. However, in that case parity symmetry is not preserved at the quantum level because $ S_{\rm CS}(A)$ is not invariant under parity symmetry whereas as it will be shown later $h(A)$ is parity invariant. This fact is on the origin of parity anomaly of three-dimensional massless fermions.
However, what is really intriguing is that in the third regime $m<2n + 1/2$ there is no parity anomaly because $k_R=0$ and the theory is at the same time invariant under global gauge transformations. This means that in fact, contrary to the common wisdom, the parity anomaly is not an unavoidable physical phenomena in a gauge invariant framework. A similar result is obtained with standard regularizations and infinite number of Pauli-Villars fields or lattice regularizations . The results are reminiscent of those obtained by Slavnov for the cancellation of the $SU(2)$ global Witten’s anomaly in four-dimensional theories with chiral fermions in the fundamental representation . The main difference between both results is that the Slavnov method requires an infinite number of Pauli-Villars regulating fields to cancel the anomaly in a gauge invariant way, whereas in this case a very simple UV modification of fermionic interactions yields a similar effect with a finite number of Pauli-Villars fields.
In the transition regime parity is also broken but the coefficient of the terms responsible for this phenomenon are different from those of the case where $m>2n + 1/2$. The ambiguity in the appearance or not of parity anomaly suggests that the effect looks more like a spontaneous symmetry breaking than a genuine anomaly breaking. Perhaps the phenomenon is nothing but a simple example of a more general feature on the breaking mechanism of discrete symmetries in three dimensions.
Indeed, the same phenomenon arises in the analysis of pure Yang-Mills theory Using a similar regularization method which includes parity odd regulating terms and Pauli-Villars fields a Chern-Simons term can be induced in pure Yang-Mills theories. The infrared behaviour is dominated by the Yang-Mills term which is parity even. The leading UV terms might be either a parity even term of the type $S^+_R(A)$ or a parity odd term like $S^-_R(A)$. The general result is $$k_{\rm R}= \begin{cases}
%\displaystyle
{{0}{\phantom{\Bigr[ \Bigl]}}} & {{\rm if}\ m > 2n+{1\over 2 }}\cr
% \displaystyle
{ - {2 N \over \pi} \arctan{\lambda_+\over \lambda_- }}& {{\rm if}\ m = 2n+{1\over 2 }}\cr
%\displaystyle
{{-N} {\phantom{\Bigr[ }}} &{{\rm if}\ m <
2n+ {1\over 2 }}.\end{cases}$$ In the first case no Chern-Simons coupling is generated whereas in the third case there is a non-trivial Chern-Simons radiative contribution with a coefficient $k_R=-N$. Both results follow from the behaviour of the flow displayed in Fig. 1. In the third case the non-trivial term generates a topological mass $$m={g^2 N \over 2\pi}$$ which is in agreement with the actual value of the mass gap in pure Yang-Mills theories . The generation of a term in the pure Yang-Mills theory points out the instability of the renormalization group flow. Moreover, it points toward a possible mechanism of generation of a mass gap in pure Yang-Mills theory. In this regime the theory is massive but parity symmetry is broken unlike the standard regime of Yang-Mills theory.
In the transition regime the theory gets a mass which depends on the relative weights of the leading parity even and parity odd terms.
The existence of non-analytic contributions to the imaginary part of the effective action ${\Gamma^{[1]}_{I}(A)}={ k_{\rm R}}
\, S_{\rm CS}(A)+ h(A)$ of massless fermionic determinants is known since the discovery of the spectral asymmetry and index theorem . In the present case they are pointed out by the existence of Chern-Simons terms with non-integer coefficients . The Pauli-Villars regularization method preserves gauge invariance and the only way to ensure the gauge invariance of the final result is by admitting the existence of a non-analytic contribution in $h(A)$ which transforms as under large gauge transformations.
The fermionic determinant $ \Dsla$ is expected to have an analytic dependence on $A$ but the effective action is the logarithm of this determinant. The existence of a zero in the determinant induces a singularity in the effective action. Thus, the effective action $\Gamma_A$ diverges for ([*nodal*]{}) gauge configurations with fermionic zero modes. Every zero of an analytic function has an integer degree, which is measured by the discontinuity of the imaginary part of the corresponding logarithm. Thus, if the fermionic determinant has an analytic dependence on the background gauge field, the only possible discontinuities at a nodal configurations must by integer multiples of $\pi$ depending on the order of the zeros. For the simple zeros the value of the discontinuity through any continuous path of gauge fields crossing the nodal field is $\pi $. For double zeros the discontinuity is $2\pi$ and so forth. If the trajectories of fields correspond to paths of three dimensional gauge fields induced by four-dimensional gauge fields with non-trivial topological charge $q$ it can be shown from the index theorem that the total discontinuity along the trajectory will be equal to $2\pi q$ .
The regularized value of the fermionic determinant in the transition regime has an imaginary component of the effective actions which undergoes non-integer discontinuities. This fact signals an extra degree of non-analyticity ([no holomorphy]{}) of the determinant in the transition regime and, thus, indicates that there is a radical difference of the transition regime with the other regimes. It has a completely different new physical behaviour which in any case cannot be interpreted in pure analytic terms.
Moreover, because of the parity symmetry of Dirac operator, if A is a nodal gauge field its transformation under parity $A^P$ is also a nodal configuration which implies that the singularities are invariant under parity transformation. Thus, the whole non-analytic component $ h(A)$ is parity preserving, i.e. The true source of parity symmetry breaking has a perturbative origin: the induced Chern-Simons term. Although the Chern-Simons radiative corrections can removed by a local counterterm, gauge invariance under large gauge transformation is broken if $k_R\notin \IR$. Therefore the theory cannot be parity invariant and gauge invariant at the same time in this case. Only in the case $k_R=2\pi n$ both symmetries can be simultaneously preserved.
In other terms, because of the hermiticity of $\Dsla$ all eigenvalues are real. Thus, the only source of imaginary terms in the effective action comes from negative eigenvalues $-1= {\rm e}^{i\pi}$ . Now at nodal points one positive eigenvalue becomes negative or one negative eigenvalue becomes positive. Thus, generically, $ h(A)$ has a $\pi$ discontinuity at configurations with fermionic zero modes. In the case of fermions in the adjoint representation, there is a level crossing at nodal points between eigenvalues becoming positive and eigenvalues becoming negative (see Figure 3). This explains why in that case the singularities of the effective action do not have any physical effect.
In all gauge invariant regularizations the non-analytic term of the effective action $h(A)$ is proportional to $k_R-k$ as a consequence of gauge symmetry. This means that the effective counting of zero-level crossings becomes regularization dependent.
In order to illustrate the phenomenon let us consider a lower dimensional example: a fermionic quantum rotor under the action of a magnetic flux . In one dimension the equivalent of Chern-Simons action is $${ k}\ S_{\rm cs}={k}\int A= { k}\, \epsilon$$ and the fermionic determinant of $\Dsla=d_\theta+A_\theta$ can be exactly computed $$\det \Dsla = {\rm e}^{- \Gamma(A)}$$ with with the renormalized coupling constant $$k_{\rm R}= \begin{cases}
{1\over 2}& {{\rm if}\ m > 2n+1/2}\cr
{ {1 \over\pi}
\arctan{\lambda_+\over\lambda_- }}& {{\rm if}\ m = 2n+1/2}\cr
{0} &{{\rm if}\ m < 2n+1/2 }\end{cases}$$ depending on regularization parameters $m,n$ and $\lambda_+,\lambda_-$. ${\rm Int}(x)$ denotes the integer part of $x$. The effective action , is gauge invariant for any value of these parameters. The Chern-Simons term of the imaginary of the effective action $ { k_R}\ S_{\rm cs}=k_R \, {\epsilon}$ is compensated by a non-analytic component which is parity invariant but transforms under global gauge transformations in a way that compensates the anomalous transformation of the Chern-Simons part. Notice that the whole imaginary part of $\Gamma(A)$ is proportional to $k_R$ in all regularization regimes. In fermionic determinants the interpretation of singularities in terms of nodal configuration is quite natural. However, in Chern-Simons theory the divergence of $\Gamma_A$ at one-loop order is more intriguing because there is not an apriori reason for singularities. In the Schrödinger representation physical states are described by functionals of gauge fields which as pointed out in in Chern-Simons theory vanish at certain nodal configurations. It is therefore not unreasonable that the effective action of the theory could diverge at some classical configurations which might be related to nodes of the vacuum state. In general, this type of singularity indicates a suppression of tunneling. One can identify some configurations where the one loop effective action diverges. In fact, it is easy to show that there is a discontinuity of $h(A)$ at the sphaleron gauge field on $S^3$. This is a gauge field which is a saddle point of Yang-Mills action, i.e. it satisfies the Euclidean Yang-Mills equations and Bianchi identity It is given explicitly by $$\left[A_{{\rm{sph}}}\right]_j={ 4R\over (x^2+4 R^2)^2}\left(
4R \epsilon^a
_{jk} x^k - 2x^a x_j+ [x^2-4 R^2]\delta^a_j\right)\sigma_a$$ for $SU(2)$ gauge fields ($R$ is the radius of the $S^ 3$ sphere). The proof that $A_{{\rm{sph}}}$ is a nodal point follows from equations which imply that $\ast F(A)$ is a zero mode of the operator $ \Delta_A$ which generates the one-loop corrections of the . There exist a similar phenomenon for massless fermions on the adjoint representation. The sphaleron is a nodal configuration of the corresponding determinant with the same spectral flow. Now, since the fermions are in the adjoint representation, two level cross the zero level at the sphaleron configuration (see Fig. 3).
In this case the fermionic determinant $\det \Dsla$ has the same properties that the vacuum state of $3+1$ dimensional gauge theories at $\theta=\pi$ and the discontinuity of $h(A)$ at sphaleron configurations on $S^3$ is a physical property which encodes the tunneling suppression due to the effect of massless fermions .
The dependence of the fermionic determinant on the background gauge field contributes to the understanding of the role of singular contributions in the effective action. The existence of zero modes determine the existence of discontinuities in the imaginary part of the effective action. In the real part the singularities are more severe. At nodal configurations the real part of the effective action becomes infinite signaling the failure of perturbation theory and the vanishing of the corresponding determinant. The analysis of the physical role of these singularities in Chern-Simons theory and its possible survival at higher orders in the coupling constant $1/k$ is an open problem.
In order to obtain a better physical picture of the transition regime let us analyse the case of pure Abelian Chern-Simons theory. Although in general there is no quantization condition for the coupling constant $k_R$, in the presence of magnetic monopoles in $M= S^1\times {\af T}^2$, consistency requires its quantization.
In temporal gauge and flat gauge fields the effective action reduces to $$%{ k_R}
S_{\rm cs}={ k_R \pi}\epsilon^{ij} \int a_i\dot{a}_j$$ and can be quantized as a quantum Hall effect in a dual torus $\widehat{\af T}^2$ with magnetic charge $k_R\in\IZ$. The number of physical states is finite and equals the value of the magnetic charge $k_R$. This explains why $k_R$ should be quantized.
In the transition regime a massless fermion induces an effective action with $k_R\notin \Z$ and extra non-analytic terms in the imaginary part. In order to analyze the physical effect of these terms let us consider a slightly different action with a similar basic behaviour $$S_{\rm cs}={ k_R \pi}\epsilon^{ij} \int [a_i]\, \dot{a}_j$$ where $[x]=x-{ }{\rm Int}({x})$ and $k_R\notin \IZ$. The system governed by such an action is equivalent to a charged particle moving in a torus under the action of two magnetic fields: one uniform magnetic field with non-integer total magnetic flux $k_R$ across the torus , and an extra magnetic field with a delta-like singularity whose magnetic flux just cancels that of the uniform magnetic field. $$F_{12}={k_R\pi}\left[2-\delta\left({a_1}\right)-\delta\left({a_2}\right)\right].$$ Thus, the total magnetic flux is null and gauge invariance under large gauge transformations is restored.
The quantum system has in this case only one vacuum state. Thus, the physical regime associated to transition regularization may be very different from the one obtained from generic regularization schemes. This would explain in physical terms the smooth interpolation between the two generic regularization regimes through the transition regime.
The fact that different regularizations of the theory give rise to different quantum theories is not so surprising. One simple but paradigmatic example is topological quantum mechanics on a Riemann surface $\Sigma$ of genus $h$ in the presence of a magnetic field $A$ with magnetic charge $k$ . In standard Hamiltonian formalism the quantum Hamiltonian is trivial ($H=0$) as corresponds to a topological theory and the dimension of the space of quantum states is finite and given by $
\dim {{\mathcal H}}_k^0= 1-h + k,$ for $k> h-1$. However, if the theory is regularized by means of a metric dependent kinetic term, $$L(x,\dot x)= {1\over 2 \Lambda} g_{ij}\dot{x}^i\dot{x}^j+
A_i \dot{x}^i,$$ the Hamiltonian becomes $H_{\Lambda}={\Lambda\over 2}\Delta_A^g$, and the topological limit $\Lambda\to \infty$ is governed by the ground states of $H_{\Lambda}$. The quantum Hilbert space of the topological field theory obtained by this method can have a dimension lower than $1-h+k$, depending on the symmetries of the background metric $g$ of the regularization . In particular, this is the case when the metric $g$ breaks the degeneracy of the ground state of the covariant Laplacian $\Delta_A^g$. The standard result is obtained by choosing only metrics which is compatible with the magnetic field $B=dA$, in the sense that they give rise to a Kähler structure on $\Sigma$.
The existence of different regimes in the ultraviolet regularization in Chern-Simons theory also has relevant implications for the induced gravitational interactions. Although Chern-Simons action is metric independent the quantum corrections generate a finite gravitational term This term which gives rise to a metric independent effective action can be canceled by the introduction of a local counterterm. But then a framing anomaly is generated as a physical effect of the theory . The novel effect is that this anomaly also become dependent on the regularization regime as the parity anomaly.
The induced gravitational term was conjectured to be of the form $\kappa= {c/24}$, where $c$ is the central charge of the conformal theory associated to Chern-Simons theory . In the present case, $ c=k(N^2-1)
/(k+N)$. In perturbation theory, this means that $$\kappa={N^2-1\over 24} \sum_{n=0}^{\infty} (-{N\over k})^n .$$ However, as anticipated $\kappa$ depends on the choice of regularization regime. The one loop contribution $$\kappa^{[1]}_{\rm R}= \begin{cases}
{N^2-1\over 24}& {{\rm if}\ m > 2n+1/2}\cr
{ {N^2-1 \over 12\pi}
\arctan{\lambda_+\over\lambda_- }} & {{\rm if}\ m = 2n+1/2}\cr
{0} &{{\rm if}\ m < 2n+1/2 }\end{cases}$$ agrees with the expected value $\kappa=(N^2-1)/24 $ only if $m> 2n+ \ha$. The vanishing of $\kappa$ in the regime with $m< 2n+ \ha$ was first anticipated by Witten . In this scheme a second order perturbative calculation was carried out in Refs. , and the result seems to agree with the standard case. In the transition regime $\kappa$ depends on the weights $\lambda_+$ and $\lambda_-$ of the parity odd and parity even regulators and does not correspond to any previously expected behaviour. In this case there is relation between the value of $\kappa$ and the renormalized coupling constant $k_R$ , $$\kappa ={(k_R-k)(N^2-1)\over 24 N}.$$ The above results suggest that this relation holds for the three regimes. It would be very interesting to investigate if the property also holds beyond one loop approximation.
Other types of gravitational terms like Einstein or cosmological constant terms can also be generated in the effective action, but they present linear or cubic UV divergences which need to be renormalized leaving an extra ambiguity in the actual values of the corresponding renormalized couplings. Metric independence requires the cancellation of both couplings. But the same gravitational term also contains some hidden Einstein and cosmological terms when the gauge field is written in terms of the [*vierbein*]{} and the spin connexion . The different values of the renormalized gravitational constant also adds an extra source of metric dependence. Although the induced Chern-Simons term can be removed by a choice of renormalization scheme its non-analytic counterpart cannot and in fact yields an extra frame dependent contribution. Only the third regime provides a fully consistent metric independent theory without parity and framing anomalies.
This connection between the renormalization of Chern-Simons coupling and the induced gravitational Chern-Simons coefficient ant its relation to the central charge of the associated conformal theory suggests a possible extension of Zamolodchikov’s c-theorem to three-dimensional systems. Topological Chern-Simons theories would correspond to two dimensional conformal theories and the interpolating regularized topologically massive theories will generate a flow from one theory with one Chern-Simons coupling to another with a different one. A c-theorem would establish the existence of a monotone function along this renormalization group flow which will coincide with the coupling of gravitational term at topological fixed points. One natural candidate for Zamolodchivov c-function, thus, can be defined in terms of the induced gravitational Chern-Simons term which is identical to $\kappa$ at the pure Chern-Simons theories and varies along renormalization group trajectories. A concrete proposal based on an version of Zamolodchikov theorem formulated in Ref. can be established from the following spectral representation of the stress tensor correlators ${ c(\lambda)}$ emerges as a natural candidate for a Zamolodchikov c-function for three-dimensional theories. Unfortunately, $c(\lambda)$ cannot be universally monotone for the same reasons as the similar spectral representation of the flow of the effective $k$ coupling cannot be monotone in all regularization regimes of pure Chern-Simons theory (see Fig.1) . This negative result does not exclude the existence of another extension of c-theorem to 2+1 dimensional theories. It merely points out that the spectral representation of the gravitational Chern-Simons term is not a good c-function. On the other hand for purely bosonic theories there are not axial Chern-Simons like interactions which could generate a simple way of describing the irreversibility of the renormalization group flow.
The presence of non-analytic terms in the effective action is fundamental for the right physical description of and massless fermions in 2+1 dimensions. The existence of such contributions is pointed out by the appearance of different perturbative corrections in different gauge invariant regularizations. The discontinuities associated to those non-analytic terms signal the presence of physical singularities associated to the zeros of the partition function in some backgrounds. The appearance of nodal configurations is also related to the quantum tunneling suppression. In massless fermions nodal contributions are associated to the existence of fermionic zero modes. However, the structure of singularities and discontinuities depends on the regularization regime and makes possible the physical differences between the corresponding theories. In regularizations with transition regimes the nature of the singularities associated to non-analytic terms suggest a non holomorphic behaviour of the effective partition function in terms of classical fields. It is remarkable that it is possible to find out gauge invariant regularization regimes where there are not parity and framing anomalies. This suggests that those anomalies can be better understood as spontaneous symmetry breaking phenomena rather than genuine anomalies. In some toy models it has been shown that the transition regularizations keep constant the number of physical states. It would be very interesting to analyze the behaviour of the number of physical states of the in the transition regime and verify if it is dramatically reduced as in the toy model. Finally, it is pointed out why the extension of Zamolodchikov c-theorem to three-dimensional theories is an interesting still open problem.
We thank Fernando Falceto, Gloria Luzón and very specially Ian Kogan for enlightening discussions during different stages of the realization of the paper. This work is partially supported by CICYT (grant FPA2004-02948).
|
---
abstract: 'It is widely believed that water and complex organic molecules (COMs) first form in the ice mantle of dust grains and are subsequently returned into the gas due to grain heating by intense radiation of protostars. Previous research on the desorption of molecules from the ice mantle assumed that grains are at rest which is contrary to the fact that grains are suprathermally rotating as a result of their interaction with an anisotropic radiation or gas flow. [To clearly understand how molecules are released in to the gas phase, the effect of grain suprathermal rotation on surface chemistry must be quantified]{}. In this paper, we study the effect of suprathermal rotation of dust grains spun-up by radiative torques on the desorption of molecules from icy grain mantles around protostars. We show that centrifugal potential energy due to grain rotation reduces the potential barrier of molecules and significantly enhances their desorption rate. We term this mechanism [*rotational-thermal*]{} or [*ro-thermal*]{} desorption. We apply the ro-thermal mechanism for studying the desorption of molecules from icy grains which are simultaneously heated to high temperatures and spun-up to suprathermal rotation by an intense radiation of protostars. We find that ro-thermal desorption is much more efficient than thermal desorption for molecules with high binding energy such as water and COMs. Our results have important implications for understanding the origin of COMs detected in star-forming regions and call for attention to the effect of suprathermal rotation of icy grains to use molecules as a tracer of physical conditions of star-forming regions.'
author:
- Thiem Hoang
- 'Ngo-Duy Tung'
bibliography:
- 'ms.bib'
title: 'Chemistry on Rotating Grain Surface: Ro-Thermal Desorption of Molecules from Ice Mantles'
---
Introduction {#sec:intro}
============
To date, more than 200 different molecules, including water and complex organic molecules (COMs, having more than six atoms), were detected in the interstellar medium (see e.g., @Caselli:2012fq for a review). It is thought that COMs form in the ice mantle of dust grains during the warming up phase induced by protostars (see @Herbst:2009go for a review). However, the question of how such molecules are returned into the gas remains unclear (@2018IAUS..332....3V).
Several desorption mechanisms have been proposed to explain the desorption of molecules from the icy grain mantle, including thermal and non-thermal mechanisms (see [@vanDishoeck:2014cu] for a review). Thermal sublimation (@1972ApJ...174..321W; ; @1992ApJS...82..167H) is the most popular mechanism to explain the detection of water and COMs in hot cores/corinos around massive/low-mass protostars because in these regions icy grains can be heated to high temperatures above 100 K (@Herbst:2009go; @Caselli:2012fq). Nevertheless, COMs are frequently detected in lukewarm envelopes around protostars where the temperature is below their sublimation threshold ($T\sim 50-100\K$; @2013ApJ...771...95O; @Fayolle:2015cu; @vanDishoeck:2013en; @Oberg:2016el). This casts doubt on the proxy of thermal sublimation. Moreover, non-thermal desorption mechanisms include desorption induced by cosmic rays such as whole grain heating or impulsive heating (CRs; ), desorption by single UV photons (photodesorption; @2009ApJ...693.1209O). The UV photodesorption is a promising mechanism to desorb molecules in cold dark clouds in which UV photons can be produced by penetration of CRs. However, these mechanisms are still difficult to quantify for astrophysical conditions, and their resulting products may be clusters rather than individual molecules.
Previous research on thermal and non-thermal desorption of molecules from the grain mantle assumed that grains are at rest, which is contrary to the fact that grains are rapidly rotating due to collisions with gas atoms and interstellar photons (@1998ApJ...508..157D; @Hoang:2010jy). [To accurately understand how molecules are released into the gas]{}, the effect of grain suprathermal rotation on gas-grain chemistry must be quantified. The goal of this paper is to quantify the effect of grain rotation on the desorption of molecules from icy grain mantles.
Interstellar dust grains are known to be rotating suprathermally, as required to reproduce starlight polarization and far-IR/submm polarized dust emission (see @Andersson:2015bq and @LAH15 for reviews). [@1979ApJ...231..404P] first suggested that dust grains can be spun-up to suprathermal rotation (with velocities larger than grain thermal velocity) by various mechanisms, including the formation of hydrogen molecules on the grain surface. Modern astrophysics establishes that dust grains of irregular shapes can rotate suprathermally due to radiative torques arising from their interaction with an anisotropic radiation field (@1996ApJ...470..551D; @2007MNRAS.378..910L; @Hoang:2008gb; @2009ApJ...695.1457H; @Herranen:2019kj) or mechanical torques induced by an anisotropic gas flow (@2007ApJ...669L..77L; @2018ApJ...852..129H). As a result, in star-forming regions and photodissociation regions (PDRs), strong radiation can both heat dust grains to high temperatures and spin them up to extremely fast rotation, such that resulting centrifugal force would have an important effect on molecule desorption.
[@Hoang:2019td] first studied the effect of suprathermal rotation induced by radiative torques on the desorption of molecules from the icy grain mantle. For a grain model made of a silicate core covered with a thick ice mantle which is expected in very dense clouds (@2010ApJ...716..825O), they discovered that the resulting centrifugal force is sufficient to disrupt the entire ice mantle into small fragments. Subsequently, molecules can evaporate from these fragments due to transient heating by UV photons or enhanced thermal sublimation.[^1] This process that can desorb the entire ice mantle is then referred to as [*rotational desorption*]{}. The rotational desorption mechanism is found to be efficient in an extended region beyond hot cores/corinos surrounding young stellar objects (YSOs). Later on, [@Le:2019wo] found that molecules can be directly ejected from ice mantles of suprathermally rotating nanoparticles in CJ-shocks.
Another popular grain model consists of a silicate core, an organic refractory layer and outer ice layer (; @1997AdSpR..19..981G; ). For this model, the ice mantle is presumably thin, of tens of monolayers of ice water, such that it is hard to disrupt the entire ice mantle because the resulting tensile stress is insufficient to separate the binding energy between the mantle and the grain core surface as we will show in Section \[sec:theory\]. In this case, the joint action of centrifugal force applied to molecules and thermal fluctuations would enhance the rate of thermal sublimation of molecules from the ice mantle, triggering desorption at temperatures below the thermal sublimation threshold. The goal of this paper is to formulate a model of thermal desorption for suprathermally rotating grains and explore its implications for astrochemistry.
The structure of our paper is as follows. In Section \[sec:theory\], we first describe the theory of thermal desorption in the presence of grain rotation, which is termed rotational-thermal or [*ro-thermal desorption*]{}. In Section \[sec:result\] we calculate the rate of thermal and ro-thermal desorption for grains spun-up by radiation torques from a strong radiation field. Section \[sec:discuss\] discusses the implications of ro-thermal desorption of molecules and polycyclic aromatic hydrocarbons (PAHs) for different astrophysical environments. A summary of our main findings is presented in Section \[sec:sum\].
Rotational-Thermal Desorption of Molecules from Icy Grain Mantle {#sec:theory}
================================================================
Here we describe our theory for rotational-thermal desorption from rotating grains of angular velocity $\omega$.
Grain model: Ice mantles on grain surface
-----------------------------------------
Ice mantles are formed on the grain surface due to accretion of gas molecules in cold and dense regions of hydrogen density $n_{\H}=n(\H)+2n(\H_{2}) \sim 10^{3}-10^{5}\cm^{-3}$ or the visual extinction $A_{V}> 3$ (@1983Natur.303..218W). Subsequently, more complex molecules, including organic molecules, are thought to form in the ice mantle when grains are being warmed up by intense radiation of protostars (see, e.g., @Herbst:2009go).
Spectral absorption features of H$_{2}$O and CO ice are highly polarized (@1996ApJ...465L..61C; @2008ApJ...674..304W) revealing that icy grain mantles have non-spherical shape and are aligned with magnetic fields (see @LAH15 for a review). [Nevertheless, we assume that the grain shape can be described by an equivalent sphere of the same volume with effective radius $a$.]{}
Figure \[fig:grain\_mod\] illustrates a grain model consisting of a silicate core, followed by a refractory carbonaceous mantle, and an outer thin ice mantle. Molecules bind to the ice mantle via binding force ($F_{b}$) arising from dipole-dipole interaction (van der Waals force) or chemical force. The grain spinning with angular velocity $\omega$ induces a centrifugal force ($F_{cen}$) on the molecule of mass $m$.
![A schematic illustration of a rapidly spinning core-mantle grain of irregular shape. The silicate core is assumed to be compact, which is covered with a refractory organic mantle and the outer, icy water-rich mantle. A molecule of mass $m$ on the ice surface experiences the binding force $F_{b}$ and centrifugal force $F_{\rm cen}$ which are in opposite directions.[]{data-label="fig:grain_mod"}](f1.pdf){width="50.00000%"}
Thermal desorption rate from non-rotating grains
------------------------------------------------
The problem of thermal desorption from a non-rotating grain is well studied in the literature (@1972ApJ...174..321W; ). The underlying physics is that when the grain is heated to high temperatures, molecules on the grain surface acquire kinetic energy from thermal fluctuations within the grain lattice and can escape from the surface.
Let $\tau_{\rm des,0}$ be the desorption rate of molecules with binding energy $E_{b}$ from a grain at rest ($\omega=0$) which is heated to temperatures $T_{d}$. Following [@1972ApJ...174..321W], one has \_[sub,0]{}\^[-1]{}=(-), where $\nu$ is the characteristic frequency given by =()\^[1/2]{} with $N_{s}$ being the surface density of binding sites [@1987ppic.proc..333T]. Typically, $N_{s}\sim 2\times 10^{15}$ site$\cm^{-2}$.
![Illustration of the potential energy of a molecule on the rotating grain. The potential barrier is reduced significantly as the angular velocity $\omega$ increases as a result of centrifugal potential ($E_{\rm cen}$).[]{data-label="fig:ETD_mod"}](f2.pdf){width="50.00000%"}
Table \[tab:Ebind\] lists the binding energy and sublimation temperature measured from experiments for popular molecules.
[l l l]{} & [$E_{b}/k$ (K)$^a$]{} & [$T_{\rm sub}$ (K)]{}\
$\rm H_{2}O$ & 5700 & 152$^b$ $\rm CH_{3}OH$ & 5530 & 99$^b$ $\rm HCOOH$ & 5570 & 155$^c$ $\rm CH_{3}CHO$ & 2775 & 30$^c$ $\rm C_{2}H_{5}OH$ & 6260 & 250$^c$ $\rm (CH_{2}OH)_2$ & 10200 & 350$^c$ $\rm NH_{3}$ & 5530 & 78$^b$$\rm CO_{2}$ & 2575 & 72$^b$ $\rm H_{2}CO$ & 2050 & 64$^b$ $\rm CH_{4}$ & 1300 & 31$^b$ $\rm CO$ & 1150 & 25$^b$ $\rm N_{2}$ & 1140 & 22$^b$
Ro-thermal desorption rate from rotating grains
-----------------------------------------------
In the presence of grain rotation, the centrifugal force acting on a molecule of mass $m$ at distance $r\sin\theta$ from the spinning axis is \_[cen]{}=m[**a**]{}\_[cen]{}=m\^[2]{}[**r**]{}=m\^[2]{}(x+y),\[eq:Fcen\] where $\ba_{\rm cen}$ is the centrifugal acceleration.
We can define centrifugal potential $\phi_{\rm cen}$ such as ${\bf a}_{\rm cen}=-\nabla\phi_{\rm cen}$. Then, the corresponding potential is \_[cen]{}=\^[2]{}\^[2]{}()=\^[2]{}\^[2]{}r\^[2]{}. \_[cen]{}==,\[eq:phi\_cen\]
As a result, the [*effective*]{} binding energy of the molecule becomes E\_[b,rot]{}=E\_[b]{}-m\_[cen]{},\[eq:Ebind\_rot\] which means that molecules only need to overcome the reduced potential barrier of $E_{b}-E_{\rm cen}$ where $E_{\rm cen}=m\langle\phi_{\rm cen}\rangle$ to be ejected from the grain surface. The rotation effect is more important for molecules with higher mass and low binding energy.
Figure \[fig:ETD\_mod\] illustrates the potential barrier of molecules on the surface of a rotating grain as a function of $\omega$. For slow rotation, the potential barrier is determined by binding force. As $\omega$ increases, the potential barrier is decreased due to the contribution of centrifugal potential.
[The molecule is instantaneously ejected from the surface if the grain is spinning sufficiently fast such that $E_{b,rot}=0$. From Equation (\[eq:Ebind\_rot\]), one can obtain the critical angular velocity for the direct ejection as follows]{}: \_[ej]{}=()\^[1/2]{}()\^[1/2]{}rad\^[-1]{}, \[eq:omega\_ej\] where $a_{-5}=a/(10^{-5}\cm)$.
The ejection angular velocity decreases with increasing grain size and molecule mass $m$, but it increases with the binding energy $E_{b}$.
The rate of ro-thermal desorption (sublimation) rate is given by \_[sub,rot]{}\^[-1]{}=(-),\[eq:tsub\_rot\] where [the subscript $\rm sub$ stands for sublimation]{}, and the second exponential term describes the probability of desorption induced by centrifugal potential.
Equation (\[eq:tsub\_rot\]) can be written as \_[sub,rot]{}\^[-1]{}=\_[sub,0]{}\^[-1]{}RD(), where the function $RD(\omega)$ describes the effect of grain rotation on the thermal desorption as given by RD()&=&( )=( )\
&&1.7which indicates the rapid increase of ro-thermal desorption rate with the grain size $a$ and angular velocity $\omega$.
Let $\tilde{\omega}=\omega/\omega_{T}$ be the suprathermal rotation parameter where $\omega_{T}=(2kT/I)^{1/2}\simeq 2\times 10^{5}a_{-5}^{-5/2}T_{2}^{1/2}\rad\s^{-1}$ with $T$ gas temperature and $T_{2}=T/100\K$, and $I=8\pi \rho a^{5}/15$ inertia moment of grains with mass volume density $\rho$. Then, one obtains RD()=( \^[2]{})=( \^[2]{}) where $M=4\pi \rho a^{3}/3$ is the grain mass.
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
Figure \[fig:frot\_omega\] shows the ratio of ro-thermal to thermal sublimation rate, $RD(\omega)$, as a function of the grain angular velocity. For a given grain temperature, the rate of ro-thermal desorption increases exponentially with the angular velocity $\omega$ when $\omega$ is approaching $\omega_{\rm ej}$ (Eq. \[eq:omega\_ej\]).
Sublimation temperatures from rotating grains
---------------------------------------------
Let $T_{\rm sub,0}$ be the sublimation temperature of grains at rest, i.e., $\omega=0$. The sublimation temperature of rotating grains is denoted by $T_{\rm sub,rot}$. To quantify the effect of grain rotation on thermal desorption, we compare the grain temperature that is required to produce the same sublimation rate from a non-rotating grain which corresponds to $\tau_{\rm sub,0}(T_{\rm sub,0})=\tau_{\rm sub,rot}(T_{\rm sub,rot})$. Thus, one obtains T\_[sub,rot]{}=(1- )T\_[sub,0]{}.\[eq:Tsub\_rot\]
The effect of grain rotation reduces the sublimation temperature as given by &=&()=()\[eq:dTsub\]\
&& 0.14a\_[-5]{}\^[2]{}()\^[2]{}()().
One can see that the the ro-thermal sublimation temperature can be decreased by $50\%$ for grains of $a=0.2\mum$ rotating at $\omega=5\times 10^{9}\rad\s^{-1}$.
Ro-thermal desorption from grains spun-up by radiative torques {#sec:result}
==============================================================
[In this section, we will apply the theory formulated in the preceding section for the situation in which grains are rotating suprathermally as a result of the spin-up by Radiative Torques (RATs, e.g., @1996ApJ...470..551D).]{}
Centrifugal potential due to radiative torques
----------------------------------------------
Following [@Hoang:2019bi], subject to a radiation field of anisotropy degree $\gamma$, mean wavelength $\bar{\lambda}$, and radiation strength $U$, dust grains of size $a$ can be spun-up to a maximum rotation rate by (RATs): \_[RAT]{}&&9.610\^[8]{}a\_[-5]{}\^[0.7]{}|\_[0.5]{}\^[-1.7]{}\
&&()()\^[-1]{}, \[eq:omega\_RAT1\] for grains with $a\lesssim \bar{\lambda}/1.8$, and \_[RAT]{}&&1.7810\^[10]{}a\_[-5]{}\^[-2]{}|\_[0.5]{}\
&&()()\^[-1]{}, \[eq:omegaRAT2\] for grains with $a> \overline{\lambda}/1.8$. Here, $n_{1}=n_{\H}/(10\cm^{-3}), \bar{\lambda}_{0.5}=\bar{\lambda}/(0.5\mum)$, $F_{\rm IR}$ is the dimensionless parameter describing the grain rotational damping by infrared emission that depends on $(n_{\H}, T_{\rm gas}, U)$ (@1998ApJ...508..157D; @Hoang:2010jy), and $U=u_{\rad}/u_{\rm ISRF}$ with $u_{\rad}$ the total radiation energy density and $u_{\rm ISRF}$ the energy density of the standard interstellar radiation field (ISRF) in the solar neighborhood (; @Hoang:2019da). The rotation rate depends on the parameter $U/n_{\H}T_{\gas}^{1/2}$ and the damping by far-infrared emission $F_{\rm IR}$.
For convenience, let $a_{\rm trans}=\bar{\lambda}/1.8$ which denotes the grain size at which the RAT efficiency changes between the power law and flat stages (see e.g., @2007MNRAS.378..910L; @Hoang:2019da), and $\omega_{\rm RAT}$ changes from Equation (\[eq:omega\_RAT1\]) to (\[eq:omegaRAT2\]).
Plugging $\omega_{\rm RAT}$ into Equation (\[eq:phi\_cen\]), one obtains the centrifugal potential due to grain rotation as follows: m\_[cen]{}&=&1.810\^[-3]{}\^[2]{} a\_[-5]{}\^[3.4]{}|\_[0.5]{}\^[-3.4]{}()\
&&()\^[2]{}()\^[2]{} eV\[eq:phi\_small\] for $a\lesssim a_{\rm trans}$ and m\_[cen]{}&=&0.6\^[2]{} a\_[-5]{}\^[-2]{}|\_[0.5]{}\^[2]{}()\
&&()\^[2]{}()\^[2]{} eV\[eq:phi\_big\] for $a>a_{\rm trans}$.
The centrifugal potential increases rapidly with the grain size as $a^{3.4}$ until $a=a_{\rm trans}$ (Eq.\[eq:phi\_small\]), and it increases with the radiation strength as $U^{2}$. Thus, this potential is important for strong radiation fields.
Using the centrifugal potentials (Eqs. \[eq:phi\_small\] and \[eq:phi\_big\]) one can calculate the rate of ro-thermal desorption (Eq. \[eq:tsub\_rot\]) and the temperature threshold for ro-thermal desorption (Eq. \[eq:Tsub\_rot\]).
Radiation strength required for rotational desorption of molecules
------------------------------------------------------------------
In addition to the ro-thermal desorption, individual molecules can be directly ejected by centrifugal forces when the rotational rate is sufficiently high. This process is termed [*rotational desorption*]{} in [@Le:2019wo]. Comparing $\omega_{\rm RAT}$ with $\omega_{\rm ej}$ (Eq. \[eq:omega\_ej\]), one can then derive the critical radiation strength at which the molecule is immediately ejected U\_[ej]{}8n\_[1]{}T\_[2]{}\^[1/2]{}(1+F\_[IR]{})()\^[1/2]{} \[eq:Uej\] for $a\lesssim a_{\rm trans}$, and U\_[ej]{}0.4n\_[1]{}T\_[2]{}\^[1/2]{}(1+F\_[IR]{})()\^[1/2]{} for $a> a_{\rm trans}$.
Radiation strength required for rotational desorption of entire ice mantles
---------------------------------------------------------------------------
As shown in [@Hoang:2019td], when the rotation rate is sufficiently high such as the tensile stress acting on the interface between a thick ice mantle and the grain core exceeds the maximum limit of the ice mantle, $S_{\rm max}\sim 10^{7}\erg\cm^{-3}$, the ice mantle is disrupted into smaller fragments. In the case of a thin ice mantle, the tensile strength is replaced by the adhesive strength, which depends on the mechanical property of the surface and grain temperature. The adhesive strength is low for clean surface, but it can reach $\sim 10^{9}\erg\cm^{-3}$ for rough surfaces [@Work:2018bu].
Let $l_{m}$ be the ice mantle thickness and $x_{0}$ be the distance from the core-mantle interface to the spinning axis. From Equation (7) in [@Hoang:2019td], one obtains the tensile stress on the ice mantle: S\_[x]{}2.510\^[9]{}\_[ice]{}\_[10]{}\^[2]{}a\_[-5]{}\^[2]{} \^[-3]{},\[eq:Sx\] where $\hat{\rho}_{\rm ice}=\rho_{\rm ice}/(1\g\cm^{-3})$ and $\omega_{10}=\omega/(10^{10}\rm rad\s^{-1})$. For a thin mantle layer of $l_{m}=a-x_{0}\ll a$, one has S\_[x]{}5 10\^[8]{}\_[ice]{}\_[10]{}\^[2]{}a\_[-5]{}l\_[-6]{}\^[-3]{},\[eq:Sx\_approx\] where $x_{0}+a\approx 2a$ is assumed, and $l_{-6}=l_{m}/(10^{-6}\cm)$. The critical rotational velocity is determined by $S_{x}=S_{\rm max}$: \_[disr]{}&=&( )\^[1/2]{}\
&& \_[ice]{}\^[-1/2]{}S\_[,9]{}\^[1/2]{} \^[-1]{},\[eq:omega\_disr\] where $S_{\max,9}=S_{\max}/(10^{9} \erg \cm^{-3})$.
The critical radiation strength to disrupt the ice mantle is then U\_[disr]{}32.5 n\_[1]{}T\_[2]{}\^[1/2]{}()S\_[max,9]{}\^[1/2]{}\[eq:Udisr\] for $a\lesssim a_{\rm trans}$, and U\_[disr]{}1.8n\_[1]{}T\_[2]{}\^[1/2]{}()S\_[max,9]{}\^[1/2]{} for $a> a_{\rm trans}$.
For grains with a compact core, the tensile strength $S_{\max}\sim 10^{9}\erg\cm^{-3}$ is expected (see e.g., @Hoang:2019da). Comparing $\omega_{\rm ej}$ from Equation (\[eq:Uej\]) with $\omega_{\rm disr}$ from Equation (\[eq:Udisr\]), one can see that ro-thermal desorption of individual molecules can occur before the disruption of ice mantle if the ice mantle thickness is below 100 monolayers of water ice (i.e., $l_{m}<200$ Å).
Radiation strength required for thermal desorption
--------------------------------------------------
Under strong radiation fields, icy grains are heated to an equilibrium temperature, which can be approximately given by $T_{d}\simeq 16.4 a_{-5}^{-1/15}U^{1/6}\K$ for silicate-core grains (@2011piim.book.....D). One can then derive the radiation strength required for the classical thermal sublimation at $T_{d}=T_{\rm sub,0}$: U\_[sub]{}610\^[11]{}a\_[-5]{}\^[6/15]{}()\^[6]{}.\[eq:Usub\]
Comparing $U_{\rm sub}$ with $U_{\rm ej}, U_{\rm disr}$ one can see that the required radiation strength for thermal desorption is many orders of magnitude higher than ro-thermal desorption as well as direct ejection. One note that, for the same radiation strength $U$, graphite grains can be heated to temperatures higher than silicates by $\sim 30\%$, but their sublimation threshold is more than two times higher than silicates. As a result, the value of $U_{\rm sub}$ for graphite grains is much larger.
Numerical results
-----------------
### Rates of ro-thermal vs. thermal desorption
To calculate the rate of ro-thermal desorption, we first compute the rotation rate spun-up by RATs as given by Equations (\[eq:omega\_RAT1\]) and (\[eq:omegaRAT2\]). Here we adopt $\gamma=0.7$ for the anisotropy degree of the radiation field as in [@1996ApJ...470..551D] for molecular clouds (see also @2007ApJ...663.1055B).[^2] We then calculate the ro-thermal desorption rate of different molecules from the surface of spinning dust grains. We consider the different gas density and radiation strengths and assume thermal equilibrium between gas and dust, i.e., $T=T_{d}$ which is valid for dense regions around protostars. Our calculations are performed for several popular molecules, including methanol, ethanol, with binding energy listed in Table \[tab:Ebind\].
Figure \[fig:rate\_U\_wave05\_a02\] shows the rate of thermal desorption (without rotation) and ro-thermal desorption (with rotation) as a function of the radiation strength $U$ assuming a typical grain size $a=0.2\mum$ and stellar radiation spectrum with $\bar{\lambda}=0.5\mum$. The corresponding grain temperatures are shown on the top horizontal axis. The ro-thermal desorption rate increases exponentially with the radiation intensity even at temperatures much below the sublimation threshold. In all realizations, the ro-thermal desorption is much faster than thermal desorption except for $CO_{2}$ with high density $n_{\rm H}=10^{5}\cm^{-3}$. The efficiency of ro-thermal desorption is stronger for lower gas density $n_{\rm H}$. This originates from the fact that grains can spin faster due to lower rotational damping by gas collisions. One can also see that for most molecules the ro-thermal desorption occurs well before the immediate ejection threshold marked by $U_{\rm ej}$.
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
Figure \[fig:rate\_U\_wave05\_a01\] shows the rate of ro-thermal vs. thermal desorption for $a=0.1\mum$. Ro-thermal desorption is still faster than thermal desorption, although the efficiency is lower that for $a=0.2\mum$ due to lower rotation rate by RATs $\omega_{\rm RAT}$ (see Eq. \[eq:omega\_RAT1\]).
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
Figure \[fig:rate\_U\_N\] shows results for N$_{2}$ and NH$_{3}$ molecules, assuming $a=0.1\mum$ and $0.2\mum$. The similar trend as other molecules (Figure \[fig:rate\_U\_wave05\_a02\]) is observed. The efficiency of ro-thermal desorption is clearly seen for NH$_3$ which has high sublimation temperature.
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
Same as Figure \[fig:rate\_U\_wave05\_a02\], but Figure \[fig:rate\_U\_wave12\_a02\] shows the results for attenuated radiation fields with $\bar{\lambda}=1.2\mum$. The efficiency of ro-thermal desorption is weaker than the case of $\bar{\lambda}=0.5\mum$, but still dominates over thermal desorption.
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
Figure \[fig:rate\_U\_N2\] shows similar results as Figure \[fig:rate\_U\_wave12\_a02\] but for $\bar{\lambda}=1.2\mum$. The results are slightly different due to the radiation field with longer mean wavelength $\bar{\lambda}$.
### Temperature threshold for ro-thermal vs. thermal desorption
Figure \[fig:deltaT\] shows the decrease of sublimation temperature, $\Delta T=|T_{\rm sub,rot}-T_{\rm sub,0}|$ due to centrifugal potential as a function of the radiation intensity ($U$) and the grain temperature, assuming the different gas density. Analytical results from Equation (\[eq:dTsub\]) are also shown for comparison. The effect of ro-thermal desorption is more important for lower density. Ro-thermal desorption is also more efficient for molecules with higher binding energy where ro-thermal desorption can occur at more than 100 K lower than the thermal desorption. The efficiency of ro-thermal desorption is more efficient for stellar photons of $\bar{\lambda}=0.5\mum$ but less efficient for reddened photons with $\bar{\lambda}=1.2\mum$.
At high densities of $n_{\rm H}=10^{5}\cm^{-3}$, ro-thermal desorption can still occur at temperatures much lower than thermal desorption for water and other molecules with high binding energy.
Note that the molecule CO$_2$ has low $E_{b}\sim 2575\K$ but its sublimation temperature is high of $T_{\rm sub,0}\sim 72\K$ (@2004MNRAS.354.1133C), which results in a slightly peaky feature $\Delta T$ in Figure \[fig:deltaT\].
{width="50.00000%"} {width="50.00000%"} {width="50.00000%"} {width="50.00000%"}
Discussion {#sec:discuss}
==========
Thermal desorption, rotational desorption, ro-thermal desorption
----------------------------------------------------------------
Thermal desorption (sublimation) is a popular mechanism to release water and complex organic molecules from the icy grain mantle in star-forming regions [@Herbst:2009go]. This desorption process ignores the fact that grains are rapidly spinning due to radiative torques when grains are subject to intense radiation field from protostars (@1996ApJ...470..551D; @Hoang:2008gb; @2009ApJ...695.1457H; @Herranen:2019kj).
The effect of suprathermal rotation on the desorption of molecules from the grain surface is first studied in [@Hoang:2019td] where the authors discovered that the resulting centrifugal force is sufficient to disrupt the ice mantle into small fragments. Subsequently, molecules can evaporate from these fragments due to transient heating by UV photons or enhanced thermal sublimation. This [*rotational*]{} desorption mechanism is found to be efficient in hot cores/corinos around young stars where the radiation strength $U$ can reach $U\sim 10^{8}-10^{9}$ (e.g., grain temperature $T\sim 500\K$). The efficiency of rotational desorption increases with the thickness of the ice mantle, so it is most efficient for grains with a thick ice mantle.
In this paper, we study the additional effect of grain rotation on thermal desorption in regions where grain temperatures are below the sublimation threshold of water and molecules, which have $U<10^{6}$ or $T<200\K$. Specifically grain rotation provides molecules on the grain surface with a centrifugal force that acts in the opposite direction from the binding force. As a result, a rather weak level of thermal excitation can help molecules to sublimate if grains are spinning rapidly. We term this mechanism [*ro-thermal desorption*]{} mechanism. [The difference between ro-thermal desorption and rotational desorption is that in the former process molecules sublimate directly from the intact icy grain mantle, whereas in the latter process, the ice mantle is first disrupted into tiny icy fragments and subsequently molecules sublimate from these icy fragments.]{}
The efficiency of the ro-thermal desorption depends both on the grain rotation rate and grain temperature, but the rotation rate plays a key role. Therefore, ro-thermal desorption can occur in weak radiation fields when grains can be spun-up by mechanical torques (@2007ApJ...669L..77L; @2018ApJ...852..129H). The ro-thermal desorption mechanism takes over rotational desorption when the ice mantle is thin (i.e., $\Delta a_{m}\ll a$) such that the tensile stress acting on the ice mantle is rather small and cannot desorb the entire mantle (see Figures \[fig:rate\_U\_wave05\_a02\]-\[fig:rate\_U\_wave05\_a01\]).
We also find that individual molecules can be directly ejected from the icy grain mantle for $\omega\gtrsim \omega_{\rm ej}$, and this rotational desorption process requires higher radiation strength than ro-thermal desorption (see Figures \[fig:rate\_U\_wave05\_a02\] and \[fig:rate\_U\_wave05\_a01\]). Compared to rotational desorption, we find that ro-thermal desorption occurs at lower radiation strength before the entire mantle can be disrupted into small fragments and efficient for thin ice mantle of thickness $l<100\AA$.
Compared to UV photodesorption that requires FUV photons between $7-10.5$ eV to be effective (@2007ApJ...662L..23O; @vanDishoeck:2013en), ro-thermal desorption can work with optical photons with the mean wavelength even at $\bar{\lambda}\gtrsim 0.5\mum$. Therefore, ro-thermal desorption can be effective in regions without FUV.
[Finally, the rot-thermal desorption mechanism is expected to be most efficient in astrophysical environments with $U/n_{\H}T_{\gas}^{1/2}> 1$ (see Eqs. \[eq:omega\_RAT1\] and \[eq:omegaRAT2\]), such as star-forming regions, the surface and intermediate layer of protoplanetary disks, reflection nebula (e.g., @2015MNRAS.448.1178H), and photodissociation regions.]{}
Ro-thermal desorption of PAHs
-----------------------------
Like other molecules, PAHs condense in the ice mantle of dust grains in cold dense clouds (@1999Sci...283.1135B; ; @2015ApJ...799...14C). Yet, the question of how PAHs are returned into the gas phase is still unclear.
Ro-thermal desorption appears to be an efficient mechanism to desorb PAHs. Since ro-thermal desorption requires lower radiation strength to desorb than rotational desorption, one can describe the efficiency of ro-thermal desorption by considering the ejection threshold. Using Equation (\[eq:omega\_ej\]), one obtains the ejection threshold of PAHs: \_[ej]{}=()\^[1/2]{}()\^[1/2]{}rad\^[-1]{},\[eq:omega\_ej\_PAH\] where the binding energy of benzene C$_{6}$H$_{6}$ and naphthalene (C$_{10}$H$_{8}$) to ice is $E_{b}/k\sim 4000\K$ and $6000\K$ (see Table 4 and 5 in @Michoulier:2018cx).
The ejection radiation strength is then U\_[ej]{}35n\_[1]{}T\_[2]{}\^[1/2]{}(1+F\_[IR]{})()\^[1/2]{} \[eq:Uej\_PAH\] for $a\lesssim a_{\rm trans}$, and U\_[ej]{}1.8n\_[1]{}T\_[2]{}\^[1/2]{}(1+F\_[IR]{})()\^[1/2]{} for $a> a_{\rm trans}$. Clearly, the ejection threshold is much lower than that of water and COMs (see Figures \[fig:rate\_U\_wave05\_a02\]-\[fig:rate\_U\_wave12\_a02\]). Therefore, the ro-thermal desorption is efficient for desorption of PAHs in star-forming regions.
Ro-thermal desorption in photodissociation regions and protoplanetary disks
---------------------------------------------------------------------------
Photodissociation regions (PDRs) are traditionally dense molecular clouds with typical gas density $n_{\H}\lesssim 10^{5}\cm^{-3}$ illuminated by O or B stars (e.g., Orion Bar) with radiation strength $U\lesssim 10^{5}-10^{6}$ (see @1999RvMP...71..173H; @Tielens:2007wo).
For a typical PDR model like Orion Bar (@2005ApJ...630..368A), the gas density and radiation fields are $n\sim 10^{4}\cm^{-3}$ and $U\sim 3\times 10^{4}$. For these physical conditions, from Figures \[fig:rate\_U\_wave05\_a02\] and \[fig:rate\_U\_wave05\_a01\], we see that ro-thermal desorption is very efficient in desorbing water and complex organic molecules. This mechanism can explain the formation of COMs and PAHs which are usually observed in PDRs (see @2017SSRv..212....1C for a review).
The surface and intermediate layer of protoplanetary disks around young stars are attractive targets for studying ro-thermal desorption due to strong radiation fields. For the same radiation intensity, the rate of ro-thermal desorption is several orders of magnitude higher than that of thermal desorption (@1995ApJ...455L.167S).
We note that in more intense radiation field of hot cores/corinos, the entire ice mantle could be desorbed via [*rotational desorption*]{} mechanism [@Hoang:2019td]. [Finally, with this paper, the effect of grain rotation on thermal desorption of molecules is complete and for the first time demonstrate the importance of accounting for grain dynamics for grain-surface chemistry.]{}
[We note that grains can also be spun-up by mechanical torques (@2007ApJ...669L..77L; @2018ApJ...852..129H). Therefore, ro-thermal desorption can occur for supersonic flows. The potential environments include icy grains in young stellar outflows (see @Hoang:2019td). ]{}
From experimental data to astrochemical modeling
------------------------------------------------
Astrochemical modeling of observational data usually takes desorption rates and chemical reaction rates measured from experiment and apply directly to interstellar dust grains (e.g., @Oberg:2009dp). Usually, the physical properties of dust grains, including grain surface, grain size, and grain dynamics, are disregarded (see @Caselli:2012fq). Recently, the effect of grain surface properties was studied by experiments in [@Potapov:2019wg], but the application to the specific grain surface is not yet available.
In light of our findings, application of experimental measurements of sublimation temperatures cannot be directly applied to model thermal sublimation of molecules from the grain mantle due to the effect of grain suprathermal rotation. We find that the effective sublimation temperature is much lower than the measured temperature for non-rotating grains in the lab, which depends on the local gas density. The difference is significant for water ice and COMs with high binding energy.
Summary {#sec:sum}
=======
We have studied the effect of grain suprathermal rotation on the thermal desorption of molecules from icy grain mantles. Our results are summarized as follows:
- We find that the centrifugal potential energy due to grain rotation acts to reduce the potential barrier of the molecule desorption and formulate a theory for thermal desorption of molecules from rapidly spinning grains. To differentiate from the classical thermal desorption mechanism, we term this mechanism rotational-thermal desorption or ro-thermal desorption.
- We apply the ro-thermal desorption theory for icy grains spun-up by radiative torques and find that the rate of ro-thermal desorption of water and COMs is much larger than that of the classical thermal sublimation.
- We derive the effective temperature threshold for ro-thermal desorption and find that this temperature is much lower than that for thermal desorption. The ro-thermal desorption temperatures decreases with increasing the radiation strength and with decreasing the gas density.
- We find that COMs can be released via ro-thermal desorption in environments with low temperatures ($T<100 \K$) provided that the gas density is not very high, i.e., $n_{\rm H}< 10^{8}\cm^{-3}$. As a result, interpretation of the detection of COMs in astrophysical conditions by means of grain heating only is likely inadequate because of centrifugal force effect.
- [To use molecules as a reliable tracer of physical and chemical properties of astrophysical environments,]{} one needs to take into account the effect of suprathermal rotation of icy grains on desorption of molecules, and chemical modeling should take into account this effect.
- Our results reveal that using experimental data for astrochemical modeling of gas-grain surface chemistry in star-forming regions would be cautious and must account for the effect of suprathermal rotation of icy grains.
We thank Le Ngoc Tram for useful comments. This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government (MSIT) (2017R1D1A1B03035359 and 2019R1A2C1087045).
[^1]: In this paper, sublimation and desorption is interchangeably used to imply the desorption of molecules from the grain surface.
[^2]: The anisotropy degree of the diffuse ISRF is lower of $\gamma=0.1$, and $\gamma=1$ for unidirectional radiation fields from a point source.
|
---
abstract: 'Exact analytical solutions for start-up and cessation flows are obtained for the affine linear Phan-Thien-Tanner fluid model. This includes the results for start-up and cessation of steady shear flow, of steady uni- and biaxial extensional flows, and of steady planar extensional flow. The solutions obtained show that at start-up of steady shear flow, the stresses go through quasi-periodic exponentially damped oscillations while approaching their steady-flow values (so that stress overshoots are present); at start-up of steady extensional flows, the stresses grow monotonically, while at cessation of steady shear and extensional flows, the stresses decay to zero quickly and non-exponentially. The steady-flow rheology of the fluid is also reviewed, the exact analytical solutions obtained in this work for steady shear and extensional flows being simpler and more detailed than the alternative formulae found in the literature. The properties of steady and transient solutions, including their asymptotic behavior at low and high Weissenberg numbers, are investigated in details. Generalization to the multimode version of the Phan-Thien-Tanner model is also discussed. Thus, this work provides a complete analytical description of the rheology of the affine linear Phan-Thien-Tanner fluid in start-up, cessation, and steady regimes of shear and extensional flows.'
author:
- 'D. Shogin'
bibliography:
- 'SLPTT.bib'
title: 'Start-up and cessation of steady shear and extensional flows: Exact analytical solutions for the affine linear Phan-Thien-Tanner fluid model'
---
\[Sec:Intro\] Introduction
===========================
Transient flow experiments are of vital imporance to rheology: they help to investigate the complex nonlinear nature of non-Newtonian fluids (in particular, of polymeric liquids), which is fully manifested in such flows.[@Bird1987a] The basic rheological experiments involving transient flows are the so-called step-rate (start-up and cessation) tests.[@Mezger2014] In start-up tests, the fluid originally at rest is set to motion by a suddenly applied constant shear or elongation rate, and the stress growth is monitored until steady flow regime is established. In cessation experiments, the fluid undergoes steady shear or extensional flow until the shear or elongation rate is removed instantaneously; the stress decay is observed while the fluid approaches equilibrium.
Possessing an exact analytical solution describing the rheological response of the fluid in such experiments is advantageous in many aspects. In particular, it facilitates fitting the fluid model to experimental data, allows to test the model for relevance and to investigate its features, reduces calculation time, and can serve as a benchmark for numerical solvers. At the same time, very few analytical solutions are known for realistic non-Newtonian fluid models, mainly because of the overall complexity of the underlying equations.[@Bird1987b]
The analytical solutions for transient flows of non-Newtonian fluids obtained during the last few years, including those for large-amplitude oscillatory flow[@Saengow2015; @Saengow2017a; @Saengow2017b; @Saengow2017c; @Saengow2018; @Poungthong2019a; @Poungthong2019b; @Saengow2019udlaos] and start-up of steady shear flow[@Saengow2019startup], are obtained either for the Oldroyd 8-constant (O8) non-Newtonian fluid model [@Oldroyd1958; @Oldroyd1961] or for its special cases. Despite its usefulness and generality, the O8 model is quite simplistic: it is constructed based on mathematical considerations only, and (albeit nonlinear in the rate-of-strain tensor components) is linear in the stress tensor components. In contrast, accurate modeling of flows of polymer solutions and melts requires use of physics-based fluid models originating from kinetic or network theories.[@Bird1987b] Such models are typically formulated through differential constitutive equations that are nonlinear in the stress tensor components.[@Bird1995] Important examples of such models are the finitely elongated nonlinear elastic dumbbell model (FENE-P) and its modifications,[@Bird1980; @Bird1983; @Chilcott1988; @Shogin2020] the Phan-Thien-Tanner (PTT) model and its variants, the Giesekus model,[@Giesekus1982; @Giesekus1983] and the more recent eXtended Pom-Pom (XPP) model.[@Verbeeten2001]
As might be expected, the literature is quite scarce when it comes to analytical and even semi-analytical results for physical non-Newtonian fluid models. Furthermore, such results are almost exclusively obtained for steady shear flows. [@Azaiez1996; @Oliveira1999; @Pinho2000; @Alves2001; @Oliveira2002; @Shogin2017; @Ribau2019] Even for steady extensional flows, the analytical descriptions found in the literature are often incomplete (only asymptotic expressions are provided)[@Bird1980; @Petrie1990; @Azaiez1996; @Tanner2003] or not “fully” analytical (the solution involves one or several parameters defined implicitly).[@Xue1998; @Ferras2019; @Shogin2020] Finally, we are aware of only two theoretical works on start-up flows involving constitutive equations that are nonlinear in stresses: the exact analytical solutions for the Giesekus model proposed by its author[@Giesekus1982] and the qualitative investigation of the PTT model by Missaghi and Petrie,[@Missaghi1982] both works being more than 35 years old.
In this work, we obtain exact analytical expressions for the material functions of the PTT model related to start-up and cessation of shear and extensional–uniaxial, biaxial, and planar–flows. Our consideration shall be restricted to the so-called “simplified”, or “affine”, version of the linear PTT model (SLPTT).[@Phan-Thien1977] In order to obtain the main result, we review the steady-flow material functions of the model and derive exact analytical expressions also for them, our formulations being simpler than those found in the literature. Thus, this work provides a complete rheological description of the SLPTT fluid properties in steady, start-up, and cessation regimes of shear and extensional flows. Furthermore, the steady-flow expressions obtained here are also valid for the FENE-P dumbbell model: in such flows, SLPTT and FENE-P models are completely equivalent; despite being quite obvious, this fact is rarely mentioned in the literature.[@Oliveira2004; @Cruz2005; @Poole2019]
This paper is organized as follows: In Sec. \[Sec:StartUpFlows\], we define the flows of interest and the material functions relevant for these flows. Then, we formulate the equations governing the dynamics of the stress tensor components in Sec. \[Sec:Formulation\]. In Sec. \[Sec:DFormulation\], these equations are reformulated in terms of dimensionless variables. The analytical expressions for the material functions for steady and step-rate flows are derived and discussed in Secs. \[Sec:SteadyFlowSolutions\] and \[Sec:StartUpSolutions\], respectively. In Sec. \[Sec:Multimode\], we briefly discuss how our results can be generalized to the multimode SLPTT model. Conclusions are presented in Sec. \[Sec:Conclusions\]. In addition, some technical details are given in Appendices \[App:Bijection\]-\[App:MainResult\] and the reader shall be referred to them when appropriate.
Throughout this paper, second-rank tensors are written with boldface Greek, vectors with boldface Latin, while scalar quantities with lightface font. The sign convention of Bird *et al.*[@Bird1987a] for the stress tensor is adopted.
Finally, it should be noted that every analytical result, exact or asymptotic, obtained in this work has been tested numerically using Wolfram Mathematica and no mismatch between analytical and numerical results were revealed in these tests.
Material functions describing start-up, cessation, and steady shear and extensional flows {#Sec:StartUpFlows}
=========================================================================================
The important rheometric flows to be considered in this work are shear flow and three special cases–uniaxial, biaxial (sometimes called equibiaxial[@Petrie2006]), and planar–of general extensional flow; for a detailed review of shear and extensional flows, we refer the reader to the book by Bird *et al.*[@Bird1987a] The general forms of the velocity field ($\boldsymbol{v}$), the rate-of-strain tensor \[$\boldsymbol{\dot{\gamma}}=(\boldsymbol{\nabla} \boldsymbol{v})+(\boldsymbol{\nabla} \boldsymbol{v})^\mathrm{T}$\], the anisotropic stress tensor ($\boldsymbol{\tau}$), and its upper-convected time derivative ($\boldsymbol{\tau}_{(1)}$) for shear and extensional flows are given in Table \[Tab:TensorFormsForDifferentFlowTypes\]. Note that we adopt the mathematically convenient approach of Bird *et al.*[@Bird1980; @Bird1987a] allowing for a unified description of uniaxial and biaxial extensional flows–namely, positive elongation rates (${\dot{\varepsilon}}>0$) in row II of Table \[Tab:TensorFormsForDifferentFlowTypes\] correspond to uniaxial extension, while allowing for negative elongation rates (${\dot{\varepsilon}}<0$) yields biaxial extension, the other aspects being exactly the same for these two flows. For the planar extension case, ${\dot{\varepsilon}}>0$.
[c c c c c]{} & $\boldsymbol{v}$ & $ \boldsymbol{\dot{\gamma}} $ & $ \boldsymbol{\tau} $ & $ \boldsymbol{\tau}_{(1)}$\
------------------------------------------------------------------------
I & $\renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
\dot{\gamma}x_2 \\ 0 \\ 0
\end{bmatrix} $ & $\renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
0 & \dot{\gamma} & 0 \\
\dot{\gamma} & 0 & 0 \\
0 & 0 & 0
\end{bmatrix} $ & $ \renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
\tau_{11} & \tau_{12} & 0 \\
\tau_{12} & \tau_{22} & 0 \\
0 & 0 & \tau_{33}
\end{bmatrix} $ & $\dfrac{\mathrm{d} \boldsymbol{\tau}}{\mathrm{d}t}-\renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
2\tau_{12} & \tau_{22} & 0 \\
\tau_{22} & 0 & 0 \\
0 & 0 & 0
\end{bmatrix}\dot{\gamma}$\
------------------------------------------------------------------------
II & $ \renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
-\dot{\varepsilon}x_1/2 \\
-\dot{\varepsilon}x_2/2 \\
\dot{\varepsilon}x_3
\end{bmatrix}$ & $ \renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
-\dot{\varepsilon} & 0 & 0 \\
0 & -\dot{\varepsilon} & 0 \\
0 & 0 & 2 \dot{\varepsilon}
\end{bmatrix}$ & $ \renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
\tau_{11} & 0 & 0 \\
0 & \tau_{22} & 0 \\
0 & 0 & \tau_{33}
\end{bmatrix} $ & $\dfrac{\mathrm{d} \boldsymbol{\tau}}{\mathrm{d}t}+\renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
\tau_{11} & 0 & 0 \\
0 & \tau_{22} & 0 \\
0 & 0 & -2\tau_{33}
\end{bmatrix}\dot{\varepsilon}$\
------------------------------------------------------------------------
III & $ \renewcommand*{\arraystretch}{0.7} \begin{bmatrix}
-\dot{\varepsilon}x_1 \\
0 \\
\dot{\varepsilon}x_3
\end{bmatrix}$ &
$\begin{bmatrix}
- 2 \dot{\varepsilon} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 2 \dot{\varepsilon}
\end{bmatrix}$ & $\renewcommand*{{0.7}}{0.7} \begin{bmatrix}
\tau_{11} & 0 & 0 \\
0 & \tau_{22} & 0 \\
0 & 0 & \tau_{33}
\end{bmatrix} $ & $\dfrac{\mathrm{d} \boldsymbol{\tau}}{\mathrm{d}t}+\renewcommand*{{0.7}}{0.7} \begin{bmatrix}
2\tau_{11} & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & -2\tau_{33}
\end{bmatrix}\dot{\varepsilon}$
In steady flows, the shear rate (${\dot{\gamma}}$) and the elongation rate (${\dot{\varepsilon}}$) are constants; the stress tensor components are time-independent. In contrast, in start-up ($+$) and cessation ($-$) flows, the shear and elongation rate take the form $$\label{Eq:Formulation:StepRate}
\begin{bmatrix}
{\dot{\gamma}}(t) \\ {\dot{\varepsilon}}(t) \end{bmatrix} = \begin{bmatrix}
{\dot{\gamma}}_0 \\ {\dot{\varepsilon}}_0 \end{bmatrix} \Theta(\pm t),$$ where ${\dot{\gamma}}_0$ and ${\dot{\varepsilon}}_0$ are constants, while $\Theta$ is the Heaviside step-function; the stress tensor components are functions of time.
The material functions describing the properties of the fluid in steady, start-up, and cessation regimes of shear and extensional flows are defined in Table \[Tab:MaterialFunctions\]. Those related to shear flow are well-known, while those introduced for extensional flows need a brief discussion. As seen from Table \[Tab:TensorFormsForDifferentFlowTypes\], uni- and biaxial extensional flows are axially symmetric; hence, $\tau_{11}=\tau_{22}$. Therefore, only one normal stress difference, namely $\tau_{33}-\tau_{11}$, needs to be specified to provide a complete rheological description of the flow. The material function associated with this stress difference is the uni- or biaxial extensional viscosity, $\bar{\eta}$. In contrast, one can see that two normal stress differences need to be specified in planar extensional flow. Following Bird *et al.*[@Bird1987a] and adopting their notations, we choose to specify $\tau_{33}-\tau_{11}$ and $\tau_{22}-\tau_{11}$ and denote the associated material functions by $\bar{\eta}_1$ and $\bar{\eta}_2$, respectively. These shall be referred to as the first and the second planar extensional viscosities. One should note that alternative choices are also possible; in particular, the “cross-viscosity” encountered in the literature[@Petrie1990; @Petrie2006] is easily identified as $\bar{\eta}_1-\bar{\eta}_2$.
[c c c]{} & Steady flow regime & Start-up ($+$) and cessation ($-$) regimes\
------------------------------------------------------------------------
I &
$\begin{array}{rl}
\eta({\dot{\gamma}})&=-\dfrac{\tau_{12}({\dot{\gamma}})}{{\dot{\gamma}}} \\
\Psi_1({\dot{\gamma}})&=-\dfrac{\tau_{11}({\dot{\gamma}})-\tau_{22}({\dot{\gamma}})}{{\dot{\gamma}}^2} \\
\Psi_2({\dot{\gamma}})&=-\dfrac{\tau_{22}({\dot{\gamma}})-\tau_{33}({\dot{\gamma}})}{{\dot{\gamma}}^2}\end{array}$ &
$\begin{array}{rl}
\eta^{\pm}(t,{\dot{\gamma}}_0)&=-\dfrac{\tau_{12}(t,{\dot{\gamma}}_0)}{{\dot{\gamma}}_0} \\
\Psi_1^\pm(t,{\dot{\gamma}}_0)&=-\dfrac{\tau_{11}(t,{\dot{\gamma}}_0)-\tau_{22}(t,{\dot{\gamma}}_0)}{{\dot{\gamma}}_0^2} \\
\Psi_2^\pm(t,{\dot{\gamma}}_0)&=-\dfrac{\tau_{22}(t,{\dot{\gamma}}_0)-\tau_{33}(t,{\dot{\gamma}}_0)}{{\dot{\gamma}}_0^2}\end{array}$\
------------------------------------------------------------------------
II & $ \bar{\eta}({\dot{\varepsilon}})=-\dfrac{\tau_{33}({\dot{\varepsilon}})-\tau_{11}({\dot{\varepsilon}})}{{\dot{\varepsilon}}}$ & $ \bar{\eta}^\pm(t,{\dot{\varepsilon}}_0)=-\dfrac{\tau_{33}(t,{\dot{\varepsilon}}_0)-\tau_{11}(t,{\dot{\varepsilon}}_0)}{{\dot{\varepsilon}}_0}$\
------------------------------------------------------------------------
III &
$\begin{array}{rl}
\bar{\eta}_1({\dot{\varepsilon}})&=-\dfrac{\tau_{33}({\dot{\varepsilon}})-\tau_{11}({\dot{\varepsilon}})}{{\dot{\varepsilon}}}\\
\bar{\eta}_2({\dot{\varepsilon}})&=-\dfrac{\tau_{22}({\dot{\varepsilon}})-\tau_{11}({\dot{\varepsilon}})}{{\dot{\varepsilon}}}
\end{array}$ &
$\begin{array}{rl}
\bar{\eta}_1^\pm(t,{\dot{\varepsilon}}_0)&=-\dfrac{\tau_{33}(t,{\dot{\varepsilon}}_0)-\tau_{11}(t,{\dot{\varepsilon}}_0)}{{\dot{\varepsilon}}_0}\\
\bar{\eta}_2^\pm(t,{\dot{\varepsilon}}_0)&=-\dfrac{\tau_{22}(t,{\dot{\varepsilon}}_0)-\tau_{11}(t,{\dot{\varepsilon}}_0)}{{\dot{\varepsilon}}_0}
\end{array}$
When it comes to the step-rate-related material functions, a convention on their names needs to be adopted. In this work, the material functions describing start-up flows shall be collectively referred to as “stress growth functions”, while those related to cessation flows as “stress relaxation functions”. Whenever a particular transient material function is to be mentioned, its symbol shall be used to avoid confusion.
Finally, one should note that $$\lim_{t\to\infty}
\begin{bmatrix}
\eta^+(t,{\dot{\gamma}}_0) \\
\Psi_{1,2}^+(t,{\dot{\gamma}}_0) \\
\bar{\eta}^+(t,{\dot{\varepsilon}}_0) \\
\bar{\eta}_{1,2}^+(t,{\dot{\varepsilon}}_0)
\end{bmatrix} =
\begin{bmatrix}
\eta^-(0,{\dot{\gamma}}_0) \\
\Psi_{1,2}^-(0,{\dot{\gamma}}_0) \\
\bar{\eta}^-(0,{\dot{\varepsilon}}_0) \\
\bar{\eta}_{1,2}^-(0,{\dot{\varepsilon}}_0)
\end{bmatrix}=
\begin{bmatrix}
\eta({\dot{\gamma}}_0) \\
\Psi_{1,2}({\dot{\gamma}}_0) \\
\bar{\eta}({\dot{\varepsilon}}_0) \\
\bar{\eta}_{1,2}({\dot{\varepsilon}}_0)
\end{bmatrix},$$ as follows from the definitions of the material functions and from the nature of start-up and cessation tests. We shall find it convenient in certain situations to normalize the stress growth and relaxation functions to their steady-flow analogs at the same shear or elongation rate. Any normalized stress growth function defined this way tends asymptotically to 1 at $t\to \infty$, and any normalized stress relaxation function equals 1 at $t=0$. In this work, the word “normalized” shall be used exclusively in this context.
\[Sec:Formulation\] Initial formulation of the start-up and cessation problems
===============================================================================
The constitutive equation for the single-mode SLPTT model can be written as $$\label{Eq:Formulation:LPTTConstitutiveEquation}
\left(1-\dfrac{\epsilon \lambda}{\eta_0} {\mathrm{tr}}{\boldsymbol{\tau}}\right) {\boldsymbol{\tau}} + \lambda {\boldsymbol{\tau}}_{(1)} =-\eta_0 {\boldsymbol{{\dot{\gamma}}}},$$ where the three positive model parameters, $\eta_0$, $\lambda$, and $\epsilon$, are the zero-shear-rate viscosity, the time constant, and the extensibility parameter, respectively.[@Phan-Thien1977] In applications, $\epsilon$ is reported to be small, e.g., of the order of $10^{-2}-10^{-1}$. Nevertheless, the solutions to be obtained in this work are valid for a wider range of $\epsilon$; hence, we assume $0<\epsilon<1/4$, which, of course, holds in practice. Finally, all the graphs to be shown in this work are plotted at $\epsilon=0.1$; this choice is made for visual purposes.
To formulate the equations governing the dynamics of stresses in step-rate tests, as described by SLPTT model, one substitutes $\boldsymbol{{\dot{\gamma}}}$, $\boldsymbol{\tau}$, and $\boldsymbol{\tau}_{(1)}$ for the chosen flow type (see Table \[Tab:TensorFormsForDifferentFlowTypes\]) along with the corresponding form of the strain rate \[Eq. (\[Eq:Formulation:StepRate\])\] into Eq. (\[Eq:Formulation:LPTTConstitutiveEquation\]). Having written the resulting tensor equation componentwise, one arrives at nonlinear dynamical systems of first-order ordinary differential equations for the non-zero stress tensor components, as shown below. In the following, the stresses shall be treated as functions of time only; their dependencies on ${\dot{\gamma}}_0$ and on the model parameters shall be considered parametric.
\[SSec:Formulation:SteSh\] Start-up of steady shear flow
---------------------------------------------------------
Substituting $\boldsymbol{{\dot{\gamma}}}$, $\boldsymbol{\tau}$, and $\boldsymbol{\tau}_{(1)}$ from row I of Table \[Tab:TensorFormsForDifferentFlowTypes\] into Eq. (\[Eq:Formulation:LPTTConstitutiveEquation\]) and using Eq. (\[Eq:Formulation:StepRate\]) with the positive sign chosen yield $$\label{Eq:Formulation:Sh:MatrixEq}
\lambda \dfrac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}
\tau_{11} \\ \tau_{12} \\ \tau_{22} \\ \tau_{33}
\end{bmatrix}+
\left ( 1- \dfrac{\epsilon \lambda}{\eta_0}{\mathrm{tr}}{\boldsymbol{\tau}}\right)
\begin{bmatrix}
\tau_{11} \\ \tau_{12} \\ \tau_{22} \\ \tau_{33}
\end{bmatrix} =
\begin{bmatrix}
2\lambda \tau_{12}{\dot{\gamma}}_0 \\ -\eta_0 {\dot{\gamma}}_0 \\ 0 \\ 0
\end{bmatrix},$$ with $\tau_{11}=\tau_{12}=\tau_{22}=\tau_{33}=0$ at $t=0$. It follows from the third and the fourth components of Eq. (\[Eq:Formulation:Sh:MatrixEq\]) that $\tau_{22}=\tau_{33}=0$ identically. This reduces the number of equations to two and implies that the material functions related to the second normal stress difference, $\Psi_2({\dot{\gamma}})$ and $\Psi_2^+(t,{\dot{\gamma}}_0)$, vanish for the SLPTT model.
\[SSec:Formulation:UBEx\] Start-up of steady uni- and biaxial extensional flow
-------------------------------------------------------------------------------
Substituting $\boldsymbol{{\dot{\gamma}}}$, $\boldsymbol{\tau}$, and $\boldsymbol{\tau}_{(1)}$ from row II of Table \[Tab:TensorFormsForDifferentFlowTypes\] into Eq. (\[Eq:Formulation:LPTTConstitutiveEquation\]) having used Eq. (\[Eq:Formulation:StepRate\]) with the positive sign leads to $$\lambda \dfrac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}
\tau_{11} \\ \tau_{22} \\ \tau_{33}
\end{bmatrix} +
\left ( 1- \dfrac{\epsilon \lambda}{\eta_0}{\mathrm{tr}}{\boldsymbol{\tau}}\right)
\begin{bmatrix}
\tau_{11} \\ \tau_{22} \\ \tau_{33}
\end{bmatrix}
+ \lambda \dot{\varepsilon}_0\begin{bmatrix}
\tau_{11} \\ \tau_{22} \\ -2\tau_{33}
\end{bmatrix} =
\begin{bmatrix}
\eta_0 \dot{\varepsilon}_0 \\ \eta_0 \dot{\varepsilon}_0 \\
-2\eta_0 \dot{\varepsilon}_0
\end{bmatrix}, \label{Eq:Formulation:UBEx:MatrixEq}$$ the initial conditions being $\tau_{11}=\tau_{22}=\tau_{33}=0$ at $t=0$. Since $\tau_{22}=\tau_{11}$ identically because of the axial symmetry of the flow, the number of equations in the system is reduced to two.
\[SSec:Formulation:PlaEx\] Start-up of planar extensional flow
---------------------------------------------------------------
Having inserted $\boldsymbol{{\dot{\gamma}}}$, $\boldsymbol{\tau}$, and $\boldsymbol{\tau}_{(1)}$ from row III of Table \[Tab:TensorFormsForDifferentFlowTypes\] into Eq. (\[Eq:Formulation:LPTTConstitutiveEquation\]) and used Eq. (\[Eq:Formulation:StepRate\]) with the positive sign, one obtains $$\lambda \dfrac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}
\tau_{11} \\ \tau_{22} \\ \tau_{33}
\end{bmatrix} +
\left ( 1- \dfrac{\epsilon \lambda}{\eta_0}{\mathrm{tr}}{\boldsymbol{\tau}}\right)
\begin{bmatrix}
\tau_{11} \\ \tau_{22} \\ \tau_{33}
\end{bmatrix} + \lambda \dot{\varepsilon}_0 \begin{bmatrix}
2\tau_{11} \\ 0 \\ -2\tau_{33}
\end{bmatrix} =
\begin{bmatrix}
2\eta_0 \dot{\varepsilon}_0 \\ 0 \\
-2\eta_0 \dot{\varepsilon}_0
\end{bmatrix}. \label{Eq:Formulation:PlaEx:MatrixEq}$$ The initial conditions are $\tau_{11}=\tau_{22}=\tau_{33}=0$ at $t=0$. The solution of the second component of Eq. (\[Eq:Formulation:PlaEx:MatrixEq\]) is trivial, $\tau_{22}=0$ at any $t$. The number of independent variables in the system is therefore two.
Cessation of steady shear and extensional flows
-----------------------------------------------
Insertion of $\boldsymbol{{\dot{\gamma}}}$, $\boldsymbol{\tau}$, and $\boldsymbol{\tau}_{(1)}$ from Table \[Tab:TensorFormsForDifferentFlowTypes\] into Eq. (\[Eq:Formulation:LPTTConstitutiveEquation\]) with the negative sign chosen in Eq. (\[Eq:Formulation:StepRate\]) results in a similar systems of ordinary differential equations that are nearly identical to each other. For the flow types described in Table \[Tab:TensorFormsForDifferentFlowTypes\], these systems can be written as $$\label{Eq:Formulation:Cessation:MatrixEq}
\lambda \dfrac{\mathrm{d}}{\mathrm{d}t}\tau_{ij}+\left ( 1- \dfrac{\epsilon \lambda}{\eta_0}{\mathrm{tr}}{\boldsymbol{\tau}}\right)\tau_{ij}=0,$$ where $(i,j) \in \left\{(1,1),(1,2),(2,2),(3,3)\right\}$ for shear flow and $(i,j) \in \left\{(1,1),(2,2),(3,3)\right\}$ for extensional flows. The initial conditions are imposed so that at all the stresses are set to their steady-flow values at $t=0$; this leads to $\tau_{22}=\tau_{33}=0$ identically for cessation of steady shear flow \[thus, $\Psi_2^-(t,{\dot{\gamma}}_0)=0$\] and to $\tau_{22}=0$ identically for cessation of planar extensional flow. Thus, the number of equations in the systems reduces to two for all cessation flows considered in this work.
\[Sec:DFormulation\] Dimensionless formulation
===============================================
Prior to solving the remaining equations of systems (\[Eq:Formulation:Sh:MatrixEq\])-(\[Eq:Formulation:Cessation:MatrixEq\]), we put these equations into dimensionless form. Regardless of the flow type and regime, we replace the time variable, $t$, by a dimensionless one, $$\label{Eq:DFormulation:DimensionlessTime}
r = t/\lambda,$$ so that for any time-dependent physical quantity, $\Lambda$, $$\label{Eq:DFormulation:TimeDifferentiationRule}
\dfrac{\mathrm{d} \Lambda}{\mathrm{d} t} =\dfrac{1}{\lambda} \dfrac{\mathrm{d} \Lambda}{\mathrm{d} r} \equiv \dfrac{1}{\lambda}\Lambda'.$$ Then, we introduce the Weissenberg number, ${\mathrm{Wi}}$, and the dimensionless stress combinations, $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{S}$, and $\mathbb{T}$, in start-up and cessation flows. The definitions of these quantities depend on the flow type and are given in Table \[Tab:DimensionlessVariables\]. We choose the positive and negative signs in these definitions so that none of the quantities ${\mathrm{Wi}}$, $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{S}$, and $\mathbb{T}$ can take negative values; this feature shall prove to be very useful. For the rest of the paper, $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{S}$, and $\mathbb{T}$ shall be treated as functions of $r$ with $\epsilon$ and ${\mathrm{Wi}}$ as parameters.
[c c c]{} I & II & III\
------------------------------------------------------------------------
${\mathrm{Wi}}= \lambda \dot{\gamma}_0$ & ${\mathrm{Wi}}= \pm\lambda \dot{\varepsilon}_0$ & ${\mathrm{Wi}}= \lambda \dot{\varepsilon}_0$\
------------------------------------------------------------------------
$\begin{bmatrix}
\mathbb{N}_1 \\
\mathbb{S}
\end{bmatrix}=-\dfrac{\epsilon}{\eta_0 {\dot{\gamma}}_0}
\begin{bmatrix}
\tau_{11} \\
\tau_{12}
\end{bmatrix}$ & $\begin{bmatrix}
\mathbb{N}_1 \\
\mathbb{T}
\end{bmatrix}=-\dfrac{\epsilon}{\eta_0 {\dot{\varepsilon}}_0}
\begin{bmatrix}
\tau_{33}-\tau_{11} \\
\tau_{11}+2\tau_{33}
\end{bmatrix}$ & $\begin{bmatrix}
\mathbb{N}_1 \\
\mathbb{N}_2 \\
\mathbb{T}
\end{bmatrix}=-\dfrac{\epsilon}{\eta_0 {\dot{\varepsilon}}_0}
\begin{bmatrix}
\tau_{33}-\tau_{11} \\
\tau_{33} \\
\tau_{11}+\tau_{33}
\end{bmatrix}$
It is also necessary to introduce the steady-flow values of $\mathbb{N}_1$, $\mathbb{N}_2$, $\mathbb{S}$, and $\mathbb{T}$, which we denote by $\delta_1$, $\delta_2$, $\sigma$, and $\tau$, respectively. These quantities, along with the steady-flow Weissenberg number, are defined in Table \[Tab:DimensionlessVariablesSteady\] and shall be considered as functions of ${\mathrm{Wi}}$ with parameter $\epsilon$, unless stated otherwise.
[c c c]{} I & II & III\
------------------------------------------------------------------------
${\mathrm{Wi}}= \lambda \dot{\gamma}$ & ${\mathrm{Wi}}= \pm\lambda \dot{\epsilon}$ & ${\mathrm{Wi}}= \lambda \dot{\epsilon}$\
------------------------------------------------------------------------
$\begin{bmatrix}
\delta_1 \\
\sigma
\end{bmatrix}=-\dfrac{\epsilon}{\eta_0 {\dot{\gamma}}}
\begin{bmatrix}
\tau_{11} \\
\tau_{12}
\end{bmatrix}$ & $\begin{bmatrix}
\delta_1 \\
\tau
\end{bmatrix}=-\dfrac{\epsilon}{\eta_0 {\dot{\varepsilon}}}
\begin{bmatrix}
\tau_{33}-\tau_{11} \\
\tau_{11}+2\tau_{33}
\end{bmatrix}$ & $\begin{bmatrix}
\delta_1 \\
\delta_2 \\
\tau
\end{bmatrix}=-\dfrac{\epsilon}{\eta_0 {\dot{\varepsilon}}}
\begin{bmatrix}
\tau_{33}-\tau_{11} \\
\tau_{33} \\
\tau_{11}+\tau_{33}
\end{bmatrix}$
The material functions of the SLPTT fluid are closely related to the dimensionless stress combinations; the corresponding relations are given in Table \[Tab:MatFunctionsVsDimlessVars\] (vanishing material functions are not shown).
[c c c]{} & &\
------------------------------------------------------------------------
$\begin{bmatrix}
\eta^\pm(r,{\mathrm{Wi}}) \\
\Psi_1^\pm(r,{\mathrm{Wi}}) \\
\eta({\mathrm{Wi}}) \\
\Psi_1({\mathrm{Wi}})
\end{bmatrix}=\dfrac{\eta_0}{\epsilon}
\begin{bmatrix}
\mathbb{S} \\
\lambda \mathbb{N}_1/{\mathrm{Wi}}\\
\sigma \\
\lambda \delta_1/{\mathrm{Wi}}\end{bmatrix}$ & $\begin{bmatrix}
\bar{\eta}^+(r,{\mathrm{Wi}}) \\
\bar{\eta}({\mathrm{Wi}})
\end{bmatrix}=\dfrac{\eta_0}{\epsilon}
\begin{bmatrix}
\mathbb{N}_1 \\
\delta_1
\end{bmatrix}$ & $\begin{bmatrix}
\bar{\eta}_{1}^\pm(r,{\mathrm{Wi}}) \\
\bar{\eta}_{2}^\pm(r,{\mathrm{Wi}}) \\
\bar{\eta}_{1}({\mathrm{Wi}}) \\
\bar{\eta}_{2}({\mathrm{Wi}})
\end{bmatrix}=\dfrac{\eta_0}{\epsilon}
\begin{bmatrix}
\mathbb{N}_1 \\
\mathbb{N}_2 \\
\delta_{1} \\
\delta_2
\end{bmatrix}$
\[SSec:DFormulation:SteSh\] Start-up of steady shear flow
----------------------------------------------------------
Using the dimensionless variables from column I of Table \[Tab:DimensionlessVariables\], one rewrites Eq. (\[Eq:Formulation:Sh:MatrixEq\]) as $$\begin{aligned}
\mathbb{N}'_1 &= -(1 + {\mathrm{Wi}}\mathbb{N}_1) \mathbb{N}_1+2 {\mathrm{Wi}}\mathbb{S}, \label{Eq:DFormulation:SteSh:1N1Evolution}\\
\mathbb{S}' &= -(1 + {\mathrm{Wi}}\mathbb{N}_1) \mathbb{S} + \varepsilon, \label{Eq:DFormulation:SteSh:2SEvolution}\end{aligned}$$ with $\mathbb{N}_1(0)=\mathbb{S}(0)=0$.
\[SSec:DFormulation:UBEx\] Start-up of steady uni- and biaxial extensional flow
--------------------------------------------------------------------------------
Inserting the dimensionless variables from column II of Table \[Tab:DimensionlessVariables\] into Eq. (\[Eq:Formulation:UBEx:MatrixEq\]), one gets $$\begin{aligned}
\mathbb{T}' &= - (1 + {\mathrm{Wi}}\mathbb{T}) \mathbb{T} + 2 {\mathrm{Wi}}\mathbb{N}_1, \label{Eq:DFormulation:UBEx:1TEvolution}\\
\mathbb{N}'_1 &= - (1 + {\mathrm{Wi}}\mathbb{T}) \mathbb{N}_1 + {\mathrm{Wi}}(\mathbb{T} \pm \mathbb{N}_1)+3 \varepsilon, \label{Eq:DFormulation:UBEx:2N1Evolution}\end{aligned}$$ with $\mathbb{T}(0) = \mathbb{N}_1(0)=0$.
\[SSec:DFormulation:PlaEx\] Start-up of steady planar extensional flow
-----------------------------------------------------------------------
One rewrites Eq. (\[Eq:Formulation:PlaEx:MatrixEq\]) in dimensionless form using column III of Table \[Tab:DimensionlessVariables\]. The result is $$\begin{aligned}
\mathbb{T}' &= - (1 + {\mathrm{Wi}}\mathbb {T}) \mathbb{T} + 2 {\mathrm{Wi}}\mathbb{N}_1, \label{Eq:DFormulation:PlaEx:1TEvolution}\\
\mathbb{N}_1' &= - (1 + {\mathrm{Wi}}\mathbb {T}) \mathbb{N}_1 +2 {\mathrm{Wi}}\mathbb{T} + 4 \varepsilon, \label{Eq:DFormulation:PlaEx:2N1Evolution}\end{aligned}$$ with $\mathbb{T}(0)=\mathbb{N}_1(0)=0$. Note that $\mathbb{N}_2$ is not an independent variable: it is easy to see that $$\mathbb{N}_2 = \dfrac{1}{2}(\mathbb{T}+\mathbb{N}_1)\label{Eq:DFormulation:PlaEx:3N2Evolution}$$ at any $r$.
Cessation of steady shear and extensional flows
-----------------------------------------------
The dimensionless form of Eq. (\[Eq:Formulation:Cessation:MatrixEq\]) is $$\begin{aligned}
\mathbb{N}'_1 &= -(1+{\mathrm{Wi}}\mathbb{N}_1)\mathbb{N}_1, \label{Eq:DFormulation:Cessation:Shear:1N1}\\
\mathbb{S}' &= -(1+{\mathrm{Wi}}\mathbb{N}_1)\mathbb{S},\label{Eq:DFormulation:Cessation:Shear:2S}\end{aligned}$$ with $\mathbb{N}_1(0)=\delta_1$ and $\mathbb{S}(0)=\sigma$, for cessation of steady shear flow and $$\begin{aligned}
\mathbb{T}' &= -(1+{\mathrm{Wi}}\mathbb{T})\mathbb{T},\label{Eq:DFormulation:Cessation:Extension:1T}\\
\mathbb{N}'_1 &= -(1+{\mathrm{Wi}}\mathbb{T})\mathbb{N}_1,\label{Eq:DFormulation:Cessation:Extension:2N1}\end{aligned}$$ with $\mathbb{T}(0)=\tau$ and $\mathbb{N}_1(0)=\delta_1$, for cessation of extensional flows. For $\mathbb{N}_2$ in cessation of planar extensional flow, the algebraic relation (\[Eq:DFormulation:PlaEx:3N2Evolution\]) still holds.
Combining Eqs. (\[Eq:DFormulation:Cessation:Shear:1N1\]) and (\[Eq:DFormulation:Cessation:Shear:2S\]) yields $$\left (\dfrac{\mathbb{N}_1}{\mathbb{S}} \right)'=0,$$ Having integrated this and used the initial conditions, one finds that $$\label{Eq:DFormulation:Cessation:MatFunShear}
\dfrac{\mathbb{N}_1}{\delta_1}=\dfrac{\mathbb{S}}{\sigma}$$ at any $r$ for cessation of steady shear flow. Similarly, for cessation of extensional flows, $$\label{Eq:DFormulation:Cessation:MatFunExtensional}
\dfrac{\mathbb{T}}{\tau}=\dfrac{\mathbb{N}_1}{\delta_1}\left(=\dfrac{\mathbb{N}_2}{\delta_2} \right)$$ at any $r$. Therefore, only one differential equation needs to be solved, namely $$\label{Eq:DFormulation:Cessation:GeneralEquation}
\mathbb{X}'=-(1+{\mathrm{Wi}}\mathbb{X})\mathbb{X},$$ with the initial condition $\mathbb{X}(0)=\chi$, where $(\mathbb{X},\chi)\equiv(\mathbb{N}_1,\delta_1)$ for shear flow and $(\mathbb{X},\chi)\equiv(\mathbb{T},\tau)$ for extensional flows.
\[Sec:SteadyFlowSolutions\] Steady-flow solutions
==================================================
Prior to solving the systems of differential equations derived in Sec. \[Sec:DFormulation\], we shall review their algebraic steady-flow variants in this section; this needs to be done for two reasons. First, we shall see that the transient material functions describing start-up and cessation flows are readily expressed in terms of their steady-flow counterparts. Second, we already mentioned that the analytical description of steady-flow material functions of the SLPTT model found in the literature is still incomplete, especially when it comes to extensional flows; this section is also meant to fill this gap.
An important remark should be made before we proceed. The constitutive equation of the SLPTT model \[Eq. (\[Eq:Formulation:LPTTConstitutiveEquation\])\] is mathematically similar to that of the FENE-P dumbbells;[@Bird1980] in steady shear and extensional flows, the similarity between the model reaches complete equivalence.[@Oliveira2004; @Cruz2005; @Poole2019] Therefore, the results obtained in this section also hold for the FENE-P dumbbells, provided one makes the following parameter replacements: $$\begin{aligned}
\eta_0 & \leftrightarrow \dfrac{b}{b+3}nk_\mathrm{B}T\lambda_H, \\
\lambda & \leftrightarrow \dfrac{b}{b+3}\lambda_H, \\
\epsilon & \leftrightarrow \dfrac{1}{b+3}.\end{aligned}$$ A detailed discussion of the parameters of the FENE-P dumbbell model ($nk_\mathrm{B}T$, $\lambda_H$, and $b$) is to be found elsewhere.[@Bird1980; @Bird1987b; @Shogin2020]
\[SSec:SteSol:ShearFlow\] Shear flow
-------------------------------------
In steady-flow regime, Eqs. (\[Eq:DFormulation:SteSh:1N1Evolution\]) and (\[Eq:DFormulation:SteSh:2SEvolution\]) become $$\begin{aligned}
(1+{\mathrm{Wi}}\delta_1) \delta_1 &= 2 {\mathrm{Wi}}\sigma, \label{Eq:SteSol:Sh:1FNSD}\\
(1+{\mathrm{Wi}}\delta_1) \sigma &= \epsilon, \label{Eq:SteSol:Sh:2Shear}\end{aligned}$$ respectively. Dividing Eq. (\[Eq:SteSol:Sh:1FNSD\]) by Eq. (\[Eq:SteSol:Sh:2Shear\]) and rearranging, one gets $$\delta_1 = \dfrac{2}{\epsilon} {\mathrm{Wi}}\sigma^2. \label{Eq:SteSol:Sh:Delta1SigmaRelation}$$ Substituting this into Eq. (\[Eq:SteSol:Sh:2Shear\]) and solving for ${\mathrm{Wi}}$ yields $$\label{Eq:SteSol:Sh:WiSigmaRelationFinal}
{\mathrm{Wi}}= \sqrt{\dfrac{\epsilon(\epsilon-\sigma)}{2\sigma^3}}, \quad 0<\sigma \leq \epsilon,$$ where the positive sign in front of the square root was chosen since ${\mathrm{Wi}}\geq 0$. Relation (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) is bijective, see Appendix \[SApp:Bijection:Wi-SigmaSteadyShearFlow\]. Hence, $\sigma({\mathrm{Wi}})$ is defined uniquely as the inverse of (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]). Alternatively, one can directly solve Eq. (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) for $\sigma$, which leads to $$\label{Eq:SteSol:Sh:WiSigmaRelationAlternative}
\sigma = \dfrac{1}{{\mathrm{Wi}}} \sqrt{\dfrac{2\epsilon}{3}}\sinh \left[\dfrac{1}{3}\mathrm{arcsinh\left(3 \sqrt{\dfrac{3\epsilon}{2}}{\mathrm{Wi}}\right)} \right].$$ The solutions given by Eqs. (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) and (\[Eq:SteSol:Sh:WiSigmaRelationAlternative\]) are, of course, equivalent. Regardless of the one preferred, $\delta_1({\mathrm{Wi}})$ is calculated using Eq. (\[Eq:SteSol:Sh:Delta1SigmaRelation\]). It should be noted that our analytical solution in form (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) can be easily used to study the asymptotic behavior of the steady-shear-flow material functions. In particular, it can be seen that at ${\mathrm{Wi}}\to 0$, $$\begin{aligned}
\sigma &\to \epsilon, \\
\delta_1 &\sim 2\epsilon {\mathrm{Wi}}\end{aligned}$$ so that (see Table \[Tab:MatFunctionsVsDimlessVars\]) $$\begin{aligned}
\eta & \to \eta_0, \\
\Psi_1 & \to 2 \eta_0 \lambda,\end{aligned}$$ while at ${\mathrm{Wi}}\to \infty$, $$\begin{aligned}
\sigma &\sim \sqrt[3]{\epsilon^2/2}{\mathrm{Wi}}^{-2/3},\\
\delta_1 & \sim \sqrt[3]{2\epsilon} {\mathrm{Wi}}^{-1/3}\end{aligned}$$ so that (see Table \[Tab:MatFunctionsVsDimlessVars\]) $$\begin{aligned}
\eta &\sim \eta_0 \sqrt[3]{1/2\epsilon}{\mathrm{Wi}}^{-2/3}, \\
\Psi_1 & \sim \eta_0 \lambda \sqrt[3]{2/\epsilon^2}{\mathrm{Wi}}^{-4/3}.\end{aligned}$$ Furthermore, it follows from Eq. (\[Eq:SteSol:Sh:Delta1SigmaRelation\]) and Table \[Tab:MatFunctionsVsDimlessVars\] that $\Psi_1$ is proportional to the square of the viscosity, $$\Psi_1=\dfrac{2\lambda}{\eta_0}\eta^2,$$ at any ${\mathrm{Wi}}$.
Our results are equivalent to those found in the literature for the SLPTT model[@Azaiez1996; @Xue1998] and for the FENE-P dumbbells[@Bird1980; @Bird1987b; @Azaiez1996; @Shogin2017] but are much simpler \[e.g., compare Eqs. (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) and (\[Eq:SteSol:Sh:Delta1SigmaRelation\]) of this work to Eqs. (37)-(40) and (48)-(51) of Azaiez *et al.*[@Azaiez1996] or to Eqs. (16)-(19) of Xue *et al.* (with $\xi=0$ and $\beta=1$)\].
\[SSec:SteSol:UBExFlow\] Uni- and biaxial extensional flow
-----------------------------------------------------------
For steady flow, Eqs. (\[Eq:DFormulation:UBEx:1TEvolution\]) and (\[Eq:DFormulation:UBEx:2N1Evolution\]) reduce to $$\begin{aligned}
(1+{\mathrm{Wi}}\tau)\tau &= 2 {\mathrm{Wi}}\delta_1, \label{Eq:SteSol:UBEx:1Trace}\\
(1+{\mathrm{Wi}}\tau)\delta_1 &= {\mathrm{Wi}}(\tau \pm \delta_1) + 3 \epsilon, \label{Eq:SteSol:UBEx:2FNSD}\end{aligned}$$ respectively, the upper signs corresponding to uniaxial and the lower to biaxial extension. From Eq. (\[Eq:SteSol:UBEx:1Trace\]), $$\delta_1 = \dfrac{(1+{\mathrm{Wi}}\tau)\tau}{2{\mathrm{Wi}}}; \label{Eq:SteSol:UBEx:DeltaTRelation}$$ having substituted this into Eq. (\[Eq:SteSol:UBEx:2FNSD\]), one arrives after rearrangements at $$\label{Eq:SteSol:UBEx:WiTQuadraticEquation}
(2\tau\pm \tau^2-\tau^3 ){\mathrm{Wi}}^2 +(6\epsilon\pm \tau-2\tau^2){\mathrm{Wi}}-\tau = 0.$$ Then, Eq. (\[Eq:SteSol:UBEx:WiTQuadraticEquation\]) is treated as quadratic in ${\mathrm{Wi}}$. Only one of its two solutions is physically meaningful, being non-negative and continuous at $\tau \geq 0$. This solution can be written as $${\mathrm{Wi}}= \dfrac{\sqrt{36\epsilon^2\pm 12\epsilon \tau+(9-24\epsilon) \tau^2}-(6\epsilon \pm\tau-2\tau^2)}{2\tau(2\mp \tau)(1 \pm \tau)} \label{Eq:SteSol:UBEx:WiTRelationFinal}$$ or, with the removable singularity at $\tau=0$ eliminated, as $${\mathrm{Wi}}=\dfrac{2\tau}{\sqrt{36\epsilon^2\pm 12\epsilon \tau+(9-24\epsilon) \tau^2}+(6\epsilon \pm\tau-2\tau^2)}, \label{Eq:SteSol:UBEx:WiTRelationFinal2}$$ the domain of the function being $0 \leq \tau <2$ for uniaxial extension and $0 \leq \tau < 1$ for biaxial extension. Equation (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) defines a bijection for both uni- and biaxial extension (see Appendix \[SApp:Bijection:Wi-TSteadyUBExFlow\]). Hence, $\tau({\mathrm{Wi}})$ is defined uniquely as the inverse of (\[Eq:SteSol:UBEx:WiTRelationFinal\]) and $\delta_1({\mathrm{Wi}})$ is found from Eq. (\[Eq:SteSol:UBEx:DeltaTRelation\]).
![\[Fig:SteadyUBEx\] The dimensionless stress combinations in steady uniaxial (positive elongation rates, right part) and biaxial (negative elongation rates, left part) extensional flow as functions of the Weissenberg number, ${\mathrm{Wi}}=\lambda \vert \dot{\epsilon} \vert$. The shape of the $\delta_1({\mathrm{Wi}})$ curve repeats that of the extensional viscosity (see Table \[Tab:MatFunctionsVsDimlessVars\]).](1.eps){width="3.37in"}
Functions $\tau({\mathrm{Wi}})$ and $\delta_1({\mathrm{Wi}})$ are shown in Fig. \[Fig:SteadyUBEx\], the right half of the plot corresponding to uniaxial extension and the left half to biaxial extension. From Eqs. (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) and (\[Eq:SteSol:UBEx:DeltaTRelation\]) it is seen that at ${\mathrm{Wi}}\to 0$, $$\begin{aligned}
\tau &\sim 6\epsilon {\mathrm{Wi}}, \\
\delta_1 & \to 3\epsilon\end{aligned}$$ for uni- and biaxial extension so that the Trouton ratio,[@Petrie2006] $$\bar{\eta} \to 3\eta_0, \label{Eq:SteSol:UBEx:TroutonRatio}$$ is recovered (see Table \[Tab:MatFunctionsVsDimlessVars\]). In both uni- and biaxial extensional flows, $\tau({\mathrm{Wi}})$ is increasing monotonically with ${\mathrm{Wi}}$ (see Appendix \[SApp:Bijection:Wi-TSteadyUBExFlow\]), but with different asymptotic behavior at ${\mathrm{Wi}}\to \infty$ \[see Eq. (\[Eq:SteSol:UBEx:WiTRelationFinal\])\]: in the uniaxial case, $$\tau \sim 2-\dfrac{1-\epsilon}{{\mathrm{Wi}}},$$ while in the biaxial case, $$\tau \sim 1-\dfrac{1-2\epsilon}{{\mathrm{Wi}}}.$$ Furthermore, $\delta_1({\mathrm{Wi}})$, and hence the extensional viscosity, increases monotonically with ${\mathrm{Wi}}$ for uniaxial extension (see Appendix \[SApp:Viscosities:UBEx\]); at ${\mathrm{Wi}}\to \infty$ \[see Eq. (\[Eq:SteSol:UBEx:DeltaTRelation\]) and Table \[Tab:MatFunctionsVsDimlessVars\]\], $$\begin{aligned}
\delta_1 & \sim 2-\dfrac{1-2\epsilon}{{\mathrm{Wi}}},\\
\bar{\eta} & \sim \dfrac{2\eta_0}{\epsilon}-\dfrac{(1-2\epsilon)\eta_0}{\epsilon {\mathrm{Wi}}}.\label{Eq:SteSol:UBEx:AsymptoticViscosityUEx}\end{aligned}$$ In contrast, for biaxial extensional flow, $\delta_1({\mathrm{Wi}})$ and the extensional viscosity are not monotonic functions but have a minimum (see Appendix \[SApp:Viscosities:UBEx\])–namely, $$\begin{aligned}
\delta_{1,\mathrm{min}} &= \dfrac{8\epsilon(1-3\epsilon)}{3-8\epsilon}, \label{Eq:SteSol:UBEx:Delta1MinimumValue}
\\
\bar{\eta}_\mathrm{min} &= \dfrac{8\eta_0(1-3\epsilon)}{3-8\epsilon}, \label{Eq:SteSol:UBEx:ExtensionalViscosityMinimumValue}\end{aligned}$$ at $$\label{Eq:SteSol:UBEx:ExtensionalViscosityMinimumPosition}
{\mathrm{Wi}}_\mathrm{min} = \dfrac{3-8\epsilon}{12(1-2\epsilon)(1-4\epsilon)};$$ at ${\mathrm{Wi}}\to \infty$ \[see Eq. (\[Eq:SteSol:UBEx:DeltaTRelation\]) and Table \[Tab:MatFunctionsVsDimlessVars\]\], $$\begin{aligned}
\delta_1 &\sim \dfrac{1}{2}-\dfrac{1-4\epsilon}{2{\mathrm{Wi}}}, \\
\bar{\eta} &\sim \dfrac{\eta_0}{2\epsilon}-\dfrac{(1-4\epsilon)\eta_0}{2\epsilon{\mathrm{Wi}}}. \label{Eq:SteSol:UBEx:AsymptoticViscosityBiEx}\end{aligned}$$
We are not aware of any analogs of Eqs. (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) and (\[Eq:SteSol:UBEx:DeltaTRelation\]) in the literature. Furthermore, to our knowledge, the character of the extensional viscosity curve (monotonic in the uniaxial case and non-monotonic in the biaxial case) was previously shown exclusively by numerical simulations and not proven analytically as in this work. Finally, we believe that we are the first to exactly describe the minimum of the biaxial extensional viscosity \[see Eqs. (\[Eq:SteSol:UBEx:ExtensionalViscosityMinimumValue\]) and (\[Eq:SteSol:UBEx:ExtensionalViscosityMinimumPosition\])\]. The asymptotic behavior of extensional viscosities at large Weissenberg numbers \[Eqs. (\[Eq:SteSol:UBEx:AsymptoticViscosityUEx\]) and (\[Eq:SteSol:UBEx:AsymptoticViscosityBiEx\])\] is in agreement with the earlier results for SLPTT and FENE-P dumbbell fluid models [@Bird1980; @Petrie1990; @Bird1987b; @Tanner2003] and thus serves as a consistency check.
\[SSec:SteSol:PlaExFlow\] Planar extensional flow
--------------------------------------------------
For steady planar extensional flow, Eqs. (\[Eq:DFormulation:PlaEx:1TEvolution\]) and (\[Eq:DFormulation:PlaEx:2N1Evolution\]) yield $$\begin{aligned}
(1+{\mathrm{Wi}}\tau)\tau &= 2 {\mathrm{Wi}}\delta_1, \label{Eq:SteSol:PlaEx:1Trace}\\
(1+{\mathrm{Wi}}\tau)\delta_1 &= 2{\mathrm{Wi}}\tau+4\epsilon, \label{Eq:SteSol:PlaEx:2FNSD}\end{aligned}$$ while Eq. (\[Eq:DFormulation:PlaEx:3N2Evolution\]) becomes $$\delta_2 = \dfrac{1}{2}(\tau+\delta_1). \label{Eq:SteSol:PlaEx:Delta2TDelta1Relation}$$ Equation (\[Eq:SteSol:PlaEx:1Trace\]) is identical in form to Eq. (\[Eq:SteSol:UBEx:1Trace\]) for uni- and biaxial extensional flows; therefore, $$\delta_1 = \dfrac{(1+{\mathrm{Wi}}\tau)\tau}{2{\mathrm{Wi}}}, \label{Eq:SteSol:PlaEx:DeltaTRelation}$$ which is of the same form as Eq. (\[Eq:SteSol:UBEx:DeltaTRelation\]). One proceeds by using Eq. (\[Eq:SteSol:PlaEx:DeltaTRelation\]) to eliminate $\delta_1$ from Eq. (\[Eq:SteSol:PlaEx:2FNSD\]). The result is $$\label{Eq:SteSol:PlaEx:WiTQuadraticEquation}
(4\tau-\tau^3){\mathrm{Wi}}^2 + 2(4\epsilon-\tau^2){\mathrm{Wi}}- \tau = 0.$$ Equation (\[Eq:SteSol:PlaEx:WiTQuadraticEquation\]) is treated as quadratic in ${\mathrm{Wi}}$. Of its two solutions, the one meeting the requirements of non-negativity and continuity can be written as $${\mathrm{Wi}}= \dfrac{2\sqrt{4\epsilon^2+(1-2\epsilon)\tau^2}-(4\epsilon-\tau^2) }{\tau(4-\tau^2)},
\label{Eq:SteSol:PlaEx:WiTRelationFinal}$$ or, with the removable singularity at $\tau=0$ eliminated, as $${\mathrm{Wi}}= \dfrac{\tau}{2\sqrt{4\epsilon^2+(1-2\epsilon)\tau^2}+(4\epsilon-\tau^2)},
\label{Eq:SteSol:PlaEx:WiTRelationFinal2}$$ with $0\leq \tau <2$. Equation (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]) defines a bijection (see Appendix \[SApp:Bijection:Wi-TSteadyPlaExFlow\]); therefore, $\tau({\mathrm{Wi}})$ is defined unambiguously as the inverse of (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]), while $\delta_1({\mathrm{Wi}})$ and $\delta_2({\mathrm{Wi}})$ are found using Eqs. (\[Eq:SteSol:PlaEx:DeltaTRelation\]) and (\[Eq:SteSol:PlaEx:Delta2TDelta1Relation\]), respectively.
![\[Fig:SteadyPlaEx\] The dimensionless stress combinations in steady planar extensional flow as functions of the Weissenberg number, ${\mathrm{Wi}}=\lambda \dot{\epsilon}$. The shapes of the $\delta_1({\mathrm{Wi}})$ and $\delta_2({\mathrm{Wi}})$ curves repeat those of the first and the second extensional viscosity, respectively (see Table \[Tab:MatFunctionsVsDimlessVars\]).](2.eps){width="3.37in"}
Functions $\delta_1({\mathrm{Wi}})$, $\delta_2({\mathrm{Wi}})$, and $\tau({\mathrm{Wi}})$ are shown in Fig. \[Fig:SteadyPlaEx\]. All of them, and therefore the planar extensional viscosities, are increasing monotonically with ${\mathrm{Wi}}$ (see Appendix \[SApp:Viscosities:PlaEx\] and Table \[Tab:MatFunctionsVsDimlessVars\]). One finds that at ${\mathrm{Wi}}\to 0$ \[see Eqs. (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]), (\[Eq:SteSol:PlaEx:DeltaTRelation\]), and (\[Eq:SteSol:PlaEx:Delta2TDelta1Relation\])\], $$\begin{aligned}
\tau & \sim 8\epsilon {\mathrm{Wi}}, \\
\delta_1 & \to 4\epsilon, \\
\delta_2 & \to 2\epsilon\end{aligned}$$ so that (see Table \[Tab:MatFunctionsVsDimlessVars\]) $$\begin{aligned}
\bar{\eta}_1 & \to 4\eta_0, \label{Eq:SteSol:PlaEx:AsymptoticEta1}\\
\bar{\eta}_2 & \to 2\eta_0, \label{Eq:SteSol:PlaEx:AsymptoticEta2}\end{aligned}$$ while at ${\mathrm{Wi}}\to \infty$ \[see Eqs. (\[Eq:SteSol:PlaEx:WiTRelationFinal\]), (\[Eq:SteSol:PlaEx:DeltaTRelation\]), and (\[Eq:SteSol:PlaEx:Delta2TDelta1Relation\])\], $$\begin{aligned}
\tau & \sim 2-\dfrac{1-\epsilon}{{\mathrm{Wi}}}\\
\delta_1& \sim 2-\dfrac{1-2\epsilon}{{\mathrm{Wi}}}\\
\delta_2& \sim 2-\dfrac{2-3\epsilon}{2{\mathrm{Wi}}}\end{aligned}$$ so that (see Table \[Tab:MatFunctionsVsDimlessVars\]) $$\begin{aligned}
\bar{\eta}_1 & \sim \dfrac{2\eta_0}{\epsilon}-\dfrac{(1-2\epsilon)\eta_0}{\epsilon {\mathrm{Wi}}},\label{Eq:SteSol:PlaEx:AsymptoticEta1HighWi}\\
\bar{\eta}_2 & \sim \dfrac{2\eta_0}{\epsilon}-\dfrac{(2-3\epsilon)\eta_0}{2\epsilon {\mathrm{Wi}}}.\label{Eq:SteSol:PlaEx:AsymptoticEta2HighWi}\end{aligned}$$
Equations (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]), (\[Eq:SteSol:PlaEx:DeltaTRelation\]), and (\[Eq:SteSol:PlaEx:Delta2TDelta1Relation\]) provide a complete description of rheological properties of the SLPTT fluid model in steady planar extensional flow. The only alternative to these equations we found in the literature \[Eqs. (23) and (24) of Xue *et al.*,[@Xue1998] with $\xi=0$ and $\beta=1$\] is both less detailed (no formula for the second planar extensional viscosity is provided) and more complicated compared to our result. The asymptotic values of the planar extensional viscosities at low and high Weissenberg numbers obtained in this work \[Eqs. (\[Eq:SteSol:PlaEx:AsymptoticEta1\]), (\[Eq:SteSol:PlaEx:AsymptoticEta2\]), (\[Eq:SteSol:PlaEx:AsymptoticEta1HighWi\]), and (\[Eq:SteSol:PlaEx:AsymptoticEta2HighWi\])\] are in agreement with general expectations[@Petrie2006] and with the earlier analytical results of Petrie.[@Petrie1990]
\[Sec:StartUpSolutions\] Step-rate solutions
=============================================
The main result of this work (for its derivation, see Appendix \[App:MainResult\]) is the exact analytical expressions for the stress growth and relaxation functions of the SLPTT fluid model. We have found that the normalized stress growth functions can be written in one of the two compact, closely related, and mathematically beautiful forms–those related to start-up of shear flow take form $\mathfrak{T}$ (for “trigonometric”), while those related to start-up of extensional flows take form $\mathfrak{H}$ (for “hyperbolic”)–while the normalized stress relaxation functions related to cessation of shear and extensional flows are of form $\mathfrak{E}$ (for “exponential”).
Forms $\mathfrak{T}$, $\mathfrak{H}$, and $\mathfrak{E}$ are presented in Table \[Tab:Forms\] together with the material functions taking these forms. The normalized trace of the stress tensor \[$\mathbb{T}(r,{\mathrm{Wi}})/\tau({\mathrm{Wi}})$\] is not traditionally considered as a material function but can be of theoretical interest; therefore, it is also included in the results for the sake of completeness.
[c c c]{} Form & Material functions & Expression\
------------------------------------------------------------------------
$\mathfrak{T}$ & $\dfrac{\eta^+(r,{\mathrm{Wi}})}{\eta({\mathrm{Wi}})}$, $\dfrac{\Psi_1^+(r,{\mathrm{Wi}})}{\Psi_1({\mathrm{Wi}})}$ & $1-\dfrac{K (\cos \omega r + a\sin \omega r)}{C \mathrm{e}^{\Omega r}+{\mathrm{Wi}}(A \cos \omega r + B \sin \omega r )}$\
------------------------------------------------------------------------
$\mathfrak{H}$ & $\dfrac{\bar{\eta}^+(r,{\mathrm{Wi}})}{\bar{\eta}({\mathrm{Wi}})}$, $\dfrac{\bar{\eta}_{1,2}^+(r,{\mathrm{Wi}})}{\bar{\eta}_{1,2}({\mathrm{Wi}})}$, $\dfrac{\mathbb{T}(r,{\mathrm{Wi}})\footnote{at start-up of extensional flows}}{\tau({\mathrm{Wi}})}$ & $1-\dfrac{K (\cosh \omega r + a \sinh \omega r)}{C \mathrm{e}^{\Omega r}+{\mathrm{Wi}}(A \cosh \omega r + B \sinh \omega r )}$\
------------------------------------------------------------------------
$\mathfrak{E}$ & $\dfrac{\eta^-(r,{\mathrm{Wi}})}{\eta({\mathrm{Wi}})}$, $\dfrac{\Psi_1^-(r,{\mathrm{Wi}})}{\Psi_1({\mathrm{Wi}})}$, $\dfrac{\bar{\eta}^-(r,{\mathrm{Wi}})}{\bar{\eta}({\mathrm{Wi}})}$, $\dfrac{\bar{\eta}_{1,2}^-(r,{\mathrm{Wi}})}{\bar{\eta}_{1,2}({\mathrm{Wi}})}$, $\dfrac{\mathbb{T}(r,{\mathrm{Wi}})\footnote{at cessation of extensional flows}}{\tau({\mathrm{Wi}})}$ & $\dfrac{1}{(1+{\mathrm{Wi}}\chi)\mathrm{e}^r-{\mathrm{Wi}}\chi}$
Forms $\mathfrak{T}$ and $\mathfrak{H}$ contain expressions $\Delta>0$, $\omega = {\mathrm{Wi}}\sqrt{\Delta}/2$, $\Omega$, $K$, $C$, $A$, and $B$, which are functions of ${\mathrm{Wi}}$. Their definitions of these functions depend on the flow type and are given in Table \[Tab:FormFunctions\]. Another expression encountered in forms $\mathfrak{T}$ and $\mathfrak{H}$ is $a$, which is defined uniquely for each material function. The definitions of $a$ corresponding to different start-up material functions are found in Table \[Tab:a\]. Finally, in form $\mathfrak{E}$, $\chi \equiv \delta_1$ for cessation of shear flow and $\chi \equiv \tau$ for cessation of extensional flows.
[r c c c]{} & I & II & III\
------------------------------------------------------------------------
$\Delta\,$ & $- \delta_1^2+8\sigma $ & $9-8\delta_1 \pm 2\tau +\tau^2$ & $16-8\delta_1+\tau^2 $\
------------------------------------------------------------------------
$\Omega\,$ & $1 + \dfrac{3}{2} {\mathrm{Wi}}\delta_1$ & $1 + \dfrac{1}{2}{\mathrm{Wi}}(\mp 1+3\tau)$ & $1 + \dfrac{3}{2}{\mathrm{Wi}}\tau$\
------------------------------------------------------------------------
$K\,$ & $1 + {\mathrm{Wi}}\delta_1 + 6 {\mathrm{Wi}}^2 \sigma$ & $1 + {\mathrm{Wi}}(\mp 1 + \tau) + 2 {\mathrm{Wi}}^2 (-1+ 3 \delta_1 \mp \tau)$ & $1 + {\mathrm{Wi}}\tau + 2 {\mathrm{Wi}}^2 (-2+3 \delta_1)$\
------------------------------------------------------------------------
$C\,$ & $1 + {\mathrm{Wi}}\delta_1 + 2 {\mathrm{Wi}}^2 \sigma $ & $1 + {\mathrm{Wi}}( \mp 1 + \tau) + {\mathrm{Wi}}^2 (-2+ 2 \delta_1 \mp \tau ) $ & $1 + {\mathrm{Wi}}\tau + 2 {\mathrm{Wi}}^2 (-2+\delta_1)$\
------------------------------------------------------------------------
$A\,$ & $4 {\mathrm{Wi}}\sigma $ & ${\mathrm{Wi}}(4\delta_1 \mp \tau)$ & $4 {\mathrm{Wi}}\delta_1$\
------------------------------------------------------------------------
$B\,$ & $\dfrac{4 \sigma}{\sqrt{\Delta}}$ & $\dfrac{4\delta_1 \mp \tau + {\mathrm{Wi}}(\mp 2\delta_1+5\tau )} {\sqrt{\Delta}}$ & $\dfrac{4 (\delta_1 + 2 {\mathrm{Wi}}\tau)} {\sqrt{\Delta}}$
[c c c c c c]{} & &\
NMF & $a$ & NMF & $a$ & NMF & $a$\
------------------------------------------------------------------------
$\dfrac{\eta^+(t,{\mathrm{Wi}})}{\eta({\mathrm{Wi}})}$ & $-\dfrac{\delta_1}{\sqrt{\Delta}}$ & $\dfrac{\bar{\eta}^+(t,{\mathrm{Wi}})}{\bar{\eta}({\mathrm{Wi}})}$ & $\dfrac{ \pm \delta_1 +2\tau - \delta_1 \tau}{\delta_1 \sqrt{\Delta}}$ & $\dfrac{\bar{\eta}_1^+(r,{\mathrm{Wi}})}{\bar{\eta}_1({\mathrm{Wi}})}$ & $\dfrac{(4-\delta_1)\tau}{\delta_1 \sqrt{\Delta}}$\
------------------------------------------------------------------------
$\dfrac{\Psi_1^+(t,{\mathrm{Wi}})}{\Psi_1({\mathrm{Wi}})}$ & $\dfrac{\delta_1 + 2 {\mathrm{Wi}}\sigma}{{\mathrm{Wi}}\delta_1 \sqrt{\Delta}}$ & $\dfrac{\mathbb{T}(t,{\mathrm{Wi}})}{\tau({\mathrm{Wi}})}$ & $\dfrac{\tau + {\mathrm{Wi}}(2\delta_1 \mp \tau)}{{\mathrm{Wi}}\tau \sqrt{\Delta}}$ & $\dfrac{\bar{\eta}_2^+(r,{\mathrm{Wi}})}{\bar{\eta}_2({\mathrm{Wi}})}$ & $\dfrac {\tau + {\mathrm{Wi}}(2\delta_1 + 4\tau -\delta_1 \tau)} {2 {\mathrm{Wi}}\delta_2 \sqrt{\Delta}}$\
------------------------------------------------------------------------
& & & & $\dfrac{\mathbb{T}(r,{\mathrm{Wi}})}{\tau({\mathrm{Wi}})}$ & $\dfrac{\tau + 2{\mathrm{Wi}}\delta_1}{{\mathrm{Wi}}\tau \sqrt{\Delta}}$
\[SSec:StartUpSolutions:ShearFlow\] Start-up of steady shear flow
------------------------------------------------------------------
![\[Fig:StartUpShearFlow\] The normalized shear stress growth function \[$\eta^+(r,{\mathrm{Wi}})/\eta({\mathrm{Wi}})$, (a)\] and first normal stress difference growth function \[$\Psi_1^+(r,{\mathrm{Wi}})/\Psi_1({\mathrm{Wi}})$, (b)\] at start-up of steady shear flow (exact analytical solutions for the SLPTT model) as functions of the dimensionless time, $r=t/\lambda$, at different Weissenberg numbers. The dotted curves show the limiting case ${\mathrm{Wi}}\to 0$.](3.eps){width="3.37in"}
The normalized stress growth functions related to start-up of steady shear flow are described by form $\mathfrak{T}$; the exact analytical solutions are shown in Fig. \[Fig:StartUpShearFlow\]. Both functions undergo quasi-periodic, exponentially damped oscillations while approaching unity.
The impact of ${\mathrm{Wi}}$ on these material functions is also seen in Fig. \[Fig:StartUpShearFlow\]. At sufficiently high Weissenberg numbers, the stress overshoots become visible. As ${\mathrm{Wi}}$ increases, these overshoots are shifted towards earlier times and become more pronounced. At any fixed ${\mathrm{Wi}}$, the relative magnitude of the shear stress overshoot is always larger than that of the first normal stress difference overshoot. Finally, the steady-flow regime is approached faster at higher ${\mathrm{Wi}}$, with the shear stress stabilizing faster than the first normal stress difference.
In the following, we shall discuss some properties of the stress growth material functions obtained using the exact analytical solution.
First, all solutions are oscillatory, the only exception being the limiting case, ${\mathrm{Wi}}\to 0$. However, the oscillations are not always easy to observe because of the exponential damping \[see, e.g., Figs. \[Fig:StartUpShearFlow\](a) and \[Fig:StartUpShearFlow\](b) at ${\mathrm{Wi}}=1$\]. In fact, only the first maximum (overshoot) is well pronounced.
Second, both material functions take their steady-flow values (unity) periodically. As seen from form $\mathfrak{T}$ (Table \[Tab:Forms\]), it occurs when $\tan \omega r = -1/a$. This leads to $$\begin{aligned}
\dfrac{\eta^+}{\eta}=1 \quad \text{at} \quad & r_k = \dfrac{1}{\omega}\arctan \dfrac{\sqrt{\Delta}}{\delta_1}+(k-1)\dfrac{\pi}{\omega}, \label{Eq:StartUpSolutions:PeriodEtaPlus} \\
\dfrac{\Psi_1^+}{\Psi_1}=1 \quad \text{at} \quad & r_k= -\dfrac{1}{\omega} \arctan\dfrac{{\mathrm{Wi}}\delta_1 \sqrt{\Delta}}{\delta_1+2{\mathrm{Wi}}\sigma}+k\dfrac{\pi}{\omega}, \label{Eq:StartUpSolutions:PeriodPsi1Plus}\end{aligned}$$ $k$ being a natural number. Thus, the corresponding “periods”, $\Delta r$, of $\eta^+$ and $\Psi_1^+$ are identical and equal $\pi /\omega$. Furthermore, let $r_1$ be the time point when a normalized stress growth function reaches unity for the first time. It is seen from Eq. (\[Eq:StartUpSolutions:PeriodEtaPlus\]) that for $\eta^+$, $0<r_1<\pi/2\omega$. This is always smaller than $r_1$ for $\Psi_1^+$, for which $\pi/2\omega<r_1<\pi/\omega$ \[see Eq. (\[Eq:StartUpSolutions:PeriodPsi1Plus\])\]; therefore, $\eta^+/\eta$ increases faster than $\Psi_1^+/\Psi_1$ at fixed ${\mathrm{Wi}}$. This can be seen from comparison of Fig. \[Fig:StartUpShearFlow\](a) to Fig. \[Fig:StartUpShearFlow\](b).
In contrast, the maxima and minima of the stress growth functions do not occur periodically. Their positions are defined by transcendental equations (not solvable analytically) containing exponential and trigonometric functions. This can be seen by taking the time derivative of form $\mathfrak{T}$ and setting it equal to zero.
Third, in the limit ${\mathrm{Wi}}\to 0$, the analytical expressions for $\eta^+$ and $\Psi_1^+$ reduce to $$\begin{aligned}
\eta^+ & \sim \eta_0(1-e^{-r}), \label{Eq:StartUpSolutions:LV}\\
\Psi_1^+ & \sim 2\eta_0\lambda[1-(1+r)e^{-r}]. \label{Eq:StartUpSolutions:LOPsi1}\end{aligned}$$ In this limiting case \[see the dotted lines in Figs. \[Fig:StartUpShearFlow\](a) and \[Fig:StartUpShearFlow\](b)\], the solutions are not oscillatory. Equation (\[Eq:StartUpSolutions:LV\]) is recognized as the so-called linear viscoelastic limit; this response, as might be expected, is identical to that of the Maxwell model.[@Bird1987b]
Finally, in the limit ${\mathrm{Wi}}\to \infty$, form $\mathfrak{T}$ yields $$\begin{aligned}
\dfrac{\eta^+}{\eta} & \sim 1-\dfrac{3\cos w-\sqrt{3}\sin w}{e^{\sqrt{3}w}+2\cos w}, \label{Eq:StartUpSolutions:HighWiEtaPlus}\\
\dfrac{\Psi_1^+}{\Psi_1} & \sim 1-\dfrac{3\cos w+\sqrt{3}\sin w}{e^{\sqrt{3}w}+2\cos w}, \label{Eq:StartUpSolutions:HighWiPsi1Plus}\end{aligned}$$ where $$w=\sqrt[3]{\dfrac{3\sqrt{3}\epsilon}{4}}{\mathrm{Wi}}^{2/3}r.$$ Having taken the derivatives of the right-hand sides of Eqs. (\[Eq:StartUpSolutions:HighWiEtaPlus\]) and (\[Eq:StartUpSolutions:HighWiPsi1Plus\]) and set them equal to zero, one can find the smallest positive roots of the resulting transcendental equations numerically, e.g., using Newton’s method. Then, it can be shown that $$\begin{aligned}
\mathrm{max}(\eta^+\!/\eta) \approx 1.114 & \text{ at } r \approx 1.468 \epsilon^{-1/3}{\mathrm{Wi}}^{-2/3}, \label{Eq:StartUpSolutions:OvershootEtaPlus}\\
\mathrm{max}(\Psi_1^+\!/\Psi_1) \approx 1.019 & \text{ at } r \approx 2.394\epsilon^{-1/3}{\mathrm{Wi}}^{-2/3}.\label{Eq:StartUpSolutions:OvershootPsi1Plus}\end{aligned}$$ This provides the asymptotic expressions for the positions of the first overshoots at high Weissenberg numbers. In addition, Eqs. (\[Eq:StartUpSolutions:OvershootEtaPlus\]) and (\[Eq:StartUpSolutions:OvershootPsi1Plus\]) set upper theoretical limits on the relative magnitudes of the stress overshoots; remarkably, these limits depend neither on ${\mathrm{Wi}}$ nor on the model parameters.
\[SSec:StartUpSolutions:ExtensionalFlows\] Start-up of steady uniaxial, biaxial, and planar extensional flows
--------------------------------------------------------------------------------------------------------------
The normalized stress growth functions related to start-up of steady extensional flows are described by form $\mathfrak{H}$; the exact analytical results are shown in Figs. \[Fig:StartUpUniExFlow\] and \[Fig:StartUpBiExFlow\] (uniaxial and biaxial extension, respectively; the results for planar extension are qualitatively similar to those for uniaxial extension). The stress growth functions approach their steady-flow values monotonically; oscillations, and therefore overshoots, are not present.
![\[Fig:StartUpUniExFlow\] The normal stress difference growth function \[$\bar{\eta}^+(r,{\mathrm{Wi}})$\] at start-up of steady uniaxial extensional flow (exact analytical solution for the SLPTT model) plotted against the dimensionless time, $r=t/\lambda$, at different Weissenberg numbers. The material function is scaled using a constant factor of $\epsilon/\eta_0$ for visual purposes. The dotted line shows the limit ${\mathrm{Wi}}\to 0$.](4.eps){width="3.37in"}
{width="6.45in"}
The functions $\bar{\eta}^+$, $\bar{\eta}_1^+$, and $\bar{\eta}_2^+$ are affected by the Weissenberg number in the same way: the shape of the curves changes gradually from smoother to more abrupt and step-like as ${\mathrm{Wi}}$ increases, the steady state being reached faster at higher ${\mathrm{Wi}}$.
At ${\mathrm{Wi}}\to 0$, the expressions for the stress growth functions reduce to $$\begin{aligned}
\bar{\eta}^+ & \sim 3\eta_0(1-e^{-r}), \\
\bar{\eta}_1^+ & \sim 4\eta_0(1-e^{-r}), \\
\bar{\eta}_2^+ & \sim 2\eta_0(1-e^{-r}),\end{aligned}$$ the form of $\bar{\eta}^+$ for uni- and biaxial extension being the same in this limit.
At ${\mathrm{Wi}}\to \infty$, $$\bar{\eta}^+ \sim \bar{\eta}_1^+ \sim \bar{\eta}_2^+ \sim \dfrac{2\eta_0}{\epsilon}\left[ 1-\left(1+\dfrac{\epsilon \mathrm{e}^{2{\mathrm{Wi}}r}}{2{\mathrm{Wi}}} \right)^{-1}\right]
\label{Eq:StartUpExtensional:Equivalence}$$ for start-up of steady uniaxial and planar extensional flows, while $$\bar{\eta}^+ \sim \dfrac{\eta_0}{2\epsilon}\left[ 1-\left(1+\dfrac{2\epsilon \mathrm{e}^{{\mathrm{Wi}}r}}{{\mathrm{Wi}}} \right)^{-1}\right]$$ for start-up of steady biaxial extensional flow.
Cessation of steady shear and extensional flows
-----------------------------------------------
The normalized stress relaxation functions related to cessation of steady shear and extensional flows are described by form $\mathfrak{E}$; the exact analytical results for shear flow (the results for extensional flows are qualitatively similar) are shown in Fig. \[Fig:CessationShearFlow\]. The stress relaxation functions decrease, approaching zero monotonically; at late times, they decay exponentially.
{width="3.37in"}
It is also seen that the normalized stress relaxation functions decrease faster at higher Weissenberg numbers. In the limit ${\mathrm{Wi}}\to 0$, $$\dfrac{\eta^-}{\eta} \sim \dfrac{\Psi_1^-}{\Psi_1} \sim \dfrac{\bar{\eta}^-}{\bar{\eta}} \sim \dfrac{\bar{\eta}_{1,2}^-}{\bar{\eta}_{1,2}}\sim \mathrm{e}^{-r}.$$ For $\eta^-/\eta$, this result is identical to the “linear viscoelastic response” of the Maxwell model.[@Bird1987a]
Generalization to multiple modes {#Sec:Multimode}
================================
The exact analytical solutions discussed so far were obtained for a single-mode SLPTT model. At the same time, the multimode versions of the PTT models are often used in applications. For the SLPTT model with $M$ modes, the stress tensor is the sum of contributions from different modes, $$\label{Eq:Formulation:Multimode}
\boldsymbol{\tau} = \sum_{m=1}^{M}\tau^{(m)},$$ where each of the contributions, $\boldsymbol{\tau}^{(m)}$, obeys the constitutive equation (\[Eq:Formulation:LPTTConstitutiveEquation\]) but is characterized by its unique zero-shear-rate viscosity, $\eta_0^{(m)}$, and time constant, $\lambda_m$.
The results for the single-mode SLPTT model are easily generalized to the case of multiple modes using Eq. (\[Eq:Formulation:Multimode\]), the relations between the material functions and the dimensionless variables (Table \[Tab:MatFunctionsVsDimlessVars\]), and the exact analytical solutions obtained in this work. For example, for the non-Newtonian viscosity, $\eta({\dot{\gamma}})$, one obtains $$\eta({\dot{\gamma}})=-\dfrac{1}{{\dot{\gamma}}}\sum_{m=1}^M\tau_{12}^{(m)}({\dot{\gamma}})=\dfrac{1}{\epsilon}\sum_{m=1}^M \eta_0^{(m)}\sigma(\lambda_m{\dot{\gamma}}),$$ while the shear stress growth function, $\eta^+(t,{\dot{\gamma}}_0)$, is $$\eta^+(t,{\dot{\gamma}}_0) = -\dfrac{1}{{\dot{\gamma}}_0}\sum_{m=1}^M\tau_{12}^{(m)}(t,{\dot{\gamma}}_0)=\dfrac{1}{\epsilon}\sum_{m=1}^M\sigma(\lambda_m{\dot{\gamma}}_0)\mathfrak{T}\left(\dfrac{t}{\lambda_m},\lambda_m {\dot{\gamma}}_0 \right).$$ The exact expressions for other steady and transient material functions considered in this work can be obtained in the same way.
Conclusions {#Sec:Conclusions}
===========
In this work, we have obtained and investigated the exact analytical solutions for start-up, cessation and steady flow regimes of shear flow and of uniaxial, biaxial, and planar extensional flows having used the single-mode and multimode versions of the SLPTT fluid model.
Our most important result is the expressions for the start-up material functions (forms $\mathfrak{T}$ and $\mathfrak{H}$ in Table \[Tab:Forms\] and the functions in these forms given in Tables \[Tab:FormFunctions\] and \[Tab:a\]), for which we have solved systems of coupled nonlinear differential equations. To our knowledge, we report the first (since Giesekus[@Giesekus1982]) exact result for the stress growth functions obtained for a physics-based non-Newtonian fluid model that is nonlinear in the stress-tensor components.
The expressions for the normalized stress relaxation functions (form $\mathfrak{E}$ in Table \[Tab:Forms\]) are much simpler to obtain; we find it surprising that they have been overlooked for many years.
Our analytical results for steady flows \[Eqs. (\[Eq:SteSol:Sh:WiSigmaRelationAlternative\]) and (\[Eq:SteSol:Sh:Delta1SigmaRelation\]) for shear flow, Eqs. (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) and (\[Eq:SteSol:UBEx:DeltaTRelation\]) for uni- and biaxial extensional flows, and Eqs. (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]), (\[Eq:SteSol:PlaEx:DeltaTRelation\]) and (\[Eq:SteSol:PlaEx:Delta2TDelta1Relation\]) for planar extensional flow along with Table \[Tab:MatFunctionsVsDimlessVars\]\] not only fill the existing gaps (mainly related to extensional flows) but also are significantly simpler than any of their available analogs we are aware of.
Notably, only basic knowledge of calculus and differential equations is required for understanding the mathematical methods we have used in this work. Therefore, we believe that our results are not only of scientific but also of pedagogical interest; this paper can be used when teaching theoretical rheology and non-Newtonian fluid mechanics to the students.
This research has been funded by VISTA–a basic research program in collaboration between The Norwegian Academy of Science and Letters and Equinor. D.S. is thankful to Tamara Shogina for suggestions on improvement of this paper.
Availability of data {#availability-of-data .unnumbered}
====================
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
\[App:Bijection\] Bijective properties of certain functions
============================================================
\[SApp:Bijection:Wi-SigmaSteadyShearFlow\] ${\mathrm{Wi}}(\sigma)$ in steady shear flow
----------------------------------------------------------------------------------------
Differentiating Eq. (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) with respect to $\sigma$ (taking into account that $\sigma>0$) yields $${ \dfrac{\mathrm{d}{\mathrm{Wi}}}{\mathrm{d}\sigma} } = \dfrac{\epsilon(2\sigma-3\epsilon)}{2\sqrt{2\epsilon(\epsilon-\sigma)\sigma^5}},$$ which is obviously negative at $0<\sigma < \epsilon$; therefore, function ${\mathrm{Wi}}(\sigma)$ defined by Eq. (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) is monotonically decreasing on its domain and thus bijective.
\[SApp:Bijection:Wi-TSteadyUBExFlow\] ${\mathrm{Wi}}(\tau)$ in steady uni- and biaxial extensional flows
---------------------------------------------------------------------------------------------------------
Taking the derivative of Eq. (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) with respect to $\tau$ yields $$\dfrac{\mathrm{d}{\mathrm{Wi}}}{\mathrm{d}\tau}=\dfrac{
4\left[3\epsilon(6\epsilon\pm \tau)+(3\epsilon+\tau^2)Y \right]
}{
Y\left[Y+(6\epsilon \pm \tau-2\tau^2)\right]^2
},
\label{AEq:UBEx:DWiDTau}$$ where $$Y=\sqrt{36\epsilon^2 \pm 12\epsilon \tau+(9-24\epsilon) \tau^2}.$$
For uniaxial extension, the upper signs in Eq. (\[AEq:UBEx:DWiDTau\]) are chosen. Then, both the numerator and the denominator are clearly positive; hence, ${\mathrm{Wi}}(\tau)$ is a monotonically increasing function and defines a bijection.
For biaxial extension \[lower signs in Eq. (\[AEq:UBEx:DWiDTau\])\], the “quick look” method is not applicable. Instead, we shall obtain the conditions at which ${ \mathrm{d} {\mathrm{Wi}}/ \mathrm{d}\tau }=0$ and show that these conditions cannot be fulfilled at allowed values of $\tau$ and $\epsilon$. This will prove that ${\mathrm{Wi}}(\tau)$ is a monotonic function on its domain and thus defines a bijection.
Setting the numerator of Eq. (\[AEq:UBEx:DWiDTau\]) with the lower signs equal to zero and solving the resulting equation for $\epsilon$ lead to $$\begin{aligned}
\epsilon^{(\pm)} &=\dfrac{\tau^2\left[9-2\tau(1+2\tau)\pm(1+2\tau)\sqrt{9+4\tau^2}\right]}{12 (-2+2\tau + 3\tau^2)}, \nonumber \\
& \quad\;\, 0\leq \tau <1. \label{AEq:UBEx:EpsilonPlusMinus}\end{aligned}$$ In the following, we shall demonstrate that $\epsilon^{(-)}$ is non-positive \[case (a)\], while $\epsilon^{(+)}$ is either non-positive \[case (b)\] or exceeds $1/4$ \[case (c)\]. Since none of these requirements can be met (recall that $0<\epsilon<1/4$), this will complete the proof.
**Case (a).** One multiplies the numerator–with the negative sign chosen–and the denominator of Eq. (\[AEq:UBEx:EpsilonPlusMinus\]) by the positive factor $9-2\tau(1+2\tau)+(1+2\tau)\sqrt{9+4\tau^2}$. After trivial algebraic manipulations, this yields $$\epsilon^{(-)}=-\dfrac{3\tau^2}{9-2\tau(1+2\tau)+(1+2\tau)\sqrt{9+4\tau^2}},$$ which is clearly non-positive at $0\leq \tau <1$.
**Case (b).** One rewrites the expression in the numerator of Eq. (\[AEq:UBEx:EpsilonPlusMinus\]), with the positive sign chosen, $$9-2\tau(1+2\tau)+(1+2\tau)\sqrt{9+4\tau^2}=9+(1+2\tau)(\sqrt{9+4\tau^2}-2\tau),$$ which shows that the numerator is non-negative at $0\leq \tau <1$. At the same time, the denominator changes its sign from negative to positive at $\tau=\tau^\ast=(\sqrt{7}-1)/3$. Therefore, $\epsilon^{(+)}$ is non-positive when $0\leq \tau < \tau^\ast$.
**Case (c).** As shown in case (b), $\epsilon^{(+)}$ is positive at $\tau^\ast<\tau<1$. However, $$\epsilon^{(+)}-\dfrac{1}{4}=\dfrac{6(1-\tau)+\tau^2(1+2\tau)(\sqrt{9+4\tau^2}-2\tau)}{12 (-2+2\tau+3 \tau^2)},
\label{AEq:UBEx:EpsMinusOneFourth}$$ which is also clearly positive; therefore, $\epsilon^{(+)}>1/4$ at $\tau^\ast<\tau<1$. This completes the last step of the proof.
\[SApp:Bijection:Wi-TSteadyPlaExFlow\] ${\mathrm{Wi}}(\tau)$ in steady planar extensional flow
-----------------------------------------------------------------------------------------------
Having differentiated Eq. (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]) with respect to $\tau$, one arrives after rearrangements at $$\label{AEq:PlaExProofs:DWiOverDTau}
\dfrac{\mathrm{d}{\mathrm{Wi}}}{\mathrm{d}\tau}=\dfrac{
8\epsilon^2+(4\epsilon+\tau^2)Y}
{Y(4\epsilon-\tau^2+Y)^2},$$ where $$Y=\sqrt{4\epsilon^2+(1-2\epsilon)\tau^2}.$$ It is seen that ${ \mathrm{d} {\mathrm{Wi}}/ \mathrm{d}\tau }>0$; thus, Eq. (\[Eq:SteSol:PlaEx:WiTRelationFinal2\]) specifies a monotonically increasing function and, therefore, defines a bijection.
\[App:Viscosities\] Shapes of the extensional viscosity curves in steady flows
===============================================================================
\[SApp:Viscosities:UBEx\] Monotonicity of the uniaxial extensional viscosity and existence of minimum in the biaxial extensional viscosity
-------------------------------------------------------------------------------------------------------------------------------------------
One starts with using Eq. (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) to eliminate ${\mathrm{Wi}}$ from Eq. (\[Eq:SteSol:UBEx:DeltaTRelation\]). The resulting equation is then differentiated with respect to $\tau$, which leads to $$\label{AEq:Viscosities:DDeltaOneOverDWi}
{ \dfrac{\mathrm{d}\delta_1}{\mathrm{d}\tau} }=\dfrac{3(3-8\epsilon)\tau\pm(Y+6\epsilon)}{4Y},$$ where $$\label{AEq:Viscosities:X}
Y = \sqrt{3\left[(\tau\pm2\epsilon)^2+2\tau^2(1-4\epsilon)+8\epsilon^2\right]}>0.$$ For uniaxial extension \[upper signs in Eqs. (\[AEq:Viscosities:DDeltaOneOverDWi\]) and (\[AEq:Viscosities:X\])\], ${ \mathrm{d} \delta_1 / \mathrm{d}\tau }$ is clearly positive (recall that $0\leq \tau<2$ and $0<\epsilon<1/4$). For biaxial extension \[lower signs in Eqs. (\[AEq:Viscosities:DDeltaOneOverDWi\]) and (\[AEq:Viscosities:X\])\], the sign of ${ \mathrm{d} \delta_1 / \mathrm{d}\tau }$ depends on the numerator of the right-hand side of Eq. (\[AEq:Viscosities:DDeltaOneOverDWi\]). The latter changes its sign from negative to positive at $$\label{AEq:UBEx:MinimumTau}
\tau = \dfrac{4\epsilon}{3-8\epsilon},$$ which belongs to the interval $[0,1)$ provided that $\epsilon<1/4$.
Having applied the chain rule to $\delta_1$, one writes $$\label{AEq:UBEx:ChainRule}
{ \dfrac{\mathrm{d}\delta_1}{\mathrm{d}{\mathrm{Wi}}} }={ \dfrac{\mathrm{d}\delta_1}{\mathrm{d}\tau} }{ \dfrac{\mathrm{d}\tau}{\mathrm{d}{\mathrm{Wi}}} }.$$ Since ${ \mathrm{d} \tau / \mathrm{d}{\mathrm{Wi}}}>0$, the sign of ${ \mathrm{d} \delta_1 / \mathrm{d}{\mathrm{Wi}}}$ is identical to that of ${ \mathrm{d} \delta_1 / \mathrm{d}\tau }$. Therefore, $\delta_1({\mathrm{Wi}})$, and hence the extensional viscosity, is increasing monotonically in uniaxial extensional flow; while for biaxial extensional flow, it goes through a minimum point. Substituting Eq. (\[AEq:UBEx:MinimumTau\]) into Eqs. (\[Eq:SteSol:UBEx:WiTRelationFinal2\]) with lower signs chosen and (\[Eq:SteSol:UBEx:DeltaTRelation\]), one obtains Eqs. (\[Eq:SteSol:UBEx:ExtensionalViscosityMinimumPosition\]) and (\[Eq:SteSol:UBEx:ExtensionalViscosityMinimumValue\]), respectively.
\[SApp:Viscosities:PlaEx\] Monotonicity of the planar extensional viscosities
------------------------------------------------------------------------------
Eliminating ${\mathrm{Wi}}$ from Eq. (\[Eq:SteSol:PlaEx:DeltaTRelation\]) using Eq. (\[Eq:SteSol:PlaEx:WiTRelationFinal\]), differentiating the result with respect to $\tau$, and rearranging, one obtains $$\label{AEq:PlaEx:DDeltaOneOverDTau}
{ \dfrac{\mathrm{d}\delta_1}{\mathrm{d}\tau} }=\dfrac{(1-2\epsilon)\tau}{\sqrt{4\epsilon^2+(1-2\epsilon)\tau^2}},$$ which is clearly non-negative. Having applied the chain rule to $\delta_1$, as in Appendix \[SApp:Viscosities:UBEx\], one demonstrates the non-negativity of ${ \mathrm{d} \delta_1 / \mathrm{d}{\mathrm{Wi}}}$. Then, one shows that ${ \mathrm{d} \delta_2 / \mathrm{d}{\mathrm{Wi}}}\geq 0$ by differentiating Eq. (\[Eq:SteSol:PlaEx:Delta2TDelta1Relation\]) with respect to ${\mathrm{Wi}}$. Therefore, both planar extensional viscosities are increasing monotonically with ${\mathrm{Wi}}$.
\[App:MainResult\] Derivation of the main results
==================================================
\[SApp:MainResult:Shear\] Start-up of steady shear flow (trigonometric form $\mathfrak{T}$)
--------------------------------------------------------------------------------------------
To solve Eqs. (\[Eq:DFormulation:SteSh:1N1Evolution\]) and (\[Eq:DFormulation:SteSh:2SEvolution\]), one introduces the new variables, $\hat{\mathbb{N}}_1$ and $\hat{\mathbb{S}}$, which are the deviations of $\mathbb{N}_1$ and $\mathbb{S}$, respectively, from their steady-flow values, $$\begin{aligned}
\hat{\mathbb{N}}_1 &= \delta_1-\mathbb{N}_1, \\
\hat{\mathbb{S}} &= \sigma - \mathbb{S}.\end{aligned}$$ After this substutution, Eqs. (\[Eq:DFormulation:SteSh:1N1Evolution\]) and (\[Eq:DFormulation:SteSh:2SEvolution\]) become, respectively, $$\begin{aligned}
\hat{\mathbb{N}}'_1 & = -(1 + 2{\mathrm{Wi}}\delta_1)\hat{\mathbb{N}}_1 + 2 {\mathrm{Wi}}\hat{\mathbb{S}} + {\mathrm{Wi}}\hat{\mathbb{N}}_1^2, \label{AEq:Solution:1N1Evolution}\\
\hat{\mathbb{S}}' &= - (1 + {\mathrm{Wi}}\delta_1)\hat{\mathbb{S}} - {\mathrm{Wi}}\sigma \hat{\mathbb{N}}_1 + {\mathrm{Wi}}\hat{\mathbb{N}}_1 \hat{\mathbb{S}}, \label{AEq:Solution:2SEvolution}\end{aligned}$$ with the initial conditions $\hat{\mathbb{N}}_1(0)=\delta_1$ and $\hat{\mathbb{S}}(0)=\sigma$. Subtracting Eq. (\[AEq:Solution:2SEvolution\]) multiplied by $\hat{\mathbb{N}}_1$ from Eq. (\[AEq:Solution:1N1Evolution\]) multiplied by $\hat{\mathbb{S}}$, dividing the result by $\hat{\mathbb{S}}^2$, and introducing $V(r)=\hat{\mathbb{N}}_1/\hat{\mathbb{S}}$ yield $$V' = 2{\mathrm{Wi}}-{\mathrm{Wi}}\delta_1 V + {\mathrm{Wi}}\sigma V^2, \label{AEq:Solution:Riccati}$$ with $V(0)=\delta_1/\sigma$. This is a Riccati differential equation, which can be solved by standard analytical methods.[@Ince2006] The solution of Eq. (\[AEq:Solution:Riccati\]) can be written as $$V(r) =\dfrac{1}{2\sigma}\left(\delta_1+\sqrt{\Delta}\dfrac{\sin \omega r+\dfrac{\delta_1}{\sqrt{\Delta}} \cos \omega r }{\cos \omega r - \dfrac{\delta_1}{\sqrt{\Delta}} \sin \omega r }\right), \label{AEq:Solution:V}$$ where $\Delta=-\delta_1^2+8\sigma$ (see column I of Table \[Tab:FormFunctions\]) and $\omega = {\mathrm{Wi}}\sqrt{\Delta}/2$. It should be noted that $\Delta>0$, meaning that all functions appearing in Eq. (\[AEq:Solution:V\]) are real-valued. The positivity of $\Delta$ can be shown by the following transformations: $$\Delta=-\delta_1^2+8\sigma = -\dfrac{4}{\epsilon^2}{\mathrm{Wi}}^2\sigma^4+8\sigma = -\dfrac{2}{\epsilon}(\epsilon-\sigma)\sigma+8\sigma = 2\sigma \left(3+\dfrac{\sigma}{\epsilon} \right)>0,$$ where Eqs. (\[Eq:SteSol:Sh:Delta1SigmaRelation\]) and (\[Eq:SteSol:Sh:WiSigmaRelationFinal\]) were used subsequently.
Then, the relation between $\hat{\mathbb{N}}_1$ and $\hat{\mathbb{S}}$, $$\hat{\mathbb{N}}_1=V(r)\hat{\mathbb{S}}, \label{AEq:Solution:NSRelation}$$ is used to eliminate $\hat{\mathbb{N}}_1$ from Eq. (\[AEq:Solution:2SEvolution\]); this leads to $$\hat{\mathbb{S}}' = - \left[1 + {\mathrm{Wi}}\delta_1+{\mathrm{Wi}}\sigma V(r)\right]\hat{\mathbb{S}} + {\mathrm{Wi}}V(r) \hat{\mathbb{S}}^2, \label{AEq:Solution:Bernoulli}$$ with $\hat{\mathbb{S}}(0)=\sigma$ and $V(r)$ given by Eq. (\[AEq:Solution:V\]). Equation (\[AEq:Solution:Bernoulli\]) is a Bernoulli differential equation; having solved it using the standard analytical methods,[@Ince2006] one obtains $$\hat{\mathbb{S}} = \dfrac{\sigma K \left(\cos \omega r -\dfrac{\delta_1}{\sqrt{\Delta}}\sin \omega r\right)}{C \mathrm{e}^{\Omega r}+{\mathrm{Wi}}(A \cos \omega r + B \sin \omega r )}, \label{AEq:Solution:SHat}$$ where $\Omega$, $K$, $C$, $A$, and $B$ are given in column I of Table \[Tab:FormFunctions\]. Finally, substituting Eqs. (\[AEq:Solution:V\]) and (\[AEq:Solution:SHat\]) into Eq. (\[AEq:Solution:NSRelation\]) and performing the multiplication, one arrives at $$\hat{\mathbb{N}}_1 = \dfrac{\delta_1 K \left(\cos \omega r + \dfrac{\delta_1 + 2 {\mathrm{Wi}}\sigma}{{\mathrm{Wi}}\delta_1 \sqrt{\Delta}}\sin \omega r\right)}{C \mathrm{e}^{\Omega r}+{\mathrm{Wi}}(A \cos \omega r + B \sin \omega r )}. \label{AEq:Solution:N1Hat}$$ It is seen that Eqs. (\[AEq:Solution:SHat\]) and (\[AEq:Solution:N1Hat\]) are of very similar form, the main difference being the coefficient in front of $\sin \omega r$ in the numerator. This coefficient is denoted by $a$; the expressions specifying $a$ corresponding to different material functions are found in Table \[Tab:a\].
Reverting to the original variables, $(\mathbb{S}, \mathbb{N}_1)$, and then expressing the normalized material functions, as described in Table \[Tab:MatFunctionsVsDimlessVars\], one obtains the main result for start-up of steady shear flow: the trigonometric form ($\mathfrak{T}$) presented in Table \[Tab:Forms\].
\[SApp:MainResult:Extensional\] Start-up of steady extensional flows (hyperbolic form $\mathfrak{H}$)
------------------------------------------------------------------------------------------------------
Derivation of the main result for start-up of steady uniaxial, biaxial, and planar extensional flows is similar to that for start-up of steady shear flow and goes through the the same steps; therefore, only the key points of the derivation shall be given here.
The new variables, $(\hat{\mathbb{T}},\hat{\mathbb{N}}_1)$, are introduced as the deviations of the old variables, $(\mathbb{T},\mathbb{N}_1)$, from their steady-flow values, $$\begin{aligned}
\hat{\mathbb{T}} &= \tau-\mathbb{T}, \\
\hat{\mathbb{N}}_1 &= \delta_1 - \mathbb{N}_1.\end{aligned}$$ With this, Eqs. (\[Eq:DFormulation:UBEx:1TEvolution\]) and (\[Eq:DFormulation:UBEx:2N1Evolution\]) become $$\begin{aligned}
\hat{\mathbb{T}}' &= - (1+{\mathrm{Wi}}\tau)\hat{\mathbb{T}} - {\mathrm{Wi}}\tau \hat{\mathbb{T}} + 2 {\mathrm{Wi}}\hat{\mathbb{N}}_1 + {\mathrm{Wi}}\hat{\mathbb{T}}^2, \\
\hat{\mathbb{N}}'_1 &= - (1+{\mathrm{Wi}}\tau) \hat{\mathbb{N}}_1 - {\mathrm{Wi}}\delta_1 \hat{\mathbb{T}} + {\mathrm{Wi}}(\hat{\mathbb{T}} \pm \hat{\mathbb{N}}_1) + {\mathrm{Wi}}\hat{\mathbb{T}}\hat{\mathbb{N}}_1, \label{AEq:SolutionB:2N1Evolution-UBEx}\end{aligned}$$ respectively, while Eqs. (\[Eq:DFormulation:PlaEx:1TEvolution\]) and (\[Eq:DFormulation:PlaEx:2N1Evolution\]) become $$\begin{aligned}
\hat{\mathbb{T}}' &= - (1+{\mathrm{Wi}}\tau)\hat{\mathbb{T}} - {\mathrm{Wi}}\tau \hat{\mathbb{T}} + 2 {\mathrm{Wi}}\hat{\mathbb{N}}_1 + {\mathrm{Wi}}\hat{\mathbb{T}}^2, \\
\hat{\mathbb{N}}'_1 &= - (1+{\mathrm{Wi}}\tau) \hat{\mathbb{N}}_1 - {\mathrm{Wi}}\delta_1 \hat{\mathbb{T}} + 2 {\mathrm{Wi}}\hat{\mathbb{T}} + {\mathrm{Wi}}\hat{\mathbb{T}} \hat{\mathbb{N}}_1, \label{AEq:SolutionB:2N1Evolution-PlaEx}\end{aligned}$$ respectively, the initial conditions being $\hat{\mathbb{T}}(0)=\tau$, $\hat{\mathbb{N}}_1(0)=0$ in both cases. Then, a new variable, $V(r)$, is defined by $$\label{AEq:SolutionB:TNRelation}
\hat{\mathbb{T}} = V(r) \hat{\mathbb{N}}_1,$$ and a Riccati equation for $V$ is constructed, as shown in Appendix \[SApp:MainResult:Shear\]. For start-up of uni- and biaxial extensional flows, this equation is $$\label{AEq:SolutionB:Riccati-UBEx}
V' =2{\mathrm{Wi}}-{\mathrm{Wi}}(\tau \pm 1)V + {\mathrm{Wi}}(\delta_1-1)V^2,$$ while for start-up of planar extensional flow, $$\label{AEq:SolutionB:Riccati-PlaEx}
V' = 2{\mathrm{Wi}}-{\mathrm{Wi}}\tau V + {\mathrm{Wi}}(\delta_1-2)V^2,$$ with $V(0)=\tau/\delta_1$ in both cases.
The solution of Eq. (\[AEq:SolutionB:Riccati-UBEx\]) is $$\label{AEq:SolutionB:V-UBEx}
V = \dfrac{1}{2(\delta_1-1)}\left(\tau \pm 1 - \sqrt{\Delta} \dfrac{\sinh \omega r + \dfrac{ \pm \delta_1 +2\tau - \delta_1 \tau}{\delta_1 \sqrt{\Delta}} \cosh \omega r}{\cosh \omega r + \dfrac{ \pm \delta_1 +2\tau - \delta_1 \tau}{\delta_1 \sqrt{\Delta}} \sinh \omega r} \right),$$ with $\Delta=9-8\delta_1\pm 2 \tau+\tau^2>0$ (see column II of Table \[Tab:FormFunctions\]), while the solution of Eq. (\[AEq:SolutionB:Riccati-PlaEx\]) is $$\label{AEq:SolutionB:V-PlaEx}
V = \dfrac{1}{2(\delta_1-2)} \left(\tau - \sqrt{\Delta} \dfrac{\sinh \omega r + \dfrac{(4-\delta_1)\tau}{\delta_1 \sqrt{\Delta}} \cosh \omega r}{\cosh \omega r + \dfrac{(4-\delta_1)\tau}{\delta_1 \sqrt{\Delta}} \sinh \omega r} \right),$$ with $\Delta=16-8\delta_1+\tau^2>0$ (see column III of Table \[Tab:FormFunctions\]). In both cases, $\omega = {\mathrm{Wi}}\sqrt{\Delta}/2$.
These solutions, together with Eq. (\[AEq:SolutionB:TNRelation\]), are used to eliminate $\hat{\mathbb{T}}$ from Eqs. (\[AEq:SolutionB:2N1Evolution-UBEx\]) and (\[AEq:SolutionB:2N1Evolution-PlaEx\]), respectively. The resulting Bernoulli equations are solved, yielding $\hat{\mathbb{N}}_1$. Then, $\hat{\mathbb{T}}$ is found from Eq. (\[AEq:SolutionB:TNRelation\]). Finally, for the planar extension case, $\mathbb{N}_2$ is obtained using Eq. (\[Eq:DFormulation:PlaEx:3N2Evolution\]) after reverting to the non-hatted variables. Calculating the normalized material functions according to Table \[Tab:MatFunctionsVsDimlessVars\] leads to the hyperbolic form $\mathfrak{H}$ presented in Table \[Tab:Forms\].
\[SApp:MainResult:Cessation\] Cessation of steady shear and extensional flows (exponential form $\mathfrak{E}$)
----------------------------------------------------------------------------------------------------------------
The derivation of the exact analytical solutions for the cessation case is trivial compared to the start-up case. Separating the variables in Eq. (\[Eq:DFormulation:Cessation:GeneralEquation\]) leads to $$\dfrac{\mathrm{d}(1+{\mathrm{Wi}}\mathbb{X}) } {1+{\mathrm{Wi}}\mathbb{X}}-\dfrac{\mathrm{d}\mathbb{X}}{\mathbb{X}}=\mathrm{d}r.$$ Integrating this with the initial condition $\mathbb{X}(0)=\chi$ yields $$\mathbb{X} = \dfrac{\chi}{(1+{\mathrm{Wi}}\chi)\mathrm{e}^r-{\mathrm{Wi}}\chi}.$$ Combining this with Eqs. (\[Eq:DFormulation:Cessation:MatFunShear\]) and (\[Eq:DFormulation:Cessation:MatFunExtensional\]), one obtains the exponential form $\mathfrak{E}$ presented in Table \[Tab:Forms\].
|
---
abstract: 'A unique sink orientation (USO) is an orientation of the $n$-dimensional cube graph ($n$-cube) such that every face (subcube) has a unique sink. The number of unique sink orientations is $n^{\Theta(2^n)}$ [@matousek2006number]. If a cube orientation is not a USO, it contains a *pseudo unique sink orientation* (PUSO): an orientation of some subcube such that every proper face of it has a unique sink, but the subcube itself hasn’t. In this paper, we characterize and count PUSOs of the $n$-cube. We show that PUSOs have a much more rigid structure than USOs and that their number is between $2^{\Omega(2^{n-\log n})}$ and $2^{O(2^n)}$ which is negligible compared to the number of USOs. As tools, we introduce and characterize two new classes of USOs: *border USOs* (USOs that appear as facets of PUSOs), and *odd USOs* which are dual to border USOs but easier to understand.'
author:
- Vitor Bosshard
- Bernd Gärtner
bibliography:
- 'puso\_arxiv.bib'
title: Pseudo Unique Sink Orientations
---
Introduction
============
#### Unique sink orientations.
Since more than 15 years, unique sink orientations (USOs) have been studied as particularly rich and appealing combinatorial abstractions of linear programming (LP) [@gartner2006linear] and other related problems [@fischer2004]. Originally introduced by Stickney and Watson in the context of the P-matrix linear complementarity problem (PLCP) in 1978 [@StiWat], USOs have been revived by Szabó and Welzl in 2001, with a more theoretical perspective on their structural and algorithmic properties [@szabo2001unique].
The major motivation behind the study of USOs is the open question whether efficient combinatorial algorithms exist to solve PLCP and LP. Such an algorithm is running on a RAM and has runtime bounded by a polynomial in the *number* of input values (which are considered to be real numbers). In case of LP, the runtime should be polynomial in the number of variables and the number of constraints. For LP, the above open question might be less relevant, since polynomial-time algorithms exist in the Turing machine model since the breakthrough result by Khachiyan in 1980 [@Kha]. For PLCP, however, no such algorithm is known, so the computational complexity of PLCP remains open.
Many algorithms used in practice for PCLP and LP are combinatorial and in fact *simplex-type* (or *Bard-type*, in the LCP literature). This means that they follow a locally improving path of candidate solutions until they either cycle (precautions need to be taken against this), or they get stuck—which in case of PLCP and LP fortunately means that the problem has been solved. The less fortunate facts are that for most known algorithms, the length of the path is exponential in the worst case, and that for no algorithm, a polynomial bound on the path length is known.
USOs allow us to study simplex-type algorithms in a completely abstract setting where cube vertices correspond to candidate solutions, and outgoing edges lead to locally better candidates. Arriving at the unique sink means that the problem has been solved. The requirement that all faces have unique sinks is coming from the applications, but is also critical in the abstract setting itself: without it, there would be no hope for nontrivial algorithmic results [@Aldous].
On the one hand, this kind of abstraction makes a hard problem even harder; on the other hand, it sometimes allows us to see what is really going on, after getting rid of the numerical values that hide the actual problem structure. In the latter respect, USOs have been very successful.
For example, in a USO we are not confined to following a path, we can also “jump around”. The fastest known deterministic algorithm for finding the sink in a USO does exactly this [@szabo2001unique] and implies the fastest known deterministic combinatorial algorithm for LP if the number of constraints is twice the number of variables [@gartner2006linear]. In a well-defined sense, this is the hardest case. Also, <span style="font-variant:small-caps;">RandomFacet</span>, the currently best randomized combinatorial simplex algorithm for LP [@Kal; @MSW] actually works on acyclic USOs (AUSOs) with the same (subexponential) runtime and a purely combinatorial analysis [@gartner2002].
The USO abstraction also helps in proving lower bounds for the performance of algorithms. The known (subexponential) lower bounds for <span style="font-variant:small-caps;">RandomFacet</span> and <span style="font-variant:small-caps;">RandomEdge</span>—the most natural randomized simplex algorithm—have first been proved on AUSOs [@Mat; @MATOUSEK2006262] and only later on actual linear programs [@Friedmann]. It is unknown which of the two algorithms is better on actual LPs, but on AUSOs, <span style="font-variant:small-caps;">RandomEdge</span> is strictly slower in the worst case [@HZ].
Finally, USOs are intriguing objects from a purely mathematical point of view, and this is the view that we are mostly adopting in in this paper.
#### Pseudo unique sink orientations.
If a cube orientation has a unique sink in every face except the cube itself, we call it a pseudo unique sink orientation (PUSO). Every cube orientation that is not a USO contains some PUSO. The study of PUSOs originates from the master’s thesis of the first author [@Bosshard] where the PUSO concept was used to obtain improved USO recognition algorithms; see Section \[sec:recognition\] below.
One might think that PUSOs have more variety than USOs: instead of exactly one sink in the whole cube, we require any number of sinks not equal to one. But this intuition is wrong: as we show, the number of PUSOs is much smaller than the number of USOs of the same dimension; in particular, only a negligible fraction of all USOs of one dimension lower may appear as facets of PUSOs. These *border USOs* and the *odd USOs*—their duals—have a quite interesting structure that may be of independent interest. The discovery of these USO classes and their basic properties, as well as the implied counting results for them and for PUSOs, are the main contributions of the paper.
#### Overview of the paper.
Section \[sec:cubes\] formally introduces cubes and orientations, to fix the language. We will define an orientation via its *outmap*, a function that yields for every vertex its outgoing edges. Section \[sec:usos\] defines USOs and PUSOs and gives some examples in dimensions two and three to illustrate the concepts. In Section \[sec:outmaps\], we characterize outmaps of PUSOs, by suitably adapting the characterization for USOs due to Szabó and Welzl [@szabo2001unique]. Section \[sec:recognition\] uses the PUSO characterization to describe a USO recognition algorithm that is faster than the one resulting from the USO characterization of Szabó and Welzl. Section \[sec:border\] characterizes the USOs that may arise as facets of PUSOs. As these are on the border between USOs and non-USOs, we call them *border USOs.* Section \[sec:odd\] introduces and characterizes the class of *odd* USOs that are dual to border USOs under inverting the outmap. Odd USOs are easier to visualize and work with, since in any face of an odd USO we again have an odd USO, a property that fails for border USOs. We also give a procedure that allows us to construct many odd USOs from a canonical one, the *Klee-Minty cube*. Based on this, Section \[sec:counting\] proves (almost matching) upper and lower bounds for the number of odd USOs in dimension $n$. Bounds on the number of PUSOs follow from the characterization of border USOs in Section \[sec:border\]. In Section \[sec:conclusion\], we mention some open problems.
Cubes and Orientations {#sec:cubes}
======================
Given finite sets $A\subseteq B$, the *cube* ${\mathcal{C}}={\mathcal{C}}^{[A,B]}$ is the graph with vertex set ${\mathrm{vert}}{{\mathcal{C}}} = [A,B]:=\{V: A\subseteq V \subseteq B\}$ and edges between any two subsets $U,V$ for which $|U\oplus V|=1$, where $U\oplus V=(U\setminus V)\cup (V\setminus U) = (U\cup V)\setminus
(U\cap V)$ is symmetric difference. We sometimes need the following easy fact. $$\label{eq:symdiff}
(U\oplus V)\cap X = (U\cap X) \oplus (V\cap X).$$
For a cube ${\mathcal{C}}={\mathcal{C}}^{[A,B]}$, ${\mathrm{dim}}{{\mathcal{C}}}:=|B\setminus A|$ is its *dimension*, ${\mathrm{carr}}{{\mathcal{C}}} := B\setminus A$ its *carrier*. A *face* of ${\mathcal{C}}$ is a subgraph of the form ${\mathcal{F}}={\mathcal{C}}^{[I,J]}$, with $A\subseteq I\subseteq J\subseteq B$. If ${\mathrm{dim}}{{\mathcal{F}}}=k$, ${\mathcal{F}}$ is a *$k$-face* or *$k$-cube*. A *facet* of an $n$-cube ${\mathcal{C}}$ is an $(n-1)$-face of ${\mathcal{C}}$. Two vertices $U,V\in{\mathrm{vert}}{{\mathcal{F}}}$ are called *antipodal* in ${\mathcal{F}}$ if $V={\mathrm{carr}}{{\mathcal{F}}}\setminus U$.
If $A=\emptyset$, we abbreviate ${\mathcal{C}}^{[A,B]}$ as ${\mathcal{C}}^B$. The *standard $n$-cube* is ${\mathcal{C}}^{[n]}$ with $[n]:=\{1,2,\ldots,n\}$.
An *orientation* ${\mathcal{O}}$ of a graph $G$ is a digraph that contains for every edge $\{U,V\}$ of $G$ exactly one directed edge $(U,V)$ or $(V,U)$. An orientation of a cube ${\mathcal{C}}$ can be specified by its *outmap* ${\phi}: {\mathrm{vert}}{{\mathcal{C}}} \rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$ that returns for every vertex the *outgoing coordinates*. On every face ${\mathcal{F}}$ of ${\mathcal{C}}$ (including ${\mathcal{C}}$ itself), the outmap induces the orientation $${\mathcal{F}}_{{\phi}} := ({\mathrm{vert}}{{\mathcal{F}}}, \{(V,V\oplus\{i\}): V\in {\mathrm{vert}}{{\mathcal{F}}},
i\in {\phi}(V)\cap {\mathrm{carr}}{{\mathcal{F}}}\}).$$ In order to actually get a proper orientation of ${\mathcal{C}}$, the outmap must be *consistent*, meaning that it satisfies $i\in {\phi}(V)\oplus{\phi}(V\oplus\{i\})$ for all $V\in{\mathrm{vert}}{{\mathcal{C}}}$ and $i\in{\mathrm{carr}}{{\mathcal{C}}}$.
Note that the outmap of ${\mathcal{F}}_{{\phi}}$ is not $\phi$ but ${\phi}_{{\mathcal{F}}}:{\mathrm{vert}}{{\mathcal{F}}}\rightarrow 2^{{\mathrm{carr}}{{\mathcal{F}}}}$ defined by $$\label{eq:faceout}
{\phi}_{{\mathcal{F}}}(V)={\phi}(V)\cap{\mathrm{carr}}{{\mathcal{F}}}.$$
In general, when we talk about a cube orientation ${\mathcal{O}}={\mathcal{C}}_{{\phi}}$, the domain of ${\phi}$ may be a supercube of ${\mathcal{C}}$ in the given context. This avoids unnecessary indices that we would get in defining ${\mathcal{O}}={\mathcal{C}}_{{\phi}_{{\mathcal{C}}}}$ via its “official” outmap ${\phi}_{{\mathcal{C}}}$. However, sometimes we want to make sure that ${\phi}$ is actually the outmap of ${\mathcal{O}}$, and then we explicitly say so.
Figure \[fig:orientation\] depicts an outmap and the corresponding 2-cube orientation.
![An outmap ${\phi}$ and the induced 2-cube orientation ${\mathcal{C}}_{{\phi}}$[]{data-label="fig:orientation"}](orientation.pdf){width="70.00000%"}
(Pseudo) Unique Sink Orientations {#sec:usos}
=================================
A unique sink orientation (USO) of a cube ${\mathcal{C}}$ is an orientation ${\mathcal{C}}_{{\phi}}$ such that every face ${\mathcal{F}}_{{\phi}}$ has a unique sink. Equivalently, every face ${\mathcal{F}}_{{\phi}}$ is a unique sink orientation.
Figure \[fig:2cubes\] shows the four combinatorially different (pairwise non-isomorphic) orientations of the 2-cube. The eye and the bow are USOs.[^1] The twin peak is not since it has two sinks in the whole cube (which is a face of itself). The cycle is not a USO, either, since it has no sink in the whole cube. The unique sink conditions for $0$- and $1$-faces (vertices and edges) are always trivially satisfied.
![The 4 combinatorially different orientations of the 2-cube[]{data-label="fig:2cubes"}](2cubes.pdf){width="\textwidth"}
If an orientation ${\mathcal{C}}_{{\phi}}$ is not a USO, there is a smallest face ${\mathcal{F}}_{{\phi}}$ that is not a USO. We call the orientation in such a face a *pseudo* unique sink orientation.
A pseudo unique sink orientation (PUSO) of a cube ${\mathcal{C}}$ is an orientation ${\mathcal{C}}_{{\phi}}$ that does not have a unique sink, but every proper face ${\mathcal{F}}_{{\phi}} \neq {\mathcal{C}}_{{\phi}}$ has a unique sink.
The twin peak and the cycle in Figure \[fig:2cubes\] are the two combinatorially different PUSOs of the 2-cube. The 3- cube has $19$ combinatorially different USOs [@StiWat], but only two combinatorially different PUSOs, see Figure \[fig:3pusos\] together with Corollary \[cor:3pusos\] below.
![The two combinatorially different PUSOs of the 3-cube[]{data-label="fig:3pusos"}](3pusos.pdf){width="65.00000%"}
We let ${\mathrm{uso}}(n)$ and ${\mathrm{puso}}(n)$ denote the number of USOs and PUSOs of the standard $n$-cube. We have ${\mathrm{uso}}(0)=1,{\mathrm{uso}}(1)=2$ as well as ${\mathrm{uso}}(2)=12$ (4 eyes and 8 bows). Moreover, ${\mathrm{puso}}(0)={\mathrm{puso}}(1)=0$ and ${\mathrm{puso}}(2)=4$ (2 twin peaks, 2 cycles).
Outmaps of (Pseudo) USOs {#sec:outmaps}
========================
Outmaps of USOs have a simple characterization [@szabo2001unique Lemma 2.3]: ${\phi}:{\mathrm{vert}}{{\mathcal{C}}}\rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$ is the outmap of a USO of ${\mathcal{C}}$ if and only if $$\label{eq:char}
({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V) \neq \emptyset$$ holds for all pairs of distinct vertices $U,V\in{\mathrm{vert}}{{\mathcal{C}}}$. This condition means the following: within the face ${\mathcal{C}}^{[U\cap V, U\cup V]}$ *spanned* by $U$ and $V$, there is a coordinate that is outgoing for exactly one of the two vertices. In particular, any two distinct vertices have different outmap values, so ${\phi}$ is injective and hence bijective.
This characterization implicitly makes a more general statement: for every face ${\mathcal{F}}$, orientation ${\mathcal{F}}_{{\phi}}$ is a USO if and only if (\[eq:char\]) holds for all pairs of distinct vertices $U,V\in{\mathcal{F}}$. The reason is that the validity of (\[eq:char\]) only depends on the behavior of ${\phi}$ within the face spanned by $U$ and $V$. Formally, for $U,V\in{\mathcal{F}}$, (\[eq:char\]) is equivalent to the USO-characterizing condition $({\phi}_{{\mathcal{F}}}(U)\oplus {\phi}_{{\mathcal{F}}}(V))\cap (U\oplus
V)\neq\emptyset$ for the orientation ${\mathcal{F}}_{{\phi}_{{\mathcal{F}}}}={\mathcal{F}}_{{\phi}}$.
\[lem:strongchar\] Let ${\mathcal{C}}$ be a cube, ${\phi}: {\mathrm{vert}}{{\mathcal{C}}} \rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$, ${\mathcal{F}}$ a face of ${\mathcal{C}}$. Then ${\mathcal{F}}_{{\phi}}$ is a USO if and only if $$({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V) \neq \emptyset$$ holds for all pairs of distinct vertices $U,V\in{\mathrm{vert}}{{\mathcal{F}}}$. In this case, the outmap ${\phi}_{{\mathcal{F}}}$ of ${\mathcal{F}}_{{\phi}}$ is bijective.
As a consequence, outmaps of PUSOs can be characterized as follows: (\[eq:char\]) holds for all pairs of non-antipodal vertices $U,V$ (which always span a proper face), but fails for some pair $U, V={\mathrm{carr}}{{\mathcal{C}}}\setminus U$ of antipodal vertices. As the validity of (\[eq:char\]) is invariant under replacing all outmap values ${\phi}(V)$ with ${\phi}'(V)={\phi}(V)\oplus R$ for some fixed $R\subseteq{\mathrm{carr}}{{\mathcal{C}}}$, we immediately obtain that PUSOs (as well as USOs [@szabo2001unique Lemma 2.1]) are closed under *flipping coordinates* (reversing all edges along some subset of the coordinates).
\[lem:puso\_flip\] Let ${\mathcal{C}}$ be a cube, ${\phi}: {\mathrm{vert}}{{\mathcal{C}}} \rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$, ${\mathcal{F}}$ a face of ${\mathcal{C}}$. Suppose that ${\mathcal{F}}_{{\phi}}$ is a PUSO and $R\subseteq {\mathrm{carr}}{{\mathcal{C}}}$. Consider the *$R$-flipped* orientation ${\mathcal{C}}_{{\phi}'}$ induced by the outmap $${\phi}'(V) := {\phi}(V) \oplus R, \quad \forall V\in {\mathrm{vert}}{{\mathcal{C}}}.$$ Then ${\mathcal{F}}_{{\phi}'}$ is a PUSO as well.
Using this, we can show that in a PUSO, (\[eq:char\]) must actually fail on *all* pairs of antipodal vertices, not just on some pair, and this is the key to the strong structural properties of PUSOs.
\[thm:char\] Let ${\mathcal{C}}$ be a cube of dimension at least $2$, ${\phi}:{\mathrm{vert}}{{\mathcal{C}}}\rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$, ${\mathcal{F}}$ a face of ${\mathcal{C}}$. Then ${\mathcal{F}}_{{\phi}}$ is a PUSO if and only if
- condition (\[eq:char\]) holds for all $U,V\in {\mathrm{vert}}{{\mathcal{F}}}, V\neq U, {\mathrm{carr}}{{\mathcal{F}}}\setminus U$ (pairs of distinct, non-antipodal vertices in ${\mathcal{F}}$), and
- condition (\[eq:char\]) fails for all $U,V\in {\mathrm{vert}}{{\mathcal{F}}}, V={\mathrm{carr}}{{\mathcal{F}}}\setminus U$ (pairs of antipodal vertices in ${\mathcal{F}}$).
In view of the above discussion, it only remains to show that (ii) holds if ${\mathcal{F}}_{{\phi}}$ is a PUSO. Let $U\in{\mathrm{vert}}{{\mathcal{F}}}$. Applying Lemma \[lem:puso\_flip\] with $R={\phi}(U)$ does not affect the validity of (\[eq:char\]), so we may assume w.l.o.g. that ${\phi}(U)=\emptyset$, hence $U$ is a sink in ${\mathcal{F}}_{{\phi}}$. For a non-antipodal $W\in{\mathrm{vert}}{{\mathcal{F}}}$, (i) implies the existence of some $i\in{\phi}(W)\cap(U\oplus W)\subseteq
{\phi}(W)\cap{\mathrm{carr}}{{\mathcal{F}}}={\phi}_{{\mathcal{F}}}(W)$, hence such a $W$ is not a sink in ${\mathcal{F}}_{{\phi}}$. But then $V={\mathrm{carr}}{{\mathcal{F}}}\setminus U$ must be a second sink in ${\mathcal{F}}_{{\phi}}$, because PUSO ${\mathcal{F}}_{{\phi}}$ does not have a unique sink. This in turn implies that (\[eq:char\]) fails for $U,V={\mathrm{carr}}{{\mathcal{F}}}\setminus U$.
\[cor:puso\_antipodals\] Let ${\mathcal{C}}_{{\phi}}$ be a PUSO with outmap ${\phi}$.
- Any two antipodal vertices $U,V={\mathrm{carr}}{{\mathcal{C}}}\setminus U$ have the same outmap value, ${\phi}(U)={\phi}(V)$.
- ${\mathcal{C}}_{{\phi}}$ either has no sink, or exactly two sinks.
For antipodal vertices, $U\oplus V={\mathrm{carr}}{{\mathcal{C}}}$, so $({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V) = \emptyset$ is equivalent to ${\phi}(U)={\phi}(V)$. In particular, the number of sinks is even but cannot exceed $2$, as otherwise, there would be two non-antipodal sinks; the proper face they span would then have more than one sink, a contradiction.
We can use the characterization of Theorem \[thm:char\] to show that PUSOs exist in every dimension $n\geq 2$.
\[puso\_existence\] Let $n \geq 2$, ${\mathcal{C}}$ the standard $n$-cube and $\pi:[n]\rightarrow[n]$ a permutation consisting of a single $n$-cycle. Consider the function ${\phi}: 2^{[n]} \mapsto 2^{[n]}$ defined by $${\phi}(V) = \{i\in [n]: |V\cap \{i, \pi(i)\}|=1\}, \quad \forall
V\subseteq[n].$$ Then ${\mathcal{C}}_{{\phi}}$ is a PUSO.
According to Theorem \[thm:char\], we need to show that condition (\[eq:char\]) fails for all pairs of antipodal vertices, but that it holds for all pairs of distinct vertices that are not antipodal.
We first consider two antipodal vertices $U$ and $V=[n]\setminus U$ in which case we get ${\phi}(U)={\phi}(V)$, so (\[eq:char\]) fails. If $U$ and $V$ are distinct and not antipodal, there is some coordinate in which $U$ and $V$ differ, *and* some coordinate in which $U$ and $V$ agree. Hence, if we traverse the $n$-cycle $(1,\pi(1),\pi(\pi(1)),\ldots)$, we eventually find two consecutive elements $i,\pi(i)$ such that $U$ and $V$ differ in coordinate $i$ but agree in coordinate $\pi(i)$, meaning that $i\in ({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V)$, so (\[eq:char\]) holds.
We conclude this section with another consequence of Theorem \[thm:char\] showing that PUSOs have a parity.
\[lem:puso\_parity\] Let ${\mathcal{C}}_{{\phi}}$ be a PUSO with outmap ${\phi}$. Then the outmap values of all vertices have the same parity, that is $$|{\phi}(U) \oplus {\phi}(V)| = 0 \mod 2, \quad \forall U,V\in{\mathrm{vert}}{{\mathcal{C}}}.$$
We call the number $|{\phi}(\emptyset)|\mod 2$ the *parity* of ${\mathcal{C}}_{{\phi}}$. By Corollary \[cor:puso\_antipodals\], a PUSO of even parity has two sinks, a PUSO of odd parity has none.
We first show that the outmap valus of any two distinct non-antipodal vertices $U$ and $V$ differ in at least two coordinates. Let $V'$ be the antipodal vertex of $V$. As $U$ is neither antipodal to $V$ nor to $V'$, Theorem \[thm:char\] along with ${\phi}(V)={\phi}(V')$ (Corollary \[cor:puso\_antipodals\]) yields $$\begin{aligned}
({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V) &\neq& \emptyset, \\
({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V') &\neq& \emptyset.
\end{aligned}$$ Since $U \oplus V$ is disjoint from $U \oplus V'$, ${\phi}(U)\oplus {\phi}(V)$ contains at least two coordinates.
Now we can prove the actual statement. Let $I$ be the image of ${\phi}$, $I := \{{\phi}(V): V\in{\mathrm{vert}}{{\mathcal{C}}}\}\subseteq
{\mathcal{C}}'={\mathcal{C}}^{{\mathrm{carr}}{{\mathcal{C}}}}$. We have $|I|\geq 2^{n-1}$, because by Lemma \[lem:strongchar\], ${\phi}_{{\mathcal{F}}}$ is bijective (and hence ${\phi}$ is injective) on each facet ${\mathcal{F}}$ of ${\mathcal{C}}$. On the other hand, $I$ forms an independent set in the cube ${\mathcal{C}}'$, as any two distinct outmap values differ in at least two coordinates; The statement follows, since the only independent sets of size at least $2^{n-1}$ in an $n$-cube are formed by all vertices of fixed parity.
Recognizing (Pseudo) USOs {#sec:recognition}
=========================
Before we dive deeper into the structure of PUSOs in the next section, we want to present a simple algorithmic consequence of the PUSO characterization provided by Theorem \[thm:char\].
Suppose that ${\mathcal{C}}$ is an $n$-cube, and that an outmap ${\phi}:{\mathrm{vert}}{{\mathcal{C}}}\rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$ is succinctly given by a Boolean circuit of polynomial size in $n$. Then it is [**coNP**]{}-complete to decide whether ${\mathcal{C}}_{{\phi}}$ is a USO [@gaertner2015complexity].[^2] ${\textbf{coNP}}$-membership is easy: every non-USO has a certificate in the form of two vertices that fail to satisfy (\[eq:char\]). Finding two such vertices is hard, though. For given vertices $U$ and $V$, let us call the computation of $({\phi}(U)\oplus {\phi}(V))\cap (U\oplus V)$ a *pair evaluation*. Then, the obvious algorithm needs $\Theta(4^n)$ pair evaluations. Using Theorem \[thm:char\], we can improve on this.
\[thm:check\] Let ${\mathcal{C}}$ be an $n$-cube, ${\phi}:{\mathrm{vert}}{{\mathcal{C}}}\rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$. Using $O(3^n)$ pair evaluations, we can check whether ${\mathcal{C}}_{{\phi}}$ is a USO.
For every face ${\mathcal{F}}$ of dimension at least $1$ (there are $3^n-2^n$ of them), we perform a pair evaluation with an arbitrary pair of antipodal vertices $U,V={\mathrm{carr}}{{\mathcal{F}}}\setminus U$. We output that ${\mathcal{C}}_{{\phi}}$ is a USO if and only if all these pair evaluations succeed (meaning that they return nonempty sets).
We need to argue that this is correct. Indeed, if ${\mathcal{C}}_{{\phi}}$ is a USO, all pair evaluations succeed by Lemma \[lem:strongchar\]. If ${\mathcal{C}}_{{\phi}}$ is not a USO, it is either not an orientation (so the pair evaluation in some 1-face fails), or it contains a PUSO ${\mathcal{F}}_{{\phi}}$ in which case the pair evaluation in ${\mathcal{F}}$ fails by Theorem \[thm:char\].
Using the same algorithm, we can also check whether ${\mathcal{C}}_{{\phi}}$ is a PUSO. Which is the case if and only if the pair evaluation succeeds on every face except ${\mathcal{C}}$ itself.
Border Unique Sink Orientations {#sec:border}
===============================
Lemma \[lem:puso\_parity\] already implies that not every USO can occur as a facet of a PUSO. For example, let us assume that an eye (Figure \[fig:2cubes\]) appears as a facet of a 3-dimensional PUSO. Then, Corollary \[cor:puso\_antipodals\] (i) completely determines the orientation in the opposite facet: we get a “mirror orientation” in which antipodal vertices have traded outgoing coordinates; see Figure \[fig:Noeye\].
![An eye is not a facet of a PUSO[]{data-label="fig:Noeye"}](NoEye.pdf){width="30.00000%"}
But now, every edge between the two facets connects two vertices with the same outmap parity within their facets, and no matter how we orient the edge, the two vertices will receive different global outmap parities. Hence, the resulting orientation cannot be a PUSO by Lemma \[lem:puso\_parity\].
It therefore makes sense to study the class of *border USOs*, the USOs that appear as facets of PUSOs.
A *border USO* is a USO that is a facet of some PUSO.
If the border USO lives on cube ${\mathcal{F}}={\mathcal{C}}^{[A,B]}$, the PUSO may live on ${\mathcal{C}}^{[A,B\cup\{n\}]}$ ($n\notin{\mathrm{carr}}{{\mathcal{F}}}$ a new coordinate), or on ${\mathcal{C}}^{[A\setminus\{n\},B]}$ ($n\in A$), but these cases lead to combinatorially equivalent situations. We will always think about extending border USOs by adding a new coordinate.
In this section, we characterize border USOs. We already know that antipodal vertices must have outmap values of different parities; a generalization of this yields a sufficient condition: if the outmap values of distinct vertices $U,V$ agree outside of the face spanned by $U$ and $V$, then the two outmap values must have different parities.
\[thm:pusof\_charact\] Let ${\mathcal{F}}_{{\psi}}$ be a USO with outmap ${\psi}$. ${\mathcal{F}}_{{\psi}}$ is a border USO if and only if the following condition holds for all pairs of distinct vertices $U,V\in{\mathrm{vert}}{{\mathcal{F}}}$: $$\label{pusof_thm_eq}
{\psi}(U)\oplus {\psi}(V) \subseteq U\oplus V \quad
\Rightarrow \quad |{\psi}(U)\oplus {\psi}(V)| = 1 \mod 2.$$
A preparatory step will be to generalize the insight gained from the case of the eye above and show that a USO can be extended to a PUSO of one dimension higher in at most two canonical ways—exactly two if the USO is actually border.
\[lem:outmap01\] Let ${\mathcal{F}}$ be a facet of ${\mathcal{C}}$, ${\mathrm{carr}}{{\mathcal{C}}}\setminus{\mathrm{carr}}{{\mathcal{F}}}=\{n\}$, and let ${\mathcal{F}}_{{\psi}}$ be a USO with outmap ${\psi}$.
- There are at most two outmaps ${\phi}:{\mathrm{vert}}{{\mathcal{C}}}\rightarrow 2^{{\mathrm{carr}}{{\mathcal{C}}}}$ such that ${\mathcal{C}}_{{\phi}}$ is a PUSO with ${\mathcal{F}}_{{\phi}}={\mathcal{F}}_{{\psi}}$. Specifically, these are ${\phi}_i,i=0,1$, with $$\label{eq:outmap01}
{\phi}_i (V) = \left\{\begin{array}{ll}
{\psi}(V), & V\in{\mathrm{vert}}{{\mathcal{F}}},~ |{\psi}(V)| = i \mod 2, \\
{\psi}(V) \cup\{n\}, & V\in{\mathrm{vert}}{{\mathcal{F}}},~ |{\psi}(V)| \neq i \mod 2, \\
{\phi}_i({\mathrm{carr}}{C}\setminus V), &V\notin{\mathrm{vert}}{{\mathcal{F}}}.
\end{array}\right.$$
- If ${\mathcal{F}}_{{\psi}}$ is a border USO, both ${\mathcal{C}}_{{\phi}_0}$ and ${\mathcal{C}}_{{\phi}_1}$ are PUSOs.
- If ${\mathcal{C}}_{{\phi}_i}$ is a PUSO for some $i\in\{0,1\}$, then ${\mathcal{C}}_{{\phi}_{1-i}}$ is a PUSO as well, and ${\mathcal{F}}_{{\psi}}$ is a border USO.
Only for ${\phi}={\phi}_i,i=0,1$, we obtain ${\mathcal{F}}_{{\phi}}={\mathcal{F}}_{{\psi}}$ and satisfy the necessary conditions of Corollary \[cor:puso\_antipodals\] (pairs of antipodal vertices have the same outmap values in a PUSO), and of Lemma \[lem:puso\_parity\] (all outmap values have the same parity in a PUSO). Hence, ${\mathcal{C}}_{{\phi}_0}$ and ${\mathcal{C}}_{{\phi}_1}$ are the only candidates for PUSOs extending ${\mathcal{F}}_{{\psi}}$. This yields (i). If ${\mathcal{F}}_{{\psi}}$ is a border USO, one of the candidates is a PUSO by definition; as the other one results from it by just flipping coordinate $n$, it is also a PUSO by Lemma \[lem:puso\_flip\]. Part (ii) follows. For part (iii), we use that ${\mathcal{F}}_{{\psi}}$ is a facet of ${\mathcal{C}}_{{\phi}_i}$, $i=0,1$, so as before, if one of the latter is a PUSO, then both are, and ${\mathcal{F}}_{{\psi}}$ is a border USO by definition.
\[cor:3pusos\] There are 2 combinatorially different PUSOs of the 3-cube (depicted in Figure \[fig:3pusos\]).
We have argued above that an eye cannot be extended to a PUSO, so let us try to extend a bow (the front facet in Figure \[fig:3pusos\]). The figure shows the two candidates for PUSOs provided by Lemma \[lem:outmap01\]. Both happen to be PUSOs, so starting from the single combinatorial type of 2-dimensional border USOs, we arrive at the two combinatorial types of 3-dimensional PUSOs.
Concluding this section, we prove the advertised characterization of border USOs.
[\[Theorem \[thm:pusof\_charact\]\]]{} Let ${\mathcal{C}}$ be a cube with facet ${\mathcal{F}}$, ${\mathrm{carr}}{{\mathcal{C}}}\setminus{\mathrm{carr}}{{\mathcal{F}}}=\{n\}$. We show that condition (\[pusof\_thm\_eq\]) fails for some pair of distinct vertices $U,V\in{\mathrm{vert}}{{\mathcal{F}}}$ if and only if ${\mathcal{C}}_{{\phi}_0}$ is not a PUSO, with ${\phi}_0$ as in (\[eq:outmap01\]). By Lemma \[lem:outmap01\], this is equivalent to ${\mathcal{F}}_{{\psi}}$ not being a border USO.
Suppose first that there are distinct $U,V\in{\mathrm{vert}}{{\mathcal{F}}}$ such that ${\psi}(U)\oplus {\psi}(V) \subseteq U\oplus V$ and $|{\psi}(U)\oplus {\psi}(V)| =0 \mod 2$, meaning that $U$ and $V$ have the same outmap parity. By definition of ${\phi}_0$, we then get $${\psi}(U)\oplus {\psi}(V)={\phi}_0(U)\oplus {\phi}_0(V) =
{\phi}_0(U)\oplus {\phi}_0(V') \subseteq U\oplus V,$$ where $V'={\mathrm{carr}}{{\mathcal{C}}}\setminus V$ is antipodal to $V$ in ${\mathcal{C}}$. Moreover, as $U\oplus V$ is also antipodal to $U\oplus V'$, the inclusion ${\phi}_0(U)\oplus {\phi}_0(V)={\phi}_0(U)\oplus
{\phi}_0(V')\subseteq U\oplus V$ is equivalent to $$\label{eq:UV'}
({\phi}_0(U)\oplus {\phi}_0(V'))\cap(U\oplus V')=\emptyset.$$ Since $U,V$ are distinct and non-antipodal (in ${\mathcal{C}}$), $U,V'$ are therefore distinct non-antipodal vertices that fail to satisfy Theorem \[thm:char\] (i), so ${\mathcal{C}}_{{\phi}_0}$ is not a PUSO.
For the other direction, we play the movie backwards. Suppose that ${\mathcal{C}}_{{\phi}_0}$ is not a PUSO. As pairs of antipodal vertices comply with Theorem \[thm:char\] (ii) by definition of ${\phi}_0$, there must be distinct and non-antipodal vertices $U,V'$ with the offending property (\[eq:UV’\]). Moreover, as ${\phi}_0$ induces USOs on both ${\mathcal{F}}$ (where we have ${\mathcal{F}}_{{\psi}}$) and its opposite facet ${\mathcal{F}}'$ (where we have a mirror image of ${\mathcal{F}}_{{\psi}}$), Lemma \[lem:strongchar\] implies that $U$ and $V'$ cannot both be in ${\mathcal{F}}$, or in ${\mathcal{F}}'$. W.l.o.g. assume that $U\in{\mathrm{vert}}{{\mathcal{F}}}, V'\in{\mathrm{vert}}{{\mathcal{F}}'}$, and let $V\in{\mathrm{vert}}{{\mathcal{F}}}$ be antipodal to $V'$. Then, as before, (\[eq:UV’\]) is equivalent to the inclusion ${\phi}_0(U)\oplus {\phi}_0(V)={\phi}_0(U)\oplus
{\phi}_0(V')\subseteq U\oplus V$. In particular, ${\phi}_0(U)$ and ${\phi}_0(V)$ must agree in coordinate $n$ which in turn implies $${\psi}(U)\oplus {\psi}(V)={\phi}_0(U)\oplus {\phi}_0(V)
\subseteq U\oplus V,$$ and since $|{\phi}_0(U)\oplus {\phi}_0(V)|=0\mod 2$ by definition of ${\phi}_0$, we have found two distinct vertices $U,V\in{\mathrm{vert}}{{\mathcal{F}}}$ that fail to satisfy (\[pusof\_thm\_eq\]).
For an example of a 3-dimensional border USO, see Figure \[fig:3pusof\]. In particular, we see that faces of border USOs are not necessarily border USOs: an eye cannot be a 2-dimensional border USO (Figure \[fig:Noeye\]), but it may appear in a facet ${\mathcal{F}}$ of a 3-dimensional border USO (for example, the bottom facet in Figure \[fig:3pusof\]), since the incident edges along the third coordinate can be chosen such that (\[pusof\_thm\_eq\]) does not impose any condition on the USO in ${\mathcal{F}}$.
![A border USO with eyes (non-border USOs) in the bottom and the top facet[]{data-label="fig:3pusof"}](3pusof.pdf){width="30.00000%"}
Let ${\mathrm{border}}(n)$ denote the number of border USOs of the standard $n$-cube. By Lemma \[lem:outmap01\], $$\label{eq:pusoborder}
{\mathrm{puso}}(n)=2{\mathrm{border}}(n-1), \quad n\geq 2.$$
Odd Unique Sink Orientations {#sec:odd}
============================
By (\[eq:pusoborder\]), counting PUSOs boils down to counting border USOs. However, as faces of border USOs are not necessarily border USOs (see the example of Figure \[fig:3pusof\]), it will be easier to work in a dual setting where we get a class of USOs that is closed under taking faces.
\[lem:duality\] Let ${\mathcal{C}}_{{\phi}}$ be a USO of ${\mathcal{C}}={\mathcal{C}}^{B}$ with outmap ${\phi}$. Then ${\mathcal{C}}_{{\phi}^{-1}}$ is a USO as well, the *dual* of ${\mathcal{C}}_{{\phi}}$.
We use the USO characterization of Lemma \[lem:strongchar\]. Since ${\mathcal{C}}_{{\phi}}$ is a USO, ${\phi}:2^B\rightarrow 2^B$ is bijective to begin with, so ${\phi}^{-1}$ exists. Now let $U',V'\in{\mathrm{vert}}{{\mathcal{C}}}$, $U'\neq V'$ and define $U:={\phi}^{-1}(U')\neq{\phi}^{-1}(V')=:V$. Then we have $$({\phi}^{-1}(U')\oplus {\phi}^{-1}(V'))\cap (U'\cap V') =
(U\oplus V) \cap ({\phi}(U)\oplus{\phi}(V) \neq \emptyset,$$ since ${\mathcal{C}}_{{\phi}}$ is a USO. Hence, ${\mathcal{C}}_{{\phi}^{-1}}$ is a USO as well.
\[def:odd\] An *odd USO* is a USO that is dual to a border USO.
Figure \[fig:duality\] shows an example of the duality with the following outmaps:
$$\begin{array}{c|c|c|c|c|c|c|c|c|c}
V & \emptyset & \{1\} & \{2\} & \{3\} & \{1,2\} & \{1,3\} & \{2,3\} &
\{1,2,3\}
& {\phi}^{-1}(V') \\ \hline
{\phi}(V) & \emptyset & \{1\} & \{1,2\} & \{2,3\} & \{2\} &
\{1,2,3\}
& \{1,3\} &
\{3\}
& V'
\end{array}$$
![The border USO ${\mathcal{C}}_{{\phi}}$ of Figure \[fig:3pusof\] (left) and its dual odd USO ${\mathcal{C}}_{{\phi}^{-1}}$ (right). The labels denote the vertices.[]{data-label="fig:duality"}](duality.pdf){width="90.00000%"}
A characterization of odd USOs now follows from Theorem \[thm:pusof\_charact\] by swapping the roles of vertices and outmaps; the proof follows the same scheme as the one of Lemma \[lem:duality\] and is omitted.
\[thm:odd\_charact\] Let ${\mathcal{C}}_{{\phi}}$ be a USO with outmap ${\phi}$. ${\mathcal{C}}_{{\phi}}$ is an odd USO if and only if the following condition holds for all pairs of distinct vertices $U,V\in{\mathrm{vert}}{{\mathcal{C}}}$: $$\label{eq:odd_thm_eq}
U\oplus V \subseteq {\phi}(U)\oplus {\phi}(V) \quad
\Rightarrow \quad |U\oplus V| = 1 \mod 2.$$
In words, if the outmap values of two distinct vertices $U,V$ differ in all coordinates within the face spanned by $U$ and $V$, then $U$ and $V$ are of odd Hamming distance.[^3] As this property also holds for any two distinct vertices within a face ${\mathcal{F}}$, this implies the following.
\[cor:odd\] Let $C_{{\phi}}$ be an odd USO, ${\mathcal{F}}$ a face of ${\mathcal{C}}$.
- ${\mathcal{F}}_{{\phi}}$ is an odd USO.
- If ${\mathrm{dim}}({\mathcal{F}})=2$, ${\mathcal{F}}_{{\phi}}$ is a bow.
Indeed, as source and sink of an eye violate (\[eq:odd\_thm\_eq\]), all 2-faces of odd USOs are bows. To make the global structure of odd USOs more transparent, we develop an alternative view on them in terms of *caps* that can be considered as “higher-dimensional bows”.
Let ${\mathcal{C}}_{{\phi}}$ be an orientation with bijective outmap ${\phi}$. For $W\in{\mathrm{vert}}{{\mathcal{C}}}$, let $\overline{W}\in{\mathrm{vert}}{{\mathcal{C}}}$ be the unique *complementary* vertex, the one whose outmap value is antipodal to ${\phi}(W)$; formally, ${\phi}(W)\oplus {\phi}(\overline{W})={\mathrm{carr}}{{\mathcal{C}}}$. ${\mathcal{C}}_{{\phi}}$ is called a *cap* if $$|W\oplus\overline{W}|=1\mod 2, \quad \forall W\in{\mathrm{vert}}{{\mathcal{C}}}.$$
Figure \[fig:oddface\] illustrates this notion on three examples.
![A 3-dimensional cap: complementary vertices (vertices that differ in all outgoing coordinates) have odd Hamming distance (left); the eye is not a cap (middle); the bow is a cap (right).[]{data-label="fig:oddface"}](oddface.pdf){width="90.00000%"}
\[lem:oddchar\] Let ${\mathcal{C}}_{{\phi}}$ be an orientation with outmap ${\phi}$. ${\mathcal{C}}_{{\phi}}$ is an odd USO if and only if all its faces are caps.
If all faces are caps, their outmaps are bijective, meaning that all faces have unique sinks. So ${\mathcal{C}}_{{\phi}}$ is a USO. It is odd, since the characterizing property (\[eq:odd\_thm\_eq\]) follows for all distinct $U,V$ via the cap spanned by $U$ and $V$.
Now suppose that ${\mathcal{C}}_{{\phi}}$ is an odd USO. Then every face ${\mathcal{F}}$ has a bijective outmap to begin with, by Lemma \[lem:strongchar\]; to show that ${\mathcal{F}}$ is a cap, consider any two complementary vertices $W,\overline{W}$ in ${\mathcal{F}}$. As $W$ and $\overline{W}$ are in particular complementary in the face that they span, they have odd Hamming distance by (\[eq:odd\_thm\_eq\]).
There is a “canonical” odd USO of the standard $n$-cube in which the Hamming distances of complementary vertices are not only odd, but in fact always equal to $1$. This orientation is known as the *Klee-Minty cube*, as it captures the combinatorial structure of the linear program that Klee and Minty used in 1972 to show for the first time that the simplex algorithm may take exponential time [@KM].
The $n$-dimensional Klee-Minty cube can be defined inductively: ${\mathrm{KM}}^{[n]}$ is obtained from ${\mathrm{KM}}^{[n-1]}$ by embedding an $[n-1]$-flipped copy of ${\mathrm{KM}}^{[n-1]}$ into the opposite facet ${\mathcal{C}}^{[\{n\},[n]]}$, with all connecting edges oriented towards ${\mathcal{C}}^{[n-1]}$; the resulting USO contains a directed Hamiltonian path; see Figure \[fig:KM\]. As a direct consequence of the construction, ${\mathrm{KM}}^{[n]}$ is a cap: complementary vertices are neighbors along coordinate $n$. Moreover, it is easy to see that each $k$-face is combinatorially equivalent to ${\mathrm{KM}}^{[k]}$, hence all faces are caps, so ${\mathrm{KM}}^{[n]}$ is an odd USO.
![The Klee-Minty cubes ${\mathrm{KM}}^{[1]},
{\mathrm{KM}}^{[2]},{\mathrm{KM}}^{[3]}$ (with pairs of complementary vertices and directed Hamiltonian path)[]{data-label="fig:KM"}](KM.pdf){width="\textwidth"}
Next, we do this more formally, as we will need the Klee-Minty cube as a starting point for generating many odd USOs.
\[lem:km\] Consider the standard $n$-cube ${\mathcal{C}}$ and the outmap ${\phi}:2^{[n]}\rightarrow 2^{[n]}$ with $${\phi}(V) = \{j\in[n]: |V\cap\{j,j+1,\ldots,n\}|=1\mod 2\}, \quad
\forall V\subseteq [n].$$ Then ${\mathrm{KM}}^{[n]}:={\mathcal{C}}_{{\phi}}$ is an odd USO that satisfies $$\label{eq:vertexflip}
{\phi}(W) \oplus {\phi}(W\oplus\{i\}) = [i], \quad \forall i\in[n].$$ for each vertex $W$.
In particular, for all $i$ and $W\in{\mathcal{C}}^{[i-1]}$, $W$ and $W\cup\{i\}$ are complementary in ${\mathcal{C}}^{[i]}$, so we recover the above inductive view of the Klee-Minty cube.
We first show that $$\label{eq:kmout}
{\phi}(U)\oplus{\phi}(V) = {\phi}(U\oplus V), \quad \forall U,V\subseteq[n].$$ Indeed, $j\in {\phi}(U)\oplus{\phi}(V)$ is equivalent to $U\cap\{j,j+1,\ldots,n\}$ and $V\cap\{j,j+1,\ldots,n\}$ having different parities, which by (\[eq:symdiff\]) is equivalent to $(U\oplus V)\cap\{j,j+1,\ldots,n\}$ having odd parity, meaning that $j\in{\phi}(U\oplus V)$.
Since ${\phi}(U)\oplus{\phi}(V)={\phi}(U\oplus V)$ contains the largest element of $U\oplus V$, (\[eq:char\]) holds for all pairs of distinct vertices, so ${\mathcal{C}}_{{\phi}}$ is a USO. Condition (\[eq:vertexflip\]) follows from $${\phi}(W) \oplus {\phi}(W\oplus\{i\}) = {\phi}(\{i\}) = [i].$$ To show that ${\mathcal{C}}_{{\phi}}$ is odd, we verify condition (\[eq:odd\_thm\_eq\]) of Theorem \[thm:odd\_charact\]. Suppose that $U\oplus V \subseteq {\phi}(U)\oplus {\phi}(V) = {\phi}(U\oplus
V)$ for two distinct vertices. Since ${\phi}(U\oplus V)$ does not contain the second-largest element of $U\oplus V$, the former inclusion can only hold if there is no such second-largest element, i.e. $U$ and $V$ have (odd) Hamming distance $1$.
The Klee-Minty cube has a quite special property: *complementing* any vertex (reversing all its incident edges) yields another odd USO.[^4] Even more is true: any set of vertices with disjoint neighborhoods can be complemented simultaneously. Thus, if we select a set of $N$ vertices with pairwise Hamming distance at least $3$, we get $2^N$ different odd USOs. We will use this in the next section to get a lower bound on the number of odd USOs. The following lemma is our main workhorse.
\[lem:vertexflip\] Let ${\mathcal{C}}_{{\phi}}$ be an odd USO of the standard $n$-cube with outmap ${\phi}$, $W\in{\mathrm{vert}}{{\mathcal{C}}}$ a vertex satisfying condition (\[eq:vertexflip\]): $${\phi}(W) \oplus {\phi}(W\oplus\{i\}) = [i], \quad \forall i\in[n].$$ Let ${\mathcal{C}}_{{\phi}'}$ be the orientation resulting from complementing (reversing all edges incident to) $W$. Formally, $$\label{eq:compl}
\begin{array}{lclcl}
{\phi}'(W) &=& {\phi}(W) &\oplus& [n], \\
{\phi}'(W\oplus\{i\}) &=& {\phi}(W\oplus\{i\}) &\oplus& \{i\}, \quad
i=1,\ldots,n,
\end{array}$$ and ${\phi}'(V)={\phi}(V)$ for all other vertices. Then ${\mathcal{C}}_{{\phi}'}$ is an odd USO as well.
We first show that every face ${\mathcal{F}}_{{\phi}'}$ has a unique sink, so that ${\phi}'$ is a USO. If $W\not\in{\mathrm{vert}}{{\mathcal{F}}}$, then ${\mathcal{F}}_{{\phi}'}={\mathcal{F}}_{{\phi}}$, so there is nothing to show. If $W\in {\mathrm{vert}}{{\mathcal{F}}}$, let ${\mathrm{carr}}{{\mathcal{F}}}=\{i_1,i_2,\ldots,i_k\}$, $i_1<i_2<\cdots<i_k$. Using (\[eq:symdiff\]), condition (\[eq:vertexflip\]) yields $$\label{eq:outprime}
{\phi}_{{\mathcal{F}}}(W) \oplus {\phi}_{{\mathcal{F}}}(W\oplus\{i_t\}) = \{i_1,i_2,\ldots,i_t\}, \quad \forall t\in[k]$$ and further $$\label{eq:outprime2}
{\phi}_{{\mathcal{F}}}(W\oplus\{i_s\} \oplus {\phi}_{{\mathcal{F}}}(W\oplus\{i_t\}) =
\{i_{s+1},i_{s+2},\ldots,i_t\}, \quad \forall s,t\in[k], s<t.$$
In particular, $W$ is complementary to $W\oplus\{i_k\}$ in ${\mathcal{F}}$, but this is the only complementary pair among the $k+1$ vertices in ${\mathcal{F}}$ that are affected by complementing $W$. From (\[eq:compl\]), it similarly follows that $$\label{eq:cyclicshift}
\begin{array}{lclcl}
{\phi}'_{{\mathcal{F}}}(W) &=& {\phi}_{{\mathcal{F}}}(W) \oplus \{i_1,i_2,\ldots,i_k\} &\stackrel{(\ref{eq:outprime})}{=}&
{\phi}_{{\mathcal{F}}}(W\oplus\{i_k\}), \\
{\phi}'_{{\mathcal{F}}}(W\oplus\{i_1\}) &=& {\phi}_{{\mathcal{F}}}(W\oplus\{i_1\})\oplus\{i_1\} &\stackrel{(\ref{eq:outprime})}{=}& {\phi}_{{\mathcal{F}}}(W), \\
{\phi}'_{{\mathcal{F}}}(W\oplus\{i_t\}) &=& {\phi}_{{\mathcal{F}}}(W\oplus\{i_t\}) \oplus\{i_{t}\}
&\stackrel{(\ref{eq:outprime2})}{=}& {\phi}_{{\mathcal{F}}}(W\oplus\{i_{t-1}\}),
\end{array}$$ for $t=2,\ldots,k$. This means that the $k+1$ affected vertices just permute their outmap values under ${\phi}_{{\mathcal{F}}}\rightarrow {\phi}'_{{\mathcal{F}}}$. This does not change the number of sinks, so $F_{{\phi}'}$ has a unique sink as well.
It remains to show that ${\mathcal{F}}_{{\phi}'}$ is a cap, so ${\mathcal{C}}_{{\phi}'}$ is an odd USO by Lemma \[lem:oddchar\]. Since ${\mathcal{F}}_{{\phi}}$ is a cap, it suffices to show that complementary vertices keep odd Hamming distance under ${\phi}_{{\mathcal{F}}}\rightarrow {\phi}'_{{\mathcal{F}}}$. This can also be seen from (\[eq:cyclicshift\]): for $t=2,\ldots,k$, the vertex of outmap value ${\phi}_{{\mathcal{F}}}(W\oplus\{i_{t-1}\})$ moves by Hamming distance $2$, namely from $W\oplus\{i_{t-1}\}$ (under ${\phi}_{{\mathcal{F}}}$) to $W\oplus\{i_{t}\}$ (under ${\phi}'_{{\mathcal{F}}}$). Hence it still has odd Hamming distance to its unaffected complementary vertex. The two complementary vertices of outmap values ${\phi}_{{\mathcal{F}}}(W)$ and ${\phi}_{{\mathcal{F}}}(W\oplus\{i_k\})$ move by Hamming distance $1$ each. Vertices of other outmap values are unaffected.
As an example, if we complement the vertex $\overline{Y}$ in the Klee-Minty cube of Figure \[fig:KM\], we obtain the odd USO in Figure \[fig:oddface\] (left); see Figure \[fig:complement\]. Vertices $\overline{X}$ and $\overline{Z}$ have moved by Hamming distance 2, while $Y$ and $\overline{Y}$ have moved by Hamming distance $1$ each. If we subsequently also complement $W$ (whose neighborhood was unaffected, so Lemma \[lem:vertexflip\] still applies), we obtain another odd USO (actually, a rotated Klee-Minty cube).
![Complementing the two vertices $\overline{Y},W$ in succession, starting from the Klee-Minty cube (left)[]{data-label="fig:complement"}](complement1.pdf "fig:"){width="30.00000%"} ![Complementing the two vertices $\overline{Y},W$ in succession, starting from the Klee-Minty cube (left)[]{data-label="fig:complement"}](complement2.pdf "fig:"){width="30.00000%"} ![Complementing the two vertices $\overline{Y},W$ in succession, starting from the Klee-Minty cube (left)[]{data-label="fig:complement"}](complement3.pdf "fig:"){width="30.00000%"}
Let ${\mathrm{odd}}(n)$ denote the number of odd USOs of the standard $n$-cube. By Definition \[def:odd\], we get $$\label{eq:oddborder}
{\mathrm{odd}}(n)={\mathrm{border}}(n), \quad \forall n\geq 0,$$ as duality (Lemma \[lem:duality\]) is a bijection on the set of all USOs.
Counting PUSOs and odd USOs {#sec:counting}
===========================
With characterizations of USOs, PUSOs, border USOs, and odd USOs available, one can explicitly enumerate these objects for small dimensions. Here are the results up to dimension $5$ (the USO column is due to Schurr [@schurr2004diss Chapter 6]). We remark that most numbers (in particular, the larger ones) have not independently been verified.
\[uso\_enum\_small\_n\]
$n$ ${\mathrm{uso}}(n)$ ${\mathrm{puso}}(n)=2{\mathrm{border}}(n-1)$ ${\mathrm{border}}(n)={\mathrm{odd}}(n)$
----- --------------------- ---------------------------------------------- ------------------------------------------
0 1 0 1
1 2 0 2
2 12 4 8
3 744 16 112
4 5’541’744 224 12’928
5 638’560’878’292’512 25’856 44’075’264
: The number of USOs, PUSOs and border / odd USOs of the standard $n$-cube, obtained through computer enumeration
The number of PUSOs appears to be very small, compared to the total number of USOs of the same dimension. In this section, we will show the following asymptotic results that confirms this impression.
\[thm:pusocount\] Let ${\mathrm{puso}}(n)$ denote the number of PUSOs of the standard $n$-cube.
- For $n\geq 2$, ${\mathrm{puso}}(n) \leq 2^{2^{n-1}}$.
- For $n\geq 6$, ${\mathrm{puso}}(n) < 1.777128^{2^{n-1}}$.
- For $n=2^k, k\geq 2$, ${\mathrm{puso}}(n) \geq 2^{2^{n-1-\log n}+1}$.
This shows that the number ${\mathrm{puso}}(n)$ is doubly exponential but still negligible compared to the number ${\mathrm{uso}}(n)$ of USOs of the standard $n$-cube: Matoušek [@matousek2006number] has shown that $${\mathrm{uso}}(n) \geq \left(\frac{n}{e}\right)^{2^{n-1}},$$ with a “matching” upper bound of ${\mathrm{uso}}(n) = n^{O(2^n)}$.
As the main technical step, we count odd USOs. We start with the upper bound.
\[lem:bound\] Let $n\geq 1$. Then
- ${\mathrm{odd}}(n) \leq 2{\mathrm{odd}}(n-1)^2$ for $n>0$.
- For $n\geq 2$ and all $k<n$, $$\label{eq:oddlower}
2{\mathrm{odd}}(n-1) \leq \left(2{\mathrm{odd}}(k)\right)^{2^{n-1-k}} =
\sqrt[2^k]{2{\mathrm{odd}}(k)}^{2^{n-1}}.$$
By Corollary \[cor:odd\] (i), every odd USO consists of two odd USOs in two opposite facets, and edges along coordinate $n$, say, that connect the two facets. We claim that for every choice of odd USOs in the two facets, there are at most two ways of connecting the facets. Indeed, once we fix the direction of some connecting edge, all the others are fixed as well, since the orientation of an edge $\{V,V\oplus\{n\}\}$ determines the orientations of all “neighboring” edges $\{V\oplus\{i\},V\oplus\{i,n\}\}$ via Corollary \[cor:odd\] (ii) (all $2$-faces are bows). Inequality (i) follows, and (ii) is a simple induction.
The three bounds on ${\mathrm{puso}}(n)$ now follow from ${\mathrm{puso}}(n)=2{\mathrm{border}}(n-1)$ (\[eq:pusoborder\]) and ${\mathrm{border}}(n-1)={\mathrm{odd}}(n-1)$ (\[eq:oddborder\]). For the bound of Theorem \[thm:pusocount\] (i), we use (\[eq:oddlower\]) with $k=0$, and for Theorem \[thm:pusocount\] (ii), we employ $k=5$ and ${\mathrm{odd}}(5)= 44'075'264$. The lower bound of Theorem \[thm:pusocount\] (iii) is a direct consequence of the following “matching” lower bound on the number of odd USOs.
\[lem:odd\_lower\] Let $n=2^k, k\geq 1$. Then ${\mathrm{odd}}(n-1) \geq 2^{2^{n-1-\log n}}$.
If $n=2^k$, there exists a perfect Hamming code of block length $n-1$ and message length $n-1-\log n$ [@Hamming]. In our language, this is a set ${\cal W}$ of $2^{n-1-\log n}$ vertices of the standard $(n-1)$-cube, with pairwise Hamming distance $3$ and therefore disjoint neighborhoods. Hence, starting from the Klee-Minty cube ${\mathrm{KM}}^{[n-1]}$ as introduced in Lemma \[lem:km\], we can apply Lemma \[lem:vertexflip\] to get a different odd USO for every subset of ${\cal W}$, by complementing all vertices in the given subset. The statement follows.
Conclusion {#sec:conclusion}
==========
In this paper, we have introduced, characterized, and (approximately) counted three new classes of $n$-cube orientations: pseudo unique sink orientations (PUSOs), border unique sink orientations (facets of PUSOs), and odd unique sink orientations (duals of border USOs). A PUSO is a dimension-minimal witness for the fact that a given cube orientation is not a USO. The requirement of minimal dimension induces rich structural properties and a PUSO frequency that is negligible compared to the frequency of USOs among all cube orientations.
An obvious open problem is to close the gap in our approximate counting results and determine the true asymptotics of $\log{\mathrm{odd}}(n)$ and hence $\log{\mathrm{puso}}(n)$. We have shown that these numbers are between $\Omega(2^{n-\log n})$ and $O(2^n)$. As our lower bound construction based on the Klee-Minty cube seems to yield rather specific odd USOs, we believe that the lower bound can be improved.
Also, border USOs and odd USOs might be algorithmically more tractable than general USOs. The standard complexity measure here is the number of outmap values[^5] that need to be inspected in order to be able to deduce the location of the sink [@szabo2001unique]. For example, in dimension $3$, we can indeed argue that border USOs and odd USOs are easier to solve than general USOs. It is known that 4 outmap values are necessary and sufficient to locate the sink in any USO of the 3-cube [@szabo2001unique]. But in border USOs and odd USOs of the 3-cube, 3 suitably chosen outmap values suffice to deduce the orientations of all edges and hence the location of the sink [@wff]; see Figure \[fig:4steps\].
![The outmap values of the 3 indicated vertices determine the orientations of the bold edges. In the case of a border USO (left), the remaing orientations are determined by the condition that antipodal vertices have different outmap parities (Theorem \[thm:pusof\_charact\]). In the case of an odd USO (right), the remaining orientations are determined by the condition that all 2-faces are bows (Corollary \[cor:odd\](ii)).[]{data-label="fig:4steps"}](4steps.pdf){width="65.00000%"}
[^1]: The naming goes back to Szabó and Welzl [@szabo2001unique].
[^2]: In fact, it is already [**coNP**]{}-complete to decide whether ${\mathcal{C}}_{{\phi}}$ is an orientation.
[^3]: Hamming distance is defined for two bit vectors, but we can also define it for two sets in the obvious way as the size of their symmetric difference.
[^4]: In general, the operation of complementing a vertex will destroy the USO property.
[^5]: provided by an oracle that can be invoked for every vertex
|
---
bibliography:
- 'Manuscript.bib'
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[**** ]{}\
J. Malo^1,2\*^, J.J. Esteve-Taboada^2^, M. Bertalmío^3^\
**1**\
**2**\
**3**\
\*
Abstract[^1] {#abstract .unnumbered}
============
Divisive Normalization and the Wilson-Cowan equations are well-known influential models of neural interaction and saturation \[Carandini and Heeger Nat.Rev.Neurosci. 2012; Wilson and Cowan Kybernetik 1973\]. However, they have been always treated as two different approaches, and have not been analytically related yet. In this work we show that Divisive Normalization can be derived from the Wilson-Cowan model. Specifically, assuming that Divisive Normalization is the steady state solution of the Wilson-Cowan differential equation, we find that the kernel that controls neural interactions in Divisive Normalization depends on the Wilson-Cowan kernel but also has a signal-dependent contribution. A standard stability analysis of a Wilson-Cowan model with the parameters obtained from our relation shows that the Divisive Normalization solution is a stable node. This stability demonstrates the consistency of our steady state assumption, and is in line with the straightforward use of Divisive Normalization with time-varying stimuli.
The proposed theory provides a physiological foundation (a relation to a dynamical network with fixed wiring among neurons) for the functional suggestions that have been done on the need of signal-dependent Divisive Normalization \[e.g. in Coen-Cagli et al., PLoS Comp.Biol. 2012\]. Moreover, this theory explains the modifications that had to be introduced ad-hoc in Gaussian kernels of Divisive Normalization in \[Martinez et al. Front. Neurosci. 2019\] to reproduce contrast responses.
The derived relation implies that the Wilson-Cowan dynamics also reproduces visual masking and subjective image distortion metrics, which up to now had been mainly explained via Divisive Normalization. Finally, this relation allows to apply to Divisive Normalization the methods which had been traditionally developed for dynamical systems such as Wilson-Cowan networks.
Introduction
============
A number of perceptual experiences in different modalities can be described with the Divisive Normalization interaction among the outputs of linear sensors [@Carandini12]. In particular, in vision, the perception of color, texture, and motion seem to be mediated by this interaction [@Brainard05; @Watson97; @Simoncelli98]. The discussion on the circuits underlying the Divisive Normalization in [@Carandini12] suggests that there may be different architectures leading to this specific computation. Recent results suggest specific mechanisms for Divisive Normalization in certain situations [@Carandini16], but the general debate on the physiological implementations that may occur is still open.
On the other hand, a number of functional advantages [@Schwartz09; @Schwartz11; @Coen12; @Coen13] suggest that the kernel that describes the interaction in Divisive Normalization should be adaptive (i.e. signal or context dependent). Moreover, the match between the linear receptive fields and the interaction kernel in the Divisive Normalization is not trivial: the conventional Gaussian kernels in [@Watson97; @Malo10] had to be tuned by hand to reproduce contrast responses [@Martinez19].
These open questions imply that it is interesting to relate Divisive Normalization to other models of neural interaction for a better understanding of its implementation, the structure of the interaction kernel, and its eventual dependence with the signal. Interesting possibilities to consider are the classical dynamic neural field models of Wilson-Cowan [@Wilson72; @Wilson73] or Amari [@Amari77], which are subtractive in nature. Subtractive and divisive adaptation models have been qualitatively related before [@Wilson93; @Cowan02]. Both models have been shown to have similar advantages in information-theoretic terms: univariate local histogram equalization in Wilson-Cowan [@Bertalmio14] and multivariate probability density factorization in Divisive Normalization [@Malo06a; @Malo10]. Additionally, both models provide similar descriptions of pattern discrimination [@Wilson93; @Bertalmio17]. However, despite all these similarities, no direct analytical correspondence has been established between these models yet.
In this paper, we assume that the psychophysical behavior described by Divisive Normalization comes from underlying neural interactions that follow the Wilson-Cowan equation. In particular, we identify the Divisive Normalization response with the stationary regime of a Wilson-Cowan system. From this identification we derive an expression for the Divisive Normalization kernel in terms of the interaction kernel of the Wilson-Cowan equation. This analytically derived relation has the following interesting consequences:
\(1) It provides a physiological foundation (a relation to a dynamical system with fixed wiring among neurons) for the functional suggestions that have been done on the need of signal-dependent Divisive Normalization, e.g. in [@Coen12].
\(2) It explains the modifications that had to be introduced ad-hoc in the kernel of Divisive Normalization in [@Martinez19] to reproduce contrast responses. This implies that the Wilson-Cowan dynamics reproduce visual masking, which up to now had been mainly explained via Divisive Normalization [@Foley94; @Watson97].
\(3) The response of Divisive Normalization to natural images using these hand-crafted kernels (needed to reproduce contrast masking) coincides with the response obtained using the theoretically-deduced kernel from the proposed relation. This implies that the Wilson-Cowan model also predicts subjective image quality, which up to now had been mainly explained via Divisive Normalization, e.g. in [@Laparra10a; @Malo10; @Laparra17].
\(4) A standard stability analysis of a Wilson-Cowan model with the parameters obtained from our derived relation shows that the Divisive Normalization solution is a stable node of this dynamic model. The robustness of Divisive Normalization found through this analysis (which up to now was only usual in dynamic models like Wilson-Cowan [@Sejnowski09]) shows the consistency of our steady state assumption. Moreover, this stability is in line with the straightforward use of Divisive Normalization with time-varying stimuli [@Simoncelli98].
The structure of the paper is as follows. The *Materials and Methods* section reviews the general context in which cortical interaction neural models will be applied: the retina-V1 neural path and the contrast perception of visual patterns. We also introduce the notation of the models we are considering: the Divisive Normalization and the Wilson-Cowan approach. Besides, we recall some experimental facts that will be used to illustrate the performance of the proposed relation: (1) contrast responses curves imply certain interactions between subbands [@Cavanaugh00; @Watson97], (2) the Divisive Normalization kernel must have a specific structure (identified in [@Martinez19]) to reproduce contrast response curves, and (3) the shape of the Divisive Normalization kernel has a specific dependence with the surrounding signal [@Cavanaugh02a; @Cavanaugh02b]. In the *Results* section we derive the analytical relation between the Divisive Normalization and the Wilson-Cowan equation. The *Discussion* section analyzes the mathematical properties and the perceptual consequences of the proposed relation. First, we check the convergence of the Wilson-Cowan solution to the Divisive Normalization response. Moreover, we demonstrate the consistency of the steady state assumption by showing that the Divisive Normalization is a stable node of the Wilson-Cowan system. Then, we address contrast perception facts using the proposed relation to build a psychophysically meaningful Wilson-Cowan model: we theoretically derive the specific structure of the kernel that was previously empirically inferred [@Martinez19], we show that the proposed interaction kernel adapts with the signal, and as a result, we reproduce general trends of contrast response curves. Finally, we discuss the use of the derived kernel in predicting the metric in the image space.
Materials and Methods
=====================
In this work the theory is illustrated in the context of models of the retina-cortex pathway. The considered framework follows the approach suggested in [@Carandini12]: a cascade of four isomorphic linear+nonlinear modules. These four modules sequentially address brightness, contrast, frequency filtered contrast masked in the spatial domain, and orientation/scale masking. An example of the transforms of the input in such models is shown in Fig. 1.
In this general context we focus on the cortical (fourth) layer: a set of linear sensors with wavelet-like receptive fields modelling simple cells in V1, and the nonlinear interaction between the responses of these linear sensors. Divisive Normalization has been the conventional model used for the nonlinearities to describe contrast perception psychophysics [@Watson97], but here we will explore the application of the Wilson-Cowan model in the contrast perception context.
Below we introduce the notation of both neural interaction models and the experimental contrast response facts that should be explained by the models.
[c]{} {height="0.90\textheight"}\
Modelling cortical interactions
-------------------------------
In the case of the V1 cortex, we refer to the set of responses of a population of simple cells as the vector ${\boldsymbol{z}}$. The considered models (Divisive Normalization and Wilson-Cowan) define a nonlinear mapping, $\mathcal{N}$, that transforms the input responses vector ${\boldsymbol{z}}$ (before the interaction among neurons) into the output responses vector ${\boldsymbol{x}}$ (after the interaction): $$\xymatrixcolsep{2pc}
\xymatrix{ {\boldsymbol{z}} \,\,\,\, \ar@/^0.7pc/[r]^{\scalebox{0.85}{$\mathcal{N}$}} & \,\,\,\, {\boldsymbol{x}}
}
\label{global_response}$$ In this setting, responses are called *excitatory* or *inhibitory*, depending on the corresponding *sign* of the signal: ${\boldsymbol{z}} = \textrm{sign}({\boldsymbol{z}}) |{\boldsymbol{z}}| $, and ${\boldsymbol{x}} = \textrm{sign}({\boldsymbol{x}}) |{\boldsymbol{x}}|$. The nonlinear mapping, $\mathcal{N}$, is an adaptive saturating transform, but it preserves the sign of the responses (i.e. $\textrm{sign}({\boldsymbol{x}})=\textrm{sign}({\boldsymbol{z}})$). Therefore, the models care about cell activation (the modulus $|\cdot|$) but not about the excitatory or inhibitory nature of the sensors (the $\textrm{sign}(\cdot)=\pm$).
The *energy* of the input responses will be given by ${\boldsymbol{e}} = |{\boldsymbol{z}}|^\gamma$, an element-wise exponentiation of the amplitudes $|z_i|$.
Given the sign-preserving nature of the nonlinear mapping, $\mathcal{N}$, and for the sake of simplicity in the notation along the paper, the variables ${\boldsymbol{z}}$ and ${\boldsymbol{x}}$ refer to the activations $|{\boldsymbol{z}}|$ and $|{\boldsymbol{x}}|$.
The Divisive Normalization model
--------------------------------
The conventional expressions of the *canonical* Divisive Normalization model [@Carandini12] use and element-wise formulation, that can be rewritten with diagonal matrices $\mathbb{D}_{(\cdot)}$ as shown in [@Martinez17].
#### Forward transform. {#forward-transform. .unnumbered}
The response transform in this model is given by:
$${\boldsymbol{x}} = \mathbb{D}_{{\boldsymbol{k}}} \cdot \mathbb{D}^{-1}_{\left( {\boldsymbol{b}} + {\boldsymbol{H}} \cdot {\boldsymbol{e}} \right)} \cdot {\boldsymbol{e}}
\label{DN_B}$$
where the output vector of nonlinear activations in V1, ${\boldsymbol{x}}$, depends on the energy of the input linear wavelet responses, ${\boldsymbol{e}}$, which are dimension-wise normalized by a sum of neighbor energies. The non-diagonal nature of the interaction kernel ${\boldsymbol{H}}$ in the denominator, ${\boldsymbol{b}} + {\boldsymbol{H}} \cdot {\boldsymbol{e}}$, implies that the $i$-th element of the response may be attenuated if the activity of the neighbor sensors, $e_j$ with $j\neq i$, is high. Each row of the kernel ${\boldsymbol{H}}$ describes how the energies of the neighbor simple cells attenuate the activity of each simple cell after the interaction. Each element of the vectors ${\boldsymbol{b}}$ and ${\boldsymbol{k}}$ represents the semisaturation and the dynamic range of the nonlinear response of each sensor, respectively.
#### Inverse transform. {#inverse-transform. .unnumbered}
In the case of the Divisive Normalization model, the analytical inverse transform, which will be used to obtain the relation between the two models, is given by [@Malo06a; @Martinez17]: $${\boldsymbol{e}} = \left( I - \mathbb{D}^{-1}_{{\boldsymbol{k}}}\cdot\mathbb{D}_{{\boldsymbol{x}}}\cdot {\boldsymbol{H}} \right)^{-1} \cdot \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot {\boldsymbol{x}}
\label{invDN}$$
The Wilson-Cowan model
----------------------
The Wilson-Cowan dynamical system was proposed to model the nonlinear interactions between the responses at specific stages in the visual pathway [@Wilson72; @Wilson73].
#### Dynamical system. {#dynamical-system. .unnumbered}
In the Wilson-Cowan model the temporal variation of the activation vector, ${\boldsymbol{\dot{x}}}$, increases with the energy of the input ${\boldsymbol{e}}$, but, for each sensor, this variation is also moderated by its own activity and by a linear combination of the activities of the neighbor sensors: $${\boldsymbol{\dot{x}}} = {\boldsymbol{e}} - \mathbb{D}_{{\boldsymbol{\alpha}}} \cdot {\boldsymbol{x}} - {\boldsymbol{W}} \cdot f({\boldsymbol{x}})
\label{EqWC}$$ where ${\boldsymbol{W}}$ is the matrix that describes the damping factor between sensors, and $f({\boldsymbol{x}})$ is a dimension-wise saturating nonlinearity. ${\boldsymbol{W}}$ is usually considered to be a fixed matrix, made of Gaussian neighborhoods, that represent the local interaction between sensors [@Faugueras09]. Note that, in Eq. \[EqWC\], both the inhibitory and the excitatory responses are considered just as negative and positive components of the same vector. Therefore, the two equations in the traditional Wilson-Cowan formulation are represented here by a single expression [@Bressloff03].
#### Steady state and inverse. {#steady-state-and-inverse. .unnumbered}
The stationary solution of the above differential equation, that can be obtained by making ${\boldsymbol{\dot{x}}} =0$ in Eq. \[EqWC\], leads to the following analytical inverse (input-from-output) relation: $${\boldsymbol{e}} = \mathbb{D}_{{\boldsymbol{\alpha}}} \cdot {\boldsymbol{x}} + {\boldsymbol{W}} \cdot f({\boldsymbol{x}})
\label{invWC}$$
As we will see later in the *Results* section, the identification of the different terms in the decoding equations corresponding to both models, Eq. \[invDN\] and Eq. \[invWC\], is the key to obtain simple analytical relations between their corresponding parameters.
Experimental facts
------------------
### Adaptive contrast response curves
In the considered spatial vision context, the models should reproduce the fundamental trends of contrast perception. Thus, the slope of the contrast response curves should depend on the spatial frequency, so that the sensitivity at threshold contrast is different for different spatial frequencies according to the Contrast Sensitivity Function (CSF) [@Campbell68]. Also, the response curves should saturate with contrast [@Legge80; @Legge81]. Finally, the responses should attenuate with the energy of the background or surround, and this additional saturation should depend on the texture of the background [@Foley94; @Watson97]: if the frequency/orientation of the test is similar to the frequency/orientation of the background, the decay should be stronger. This background-dependent adaptive saturation, or *masking*, is mediated by cortical sensors tuned to spatial frequency with responses that saturate depending on the background, as illustrated in Fig. \[facts\].
The above trends are key to discard too simple models, and also to propose the appropriate modifications in the model architecture to get reasonable results [@Martinez19].
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![. []{data-label="facts"}](./Figure2.jpg "fig:"){width="95.00000%"}
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### Unexplained kernel structure in Divisive Normalization
In the Divisive Normalization setting, the masking interaction between tests and backgrounds of different textures is classically described by using a Gaussian kernel in the denominator of Eq. \[DN\_B\] in wavelet-like domains: the effect of the $j$-th wavelet sensor on the attenuation of the $i$-th wavelet sensor decays with the distance in space between the $i$-th and $j$-th sensors, but also with the spatial frequency and orientation [@Watson97]. We will refer to these unit-norm Gaussian kernels as Watson and Solomon kernels [@Watson97], and will be represented by ${\boldsymbol{H}}^{{\boldsymbol{ws}}}$. Gaussian kernels are useful to describe the general behavior shown in Fig. \[facts\]: activity in close neighbors lead to strong decays in the response, while activity in neighbors tuned to more distant features has smaller effect.
However, in order to have well behaved responses in every subband with every possible background, a *special balance* between the wavelet representation and the Gaussian kernels is required. When using reasonable log-polar Gabor basis or steerable filters to model V1 receptive fields, as in [@Watson97; @Schwartz01], the energies of the sensors tuned to low frequencies is notably higher than the energy of high-frequency sensors. Moreover, the smaller number of sensors in low frequency subbands in this kind of wavelet representations implies that unit-norm Gaussian kernels have bigger values in coarse subbands. These two facts overemphasize the impact of low-frequency responses on high-frequency responses. Thus, in [@Martinez19] we found that classical unit-norm Gaussian kernels require *ad-hoc* extra modulation to avoid excessive effect of low frequency backgrounds on high frequency tests. The appropriate wavelet-kernel balance was then reestablished by introducing extra high-pass filters in the Gaussian kernel ${\boldsymbol{H}}^{{\boldsymbol{ws}}}$, with the aim to moderate the effect of low frequencies [@Martinez19]: $${\boldsymbol{H}} = \mathbb{D}_{{\boldsymbol{l}}} \cdot {\boldsymbol{H}}^{{\boldsymbol{ws}}} \cdot \mathbb{D}_{{\boldsymbol{r}}}
\label{new_kernel_eq}$$
In this new definition of the kernel: (1) the diagonal matrix at the right, $\mathbb{D}_{{\boldsymbol{r}}}$, pre-weights the subbands of ${\boldsymbol{e}}$ to moderate the effect of low frequencies before computing the interaction; and (2) the diagonal matrix at the left, $\mathbb{D}_{{\boldsymbol{l}}}$, sets the relative weight of the masking for each sensor, moderating low frequencies again. The vectors ${\boldsymbol{r}}$ and ${\boldsymbol{l}}$ were tuned *ad-hoc* in [@Martinez19] to get reasonable contrast response curves, both for low and high frequency tests.
Hoever, *what is the explanation for this specific structure of the kernel matrix in Eq. \[new\_kernel\_eq\]?* *And where do these two high-pass diagonal matrices come from?*
### Adaptive nature of kernel in Divisive Normalization
Previous physiological experiments on cats and macaques demonstrated that the effect of the surround on each cell does not come equally from all peripheral regions, showing up the existence of a spatially asymmetric surround [@Nelson1985; @Deangelis94; @Walker99; @Cavanaugh02a; @Cavanaugh02b]. As shown in Fig. \[fig\_cohen\] (top-left), the experimental cell response is suppressed and this attenuation due to the surround is greater when the grating patches are *iso-oriented* and at the ends of the receptive field (as defined by the axis of preferred orientation) [@Cavanaugh02b].
In the Divisive Normalization context, this asymmetry could be explained with non-isotropic interaction kernels. Depending on the texture of the surround, the interaction strength in certain direction may change. This would change the denominator, and hence the gain in the response.
Coen-Cagli et al. [@Coen-Cagli12] proposed a specific statistical model to account for these contextual dependencies. This model includes grouping and segmentation of neighboring oriented features, and leads to a flexible generalization of the Divisive Normalization. Representative center-surround configurations considered in the statistical model are shown in Fig. \[fig\_cohen\] (bottom-left). A surround orientation can be either *co-assigned* with the center group or *not co-assigned*. In the first case, the model assumes dependence between center and surround, and includes them both in the normalization pool for the center. In the second case, the model assumes center-surround independence, and does not include the surround in the normalization pool. Fig. \[fig\_cohen\] (bottom-right) shows the covariance matrices learned from natural images between the variables associated with center and surround in the proposed statistical model. As expected, the variances of the center and its *co-linear* neighbors, and also the covariance between them, are larger, due to the predominance of *co-linear* structures in natural images. The cell response that is computationally obtained assuming their statistical model is shown in Fig. \[fig\_cohen\] (top-right), together with the probability that center and surround receptive fields are *co-assigned* to the same normalization pool, and contribute then to the divisive normalization of the model response. Note that the higher the probability of *co-assignment* between the center and surround, the higher the suppression (or the lower the signal) in the cell response.
This flexible (or adaptive) Divisive Normalization model based on image statistics [@Coen-Cagli12] allows to explain the experimental asymmetry in the center-surround modulation [@Cavanaugh02b]. However, no direct mechanistic approach has been proposed yet to describe how this adaptation in the Divisive Normalization kernel may be implemented.
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![[]{data-label="fig_cohen"}](./Figure3.jpg "fig:"){width="130.00000%"}
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Results: analytical equivalence between models
==============================================
The kernels that describe the relation between sensors in the Divisive Normalization and the Wilson-Cowan models, ${\boldsymbol{H}}$ and ${\boldsymbol{W}}$, have similar qualitative roles: both moderate the response, either by division or subtraction, taking into account the activity of the neighbor sensors.
In this section, we derive the equivalence between both models assuming that the Divisive Normalization behavior corresponds to the steady state solution of the Wilson-Cowan dynamics. This leads to an interesting analytical relation between both kernels, ${\boldsymbol{H}}$ and ${\boldsymbol{W}}$.
Under the steady state assumption, it is possible to identify the different terms in the decoding equations in both cases (Eq. \[invDN\] and Eq. \[invWC\]). However, just to get a simpler analytical relation between both kernels, we make one extra assumption on each model. Next section will confirm that these extra assumptions are valid in practice.
First, in the Divisive Normalization model, the identification may be simpler by taking the series expansion of the inverse in Eq. \[invDN\]. This expansion was used in [@Malo06a] because it clarifies the condition for invertibility: $$\left( I - \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}} \right)^{-1}
= I + \sum_{n=1}^{\infty} \left( \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}} \right)^n
\nonumber$$
The inverse exist if the eigenvalues of $\mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}}$ are smaller than one so that the series converges. In fact, if the eigenvalues are small, the inverse can be well approximated by a small number of terms in the Maclaurin series. Taking into account this approximation, Eq. \[invDN\] may be written as:
$$\begin{aligned}
{\boldsymbol{e}} & = & \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot {\boldsymbol{x}} + \left( \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}} \right) \cdot \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot {\boldsymbol{x}} + \nonumber \\
& & \kern 1.7cm + \left( \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}} \right)^2 \cdot \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot {\boldsymbol{x}} + \nonumber \\
& & \kern 1.7cm + \left( \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}} \right)^3 \cdot \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot {\boldsymbol{x}} + \cdots \nonumber \\[0.4cm]
{\boldsymbol{e}} &\approx& \left( \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} + \mathbb{D}^{-1}_{{\boldsymbol{k}}} \cdot \mathbb{D}_{{\boldsymbol{x}}} \cdot {\boldsymbol{H}} \cdot \mathbb{D}_{{\boldsymbol{b}}} \cdot \mathbb{D}^{-1}_{{\boldsymbol{k}}} \right) \cdot {\boldsymbol{x}}
\label{approx_invDN}\end{aligned}$$
Second, in the case of the Wilson-Cowan model, in Eq. \[invWC\] we also approximate the saturation function $f({\boldsymbol{x}})$ by means of a Maclaurin series using its first derivative (green function in Fig. \[f\_x\], left): $f({\boldsymbol{x}}) \approx \frac{df}{dx} \cdot {\boldsymbol{x}}$. As a result, $f({\boldsymbol{x}}) \approx \mathbb{D}_{\frac{df}{dx}} \cdot {\boldsymbol{x}}$, Eq. \[invWC\] may be written as: $${\boldsymbol{e}} \approx \left( \mathbb{D}_{{\boldsymbol{\alpha}}} + {\boldsymbol{W}} \cdot \mathbb{D}_{\frac{df}{dx}} \right) \cdot {\boldsymbol{x}}
\label{invWC2}$$
{height="5cm"} \[f\_x\]
Now, the identification between the approximated versions of the decoding equations, Eq. \[approx\_invDN\] and Eq. \[invWC2\], is straightforward. As a result, we get the following relations between the parameters of both models: $$\begin{aligned}
{\boldsymbol{b}} &=& {\boldsymbol{k}} \odot {\boldsymbol{\alpha}} \nonumber \\
{\boldsymbol{H}} &=& \mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{x}}}\right)} \cdot {\boldsymbol{W}} \cdot \mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{b}}} \odot \frac{df}{dx}\right)}
\label{relation_W_H}\end{aligned}$$ where the symbol $\odot$ denotes the element-wise (or Hadamard) product.
Therefore, both models are equivalent if the Divisive Normalization kernel *inherits* the structure from the Wilson-Cowan kernel modified by these pre- and post- diagonal matrices, $\mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{x}}}\right)}$ and $\mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{b}}} \odot \frac{df}{dx}\right)}$, respectively.
Note that the resulting kernel ${\boldsymbol{H}}$ in Eq. \[relation\_W\_H\] has exactly the same structure as the one in Eq. \[new\_kernel\_eq\]. This theoretical result suggests an explanation for the structure that had to be introduced *ad-hoc* in [@Martinez19] just to reproduce contrast masking. Note that interaction in the Wilson-Cowan case may be understood as wiring between sensors tuned to similar features, so unit-norm Gaussian, ${\boldsymbol{W}} = {\boldsymbol{H}}^{{\boldsymbol{ws}}}$, is a reasonable choice [@Wilson73; @Faugueras09]. Note also that the weights before and after the Gaussian interaction (the diagonal matrices) are signal dependent, which implies that the interaction kernel in Divisive Normalization should be *adaptive*. The one in the left, $\mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{x}}}\right)}$, has a direct dependence on the inverse of the signal, while the one in the right, $\mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{b}}} \odot \frac{df}{dx}\right)}$, depends on the derivative of the saturation function $f({\boldsymbol{x}})$, which also depends on the signal as shown in Fig. \[f\_x\] (right). In the next Section we show that the vectors (Hadamard quotients) ${\boldsymbol{k}}/{\boldsymbol{x}}$ and $\frac{{\boldsymbol{k}}}{{\boldsymbol{b}}} \odot \frac{df}{dx}$ do have the high-pass frequency nature that explains why the low frequencies in ${\boldsymbol{e}}$ had to be attenuated *ad-hoc* by introducing $\mathbb{D}_{{\boldsymbol{l}}}$ and $\mathbb{D}_{{\boldsymbol{r}}}$. We also show that the term of the right, $\mathbb{D}_{\left(\frac{{\boldsymbol{k}}}{{\boldsymbol{b}}} \odot \frac{df}{dx}\right)}$, produces the shape changes needed on the interactions.
It is important to stress that the assumptions made to get the simplified versions of the decoding equations that lead to the analytical relations in Eq. \[relation\_W\_H\], were done only for the sake of simplicity in the final relations obtained. Actually, once these relations were obtained, the simulations in the following sections use the full expressions of the models (i.e. no linearization or truncation is assumed any more in Eqs. \[invDN\] and \[invWC\]). In summary, the relations in Eq. \[relation\_W\_H\] are exact for the simplified versions of the models. Considering the full version of the models, Eq. \[relation\_W\_H\] would be an approximation. However, the discussion below points out the validity of this approximation since plugging these expressions into the full versions of the models also leads to consistent results.
Discussion
==========
In this section we analyze the *mathematical properties* and the *consequences on contrast perception* of the above result: the relation between models that was obtained in Eq. \[relation\_W\_H\].
Regarding the *mathematical properties*, we first prove in Section \[math\_properties\] the consistency of the steady state assumption by showing that: (a) the integration of the Wilson-Cowan equation converges to the Divisive Normalization solution; and (b) the Divisive Normalization solution is a stable node of the Wilson-Cowan system. Convergence and stability results are obtained with sensible parameters for the visual cortex, since they were psychophysically tuned in [@Malo15; @Martinez17; @Martinez19].
Then, in Section \[consequences\] we address different *consequences on contrast perception* using the proposed relation: (a) we analyze the signal-dependent behavior of the theoretically derived kernel, and the benefits of the high-pass behavior to moderate the weight of the low-frequency components; (b) we show that the shape of the interactions between sensors changes depending on the surround; (c) we reproduce the contrast response curves with the proposed signal-dependent kernel; and (d) we discuss the use of the derived kernel in predicting the subjective metric of the image space.
All these results can be reproduced through the code described in the Supplementary Material S1 (section \[code\]).
Mathematical properties {#math_properties}
-----------------------
### Wilson-Cowan converges to the Divisive Normalization solution
The Wilson-Cowan expression (Eq. \[EqWC\]) defines an initial value problem where the response at time zero evolves (or is updated) according to the right hand side of the differential equation. In our case, we assume that the initial value of the output response is just the input ${\boldsymbol{e}}$: $$\begin{aligned}
{\boldsymbol{\dot{x}}} &=& {\boldsymbol{e}} - \mathbb{D}_{{\boldsymbol{\alpha}}} \cdot {\boldsymbol{x}} - {\boldsymbol{W}} \cdot f({\boldsymbol{x}}) \nonumber \\
{\boldsymbol{x}}(0) &=& {\boldsymbol{e}}
\label{initial_solution_Euler}\end{aligned}$$ and we then solve this first degree differential equation, given this initial value, by simple Euler’s integration.
Figure \[fig\_conv\] shows the evolution of the response obtained from this integration, applied to 35 natural images taken from calibrated databases [@VanHateren98; @Laparra12], and using proper values for the auto-attenuation factor ${\boldsymbol{\alpha}}$ and interaction factor ${\boldsymbol{W}}$ compatible with the psychophysical experiments detailed in [@Malo15; @Martinez17; @Martinez19]. As can be seen, the solution of the Wilson-Cowan integration converges to the Divisive Normalization solution because: (1) the difference between both solutions decreases as it is updated (Fig. \[fig\_conv\], left); and (2) this result is the steady state because the update in the solution tends to zero (Fig. \[fig\_conv\], right). The final relative difference between the steady state of the Wilson-Cowan integration and the Divisive Normalization solution is $\frac{|{\boldsymbol{x}}_{\textrm{WC}}-{\boldsymbol{x}}_{\textrm{DN}}|^2}{|{\boldsymbol{x}}_{\textrm{DN}}|^2} = 0.0011 \pm 0.0004$.
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{height="0.56\linewidth"} {height="0.56\linewidth"}
\[-0.0cm\]
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\[fig\_conv\]
### Stability analysis of Divisive Normalization solution
The stability of a dynamical system at a steady state is determined by the Jacobian with regard to perturbations in the response: if the eigenvalues of this Jacobian are all negative, the considered response is a stable node [@Logan15]. Thus, the Jacobian with regard to the output signal of the right hand side of the Wilson-Cowan differential equation (Eq. \[EqWC\]) is: $$J = - (\mathbb{D}_{\alpha} + {\boldsymbol{W}} \cdot \mathbb{D}_{\frac{df}{dx}})
\label{eq_stabil}$$ In this Jacobian one may use values for the Wilson-Cowan parameters derived from experimentally fitted Divisive Normalization (in our case fitted in [@Malo15; @Martinez17; @Martinez19]).
The stability of the system is shown below in two situations: (1) an illustrative reduced-scale simple model for 3-pixel images (fully specified in the Supplementary Material included in Section \[small\_DN\]) that allows full visualization of the vector field of perturbations in the phase space of the system; and (2) the full-scale model for actual images, using parameters fitted in [@Malo15; @Martinez17; @Martinez19].
In the reduced-scale model, perturbations of the response leads to the dynamics shown in the phase space of Fig. \[fig\_stab\]. The vector field induced by the Jacobian implies that any perturbation is sent back to the origin (no-perturbation) point, which is a stable node of the system.
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{height="0.65\linewidth"} {height="0.65\linewidth"}
\[-0.0cm\]
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\[fig\_stab\]
This behavior is consistent for any response ${\boldsymbol{x}}$, as can be seen by plotting the equivalent results using the signal dependent Jacobian (the one that depends on the Divisive Normalization parameters in the right hand side of Eq. \[eq\_stabil\]). Thus, Fig. \[fig\_stab2\] shows the dynamics around a range of responses in the non-zero frequency plane for a constant value of the sensor tuned to brightness (zero-frequency). In every case, eventual oscillations are attenuated and the response returns to the Divisive Normalization solution highlighted in red. The behavior at other brightness levels is equivalent, and perturbations not restricted to the non-zero frequency plane are attenuated as well.
{height="17cm"} \[fig\_stab2\]
More interestingly, a meaningful full-scale cortical model fitted using visual psychophysics [@Malo15; @Martinez17; @Martinez19] also leads to the same result: the Divisive Normalization solution is a stable node of the equivalent dynamical Wilson-Cowan network. In this full-scale case the phase space cannot be visualized as above. Therefore, we rely on the analysis of the eigenvalues of the Jacobian computed using Eq. \[eq\_stabil\] with the experimental parameters for Divisive Normalization [@Malo15; @Martinez17; @Martinez19]. In this experiment we evaluated the eigenvalues with responses to $10^5$ patches of $40\times40$ images taken from the calibrated Van Hateren natural image database [@VanHateren98]. Figure \[fig\_stab4\] shows that *all* the eigenvalues are negative, indicating that the Divisive Normalization solution is a stable node of the dynamical system, and that this behavior is consistent (small variance) for the range of responses elicited by natural images.
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{height="0.50\linewidth"} {height="0.50\linewidth"}
\[-0.0cm\]
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\[fig\_stab4\]
The conclusion of this analysis is that realistic Divisive Normalization solutions are stable nodes of the equivalent Wilson-Cowan systems. This conclusion confirms the assumption under the proposed relation: Divisive Normalization as a steady state of the Wilson-Cowan dynamics.
Consequences on contrast perception {#consequences}
-----------------------------------
The proposed relation implies that the Divisive Normalization kernel *inherits* the structure of the Wilson-Cowan interaction matrix (typically Gaussian [@Wilson73; @Faugueras09]), modified by some specific signal dependent diagonal matrices, as seen in Eq. \[relation\_W\_H\], and allows to explain a range of contrast perception phenomena.
First, regarding the structure of the kernel, we show that our prediction is consistent with previously required modifications of the Gaussian kernel in Divisive Normalization to reproduce contrast perception [@Martinez19]. Second, we show that this non-Gaussian kernel modifies its shape following a signal-dependent behavior, thus explaining the experiments reported in [@Cavanaugh02b; @Coen12]. Third, we use the predicted signal-dependent kernel to simulate contrast response curves consistent with [@Foley94; @Watson97]. And finally, the proposed relation is also applied to reproduce the experimental visibility of spatial patterns in more general contexts as subjective image quality assessment [@LIVE6; @ponomarenko08].
### Structure of the kernel in Divisive Normalization
In this section we analyze the effect of the signal in the Divisive Normalization kernel according to Eq. \[relation\_W\_H\], by using an illustrative stimulus and psychophysically sensible values for the parameters ${\boldsymbol{k}}$, ${\boldsymbol{b}}$, and ${\boldsymbol{H}}^{{\boldsymbol{ws}}}$ (or ${\boldsymbol{W}}$). Specifically, we compare the empirical filters $\mathbb{D}_{{\boldsymbol{l}}}$ and $\mathbb{D}_{{\boldsymbol{r}}}$, that had to be introduced *ad-hoc* in [@Martinez19], with the theoretical ones obtained through Eq. \[relation\_W\_H\].
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![[]{data-label="explanation"}](./Figure9.jpg "fig:"){width="130.00000%"}
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Before going into the details of the kernel, lets have a look at the response ${\boldsymbol{x}}$ for an illustrative input image. Fig. \[explanation\] shows the corresponding responses of linear and nonlinear V1-like sensors based on steerable wavelets. Typical responses for natural images are low-pass signals (see the vectors at the right of the corresponding wavelet-like representations). The response in each subband is an adaptive (context dependent) nonlinear transduction. Each point at the black-to-red plots at the bottom represents the input-output relation for each neuron in the subbands of the different scales (from coarse to fine). As each neuron has a different neighborhood, there is no simple input-output transduction function, but a scatter plot representing different instances of an adaptive transduction.
The considered image is designed to lead to specific excitations in certain sensors (subbands and locations in the wavelet domain). Note, for instance, the high and low frequency synthetic patterns (12 and 6 cycles per degree, cpd, horizontal and vertical, respectively) in the image regions highlighted with the red and blue dots. In the wavelet representations we also highlighted some specific sensors in red and blue corresponding to the same spatial locations and the horizontal subband tuned to 12 cpd. Given the tuning properties of the neurons highlighted in red and blue, it makes sense that wavelet sensor in red has bigger response than the sensor in blue.
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![[]{data-label="Fig_filters"}](./Figure10.jpg "fig:"){width="95.00000%"}
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With this knowledge of the signal in mind (low-pass trend in ${\boldsymbol{x}}$ shown in Fig. \[explanation\]), and considering that the derivative $\frac{df}{dx}$ decreases with the signal (see Fig. \[f\_x\]), so it will be bigger for high frequencies where the signal is smaller, and that the vector ${\boldsymbol{b}}$ is bigger for low-frequencies [@Martinez19], we can imagine the high-pass nature of the vectors that depend on $\frac{1}{{\boldsymbol{x}}}$ and $\frac{df}{dx}$ included in the diagonal matrices that appear at the left and right sides of the kernel ${\boldsymbol{W}}$ in Eq. \[relation\_W\_H\].
Fig. \[Fig\_filters\] compares the empirical *left* and *right* vectors, ${\boldsymbol{l}}$ and ${\boldsymbol{r}}$ that were adjusted *ad-hoc* to reproduce contrast curves in [@Martinez19], with those based on the proposed relation with the Wilson-Cowan model (vectors based on $\frac{1}{{\boldsymbol{x}}}$ and $\frac{df}{{\boldsymbol{dx}}}$). Interestingly, both empirical and theoretical filters show similar high-pass nature and coincide in order of magnitude.
Consistency of the structure of the empirical and theoretical interaction matrices (Eq. \[new\_kernel\_eq\] and Eq. \[relation\_W\_H\]), and coincidence of empirical and theoretical filters (Fig. \[Fig\_filters\]) suggests that the proposed theory explains the modifications that had to be introduced in classical unit-norm kernels in Divisive Normalization to explain contrast response.
### Shape adaptation of the kernel depending on the signal
Once we have shown the high-pass nature of the vectors $\frac{{\boldsymbol{k}}}{{\boldsymbol{x}}}$ and $\frac{{\boldsymbol{k}}}{{\boldsymbol{b}}}\odot \frac{df}{dx}$, lets see in more detail the signal-dependent adaptivity of the kernel. In order to do so, lets consider the interaction neighborhood of two particular sensors in the wavelet representation: specifically, the sensors highlighted in red and blue in Fig. \[explanation\].
Fig. \[kernels\] compares different versions of the two individual neighborhoods displayed in the same wavelet representation: *left* the unit-norm Gaussian kernels, ${\boldsymbol{H}}^{ws}$, and *right* the empirical kernel modulated by *ad-hoc* pre- and post-filters, Eq. \[new\_kernel\_eq\]. In these diagrams lighter gray in each $j$-th sensor corresponds to bigger interaction with the considered $i$-th sensor (highlighted in color). The gray values are normalized to the global maximum in each case. Each subband displays two Gaussians. Obviously, each Gaussian corresponds to only one of the sensors (the one highlighted in red or in blue, depending on the spatial location of the Gaussian). We used a single wavelet diagram since the two neighborhoods do not overlap and there is no possible confusion between them.
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![**Gaussian and empirical interaction kernels for the sensors highlighted in red and light blue in Fig. \[explanation\]**. Gaussian kernel (left) with overestimated contribution of low-frequency subbands (highlighted in orange). Hand-crafted kernel (right) to reduce the influence of low-frequencies subbands (highlighted in green).](./Figure11.jpg "fig:"){width="120.00000%"}
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. \[kernels\]
In the base-line unit-norm Gaussian case, ${\boldsymbol{H}}^{ws}$, a unit-volume Gaussian in space is defined centered in the spatial location preferred by the $i$-th sensor. Then, the corresponding Gaussians at every subband are weighted by a factor that decays as a Gaussian over scale and orientation from the maximum, centered at the subband of the $i$-th sensor. The *problem* with the unit-norm Gaussian in every scale is that the reduced set of sensors for low-frequency scales lead to higher values of the kernel so that it has the required volume. In that situation the impact of activity in low-frequency subbands is substantially higher. This fact, combined with the low-pass trend of wavelet signals, implies a strong bias of the response and ruins the contrast masking curves. This *problem* is represented by the relatively high values of the neighborhoods in the low-frequency subbands highlighted in orange.
This overemphasis in the low-frequency scales was corrected *ad-hoc* using right- and left- multiplication in Eq. \[new\_kernel\_eq\] by hand-crafted high-pass filters. The effect of these filters is to reduce the values for the Gaussian neighborhoods at the low-frequency scales, as seen in the empirical kernel at Fig. \[kernels\]-right. The positive effect of the high-pass filters is reducing the impact of the neighborhoods at low-frequency subbands (highlighted in green).
In both cases (the classical ${\boldsymbol{H}}^{{\boldsymbol{ws}}}$, and the hand-crafted ${\boldsymbol{H}} = \mathbb{D}_{{\boldsymbol{l}}} \cdot {\boldsymbol{H}}^{{\boldsymbol{ws}}} \cdot \mathbb{D}_{{\boldsymbol{r}}}$) the size of the interaction neighborhood (the interaction length) is signal independent. Note that the neighborhoods for both sensors (red and blue) are the same, regardless of the different stimulation that can be seen in Fig. \[explanation\].
--------------------------------------------------------------------------
![[]{data-label="lo_nuestro"}](./kernels.JPG "fig:"){width="145.00000%"}
--------------------------------------------------------------------------
Fig. \[lo\_nuestro\] shows the kernels obtained from Eq. \[relation\_W\_H\]. The top row shows the three components of ${\boldsymbol{H}}$: (1) the top-left term which is proportional to $\frac{1}{{\boldsymbol{x}}}$, (2) the top-center term which consists of the invariant Gaussian neighborhoods ${\boldsymbol{W}}$, and (3) the top-right term which is proportional to $\frac{df}{dx}$. And the bottom row shows the result of the product of the three terms: the bottom-left shows the global result and the bottom-right zooms on the high-frequency horizontal subband.
These three terms have the following positive results: (1) the product by the high-pass terms moderates the effect of the unit-norm Gaussian at low-frequency subbands as in the empirical kernel tuned in [@Martinez19] ahown in Fig. \[kernels\]-right, (2) the term proportional to $\frac{1}{{\boldsymbol{x}}}$ scales the interaction length according to the signal, and (3) the shape of the kernel depends on the signal because $H_{ij}$ is modulated by $(\frac{df}{dx})_j$, and this implies that when the surround is aligned with the sensor, the kernel elongates in that direction (as the probability of co-assignment in Fig. \[fig\_cohen\]). This will lead to smaller responses when the sensor is flanked by co-linear stimuli (as in Cavanaugh results [@Cavanaugh02b].
In summary, deriving the Divisive Normalization as the steady state of a Wilson-Cowan system with Gaussian unit-norm wiring explains two experimental facts: (1) the high-pass filters that had to be added to the structure of the kernel in Divisive Normalization to reproduce contrast responses [@Martinez19], and (2) the adaptive asymmetry of the kernel that changes its shape depending on the background texture [@Nelson1985; @Deangelis94; @Walker99; @Cavanaugh02a; @Cavanaugh02b].
### Contrast response curves from the Wilson-Cowan model
The above results suggest that the Wilson-Cowan model could successfully reproduce contrast response curves and masking, which have not yet been addressed through this model. Here we explicitly check this hypothesis.
We can use the proposed relation, Eq. \[relation\_W\_H\], to plug successful parameters of Divisive Normalization fitted for contrast perception into the equivalent Wilson-Cowan model. We can avoid the integration of the differential equation using the knowledge of the steady state. The only problem to compute the response through the steady state solution is that the kernel of the Divisive Normalization depends on the (still unknown) response.
In this case we compute a first guess of the response, $\hat{{\boldsymbol{x}}}$, using the fixed hand-crafted kernel tuned in [@Martinez19], and then, this first guess is used to compute the proposed signal-dependent kernel, which in turn is used to compute the actual response, ${\boldsymbol{x}}$. We consistently got compatible responses (e.g. small relative differences of $|x-\hat{x}|/|x| = 0.0022 \pm 0.0001$ over the TID dateset of natural images), so we used this method to compute the responses to synthetic patterns of controlled spatial frequency and contrast.
Fig. \[contrast\_curves\] shows the results obtained for the response curves corresponding to neurons that are tuned to low and high spatial frequency tests, as a function of the contrast of these tests located on top of backgrounds of different contrast, spatial frequency, and orientation. In each case we considered for the background four different contrasts (represented by the different line styles). The results in this figure display the expected qualitative properties of contrast perception:
#### Frequency selectivity. {#frequency-selectivity. .unnumbered}
The magnitude of the response depends on the frequency of the test: responses for the low-frequency test are bigger than the responses for the high-frequency test, as expected from the Contrast Sensitivity Function [@Campbell68].
#### Saturation. {#saturation. .unnumbered}
The responses increase with the contrast of the test, but this increase is non-linear and the responses decrease with the contrast of the background [@Legge80; @Legge81].
#### Cross-masking. {#cross-masking. .unnumbered}
The decrease depends on the spatio-frequency similarities between the test and the background. Note that the low-frequency test is more attenuated by the low-frequency background of the same orientation than by the high-frequency background of orthogonal orientation. Similarly, the high-frequency test is more affected by the high-frequency background of the same orientation [@Foley94; @Watson97].
------------------------------------------------------------------------------------------------
![[]{data-label="contrast_curves"}](./contrast_response_curves.JPG "fig:"){width="120.00000%"}
------------------------------------------------------------------------------------------------
### Metric in the image space from the Wilson-Cowan model
As a result of the consistency of the derived relation between models, Eq. \[relation\_W\_H\], the Wilson-Cowan model may also be used to predict subjective image distortion scores. In this section we explicitly check the performance of the Wilson-Cowan response to compute the visibility of distortions from neural differences following the same approach detailed in the previous section regarding the computation of the signal-dependent kernel and its use to obtain the steady state.
The TID database [@TID2008; @TID2013], which contains natural images modified with several kinds of distortions (of different nature and with different intensity levels), was used for the purpose of this section. Responses with a fixed interaction kernel (the conventional Divisive Normalization approach) and with the adaptive (Wilson-Cowan model) kernel, were correlated with the subjective image distortion mean opinion scores (MOS) experimentally obtained for all distorted images. Figure \[distortion\] shows the corresponding scatter plots: (left) subjective image distortion MOS compared to the predictions obtained by Divisive Normalization using fixed kernel; (right) subjective image distortion MOS compared to the predictions obtained through the equivalent adaptive Wilson-Cowan kernel. Note the high values obtained for the Pearson’s correlation coefficients in both cases: $0.815$ for the Divisive Normalization approach and $0.816$ for the Wilson-Cowan model, and the close similarities between them, proving thus the consistency of the proposed relation between models.
-----------------------------------------------------------------------------------------------
![[]{data-label="distortion"}](./correlation_TID.JPG "fig:"){width="120.00000%" height="8cm"}
-----------------------------------------------------------------------------------------------
Concluding remarks
==================
In this paper we derived an analytical relation between two well-known models of neural interaction: the Wilson-Cowan model [@Wilson72; @Wilson73] and the Divisive Normalization [@Carandini94; @Carandini12]. Specifically, assuming that the Divisive Normalization is the steady state solution of the Wilson-Cowan dynamic differential equation, the Divisive Normalization interaction kernel may be derived from the Wilson-Cowan kernel weighted by two signal-dependent contributions.
We showed the mathematical consistency of the proposed relation by showing that the integration of the Wilson-Cowan equation converges to the Divisive Normalization solution, and proving that the Divisive Normalization solution is a stable node of the Wilson-Cowan system.
Moreover, the derived relation has the following implications in contrast perception: (a) the specific structure obtained for the interaction kernel of Divisive Normalization explains the need of high-pass filters for unit-norm Gaussian interactions to describe contrast masking found in [@Martinez19]; (b) the signal-dependent kernel predicts elongations of the interaction neighborhood in backgrounds aligned with the sensor, thus providing a mechanistic explanation to the adaptation to background patterns found in [@Cavanaugh02a; @Cavanaugh02b]; and (c) low-level Wilson-Cowan dynamics may also explain behavioral aspects that have been classically explained through Divisive Normalization, such as contrast response curves [@Foley94; @Watson97], or image distortion metrics [@Laparra10a; @Berardino17].
Finally, the equivalence between models proposed here opens the possibility to analyze Divisive Normalization from new perspectives, following methods that have been developed for Wilson-Cowan systems [@Sejnowski09].
### Acknowledgments {#acknowledgments .unnumbered}
This work was partially funded by the Spanish Government and EU-FEDER fund through the MINECO grants TIN2015-71537-P and DPI2017-89867-C2-2-R; and by the European Union’s Horizon 2020 research and innovation programme under grant agreement number 761544 (project HDR4EU) and under grant agreement number 780470 (project SAUCE).
Supplementary Material S1: Matlab code {#code}
======================================
This appendix lists the main Matlab routines associated to each experiment described in the main text. All the material is in the file `DivNorm_from_Wilson_Cowan.zip`. Detailed parameters of the models and the instructions on how to use these routines are given in the corresponding `*.m` files.
- **The retina-cortex model:** The file [`BioMultiLayer_L_NL_color.zip`]{} contains the [`Matlab`]{} toolbox that implements the 4-layer network based on Divisive Normalization for spectral or color images considered in Fig. 1 of the main text. This toolbox includes the model, its inverse and Jacobians, and a distortion metric based on the model. The demo function [`demo_deep_DN_iso_color_spectral.m`]{} shows how to choose the parameters of the model, how to apply it to spectral images and images in opponent color representations, and how to compute the responses, the Jacobians and the inverse. The demo function [`demo_metric_deep_DN_iso_color.m`]{} shows how to represent conventional digital images in the appropriate opponent color representation.
- **Experiments on convergence:** The toolbox includes the functions and that compute and invert the dimension-wise saturating response of the Wilson-Cowan model depicted in Fig. \[f\_x\] of the main text. These functions also compute the corresponding derivative with regard to the stimuli. These functions are applied together with the large-scale interaction kernel for the Wilson-Cowan model to check the convergence of the system in . That script applies Euler integration and shows the convergence of the dynamic solution to the equivalent Divisive Normalization solution. The dynamic response is also checked in , where a small-scale Wilson-Cowan network for 3-pixel images (fully described in the Supplementary Material S2, with parameters computed in ), is excited with 10000 stimuli and the corresponding responses are computed using the function .
- **Experiments on stability:** The stability of the dynamic Wilson-Cowan system is studied through the eigen-decomposition of the Jacobian that controls the amplification of the perturbations in the small-scale example (), which includes visualizations of the phase diagram; and in the full scale example ().
- **Signal-dependent kernel:** The script generates an illustrative image made of high contrast patterns with selected frequencies to stimulate specific subbands of the models. Then, it computes the responses to such stimulus and the corresponding signal dependent-filters according to the relations derived in the main text, Eq. \[relation\_W\_H\]. These theoretical filters are compared to the empirical filters found in [@Martinez19]. Finally, in environments where the surround is aligned with the wavelet sensors, the shape of the interaction kernel is found to change as in [@Cavanaugh02b].
- **Contrast response curves:** The script generates a series of noisy Gabor patterns of controlled frequency and contrast displayed on top of noisy sinusoids of different frequencies, orientations and contrasts. it computes the visibility of these patterns seen on top of the backgrounds by applying the Divisive Normalization model with the signal-dependent lernel derived from the Wilson-Cowan model. The visibility was computed from the response of the neurons tuned to the tests.
- **Image distortion metric:** The series of scripts compute the Divisive Normalization response with the signal-dependent kernel derived from the Wilson-Cowan model for the original and distorted images of the TID database (previously expressed in the appropriate ATD color space). Then, the Euclidean distance is applied to compute the visibility of the distortions.
The distances are computed by applying that computes the responses by calling .
Supplementary Material S2:\
Small-scale Divisive Normalization {#small_DN}
==================================
#### Overview. {#overview. .unnumbered}
The reduced-scale model consist of two *linear+nonlinear* layers: (1) a linear *radiance-to-luminance* transform using a standard Spectral Sensitivity Function, $V_\lambda$, in the spectral integration [@Stiles82], followed by a simple exponential for the *luminance-to-brightness* nonliniearity applied pixel-wise in the spatial domain, that simulates the Weber-Fechner response to luminance [@Fairchild13], and (2) a *linear+nonlinear* layer in which the linear transform is a discrete cosine transform (a orthonormal rotation) followed by a low-pass weighting function that simulate frequency-tuned sensors and the Contrast Sensitivity Function (CSF) [@Campbell68]. Then, the outputs of the frequency sensors undergo a nonlinear interaction that may be a Divisive Normalization [@Carandini94; @Carandini12; @Martinez17; @Martinez19], or its equivalent Wilson-Cowan network, with parameters computed according to Eq. \[relation\_W\_H\].
$$\xymatrixcolsep{2pc}
\xymatrix{ {\boldsymbol{x}}^0 \ar@/^1pc/[r]^{\scalebox{1.00}{$\mathcal{L}^{(1)}$}} & {\boldsymbol{r}}^1 \ar@/^1pc/[r]^{\scalebox{1.00}{$\mathcal{N}^{(1)}$}} & {\boldsymbol{x}}^1 \ar@/^1pc/[r]^{\scalebox{1.00}{$\mathcal{L}^{(2)}$}} & {\boldsymbol{r}}^2 \ar@/^1pc/[r]^{\scalebox{1.00}{$\mathcal{N}^{(2)}$}} & {\boldsymbol{x}}^2
}
\label{modular}$$
#### Transform. {#transform. .unnumbered}
The actual inputs of our code are the responses of the linear photoreceptors: 3-pixel image vectors with normalized luminance values, i.e. ${\boldsymbol{r}}^1 \in \mathbb{R}^3$. The normalized luminance was computing dividing the absolute luminance in $cd/m^2$ by the value corresponding to the 95% percentile of the luminance, in our case 260 $cd/m^2$.
- The *luminance-to-brightness* transform, $\mathcal{N}^{(1)}$, is just: $${\boldsymbol{x}}^1 = ({\boldsymbol{r}}^1)^\gamma \,\,\,\,\,\,\,\,\,\, \textrm{where} \,\,\,\, \gamma = 0.6$$
- The linear transform of frequency-tuned sensors with CSF gain, $\mathcal{L}^{(2)}$, is: $${\boldsymbol{r}}^2 = G_{\textrm{CSF}} \cdot F \cdot {\boldsymbol{x}}^1 \,\,\,\,\,\,\,\,\,\, \textrm{where}$$ $$\begin{aligned}
F &=& \left(
\begin{array}{ccc}
\sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} & \sqrt{\frac{1}{3}} \\[0.3cm]
\sqrt{\frac{1}{2}} & 0 & - \sqrt{\frac{1}{2}} \\[0.3cm]
-\sqrt{\frac{1}{6}} & \sqrt{\frac{2}{3}} & -\sqrt{\frac{1}{6}} \\
\end{array}
\right)\\[0.3cm]
G_{\textrm{CSF}} &=& \left(
\begin{array}{ccc}
\,\, 1 \,\,\,\, & \,\,\,\, 0 \,\,\,\,& \,\,\,\, 0 \,\, \\
\,\, 0 \,\,\,\, & \,\,\,\, 0.5 \,\,\,\,& \,\,\,\, 0 \,\, \\
\,\, 0 \,\,\,\, & \,\,\,\, 0 \,\,\,\,& \,\,\,\, 0.3 \,\,
\end{array}
\right)\end{aligned}$$
- The *Divisive Normalization* of the frequency-tuned sensors, $\mathcal{N}^{(2)}_{\textrm{DN}}$, is: $${\boldsymbol{x}}^2 = sign({\boldsymbol{x}}^2) \odot \mathbb{D}_{{\boldsymbol{k}}} \cdot \mathbb{D}^{-1}_{\left( {\boldsymbol{b}} + {\boldsymbol{H}} \cdot |{\boldsymbol{r}}^2|^\gamma \right)} \cdot |{\boldsymbol{r}}^2|^\gamma \,\,\,\,\,\, \textrm{where} \,\,\,\,\,\, \gamma = 0.7, \,\,\,\,\,\, \textrm{and,}
\label{DN_B2}$$
[ ]{}
and the vector of semisaturations, ${\boldsymbol{b}}$, is: $${\boldsymbol{b}} = \left(
\begin{array}{c}
0.08 \\ 0.03 \\ 0.01 \\ \end{array}
\right)$$
- The equivalent Wilson-Cowan interaction, $\mathcal{N}^{(2)}_{\textrm{WC}}$, is defined by the differential equation \[EqWC\], where the auto-attenuation, ${\boldsymbol{\alpha}}$, and the interaction matrix, ${\boldsymbol{W}}$, are: $$\begin{aligned}
{\boldsymbol{\alpha}} &=& \left(
\begin{array}{c}
0.41 \\ 1.10 \\
1.30 \\
\end{array}
\right)\\
W &=& \left(
\begin{array}{ccc}
\, 0.93 \,\,\, & \,\,\, 0.06 \,\,\,& \,\,\, 0.01 \, \\
\, 0.04 \,\,\, & \,\,\, 0.93 \,\,\,& \,\,\, 0.05 \, \\
\, 0 \,\,\, & \,\,\, 0.02 \,\,\,& \,\,\, 0.98 \,
\end{array}
\right)\end{aligned}$$ and the saturation function is: $$f({\boldsymbol{x}}) = c \, {\boldsymbol{x}}^\gamma \,\,\,\,\,\, \textrm{where} \,\,\,\,\,\, \gamma = 0.4, \,\,\,\,\,\, \textrm{and,}$$ the scaling constant is, $c = \hat{{\boldsymbol{x}}}^{1-\gamma}$, and $\hat{{\boldsymbol{x}}}$ is the average response over natural images (for the Divisive Normalization transform): $$\hat{{\boldsymbol{x}}} = \left(
\begin{array}{c}
1.12 \\ 0.02 \\
0.01 \\
\end{array}
\right)$$ This exponent is also used for the definition of energy in Wilson-Cowan, ${\boldsymbol{e}} = |{\boldsymbol{r}}|^\gamma$.
Note that the interaction neighborhoods have unit volume, $\sum_j W_{ij} = 1 \,\,\, \forall j$, as suggested in [@Watson97], and then, the Divisive Normalization kernel is given by the product of this unit-volume neighborhood and two left and right filters in the diagonal matrices, $\mathbb{D}_l$ and $\mathbb{D}_r$ [@Martinez19]. The values for the semisaturation, ${\boldsymbol{b}}$, and the diagonal matrices $\mathbb{D}_l$ and $\mathbb{D}_r$ were inspired by the contrast response results in [@Martinez19]: we set the semisaturation according to the average response of natural images (low-pass in nature), and we initialized the left and right filters to high-pass. However, afterwards, in order to make $\mathcal{N}_{\textrm{DN}}$ and $\mathcal{N}_{\textrm{WC}}$ consistent, we applied the Divisive Normalization over natural images and we iteratively updated the values of the right and left filters according to Eq. \[relation\_W\_H\]. In the end, we arrived to the values in the above expressions (where the filter at the left is high-pass, but the filter at the right is not). Note that the attenuation in Wilson-Cowan is computed using Eq. \[relation\_W\_H\].
#### Jacobian. {#jacobian. .unnumbered}
The information theoretic computations strongly depend on how the system (locally) deforms the signal representation (e.g. Eq. \[analitic\]). This is described by the Jacobian of the transform with regard to the signal, $\nabla_{{\boldsymbol{r}}^1} S = \nabla_{{\boldsymbol{r}}^2} \mathcal{N}^{(2)} \cdot \nabla_{{\boldsymbol{x}}^1} \mathcal{L}^{(2)} \cdot \nabla_{{\boldsymbol{r}}^1} \mathcal{N}^{(1)}$. In this reduced-scale model, this Jacobian (for the Wilson-Cowan case) is: $$\nabla_{{\boldsymbol{r}}^1} S = \gamma_2 \left(\mathbb{D}_{{\boldsymbol{\alpha}}} + {\boldsymbol{W}} \cdot \mathbb{D}_{\frac{df}{dx}}\right)^{-1}
\cdot \left( \mathbb{D}_{\left( \mathbb{D}_{{\boldsymbol{\alpha}}} \cdot {\boldsymbol{x}}^2 + {\boldsymbol{W}} \cdot f({\boldsymbol{x}}^2) \right)}\right)^{1-\frac{1}{\gamma_2}} \cdot G_{\textrm{CSF}} \cdot F \cdot \mathbb{D}_{\gamma_1 ({\boldsymbol{r}}^1)^{\gamma_1-1}}$$
[^1]: This work substantially expands the original report presented at MODVIS 2018: “Appropriate kernels for Divisive Normalization explained by Wilson-Cowan equations”, https://arxiv.org/abs/1804.05964
|
---
author:
- |
Maciej Dunajski\
Department of Applied Mathematics and Theoretical Physics,\
University of Cambridge,\
Wilberforce Road,\
Cambridge CB3 0WA, UK
date: 'November 15, 2004'
title: Oriented straight lines and twistor correspondence
---
Oriented lines in ${\mathbb{R}}^{n+1}$
======================================
Oriented geodesics in ${\mathbb{R}}^{n+1}$ are straight lines. They can be parametrised by choosing a unit vector $\u$ giving a direction, and taking the position vector $\v$ of the point on the geodesic nearest to the chosen origin. A pair of vectors $(\u, \v)$ corresponds to the oriented line ${\v}+t\u$, where $t\in{\mathbb{R}}$. The space of oriented geodesics is then given by \[twistor\_sp\] ={([u]{}, [v]{} )S\^[n]{}\^[n+1]{}, =0 }. For each fixed $\u$ this space restricts to a tangent plane to a unit $n$–sphere, and so ${{\mathbb{T}}}$ is just the tangent bundle $TS^{n}$. We shall call ${{\mathbb{T}}}$ the twistor space. There exists a fix-point-free map $\tau:{{\mathbb{T}}}\longrightarrow{{\mathbb{T}}}$, such that $\tau^2=1$, obtained by reversing the orientation of each geodesic, i.e. $\tau(\u, \v)=(-\u, \v)$.
Let $p$ be a point in ${\mathbb{R}}^{n+1}$ with a position vector ${\bf p}$. The oriented lines through ${p}$ are parametrised by the unit $n$–sphere in $T_p{\mathbb{R}}^{n+1}$, and therefore each $p$ corresponds to a section $L_p:S^{n}\longrightarrow TS^{n}$ given by \[quad\_sec\] u([, [**s(u)**]{}), =[**p**]{}]{}-[([**p**]{}.)u]{}. Note that these sections are preserved by $\tau$. Each section vanishes at two points, where $\pm{\bf p}$ is normal to the sphere.
Laplace sections
----------------
The Euclidean group $E(n+1)$ acts on ${\mathbb{R}}^{n+1}$ and on $TS^{n}$, and $E(n+1)/so(n+1)={\mathbb{R}}^{n+1}$, so the preferred sections are orbits of $so(n+1)$. The ($n+1)$-dimensional space of preferred sections of ${\mathbb{T}}$ corresponding to points in ${\mathbb{R}}^{n+1}$ can be characterised as the eigenspace of the Laplacian on the n-sphere with eigenvalue $n$, with the vector fields being the gradients for the eigenfunctions.
\[laplace\_sec\] The gradients of eigenfunctions of the Laplacian on $S^n$ with eigenvalue $n$ are called the Laplace sections of $TS^n$.
\[twistor\_theorem\] There is a one-to-one correspondence between (Fig. \[Ntwistor\_fig\])
![Twistor Correspondence[]{data-label="Ntwistor_fig"}](Ntwistor.eps){width="8cm" height="5cm"}
$$\begin{aligned}
{\mathbb{R}}^{n+1}&\longleftrightarrow& TS^n\\
\mbox{Points} &\longleftrightarrow& \mbox{Laplace sections} \\
\mbox{Oriented lines}&\longleftrightarrow&\mbox{Points.}\end{aligned}$$
[**Proof.**]{} To complete the proof we need to show that all Laplace sections are of the form (\[quad\_sec\]) in some coordinates. To see it consider a unit sphere $S^n$ isometrically immersed in ${\mathbb{R}}^{n+1}$, and identify a point of $S^n$ with a unit position vector $\u$. Let $h$ be the Riemannian metric on $S^n$ induced by the Euclidean inner product on ${\mathbb{R}}^{n+1}$, and let $X, Y\in T_{\u} S^{n}$. Then $${\nabla'_X}Y=\nabla_X Y+h(X, Y)\u$$ where $\nabla'$ is the flat connection on ${\mathbb{R}}^{n+1}$, and $\nabla$ is the induced connection on the sphere. If $F:{\mathbb{R}}^{n+1}\rightarrow {\mathbb{R}}$, then \[lap\_formula\] \_[\^[n+1]{}]{}(F)=-r\^[-n]{}(r\^n) +r\^[-2]{}\_[S\^n]{}(F|\_[S\^n(r)]{}), where $\triangle_{{\mathbb{R}}^{n+1}}=-\nabla'\cdot \nabla'$ is the Laplacian on ${\mathbb{R}}^{n+1}$, $\triangle_{S^n}$ is the Laplacian on the unit $n$–sphere, and $F|_{S^n(r)}$ is the restriction of $F$ to an $n$–sphere of radius $r$.
For any constant vector ${\bf p}\in {\mathbb{R}}^{n+1}$ consider a function $\chi({\bf u})=\u\cdot{\bf p}$ on $S^n$. We verify that \[gradient\] ’()=-r\^[-1]{}([**p**]{}- ([u]{})[u]{}), \_[\^[n+1]{}]{}()=. Restricting the Laplacian to the unit sphere with $r=1$ we deduce that \[laplacian\] \_[S\^n]{} ()=n. In particular each coordinate function in ${\mathbb{R}}^{n+1}$ regarded as a function on $S^n$ is an eigenfunction of $\triangle_{S^n}$ with an eigenvalue $n$.
The space of solutions to (\[laplacian\]) is $n+1$ dimensional and the bijection between linear functions on ${\mathbb{R}}^{n+1}$ and solutions to (\[laplacian\]) can be established as follows: We have already verified that restrictions of linear functions from ${\mathbb{R}}^{n+1}$ to $S^n$ satisfy (\[laplacian\]). Conversely, let $\chi:S^n\longrightarrow {\mathbb{R}}$ satisfy (\[laplacian\]). Using the representation (\[lap\_formula\]) we deduce that $r\chi$ is a harmonic function homogeneous of degree one on ${\mathbb{R}}^{n+1}$. Let $x_i$ be local coordinates on ${\mathbb{R}}^{n+1}$ with $|x|=r$. Therefore for each $i=1, ..., n+1,$ $\p (r\chi)/\p x_i$ is harmonic and homogeneous of degree $0$, and so it descends to a harmonic function on $S^n$. There are no such functions apart form the constants, so we deduce that $r\chi={\bf x}\cdot{\bf p}$, thus establishing the bijection[^1].
$\Box $
This argument can be extended to show that the space of homogeneous harmonic polynomials on ${\mathbb{R}}^{n+1}$ of degree $k>1$, when restricted to $S^n$ constitute the eigenspace of $\triangle_{S^n}$ with eigenvalue $k(k+n-1)$. The multiplicity of this eigenvalue is (consult [@BGM71] for details) $${n+k\choose k}-{n+k-2 \choose k-2}.$$
Let us list the properties of the Laplace sections which follow from Theorem \[twistor\_theorem\]
- Laplace sections are invariant under a map $\tau:TS^n\longrightarrow TS^n$ given by reversing orientations of lines in ${\mathbb{R}}^{n+1}$.
- Each non-zero Laplace section vanishes at exactly two points on $S^n$. Two distinct non-zero Laplace sections $L_p$ and $L_q$ intersect at two points in $TS^n$. These points correspond to two oriented lines joining $p,q \in {\mathbb{R}}^{n+1}$ $$\u=\pm\frac{{\bf p}-{\bf q}}{|{\bf p}-{\bf q}|},\qquad
{\bf v}=\frac{{\bf p}\cdot{\bf q}-|{\bf q}|^2}{|{\bf p}-{\bf q}|}{\bf
p}+\frac{{\bf p}\cdot{\bf q}-|{\bf p}|^2}{|{\bf p}-{\bf q}|}
{\bf q}.$$ Three (or more) Laplace sections generically don’t meet.
To make the whole construction independent on the choice of the origin in ${\mathbb{R}}^{n+1}$, we should regard the twistor space as an affine vector bundle over $S^n$ with no preferred zero section.
The twistor space ${\mathbb{T}}$ can also be obtained by factoring the correspondence space $S^n\times{\mathbb{R}}^{n+1}$ by the action $({\u}, {\bf v})\longrightarrow ({\u}, t{\u}+ {\bf v})$ for $t\in{\mathbb{R}}$. This action is generated by the geodesic flow $X$, and leads to a double fibration $$\begin{array}{rcccl}
&&S^n\times{\mathbb{R}}^{n+1}&&\\
&p_2\swarrow&&\searrow p_1&\\
&{{\mathbb{R}}^{n+1}}&&{\mathbb{T}}&
\end{array}$$ given by $$p_2({\u}, {\bf v})={\bf v}, \qquad
p_1({\u}, {\bf v})=(\u, {\bf v}-({\bf v}\cdot {\u}){\u}).$$
Let us look at some special cases: (here $\nabla=\p/\p\u$)
- For $n=1$ the unit circle $S^1$ is parametrised by $\phi\in[0,
2\pi]$, ${\bf p}=(x_1, x_2)$, and $${\bf s(u)}\cdot
\nabla=\mbox{Re}\;\Big((x_1+ix_2)\exp{(i\phi)}\frac{{\mathrm{d}}}{{\mathrm{d}}\phi}\Big).$$
- For $n=2$ one easily verifies $${\bf s(u)}\cdot \nabla=\mbox{Re}\;\Big(((x_1+ix_2)+2\ll x_3-\ll^2(x_1-ix_2))
\frac{{\mathrm{d}}}{{\mathrm{d}}\ll}\Big),$$ where $\ll =(u_1+iu_2)/(1-u_3)$ is a holomorphic coordinate on ${\mathbb{CP}}^1=S^2$, and ${\bf p}=(x_1, x_2, x_3)$. The Laplace sections are in this case holomorphic sections of $T{\mathbb{CP}}^1$ preserved by $\tau$. This is the original twistor correspondence established by Hitchin [@H82] in his construction of magnetic monopoles, and recently used in [@GK04] in a study of generalised surfaces in ${{\mathbb{R}}^3}$.
A much older application goes back to Whittaker [@W03]. We shall explain it in a modern language of Hitchin: Given an element of $f\in H^1(T{\mathbb{CP}}^1, {\OO}(-2))$) restrict it to a Laplace section. The general harmonic function on ${\mathbb{R}}^3$ is then given by $$V(x_1, x_2, x_3)=\oint_{\Gamma} f(\ll, (x_1+ix_2)+2\ll x_3-\ll^2(x_1-ix_2)){\mathrm{d}}\ll,$$ where $\Gamma\subset L_p\cong{\mathbb{CP}}^1$ is a real closed contour.
A different integral transform (the X-ray transform introduced by John [@J38]) can be used to construct solutions to ultra-hyperbolic wave equation on the twistor space. This takes a smooth function on ${\mathbb{RP}}^3$ (a compactification of ${\mathbb{R}}^3$) and integrates it over an oriented geodesic. The resulting function is defined on the Grassmannian $\mbox{Gr}_2({\mathbb{R}}^4)$ of two-planes in ${\mathbb{R}}^4$ and satisfies the wave equation for a flat metric in $(++--)$ signature.
Almost complex structure and $TS^6$
===================================
The Riemannian connection $\nabla$ on $S^n$ can be used to define an almost complex structure on $TS^n$ for any $n$. Let $T(TS^n)=V\oplus H$ be the splitting of the tangent space to $TS^n$ into vertical and horizontal components. Define $J_D: TS^n\longrightarrow TS^n$ by $$J_D(X_H)=X_V, \qquad J_D(X_V)=-X_H,$$ where $X_V$ and $X_H$ are the vertical and horizontal parts of a vector on $TS^n$. This structure was studied by Dombrowski [@D62] who showed that the torsion of $J_D$ does not vanish unless both the torsion and the curvature of $\nabla$ are zero. This almost complex structure has nothing to do with the Laplace sections defined in Def. \[laplace\_sec\]. From now on we shall restrict to the case $n=6$ where another (inequivalent) almost complex structure can be defined on ${\mathbb{T}}$. The basic facts about the cross products on ${\mathbb{R}}^7$ will be recalled, and used to show that the Laplace sections are almost complex.
Cross product in ${\mathbb{R}}^7$ and the group $G_2$
-----------------------------------------------------
Let $(x_1, ..., x_7)$ be coordinates on ${\mathbb{R}}^7$, and let ${\mathrm{d}}x_{ijk}$ be a shorthand notation for ${\mathrm{d}}x_i\wedge {\mathrm{d}}x_j\wedge {\mathrm{d}}x_k$. Following Bryant [@B87] we define the exceptional group $G_2$ as $$G_2=\{\rho\in GL(7, {\mathbb{R}})| \rho^*(\phi)=\phi\},$$ where $$\phi={\mathrm{d}}x_{123}+ {\mathrm{d}}x_1\wedge({\mathrm{d}}x_{45}+{\mathrm{d}}x_{67})+{\mathrm{d}}x_2\wedge({\mathrm{d}}x_{46}-{\mathrm{d}}x_{57})
-{\mathrm{d}}x_3\wedge({\mathrm{d}}x_{47}+{\mathrm{d}}x_{56}).$$ It is a compact, connected, and simply connected Lie group of dimension $14$. It also preserves the Euclidean metric $g={\mathrm{d}}x_1^2+...+{\mathrm{d}}x_7^2$, the orientation ${\mathrm{d}}x_{1234567}$, and the four-form $$*\phi= {\mathrm{d}}x_{4567}+ {\mathrm{d}}x_{23}\wedge({\mathrm{d}}x_{45}+ {\mathrm{d}}x_{67})
-{\mathrm{d}}x_{13}\wedge({\mathrm{d}}x_{46} -{\mathrm{d}}x_{57})
-{\mathrm{d}}x_{12}\wedge({\mathrm{d}}x_{47}+{\mathrm{d}}x_{56}).$$ The group $G_2$ acts transitively on a unit sphere $S^6\subset {\mathbb{R}}^7$ with a stabiliser $SU(3)$.
A cross product $\times:{\mathbb{R}}^7\times{\mathbb{R}}^7\longrightarrow {\mathbb{R}}^7$ can be defined by $$g(X\times Y, Z)=\phi(X, Y, Z).$$ This cross product has the same properties as the one induced by the octonion multiplication, which leads to a more standard definition of $G_2$ as the group of automorphisms of the octonions. The induced cross product satisfies the identities analogous to those in three-dimensions \[identities7\] g(XY, XY)=g(X, X)g(Y, Y)-g(X, Y)\^2, X(XY)=g(X, Y)X-g(X, X)Y.
Pseudoholomorphic sections of $TS^6$
------------------------------------
Consider a curve $\gamma(s, t)$ of oriented lines in ${\mathbb{R}}^7$ parametrised by $s\in{\mathbb{R}}$, and given by $${\bf\gamma}(s, t)={\v} (s)+t{\u} (s).$$ A ${\u}$-orthogonal projection of tangent vector $t\dot{\u}+\dot{\v}$ gives rise to a normal Jacobi field \[split\_jacobi\] V=(-(. [u]{})[u]{}+t)|\_[s=0]{} =(, -(. [u]{})[u]{}), =. All vectors tangent to a space of oriented geodesics are of this form.
Let us define a map $\widetilde{J}:T{{\mathbb{T}}}\longrightarrow T{{\mathbb{T}}}$ by \[def\_structure\] V(V)=[u]{}V, VT\_[([, v]{})]{}. From the properties (\[identities7\]) of cross-product $\times$ in ${\mathbb{R}}^7$ it follows that $\widetilde{J}$ is an almost complex structure. Indeed, $$\widetilde{J}^2(V)={\u}\times({\u}\times V)=({\u .}V){\u}-({\u .\u})V=-V.$$ Note that $\tau(\widetilde{J})=-\widetilde{J}$.
This almost complex structure is related to a standard almost complex structure $J$ on $S^6$ defined by $J(\v)=\u\times \v$. To see this consider the restriction of the Euclidean scalar product from ${\mathbb{R}}^7$ to $S^6$. This gives the unique nearly Kähler metric $h$ on $S^6$ compatible with $J$ [@FI55] in a sense that $$h(X, Y)=h(JX, JY), \qquad \nabla_{X}J(X)=0, \qquad \forall X, Y\in TS^6,$$ where $\nabla$ is the Levi–Civita connection of $h$. Let $$T(TS^6)=V\oplus H$$ be the splitting of the tangent space to $TS^6$ into vertical and horizontal components with respect to $\nabla$. The almost complex structure on $TS^6$ defined by taking the standard almost complex structure $J$ on each factor $H$ and $V$ coincides with the almost–complex structure (\[def\_structure\]), because the splitting (\[split\_jacobi\]) coincides with the splitting $T(TS^6)$ induced by $\nabla$ (which is a projection of splitting given by restricting $\nabla'$ to a tangent space). In particular $\widetilde{J}$ is not integrable, since $J$ isn’t.
Let $\rho:S^6\rightarrow S^6$ be an element of $G_2$, and let ${\v}\in T_{\u} S^6$. Then $$\rho_*(J({\v}))=\rho({\u})\times\rho_*({\v})=\widetilde{J}(\rho_*({\v}))\in T_{\rho({\u})}S^6.$$ and we deduce that the Laplace sections $L_p$ of ${{\mathbb{T}}}\longrightarrow S^6$ which correspond to points in ${\mathbb{R}}^7$ are $G_2$–invariant in a sense that $$\rho_*(L_{p}({\u}))=L_{\rho({\bf p})}\rho(\u).$$
Now we want to argue that the Laplace sections are also almost complex in the sense that $$\widetilde{J}\circ (L_p)_*=(L_p)_*\circ J.$$ This follows directly from the geometrical construction because $\u$ is a unit normal to a sphere of geodesics $L_p$ through $p$, and the cross product preserves the almost complex structure on $S^6$ (the almost complex structure on the space of lines is a rotation in ${\mathbb{R}}^7$ through $90$ degrees about the direction of the line which preserves the tangent spaces of $L_p$).
It can also be seen by applying $\widetilde{J}$ to (\[split\_jacobi\]) and performing a direct calculation. This leads to an overdetermined system of equations for $L:S^6\rightarrow TS^6, L({\u})=(u^j, L^j(u))$ $$\Big(\phi_{ljm}u^j\Sm_{pk}+\phi_{kjp}u^j\Sm_{ml}\Big)\frac{\p L^m}{\p
u^p}=0,$$ where $\Sm_{ij}=\delta_{ij}-u_iu_j$. These equations are satisfied by the Laplace sections.
Other twistor correspondences
=============================
In this final section we shall mention two other generalisations of the Hitchin correspondence. The first one (due to Study [@S03] for $n=2$) is more than hundred years old. The second one (due to Murray [@M85]) gives a way of solving the Laplace equation.
[**Study’s correspondence.**]{} The correspondence between oriented lines in ${\mathbb{R}}^{n+1}$ and points in $TS^n$ can be re-expressed in terms of the dual numbers of the form $$a+\tau b$$ where $a, b \in {\mathbb{R}}$, and $\tau^2=0$. Let ${\mathbb{D}}$ denote the space of the dual numbers. Any oriented line in ${\mathbb{R}}^{n+1}$ can be represented by a vector in ${\mathbb{D}}^{n+1}$ $${\bf A}=\u +\tau \v$$ which is of unit length with respect to an Euclidean norm in ${\mathbb{D}}^{n+1}$ induced from ${\mathbb{R}}^{n+1}$. This gives an analogue of Study’s result [@S03]: There is a one to one correspondence between oriented lines in ${\mathbb{R}}^{n+1}$ and points on the dual unit sphere in ${\mathbb{D}}^{n+1}$. Comparing this with (\[twistor\_sp\]), we see that the dual unit sphere in ${\mathbb{D}}^{n+1}$ is equivalent to $TS^n$ with an additional structure (that of dual numbers) selected on the fibres.
Let $\theta$ and $\rho$ be the angle and the distance between two oriented lines represented by ${\bf A}$ and ${\bf B}$. Define a dual angle by $$\Theta=\theta+\tau \rho.$$ Using a formal definition $$\cos{\Theta}=1-\frac{1}{2!}\Theta^2+\frac{1}{4!}\Theta^4 +...=
\cos{\theta}-\tau\sin{\theta},$$ one can verify an attractive looking formula $${\bf A}\cdot{\bf B}=\cos{\Theta},$$ and deduce that group of Euclidean motions in ${\mathbb{R}}^{n+1}$ is equivalent to $O(n+1, {\mathbb{D}})$.
[**Murray’s correspondence.**]{} Let $[z_0, z_1, ..., z_n]$ be homogeneous coordinates on ${\mathbb{CP}}^n$, and let $f=z_0^2+z_1^2+...+z_n^2$ define a section of ${\OO}(2)\longrightarrow {\mathbb{CP}}^n$. This section vanishes on a hyper-quadric $$X=\{f=0, [z]\in {\mathbb{CP}}^n\}\subset{\mathbb{CP}}^n.$$ Murray [@M85] defines a twistor space $Z$ to be a restriction of the total space of ${\OO}(1)\longrightarrow {\mathbb{CP}}^n$ to $X$. This leads to a double fibration $$\begin{array}{rcccl}
&&X\times{\mathbb{R}}^{n+1}&&\\
&m_2\swarrow&&\searrow m_1&\\
&{{\mathbb{R}}^{n+1}}&&Z.&
\end{array}$$ The canonical bundle of $K_X$ of $X$ in $Z$ is ${\OO}(-n+1)$.
Let $\triangle_{{\mathbb{R}}^{n+1}}$ be the Laplacian on ${\mathbb{R}}^{n+1}$. There exists an isomorphism $$T: H^{n-1}(Z, K_X)\longrightarrow Ker\; (\triangle_{{\mathbb{R}}^{n+1}})$$ given by $$T(\om)(z)=\int_{X_z}\om,$$ where $(\om)$ is a $K_X$-valued $(0, n)$ form on $Z$ pulled back to $X\times{\mathbb{R}}^{n+1}$.
The twistor spaces ${\mathbb{T}}$ and $Z$ have the same dimensions, but the connection between Theorem \[twistor\_theorem\] and the Murray correspondence is not clear.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank Michael Eastwood, Nigel Hitchin and Simon Salamon for useful discussions, and Marc Lachièze-Rey for pointing out some errors in an earlier version of this paper.
[jafsdl]{}
Berger, M. Gauduchon, P., & Mazet, E. (1971) [*Le spectre d’une variété riemannienne.*]{} Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York.
Bryant, R. (1987) Metrics with exceptional holonomy. Ann. of Math. (2) 126, no. 3, 525–576.
Dombrowski, P. (1962) On the geometry of the tangent bundle. J. Reine Angew. Math. 210, 73–88.
Fukami, T & Ishihara, S. (1955) Almost Hermitian Structure in $S^6$ Tohoku Math. J. (2), 151–156.
Guilfoyle, B. & Klingenberg, W. (2004) Generalised Surfaces in ${\mathbb{R}}^3$. Math. Proc. of the R.I.A. [**104A**]{}. \[math.DG/0406185\]
Hitchin, N.J. (1982) Monopoles and Geodesics, Commun. Math. Phys. [**83**]{} 579-602.
John, F. (1938) The ultrahyperbolic differential equation with four independent variables. Duke Math. Journ [**4**]{} 300-322.
Murray, M. K. (1985) A twistor correspondence for homogeneous polynomial differential operators. Math. Ann. 272, no. 1, 99–115.
Study, E. (1903) [*Geometrie der Dynamen*]{} B.G. Teubner Verlagsgesellschaft, mbH, Leipzig.
Whittaker E.T. (1903). On the partial differential equations of mathematical physics. Math. Ann. [**70**]{} 333-355.
[^1]: Another (equivalent) characterisation of the preferred sections (\[quad\_sec\]) is a direct consequence of (\[gradient\]). Consider the infinitesimal generators ${\bf s}$ of non-homothetic conformal transformations, such that ${\bf s}=\nabla\chi$. The equation ${\cal L}_sh=2\chi h$ will then imply that $\chi$ satisfies (\[laplacian\]).
|
---
author:
- |
\
Jakob Erik Björnberg\
\
bibliography:
- 'thesis.bib'
title: |
Graphical representations\
of Ising and Potts models\
---
Preface {#preface .unnumbered}
=======
This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text.
I would like to thank my PhD supervisor Geoffrey Grimmett. Chapter \[qim\_ch\] and Section \[sec\_1d\] were done in collaboration with him. We have agreed that $65\%$ of the work is mine. This work has appeared in a journal as a joint publication [@bjogr2]. I am the sole author of the remaining material. Section \[starlike\_sec\] has been published in a journal [@bjo0].
I would also like to thank the following. Anders Björner and the Royal Institute of Technology (KTH) in Stockholm, Sweden, made this work possible through extremely generous support and funding. The House of Knights (Riddarhuset) in Stockholm, Sweden, has supported me very generously throughout my studies. I have received further generous support from the Engineering and Physical Sciences Research Council under a Doctoral Training Award to the University of Cambridge. The final writing of this thesis took place during a very stimulating stay at the Mittag-Leffler Institute for Research in Mathematics, Djursholm, Sweden, during the spring of 2009.
Summary {#summary .unnumbered}
=======
Statistical physics seeks to explain macroscopic properties of matter in terms of microscopic interactions. Of particular interest is the phenomenon of phase transition: the sudden changes in macroscopic properties as external conditions are varied. Two models in particular are of great interest to mathematicians, namely the Ising model of a magnet and the percolation model of a porous solid. These models in turn are part of the unifying framework of the random-cluster representation, a model for random graphs which was first studied by Fortuin and Kasteleyn in the 1970’s. The random-cluster representation has proved extremely useful in proving important facts about the Ising model and similar models.
In this work we study the corresponding graphical framework for two related models. The first model is the transverse field quantum Ising model, an extension of the original Ising model which was introduced by Lieb, Schultz and Mattis in the 1960’s. The second model is the space–time percolation process, which is closely related to the contact model for the spread of disease. In Chapter \[st\_ch\] we define the appropriate ‘space–time’ random-cluster model and explore a range of useful probabilistic techniques for studying it. The space–time Potts model emerges as a natural generalization of the quantum Ising model. The basic properties of the phase transitions in these models are treated in this chapter, such as the fact that there is at most one unbounded [[fk]{}]{}-cluster, and the resulting lower bound on the critical value in ${\mathbb{Z}}$.
In Chapter \[qim\_ch\] we develop an alternative graphical representation of the quantum Ising model, called the random-parity representation. This representation is based on the random-current representation of the classical Ising model, and allows us to study in much greater detail the phase transition and critical behaviour. A major aim of this chapter is to prove sharpness of the phase transition in the quantum Ising model—a central issue in the theory—and to establish bounds on some critical exponents. We address these issues by using the random-parity representation to establish certain differential inequalities, integration of which give the results.
In Chapter \[appl\_ch\] we explore some consequences and possible extensions of the results established in Chapters \[st\_ch\] and \[qim\_ch\]. For example, we determine the critical point for the quantum Ising model in ${\mathbb{Z}}$ and in ‘star-like’ geometries.
Introduction and background
===========================
Many physical and mathematical systems undergo a *phase transition*, of which some of the following examples may be familiar to the reader: water boils at $100^\circ$C and freezes at $0^\circ$C; Erdős-Rényi random graphs produce a ‘giant component’ if and only if the edge-probability $p>1/n$; and magnetic materials exhibit ‘spontaneous magnetization’ at temperatures below the Curie point. In physical terminology, these phenomena may be unified by saying that there is an ‘order parameter’ $M$ (density, size of largest component, magnetization) which behaves non-analytically on the parameters of the system at certain points. In the words of Alan Sokal: “at a phase transition $M$ may be discontinuous, or continuous but not differentiable, or 16 times differentiable but not 17 times”—any behaviour of this sort qualifies as a phase transition.
Since it is the example closest to the topic of this work, let us look at the case of spontaneous magnetization. For the moment we will stay on an entirely intuitive level of description. If one takes a piece of iron and places it in a magnetic field, one of two things will happen. When the strength of the external field is decreased to nought, the iron piece may retain magnetization, or it may not. Experiments confirm that there is a critical value $T_{\mathrm{c}}$ of the temperature $T$ such that: if $T<T_{\mathrm{c}}$ there is a residual (‘spontaneous’) magnetization, and if $T>T_{\mathrm{c}}$ there is not. See Figure \[mag\_fig\].
![Magnetization $M$ when $T>T_{\mathrm{c}}$ (left) and when $T<T_{\mathrm{c}}$ (right). The residual magnetization $M_0$ is zero at high temperature and positive at low temperature.[]{data-label="mag_fig"}](thesis.31 "fig:") ![Magnetization $M$ when $T>T_{\mathrm{c}}$ (left) and when $T<T_{\mathrm{c}}$ (right). The residual magnetization $M_0$ is zero at high temperature and positive at low temperature.[]{data-label="mag_fig"}](thesis.32 "fig:")
Thus the order parameter $M_0(T)$ (residual magnetization) is non-analytic at $T=T_{\mathrm{c}}$ (and it turns out that the phase transition is of the ‘continuous but not differentiable’ variety, see Theorem \[crit\_val\_cor\]). Can we account for this behaviour in terms of the ‘microscopic’ properties of the material, that is in terms of individual atoms and their interactions?
Considerable ingenuity has, since the 1920’s and earlier, gone in to devising mathematical models that strike a good balance between three desirable properties: physical relevance, mathematical (or computational) tractability, and ‘interesting’ critical behaviour. A whole arsenal of mathematical tools, rigorous as well as non-rigorous, have been developed to study such models. One of the most exciting aspects of the mathematical theory of phase transition is the abundance of amazing conjectures originating in the physics literature; attempts by mathematicians to ‘catch up’ with the physicists and rigorously prove some of these conjectures have led to the development of many beautiful mathematical theories. As an example of this one can hardly at this time fail to mention the theory of [[[sle]{}]{}]{} which has finally established some long-standing conjectures in two-dimensional models [@smirnov01; @smirnov_ising].
This work is concerned with the representation of physical models using stochastic geometry, in particular what are called percolation-, [[fk]{}]{}-, and random-current representations. A major focus of this work is on the *quantum Ising model* of a magnet (described below); on the way to studying this model we will also study ‘space–time’ random-cluster (or [[fk]{}]{}) and Potts models. Although a lot of attention has been paid to the graphical representation of classical Ising-like models, this is less true for quantum models, hence the current work. Our methods are rigorous, and mainly utilize the mathematical theory of probability. Although graphical methods may give less far-reaching results than the ‘exact’ methods favoured by mathematical physicists, they are also more robust to changes in geometry: towards the end of this work we will see some examples of results on high-dimensional, and ‘complex one-dimensional’, models where exact methods cannot be used.
Classical models
----------------
### The Ising model
The best-known, and most studied, model in statistical physics is arguably the Ising model of a magnet, given as follows. One represents the magnetic material at hand by a finite graph $L=(V,E)$ where the vertices $V$ represent individual particles (or atoms) and an edge is placed between particles that interact (‘neighbours’). A ‘state’ is an assignment of the numbers $+1$ and $-1$ to the vertices of $L$; these numbers are usually called ‘spins’. The set $\{-1,+1\}^V$ of such states is denoted ${\Sigma}$, and an element of ${\Sigma}$ is denoted ${\sigma}$. The model has two parameters, namely the temperature $T\geq 0$ and the external magnetic field $h\geq0$. The probability of seeing a particular configuration ${\sigma}$ is then proportional to the number $$\label{a1}
\exp\Big({\beta}\sum_{e=xy\in E}{\sigma}_x{\sigma}_y+{\beta}h\sum_{x\in V}{\sigma}_x\Big).$$ Here ${\beta}=(k_{\mathrm{B}}T)^{-1}>0$ is the ‘inverse temperature’, where $k_{\mathrm{B}}$ is a constant called the ‘Boltzmann constant’. Intuitively, the number is bigger if more spins agree, since ${\sigma}_x{\sigma}_y$ equals $+1$ if ${\sigma}_x={\sigma}_y$ and $-1$ otherwise; similarly it is bigger if more spins ‘align with the external field’ in that ${\sigma}_x=+1$. In particular, the spins at different sites are not in general statistically independent, and the structure of this dependence is subtly influenced by the geometry of the graph $L$. This is what makes the model interesting.
The Ising model was introduced around 1925 (not originally by but *to* Ising by his thesis advisor Lenz) as a candidate for a model that exhibits a phase transition [@ising1925]. It turns out that the magnetization $M$, which is by definition the expected value of the spin at some given vertex, behaves (in the limit as the graph $L$ approaches an infinite graph ${\mathbb{L}}$) non-analytically on the parameters ${\beta},h$ at a certain point $({\beta}={\beta}_{\mathrm{c}},h=0)$ in the $({\beta},h)$-plane.
The Ising model is therefore the second-simplest physical model with an interesting phase transition; the simplest such model is the following. Let ${\mathbb{L}}=({\mathbb{V}},{\mathbb{E}})$ be an infinite, but countable, graph. (The main example to bear in mind is the lattice ${\mathbb{Z}}^d$ with nearest-neighbour edges.) Let $p\in[0,1]$ be given, and examine each edge in turn, keeping it with probability $p$ and deleting it with probability $1-p$, these choices being independent for different edges. The resulting subgraph of ${\mathbb{L}}$ is typically denoted ${\omega}$, and the set of such subgraphs is denoted ${\Omega}$. The graph ${\omega}$ will typically not be connected, but will break into a number of connected components. Is one of these components infinite? The model possesses a phase transition in the sense that the probability that there exists an infinite component jumps from 0 to 1 at a critical value $p_{\mathrm{c}}$ of $p$.
This model is called *percolation*. It was introduced by Broadbent and Hammersley in 1957 as a model for a porous material immersed in a fluid [@broadbent_hammersley]. Each edge in ${\mathbb{E}}$ is then thought of as a small hole which may be open (if the corresponding edge is present in ${\omega}$) or closed to the passage of fluid. The existence of an infinite component corresponds to the fluid being able to penetrate from the surface to the ‘bulk’ of the material. Even though we are dealing here with a countable set of independent random variables, the theory of percolation is a genuine departure from the traditional theory of sequences of independent variables, again since geometry plays such a vital role.
### The random-cluster model
At first sight, the Ising- and percolation models seem unrelated, but they have a common generalization. On a finite graph $L=(V,E)$, the percolation configuration ${\omega}$ has probability $$\label{a2}
p^{|{\omega}|}(1-p)^{|E{\setminus}{\omega}|},$$ where $|\cdot|$ denotes the number of elements in a finite set, and we have identified the subgraph ${\omega}$ with its edge-set. A natural way to generalize is to consider absolutely continuous measures, and it turns out that the distributions defined by $$\label{a3}
\phi({\omega}):=p^{|{\omega}|}(1-p)^{|E{\setminus}{\omega}|}\frac{q^{k({\omega})}}{Z}$$ are particularly interesting. Here $q>0$ is an additional parameter, $k({\omega})$ is the number of connected components in ${\omega}$, and $Z$ is a normalizing constant. The ‘cluster-weighting factor’ $q^{k({\omega})}$ has the effect of skewing the distribution in favour of few large components (if $q<1$) or many small components (if $q>1$), respectively. This new model is called the random-cluster model, and it contains percolation as the special case $q=1$. By considering limits as $L\uparrow{\mathbb{L}}$, one may see that the random-cluster models (with $q\geq 1$) also have a phase transition in the same sense as the percolation model, with associated critical probability $p_{\mathrm{c}}=p_{\mathrm{c}}(q)$.
There is also a natural way to generalize the Ising model. This is easiest to describe when $h=0$, which we assume henceforth. The relative weights depend (up to a multiplicative constant) only on the number of adjacent vertices with equal spin, so the same model is obtained by using the weights $$\label{a4}
\exp\Big(2{\beta}\sum_{e=xy\in E}{\delta}_{{\sigma}_x,{\sigma}_y}\Big),$$ where ${\delta}_{a,b}$ is 1 if $a=b$ and 0 otherwise. (Note that ${\delta}_{{\sigma}_x,{\sigma}_y}=({\sigma}_x{\sigma}_y+1)/2$.) In this formulation it is natural to consider the more general model when the spins ${\sigma}_x$ can take not only two, but $q=2,3,\dotsc$ different values, that is each ${\sigma}_x\in\{1,\dotsc,q\}$. Write $\pi$ for the corresponding distribution on spin configurations; the resulting model is called the $q$-state Potts model. It turns out that the $q$-state Potts models is closely related to the random-cluster model, one manifestation of this being the following. (See [@fortuin_kasteleyn_1972], or [@grimmett_rcm Chapter 1] for a modern proof.)
\[thm1\] If $q\geq2$ is an integer and $p=1-e^{-2{\beta}}$ then for all $x,y\in V$ $$\pi({\sigma}_x={\sigma}_y)-\frac{1}{q}=
\Big(1-\frac{1}{q}\Big)\phi(x{\leftrightarrow}y)$$
Here $\pi({\sigma}_x={\sigma}_y)$ denotes the probability that, in the Potts model, the spin at $x$ takes the same value as the spin at $y$. Similarly, $\phi(x{\leftrightarrow}y)$ is the probability that, in the random-cluster model, $x$ and $y$ lie in the same component of ${\omega}$. Since the right-hand-side concerns a typical graph-theoretic property (connectivity), the random-cluster model is called a ‘graphical representation’ of the Potts model. The close relationship between the random-cluster and Potts models was unveiled by Fortuin and Kasteleyn during the 1960’s and 1970’s in a series of papers including [@fortuin_kasteleyn_1972]. The random-cluster model is therefore sometimes called the ‘[[fk]{}]{}-representation’. In other words, Theorem \[thm1\] says that the correlation between distant spins in the Potts model is translated to the existence of paths between the sites in the random-cluster model. Using this and related facts one can deduce many important things about the phase transition of the Potts model by studying the random-cluster model. This can be extremely useful since the random-cluster formulation allows geometric arguments that are not present in the Potts model. Numerous examples of this may be found in [@grimmett_rcm]; very recently, in [@smirnov_ising], the ‘loop’ version of the random-cluster model was also used to prove conformal invariance for the two-dimensional Ising model, a major breakthrough in the theory of the Ising model.
### Random-current representation {#intro_rcr}
For the Ising model there exists also another graphical representation, distinct from the random-cluster model. This is called the ‘random-*current* representation’ and was developed in a sequence of papers in the late 1980’s [@aiz82; @abf; @af], building on ideas in [@ghs]. These papers answered many questions for the Ising model on ${\mathbb{L}}={\mathbb{Z}}^d$ with $d\geq 2$ that are still to this day unanswered for general Potts models. Cast in the language of the $q=2$ random-cluster model, these questions include the following \[answers in square brackets\].
- If $p<p_{\mathrm{c}}$, is the expected size of a component finite or infinite? \[Finite.\]
- If $p<p_{\mathrm{c}}$, do the connection probabilities $\phi(x{\leftrightarrow}y)$ go to zero exponentially fast as $|x-y|\rightarrow{\infty}$? \[Yes.\]
- At $p=p_{\mathrm{c}}$, does $\phi(x{\leftrightarrow}y)$ go to zero exponentially fast as $|x-y|\rightarrow{\infty}$? \[No.\]
In fact, even more detailed information could be obtained, especially in the case $d\geq4$, giving at least partial answer to the question
- How does the magnetization $M=M({\beta},h)$ behave as the critical point $({\beta}_{\mathrm{c}},0)$ is approached?
It is one of the main objectives of this work to develop a random-current representation for the *quantum* Ising model (introduced in the next section), and answer the above questions also for that model.
Here is a very brief sketch of the random-current representation of the classical Ising model. Of particular importance is the normalizing constant or ‘partition function’ that makes a probability distribution, namely $$\label{a5}
\sum_{{\sigma}\in{\Sigma}}\exp\Big({\beta}\sum_{e=xy\in E}{\sigma}_x{\sigma}_y\Big)$$ (we assume that $h=0$ for simplicity). We rewrite using the following steps. Factorize the exponential in as a product over $e=xy\in E$, and then expand each factor as a Taylor series in the variable ${\beta}{\sigma}_x{\sigma}_y$. By interchanging sums and products we then obtain a weighted sum over vectors ${\underline}m$ indexed by $E$ of a quantity which (by $\pm$ symmetry) is zero if a certain condition on ${\underline}m$ fails to be satisfied, and a positive constant otherwise. The condition on ${\underline}m$ is that: for each $x\in V$ the sum over all edges $e$ adjacent to $x$ of $m_e$ is a multiple of 2.
Once we have rewritten the partition function in this way, we may interpret the weights on ${\underline}m$ as probabilities. It follows that the partition function is (up to a multiplicative constant) equal to the probability that the random graph $G_{{\underline}m}$ with each edge $e$ replaced by $m_e$ parallel edges is *even* in that each vertex has even total degree. Similarly, other quantities of interest may be expressed in terms of the probability that only a given set of vertices fail to have even degree in $G_{{\underline}m}$; for example, the correlation between ${\sigma}_x$ and ${\sigma}_y$ for $x,y\in V$ is expressed in terms of the probability that only $x$ and $y$ fail to have even degree. By elementary graph theory, the latter event implies the existence of a path from $x$ to $y$ in $G_{{\underline}m}$. By studying connectivity in the above random graphs with restricted degrees one obtains surprisingly detailed information about the Ising model. Much more will be said about this method in Chapter \[qim\_ch\], see for example the Switching Lemma (Theorem \[sl\]) and its applications in Section \[sw\_appl\_sec\].
Quantum models and space–time models
------------------------------------
There is a version of the Ising model formulated to meet the requirements of quantum theory, introduced in [@lieb]. We will only be concerned with the *transverse field* quantum Ising model. Its definition and physical motivation bear a certain level of complexity which it is beyond the scope of this work to justify in an all but very cursory manner. One is given, as before, a finite graph $L=(V,E)$, and one is interested in the properties of certain matrices (or ‘operators’) acting on the Hilbert space ${\mathcal{H}}=\bigotimes_{v\in V}{\mathbb{C}}^2$. The set ${\Sigma}=\{-1,+1\}^V$ may now be identified with a basis for ${\mathcal{H}}$, defined by letting each factor ${\mathbb{C}}$ in the tensor product have basis consisting of the two vectors ${|+\rangle}:=\big(\begin{smallmatrix} 1 \\ 0\end{smallmatrix}\big)$ and ${|-\rangle}:=\big(\begin{smallmatrix} 0 \\ 1\end{smallmatrix}\big)$. We write ${|{\sigma}\rangle}=\bigotimes_{v\in V}{|{\sigma}_v\rangle}$ for these basis vectors. In addition to the inverse temperature ${\beta}>0$, one is given parameters ${\lambda},{\delta}>0$, interpreted as spin-coupling and transverse field intensities, respectively. The latter specify the *Hamiltonian* $$\label{qi_ham_eq}
H=-\tfrac{1}{2}{\lambda}\sum_{e=uv\in E}{\sigma}_u^{(3)}{\sigma}_v^{(3)}-
{\delta}\sum_{v\in V}{\sigma}_v^{(1)},$$ where the ‘Pauli spin-$\frac12$ matrices’ are given as $${\sigma}^{(3)}=
\Bigg(\begin{matrix}
1 & 0 \\
0 & -1
\end{matrix}\Bigg),
\qquad
{\sigma}^{(1)}=
\Bigg(\begin{matrix}
0 & 1 \\
1 & 0
\end{matrix}\Bigg),$$and ${\sigma}^{(i)}_v$ acts on the copy of ${\mathbb{C}}^2$ in ${\mathcal{H}}$ indexed by $v\in V$. Intuitively, the matrices ${\sigma}^{(1)}$ and ${\sigma}^{(3)}$ govern spins in ‘directions’ $1$ and $3$ respectively (there is another matrix ${\sigma}^{(2)}$ which does not feature in this model). The external field is called ‘transverse’ since it acts in a different ‘direction’ to the internal interactions. When ${\delta}=0$ this model therefore reduces to the (zero-field) classical Ising model (this will be obvious from the space–time formulation below).
The basic operator of interest is $e^{-{\beta}H}$, which is thus a (Hermitian) matrix acting on ${\mathcal{H}}$; one usually normalizes it and studies instead the matrix $e^{-{\beta}H}/{\mathrm{tr}}(e^{-{\beta}H})$. Here the *trace* of the Hermitian matrix $A$ is defined as $${\mathrm{tr}}(A) = \sum_{{\sigma}\in{\Sigma}} {\langle{\sigma}|}A{|{\sigma}\rangle},$$where ${\langle{\sigma}|}$ is the adjoint, or conjugate transpose, of the column vector ${|{\sigma}\rangle}$, and we are using the usual matrix product. An eigenvector of $e^{-{\beta}H}/{\mathrm{tr}}(e^{-{\beta}H})$ may be thought of as a ‘state’ of the system, and is now a ‘mixture’ (linear combination) of classical states in ${\Sigma}$; the corresponding eigenvalue (which is real since the matrix is Hermitian) is related to the ‘energy level’ of the state.
In this work we will not be working directly with this formulation of the quantum Ising model, but a (more probabilistic) ‘space–time’ formulation, which we describe briefly now. It is by now standard that many properties of interest in the transverse field quantum Ising model may be studied by means of a ‘path integral’ representation, which maps the model onto a type of classical Ising model on the continuous space $V\times[0,{\beta}]$. (To be precise, the endpoints of the interval $[0,{\beta}]$ must be identified for this mapping to hold.) This was first used in [@ginibre69], but see also for example [@akn; @aizenman_nacht; @campanino_klein_perez; @chayes_ioffe_curie-weiss; @GOS; @nachtergaele93] and the recent surveys to be found in [@G-pgs; @ioffe_geom]. Precise definitions will be given in Chapter \[st\_ch\], but in essence we must consider piecewise constant *functions* ${\sigma}:V\times[0,{\beta}]\rightarrow\{-1,+1\}$, which are random and have a distribution reminiscent of . The resulting model is called the ‘space–time Ising model’. As for the classical case, it is straightforward to generalize this to a space–time *Potts* model with $q\geq 2$ possible spin values, and also to give a graphical representation of these models in terms of a space–time random-cluster model. Although the partial continuity of the underlying geometry poses several technical difficulties, the corresponding theory is very similar to the classical random-cluster theory. The most important basic properties of the models are developed in detail in Chapter \[st\_ch\]. On taking limits as $L$ and/or ${\beta}$ become infinite, one may speak of the existence of unbounded connected components, and one finds (when ${\beta}={\infty}$) that there is a critical dependence on the ratio $\rho={\lambda}/{\delta}$ of the probability of seeing such a component. One may also develop, as we do in Chapter \[qim\_ch\], a type of random-current representation of the space–time Ising model which allows us to deduce many facts about the critical behaviour of the *quantum* Ising model.
Other models of space–time type have been around for a long time in the probability literature. Of these the most relevant for us is the *contact process* (more precisely, its graphical representation), see for example [@liggett85; @liggett99] and references therein. In the contact process, one imagines individuals placed on the vertices of a graph, such as ${\mathbb{Z}}^2$. Initially, some of these individuals may be infected with a contagious disease. As time passes, the individuals themselves stay fixed but the disease may spread: individuals may be infected by their neighbours, or by a ‘spontaneous’ infection. Infected individuals may recover spontaneously. Infections and recoveries are governed by Poisson processes, and depending on the ratio of infection rate to recovery rate the infection may or may not persist indefinitely. The contact model may be regarded as the $q=1$ or ‘independent’ case of the space–time random-cluster model (one difference is that we in the space–time model regard time as ‘undirected’). Thus one may get to general space–time random-cluster models in a manner reminiscent of the classical case, by skewing the distribution by an appropriate ‘cluster weighting factor’. This approach will be treated in detail in Section \[basics\_sec\].
Outline
-------
A brief outline of the present work follows. In Chapter \[st\_ch\], the space–time random-cluster and Potts models are defined. As for the classical theory, one of the most important tools is *stochastic comparison*, or the ability to compare the probabilities of certain events under measures with different parameters. A number of results of this type are presented in Section \[stoch\_ineq\_sect\]. We then consider the issue of defining random-cluster and Potts measures on infinite graphs, and of their phase transitions. We etablish the existence of weak limits of Potts and random-cluster measures as $L\uparrow{\mathbb{L}}$, and introduce the central question of when there is a unique such limit. It turns out that this question is closely related to the question if there can be an unbounded connected component; this helps us to define a critical value $\rho_{\mathrm{c}}(q)$. In general not a lot can be said about the precise value of $\rho_{\mathrm{c}}(q)$, but in the case when ${\mathbb{L}}={\mathbb{Z}}$ there are additional geometric (duality) arguments that can be used to show that $\rho_{\mathrm{c}}(q)\geq q$.
Chapter \[qim\_ch\] deals exclusively with the quantum Ising model in its space–time formulation. We develop the ‘random parity representation’, which is the space–time analog of the random-current representation, and the tools associated with it, most notably the switching lemma. This representation allows us to represent truncated correlation functions in terms of single geometric events. Since truncated correlations are closely related to the derivatives of the magnetization $M$, we can use this to prove a number of inequalities between the different partial derivatives of $M$, along the lines of [@abf]. Integrating these differential inequalities gives the information on the critical behaviour that was referred to in Section \[intro\_rcr\], namely the sharpness of the phase transition, bounds on critical exponents, and the vanishing of the mass gap. Chapter \[qim\_ch\] (as well as Section \[sec\_1d\]) is joint work with Geoffrey Grimmett, and appears in the article *The phase transition of the quantum Ising model is sharp* [@bjogr2], published by the Journal of Statistical Physics.
Finally, in Chapter \[appl\_ch\], we combine the results of Chapter \[qim\_ch\] with the results of Chapter \[st\_ch\] in some concrete cases. Using duality arguments we prove that the critical ratio $\rho_{\mathrm{c}}(2)=2$ in the case ${\mathbb{L}}={\mathbb{Z}}$. We then develop some further geometric arguments for the random-cluster representation to deduce that the critical ratio is the same as for ${\mathbb{Z}}$ on a much larger class of ‘${\mathbb{Z}}$-like’ graphs. These arguments (Section \[starlike\_sec\]) appear in the article *Critical value of the quantum Ising model on star-like graphs* [@bjo0], published in the Journal of Statistical Physics. We conclude by describing some future directions for research in this area.
Space–time models:\
random-cluster, Ising, and Potts {#st_ch}
================================
> [*Summary.*]{} We provide basic definitions and facts pertaining to the space–time random-cluster and -Potts models. Stochastic inequalities, a major tool in the theory, are proved carefully, and the notion of phase transition is defined. We also introduce the notion of graphical duality.
Definitions and basic facts {#basics_sec}
---------------------------
The space–time models we consider live on the product of a graph with the real line. To define space–time random-cluster and Potts models we first work on bounded subsets of this product space, and then pass to a limit. The continuity of ${\mathbb{R}}$ makes the definitions of boundaries and boundary conditions more delicate than in the discrete case.
### Regions and their boundaries
Let ${\mathbb{L}}=({\mathbb{V}},{\mathbb{E}})$ be a countably infinite, connected, undirected graph, which is *locally finite* in that each vertex has finite degree. Here ${\mathbb{V}}$ is the vertex set and ${\mathbb{E}}$ the edge set. For simplicity we assume that ${\mathbb{L}}$ does not have multiple edges or loops. An edge of ${\mathbb{L}}$ with endpoints $u$, $v$ is denoted by $uv$. We write $u \sim v$ if $uv \in {\mathbb{E}}$. The main example to bear in mind is when ${\mathbb{L}}={\mathbb{Z}}^d$ is the $d$-dimensional lattice, with edges between points that differ by one in exactly one coordinate.
Let $$\begin{gathered}
{\mathbb{K}}:=\bigcup_{v\in{\mathbb{V}}} (v\times {\mathbb{R}}),{\quad\quad}{\mathbb{F}}:=\bigcup_{e\in{\mathbb{E}}} (e\times {\mathbb{R}}),
\label{o8}\\
{\mathbf{\Theta}}:=({\mathbb{K}},{\mathbb{F}}).\end{gathered}$$Let $L=(V,E)$ be a finite connected subgraph of ${\mathbb{L}}$. In the case when ${\mathbb{L}}={\mathbb{Z}}^d$, the main example for $L$ is the ‘box’ $[-n,n]^d$. For each $v\in V$, let $K_v$ be a finite union of (disjoint) bounded intervals in ${\mathbb{R}}$. No assumption is made whether the constituent intervals are open, closed, or half-open. For $e=uv\in E$ let $F_e:=K_u\cap K_v{\subseteq}{\mathbb{R}}$. Let $$K:=\bigcup_{v\in V}(v\times K_v),\quad
F:=\bigcup_{e\in E}(e\times F_e).$$We define a *region* to be a pair $$\label{def-newL}
{\Lambda}=(K,F)$$for $L$, $K$ and $F$ defined as above. We will often think of ${\Lambda}$ as a subset of ${\mathbf{\Theta}}$ in the natural way, see Figure \[region\_fig\].
![A region ${\Lambda}=(K,F)$ as a subset of ${\mathbf{\Theta}}$ when ${\mathbb{L}}={\mathbb{Z}}$. Here ${\mathbb{K}}$ is drawn dashed, $K$ is drawn bold black, and $F$ is drawn bold grey. An endpoint of an interval in $K$ (respectively, $F$) is drawn as a square bracket if it is included in $K$ (respectively, $F$) or as a rounded bracket if it is not.[]{data-label="region_fig"}](thesis.33)
Since a region ${\Lambda}=(K,F)$ is completely determined by the set $K$, we will sometimes abuse notation by writing $x\in{\Lambda}$ when we mean $x\in K$, and think of subsets of $K$ (respectively, ${\mathbb{K}}$) as subsets of ${\Lambda}$ (respectively, ${\mathbf{\Theta}}$).
An important type of a region is a *simple region*, defined as follows. For $L$ as above, let ${\beta}>0$ and let $K$ and $F$ be given by letting each $K_v=[-{\beta}/2,{\beta}/2]$. Thus $$\begin{gathered}
K=K(L,{\beta}):=\bigcup_{v\in V} (v\times [-{\beta}/2,{\beta}/2]),\\
F=F(L,{\beta}):=\bigcup_{e\in E} (e\times [-{\beta}/2,{\beta}/2]),\\
{\Lambda}={\Lambda}(L,{\beta}):=(K,F).
\label{def-oldL}\end{gathered}$$ Note that in a simple region, the intervals constituting $K$ are all closed. (Later, in the quantum Ising model of Chapter \[qim\_ch\], the parameter ${\beta}$ will be interpreted as the ‘inverse temperature’.)
Introduce an additional point ${\Gamma}$ external to ${\mathbf{\Theta}}$, to be interpreted as a ‘ghost-site’ or ‘point at infinity’; the use of ${\Gamma}$ will be explained below, when the space–time random-cluster and Potts models are defined. Write ${\mathbf{\Theta}}^{\Gamma}={\mathbf{\Theta}}\cup\{{\Gamma}\}$, ${\mathbb{K}}^{\Gamma}={\mathbb{K}}\cup\{{\Gamma}\}$, and similarly for other notation.
We will require two distinct notions of boundary for regions ${\Lambda}$. For $I\subseteq{\mathbb{R}}$ we denote the closure and interior of $I$ by ${\overline}I$ and $I^\circ$, respectively. For ${\Lambda}$ a region as in , define the *closure* to be the region ${\overline}{\Lambda}=({\overline}K,{\overline}F)$ given by $${\overline}K:=\bigcup_{v\in V}(v\times {\overline}K_v),\quad
{\overline}F:=\bigcup_{e\in E}(e\times {\overline}F_e);$$similarly define the *interior* of ${\Lambda}$ to be the region ${\Lambda}^\circ=(K^\circ,F^\circ)$ given by $$K^\circ:=\bigcup_{v\in V}(v\times K_v^\circ),\quad
F^\circ:=\bigcup_{e\in E}(e\times F_e^\circ).$$Define the *outer boundary* $\partial {\Lambda}$ of ${\Lambda}$ to be the union of ${\overline}K\setminus K^\circ$ with the set of points $(u,t)\in K$ such that $u\sim v$ for some $v\in{\mathbb{V}}$ such that $(v,t)\not\in K$. Define the *inner boundary* ${\hat\partial}{\Lambda}$ of ${\Lambda}$ by ${\hat\partial}{\Lambda}:=(\partial{\Lambda})\cap K$. The inner boundary of ${\Lambda}$ will often simply be called the *boundary* of ${\Lambda}$. Note that if $x$ is an endpoint of a closed interval in $K_v$, then $x\in\partial{\Lambda}$ if and only if $x\in{\hat\partial}{\Lambda}$, but if $x$ is an endpoint of an open interval in $K_v$, then $x\in\partial{\Lambda}$ but $x\not\in{\hat\partial}{\Lambda}$. In particular, if ${\Lambda}$ is a simple region then $\partial{\Lambda}={\hat\partial}{\Lambda}$. A word of caution: this terminology is nonstandard, in that for example the interior and the boundary of a region, as defined above, need not be disjoint. See Figure \[boundary\_fig\]. We define $\partial{\Lambda}^{\Gamma}=\partial{\Lambda}\cup\{{\Gamma}\}$ and ${\hat\partial}{\Lambda}^{\Gamma}={\hat\partial}{\Lambda}\cup\{{\Gamma}\}$.
![The (inner) boundary ${\hat\partial}{\Lambda}$ of the region ${\Lambda}$ of Figure \[region\_fig\] is marked black, and $K\setminus{\hat\partial}{\Lambda}$ is marked grey. An endpoint of an interval in ${\hat\partial}{\Lambda}$ is drawn as a square bracket if it lies in ${\hat\partial}{\Lambda}$ and as a round bracket otherwise.[]{data-label="boundary_fig"}](thesis.34)
A subset $S$ of ${\mathbb{K}}$ will be called *open* if it equals a union of the form $$\bigcup_{v\in{\mathbb{V}}}(v\times U_v),$$ where each $U_v{\subseteq}{\mathbb{R}}$ is an open set. Similarly for subsets of ${\mathbb{F}}$. The ${\sigma}$-algebra generated by this topology on ${\mathbb{K}}$ (respectively, on ${\mathbb{F}}$) will be denoted ${\mathcal{B}}({\mathbb{K}})$ (respectively, ${\mathcal{B}}({\mathbb{F}})$) and will be referred to as the Borel ${\sigma}$-algebra.
Occasionally, especially in Chapter \[qim\_ch\], we will in place of ${\mathbf{\Theta}}$ be using the finite ${\beta}$ space ${\mathbf{\Theta}}_{\beta}=({\mathbb{K}}_{\beta},{\mathbb{F}}_{\beta})$ given by $$\label{def-bL}
{\mathbb{K}}_{\beta}:=\bigcup_{v\in{\mathbb{V}}} (v\times [-{\beta}/2,{\beta}/2]),{\quad\quad}{\mathbb{F}}_{\beta}:=\bigcup_{e\in{\mathbb{E}}} (e\times [-{\beta}/2,{\beta}/2]).$$This is because in the quantum Ising model ${\beta}$ is thought of as ‘inverse temperature’, and then both ${\beta}<{\infty}$ (positive temperature) and ${\beta}={\infty}$ (ground state) are interesting.
In what follows, proofs will often, for simplicity, be given for simple regions only; proofs for general regions will in these cases be straightforward adaptations. We will frequently be using integrals of the forms $$\int_K f(x)\,dx\qquad\text{and}\qquad\int_F g(e)\, de.$$ These are to be interpreted, respectively, as $$\label{KF_int}
\sum_{v\in V}\int_{K_v} f(v,t)\,dt,
{\quad\quad}\sum_{e\in E} \int_{F_e} g(e,t) \,dt.$$ If $A$ is an event, we will write ${\hbox{\rm 1\kern-.27em I}}_A$ or ${\hbox{\rm 1\kern-.27em I}}\{A\}$ for the indicator function of $A$.
### The space–time percolation model
Write ${{\mathbb{R}}_+}=[0,{\infty})$ and let ${\lambda}:{\mathbb{F}}\rightarrow{{\mathbb{R}}_+}$, ${\delta}:{\mathbb{K}}\rightarrow{{\mathbb{R}}_+}$, and ${\gamma}:{\mathbb{K}}\rightarrow{{\mathbb{R}}_+}$ be bounded functions. We assume throughout that ${\lambda},{\delta},{\gamma}$ are all Borel-measurable. We retain the notation ${\lambda}$, ${\delta}$, ${\gamma}$ for the restrictions of these functions to ${\Lambda}$, given in . Let ${\Omega}$ denote the set of triples ${\omega}=(B,D,G)$ of countable subsets $B{\subseteq}{\mathbb{F}}$, $D,G{\subseteq}{\mathbb{K}}$; these triples will often be called *configurations*. Let $\mu_{\lambda}$, $\mu_{\delta}$, $\mu_{\gamma}$ be the probability measures associated with independent Poisson processes on ${\mathbb{K}}$ and ${\mathbb{F}}$ as appropriate, with respective intensities ${\lambda}$, ${\delta}$, ${\gamma}$. Let $\mu$ denote the probability measure $\mu_{\lambda}\times\mu_{\delta}\times\mu_{\gamma}$ on ${\Omega}$. Note that, with $\mu$-probability 1, each of the countable sets $B,D,G$ contains no accumulation points; we call such a set *locally finite*. We will sometimes write $B({\omega}),D({\omega}),G({\omega})$ for clarity.
\[rem-as\] For simplicity of notation we will frequently overlook events of probability zero, and will thus assume for example that ${\Omega}$ contains only triples $(B,D,G)$ of locally finite sets, such that no two points in $B\cup D\cup G$ have the same ${\mathbb{R}}$-coordinates.
For the purpose of defining a metric and a ${\sigma}$-algebra on ${\Omega}$, it is convenient to identify each ${\omega}\in{\Omega}$ with a collection of step functions. To be definite, we then regard each ${\omega}\cap(v\times{\mathbb{R}})$ and each ${\omega}\cap(e\times{\mathbb{R}})$ as an *increasing, right-continuous* step function, which equals 0 at $(v,0)$ or $(e,0)$ respectively. There is a metric on the space of right-continuous step functions on ${\mathbb{R}}$, called the Skorokhod metric, which may be extended in a straightforward manner to a metric on ${\Omega}$. Details may be found in Appendix \[skor\_app\], alternatively see [@bezuidenhout_grimmett], and [@ethier_kurtz Chapter 3] or [@lindvall Appendix 1]. We let ${\mathcal{F}}$ denote the ${\sigma}$-algebra on ${\Omega}$ generated by the Skorokhod metric. Note that the metric space ${\Omega}$ is *Polish*, that is to say separable (it contains a countable dense subset) and complete (Cauchy sequences converge).
However, in the context of percolation, here is how we usually want to think about elements of ${\Omega}$. Recall the ‘ghost site’ or ‘point at infinity’ ${\Gamma}$. Elements of $D$ are thought of as ‘deaths’, or missing points; elements of $B$ as ‘bridges’ or line segments between points $(u,t)$ and $(v,t)$, $uv\in{\mathbb{E}}$; and elements of $G$ as ‘bridges to ${\Gamma}$’. See Figure \[sample\_fig\] for an illustration of this. Elements of $B$ will sometimes be referred to as *lattice bonds* and elements of $G$ as *ghost bonds*. A lattice bond $(uv,t)$ is said to have *endpoints* $(u,t)$ and $(v,t)$; a ghost bond at $(v,t)$ is said to have endpoints $(v,t)$ and ${\Gamma}$.
For two points $x,y\in{\mathbb{K}}$ we say that there is a *path*, or an *open path*, in ${\omega}$ between $x$ and $y$ if there is a sequence $(x_1,y_1),\dotsc,(x_n,y_n)$ of pairs of elements of ${\mathbb{K}}$ satisfying the following:
- Each pair $(x_i,y_i)$ consists either of the two endpoints of a single lattice bond (that is, element of $B$) or of the endpoints in ${\mathbb{K}}$ of two distinct ghost bonds (that is, elements of $G$),
- Writing $y_0=x$ and $x_{n+1}=y$, we have that for all $0\leq i\leq n$, there is a $v_i\in{\mathbb{V}}$ such that $y_{i},x_{i+1}\in (v_i\times{\mathbb{R}})$,
- For each $0\leq i\leq n$, the (closed) interval in $v_i\times{\mathbb{R}}$ with endpoints $y_i$ and $x_{i+1}$ contains no elements of $D$.
In words, there is a path between $x$ and $y$ if $y$ can be reached from $x$ by traversing bridges and ghost-bonds, as well as subintervals of ${\mathbb{K}}$ which do not contain elements of $D$. For example, in Figure \[sample\_fig\] there is an open path between any two points on the line segments that are drawn bold. By convention, there is always an open path from $x$ to itself. We say that there is a path between $x\in{\mathbb{K}}$ and ${\Gamma}$ if there is a $y\in G$ such that there is a path between $x$ and $y$. Sometimes we say that $x,y\in{\mathbb{K}}^{\Gamma}$ are *connected* if there is an open path between them. Intuitively, elements of $D$ break connections on vertical lines, and elements of $B$ create connections between neighbouring lines. The use of ${\Gamma}$, and the process $G$, is to provide a ‘direct link to $\infty$’; two points that are joined to ${\Gamma}$ are automatically joined to eachother.
We write $\{x{\leftrightarrow}y\}$ for the event that there is an open path between $x$ and $y$. We say that two subsets $A_1,A_2{\subseteq}{\mathbb{K}}$ are connected, and write $A_1{\leftrightarrow}A_2$, if there exist $x\in A_1$ and $y\in A_2$ such that $x{\leftrightarrow}y$. For a region ${\Lambda}$, we say that there is an open path between $x,y$ *inside ${\Lambda}$* if $y$ can be reached from $x$ by traversing death-free line segments, bridges, and ghost-bonds that all lie in ${\Lambda}$. Open paths *outside* ${\Lambda}$ are defined similarly.
With the above interpretation, the measure $\mu$ on $({\Omega},{\mathcal{F}})$ is called the space–time percolation measure on ${\mathbf{\Theta}}$ with parameters ${\lambda},{\delta},{\gamma}$.
![Part of a configuration ${\omega}$ when ${\mathbb{L}}={\mathbb{Z}}$. Deaths are marked as crosses and bridges as horizontal line segments; the positions of ghost-bonds are marked as small circles. One of the connected components of ${\omega}$ is drawn bold.[]{data-label="sample_fig"}](thesis.1)
The measure $\mu$ coincides with the law of the graphical representation of a contact process with spontaneous infections, see [@aizenman_jung; @bezuidenhout_grimmett]. In this work, however, we regard ‘time’ as undirected, and thus think of ${\omega}$ as a geometric object rather than as a process evolving in time.
### Boundary conditions
Any ${\omega}\in{\Omega}$ breaks into *components*, where a component is by definition the maximal subset of ${\mathbb{K}}^{\Gamma}$ which can be reached from a given point in ${\mathbb{K}}^{\Gamma}$ by traversing open paths. See Figure \[sample\_fig\]. One may imagine ${\mathbb{K}}$ as a collection of infinitely long strings, which are cut at deaths, tied together at bridges, and also tied to ${\Gamma}$ at ghost-bonds. The components are the pieces of string that ‘hang together’. The random-cluster measure, which is defined in the next subsection, is obtained by ‘skewing’ the percolation measure $\mu$ in favour of either many small, or a few big, components. Since the total number of components in a typical ${\omega}$ is infinite, we must first, in order to give an analytic definition, restrict our attention to the number of components which intersect a fixed region ${\Lambda}$. We consider a number of different rules for counting those components which intersect the boundary of ${\Lambda}$. Later we will be interested in limits as the region ${\Lambda}$ grows, and whether or not these ‘boundary conditions’ have an effect on the limit.
Let ${\Lambda}=(K,F)$ be a region. We define a *random-cluster boundary condition* $b$ to be a finite nonempty collection $b=\{P_1,\dotsc,P_m\}$, where the $P_i$ are disjoint, nonempty subsets of ${\hat\partial}{\Lambda}^{\Gamma}$, such that each $P_i\setminus\{{\Gamma}\}$ is a finite union of intervals. (These intervals may be open, closed, or half-open, and may consist of a single point.) We require that ${\Gamma}$ lies in one of the $P_i$, and by convention we will assume that ${\Gamma}\in P_1$. Note that the union of the $P_i$ will in general be a proper subset of ${\hat\partial}{\Lambda}^{\Gamma}$. For $x,y\in{\Lambda}^{\Gamma}$ we say that $x{\leftrightarrow}y$ *with respect to $b$* if there is a sequence $x_{1},\dotsc,x_{l}$ (with $0\leq l\leq m$) such that
- Each $x_{j}\in P_{i_j}$ for some $0\leq i_j\leq m$;
- There are open paths inside ${\Lambda}$ from $x$ to $x_1$ and from $x_l$ to $y$;
- For each $j=1,\dotsc,l-1$ there is some point $y_{j}\in P_{i_j}$ such that there is a path inside ${\Lambda}$ from $y_j$ to $x_{j+1}$.
See Figure \[bc\_fig\] for an example.
![Connectivities with respect to the boundary condition $b=\{P_1\}$, where $P_1\setminus\{{\Gamma}\}$ is the subset drawn bold. The following connectivities hold: $a{\leftrightarrow}b$, $a{\leftrightarrow}c$, $a\not{\leftrightarrow}d$. (This picture does not specify which endpoints of the subintervals of $P_1$ lie in $P_1$.)[]{data-label="bc_fig"}](thesis.35)
When ${\Lambda}$ and $b$ are fixed and $x,y\in{\Lambda}^{\Gamma}$, we will typically without mention use the symbol $x{\leftrightarrow}y$ to mean that there is a path between $x$ and $y$ in ${\Lambda}$ with respect to $b$. Intuitively, each $P_i$ is thought of as *wired together*; as soon as you reach one point $x_j\in P_{i_j}$ you automatically reach all other points $y_j\in P_{i_j}$. It is important in the definition that each $P_i$ is a subset of the *inner* boundary ${\hat\partial}{\Lambda}^{\Gamma}$ and not $\partial{\Lambda}^{\Gamma}$.
Here are some important examples of random-cluster boundary conditions.
- If $b=\{{\hat\partial}{\Lambda}^{\Gamma}\}$ then the entire boundary ${\hat\partial}{\Lambda}$ is wired together; we call this the *wired* boundary condition and denote it by $b={\mathrm{w}}$;
- If $b=\{\{{\Gamma}\}\}$ then $x{\leftrightarrow}y$ with respect to $b$ if and only if there is an open path between $x,y$ inside ${\Lambda}$; we call this the *free* boundary condition, and denote it by $b={\mathrm{f}}$.
- Given any $\tau\in{\Omega}$, the boundary condition $b=\tau$ is by definition obtained by letting the $P_i$ consist of those points in ${\hat\partial}{\Lambda}^{\Gamma}$ which are connected by open paths of $\tau$ *outside* ${\Lambda}$.
- We may also impose a number of *periodic* boundary conditions on simple regions. One may then regard $[-{\beta}/2,{\beta}/2]$ as a circle by identifying its endpoints, and/or in the case $L=[-n,n]^d$ identify the latter with the torus $({\mathbb{Z}}/[-n,n])^d$. Notation for periodic boundary conditions will be introduced when necessary. Periodic boundary conditions will be particularly important in the study of the quantum Ising model in Chapter \[qim\_ch\].
For each boundary condition $b$ on ${\Lambda}$, define the function $k^b_{\Lambda}:{\Omega}\rightarrow\{1,2,\dotsc,{\infty}\}$ to count the number of components of ${\omega}$ in ${\Lambda}$, counted with respect to the boundary condition $b$. There is a natural partial order on boundary conditions given by: $b'\geq b$ if $k^{b'}_{\Lambda}({\omega})\leq k^{b}_{\Lambda}({\omega})$ for all ${\omega}\in{\Omega}$. \[bc\_po\] For example, for any boundary condition $b$ we have $k^{\mathrm{w}}_{\Lambda}\leq k^b_{\Lambda}\leq k^{\mathrm{f}}_{\Lambda}$ and hence ${\mathrm{w}}\geq b\geq{\mathrm{f}}$. (Alternatively, $b'\geq b$ if $b$ is a refinement of $b'$. Note that for $b=\tau\in{\Omega}$, this partial order agrees with the natural partial order on ${\Omega}$, defined in Section \[stoch\_ineq\_sect\].)
### The space–time random-cluster model {#strcm_subsec}
For $q>0$ and $b$ a boundary condition, define the (random-cluster) *partition functions* $$Z^b_{\Lambda}=Z^b_{\Lambda}({\lambda},{\delta},{\gamma},q):=\int_{\Omega}q^{k^b_{\Lambda}({\omega})}\:d\mu({\omega}).$$It is not hard to see that each $Z^b_{\Lambda}<\infty$.
\[rc\_def\] We define the *finite-volume random-cluster measure* $\phi^b_{\Lambda}=\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}$ on ${\Lambda}$ to be the probability measure on $({\Omega},{\mathcal{F}})$ given by $$\frac{d\phi^b_{\Lambda}}{d\mu}({\omega}):=\frac{q^{k^b_{\Lambda}({\omega})}}{Z^b_{\Lambda}}.$$
Thus, for any bounded, ${\mathcal{F}}$-measurable $f:{\Omega}\rightarrow{\mathbb{R}}$ we have that $$\phi^b_{\Lambda}(f)=\frac{1}{Z^b_{\Lambda}}\int_{\Omega}f({\omega})q^{k^b_{\Lambda}({\omega})}
\, d\mu({\omega}).$$
We say that an event $A\in{\mathcal{F}}$ is *defined* on a pair $(S,T)$ of subsets $S\subseteq{\mathbb{K}}$ and $T{\subseteq}{\mathbb{F}}$ if whenever ${\omega}\in A$, and ${\omega}'\in{\Omega}$ is such that $B({\omega})\cap T=B({\omega}')\cap T$, $D({\omega})\cap S=D({\omega}')\cap S$ and $G({\omega})\cap S=G({\omega}')\cap S$, then also ${\omega}'\in A$. Let ${\mathcal{F}}_{(S,T)}\subseteq{\mathcal{F}}$ be the ${\sigma}$-algebra of events defined on $(S,T)$. For ${\Lambda}=(K,F)$ a region we write ${\mathcal{F}}_{{\Lambda}}$ for ${\mathcal{F}}_{(K,F)}$; we abbreviate ${\mathcal{F}}_{(S,\varnothing)}$ and ${\mathcal{F}}_{(\varnothing,T)}$ by ${\mathcal{F}}_S$ and ${\mathcal{F}}_T$, respectively. Let ${\mathcal{T}}_{(S,T)}={\mathcal{F}}_{({\mathbb{K}}{\setminus}S,{\mathbb{F}}{\setminus}T)}$ denote the ${\sigma}$-algebra of events defined *outside* $S$ and $T$. We call $A\in{\mathcal{F}}$ a *local* event if there is a region ${\Lambda}$ such that $A\in{\mathcal{F}}_{\Lambda}$ (this is sometimes also called a *finite-volume* event or a *cylinder* event).
Note that the version of $d\phi^b_{\Lambda}/d\mu$ given in Definition \[rc\_def\] is ${\mathcal{F}}_{\Lambda}$-measurable; thus we may either regard $\phi^b_{\Lambda}$ as a measure on the full space $({\Omega},{\mathcal{F}})$, or, by restricting consideration to events in ${\mathcal{F}}_{\Lambda}$, as a measure on $({\Omega},{\mathcal{F}}_{\Lambda})$.
For ${\Delta}=(K,F)$ a region and ${\omega},\tau\in{\Omega}$, let $$\begin{split}
B_{\Delta}({\omega},\tau)&=(B({\omega})\cap F)\cup (B(\tau)\cap ({\mathbb{F}}{\setminus}F)),\\
D_{\Delta}({\omega},\tau)&=(D({\omega})\cap K)\cup (D(\tau)\cap ({\mathbb{K}}{\setminus}K)),\\
G_{\Delta}({\omega},\tau)&=(G({\omega})\cap K)\cup (G(\tau)\cap ({\mathbb{K}}{\setminus}K)).
\end{split}$$ We write $$({\omega},\tau)_{\Delta}=(B_{\Delta}({\omega},\tau),D_{\Delta}({\omega},\tau),G_{\Delta}({\omega},\tau))$$ for the configuration that agrees with ${\omega}$ in ${\Delta}$ and with $\tau$ outside ${\Delta}$. The following result is a very useful ‘spatial Markov’ property of random-cluster measures; it is sometimes referred to as the [[dlr]{}]{}-, or Gibbs-, property. The proof follows standard arguments and may be found in Appendix \[pfs\_app\].
\[cond\_meas\_rcm\] Let ${\Lambda}\subseteq{\Delta}$ be regions, $\tau\in{\Omega}$, and $A\in{\mathcal{F}}$. Then $$\phi^\tau_{\Delta}(A\mid{\mathcal{T}}_{\Lambda})({\omega})=\phi_{\Lambda}^{({\omega},\tau)_{\Delta}}(A),
\qquad \phi^\tau_{\Delta}\mbox{-a.s.}$$
Analogous results hold for $b\in\{{\mathrm{f}},{\mathrm{w}}\}$. The following is an immediate consequence of Proposition \[cond\_meas\_rcm\].
\[del\_contr\] Let ${\Lambda}\subseteq{\Delta}$ be regions such that ${\hat\partial}{\Lambda}\cap{\hat\partial}{\Delta}=\varnothing$, and let $b$ be a boundary condition on ${\Delta}$. Let ${\mathcal{C}}$ be the event that all components inside ${\Lambda}$ which intersect ${\hat\partial}{\Lambda}$ are connected in ${\Delta}{\setminus}{\Lambda}$; let ${\mathcal{D}}$ be the event that *none* of these components are connected in ${\Delta}{\setminus}{\Lambda}$. Then $$\phi^b_{\Delta}(\cdot\mid {\mathcal{C}})=\phi_{\Lambda}^{\mathrm{w}}(\cdot)\quad\text{and}\quad
\phi^b_{\Delta}(\cdot\mid {\mathcal{D}})=\phi_{\Lambda}^{\mathrm{f}}(\cdot).$$
### The space–time Potts model {#stp_sec}
The classical random-cluster model is closely related to the Potts model of statistical mechanics. Similarly there is a natural ‘space–time Potts model’ which may be coupled with the space–time random-cluster model. A realization of the space–time Potts measure is a piecewise constant ‘colouring’ of ${\mathbb{K}}^{\Gamma}$. As for the random-cluster model, we will be interested in specifying different boundary conditions, and these will not only tell us which parts of the boundary are ‘tied together’, but may also specify the precise colour on certain parts of the boundary.
Let us fix a region ${\Lambda}$ and $q\geq 2$ an *integer*. Let ${\mathcal{N}}={\mathcal{N}}_q$ be the set of functions $\nu:{\mathbb{K}}^{\Gamma}\rightarrow\{1,\dotsc,q\}$ which have the property that their restriction to any $v\times{\mathbb{R}}$ is piecewise constant and right-continuous. Let ${\mathcal{G}}$ be the ${\sigma}$-algebra on ${\mathcal{N}}$ generated by all the functions $\nu\mapsto(\nu(x_1),\dotsc,\nu(x_N))\in{\mathbb{R}}^N$ as $N$ ranges through the integers and $x_1,\dotsc,x_N$ range through ${\mathbb{K}}^{\Gamma}$ (this coincides with the ${\sigma}$-algebra generated by the Skorokhod metric, see Appendix \[skor\_app\] and [@ethier_kurtz Proposition 3.7.1]). For $S\subseteq{\mathbb{K}}$ define the ${\sigma}$-algebra ${\mathcal{G}}_S\subseteq{\mathcal{G}}$ of events defined on $S^{\Gamma}$. Although we canonically let $\nu\in{\mathcal{N}}$ be right-continuous, we will usually identify such $\nu$ which agree off sets of Lebesgue measure zero, compare Remark \[rem-as\]. Thus we will without further mention allow $\nu$ to be any piecewise constant function with values in $\{1,\dotsc,q\}$, and we will frequently even allow $\nu$ to be undefined on a set of measure zero. We call elements of ${\mathcal{N}}$ ‘spin configurations’ and will usually write $\nu_x$ for $\nu(x)$.
Let $b=\{P_1,\dotsc,P_m\}$ be any random-cluster boundary condition and let ${\alpha}:\{1,\dotsc,m\}\rightarrow\{0,1,\dotsc,q\}$. We call the pair $(b,{\alpha})$ a *Potts boundary condition*. We assume that ${\Gamma}\in P_1$, and write ${\alpha}_{\Gamma}$ for ${\alpha}(1)$; we also require that ${\alpha}_{\Gamma}\neq 0$. Let $D{\subseteq}K$ be a finite set, and let ${\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D)$ be the set of $\nu\in{\mathcal{N}}$ with the following properties.
- For each $v\in V$ and each interval $I{\subseteq}K_v$ such that $I\cap D=\varnothing$, $\nu$ is constant on $I$,
- if $i\in\{1,\dotsc,m\}$ is such that ${\alpha}(i)\neq 0$ then $\nu_x={\alpha}(i)$ for all $x\in P_i$,
- if $i\in\{1,\dotsc,m\}$ is such that ${\alpha}(i)=0$ and $x,y\in P_i$ then $\nu_x=\nu_y$,
- if $x\not\in{\Lambda}$ then $\nu_x={\alpha}_{\Gamma}$.
Intuitively, the boundary condition $b$ specifies which parts of the boundary are forced to have the same spin, and the function ${\alpha}$ specifies the *value* of the spin on some parts of the boundary; ${\alpha}(i)=0$ is taken to mean that the value on $P_i$ is not specified. (The value of ${\alpha}$ at ${\Gamma}$ is special, in that it takes on the role of an external field, see .)
Let ${\lambda}:{\mathbb{F}}\rightarrow{\mathbb{R}}$, ${\gamma}:{\mathbb{K}}\rightarrow{\mathbb{R}}$ and ${\delta}:{\mathbb{K}}\rightarrow{\mathbb{R}}_+$ be bounded and Borel-measurable; note that ${\lambda}$ and ${\gamma}$ are allowed to take negative values. For $a,b\in{\mathbb{R}}$, let ${\delta}_{a,b}={\hbox{\rm 1\kern-.27em I}}_{\{a=b\}}$, and for $\nu\in{\mathcal{N}}$ and $e=xy\in{\mathbb{E}}$, let ${\delta}_\nu(e)={\delta}_{\nu_x,\nu_y}$. Let $\pi_{\Lambda}^{b,{\alpha}}$ denote the probability measure on $({\mathcal{N}},{\mathcal{G}})$ defined by, for each bounded and ${\mathcal{G}}$-measurable $f:{\mathcal{N}}\rightarrow{\mathbb{R}}$, letting $\pi^{b,{\alpha}}_{\Lambda}(f(\nu))$ be a constant multiple of $$\label{st_potts_def_eq}
\int d\mu_{\delta}(D)\,\sum_{\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D)} f(\nu)
\exp\Big(\int_F{\lambda}(e){\delta}_\nu(e)de+
\int_K{\gamma}(x) {\delta}_{\nu_x,{\alpha}_{\Gamma}}dx\Big)$$ (with constant determined by the requirement that $\pi_{\Lambda}^{b,{\alpha}}$ be a probability measure). The integrals in are to be interpreted as in .
\[potts\_def\] The probability measure $\pi^{b,{\alpha}}_{\Lambda}=\pi^{b,{\alpha}}_{{\Lambda};q,{\lambda},{\gamma},{\delta}}$ on $({\mathcal{N}},{\mathcal{G}})$ defined by is called the space–time Potts measure with $q$ states on ${\Lambda}$.
Note that, as with $\phi^b_{\Lambda}$, we may regard $\pi^{b,{\alpha}}_{{\Lambda}}$ as a measure on $({\mathcal{N}},{\mathcal{G}}_{\Lambda})$. Here is a word of motivation for in the case $b={\mathrm{f}}$ and ${\alpha}_{\Gamma}=q$; similar constructions hold for other $b,{\alpha}$. See Figure \[bridge\_part\_fig\] in Section \[ssec-rpr\], and also [@GOS]. The set $(v\times K_v){\setminus}{D}$ is a union of maximal death-free intervals $v\times J_v^k$, where $k=1,2,\dotsc,n$ and $n=n(v,{D})$ is the number of such intervals. We write $V({D})$ for the collection of all such intervals as $v$ ranges over $V$, together with the ghost-vertex ${{\Gamma}}$, to which we assign spin $\nu_{{\Gamma}}=q$. The set ${\mathcal{N}}^{{\mathrm{f}},{\alpha}}_{\Lambda}({D})$ may be identified with $\{1,\dotsc,q\}^{V({D})}$, and we may think of $V(D)$ as the set of vertices of a graph with edges given as follows. An edge is placed between ${{\Gamma}}$ and each $\bar v\in V(D)$. For $\bar u, \bar v\in V(D)$, with $\bar u=u\times I_1$ and $\bar v=v\times I_2$ say, we place an edge between $\bar u$ and $\bar v$ if and only if: (i) $uv$ is an edge of $L$, and (ii) $I_1\cap I_2\neq{\varnothing}$. Under the space–time Potts measure *conditioned on $D$*, a spin-configuration $\nu\in {\mathcal{N}}^{{\mathrm{f}},{\alpha}}_{\Lambda}(D)$ on this graph receives a (classical) Potts weight $$\exp\left\{\sum_{\bar u\bar v}J_{\bar u\bar v}{\delta}_{\nu}(\bar u\bar v)
+ \sum_{\bar v} h_{\bar v}{\delta}_{\nu_{\bar v},q}\right\},$$ where $\nu_{\bar v}$ denotes the common value of $\nu$ along $\bar v$, and where $$J_{\bar u\bar v}=\int_{I_1\cap I_2}{\lambda}(uv,t)\,dt\qquad\text{and}
\qquad h_{\bar v}=\int_{\bar v}{\gamma}(x)\,dx.$$ This observation will be pursued further for the Ising model in Section \[ssec-rpr\].
The space–time Potts measure may, for special boundary conditions, be coupled to the space–time random-cluster measure, as follows. For ${\alpha}$ of the form $({\alpha}_{\Gamma},0,\dotsc,0)$, we call $(b,{\alpha})$ a *simple* Potts boundary condition. Thus, under a simple boundary condition, the only spin value which is specified in advance is that of ${\Gamma}$. Let ${\omega}=(B,D,G)\in{\Omega}$ be sampled from $\phi^b_{\Lambda}$ and write ${\mathcal{N}}^{b,{\alpha}}_{\Lambda}({\omega})$ for the set of $\nu\in{\mathcal{N}}$ such that (i) $\nu_x={\alpha}_{\Gamma}$ for $x\not\in{\Lambda}$, and (ii) if $x,y\in{\Lambda}$ and $x{\leftrightarrow}y$ in ${\omega}$ under the boundary condition $b$ in ${\Lambda}$ then $\nu_x=\nu_y$. In particular, since ${\Gamma}\not\in{\Lambda}$ we have that $\nu_{\Gamma}={\alpha}_{\Gamma}$. Note that each ${\mathcal{N}}^{b,{\alpha}}_{\Lambda}({\omega})$ is a finite set. With ${\omega}$ given, we sample $\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}({\omega})$ as follows. Set $\nu_{\Gamma}:={\alpha}_{\Gamma}$ and set $\nu_x={\alpha}_{\Gamma}$ for all $x\not\in{\Lambda}^{\Gamma}$; then choose the spins of the other components of ${\omega}$ in ${\Lambda}$ uniformly and independently at random. The resulting pair $({\omega},\nu)$ has a distribution ${\mathbb{P}}^{b,{\alpha}}_{\Lambda}$ on $({\Omega},{\mathcal{F}})\times({\mathcal{N}},{\mathcal{G}})$ given by $$\label{es_def}
\begin{split}
{\mathbb{P}}^{b,{\alpha}}_{\Lambda}(f({\omega},\nu))&= \int_{\Omega}d\phi^b_{\Lambda}({\omega})
\,\frac{1}{q^{k^b_{\Lambda}({\omega})-1}}\sum_{\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}({\omega})} f({\omega},\nu)\\
&\propto \int_{\Omega}d\mu({\omega}) \,\sum_{\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}({\omega})} f({\omega},\nu),
\end{split}$$ for all bounded $f:{\Omega}\times{\mathcal{N}}\rightarrow{\mathbb{R}}$, measurable in the product ${\sigma}$-algebra ${\mathcal{F}}\times{\mathcal{G}}$. We call the measure ${\mathbb{P}}^{b,{\alpha}}_{\Lambda}$ of the Edwards–Sokal measure. This definition is completely analogous to a coupling in the discrete model, which was was found in [@edwards_sokal]. Usually we take ${\alpha}_{\Gamma}=q$ and in this case we will often suppress reference to ${\alpha}$, writing for example ${\mathcal{N}}^b_{\Lambda}({\omega})$ and similarly for other notation.
The marginal of ${\mathbb{P}}^{b,{\alpha}}_{\Lambda}$ on $({\mathcal{N}},{\mathcal{G}})$ is computed as follows. Assume that $f({\omega},\nu)\equiv f(\nu)$ depends only on $\nu$, and let $D\subseteq K$ be a finite set. For $\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D),$ let $\{\nu\sim{\omega}\}$ be the event that ${\omega}$ has no open paths *inside* ${\Lambda}$ that violate the condition that $\nu$ be constant on the components of ${\omega}$. We may rewrite as $${\mathbb{P}}^{b,{\alpha}}_{\Lambda}(f(\nu))
\propto \int d\mu_{\delta}(D)\int d(\mu_{\lambda}\times\mu_{\gamma})(B,G)
\sum_{\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D)} f(\nu){\hbox{\rm 1\kern-.27em I}}\{\nu\sim{\omega}\}.$$ With $D$ fixed, the probability under $\mu_{\lambda}\times\mu_{\gamma}$ of the event $\{\nu\sim{\omega}\}$ is $$\label{i5}
\exp\Big(-\int_F{\lambda}(e)(1-{\delta}_\nu(e))de
-\int_K{\gamma}(x)(1-{\delta}_{\nu_x,{\alpha}_{\Gamma}})dx\Big).$$ Taking out a constant, it follows that ${\mathbb{P}}^{b,{\alpha}}_{\Lambda}(f(\nu))$ is proportional to $$\begin{aligned}
\int d\mu_{\delta}(D)\,\sum_{\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D)} f(\nu)
\exp\Big(\int_F{\lambda}(e){\delta}_\nu(e)de+
\int_K{\gamma}(x) {\delta}_{\nu_x,{\alpha}_{\Gamma}}dx\Big).\end{aligned}$$ Comparing this with , and noting that both equations define probability measures, it follows that ${\mathbb{P}}^{b,{\alpha}}_{\Lambda}(f(\nu))=\pi^{b,{\alpha}}_{\Lambda}(f)$.
We may ask for a description of how to obtain an ${\omega}$ with law $\phi^b_{\Lambda}$ from a $\nu$ with law $\pi^{b,{\alpha}}_{\Lambda}$. In analogy with the discrete case this is as follows:
> Given $\nu\sim\pi^{b,{\alpha}}_{\Lambda}(\cdot)$, place a death wherever $\nu$ changes spin in ${\Lambda}$, and also place additional deaths elsewhere in ${\Lambda}$ at rate ${\delta}$; place bridges between intervals in ${\Lambda}$ of the same spin at rate ${\lambda}$; and place ghost-bonds in intervals in ${\Lambda}$ of spin ${\alpha}$ at rate ${\gamma}$. The outcome ${\omega}$ has law $\phi^b_{\Lambda}(\cdot)$.
It follows that we have the following correspondence between $\phi=\phi^b_{\Lambda}$ and $\pi=\pi^{b,{\alpha}}_{{\Lambda},q}$ when $(b,{\alpha})$ is simple. The result is completely analogous to the corresponding result for the discrete Potts model (Theorem \[thm1\]), and the proof is included only for completeness.
\[corr\_conn\_prop\] Let $x,y\in{\Lambda}^{\Gamma}$. Then $$\pi(\nu_x=\nu_y)=\Big(1-\frac{1}{q}\Big)\phi(x\leftrightarrow y)+\frac{1}{q}.$$
Writing ${\mathbb{P}}$ for the Edwards–Sokal coupling, we have that $$\begin{aligned}
q\pi(\nu_x=\nu_y)-1&={\mathbb{P}}(q\cdot{\mathbb{P}}(\nu_x=\nu_y\mid{\omega})-1)\\
&={\mathbb{P}}\Big(q\big({\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }{\omega}\}+
\frac{1}{q}{\hbox{\rm 1\kern-.27em I}}\{x\not{\leftrightarrow}y\mbox{ in }{\omega}\}\big)-1\Big)\\
&={\mathbb{P}}((q-1)\cdot{\hbox{\rm 1\kern-.27em I}}\{x\leftrightarrow y\mbox{ in }{\omega}\})\\
&=(q-1)\phi(x\leftrightarrow y).\end{aligned}$$
The case $q=2$ merits special attention. In this case it is customary to replace the states $\nu_x=1,2$ by $-1,+1$ respectively, and we thus define ${\sigma}_x=2\nu_x-3$. For ${\alpha}$ taking values in $\{0,-1,+1\}$, we let ${\Sigma},{\Sigma}^{b,{\alpha}}_{\Lambda}({\omega}),{\Sigma}^{b,{\alpha}}_{\Lambda}(D)$ denote the images of ${\mathcal{N}},{\mathcal{N}}^{b,{\alpha}}_{\Lambda}({\omega}),{\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D)$ respectively under the map $\nu\mapsto{\sigma}$. Reference to ${\alpha}$ may be suppressed if $(b,{\alpha})$ is simple and ${\alpha}_{\Gamma}=+1$.
We have that $$\label{sigma_nu}
{\hbox{\rm 1\kern-.27em I}}\{{\sigma}_x={\sigma}_y\}=\frac{1}{2}({\sigma}_x{\sigma}_y+1),\qquad
{\hbox{\rm 1\kern-.27em I}}\{{\sigma}_x={\alpha}_{\Gamma}\}=\frac{1}{2}({\alpha}_{\Gamma}{\sigma}_x+1).$$ Consequently, $\pi^{b,{\alpha}}_{{\Lambda};q=2}(f({\sigma}))$ is proportional to $$\label{ising_def_eq}
\int d\mu_{\delta}(D)\,\sum_{{\sigma}\in{\Sigma}^{b,{\alpha}}_{\Lambda}(D)} f({\sigma})
\exp\Big(\frac{1}{2}\int_F{\lambda}(e){\sigma}_e\,de+
\frac{1}{2}\int_K{\gamma}(x){\alpha}_{\Gamma}{\sigma}_x \,dx\Big),$$ where we have written ${\sigma}_e$ for ${\sigma}_x{\sigma}_y$ when $e=xy$. In this formulation, we call the measure of the *Ising measure*. Expected values with respect to this measure will typically be written ${\langle}\cdot{\rangle}^{b,{\alpha}}_{\Lambda}$; thus for example Proposition \[corr\_conn\_prop\] says that when $q=2$ and $(b,{\alpha})$ is simple, then $$\label{ising_corr_conn}
{\langle}{\sigma}_x{\sigma}_y{\rangle}^{b,{\alpha}}_{\Lambda}=\phi^b_{\Lambda}(x{\leftrightarrow}y).$$
For later reference, we make a note here of the constants of proportionality in the above definitions. Let $$Z^b_{\mathrm{RC}}=Z^b_{\mathrm{RC}}(q)=\int_{{\Omega}} q^{k^b_{\Lambda}({\omega})}\,d\mu({\omega})$$ denote the partition function of the random-cluster model, and $$Z_{\mathrm{Potts}}^{b,{\alpha}}(q)=\int d\mu_{\delta}(D)\,\sum_{\nu\in{\mathcal{N}}^{b,{\alpha}}_{\Lambda}(D)}
\exp\Big(\int_F {\delta}_\nu(e){\lambda}(e)\,de+\int_K {\delta}_{\nu_x,{\alpha}_{\Gamma}}{\gamma}(x)\, dx\Big)$$ that of the $q$-state Potts model. Also, let $$\label{ising_pf}
Z^{b,{\alpha}}_{\mathrm{Ising}}=\int d\mu_{\delta}(D)\,\sum_{{\sigma}\in{\Sigma}^{b,{\alpha}}_{\Lambda}(D)}
\exp\Big(\frac{1}{2}\int_F{\lambda}(e){\sigma}_e\,de+
\frac{1}{2}\int_K{\gamma}(x){\alpha}_{\Gamma}{\sigma}_x\, dx\Big)$$ be the partition function of the Ising model. By keeping track of the constants in the above calculations we obtain the following result, which for simplicity is stated only for ${\alpha}_{\Gamma}=q$.
\[pfs\] Let $b$ be a random-cluster boundary condition. Then $$\begin{aligned}
Z^{b}_{\mathrm{Potts}}(q)&
=\frac{1}{q}Z^b_{\mathrm{RC}}(q)\cdot
\exp\Big(\int_F{\lambda}(e)\,de+\int_K{\gamma}(x)\,dx\Big)\\
Z^{b}_{\mathrm{Ising}}&=Z^{b}_{\mathrm{Potts}}(2)\cdot
\exp\Big(-\frac{1}{2}\int_F{\lambda}(e)\,de-\frac{1}{2}\int_K{\gamma}(x)\,dx\Big)\\
&=\frac{1}{2}Z^b_{\mathrm{RC}}(2)\cdot
\exp\Big(\frac{1}{2}\int_F{\lambda}(e)\,de+\frac{1}{2}\int_K{\gamma}(x)\,dx\Big).\nonumber\end{aligned}$$
It is easy to check, by a direct computation, that the Potts model behaves in a similar manner to the random-cluster model upon conditioning on the value of $\nu$ in part of a region, i.e. that analogs of Proposition \[cond\_meas\_rcm\] and Corollary \[del\_contr\] hold. We will not state these results explicitly in full generality, but will record here the following special case for later reference.
\[potts\_cond\] Let ${\Lambda}\subseteq{\Delta}$ denote two regions, and consider the boundary condition $({\mathrm{w}},{\alpha})$. Then for all ${\mathcal{G}}_{\Lambda}$-measurable $f$ we have that $$\pi_{\Lambda}^{{\mathrm{w}},{\alpha}}(f(\nu))=
\pi_{\Delta}^{{\mathrm{w}},{\alpha}}(f(\nu)\mid{\sigma}\equiv {\alpha}_{\Gamma}\mbox{ on }{\Delta}{\setminus}{\Lambda}).$$
Stochastic comparison {#stoch_ineq_sect}
---------------------
The ability to compare the probabilities of events under a range of different measures is extremely important in the theory of random-cluster measures. In this section we develop in detail the basis for such a methodology in the space–time setting. We also prove versions of the [[[gks]{}]{}-]{} and [[[fkg]{}]{}]{} inequalities suitable for the space–time Potts and Ising measures, respectively.
Let ${\Lambda}$ be a region. Let the pair $(E,{\mathcal{E}})$ denote one of $({\Omega},{\mathcal{F}})$, $({\Omega},{\mathcal{F}}_{\Lambda})$, $({\Sigma},{\mathcal{G}})$ and $({\Sigma},{\mathcal{G}}_{\Lambda})$. Thus $E$, equipped with the Skorokhod metric, is a Polish metric space. Given a partial order $\geq$ on $E$, a measurable function $f:E\rightarrow{\mathbb{R}}$ is called *increasing* if for all ${\omega},\xi\in E$ such that ${\omega}\geq\xi$ we have $f({\omega})\geq f(\xi)$. An event $A\in{\mathcal{E}}$ is increasing if the indicator function ${\hbox{\rm 1\kern-.27em I}}_A$ is. We assume that the set $\{({\omega},\xi)\in E^2:{\omega}\geq\xi\}$ is closed in the product topology; this will hold automatically in our applications.
Let $\psi_1,\psi_2$ be two probability measures on $(E,{\mathcal{E}})$.
\[stoch\_dom\_def\] We say that $\psi_1$ *stochastically dominates* $\psi_2$, and we write $\psi_1\geq\psi_2$, if $\psi_1(f)\geq\psi_2(f)$ for all bounded, increasing local functions $f$.
By a standard approximation argument using the monotone convergence theorem, $\psi_1\geq\psi_2$ holds if for all increasing local *events* $A$ we have $\psi_1(A)\geq\psi_2(A)$.
The following general result lies at the heart of stochastic comparison and will be used repeatedly. It goes back to [@strassen]; see also [@lindvall Theorem IV.2.4] and [@ghm Theorem 4.6].
\[strassen\_thm\] Let $\psi_1,\psi_2$ be probability measures on $(E,{\mathcal{E}})$. The following statements are equivalent.
1. $\psi_1\geq\psi_2$;
2. For all *continuous* bounded increasing local functions $f:E\rightarrow{\mathbb{R}}$ we have $\psi_1(f)\geq\psi_2(f)$;
3. There exists a probability measure $P$ on $(E^2,{\mathcal{E}}^2)$ such that $$P(\{({\omega}_1,{\omega}_2):{\omega}_1\geq{\omega}_2\})=1.$$
Note that the equivalence of (1) and (3) extends to *countable* sequences $\psi_1,\psi_2,\psi_3,\dotsc$; see [@lindvall Theorem IV.6.1].
\[pos\_assoc\_def\] A measure $\psi$ is on $(E,{\mathcal{E}})$ is called *positively associated* if for all local increasing events $A,B$ we have that $\psi(A\cap B)\geq\psi(A)\psi(B)$.
The inequality $\psi(A\cap B)\geq\psi(A)\psi(B)$ for local increasing events is sometimes referred to as the [[fkg]{}]{}-inequality as the systematic study of such inequalities was initiated by Fortuin, Kasteleyn and Ginibre [@fkg].
### Stochastic inequalities for the random-cluster model {#rcm_si}
The results in this section are applications, and slight modifications, of stochastic comparison results for point processes that appear in [@preston75] and [@georgii_kuneth]. See also [@ghm Theorem 10.4]. Some of the results, such as positive association in the space–time random-cluster model, have been stated before, sometimes with additional assumptions; see for example [@akn; @aizenman_nacht; @bezuidenhout_grimmett]. We do not believe detailed proofs for space–time models have appeared before. The results presented are satisfyingly similar to those for the discrete case, compare [@grimmett_rcm Chapter 3] and [@grimmett_gks].
We will follow the method of [@preston75] rather than the later (and more general) [@georgii_kuneth]. This is because the former method avoids discretization and is closer to the standard approach of [@holley74] (also [@grimmett_rcm Chapter 2]) for the classical random-cluster model. The method makes use of coupled Markov chains on ${\Omega}$ (specifically, jump-processes, see [@feller71_vol2 Chapter X]).
For ${\omega}\in{\Omega}$, write $B({\omega}),D({\omega}),G({\omega})$ for the sets of bridges, deaths and ghost-bonds in ${\omega}$, respectively. We define a partial order on ${\Omega}$ by saying that ${\omega}\geq\xi$ if $B({\omega})\supseteq B(\xi)$, $D({\omega})\subseteq D(\xi)$ and $G({\omega})\supseteq G(\xi)$.
We will in this section only consider measures on ${\mathcal{F}}_{\Lambda}$, that is we take $(E,{\mathcal{E}})=({\Omega},{\mathcal{F}}_{\Lambda})$. We will regard $B,G,D$ as subsets of $K$ and $F$ as appropriate. The symbol $x$ will be used to denote a generic point of ${\Lambda}\equiv K\cup F$, interpreted either as a bridge, a ghost-bond, or a death, as specified. More formally, we may regard $x$ as an element of $F\cup(K\times\{\mathrm{d}\})\cup(K\times\{\mathrm{g}\})$, where the labels $\mathrm{d},\mathrm{g}$ allow us to distinguish between deaths and ghost-bonds, respectively. We let $X=(X_t:t\geq 0)$ be a continuous-time stochastic process with state space ${\Omega}$, defined as follows. If $X_t=(B,G,D)$, there are 6 possible transitions. The process can either jump to one of $$\label{iml1}
(B\cup\{x\},G,D),\quad\mbox{or }
(B,G\cup\{x\},D),\quad\mbox{or }
(B,G,D\cup\{x\}),$$ where $x\in{\Lambda}$; the corresponding move is called a *birth* at $x$. Alternatively, in the case where $x\in B$, the process can jump to $$(B{\setminus}\{x\},G,D),$$ and similarly for $x\in G$ or $x\in D$; the corresponding move is called a *demise* at $x$. If ${\omega}=(B,G,D)\in{\Omega}$, we will often abuse notation and write ${\omega}^x$ for the configuration with a point at $x$ added, making it clear from the context whether $x$ is a bridge, ghost-bond, or death. Similarly, if $x\in B\cup G\cup D$, we will write ${\omega}_x$ for the configuration with the bridge, ghost-bond or death at $x$ removed.
The transitions described above happen at the following rates. Let ${\mathcal{L}}$ denote the Borel ${\sigma}$-algebra on ${\Lambda}\equiv F\cup(K\times\{\mathrm{d}\})\cup(K\times\{\mathrm{g}\})$, and let ${\mathcal{B}}:{\Omega}\times{\mathcal{L}}\rightarrow{\mathbb{R}}$ be a given function, such that for each ${\omega}\in{\Omega}$, ${\mathcal{B}}({\omega};\cdot)$ is a finite measure on $({\Lambda},{\mathcal{L}})$. Also let ${\mathcal{D}}:{\Omega}\times{\Lambda}\rightarrow{\mathbb{R}}$ be such that for all ${\omega}\in{\Omega}$ we have that ${\mathcal{D}}({\omega};x)$ is a non-negative measurable function of $x$. If for some $t\geq 0$ we have that $X_t={\omega}$, then there is a birth in the measurable set $H\subseteq{\Lambda}$ before time $t+s$ with probability ${\mathcal{B}}({\omega}; H)s+o(s)$. Alternatively, there is a demise at the point $x\in{\omega}$ before time $t+s$ with probability ${\mathcal{D}}({\omega}_x;x)s+o(s)$.
We may give an equivalent ‘jump-hold’ description of the chain, as follows. Let $$\label{car1}
{\mathcal{A}}({\omega}):={\mathcal{B}}({\omega};{\Lambda})+\sum_{x\in{\omega}}{\mathcal{D}}({\omega}_x;x).$$ For $A\in{\mathcal{F}}_{\Lambda}$ let $$\label{car2}
{\mathcal{K}}({\omega},A):=\frac{1}{{\mathcal{A}}({\omega})}\Big({\mathcal{B}}({\omega};\{x\in{\Lambda}:{\omega}^x\in A\})
+\sum_{\substack{x\in{\omega}\\{\omega}_x\in A}}{\mathcal{D}}({\omega}_x;x)\Big).$$ Then given that $X_t={\omega}$, the holding time until the next transition has the exponential distribution with parameter ${\mathcal{A}}({\omega})$; once the process jumps it goes to some state $\xi\in A$ with probability ${\mathcal{K}}({\omega},A)$. Existence and basic properties of such Markov chains are discussed in [@preston75].
We will aim to construct such chains $X$ which are in detailed balance with a given probability measure $\psi$ on $({\Omega},{\mathcal{F}}_{\Lambda})$. It will be necessary to make some assumptions on $\psi$, and these will be stated when appropriate. For now the main assumption we make is the following. Let ${\kappa}=\mu_{1,1,1,}$ denote the probability measure on $({\Omega},{\mathcal{F}}_{\Lambda})$ given by letting $B,G,D$ all be independent Poisson processes of constant intensity $1$.
\[MC\_ass\_1\] The probability measure $\psi$ is absolutely continuous with respect to ${\kappa}$; there exists a version of the density $$f=\frac{d\psi}{d{\kappa}}$$ which has the property that for all ${\omega}\in{\Omega}$ and $x\in{\Lambda}$, if $f({\omega})=0$ then $f({\omega}^x)=0$.
The space–time percolation measures (restricted to ${\Lambda}$) satisfy Assumption \[MC\_ass\_1\], because by standard properties of Poisson processes, if $\mu=\mu_{{\lambda},{\delta},{\gamma}}$ then a version of the density is given by $$\frac{d\mu}{d{\kappa}}({\omega})\propto \prod_{x\in B}{\lambda}(x)
\prod_{y\in D}{\delta}(y)\prod_{z\in G}{\gamma}(z).$$ Moreover, the random-cluster measure $\phi^b_{\Lambda}=\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}$ also satisfies Assumption \[MC\_ass\_1\], having density $$\frac{d\phi^b_{\Lambda}}{d{\kappa}}({\omega})=
\frac{d\phi^b_{\Lambda}}{d\mu}({\omega})\frac{d\mu}{d{\kappa}}({\omega})
\propto q^{k^b_{\Lambda}({\omega})}\prod_{x\in B}{\lambda}(x)
\prod_{y\in D}{\delta}(y)\prod_{z\in G}{\gamma}(z)$$ against ${\kappa}$.
The *Papangelou intensity* of $\psi$ is the function ${\iota}:{\Omega}\times{\Lambda}\rightarrow{\mathbb{R}}$ given by $${\iota}({\omega},x)=\frac{f({\omega}^x)}{f({\omega})}$$ (where we take $0/0$ to be 0).
The following construction will not itself be used, but serves as a helpful illustration. To construct a birth-and-death chain which has equilibrium distribution $\psi$ we would simply take ${\mathcal{D}}\equiv 1$ and ${\mathcal{B}}({\omega};dx)={\iota}({\omega},x)dx$. (Here $dx$ denotes Lebesgue measure on $F\cup(K\times\{\mathrm{d}\})\cup(K\times\{\mathrm{g}\})$.) The corresponding chain $X$ is in detailed balance with $\psi$, since $d\psi({\omega}_x)\cdot{\mathcal{B}}({\omega}_x; dx)=d{\kappa}({\omega}_x)f({\omega}^x)dx=d\psi({\omega}^x)\cdot 1$. In light of this one may may think of ${\iota}({\omega},x)$ as the intensity with which the chain $X$, in equilibrium with $\psi$, attracts a birth at $x$.
For the random-cluster measure $\phi^b_{\Lambda}$, $$\label{rcm_intensity}
{\iota}({\omega},x)=
q^{k^b_{\Lambda}({\omega}^x)-k^b_{\Lambda}({\omega})}\cdot
\left\{\begin{array}{ll}
{\lambda}(x), & \mbox{for $x$ a bridge} \\
{\delta}(x), & \mbox{for $x$ a death} \\
{\gamma}(x), & \mbox{for $x$ a ghost-bond}.
\end{array}\right.$$
In the rest of this section we let $\psi,\psi_1,\psi_2$ be three probability measures satisfying Assumption \[MC\_ass\_1\], and let $f,f_1,f_2$ and ${\iota},{\iota}_1,{\iota}_2$ denote their density functions against ${\kappa}$ and their Papangelou intensities, respectively.
\[preston\_lattice\] We say that the pair $(\psi_1,\psi_2)$ satisfies the *lattice condition* if the following hold whenever ${\omega}\geq\xi$:
1. ${\iota}_1({\omega},x)\geq {\iota}_2(\xi,x)$ whenever $x$ is a bridge or ghost-bond such that $\xi^x\not\leq{\omega}$;
2. ${\iota}_2(\xi,x)\geq {\iota}_1({\omega},x)$ whenever $x$ is a death such that $\xi\not\leq{\omega}^x$.
We say that $\psi$ has the *lattice property* if the following hold whenever ${\omega}\geq\xi$:
1. ${\iota}({\omega},x)\geq {\iota}(\xi,x)$ whenever $x$ is a bridge or ghost-bond such that $\xi^x\not\leq{\omega}$;
2. ${\iota}(\xi,x)\geq {\iota}({\omega},x)$ whenever $x$ is a death such that $\xi\not\leq{\omega}^x$.
(We use the term ‘lattice’ in the above definition in the same sense as [@fkg]; ‘lattice’ is the name for any partially ordered set in which any two elements have a least upper bound and greatest lower bound.)
The next result states that ‘well-behaved’ measures $\psi_1,\psi_2$ which satisfy the lattice condition are stochastically ordered, in that $\psi_1\geq\psi_2$. Intuitively, the lattice condition implies that a chain with equilibrium distribution $\psi_1$ acquires bridges and ghost-bonds faster than, but deaths slower than, the chain corresponding to $\psi_2$. Similarly, we will see that measures with the lattice property are positively associated; a similar intuition holds in this case.
\[preston\_thm\_1\] Suppose $\psi_1,\psi_2$ satisfy the lattice condition, and that the Papangelou intensities ${\iota}_1,{\iota}_2$ are bounded. Then $\psi_1\geq\psi_2$.
\[FKG\] Suppose $\psi$ has the lattice property, and that ${\iota}$ is bounded. Then $\psi$ is positively associated.
This essentially follows from [@preston75], the main difference being that our order on ${\Omega}$ is different, in that ‘deaths count negative’. The method of [@preston75] is to couple two jump-processes $X$ and $Y$, which have the respective equilibrium distributions $\psi_1$ and $\psi_2$. One may define a jump process on the product space ${\Omega}\times{\Omega}$ in the same way as described in and ; here is the specific instance we require.
Let $T:=\{({\omega},\xi)\in{\Omega}^2:{\omega}\geq\xi\}$, and for $a,b\in{\mathbb{R}}$ write $a\vee b$ and $a\wedge b$ for the maximum and minimum of $a$ and $b$, respectively. We let $Z=(X,Y)$ be the birth-and-death process on $T$ started at $(\varnothing,\varnothing)$ and given by the ${\mathcal{A}}$ and ${\mathcal{K}}$ defined below. First, $$\begin{gathered}
{\mathcal{A}}({\omega},\xi):=\int_{\Lambda}({\iota}_1({\omega},x)\vee {\iota}_2(\xi,x))\:dx+\\
+(|B({\omega})|\vee|B(\xi)|)+(|D({\omega})|\vee|D(\xi)|)+(|G({\omega})|\vee|G(\xi)|).\end{gathered}$$ Write ${\omega}\cap\xi$ for the element $(B({\omega})\cap B(\xi),D({\omega})\cap D(\xi),G({\omega})\cap G(\xi))$ of ${\Omega}$; similarly let ${\omega}\setminus\xi=(B({\omega}){\setminus}B(\xi),D({\omega}){\setminus}D(\xi),G({\omega}){\setminus}G(\xi))$. For $A\subseteq T$ measurable in the product topology, let $${\mathcal{K}}({\omega},\xi;A):=\frac{1}{{\mathcal{A}}({\omega},\xi)}
\big({\mathcal{K}}_{\mathrm{b}}({\omega},\xi;A)+{\mathcal{K}}_{\mathrm{d}}({\omega},\xi;A)\big)$$ where $$\begin{gathered}
{\mathcal{K}}_{\mathrm{d}}({\omega},\xi;A):=|\{x\in{\omega}\cap\xi:({\omega}_x,\xi_x)\in A\}|+\\
+|\{x\in{\omega}{\setminus}\xi:({\omega}_x,\xi)\in A\}|
+|\{x\in\xi{\setminus}{\omega}:({\omega},\xi_x)\in A\}|\end{gathered}$$ and $$\begin{gathered}
{\mathcal{K}}_{\mathrm{b}}({\omega},\xi;A):=
\int_{\Lambda}{\hbox{\rm 1\kern-.27em I}}_A({\omega}^x,\xi^x)({\iota}_1({\omega},x)\wedge {\iota}_2(\xi,x))\:dx+\\
+\int_{\Lambda}{\hbox{\rm 1\kern-.27em I}}_A({\omega}^x,\xi)[{\iota}_1({\omega},x)-({\iota}_1({\omega},x)\wedge {\iota}_2(\xi,x))]\:dx+\\
+\int_{\Lambda}{\hbox{\rm 1\kern-.27em I}}_A({\omega},\xi^x)[{\iota}_2(\xi,x)-({\iota}_1({\omega},x)\wedge {\iota}_2(\xi,x))]\:dx.\end{gathered}$$ Thanks to the lattice condition, $Z$ is indeed a process on $T$. In other words, if ${\omega}\geq\xi$ then ${\mathcal{K}}({\omega},\xi;T)=1$. It is also not hard to see that $X$ and $Y$ are birth-and-death processes on ${\Omega}$ with transition intensities ${\mathcal{B}}_1,{\mathcal{D}}_1$ and ${\mathcal{B}}_2,{\mathcal{D}}_2$ respectively, where ${\mathcal{D}}_k\equiv 1$ and ${\mathcal{B}}_k({\omega};dx)={\iota}_k({\omega},x)dx$, for $k=1,2$.
Define, for $n\geq 0$ and $k\in\{1,2\}$, $${\mathcal{B}}^{(n)}_k=\sup_{|{\omega}|=n}{\mathcal{B}}_k({\omega};{\Lambda}),$$ where $|{\omega}|$ is the total number of bridges, ghost-bonds and deaths in ${\omega}$. The boundedness of ${\iota}_1,{\iota}_2$ ensures that the following properties, which appear as conditions in [@preston75], hold. First, the expectation $$\label{iml2}
{\kappa}({\mathcal{B}}_k(\cdot;{\Lambda}))<{\infty},$$ and second, $$\label{iml3}
\sum_{n=1}^\infty\frac{{\mathcal{B}}_k^{(0)}\dotsb{\mathcal{B}}_k^{(n-1)}}{n!}
<\infty.$$ Theorems 7.1 and 8.1 of [@preston75] therefore combine to give that the chain $Z$ has a unique invariant distribution $P$ such that $Z_t\Rightarrow P$, and such that $P(F\times{\Omega})=\psi_1(F)$ and $P({\Omega}\times F)=\psi_2(F)$. Since $P(T)=1$, the result follows: if $A\in{\mathcal{F}}_{\Lambda}$ is increasing then $$\psi_1(A)=P({\omega}\in A,\,{\omega}\geq\xi)\geq
P(\xi\in A,\,{\omega}\geq\xi)=\psi_2(A).$$
The two technical properties and are not strictly necessary for the main results of [@preston75], as shown in [@georgii_kuneth], but they do seem necessary for the proof method in [@preston75]. See [@georgii_kuneth Remark 1.6].
Theorem \[FKG\] follows from Theorem \[preston\_thm\_1\] using the following standard argument [@holley74].
Let $g,h$ be two bounded, increasing and ${\mathcal{F}}_{\Lambda}$-measurable functions. By adding constants, if necessary, we may assume that $g,h$ are strictly positive. Let $\psi_2=\psi$ and let $\psi_1$ be given by $$f_1({\omega})=\frac{d\psi_1}{d{\kappa}}({\omega}):=\frac{h({\omega})f({\omega})}{\psi(h)}.$$ We have that $${\iota}_1({\omega},x)=\frac{h({\omega}^x)f({\omega}^x)}{h({\omega})f({\omega})},
\qquad{\iota}_2(\xi,x)=\frac{f(\xi^x)}{f(\xi)}.$$ Clearly ${\iota}_1,{\iota}_2$ are uniformly bounded; we check that $\psi_1,\psi_2$ satisfy the lattice condition. Let ${\omega}\geq\xi$. If $x$ is a bridge or a ghost-bond then $h({\omega}^x)/h({\omega})\geq1$, so by the lattice property of $\psi$ we have that ${\iota}_1({\omega},x)\geq{\iota}_2(\xi,x)$. Similarly, if $x$ is a death then $h(\xi^x)/h(\xi)\leq1$ so ${\iota}_1({\omega},x)\leq{\iota}_2(\xi,x)$, as required.
We thus have that $$\psi(gh)=\psi(h)\psi_1(g)\geq\psi(h)\psi_2(g)=
\psi(h)\psi(g).$$
For the next result we let ${\lambda},{\delta},{\gamma},{\lambda}',{\delta}',{\gamma}'$ be non-negative, bounded and Borel-measurable, and write ${\lambda}'\geq{\lambda}$ if ${\lambda}'$ is pointwise no less that ${\lambda}$ (and similarly for other functions). For $a\in{\mathbb{R}}$, write $a{\lambda}$ or ${\lambda}a$ for the function $x\mapsto a\cdot{\lambda}(x)$ (and similarly for other functions). Recall also the ordering of boundary conditions defined in Section \[basics\_sec\] (page ).
\[correlation\] If $q\geq1$ and $0<q'\leq q$ then for any boundary condition $b$ we have that $$\begin{split}
\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}&\leq \phi^b_{{\Lambda};q',{\lambda}',{\delta}',{\gamma}'},\qquad
\mbox{if } {\lambda}'\geq{\lambda},\:{\delta}'\leq{\delta}\mbox{ and } {\gamma}'\geq{\gamma}\\
\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}&\geq \phi^b_{{\Lambda};q',{\lambda}',{\delta}',{\gamma}'},\qquad
\mbox{if } {\lambda}'\leq{\lambda}q'/q,\:{\delta}'\geq{\delta}q/q', \mbox{ and }
{\gamma}'\leq{\gamma}q'/q.
\end{split}$$ Moreover, if $b'\geq b$ are two boundary conditions, then $$\phi^{b'}_{{\Lambda};q,{\lambda},{\delta},{\gamma}}\geq \phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}.$$
\[comp\_perc\] Let $b$ be any boundary condition. If $q\geq 1$ then $$\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}\leq \mu_{{\lambda},{\delta},{\gamma}}\quad\mbox{and}\quad
\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}\geq \mu_{{\lambda}/q,q{\delta},{\gamma}/q}$$ and if $0<q<1$ then $$\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}\geq \mu_{{\lambda},{\delta},{\gamma}}\quad\mbox{and}\quad
\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}\leq \mu_{{\lambda}/q,q{\delta},{\gamma}/q}.$$
We prove the first inequality; the rest are similar. The proof (given Theorem \[preston\_thm\_1\]) is completely analogous to the one for the discrete random-cluster model, see [@grimmett_rcm Theorem 3.21]. Recall the formula for ${\iota}(\cdot,\cdot)$ in the random-cluster case. Let $\psi_1=\phi^b_{{\Lambda};q',{\lambda}',{\delta}',{\gamma}'}$ and $\psi_2=\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}$. Clearly ${\iota}_1,{\iota}_2\leq qr$ for all ${\omega},x$, where $r$ is an upper bound for all of ${\lambda},{\delta},{\gamma},{\lambda}',{\delta}',{\gamma}'$. Let us check the lattice conditions of Definition \[preston\_lattice\]. Let ${\omega}\leq\xi$ and let $x$ be a bridge such that $\xi^x\not\leq{\omega}$. Then ${\iota}_1({\omega},x)={\lambda}'(x)(q')^{k^b_{\Lambda}({\omega}^x)-k^b_{\Lambda}({\omega})}$ and ${\iota}_2(\xi,x)={\lambda}(x)q^{k^b_{\Lambda}(\xi^x)-k^b_{\Lambda}(\xi)}$. Note that the powers of $q,q'$ both take values in $\{0,-1\}$. Since ${\lambda}'\geq{\lambda}$ and $q'\leq q$, we are done if we show that $k^b_{\Lambda}({\omega}^x)-k^b_{\Lambda}({\omega})\geq k^b_{\Lambda}(\xi^x)-k^b_{\Lambda}(\xi)$. The left-hand-side is $-1$ if and only if $x$ ties together two different components of ${\omega}$. But if it does, then certainly it does the same to $\xi$ since $\xi\leq{\omega}$; so then also the right-hand-side is $-1$, as required. It follows that ${\iota}_1({\omega},x)\geq {\iota}_2(\xi,x)$. The cases when $x$ is a death or a ghost-bond are similar.
\[rc\_fkg\] Let $q\geq1$. The random-cluster measure $\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}$ is positively associated.
Presumably positive association fails when $q<1$, as it does in the discrete random-cluster model.
We only have to verify that $\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}$ has the lattice property. Since $q\geq1$ this follows from the fact that $k^b_{\Lambda}({\omega}^x)-k^b_{\Lambda}({\omega})\geq k^b_{\Lambda}(\xi^x)-k^b_{\Lambda}(\xi)$ if ${\omega}\geq\xi$ and $x$ is a bridge or ghost-bond, and the other way around if $x$ is a death, as in the proof of Theorem \[correlation\].
The next result is a step towards the ‘finite energy property’ of Lemma \[fin\_en\]; it provides upper and lower bounds on the probabilities of seeing or not seeing any bridges, deaths or ghost-bonds in small regions. These bounds are useful because they are uniform in ${\Lambda}$. For the statement of the result, we let $q>0$, let ${\Lambda}=(K,F)$ be a region and $I\subseteq K$ and $J\subseteq F$ intervals. Define $${\overline}{\lambda}=\sup_{x\in J}{\lambda}(x),\qquad {\underline}{\lambda}=\inf_{x\in J}{\lambda}(x)$$ and similarly for ${\overline}{\delta},{\underline}{\delta},{\overline}{\gamma},{\underline}{\gamma}$ with $J$ replaced by $I$. Write $$\begin{aligned}
\eta_{\lambda}&=\min\{e^{-{\overline}{\lambda}|J|},e^{-{\overline}{\lambda}|J|/q}\},&
\qquad \eta^{\lambda}&={\mathrm{max}}\{e^{-{\underline}{\lambda}|J|},e^{-{\underline}{\lambda}|J|/q}\},
\nonumber\\
\eta_{\delta}&=\min\{e^{-{\overline}{\delta}|I|},e^{-q{\overline}{\delta}|I|}\},&
\qquad \eta^{\delta}&={\mathrm{max}}\{e^{-{\underline}{\delta}|I|},e^{-q{\underline}{\delta}|I|}\},
\nonumber\\
\eta_{\gamma}&=\min\{e^{-{\overline}{\gamma}|I|},e^{-{\overline}{\gamma}|I|/q}\},&
\qquad \eta^{\gamma}&={\mathrm{max}}\{e^{-{\underline}{\gamma}|I|},e^{-{\underline}{\gamma}|I|/q}\}.
\nonumber\end{aligned}$$ These are to be interpreted as six distinct quantities.
\[fin\_en1\] For any boundary condition $b$ we have that $$\begin{aligned}
\eta_{\lambda}&\leq\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}(|B\cap J|=0\mid{\mathcal{F}}_{{\Lambda}{\setminus}J})
\leq \eta^{\lambda}\nonumber\\
\eta_{\delta}&\leq\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}(|D\cap I|=0\mid{\mathcal{F}}_{{\Lambda}{\setminus}I})
\leq \eta^{\delta}\nonumber\\
\eta_{\gamma}&\leq\phi^b_{{\Lambda};q,{\lambda},{\delta},{\gamma}}(|G\cap I|=0\mid{\mathcal{F}}_{{\Lambda}{\setminus}I})
\leq \eta^{\gamma}\nonumber\end{aligned}$$
Follows from Proposition \[cond\_meas\_rcm\] and Corollary \[comp\_perc\].
It is convenient, but presumably not optimal, to deduce finite energy from stochastic ordering as we have done here. For discrete models it is straightforward to prove the analog of Proposition \[fin\_en1\] without using stochastic domination, see [@grimmett_rcm Theorem 3.7].
### The FKG-inequality for the Ising model {#ising_fkg_sec}
There is a natural partial order on the set ${\Sigma}^{b,{\alpha}}_{\Lambda}$ of space–time Ising configurations, given by: ${\sigma}\geq\tau$ if ${\sigma}_x\geq\tau_x$ for all $x\in K$. In Section \[ising\_uniq\_sec\] we will require a [[fkg]{}]{}-inequality for the Ising model, and we prove such a result in this section. It will be important to have a result that is valid for all boundary conditions $(b,{\alpha})$ of Ising type, and when the function ${\gamma}$ is allowed to take negative values. The result will be proved by expressing the space–time Ising measure as a weak limit of discrete Ising measures, for which the [[fkg]{}]{}-inequality is known. The same approach was used for the space–time percolation model in [@bezuidenhout_grimmett]. We let ${\lambda},{\delta}$ denote non-negative functions, as before, and we let $b=\{P_1,\dotsc,P_m\}$ and ${\alpha}$ be fixed.
Recall that $K$ consists of a collection of disjoint intervals $I^v_i$. Write ${\mathcal{E}}$ for the set of endpoints $x$ of the $I^v_i$ for which $x\in K$. Similarly, each $P_i\setminus\{{\Gamma}\}$ is a finite union of disjoint intervals; write ${\mathcal{B}}$ for the set of endpoints $y$ of these intervals for which $y\in K$. For ${\varepsilon}>0$, let $$K^{\varepsilon}={\mathcal{E}}\cup{\mathcal{B}}\cup\{(v,{\varepsilon}k)\in K: k\in{\mathbb{Z}}\}.$$ Let ${\Sigma}^{\varepsilon}$ denote the set of vectors ${\sigma}'\in\{-1,+1\}^{K^{\varepsilon}\cup\{{\Gamma}\}}$ that respect the boundary condition $(b,{\alpha})$; that is, (i) if $x,y\in K^{\varepsilon}\cup\{{\Gamma}\}$ are such that $x,y\in P_i$ for some $i$, then ${\sigma}'_x={\sigma}'_y$, and (ii) if in addition ${\alpha}(i)\neq 0$ then ${\sigma}'_x={\alpha}(i)$. For each $x=(v,t)\in K^{\varepsilon}$, let $t'>t$ be maximal such that the interval $I_{\varepsilon}(x):=v\times[t,t')$ lies in $K$ but contains no other element of $K^{\varepsilon}$; if no such $t'$ exists let $I_{\varepsilon}(x):=\{x\}$. See Figure \[discr\_fig\].
![Discretized Ising model. $K$ is drawn as solid vertical lines, and is the union of four closed, disjoint intervals. Dotted lines indicate the levels $k{\varepsilon}$ for $k\in{\mathbb{Z}}$. Elements of $K^{\varepsilon}$ are drawn as black dots. The interval $J=uv\times\{[s,s')\cap[t,t')\}$, which appears in the integral in , is drawn grey. In this illustration $b={\mathrm{f}}$.[]{data-label="discr_fig"}](thesis.38)
We now define the appropriate coupling constants for the discretized model. Let $x,y\in K^{\varepsilon}$, $x\neq y$. First suppose $I_{\varepsilon}(x)$ and $I_{\varepsilon}(y)$ share an endpoint, which we may assume to be the right endpoint of $I_{\varepsilon}(x)$. Then define $$\label{pp1}
p^{\varepsilon}_{xy}=1-\int_{I_{\varepsilon}(x)}{\delta}(z)\, dz.$$ Next, suppose $x=(u,s)$ and $y=(v,t)$ are such that $uv\in E$, and such that $I_{\varepsilon}(x)=\{u\}\times[s,s')$ and $I_{\varepsilon}(y)=\{v\}\times[t,t')$ satisfy $[s,s')\cap[t,t')\neq{\varnothing}$. Then let $J=uv\times\{[s,s')\cap[t,t')\}$ and define $$\label{pp2}
p^{\varepsilon}_{xy}=\int_J{\lambda}(e)\, de.$$ For all other $x,y\in K^{\varepsilon}$ we let $p^{\varepsilon}_{xy}=0$. Finally, for all $x\in K^{\varepsilon}$ define $$\label{pp3}
p^{\varepsilon}_{x{\Gamma}}=\int_{I_{\varepsilon}(x)}{\gamma}(z)\, dz.$$ Note that $p^{\varepsilon}_{x{\Gamma}}$ can be negative.
Let $J^{\varepsilon}_{xy}$ and $h^{\varepsilon}_x$ ($x,y\in K^{\varepsilon}$) be defined by $$\label{js}
1-p^{\varepsilon}_{xy}=e^{-2J^{\varepsilon}_{xy}},\qquad
1-p^{\varepsilon}_{x{\Gamma}}=e^{-2h^{\varepsilon}_x}.$$ Let $\pi'_{\varepsilon}$ be the Ising measure on ${\Sigma}^{\varepsilon}$ with these coupling constants, that is $$\label{mrk2}
\pi'_{\varepsilon}({\sigma}')=\frac{1}{Z^{\varepsilon}}\exp\Big(
\frac{1}{2}\sum_{x,y\in K^{\varepsilon}}J^{\varepsilon}_{xy}{\sigma}'_x{\sigma}'_y+
\sum_{x\in K^{\varepsilon}}h^{\varepsilon}_x{\sigma}'_x{\alpha}_{\Gamma}\Big),$$ where $Z^{\varepsilon}$ is the appropriate normalizing constant. In the special case when ${\gamma}\geq 0$ and $(b,{\alpha})$ is simple, all the $p^{\varepsilon}_{xy}$ and $p^{\varepsilon}_{x{\Gamma}}$ lie in $[0,1]$ for ${\varepsilon}$ sufficiently small, and $\pi'_{\varepsilon}$ is coupled via the standard Edwards–Sokal measure [@grimmett_rcm Theorem 1.10] to the $q=2$ random-cluster measure with these edge-probabilities.
There is a natural way to map each element ${\sigma}'\in{\Sigma}^{\varepsilon}$ to an element ${\sigma}$ of ${\Sigma}^{\mathrm{f}}_{\Lambda}$, namely by letting ${\sigma}$ take the value ${\sigma}'_x$ throughout $I_{\varepsilon}(x)$. Let $\pi_{\varepsilon}$ denote the law of ${\sigma}$ under this mapping. By a direct computation using (for example by splitting off the factor corresponding to ‘vertical’ interactions in the sum over $x,y$) one may see that $$\label{mrk3}
\pi_{\varepsilon}\Rightarrow{\langle}\cdot{\rangle}^{b,{\alpha}}_{\Lambda}\quad\mbox{as }{\varepsilon}\downarrow 0,$$ where ${\langle}\cdot{\rangle}^{b,{\alpha}}_{\Lambda}$ is the space–time Ising measure defined at .
For $S\in{\mathcal{G}}_{\Lambda}$ an event, we write $\partial S$ for the boundary of $S$ in the Skorokhod metric. We say that $S$ is a *continuity set* if ${\langle}{\hbox{\rm 1\kern-.27em I}}_{\partial S}{\rangle}^{b,{\alpha}}_{\Lambda}=0$. By standard facts about weak convergence, implies that $\pi_{\varepsilon}(S)\rightarrow{\langle}{\hbox{\rm 1\kern-.27em I}}_S{\rangle}^{b,{\alpha}}_{\Lambda}$ for each continuity set $S$. Note that $\partial(S\cap T){\subseteq}\partial S\cup \partial T$, so if $S,T\in{\mathcal{G}}_{\Lambda}$ are continuity sets then so is $S\cap T$.
\[ising\_fkg\] Let $S,T\in{\mathcal{G}}_{\Lambda}$ be increasing continuity sets. Then $${\langle}{\hbox{\rm 1\kern-.27em I}}_{S\cap T}{\rangle}^{b,{\alpha}}_{\Lambda}\geq
{\langle}{\hbox{\rm 1\kern-.27em I}}_{S}{\rangle}^{b,{\alpha}}_{\Lambda}{\langle}{\hbox{\rm 1\kern-.27em I}}_{T}{\rangle}^{b,{\alpha}}_{\Lambda}.$$
By the standard [[fkg]{}]{}-inequality for the classical Ising model, we have for each ${\varepsilon}>0$ that $$\pi_{\varepsilon}(S\cap T)\geq\pi_{\varepsilon}(S)\pi_{\varepsilon}(T).$$ The result follows from .
In the next result, we write ${\langle}\cdot{\rangle}_{\gamma}$ for the space–time Ising measure ${\langle}\cdot{\rangle}^{b,{\alpha}}_{\Lambda}$ with ghost-field ${\gamma}$.
\[ising\_mon\] Let $S$ be an increasing continuity set, and let ${\gamma}_1\geq{\gamma}_2$ pointwise. Then ${\langle}{\hbox{\rm 1\kern-.27em I}}_S{\rangle}_{{\gamma}_1}\geq{\langle}{\hbox{\rm 1\kern-.27em I}}_S{\rangle}_{{\gamma}_2}$.
Follows from and the fact that $\pi'_{\varepsilon}$ is increasing in ${\gamma}$.
\[cty\_ex\] Here is an example of a continuity set. Let $R{\subseteq}K$ be a finite union of intervals, some of which may consist of a single point. Let $a\in\{-1,+1\}$. Then the event $$S=\{{\sigma}\in{\Sigma}:{\sigma}_x=a\mbox{ for all }x\in R\}$$ is a continuity set, since ${\sigma}\in\partial S$ only if ${\sigma}$ changes value exactly on an endpoint of one of the intervals constituting $R$.
The assumption above that $S,T$ be continuity sets is an artefact of the proof method and can presumably be removed. It should be possible to establish versions of Theorems \[preston\_thm\_1\] and \[FKG\] also for Ising spins, using a Markov chain approach. The auxiliary process $D$ complicates this. The author would like to thank Jeffrey Steif for pointing out an error in an earlier version of this subsection.
### Correlation inequalities for the Potts model {#gks_sec}
A cornerstone in the study of the classical Ising model is provided by the so-called [[gks]{}]{}- or Griffiths’ inequalities (see [@griffiths67_I; @griffiths67_II; @kelly_sherman]) which state that certain covariances are non-negative. Recently, in [@ganikhodjaev_razak] and [@grimmett_gks], it was demonstrated that these inequalities follow from the [[fkg]{}]{}-inequality for the random-cluster representation, using an argument that also extends to the Potts models. In this section we adapt the methods of [@grimmett_gks] to the space–time setting.
Let $q\geq 2$ be fixed, ${\Lambda}$ a fixed region, and $b$ a fixed random-cluster boundary condition. We let ${\alpha}$ be such that $(b,{\alpha})$ is a simple boundary condition with ${\alpha}_{\Gamma}=q$. It is important to note that the proofs in this section are only valid for this choice of ${\alpha}$. Therefore, some of the results here are less general than what we require for detailed study of the space–time Ising model, and we will then resort to the results of the previous subsection.
Let $\pi,\phi$ denote the Potts- and random-cluster measures with the given parameters, respectively. We will be using the complex variables $${\sigma}_x=\exp\Big(\frac{2\pi i\nu_x}{q}\Big),$$where $i=\sqrt{-1}$. Note that when $q=2$ this agrees with the previous definition on page . (In [@grimmett_gks] many alternative possibilities for ${\sigma}$ are explored; similar results hold at the same level of generality here, but we refrain from treating this added generality for simplicity of presentation.)
Define for $A\subseteq K$ a finite set $${\sigma}_A:=\prod_{x\in A}{\sigma}_x.$$ More generally, if ${\underline}r=(r_x:x\in A)$ is a vector of integers indexed by $A$, define $${\sigma}^{{\underline}r}_A:=\prod_{x\in A}{\sigma}^{r_x}_x.$$ Thus ${\sigma}_A\equiv{\sigma}^{{\underline}1}_A$ where ${\underline}1$ is a constant vector of $1$’s. The set $B$ in the following should not be confused with the bridge-set $B=B({\omega})$.
\[gks\_lem\] Let $A,B\subseteq K$ be finite sets, not necessarily disjoint, and let ${\underline}r=(r_x:x\in A)$ and ${\underline}s=(s_y:y\in B)$. Then $$\label{gks_1_eq}
\pi({\sigma}_A^{{\underline}r})\geq 0$$ and $$\label{gks_2_eq}
\pi({\sigma}_A^{{\underline}r};{\sigma}_B^{{\underline}s})
:= \pi({\sigma}_A^{{\underline}r}{\sigma}_B^{{\underline}s}) -
\pi({\sigma}_A^{{\underline}r})\pi({\sigma}_B^{{\underline}s}) \geq 0.$$ In particular, $\pi({\sigma}_A)\geq 0$ and $\pi({\sigma}_A;{\sigma}_B) \geq 0$.
A result similar to Lemma \[gks\_lem\] holds for $A,B\subseteq{\overline}K$, but then care must be taken to define ${\sigma}_x$ appropriately for points $x\in\partial{\Lambda}$ that do not lie in ${\Lambda}$. For example, if $x=(v,t)$ is an isolated point in ${\mathbb{K}}\setminus K$ then the corresponding result holds if we replace ${\sigma}_x$ by one of ${\sigma}_{x+}$ or ${\sigma}_{x-}$, where ${\sigma}_{x+}=\lim_{{\varepsilon}\downarrow0}{\sigma}_{(v,t+{\varepsilon})}$ and ${\sigma}_{x-}=\lim_{{\varepsilon}\downarrow0}{\sigma}_{(v,t-{\varepsilon})}$ (these limits exist almost surely but are in general different for such $x$).
For ${\omega}\in{\Omega}$ let $k=k^b_{\Lambda}({\omega})$, and let $C_1({\omega}),\dotsc,C_{k}({\omega})$ denote the components of ${\omega}$ in ${\Lambda}$, defined according to the boundary condition $b$. We assume that ${\Gamma}\in C_k({\omega})$, and thus $C_1({\omega}),\dotsc,C_{k-1}({\omega})$ are the ‘${\Gamma}$-free’ components of ${\omega}$. Lemma \[gks\_lem\] will follow from Theorems \[correlation\] and \[rc\_fkg\] using the following representation.
\[gks\_fkg\_lem\] Let ${\underline}r=(r_x:x\in A)$ and write $r_j=\sum_{x\in A\cap C_j}r_x$ (for $j=1,\dotsc,k-1$). Then $$\pi({\sigma}^{{\underline}r}_A)=\phi(r_j\equiv 0
\mbox{ (mod $q$), for } j=1,\dotsc,k-1).$$
Note that the event on the right-hand-side is increasing; also note that if $r_x=1$ for all $x$ then $r_j=|A\cap C_j|$.
Let $U_1,U_2,\dotsc$ be independent random variables with the uniform distribution on $\{e^{2\pi im/q}:m=1,\dotsc,q\}$, and let ${\mathbb{P}}$ denote the Edwards–Sokal coupling of $\pi$ and $\phi$. We have that $$\label{hej2}
{\mathbb{P}}({\sigma}_A^{{\underline}r}\mid{\omega})=E\Big(1\cdot\prod_{j=1}^{k-1}U_j^{r_j}\Big)
=\prod_{j=1}^{k-1}E(U_j^{r_j}),$$ where $E$ denotes expectation over the $U_j$ (recall that $\nu_{\Gamma}=q$, so ${\sigma}_{\Gamma}=1$). Since $U_j$ is uniform we have that $$E(U_j^r)=\frac{1}{q}\sum_{m=1}^q \big(e^{2\pi i m/q}\big)^r
=\left\{
\begin{array}{ll}
1, & \mbox{if $r\equiv 0$ (mod $q$)}, \\
0, & \mbox{otherwise}.
\end{array}\right.$$ The result follows on taking the expectation of .
It is immediate from Lemma \[gks\_fkg\_lem\] that $\pi({\sigma}^{{\underline}r}_A)\geq0$, which is . For we note that ${\sigma}_A^{{\underline}r}{\sigma}_B^{{\underline}s}={\sigma}_{A\cup B}^{{\underline}t}$, where ${\underline}t$ is the vector indexed by $A\cup B$ given by $t_x=r_x+s_x$ if $x\in A\cap B$, $t_x=r_x$ if $x\in A\setminus B$, and $t_x=s_x$ if $x\in B\setminus A$. Thus, with the obvious abbreviations, $$\begin{aligned}
\pi({\sigma}^{{\underline}r}_A{\sigma}^{{\underline}s}_B)&=\phi(t_j\equiv 0\;\forall j)\\
&\geq\phi(r_j\equiv 0\;\forall j\mbox{ and }
s_j\equiv 0\;\forall j)\\
&\geq \phi(r_j\equiv 0\;\forall j)\phi(s_j\equiv 0\;\forall j)\\
&=\pi({\sigma}^{{\underline}r}_A)\pi({\sigma}^{{\underline}s}_B),\end{aligned}$$ where the second inequality follows from positive association of $\phi$, Theorem \[rc\_fkg\].
In the Ising model, the covariance is related to the derivative of ${\langle}{\sigma}_A{\rangle}$ with respect to the coupling strengths; thus it follows from that ${\langle}{\sigma}_A{\rangle}$ is increasing in these quantities. Here is the corresponding result for the Potts model.
Let $A\subseteq K$ be a finite set, and let $R\subseteq K$ be a finite union of positive length intervals whose interiors are disjoint from $A$. We write ${\Lambda}'$ for the region corresponding to $K'=K{\setminus}R$. If $b=(P_1,\dotsc,P_m)$ we define the boundary condition $b'=(P'_1,\dotsc,P'_m)$, where $P'_i=P_i\setminus R$. Thus $b'$ agrees with $b$ on ${\hat\partial}{\Lambda}$, but is ‘free’ on ${\hat\partial}{\Lambda}'{\setminus}{\hat\partial}{\Lambda}$. See Figure \[R\_fig\]. Similar results hold for other $b'$.
![Left: a region ${\Lambda}$ with the boundary condition $b=\{P_1\}$, where $P_1\setminus\{{\Gamma}\}$ is drawn bold. Right: the corresponding region ${\Lambda}'$ when the set $R$, drawn dashed, has been removed; the boundary condition is $b'=\{P'_1\}$ where $P'_1=P_1\setminus R$ and $P_1'\setminus\{{\Gamma}\}$ is drawn bold. In this picture we have not specified which endpoints of $R$ belong to $R$.[]{data-label="R_fig"}](thesis.36 "fig:") ![Left: a region ${\Lambda}$ with the boundary condition $b=\{P_1\}$, where $P_1\setminus\{{\Gamma}\}$ is drawn bold. Right: the corresponding region ${\Lambda}'$ when the set $R$, drawn dashed, has been removed; the boundary condition is $b'=\{P'_1\}$ where $P'_1=P_1\setminus R$ and $P_1'\setminus\{{\Gamma}\}$ is drawn bold. In this picture we have not specified which endpoints of $R$ belong to $R$.[]{data-label="R_fig"}](thesis.37 "fig:")
\[cor\_mon\_lem\] The average $\pi^b_{\Lambda}({\sigma}^{{\underline}r}_A)$ is increasing in ${\lambda}$ and ${\gamma}$ and decreasing in ${\delta}$. Moreover, $$\label{cor_mon_eq}
\pi^{b'}_{{\Lambda}'}({\sigma}^{{\underline}r}_A)\leq \pi^b_{\Lambda}({\sigma}^{{\underline}r}_A).$$
We interpret $\pi^{b'}_{{\Lambda}'}({\sigma}^{{\underline}r}_A)$ as $0$ when $A$ intersects the interior of $R$.
The claim about monotonicity in ${\lambda},{\gamma},{\delta}$ follows from the stochastic ordering of random-cluster measures, Theorem \[correlation\], and the representation in Lemma \[gks\_fkg\_lem\]. Let us prove . It suffices to consider the case when $R=I$ is a single interval. First note that $$\pi^b_{\Lambda}({\sigma}^{{\underline}r}_A)=\phi^b_{\Lambda}(T)\geq\tilde\phi^b_{\Lambda}(T),$$ where $T$ is the event on the right-hand-side of Lemma \[gks\_fkg\_lem\], and $\tilde\phi^b_{\Lambda}$ is the measure $\phi^b_{\Lambda}$ with ${\gamma}$ set to zero on $I$, and ${\lambda}(e)$ set to zero whenever $e\not\in F'$. Hence, using also Corollary \[del\_contr\], $$\label{car3}
\pi^{b'}_{{\Lambda}'}({\sigma}^{{\underline}r}_A)=\phi^{b'}_{{\Lambda}'}(T)=
\tilde\phi^{b}_{\Lambda}(T\mid D\cap I\neq\varnothing)\leq
\frac{\tilde\phi^b_{\Lambda}(T)}{1-e^{-{\delta}(I)}}\leq
\frac{\pi^b_{\Lambda}({\sigma}^{{\underline}r}_A)}{1-e^{-{\delta}(I)}},$$ where $${\delta}(I)=\int_I{\delta}(x)dx.$$ The left-hand-side of does not depend on the value of ${\delta}$ on $I$, so we may let ${\delta}\rightarrow\infty$ on $I$ to deduce the result.
Here is a consequence of Lemma \[gks\_lem\] when ${\underline}r$ is not constant. Let $x,y\in K$, and write $\tau_{xy}={\sigma}_x{\sigma}_y^{-1}$. Then $\tau_{xy}$ is a $q$th root of unity, and it follows that $${\hbox{\rm 1\kern-.27em I}}\{\nu_x=\nu_y\}={\hbox{\rm 1\kern-.27em I}}\{{\sigma}_x={\sigma}_y\}=\frac{1}{q}\sum_{r=0}^{q-1}\tau_{xy}^r.$$ So if $z,w\in K$ too then $$\label{tjo4}
\begin{split}
\pi^b_{\Lambda}(\nu_x=\nu_y,\,\nu_z=\nu_w)&=
\frac{1}{q^2}\sum_{r,s=0}^{q-1}\pi^b_{\Lambda}(\tau_{xy}^r\tau_{zw}^s)\\
&\geq\frac{1}{q^2}\sum_{r,s=0}^{q-1}
\pi^b_{\Lambda}(\tau_{xy}^r)\pi^b_{\Lambda}(\tau_{zw}^s)\\
&=\pi^b_{\Lambda}(\nu_x=\nu_y)\pi^b_{\Lambda}(\nu_z=\nu_w).
\end{split}$$ This inequality does not quite follow from the correlation/connection property of Proposition \[corr\_conn\_prop\] when $q>2$. In the case when ${\gamma}=0$ it follows straight away from the Edwards–Sokal coupling, without using stochastic domination properties of the random-cluster model; see [@ghm Corollary 6.5].
Infinite-volume random-cluster measures {#inf_rc_sec}
---------------------------------------
In this section we define random-cluster measures on the *unbounded* spaces ${\mathbf{\Theta}},{\mathbf{\Theta}}_{\beta}$ of and , for which Definition \[rc\_def\] cannot make sense (since $k$ will be infinite). One standard approach in statistical physics is to study the class of measures which satisfy a conditioning property similar to that of Proposition \[cond\_meas\_rcm\] for all bounded regions; the first task is then to show that this class is nonempty. The book [@georgii88] is dedicated to this approach for classical models. We will instead follow the route of proving weak convergence as the bounded regions ${\Lambda}$ grow. In doing so we follow standard methods (see [@grimmett_rcm Chapter 4]), adapted to the current setting. See also [@aizenman_nacht] for results of this type.
Central to the topic of infinite-volume measures is the question when there is a unique such measure. There may in general be multiple such measures, obtainable by passing to the limit using different boundary conditions. Non-uniqueness of infinite-volume measures is intimately related to the concept of phase transition described in the Introduction. Intuitively, if there is not a unique limiting measure this means that the boundary conditions have an ‘infinite range’ effect, and that the system does not know what state to favour, indicating a transition from one preferred state to another.
### Weak limits {#rc_wl_sec}
We fix $q\geq1$ and non-negative bounded measurable functions ${\lambda},{\delta},{\gamma}$. Let $L_n$ be a sequence of subgraphs of ${\mathbb{L}}$ and ${\beta}_n$ a sequence of positive numbers. Writing ${\Lambda}_n$ for the simple region given by $L_n$ and ${\beta}_n$ as in , we say that ${\Lambda}_n\uparrow{\mathbf{\Theta}}$ if $L_n\uparrow{\mathbb{L}}$ and ${\beta}_n\rightarrow{\infty}$. We assume throughout that $L_n$ and ${\beta}_n$ are strictly increasing. Versions of the results in this section are valid also when ${\beta}<{\infty}$ is kept fixed as $L_n\uparrow{\mathbb{L}}$ so that ${\Lambda}_n\uparrow{\mathbf{\Theta}}_{\beta}$ given in . We will only supply proofs in the ${\beta}_n\rightarrow{\infty}$ case as the ${\beta}<{\infty}$ case is similar.
Recall that a sequence $\psi_n$ of probability measures on $({\Omega},{\mathcal{F}})$ is *tight* if for each ${\varepsilon}>0$ there is a compact set $A_{\varepsilon}$ such that $\psi_n(A_{\varepsilon})\geq1-{\varepsilon}$ for all $n$. Here compactness refers, of course, to the Skorokhod topology outlined in Section \[basics\_sec\] and defined in detail in Appendix \[skor\_app\].
Let $\phi^b_n:=\phi^b_{{\Lambda}_n}$. The proof of the following result is given in Appendix \[skor\_app\].
\[tight\_lem\] For any sequence of boundary conditions $b_n$ on ${\Lambda}_n$, the sequence of measures $\{\phi^{b_n}_n:n\geq 1\}$ is tight.
For $x=(e,t)\in{\mathbb{F}}$ with $t\geq0$ (respectively $t<0$), let $V_x({\omega})$ denote the number of elements of the set $B\cap(\{e\}\times[0,t])$ (respectively $B\cap(\{e\}\times(-t,0])$). Similarly, for $x\in{\mathbb{K}}\times\{\mathrm{d}\}$ and $x\in{\mathbb{K}}\times\{\mathrm{g}\}$, define $V_x$ to count the number of deaths and ghost-bonds between $x$ and the origin, respectively. An event of the form $$R=\{{\omega}\in{\Omega}: V_{x_1}({\omega})\in A_1,\dotsc,
V_{x_m}({\omega})\in A_m\}\in{\mathcal{F}}$$ for $m\geq 1$ and the $A_i\subseteq{\mathbb{Z}}$ is called a *finite-dimensional cylinder event*. For $z=(z_1,\dotsc,z_m)$ and $z'=(z'_1,\dotsc,z'_m)$ elements of ${\mathbb{Z}}^m$, we write $z'\geq z$ if $z'_i\geq z_i$ for all $i=1,\dotsc, m$; we write $z'> z$ if $z'\geq z$ and $z'\neq z$.
\[conv\_lem\] Let $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ and $q\geq1$. The sequence of measures $\phi^b_n$ converges weakly to a probability measure. The limit measure does not depend on the choice of sequence ${\Lambda}_n\uparrow{\mathbf{\Theta}}$.
The limiting measure in Theorem \[conv\_lem\] will be denoted $\phi^b$, or $\phi^{b,{\beta}}_{q,{\lambda},{\delta},{\gamma}}$ if the parameters need to be emphasized; here ${\beta}\in(0,{\infty}]$.
Consider the case $b={\mathrm{w}}$. Let ${\Lambda}$ be a simple region and $f:{\Omega}\rightarrow{\mathbb{R}}$ an increasing, ${\mathcal{F}}_{\Lambda}$-measurable function. Let $n$ be large enough so that ${\Lambda}_n\supseteq{\Lambda}$ and let ${\mathcal{C}}$ be the event that all components inside ${\Lambda}_n$ which intersect ${\hat\partial}{\Lambda}_n$ are connected in ${\Lambda}_{n+1}$. Then by Corollary \[del\_contr\] and the [[fkg]{}]{}-property we have that $$\label{wcs1}
\phi^{\mathrm{w}}_n(f)=\phi^{\mathrm{w}}_{n+1}(f\mid {\mathcal{C}})\geq \phi^{\mathrm{w}}_{n+1}(f),$$ which is to say that $\phi^{\mathrm{w}}_n\geq\phi^{\mathrm{w}}_{n+1}$. At this point we could appeal to Corollary IV.6.4 of [@lindvall], which proves that a sequence of probability measure which is tight and stochastically ordered as in necessarily converges weakly. However, we shall later need to know that the finite dimensional distributions converge, so we prove this now; it then follows from tightness and standard properties of the Skorokhod topology that the sequence converges weakly.
Let $x_1,\dotsc,x_k\in F\cup(K\times\{\mathrm{g}\})$ and let $x_{k+1},\dotsc,x_m\in K\times\{\mathrm{d}\}$. For $z=(z_1,\dotsc,z_m)\in{\mathbb{Z}}^m$, write $$\tilde z=(z_1,\dotsc,z_k,-z_{k+1},\dotsc,-z_m).$$ Let $V=V({\omega})=(V_{x_1}({\omega}),\dotsc,V_{x_m}({\omega}))$ and for $A{\subseteq}{\mathbb{Z}}^m$ consider the finite-dimensional cylinder event $R=\{V\in A\}$. We have that $$\label{wcs2}
\begin{split}
\phi^{\mathrm{w}}_n(R)&=
\sum_{z\in A}\phi^{\mathrm{w}}_n(V=z)
=\sum_{z\in A}\phi^{\mathrm{w}}_n(\tilde V=\tilde z)\\
&=\sum_{z\in A}[\phi^{\mathrm{w}}_n(\tilde V\geq\tilde z)
-\phi^{\mathrm{w}}_n(\tilde V>\tilde z)].
\end{split}$$ The events $\{\tilde V\geq\tilde z\}$ and $\{\tilde V>\tilde z\}$ are both increasing, so by the limits $${\overline}\phi(\tilde V\geq\tilde z)=\lim_{n\rightarrow{\infty}}
\phi^{\mathrm{w}}_n(\tilde V\geq\tilde z)\quad\text{and}\quad
{\overline}\phi(\tilde V>\tilde z)=\lim_{n\rightarrow{\infty}}
\phi^{\mathrm{w}}_n(\tilde V>\tilde z)$$ exist. Define ${\overline}\phi$ by $${\overline}\phi(R):=\sum_{z\in A}[{\overline}\phi(\tilde V\geq\tilde z)
-{\overline}\phi(\tilde V>\tilde z)].$$ Then, by the bounded convergence theorem, ${\overline}\phi$ defines a probability measure on the algebra of finite-dimensional cylinder events in ${\mathcal{F}}_{\Lambda}$. Thus ${\overline}\phi$ extends to a unique probability measure $\phi^{\mathrm{w}}$ on ${\mathcal{F}}_{\Lambda}$ (see [@billingsley_probmeas Theorem 3.1]). Since $\phi^{\mathrm{w}}_n(R)\rightarrow\phi^{\mathrm{w}}(R)$ for all finite-dimensional cylinder events in ${\mathcal{F}}_{\Lambda}$ and since the sequence $(\phi^{\mathrm{w}}_n:n\geq1)$ is tight, it follows that $\phi^{\mathrm{w}}_n\Rightarrow\phi^{\mathrm{w}}$ on $({\Omega},{\mathcal{F}}_{\Lambda})$. Since ${\Lambda}$ was arbitrary and the ${\mathcal{F}}_{\Lambda}$ generate ${\mathcal{F}}$, the convergence for $b={\mathrm{w}}$ follows.
For the independence of the choice of sequence ${\Lambda}_n$, let also ${\Delta}_n\uparrow{\mathbf{\Theta}}$. Let $m$ be an integer, and choose $l=l(m)$ and $n=n(m)$ so that ${\Lambda}_l\subseteq{\Delta}_m\subseteq{\Lambda}_n$. We have that $$\phi^{\mathrm{w}}_{{\Lambda}_l}\geq\phi^{\mathrm{w}}_{{\Delta}_m}\geq\phi^{\mathrm{w}}_{{\Lambda}_n},$$ so letting $m\rightarrow\infty$ tells us that the limits are equal (see Remark \[po\_rk\]).
The arguments for $b={\mathrm{f}}$ are similar.
\[po\_rk\] If $\psi_1,\psi_2$ are two probability measures on $({\Omega},{\mathcal{F}})$ such that both $\psi_1\geq\psi_2$ and $\psi_2\geq\psi_1$ then $\psi_1=\psi_2$. To see this, note that for $R$ any finite-dimensional cylinder event, we may as in write $$\psi_j(R)=\sum_{z\in A} [\psi_j(\tilde V\geq\tilde z)
-\psi_j(\tilde V>\tilde z)],\quad j=1,2.$$ It follows that $\psi_1(R)=\psi_2(R)$ for all such $R$, and hence that $\psi_1=\psi_2$ (see Appendix \[skor\_app\]).
For any sequence $b_n$ of boundary conditions, *if* the sequence of measures $(\phi^{b_n}_n:n\geq1)$ has a weak limit $\phi$, then $\phi^{\mathrm{f}}\leq\phi\leq\phi^{\mathrm{w}}$; this follows from the second part of Theorem \[strassen\_thm\]. Hence there is a unique random-cluster measure if and only if $\phi^{\mathrm{f}}=\phi^{\mathrm{w}}$. It turns out that the set of real triples $({\lambda},{\delta},{\gamma})$ such that there is *not* a unique random-cluster measure has Lebesgue measure zero, see Theorem \[rc\_uniq\_thm\].
### Basic properties
Some further properties of the measures $\phi^b$, for $b\in\{{\mathrm{f}},{\mathrm{w}}\}$, follow, all being straightforward adaptations of standard results, as summarized in [@grimmett_rcm Section 4.3]. First, recall the upper and lower bounds on the probabilities of seeing no bridges, deaths or ghost-bonds in small regions which is provided by Proposition \[fin\_en1\], as well as the notation introduced there.
\[fin\_en\] Let $q\geq1$ and let $I\subseteq{\mathbb{K}}$ and $J\subseteq{\mathbb{F}}$ be bounded intervals. Then for $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ we have that $$\begin{aligned}
\eta_{\lambda}&\leq\phi^b(|B\cap J|=0\mid{\mathcal{T}}_{J})
\leq \eta^{\lambda}\nonumber\\
\eta_{\delta}&\leq\phi^b(|D\cap I|=0\mid{\mathcal{T}}_{I})
\leq\eta^{\delta}\nonumber\\
\eta_{\gamma}&\leq\phi^b(|G\cap I|=0\mid{\mathcal{T}}_{I})
\leq \eta^{\gamma}\nonumber\end{aligned}$$
The same result holds for any weak limit of random-cluster measures with $q>0$; we assume that $q\geq1$ and $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ only because then we know that the measures $\phi^b_{\Lambda}$ converge.
Recall the notation $V_x({\omega})$ introduced before Theorem \[conv\_lem\], and note that the event $\{|B\cap J|=0\}$ is a finite-dimensional cylinder event. For $J\subseteq{\mathbb{F}}$ as in the statement, let $x_1,x_2,\dotsc$ be an enumeration of the points in $({\mathbb{K}}\times\{\mathrm{d}\})\cup({\mathbb{K}}\times\{\mathrm{g}\})\cup({\mathbb{F}}{\setminus}J)$ with rational ${\mathbb{R}}$-coordinate. We have that ${\mathcal{T}}_J={\sigma}(V_{x_1},V_{x_2},\dotsc)$ so by the martingale convergence theorem $$\phi^b(|B\cap J|=0\mid{\mathcal{T}}_{J})=
\lim_{n\rightarrow{\infty}}\phi^b(|B\cap J|=0\mid V_{x_1},\dotsc,V_{x_n}).$$ For ${\underline}z\in{\mathbb{Z}}^n$, let $A_{{\underline}z}=\{(V_{x_1},\dotsc,V_{x_n})={\underline}z\}$. Then $$\begin{split}
\phi^b(|B\cap J|=0\mid {\mathcal{F}}_n)&=
\sum_{{\underline}z\in {\mathbb{Z}}^n}\frac{\phi^b(A_{{\underline}z},\{|B\cap J|=0\})}
{\phi^b(A_{{\underline}z})}{\hbox{\rm 1\kern-.27em I}}_{A_{{\underline}z}}\\
&=\lim_{{\Delta}}\sum_{{\underline}z\in{\mathbb{Z}}^n}
\frac{\phi_{\Delta}^b(A_{{\underline}z},\{|B\cap J|=0\})}
{\phi_{\Delta}^b(A_{{\underline}z})}{\hbox{\rm 1\kern-.27em I}}_{A_{{\underline}z}}\\
&=\lim_{{\Delta}}\phi^b_{\Delta}(|B\cap J|=0\mid {\mathcal{F}}_n).
\end{split}$$ The result now follows from Proposition \[fin\_en1\]. A similar argument holds for $\{|D\cap I|=0\}$ and $\{|G\cap I|=0\}$.
Define an *automorphism* on ${\mathbf{\Theta}}$ to be a bijection $T:{\mathbf{\Theta}}\rightarrow{\mathbf{\Theta}}$ of the form $T=({\alpha},g):(x,t)\mapsto({\alpha}(x),g(t))$ where ${\alpha}:{\mathbb{V}}\rightarrow{\mathbb{V}}$ is an automorphism of the graph ${\mathbb{L}}$, and $g:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is a continuous bijection. Thus ${\alpha}$ has the property that ${\alpha}(x){\alpha}(y)\in{\mathbb{E}}$ if and only if $xy\in{\mathbb{E}}$. For $T$ an automorphism and ${\omega}=(B,D,G)\in{\Omega}$, let $T({\omega})=(T(B),T(D),T(G))$. For $f:{\Omega}\rightarrow{\mathbb{R}}$ measurable, let $(f\circ T)({\omega})=f(T({\omega}))$, and for $\phi$ a measure on $({\Omega},{\mathcal{F}})$ define $\phi\circ T(f)=\phi(T(f))$.
\[invar\_lem\] Let $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ and let $T$ be an automorphism of ${\mathbf{\Theta}}$ such that ${\lambda}={\lambda}\circ T$, ${\gamma}={\gamma}\circ T$ and ${\delta}={\delta}\circ T$. Then $\phi^b$ is invariant under $T$, that is $\phi^b=\phi^b\circ T$.
Let $f$ be a measurable function. Under the given assumptions, we have that for any region ${\Lambda}$, $$\phi^b_{{\Lambda}}(f\circ T)=\int f(T({\omega}))\,d\phi^b_{{\Lambda}}({\omega})
=\int f({\omega})\,d\phi^b_{T^{-1}({\Lambda})}({\omega})=\phi^b_{T^{-1}({\Lambda})}(f).$$ The result now follows from Theorem \[conv\_lem\].
\[tail\_triv\] The tail ${\sigma}$-algebra ${\mathcal{T}}$ is trivial under the measures $\phi^{\mathrm{f}}$ and $\phi^{\mathrm{w}}$, in that $\phi^b(A)\in\{0,1\}$ for all $A\in{\mathcal{T}}$.
Let ${\Lambda}\subseteq{\Delta}$ be two regions. We treat the case when $b={\mathrm{f}}$, the case $b={\mathrm{w}}$ follows similarly on reversing several of the inequalities below. Let $A\in{\mathcal{F}}_{\Lambda}$ be an increasing finite-dimensional cylinder event, and let $B\in{\mathcal{F}}_{{\Delta}{\setminus}{\Lambda}}\subseteq{\mathcal{T}}_{\Lambda}$ be an arbitrary finite-dimensional cylinder event. We may assume without loss of generality that $\phi^{\mathrm{f}}_{\Delta}(B)>0$. By the conditioning property Proposition \[cond\_meas\_rcm\] and the stochastic ordering of Theorem \[correlation\], we have that $$\phi^{\mathrm{f}}_{\Delta}(A\cap B)=\phi^{\mathrm{f}}_{\Delta}(A\mid B)\phi^{\mathrm{f}}_{\Delta}(B)
\geq \phi^{\mathrm{f}}_{\Lambda}(A)\phi^{\mathrm{f}}_{\Delta}(B).$$ Let ${\mathcal{R}}$ denote the set of finite-dimensional cylinder events in ${\mathcal{T}}_{\Lambda}$. Letting ${\Delta}\uparrow{\mathbf{\Theta}}$ implies that $$\label{hej3}
\phi^{\mathrm{f}}(A\cap B)\geq \phi^{\mathrm{f}}_{\Lambda}(A)\phi^{\mathrm{f}}(B)$$ for all $B\in{\mathcal{R}}$ and all increasing finite-dimensional cylinder events $A\in{\mathcal{F}}_{\Lambda}$. The set ${\mathcal{R}}$ is an algebra, so for fixed $A$ the difference between the left and right sides of extends to a finite measure $\psi$ on ${\mathcal{T}}_{\Lambda}$, and by the uniqueness of this extension it follows that $0\leq\psi(B)=\phi^{\mathrm{f}}(A\cap B)- \phi^{\mathrm{f}}_{\Lambda}(A)\phi^{\mathrm{f}}(B)$ for all $B\in{\mathcal{T}}_{\Lambda}{\subseteq}{\mathcal{T}}$. Thus we may let ${\Lambda}\uparrow{\mathbf{\Theta}}$ to deduce that $$\label{hej4}
\phi^{\mathrm{f}}(A\cap B)\geq \phi^{\mathrm{f}}(A)\phi^{\mathrm{f}}(B)$$ for all increasing finite-dimensional cylinder events $A\in{\mathcal{F}}_{\Lambda}$ and all $B\in{\mathcal{T}}$. However, also holds with $B$ replaced by its complement $B^c$; since $$\phi^{\mathrm{f}}(A\cap B)+\phi^{\mathrm{f}}(A\cap B^c)=
\phi^{\mathrm{f}}(A)\phi^{\mathrm{f}}(B)+\phi^{\mathrm{f}}(A)\phi^{\mathrm{f}}(B^c)$$ it follows that $$\label{hej5}
\phi^{\mathrm{f}}(A\cap B)= \phi^{\mathrm{f}}(A)\phi^{\mathrm{f}}(B)$$ for all increasing finite-dimensional cylinder events $A\in{\mathcal{F}}_{\Lambda}$ and all $B\in{\mathcal{T}}$. For fixed $B$, the left and right sides of are finite measures which agree on all increasing events $A\in{\mathcal{F}}_{\Lambda}$. Using the reasoning of Remark \[po\_rk\], it follows that holds for all $A\in{\mathcal{F}}_{\Lambda}$, and hence also for all $A\in{\mathcal{F}}$. Setting $A=B\in{\mathcal{T}}$ gives the result.
In the case when ${\mathbb{L}}={\mathbb{Z}}^d$ and ${\lambda},{\delta},{\gamma}$ are constant, define the automorphisms $T_x$, for $x\in{\mathbb{Z}}^d$, by $$T_x(y,t)=(y+x,t).$$ The $T_x$ are called *translations*. An event $A\in{\mathcal{F}}$ is called $T_x$-invariant if $A=T_x^{-1}A$. The following ergodicity result is a standard consequence of Proposition \[tail\_triv\], see for example [@georgii88 Proposition 14.9] (here $0$ denotes the element $(0,\dotsc,0)$ of ${\mathbb{Z}}^d$).
\[ergod\_lem\] Let $x\in{\mathbb{Z}}^d\setminus\{0\}$ and $b\in\{{\mathrm{f}},{\mathrm{w}}\}$. If $A\in{\mathcal{F}}$ is $T_x$-invariant then $\phi^b(A)\in\{0,1\}$.
### Phase transition
In the random-cluster model, the probability that there is an unbounded connected component serves as ‘order parameter’: depending on the values of the parameters ${\lambda},{\delta},{\gamma}$ this probability may be zero or positive. We show in this section that one may define a critical point for this probability, and then establish some very basic facts about the phase transition. We assume throughout this section that ${\gamma}=0$, that $q\geq1$, that ${\lambda}\geq0$, ${\delta}>0$ are constant, and that ${\mathbb{L}}={\mathbb{Z}}^d$ for some $d\geq1$. Some of the results hold for more general ${\mathbb{L}}$, but we will not pursue this here. The boundary condition $b$ will denote either ${\mathrm{f}}$ or ${\mathrm{w}}$ throughout.
Let $\{0{\leftrightarrow}{\infty}\}$ denote the event that the origin lies in an unbounded component. Define for $0<{\beta}\leq{\infty}$, $$\theta^{b,{\beta}}({\lambda},{\delta},q):=\phi^{b,{\beta}}_{q,{\lambda},{\delta}}(0\leftrightarrow{\infty}).$$When ${\beta}={\infty}$ a simple rescaling argument implies that $\theta^{b,{\infty}}({\lambda},{\delta},q)$ depends on ${\lambda},{\delta}$ through the ratio $\rho={\lambda}/{\delta}$ only. Hence we will often in what follows set ${\delta}=1$ and ${\lambda}=\rho$, and define for $0<{\beta}\leq{\infty}$ $$\theta^{b,{\beta}}(\rho)=
\theta^{b,{\beta}}(\rho,q):=\phi^{b,{\beta}}_{q,\rho,1}(0\leftrightarrow{\infty}).$$ By the stochastic monotonicity of Theorem \[correlation\], and a small argument justifying its application to the event $\{0{\leftrightarrow}{\infty}\}$, the quantity $\theta^b(\rho)$ is increasing in $\rho$.
\[crit\_def\] For $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ and $0<{\beta}\leq{\infty}$ we define the *critical value* $$\rho^{b,{\beta}}_{\mathrm{c}}(q):=\sup\{\rho\geq 0:\theta^{b,{\beta}}(q,\rho)=0\}.$$
In what follows we will usually suppress reference to ${\beta}$. We will see in Section \[press\_sec\] that $\rho^{\mathrm{f}}(q)=\rho^{\mathrm{w}}(q)$ for all $q\geq1$. Therefore we will write $\rho_{\mathrm{c}}(q)$ for their common value. We write $\phi^{b}_\rho$ for $\phi^{b,{\beta}}_{q,\rho,1}$.
One may adapt standard methods (see [@grimmett_rcm Theorem 5.5]) to prove the following:
\[crit\_nontriv\] Unless $d=1$ and ${\beta}<\infty$ we have that $$0<\rho_c(q)<\infty.$$
(If $d=1$ and ${\beta}<{\infty}$ then a standard zero-one argument, involving comparison to percolation and the second Borel–Cantelli lemma, implies that $\rho_{\mathrm{c}}=0$.)
Fix $\rho>0$ and for ${\omega}\in{\Omega}$ let $N=N({\omega})$ denote the number of distinct unbounded components in ${\omega}$. By Lemma \[ergod\_lem\], using for example the translation $T:(x,t)\mapsto(x+1,t)$, we have that $N$ is almost surely constant under the measures $\phi^b_\rho(\cdot)$, $b\in\{{\mathrm{f}},{\mathrm{w}}\}$.
\[inf\_clust\_uniq\] The number $N$ of unbounded components is either $0$ or $1$ almost surely under $\phi^b_\rho$.
We follow the strategy of [@burton_keane], and as previously we provide details only in the ${\beta}={\infty}$ case. We first show that $N\in\{0,1,{\infty}\}$ almost surely. Suppose to the contrary that there exists $2\leq m<{\infty}$ such that $N=m$ almost surely. Then we may choose (deterministic) $n,{\beta}$ sufficiently large that the corresponding simple region ${\Lambda}_n={\Lambda}_n({\beta})$, regarded as a subset of ${\mathbf{\Theta}}$, has the property that $\phi^b_\rho(A)>0$, where $A$ is the event that the $m$ distinct unbounded components all meet $\partial{\Lambda}_n$. Let $C$ be the event that all points in $\partial{\Lambda}_n$ are connected inside ${\Lambda}_n$. By the finite energy property, Lemma \[fin\_en\], we have that $\phi^b_\rho(C\mid A)>0$, and hence $\phi^b_\rho(C\cap A)>0$. But on $\{C\cap A\}$ we have $N=1$, a contradiction. Thus $N\in\{0,1,{\infty}\}$.
Now suppose that $N={\infty}$ almost surely. Let ${\beta}=2n$, and for $v\in{\mathbb{V}}$ and $r\in{\mathbb{Z}}$ let $$I_{v,r}=\{v\}\times[r,r+1]\subseteq {\mathbb{K}}.$$ We call $I_{v,r}$ a *trifurcation* if (i) it is contained in exactly one unbounded component, and (ii) if one removes all bridges incident on $I_{v,r}$ and places a least one death in $I_{v,r}$, then the unbounded component containing it breaks into three distinct unbounded components. See Figure \[trif\_fig\].
![A trifurcation interval (left); upon removing all incident bridges and placing a death in the interval, the unbounded cluster breaks in three (right).[]{data-label="trif_fig"}](thesis.3 "fig:")![A trifurcation interval (left); upon removing all incident bridges and placing a death in the interval, the unbounded cluster breaks in three (right).[]{data-label="trif_fig"}](thesis.4 "fig:")
We claim that $$\label{pos1}
\phi^b_\rho(I_{0,0}\mbox{ is a trifurcation})>0.$$ To see this let $n$ be large enough so that $\partial{\Lambda}_n$ meets three distinct unbounded components with positive probability. Conditional on ${\mathcal{T}}_{{\Lambda}_n}$, the finite energy property Lemma \[fin\_en\] allows us to modify the configuration inside ${\Lambda}_n$ so that, with positive probability, $I_{0,0}$ is a trifurcation.
We note from translation invariance, Lemma \[invar\_lem\], that the number $T_n$ of trifurcations in ${\Lambda}_n$ satisfies $$\label{pos2}
\begin{split}
\phi^b_\rho(T_n)&=\sum_{\substack{v\in[-n,n]^d\\r=-n,\dotsc,n-1}}
\phi^b_\rho(I_{v,r}\mbox{ is a trifurcation})\\
&=2n(2n+1)^d\phi^b_\rho(I_{0,0}\mbox{ is a trifurcation}).
\end{split}$$ Define the *sides* of ${\Lambda}_n$ to be the union of all intervals $v\times[-n,n]$ where $v$ has at least one coordinate which is $\pm n$. Topological considerations imply that $T_n$ is bounded from above by the total number of deaths on the sides of ${\Lambda}_n$ plus twice the number of vertices in $[-n,n]^d$. (Each trifurcation needs at least one unique point of exit from ${\Lambda}_n$). Using the stochastic domination in Corollary \[comp\_perc\] or otherwise, it follows that $\phi^b_\rho(T_n)\leq 2(2n+1)^d+{\delta}\cdot 4dn(2n+1)^{d-1}$. In view of and this is a contradiction. See [@br_perc Chapter 5] for more details on the topological aspects of this argument.
It follows from Theorem \[inf\_clust\_uniq\] that $N=0$ almost surely under $\phi^b_{\mathrm{c}}$ if $\rho<\rho_c$ and that $N=1$ almost surely if $\rho>\rho_c$. It is crucial for the proof that ${\mathbb{L}}={\mathbb{Z}}^d$ is ‘amenable’ in the sense that the boundary of $[-n,n]^d$ is an order of magnitude smaller than the volume. The result fails, for example, when ${\mathbb{L}}$ is a tree, in which case $N={\infty}$ may occur; see [@pemantle92] for the corresponding phenomenon in the contact process.
### Convergence of pressure {#press_sec}
In this section we adapt the well-known ‘convergence of pressure’ argument to the space–time random-cluster model. By relating the question of uniqueness of measures to that of the existence of certain derivatives, we are able to deduce that there is a unique infinite-volume measure at almost every $({\lambda},{\delta},{\gamma})$, see Theorem \[rc\_uniq\_thm\] below. Arguments of this type are ‘folklore’ in statistical physics, and appear in many places such as [@ellis85:LD; @georgii88; @israel79]. We follow closely the corresponding method for the discrete random-cluster model given in [@grimmett_rcm Chapter 4].
Let ${\lambda},{\delta},{\gamma}>0$ be constants. We will for simplicity of presentation be treating only the case when ${\gamma}>0$ and $q\geq 1$, though similar arguments hold when ${\gamma}=0$ and when $0<q<1$. The partition function $$Z^b_{\Lambda}({\lambda},{\delta},{\gamma},q)=\int_{{\Omega}} q^{k^b_{\Lambda}({\omega})}
\,d\mu_{{\lambda},{\delta},{\gamma}}({\omega})$$ is now a function ${\mathbb{R}}_+^4\rightarrow{\mathbb{R}}$. In this section we will study the related *pressure functions* $$\label{pr_eq}
P^b_{\Lambda}({\lambda},{\delta},{\gamma},q)=\frac{1}{|{\Lambda}|}\log Z^b_{\Lambda}({\lambda},{\delta},{\gamma},q).$$ Here, and in what follows, we have abused notation by writing $|{\Lambda}|$ for the (one-dimensional) Lebesgue measure $|K|$ of $K$, where ${\Lambda}=(K,F)$. We will be considering limits of $P^b_{\Lambda}$ as the region ${\Lambda}$ grows. To be concrete we will be considering regions of the form $$\label{pr_reg}
{\Lambda}={\Lambda}_{{\underline}n,{\beta}}\equiv\{1,\dotsc,n_1\}\times\dotsb\{1,\dotsc,n_d\}
\times[0,{\beta}]$$ and limits when ${\Lambda}\uparrow{\mathbf{\Theta}}$, that is to say all $n_1,\dotsc,n_d,{\beta}\rightarrow{\infty}$ (simultaneously). Strictly speaking such regions do not tend to ${\mathbf{\Theta}}$, but the $P^b_{\Lambda}$ are not affected by translating ${\Lambda}$. It will be clear from the arguments that one may deal in the same way with limits as ${\Lambda}\uparrow{\mathbf{\Theta}}_{\beta}$ with ${\beta}<{\infty}$ fixed. When ${\underline}n$ and ${\beta}$ need to be emphasized we will write ${\Lambda}_{{\underline}n,{\beta}}=(K_{{\underline}n,{\beta}},F_{{\underline}n,{\beta}})$.
Here is a simple observation about $Z^b_{\Lambda}$. Writing $$\label{seb3}
r=\log{\lambda},\quad s=\log {\delta}, \quad t=\log {\gamma},\quad
u=\log q,$$ and $$D_{\Lambda}=D\cap K, \quad G_{\Lambda}=G\cap K, \quad B_{\Lambda}=B\cap F,$$ we have that $$\begin{split}
Z^b_{\Lambda}(r,s,t,u)&\equiv Z^b_{\Lambda}({\lambda},{\delta},{\gamma},q)\\
&=\int_{\Omega}d\mu_{1,1,1}({\omega})
\exp\big(r|B_{\Lambda}|+s|D_{\Lambda}|+t|G_{\Lambda}|+u k^b_{\Lambda}\big).
\end{split}$$ (Where $\mu_{1,1,1}$ is the percolation measure where $B,D,G$ all have rate 1.) This follows from basic properties of Poisson processes. It will sometimes be more convenient to work with $Z^b_{\Lambda}(r,s,t,u)$ in this form. We will also write $P^b_{\Lambda}(r,s,t,u)$ for the pressure using these parameters .
Let ${\underline}h=(h_1,\dotsc,h_4)$ be a unit vector in ${\mathbb{R}}^4$, and let $y\in{\mathbb{R}}$. It follows from a simple computation that the function $f(y)=P^b_{\Lambda}((r,s,t,u)+y h)$ has non-negative second derivative. Indeed, $f''(y)$ is the variance under the appropriate random-cluster measure of the quantity $$h_1|B_{\Lambda}|+h_2|D_{\Lambda}|+h_3|G_{\Lambda}|+h_4 k^b_{\Lambda}.$$ Since variances are non-negative, have proved
\[press\_convex\_lem\] Each $P^b_{\Lambda}(r,s,t,u)$ is a convex function ${\mathbb{R}}^4\rightarrow{\mathbb{R}}$.
Our first objective in this section is the following result.
\[press\_converge\_thm\] The limit $$P(r,s,t,u)=\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}} P^b_{\Lambda}(r,s,t,u)$$ exists for all $r,s,t,u\in{\mathbb{R}}$ and all sequences ${\Lambda}\uparrow{\mathbf{\Theta}}$ of the form , and is independent of the boundary condition $b$.
The function $P$ is usually called the *specific Gibbs free energy*, or *free energy* for short. It follows that $P$ is a convex function ${\mathbb{R}}^4\rightarrow{\mathbb{R}}$, and hence that the set ${\mathcal{D}}$ of points in ${\mathbb{R}}^4$ at which one or more partial derivative of $P$ fails to exist has zero Lebesgue measure. We will return to this observation after the proof of Theorem \[press\_converge\_thm\].
We first prove convergence of $P^{\mathrm{f}}_{\Lambda}$ with free boundary, and then deduce the result for general $b$. For each $i=1,\dotsc,d$ let $0<m_i\leq n_i$ and also let $0<{\alpha}<{\beta}$. Write $|{\underline}m|=m_1\dotsb m_d$. We may regard the region ${\Lambda}_{{\underline}m,{\alpha}}$ as a subset of ${\Lambda}_{{\underline}n,{\beta}}$. Write $T^{{\underline}n,{\beta}}_{{\underline}m,{\alpha}}$ for the set of points in $F_{{\underline}n,{\beta}}\setminus F_{{\underline}m,{\alpha}}$ adjacent to at least one point in $K_{{\underline}m,{\alpha}}$. We have that $$k^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}}
\left\{\begin{array}{l}
\leq k^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}+
k^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}\setminus{{\Lambda}_{{\underline}m,{\alpha}}}} \\
\geq k^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}+
k^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}\setminus{{\Lambda}_{{\underline}m,{\alpha}}}}
-|{\underline}m|-|B\cap T^{{\underline}n,{\beta}}_{{\underline}m,{\alpha}}| -1.
\end{array}\right.$$ The lower bound follows because the number of ‘extra’ components created by ‘cutting out’ ${\Lambda}_{{\underline}m,{\alpha}}$ from ${\Lambda}_{{\underline}n,{\beta}}$ is bounded by the number of intervals constituting $K_{{\underline}m,{\alpha}}$, plus the number of bridges that are cut, plus 1 (for the component of ${\Gamma}$). The upper bound is similar but simpler. Thus $$\label{seb1}
\begin{split}
\log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}}&=
\log\mu_{{\lambda},{\delta},{\gamma}}(q^{k^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}}})\\
&\left\{\begin{array}{l}
\leq \log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}
+\log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}\setminus {\Lambda}_{{\underline}m,{\alpha}}}\\
\geq \log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}
+\log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}\setminus {\Lambda}_{{\underline}m,{\alpha}}}-\\
\quad-(\log q) |{\underline}m|-
{\lambda}(1-1/q){\alpha}d|{\underline}m|\sum_{i=1}^d\frac{1}{m_i}-\log q.
\end{array}\right.
\end{split}$$ We have used the fact that $$|T^{{\underline}n,{\beta}}_{{\underline}m,{\alpha}}|\leq
{\alpha}d|{\underline}m|\sum_{i=1}^d\frac{1}{m_i}.$$
There are $\prod_{i=1}^d\lfloor n_i/m_i \rfloor\cdot \lfloor{\beta}/{\alpha}\rfloor$ ‘copies’ of ${\Lambda}_{{\underline}m,{\alpha}}$ in ${\Lambda}_{{\underline}n,{\beta}}$, each being a translation of ${\Lambda}_{{\underline}m,{\alpha}}$ by a vector $${\underline}{l}\in\{(b_1m_1,\dotsc,b_dm_d,c{\alpha}):
b_i=1,\dotsc,\lfloor n_i/m_i \rfloor,\:
c=1,\dotsc,\lfloor {\beta}/{\alpha}\rfloor \}.$$ Write $${\Lambda}=\Big(\bigcup_{{\underline}{l}} ({\Lambda}_{{\underline}{m},{\alpha}}+{\underline}{l})\Big)\cup {\Lambda}';$$ this union is disjoint up to a set of measure zero. Let ${\Lambda}'=(K',F')$. Repeating the argument leading up to once for each ‘copy’ of ${\Lambda}_{{\underline}m,{\beta}}$ we deduce that $Z^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}}$ is bounded above by $$\label{abc1}
\Big(\prod_{i=1}^d\lfloor n_i/m_i \rfloor\cdot \lfloor{\beta}/{\alpha}\rfloor
\Big)\log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}+\log Z^{\mathrm{f}}_{{\Lambda}'}$$ and below by the same quantity minus $$\prod_{i=1}^d\lfloor n_i/m_i \rfloor\cdot \lfloor{\beta}/{\alpha}\rfloor
\Big((\log q) |{\underline}m|+
{\lambda}(1-1/q){\alpha}d|{\underline}m|\sum_{i=1}^d\frac{1}{m_i}+\log q\Big).$$ We will prove shortly that $$\label{seb2}
\lim_{n_i,{\beta}\rightarrow{\infty}} \frac{1}{|{\Lambda}_{{\underline}n,{\beta}}|}
\log Z^{\mathrm{f}}_{{\Lambda}'}=0;$$ once this is done it follows on dividing by $|{\Lambda}_{{\underline}n,{\beta}}|={\beta}\cdot|{\underline}n|$ and letting all $n_i,{\beta}\rightarrow{\infty}$ that $$\begin{split}
\frac{1}{|{\Lambda}_{{\underline}m,{\alpha}}|}\log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}&\leq
\liminf_{n_i,{\beta}\rightarrow{\infty}} P^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}}
\leq \limsup_{n_i,{\beta}\rightarrow{\infty}} P^{\mathrm{f}}_{{\Lambda}_{{\underline}n,{\beta}}}\\
&\leq \frac{1}{|{\Lambda}_{{\underline}m,{\alpha}}|}\log Z^{\mathrm{f}}_{{\Lambda}_{{\underline}m,{\alpha}}}
+\frac{1}{{\alpha}}\log q+\\
&\qquad+{\lambda}(1-1/q)d\sum_{i=1}^d\frac{1}{m_i}
+\frac{1}{|{\Lambda}_{{\underline}m,{\alpha}}|}\log q,
\end{split}$$ and hence that $\lim_{\Lambda}P^{\mathrm{f}}_{\Lambda}$ exists and is finite.
Let us prove the claim . The set $K_{{\Lambda}'}$ consists of a number of disjoint intervals, of which $$\prod_{i=1}^d m_i\lfloor n_i/m_i\rfloor$$ have length ${\beta}-{\alpha}\lfloor{\beta}/{\alpha}\rfloor$, and $$\prod_{i=1}^d n_i-
\prod_{i=1}^d m_i\lfloor n_i/m_i\rfloor$$ have length ${\beta}$. The number $k^{\mathrm{f}}_{{\Lambda}'}$ of components is bounded above by the sum over all such intervals $L$ of $|D\cap L|+2$ (we have added $1$ for the component of ${\Gamma}$). Hence $$\begin{split}
0\leq \log Z^{\mathrm{f}}_{{\Lambda}'}&=\mu_{{\lambda},{\delta},{\gamma}}(q^{k^{\mathrm{f}}_{{\Lambda}'}})\\
&\leq \Big(\prod_{i=1}^d m_i\lfloor n_i/m_i\rfloor\Big)\cdot
(q-1){\delta}({\beta}-{\alpha}\lfloor {\beta}/{\alpha}\rfloor)+\\
&\qquad+\Big(\prod_{i=1}^d n_i-
\prod_{i=1}^d m_i\lfloor n_i/m_i\rfloor\Big)\cdot
(q-1){\delta}{\beta}+2\log q.
\end{split}$$ Equation follows.
Finally, we must prove convergence with arbitrary boundary condition. It is clear that for any boundary condition $b$ we have $$k^{\mathrm{w}}_{\Lambda}\leq k^b_{\Lambda}\leq k^{\mathrm{f}}_{\Lambda}.$$ On the other hand $$k^{\mathrm{w}}_{\Lambda}\geq k^{\mathrm{f}}_{\Lambda}-2|{\underline}n|-|D\cap \partial{\Lambda}|-1.$$ The result follows.
We now switch parameters to $r,s,t,u$, given in . For fixed $u$ (i.e. fixed $q$) let ${\mathcal{D}}_u={\mathcal{D}}_q$ be the set of points $(r,s,t)\in{\mathbb{R}}^3$ at which at least one of the partial derivatives $$\frac{\partial P}{\partial r}, \quad
\frac{\partial P}{\partial s}, \quad
\frac{\partial P}{\partial t}$$ fails to exist. Since $P$ is convex, ${\mathcal{D}}_q$ has zero (three-dimensional) Lebesgue measure. By general properties of convex functions, the partial derivatives $$\frac{\partial P^b_{\Lambda}}{\partial r}, \quad
\frac{\partial P^b_{\Lambda}}{\partial s}, \quad
\frac{\partial P^b_{\Lambda}}{\partial t}$$ converge to the corresponding derivatives of $P$ whenever $(r,s,t)\not\in{\mathcal{D}}_q$, for any $b$. Now observe that $$\begin{split}
\frac{\partial P^{\mathrm{f}}_{\Lambda}}{\partial r}&=
\frac{1}{|{\Lambda}|}\phi^{\mathrm{f}}_{\Lambda}(|B_{\Lambda}|)\leq
\frac{1}{|{\Lambda}|}\phi^{\mathrm{f}}(|B_{\Lambda}|)\\
&\leq\frac{1}{|{\Lambda}|}\phi^{\mathrm{w}}(|B_{\Lambda}|)\leq
\frac{1}{|{\Lambda}|}\phi^{\mathrm{w}}_{\Lambda}(|B_{\Lambda}|)=
\frac{\partial P^{\mathrm{w}}_{\Lambda}}{\partial r},
\end{split}$$ so if $(r,s,t)\not\in{\mathcal{D}}_q$ then $$\label{seb4}
\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}}\frac{1}{|{\Lambda}|}\phi^{\mathrm{f}}(|B_{\Lambda}|)=
\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}}\frac{1}{|{\Lambda}|}\phi^{\mathrm{w}}(|B_{\Lambda}|)=
\frac{\partial P}{\partial r}.$$ Recall from Lemma \[invar\_lem\] that $\phi^{\mathrm{f}}$ and $\phi^{\mathrm{w}}$ are both invariant under translations. The set $B$ is a point process on ${\mathbb{F}}$, which is therefore stationary under both $\phi^{\mathrm{f}}$ and $\phi^{\mathrm{w}}$, and hence has constant intensities under these measures. Said another way, the *mean measures* $m^{\mathrm{f}},m^{\mathrm{w}}$ on $({\mathbb{F}},{\mathcal{B}}({\mathbb{F}}))$, given respectively by $$m^{\mathrm{f}}(F):=\phi^{\mathrm{f}}(|B\cap F|),\quad\text{and}\quad
m^{\mathrm{w}}(F):=\phi^{\mathrm{w}}(|B\cap F|)$$ are translation invariant measures. It is therefore a general fact that there are constants $c^{\mathrm{f}}_{\mathrm{b}}$ and $c^{\mathrm{w}}_{\mathrm{b}}$ such that for all regions ${\Lambda}=(K,F)$, $$m^{\mathrm{f}}(F)=\phi^{\mathrm{f}}(|B_{\Lambda}|)=c^{\mathrm{f}}_{\mathrm{b}}|F|,\quad\text{and}\quad
m^{\mathrm{w}}(F)=\phi^{\mathrm{w}}(|B_{\Lambda}|)=c^{\mathrm{w}}_{\mathrm{b}}|F|,$$ where $|\cdot|$ denotes Lebesgue measure. Similarly, there are constants $c^{\mathrm{f}}_{\mathrm{d}}$, $c^{\mathrm{w}}_{\mathrm{d}}$, $c^{\mathrm{f}}_{\mathrm{g}}$ and $c^{\mathrm{w}}_{\mathrm{g}}$ such that $$\phi^{\mathrm{f}}(|D_{\Lambda}|)=c^{\mathrm{f}}_{\mathrm{d}}|K|,\quad\text{and}\quad
\phi^{\mathrm{w}}(|D_{\Lambda}|)=c^{\mathrm{w}}_{\mathrm{d}}|K|,$$ and $$\phi^{\mathrm{f}}(|G_{\Lambda}|)=c^{\mathrm{f}}_{\mathrm{g}}|K|,\quad\text{and}\quad
\phi^{\mathrm{w}}(|G_{\Lambda}|)=c^{\mathrm{w}}_{\mathrm{g}}|K|,$$ for all regions ${\Lambda}=(K,F)$.
Note that $$\lim_{n_i,{\beta}\rightarrow{\infty}} \frac{|F_{{\underline}n,{\beta}}|}{|K_{{\underline}n,{\beta}}|}=d.$$ It follows from , and similar calculations for $D$ and $G$, that $$c^{\mathrm{f}}_{\mathrm{b}}=c^{\mathrm{w}}_{\mathrm{b}},\quad
c^{\mathrm{f}}_{\mathrm{d}}=c^{\mathrm{w}}_{\mathrm{d}},\quad\text{and}
\quad c^{\mathrm{f}}_{\mathrm{g}}=c^{\mathrm{w}}_{\mathrm{g}}
\quad\text{whenever } (r,s,t)\not\in{\mathcal{D}}_q.$$ Recall the condition given at the end of Section \[rc\_wl\_sec\] for the uniqueness of the infinite-volume random-cluster measures, namely that $\phi^{\mathrm{f}}=\phi^{\mathrm{w}}$. We will use the facts listed above to prove
\[rc\_uniq\_thm\] There is a unique random-cluster measure, in that $\phi^{\mathrm{f}}=\phi^{\mathrm{w}}$, whenever $(r,s,t)\not\in{\mathcal{D}}_q$.
The corresponding results holds when ${\gamma}\geq0$ is fixed, in that $\phi^{\mathrm{f}}=\phi^{\mathrm{w}}$ except on a set of points $(r,s)$ of zero (two-dimensional) Lebesgue measure. For also ${\delta}>0$ fixed, the corresponding set of ${\lambda}$ where uniqueness fails is countable, again by general properties of convex functions. Presumably this latter set consists of a single point, namely the point corresponding to $\rho=\rho_{\mathrm{c}}$, but this has not been proved even for the discrete models.
Since $\phi^{\mathrm{w}}\geq\phi^{\mathrm{f}}$, there is by Theorem \[strassen\_thm\] a coupling ${\mathbb{P}}$ of the two measures such that $${\mathbb{P}}(\{({\omega}^{\mathrm{w}},{\omega}^{\mathrm{f}})\in{\Omega}^2:{\omega}^{\mathrm{w}}\geq{\omega}^{\mathrm{f}}\})=1,$$ and such that ${\omega}^{\mathrm{w}}$ and ${\omega}^{\mathrm{f}}$ have marginal distributions $\phi^{\mathrm{w}}$ and $\phi^{\mathrm{f}}$ under ${\mathbb{P}}$, respectively. Write $B^b$, $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ for the bridges of ${\omega}^b$, and similarly for deaths and ghost-bonds. Let $A\in{\mathcal{F}}_{\Lambda}$ be an increasing event. Then $$\begin{split}
0\leq \phi^{\mathrm{w}}(A)-\phi^{\mathrm{f}}(A)&\leq
{\mathbb{P}}({\omega}^{\mathrm{w}}\in A,{\omega}^{\mathrm{w}}\neq{\omega}^{\mathrm{f}}\text{ in }{\Lambda})\\ &\leq
{\mathbb{P}}(|B^{\mathrm{w}}_{\Lambda}\setminus B^{\mathrm{f}}_{\Lambda}|+|D^{\mathrm{f}}_{\Lambda}\setminus D^{\mathrm{w}}_{\Lambda}|
+|G^{\mathrm{w}}_{\Lambda}\setminus G^{\mathrm{f}}_{\Lambda}|)\\
&=\phi^{\mathrm{w}}(|B_{\Lambda}|)-\phi^{\mathrm{f}}(|B_{\Lambda}|)+
\phi^{\mathrm{f}}(|D_{\Lambda}|)-\phi^{\mathrm{w}}(|D_{\Lambda}|)+\\
&\qquad +\phi^{\mathrm{w}}(|G_{\Lambda}|)-\phi^{\mathrm{f}}(|G_{\Lambda}|)\\
&=|{\Lambda}|(c^{\mathrm{w}}_{\mathrm{b}}-c^{\mathrm{f}}_{\mathrm{b}}+
c^{\mathrm{f}}_{\mathrm{d}}-c^{\mathrm{w}}_{\mathrm{d}}+
c^{\mathrm{w}}_{\mathrm{g}}-c^{\mathrm{f}}_{\mathrm{g}})\\
&=0,
\end{split}$$ so $\phi^{\mathrm{w}}=\phi^{\mathrm{f}}$ as required.
Here is a consequence when ${\gamma}=0$. Recall that we set ${\lambda}=\rho$ and ${\delta}=1$. Suppose $0<\rho<\rho'$ are given. We may pick ${\lambda}_1=\rho_1$ so that $\rho<\rho_1<\rho'$ and so that there is a unique infinite-volume measure with parameters ${\lambda}_1=\rho_1,{\delta}_1=1$ and ${\gamma}=0$. Hence $$\label{seb5}
\phi^{\mathrm{w}}_\rho\leq\phi^{\mathrm{w}}_{\rho_1}=
\phi^{\mathrm{f}}_{\rho_1}\leq\phi^{\mathrm{f}}_{\rho'}.$$ It follows that the critical values $\rho^{\mathrm{f}}_{\mathrm{c}}(q)$ and $\rho^{\mathrm{w}}_{\mathrm{c}}(q)$ of Definition \[crit\_def\] are equal for all $q\geq1$.
Duality in ${\mathbb{Z}}\times{\mathbb{R}}$ {#duality_sec}
-------------------------------------------
In this section we let ${\mathbb{L}}={\mathbb{Z}}$. Thanks to the notion of planar duality for graphs, much more is known about the discrete random-cluster model in two dimensions than in general dimension. In particular, the critical value for $q=1,2$ and $q\geq 25.72$ has been calculated in two dimensions, see [@abf; @kesten_half; @laanait_etal_91; @laanait_etal_86]. In the space-time setting, the $d=1$ model occupies the two-dimensional space ${\mathbb{Z}}\times{\mathbb{R}}$, so we may adapt duality arguments to this case; that is the objective of this section. Such arguments have been applied when $q=1$ to prove that $\rho_{\mathrm{c}}(1)=1$, see [@bezuidenhout_grimmett]. We will see in Chapter \[appl\_ch\] that $\rho_{\mathrm{c}}(2)=2$, and Theorem \[zhang\_thm\] in the present section is a first step towards this result.
Throughout this section we assume that ${\gamma}=0$, and hence suppress reference to both ${\gamma}$ and $G$. We also assume that $q\geq1$ and that ${\lambda},{\delta}$ are positive constants. In light of Theorem \[crit\_nontriv\] we may disregard the ${\beta}<{\infty}$ case, hence we deal in this section only with the ${\beta}\rightarrow{\infty}$ case. We think of ${\mathbf{\Theta}}\equiv{\mathbb{Z}}\times{\mathbb{R}}$ as embedded in ${\mathbb{R}}^2$ in the natural way.
We write ${\mathbb{L}}_{\mathrm{d}}$ for ${\mathbb{Z}}+1/2$; of course ${\mathbb{L}}$ and ${\mathbb{L}}_{\mathrm{d}}$ are isomorphic graphs. With any ${\omega}=(B,D)\in{\Omega}$ we associate the ‘dual’ configuration ${\omega}_{\mathrm{d}}:=(D,B)$ regarded as a configuration in ${\mathbf{\Theta}}_{\mathrm{d}}={\mathbb{L}}_{\mathrm{d}}\times{\mathbb{R}}$. Thus each bridge in ${\omega}$ corresponds to a death in ${\omega}_{\mathrm{d}}$, and each death in ${\omega}$ corresponds to a bridge in ${\omega}_{\mathrm{d}}$. This correspondence is illustrated in Figure \[duality\_fig\]. We identify ${\omega}_{\mathrm{d}}=(D,B)$ with the element $(D-1/2,B-1/2)$ of ${\Omega}$. Under this identification we may for any measurable $f:{\Omega}\rightarrow{\mathbb{R}}$ define $f_{\mathrm{d}}:{\Omega}\rightarrow{\mathbb{R}}$ by $f_{\mathrm{d}}({\omega})=f({\omega}_{\mathrm{d}})$.
![An illustration of duality. The primal configuration ${\omega}$ is drawn solid black, the dual ${\omega}_{\mathrm{d}}$ dashed grey.[]{data-label="duality_fig"}](thesis.5)
In the case when $q=1$ it is clear that for any measurable function $f:{\Omega}\rightarrow{\mathbb{R}}$, we have the relation $\mu_{{\lambda},{\delta}}(f_{\mathrm{d}})=\mu_{{\delta},{\lambda}}(f)$, since the roles of ${\lambda}$ and ${\delta}$ are swapped under the duality transformation. We will see that a similar result holds when $q>1$.
Let $\psi_1,\psi_2$ be probability measures on $({\Omega},{\mathcal{F}})$. We say that $\psi_2$ is dual to $\psi_1$ if for all measurable $f:{\Omega}\rightarrow{\mathbb{R}}$ we have that $$\label{dual_meas_cond}
\psi_1(f_{\mathrm{d}})=\psi_2(f).$$
Thus the dual of $\mu_{{\lambda},{\delta}}$ is $\mu_{{\delta},{\lambda}}$. Clearly it is enough to check on some determining class of functions, such as the local functions.
It will be convenient in what follows to denote the free and wired random-cluster measures on a region ${\Lambda}$ by $\phi^0_{{\Lambda};q,{\lambda},{\delta}}$ and $\phi^1_{{\Lambda};q,{\lambda},{\delta}}$ respectively, instead of $\phi^{\mathrm{f}}_{{\Lambda};q,{\lambda},{\delta}}$ and $\phi^{\mathrm{w}}_{{\Lambda};q,{\lambda},{\delta}}$. The following result is stated in terms of infinite-volume measures, but from the proof we see that an analogous result holds also in finite volume.
\[duality\_thm\] Let $b\in\{0,1\}$. The dual of the measure $\phi^b_{q,{\lambda},{\delta}}$ is $\phi^{1-b}_{q,q{\delta},{\lambda}/q}$.
Fix ${\beta}>0$ and $q\geq 1$; later we will let ${\beta}\rightarrow\infty$. We write $[m,n]$ for the graph $L\subseteq{\mathbb{L}}$ induced by the set $\{m,m+1,\dotsc,n\}\subseteq{\mathbb{Z}}$ and ${\Lambda}_{m,n}=(K_{m,n},F_{m,n})$ for the corresponding simple region. We write $\phi^b_{m,n;{\lambda},{\delta}}$ for the random-cluster measure on the region ${\Lambda}_{m,n}$, with similar adjustments to other notation.
In what follows it will be useful to restrict attention to the bridges and deaths of ${\omega}\in{\Omega}$ that fall in ${\Lambda}_{m,n}$ only. It is then most natural to consider only those (dual) bridges and deaths of ${\omega}_{\mathrm{d}}$ that fall in ${\Lambda}_{m,n-1}+1/2$. In line with this we define $$B_{m,n}({\omega}):=B({\omega})\cap F_{m,n},\qquad
D_{m,n}({\omega}):=D({\omega})\cap K_{m,n};$$ and for the dual $$B_{m,n-1}({\omega}_{\mathrm{d}}):=D({\omega})\cap K_{m+1,n-1},\qquad
D_{m,n-1}({\omega}_{\mathrm{d}}):=B({\omega})\cap F_{m,n}.$$
The first step is to establish an analog of the Euler equation for planar graphs. We claim that $$\label{tjo1}
k^1_{m,n}({\omega})-k^0_{m,n-1}({\omega}_{\mathrm{d}})+
|B_{m,n}({\omega})|-|D_{m,n}({\omega})|=1-n+m.$$ (A similar result was obtained in [@aizenman_nacht Lemma 3.3].) This is best proved inductively by successively adding elements to the sets $B_{m,n}({\omega})$ and $D_{m,n}({\omega})$. If both sets are empty, the claim follows on inspection. For each bridge you add to $B_{m,n}({\omega})$, either $k^1_{m,n}({\omega})$ decreases by one or $k^0_{m,n-1}({\omega}_{\mathrm{d}})$ increases by one, but never both. Similarly when you add deaths to $D_{m,n}({\omega})$, either $k^1_{m,n}({\omega})$ increases by one or $k^0_{m,n-1}({\omega}_{\mathrm{d}})$ decreases by one for each death, but never both. That establishes .
Let $\mu_{m,n;{\lambda},{\delta}}$ denote the percolation measure restricted to ${\Lambda}_{m,n}$. For $f:{\Omega}\rightarrow{\mathbb{R}}$ any ${\mathcal{F}}_{{\Lambda}_{m,n-1}}$-measurable, bounded and continuous function, we have, using , that $$\label{tjo2}
\begin{split}
\phi^1_{m,n;{\lambda},{\delta}}(f_{\mathrm{d}})&\propto
\int d\mu_{m,n;{\lambda},{\delta}}({\omega}) q^{k^1_{m,n}({\omega})}f({\omega}_{\mathrm{d}})\\
&\propto \int d\mu_{m,n;{\lambda},{\delta}}({\omega}) q^{k^0_{m,n-1}({\omega}_{\mathrm{d}})}
q^{|D_{m,n}({\omega})|}q^{-|B_{m,n}({\omega})|}f({\omega}_{\mathrm{d}})\\
&\propto \int d\mu_{m,n-1;{\delta},{\lambda}}({\omega}_{\mathrm{d}}) q^{k^0_{m,n-1}({\omega}_{\mathrm{d}})}
q^{|B_{m,n-1}({\omega}_{\mathrm{d}})|}q^{-|D_{m,n-1}({\omega}_{\mathrm{d}})|}f({\omega}_{\mathrm{d}})\\
&\propto \int d\mu_{m,n-1;q{\delta},{\lambda}/q}({\omega}_{\mathrm{d}})
q^{k^0_{m,n-1}({\omega}_{\mathrm{d}})}f({\omega}_{\mathrm{d}})\\
&\propto\phi^0_{m,n-1;q{\delta},{\lambda}/q}(f).
\end{split}$$ We have used the fact that $$\frac{d\mu_{m,n-1;q{\delta},{\lambda}/q}}{d\mu_{m,n-1;{\delta},{\lambda}}}({\omega})
\propto q^{|B_{m,n-1}({\omega})|}q^{-|D_{m,n-1}({\omega})|},$$ a simple statement about Poisson processes.
Since both sides of are probability measures, it follows that $$\label{tjo3}
\phi^1_{m,n;{\lambda},{\delta}}(f_{\mathrm{d}})=\phi^1_{m,n-1;q{\delta},{\lambda}/q}(f).$$ Letting $m,n,{\beta}\rightarrow\infty$ in and using Theorem \[conv\_lem\], the result follows.
Note that if ${\lambda}/{\delta}=\rho$ then the corresponding ratio for the dual measure is $q{\delta}/({\lambda}/q)=q^2/\rho$. We therefore say that the space–time random-cluster model is *self-dual* if $\rho=q$. This self-duality was referred to in [@aizenman_nacht Proposition 3.4].
### A lower bound on $\rho_c$ when $d=1$
In this section we adapt Zhang’s famous and versatile argument (published in [@grimmett_rcm Chapter 6]) to the space-time setting. See [@bezuidenhout_grimmett] for the special case of this argument when $q=1$.
\[zhang\_thm\] If $d=1$ and $\rho=q$ then $\theta^{\mathrm{f}}(\rho,q)=0$; hence the critical ratio $\rho_{\mathrm{c}}\geq q$.
Assume for a contradiction that with $\rho=q$ we have that $\theta^{\mathrm{f}}(\rho,q)>0$. Then by Theorem \[inf\_clust\_uniq\] there is almost surely a unique unbounded component in ${\omega}$ under $\phi^{\mathrm{f}}$. It follows from self-duality and the fact that $\theta^{\mathrm{w}}\geq\theta^{\mathrm{f}}$ that there is almost surely also a unique unbounded component in ${\omega}_{\mathrm{d}}$. Let $D_n=\{(x,t)\in{\mathbb{R}}^2:|x+1/2|+|t|\leq n\}$ be the ‘lozenge’, and $D_n^{\mathrm{d}}=\{(x,t)\in{\mathbb{R}}^2:|x|+|t|\leq n\}$ its ‘dual’, as in Figure \[zhang\_fig\].
![On the event $A_2\cap A_4\cap A^{\mathrm{d}}_1\cap A^{\mathrm{d}}_3$ either the unbounded primal cluster breaks into 2 parts, or the dual one does.[]{data-label="zhang_fig"}](thesis.2)
Number the four sides of each of $D_n$ and $D_n^{\mathrm{d}}$ counterclockwise, starting in each case with the north-east side. For $i=1,\dotsc,4$ let $A_i$ be the event that the $i$th side of $D_n$ is attached to an unbounded path of ${\omega}$, which does not otherwise intersect $D_n$. Similarly let $A^{\mathrm{d}}_i$ be the event that the $i$th side of the dual $D_n^{\mathrm{d}}$ is attached to an unbounded path of ${\omega}_{\mathrm{d}}$. Clearly $\phi^{\mathrm{f}}(\cup_{i=1}^4 A_i)\rightarrow1$ as $n\rightarrow\infty$. However, all the $A_i$ are increasing, and by symmetry under reflection they carry equal probability. It follows from positive association, Theorem \[rc\_fkg\], that $$\phi^{\mathrm{f}}(\cup_{i=1}^4 A_i)\leq 1-\phi^{\mathrm{f}}(A_2^c)^4=1-(1-\phi^{\mathrm{f}}(A_2))^4,$$ and hence $\phi^{\mathrm{f}}(A_2)\rightarrow1$ too. Hence for $n$ large enough we have that $\phi^{\mathrm{f}}(A_2)=\phi^{\mathrm{f}}(A_4)\geq 5/6$, so by positive association again $\phi^{\mathrm{f}}(A_2\cap A_4)\geq (5/6)^2>5/8$ for $n$ large enough. In the same way it follows that for large $n$ we have $\phi^{\mathrm{f}}(A^{\mathrm{d}}_1\cap A^{\mathrm{d}}_3)>5/8$. But then $$\phi^{\mathrm{f}}(A_2\cap A_4\cap A^{\mathrm{d}}_1\cap A^{\mathrm{d}}_3)
\geq\frac{10}{8}-1=\frac{1}{4}.$$ Now a glance at Figure \[zhang\_fig\] should convince the reader that this contradicts the uniqueness of the unbounded cluster, either in ${\omega}$ or ${\omega}_{\mathrm{d}}$. This contradiction shows that $\theta^{\mathrm{f}}(\rho,q)=0$ as required.
It is natural to suppose that the critical value equals the self-dual value ${\lambda}/{\delta}=q$. For $q=1$ this is proved in [@bezuidenhout_grimmett] and in [@aizenman_jung]; for $q=2$ it is proved in Theorem \[crit\_val\_cor\] (see also [@bjogr2]).
Infinite-volume Potts measures {#inf_potts_sec}
------------------------------
Using the convergence results in Section \[inf\_rc\_sec\], we will in this section construct infinite-volume weak limits of Potts measures. We will also provide more details about uniqueness of infinite-volume measure in the space-time Ising model, extending in that case the arguments of Section \[press\_sec\]. The results in this section will form the foundation for our study of the quantum Ising model in Chapter \[qim\_ch\].
### Weak limits of Potts measures
Let $q\geq 2$ be an integer, and let ${\alpha}_{\Gamma}=q$; we will suppress reference to the simple boundary condition $(b,{\alpha})$ throughout this subsection. Recall the two random-cluster measures $\phi_{\Lambda}^{\mathrm{w}}$ and $\phi_{\Lambda}^{\mathrm{f}}$ as well as their Potts counterparts $\pi_{\Lambda}^{\mathrm{w}}$ and $\pi_{\Lambda}^{\mathrm{f}}$, connected via the coupling . For simplicity we assume in this section that ${\mathbb{L}}={\mathbb{Z}}^d$ for some $d\geq 1$; similar arguments are valid in greater generality, but we do not pursue this here. All regions in this section will be simple, as in . We let ${\Lambda}_n=(K_n,F_n)$ denote a strictly increasing sequence of simple regions, containing the origin and increasing to either ${\mathbf{\Theta}}$ or ${\mathbf{\Theta}}_{\beta}$. Denote by $\phi_n^{\mathrm{w}}$, $\phi_n^{\mathrm{f}}$, $\pi_n^{\mathrm{w}}$ and $\pi_n^{\mathrm{f}}$ the corresponding random-cluster and Potts measures. Proofs will be given for the ${\beta}=\infty$ case, the case ${\beta}<{\infty}$ is similar.
Throughout this subsection we will be making use of the concept of *lattice components*: given ${\omega}=(B,D,G)$ the lattice components of ${\omega}$ are the connected components in ${\mathbb{K}}$ of the configuration $(B,D,\varnothing)$. We will think of the points in $G$ as green points, and of any lattice component containing an element of $G$ as *green*. In this subsection we will only use the notation $x{\leftrightarrow}y$ to mean that $x,y$ lie in the same *lattice* component. We write $C_x({\omega})$ for the lattice component of $x$ in ${\omega}$.
The following convergence result is an adaptation of arguments in [@ACCN], see also [@grimmett_rcm Theorem 4.91].
\[potts\_lim\_thm\] The weak limits $$\pi^{\mathrm{f}}=\lim_{n\rightarrow{\infty}}\pi^{\mathrm{f}}_n
\qquad\mbox{and}\qquad
\pi^{\mathrm{w}}=\lim_{n\rightarrow{\infty}}\pi^{\mathrm{w}}_n$$ exist and are independent of the manner in which ${\Lambda}_n\uparrow{\mathbf{\Theta}}$. Moreover, $\pi^{\mathrm{f}}$ and $\pi^{\mathrm{w}}$ are given as follows:
- Let ${\omega}\sim\phi^{\mathrm{f}}$ and assign to each green component of ${\omega}$ spin $q$, and assign to the remaining components uniformly independent spins from $1,\dotsc,q$; then the resulting spin configuration has law $\pi^{\mathrm{f}}$.
- Let ${\omega}\sim\phi^{\mathrm{w}}$ and assign to each unbounded component and each green component of ${\omega}$ spin $q$, and assign to the remaining components uniformly independent spins from $1,\dotsc,q$; then the resulting spin configuration has law $\pi^{\mathrm{w}}$.
We will make use of a certain total order on ${\mathbb{K}}={\mathbb{Z}}^d\times{\mathbb{R}}$. The precise details are not important, except that the ordering be such that every (topologically) closed set contains an earliest point. We define such an ordering as follows. We say that $x=(x_1,\dotsc,x_d,t)<(x_1',\dotsc,x_d',t')=x'$ if (a) for $k\in\{1,\dotsc,d\}$ minimal with $x_kx'_k<0$ we have $x_k>0$; or if (a) fails but (b) $tt'<0$ with $t>0$; or if (a) and (b) fail but (c) $|x|<|x'|$ lexicographically, where $|x|=(|x_1|,\dotsc,|x_d|,|t|)$.
Slightly different arguments are required for the two boundary conditions. We give the argument only for free boundary. It will be necessary to modify the probability space $({\Omega},{\mathcal{F}})$, as follows (we omit some details). For each $n\geq 1$ and each ${\omega}=(B,D,G)\in{\Omega}$, let $\tilde{\omega}_n=(\tilde B_n,\tilde D_n,\tilde G_n)$ be given by $$\tilde B_n=B\cap F_n, \quad
\tilde D_n=(D\cap K_n)\cup({\mathbb{K}}{\setminus}K_n), \quad
\tilde G_n=G\cap K_n.
$$ Thus, in $\tilde{\omega}_n$, no two points in ${\mathbb{K}}\setminus K_n$ are connected. Let $\tilde{\Omega}={\Omega}\cup\{\tilde{\omega}_n:{\omega}\in{\Omega},n\geq 1\}$, and define connectivity in elements of $\tilde{\Omega}$ in the obvious way. Define the functions $V_x$ as before Theorem \[conv\_lem\]; if $x\in{\mathbb{K}}\times\{\mathrm{d}\}$ then $V_x$ may now take the value $+{\infty}$. Let $\tilde{\mathcal{F}}$ denote the ${\sigma}$-algebra generated by the $V_x$’s. (Alternatively, $\tilde{\mathcal{F}}$ is the ${\sigma}$-algebra generated by the appropriate Skorokhod metric when the associated step functions are allowed to take the values $\pm{\infty}$.) Let $\tilde\phi^{\mathrm{f}}_n$ denote the law of $\tilde{\omega}_n$ when ${\omega}$ has law $\phi^{\mathrm{f}}_n$. Note that the number of components of $\tilde{\omega}_n$ equals $k^{\mathrm{f}}_n({\omega})$.
Extending the partial order on ${\Omega}$ to $\tilde{\Omega}$ in the natural way, we see that for each $n$ we have $\tilde\phi^{\mathrm{f}}_n\leq\tilde\phi^{\mathrm{f}}_{n+1}$. (It is here that we need to use $\tilde{\Omega}$, since the stochastic ordering $\phi^{\mathrm{f}}_n\leq\phi^{\mathrm{f}}_{n+1}$ holds only on ${\mathcal{F}}_n$, not on the full ${\sigma}$-algebra ${\mathcal{F}}$.) Hence there exists by Strassen’s Theorem \[strassen\_thm\] a probability measure $P$ on $(\tilde{\Omega}^{\mathbb{N}},\tilde{\mathcal{F}}^{\mathbb{N}})$ such that in the sequence $(\tilde{\omega}_1,\tilde{\omega}_2,\dotsc)$ the $n$th component has marginal distribution $\tilde\phi^{\mathrm{f}}_n$, and such that $\tilde{\omega}_n\leq\tilde{\omega}_{n+1}$ for all $n$, with $P$-probability one. The sequence $\tilde{\omega}_n$ increases to a limiting configuration $\tilde{\omega}_\infty$, which has law $\phi^{\mathrm{f}}$. We have that $\phi^{\mathrm{f}}({\Omega})=1$.
For each fixed (bounded) region ${\Delta}$, if $n$ is large enough then $\tilde{\omega}_n$ agrees with $\tilde{\omega}_\infty$ throughout ${\Delta}$. Let ${\Lambda}$ be a fixed region, and let ${\Delta}={\Delta}(\tilde{\omega}_{\infty})\supset{\Lambda}$ be large enough so that the following hold:
1. Each bounded lattice-component of $\tilde{\omega}_\infty$ which intersects ${\Lambda}$ is entirely contained in ${\Delta}$;
2. Any two points $x,y\in{\Lambda}$ which are connected in $\tilde{\omega}_{\infty}$ are connected inside ${\Delta}$;
3. Any lattice-component of $\tilde{\omega}_{\infty}$ which is green has a green point inside ${\Delta}$.
It is (almost surely) possible to choose such a ${\Delta}$ because only finitely many lattice components intersect ${\Lambda}$. We choose $n=n(\tilde{\omega}_{\infty})$ large enough so that $\tilde{\omega}_n,\tilde{\omega}_{n+1},\dotsc$ all agree with $\tilde{\omega}_{\infty}$ throughout ${\Delta}$.
*Claim:* for all $x,y\in{\Lambda}$, we have that $x{\leftrightarrow}y$ in $\tilde{\omega}_n$ if and only if $x{\leftrightarrow}y$ in $\tilde{\omega}_{\infty}$. To see this, first note that $C_x(\tilde{\omega}_n)\subseteq C_x(\tilde{\omega}_{\infty})$ since $\tilde{\omega}_n\leq\tilde{\omega}_{\infty}$, proving one of the implications. Suppose now that $x{\leftrightarrow}y$ in $\tilde{\omega}_{\infty}$. Then by our choice of ${\Delta}$, there is a path from $x$ to $y$ inside ${\Delta}$. But $\tilde{\omega}_{\infty}$ and $\tilde{\omega}_n$ agree on ${\Delta}$, so it follows that also $x{\leftrightarrow}y$ in $\tilde{\omega}_n$.
Let $\tilde{\omega}\in\tilde{\Omega}$, and let $C$ be a lattice component of $\tilde{\omega}$. The (topological) closure of $C$ contains an earliest point in the order defined above. Order the lattice components $C_1(\tilde{\omega}),C_2(\tilde{\omega}),\dotsc$ according to the earliest point in their closure; this ordering is almost surely well-defined under any of $\tilde\phi^{\mathrm{f}}_n,\tilde\phi^{\mathrm{f}}$. Note that the claim above implies that this ordering agrees for those lattice components of $\tilde{\omega}_n$ and $\tilde{\omega}_{\infty}$ which intersect ${\Lambda}$.
Let $S_1,S_2,\dotsc$ be independent and uniform on $\{1,\dotsc,q\}$, and define for $x\in{\mathbf{\Theta}}$, $$\tau_x(\tilde{\omega})=\left\{
\begin{array}{ll}
q, & \mbox{if } C_x(\tilde{\omega})\mbox{ is green}, \\
S_i, & \mbox{otherwise, where } C_x(\tilde{\omega})=C_i.
\end{array}\right.$$ Then $\tau(\tilde{\omega}_{\infty})$ has the law $\pi^{\mathrm{f}}$ described in the statement of the theorem, and $\tau(\tilde{\omega}_n)$ has the law $\pi^{\mathrm{f}}_n$ on events in ${\mathcal{G}}_{\Lambda}$. Moreover, from the claim it follows that $\tau_x(\tilde{\omega}_{\infty})=\tau_x(\tilde{\omega}_n)$ for any $x\in{\Lambda}$. Hence for all continuous, bounded $f$, measurable with respect to ${\mathcal{G}}_{\Lambda}$, we have that $f(\tau(\tilde{\omega}_n))\rightarrow f(\tau(\tilde{\omega}_{\infty}))$ almost surely. It follows from the bounded convergence theorem that $$\pi^{\mathrm{f}}_n(f)=E(f(\tau(\tilde{\omega}_n)))
\rightarrow E(f(\tau(\tilde{\omega}_{\infty})))
=\pi^{\mathrm{f}}(f).$$ Since such $f$ are convergence determining it follows that $\pi^{\mathrm{f}}_n\Rightarrow\pi^{\mathrm{f}}$.
\[inf\_vol\_rk\] From the representation given in Theorem \[potts\_lim\_thm\] it follows that the correlation/connectivity relation of Proposition \[corr\_conn\_prop\] holds also for infinite-volume random-cluster and Potts measures. In particular, when $q=2$, it follows (using the obvious notation) that the analogue of holds, namely $${\langle}{\sigma}_x{\sigma}_y{\rangle}^b=\phi^b(x{\leftrightarrow}y),$$ for $b\in\{{\mathrm{f}},{\mathrm{w}}\}$. Note also that when ${\gamma}=0$ then, as in Proposition \[corr\_conn\_prop\], we have for for $b\in\{{\mathrm{f}},{\mathrm{w}}\}$ that $$\label{corr_conn_inf}
{\langle}{\sigma}_x{\rangle}^b=\phi^b(x{\leftrightarrow}{\infty}).$$
### Uniqueness in the Ising model {#ising_uniq_sec}
We turn our attention now to the space–time Ising model on ${\mathbb{L}}={\mathbb{Z}}^d$ with constant ${\lambda},{\delta},{\gamma}$. In this section we continue our discussion, started in Section \[press\_sec\], about uniqueness of infinite-volume measures. More information can be obtained in the case of the Ising model, partly thanks to the so-called [[ghs]{}]{}-inequality which allows us to show the absence of a phase transition when ${\gamma}\neq0$. In contrast, using only results obtained via the random-cluster representation one can say next to nothing about uniqueness when ${\gamma}\neq0$ since there is no useful way of combining a $+1$ external field with a $-1$ ‘lattice-boundary’. The arguments in this section follow very closely those for the classical Ising model, as developed in [@lebowitz_martin-lof] and [@preston_ghs] (see also [@ellis85:LD Chapters IV and V]). We provide full details for completeness.
As remarked earlier, the Ising model admits more boundary conditions than the corresponding random-cluster model. It will therefore seem like some of the arguments presented below repeat what was said in Section \[press\_sec\]. It should be noted, however, that the arguments in this section can deal with all boundary conditions that occur in the Ising model. It will be particularly useful to consider the $+$ and $-$ boundary conditions, defined as follows. Let $b=\{P_1,P_2\}$ where $P_1=\{{\Gamma}\}$ and $P_2={\hat\partial}{\Lambda}$. We define the *$+$ boundary condition* by letting ${\alpha}_1={\alpha}_2=+1$; when ${\gamma}\geq 0$ this equals the wired random-cluster boundary condition with ${\alpha}_{\Gamma}=+1$. We define the *$-$ boundary condition* by letting ${\alpha}_1=+1$ and ${\alpha}_2=-1$. The measure ${\langle}\cdot{\rangle}^-_{\Lambda}$ does not have a satisfactory random-cluster representation when ${\gamma}>0$. (See [@chayes_machta_redner] for an in-depth treatment of some difficulties associated with the graphical representation of the Ising model in an arbitrary external field.) In line with physical terminology we will sometimes in this section refer to the measures ${\langle}\cdot{\rangle}^{b,{\alpha}}_{\Lambda}$ as ‘states’.
For simplicity of notation we will in this section replace ${\lambda}$ and ${\gamma}$ by $2{\lambda}$ and $2{\gamma}$ throughout. We will be writing $Z^{b,{\alpha}}_{\Lambda}$ for the Ising partition function , which therefore becomes $$Z^{b,{\alpha}}_{\Lambda}=\int d\mu_{\delta}(D)\,\sum_{{\sigma}\in{\Sigma}^{b,{\alpha}}_{\Lambda}(D)}
\exp\Big(\int_F{\lambda}(e){\sigma}_ede+\int_K{\gamma}(x){\sigma}_x dx\Big).$$ We will similarly write $P^{b,{\alpha}}_{\Lambda}=(\log Z^{b,{\alpha}}_{\Lambda})/|{\Lambda}|$. Thanks to Proposition \[pfs\], the $P^{b,{\alpha}}_{\Lambda}$ thus defined converge to a function $P$ which is a multiple of the original $P$ in Theorem \[press\_converge\_thm\]. Straightforward modifications of the argument in Theorem \[press\_converge\_thm\] let us deduce that this convergence holds for all boundary conditions $b$ of Ising-type.
We assume throughout this section that ${\Lambda}={\Lambda}_n\uparrow{\mathbf{\Theta}}$ in such a way that $$\label{vanhove}
\frac{|K_n{\setminus}K_{n-1}|}{|K_n|}\rightarrow 0,
{\quad\quad}\mbox{as }n\rightarrow\infty,$$ where ${\Lambda}_n=(K_n,F_n)$ and $|\cdot|$ denotes Lebesgue measure. As previously, straightforward modifications of the argument are valid when ${\beta}<{\infty}$ is fixed and ${\Lambda}\uparrow{\mathbf{\Theta}}_{\beta}$.
Here are some general facts about convex functions; some facts like these were already used in Section \[press\_sec\]. See e.g. [@ellis85:LD Chapter IV] for proofs. Recall that for a function $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$, the left and right derivatives of $f$ are given respectively by $$\frac{\partial f}{\partial {\gamma}^+}:=
\lim_{h\downarrow 0}\frac{f({\gamma}+h)-f({\lambda})}{h}
\quad\mbox{and}\quad
\frac{\partial f}{\partial {\gamma}^-}:=
\lim_{h\downarrow 0}\frac{f({\gamma}-h)-f({\lambda})}{-h}$$ provided these limits exist.
\[conv\_prop\] Let $I\subseteq{\mathbb{R}}$ be an open interval and $f:I\rightarrow{\mathbb{R}}$ a convex function; also let $f_n:I\rightarrow{\mathbb{R}}$ be a sequence of convex functions. Then
- The left and right derivatives of $f$ exist throughout $I$; the right derivative is right-continuous and the left derivative is left-continuous.
- The derivative $f'$ of $f$ exists at all but countably many points in $I$.
- If all the $f_n$ are differentiable and $f_n\rightarrow f$ pointwise then the derivatives $f_n'$ converge to $f'$ whenever the latter exists.
- If the $f_n$ are uniformly bounded above and below then there exists a sub-sequence $f_{n_k}$ and a (necessarily convex) function $f$ such that $f_{n_k}\rightarrow f$ pointwise.
We will usually keep ${\lambda},{\delta}$ fixed and regard $P=P({\gamma})$ as a function of ${\gamma}$, and similarly for other functions. Note that $P$ is an even function of ${\gamma}$: we have for all ${\gamma}>0$ that $P^+_{\Lambda}(-{\gamma})=P^-_{\Lambda}({\gamma})$, and since the limit $P$ is independent of boundary condition it follows that $P(-{\gamma})=P({\gamma})$.
Let $$\label{diff1}
\bar M^{b,{\alpha}}_{\Lambda}:=\frac{\partial P^{b,{\alpha}}_{\Lambda}}{\partial {\gamma}}=
\frac{1}{|{\Lambda}|}\int_{\Lambda}dx{\langle}{\sigma}_x{\rangle}^{b,{\alpha}}_{\Lambda},$$ where we abuse notation to write $x\in{\Lambda}$ (respectively, $|{\Lambda}|$) in place of the more accurate $x\in K$ (respectively, $|K|$). Also let $$M^{b,{\alpha}}_{\Lambda}:={\langle}{\sigma}_0{\rangle}^{b,{\alpha}}_{\Lambda}.$$ Note that together with the first [[gks]{}]{}-inequality imply that $P^{\mathrm{w}}_{\Lambda}$, and hence also $P$, is increasing for ${\gamma}>0$ (and hence decreasing for ${\gamma}<0$). Moreover, we see that $$\label{diff2}
\frac{\partial^2 P^{\mathrm{w}}_{\Lambda}}{\partial {\gamma}^2}=
\frac{1}{|{\Lambda}|}\int_{\Lambda}\int_{\Lambda}dxdy{\langle}{\sigma}_x;{\sigma}_y{\rangle}^{\mathrm{w}}_{\Lambda}\geq0,$$ from the second [[gks]{}]{}-inequality . Thus $P$ is convex in ${\gamma}$.
\[pm\_states\] The states ${\langle}\cdot{\rangle}^+_{\Lambda}$ and ${\langle}\cdot{\rangle}^-_{\Lambda}$ converge weakly as ${\Lambda}\uparrow{\mathbf{\Theta}}$. The limiting states ${\langle}\cdot{\rangle}^+$ and ${\langle}\cdot{\rangle}^-$ are independent of the way in which ${\Lambda}\uparrow{\mathbf{\Theta}}$ and are translation invariant.
The convergence result for $+$ boundary follows from Theorem \[potts\_lim\_thm\] and Remark \[inf\_vol\_rk\], since when $q=2$ the measure $\pi^{\mathrm{w}}_{\Lambda}$ there is precisely the state ${\langle}\cdot{\rangle}_{\Lambda}^+$. However, the result for $-$ boundary does not follow from that result since the random-cluster representation as employed there does not admit the spin at ${\Gamma}$ to be different from that at $\partial{\Lambda}$. (One would have to condition on the event that, in the random-cluster model, the boundary is disconnected from ${\Gamma}$, and then one loses desired monotonicity properties.)
In the proof of Lemma \[pm\_states\] we will be applying the [[fkg]{}]{}-inequality, Lemma \[ising\_fkg\]. For each $x\in{\mathbb{K}}$, let $\nu'_x=({\sigma}_x+1)/2$ and for $A{\subseteq}{\mathbb{K}}$ finite, write $$\label{mrk1}
\nu'_A=\prod_{x\in A} \nu'_x.$$ Note that $\nu'_A={\hbox{\rm 1\kern-.27em I}}_S$, where $S$ is the event that ${\sigma}_x=+1$ for all $x\in A$. This is an increasing event, and a continuity set by Example \[cty\_ex\]. Similarly, if ${\Lambda}{\subseteq}{\Delta}$ are regions and $T$ is the event that ${\sigma}=+1$ on ${\Delta}\setminus {\Lambda}$, then $T$ is an increasing event and a continuity set, also by Example \[cty\_ex\].
It is easy to check that the variables $\nu'_A$, as $A$ ranges over the finite subsets of ${\mathbb{K}}$, form a convergence determining class. By Lemma \[potts\_cond\] and Lemma \[ising\_fkg\] we therefore see that for any regions ${\Lambda}\subseteq{\Delta}$ we have that $${\langle}\nu'_A{\rangle}^+_{\Lambda}={\langle}\nu'_A\mid{\sigma}\equiv+1\mbox{ on } {\Delta}{\setminus}{\Lambda}{\rangle}^+_{\Delta}\geq {\langle}\nu'_A{\rangle}^+_{\Delta}$$ and $${\langle}\nu'_A{\rangle}^-_{\Lambda}={\langle}\nu'_A\mid{\sigma}\equiv-1\mbox{ on } {\Delta}{\setminus}{\Lambda}{\rangle}^-_{\Delta}\leq {\langle}\nu'_A{\rangle}^-_{\Delta}.$$ Hence ${\langle}\nu'_A{\rangle}^+_{\Lambda}$ and ${\langle}\nu'_A{\rangle}^-_{\Lambda}$ converge for all finite $A\subseteq{\mathbb{K}}$, as required.
The proof of Lemma \[pm\_states\] shows in particular that $$\label{mo1}
{\langle}{\sigma}_0{\rangle}^+_{\Lambda}\downarrow{\langle}{\sigma}_0{\rangle}^+
\qquad\mbox{and}\qquad {\langle}{\sigma}_0{\rangle}^-_{\Lambda}\uparrow{\langle}{\sigma}_0{\rangle}^-,$$ and indeed that all the ${\langle}{\sigma}_A{\rangle}^\pm_{\Lambda}$ converge to the corresponding ${\langle}{\sigma}_A{\rangle}^\pm$. Recall that by convexity, the left and right derivatives of $P$ exist at all ${\gamma}\in{\mathbb{R}}$.
\[press\_der\_lem1\] For all ${\gamma}\in{\mathbb{R}}$ we have that $$\frac{\partial P}{\partial {\gamma}^+}={\langle}{\sigma}_0{\rangle}^+
\qquad\mbox{and}\qquad
\frac{\partial P}{\partial {\gamma}^-}={\langle}{\sigma}_0{\rangle}^-.$$
As a preliminary step we first show that $\bar M^\pm_{\Lambda}$ has the same infinite-volume limit as $M^\pm_{\Lambda}$, that is to say $$\label{s1}
\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}}\bar M^\pm_{\Lambda}={\langle}{\sigma}_0{\rangle}^\pm.$$ We prove this in the case of $+$ boundary, the case of $-$ boundary being similar. First note that $$\bar M^+_{\Lambda}=\frac{1}{|{\Lambda}|}\int_{\Lambda}dx{\langle}{\sigma}_x{\rangle}^+_{\Lambda}\geq
\frac{1}{|{\Lambda}|}\int_{\Lambda}dx{\langle}{\sigma}_x{\rangle}^+={\langle}{\sigma}_0{\rangle}^+,$$ by and translation invariance. Thus $\liminf_{\Lambda}\bar M^+_{\Lambda}\geq{\langle}{\sigma}_0{\rangle}^+$. Next let ${\varepsilon}>0$ and let ${\Lambda}$ be large enough so that ${\langle}{\sigma}_0{\rangle}^+_{\Lambda}\leq{\langle}{\sigma}_0{\rangle}^++{\varepsilon}$. If $x\in{\mathbb{K}}$ and ${\Delta}$ is large enough that the translated region ${\Lambda}+x\subseteq{\Delta}$ then $${\langle}{\sigma}_x{\rangle}^+_{\Delta}\leq{\langle}{\sigma}_x{\rangle}^+_{{\Lambda}+x}={\langle}{\sigma}_0{\rangle}^+_{\Lambda}\leq{\langle}{\sigma}_0{\rangle}^++{\varepsilon}.$$ Let ${\Delta}':=\{x\in{\Delta}:{\Lambda}+x\in{\Delta}\}$. Then $$\begin{split}
\bar M^+_{\Delta}&=\frac{1}{|{\Delta}|}\int_{\Delta}dx{\langle}{\sigma}_x{\rangle}^+_{\Delta}\leq
\frac{1}{|{\Delta}|}\Big(\int_{{\Delta}'} dx{\langle}{\sigma}_x{\rangle}^+_{\Delta}+|{\Delta}{\setminus}{\Delta}'|\Big)\\& \leq
\frac{1}{|{\Delta}|}\Big(|{\Delta}'|\big({\langle}{\sigma}_x{\rangle}^++{\varepsilon}\big)+|{\Delta}{\setminus}{\Delta}'|\Big).
\end{split}$$ It therefore follows from the assumption that $\limsup_{\Lambda}\bar M^+_{\Lambda}\leq{\langle}{\sigma}_0{\rangle}^++{\varepsilon}$, which gives .
Next we claim that ${\langle}{\sigma}_0{\rangle}^+$ and ${\langle}{\sigma}_0{\rangle}^-$ are right- and left continuous in ${\gamma}$, respectively. First consider $+$ boundary. Then for ${\gamma}'>{\gamma}$, we have for any ${\Lambda}$ from Lemma \[ising\_mon\] that ${\langle}{\sigma}_0{\rangle}^+_{{\Lambda},{\gamma}'}\geq{\langle}{\sigma}_0{\rangle}^+_{{\Lambda},{\gamma}}$. Thus $$\label{mo3}
\begin{split}
{\langle}{\sigma}_0{\rangle}^+_{\gamma}&\leq\liminf_{{\gamma}'\downarrow{\gamma}}{\langle}{\sigma}_0{\rangle}^+_{{\gamma}'}
\leq\limsup_{{\gamma}'\downarrow{\gamma}}{\langle}{\sigma}_0{\rangle}^+_{{\gamma}'}\\
&\leq\limsup_{{\gamma}'\downarrow{\gamma}}{\langle}{\sigma}_0{\rangle}^+_{{\Lambda},{\gamma}'}=
{\langle}{\sigma}_0{\rangle}^+_{{\Lambda},{\gamma}}\xrightarrow[{\Lambda}\uparrow{\mathbf{\Theta}}]{}{\langle}{\sigma}_0{\rangle}^+_{\gamma}.
\end{split}$$ (We have used the fact that ${\langle}{\sigma}_0{\rangle}^+_{\Lambda}$ is continuous in ${\gamma}$.) A similar calculation holds for $-$ boundary.
Now, by convexity of $P$, the right derivative $\frac{\partial P}{\partial{\gamma}^+}$ is right-continuous, and also $\lim_{\Lambda}\bar M^\pm_{\Lambda}=\frac{\partial P}{\partial{\gamma}}$ whenever the right side exists. But it exists for all but countably many ${\gamma}$, so given ${\gamma}$ there is a sequence ${\gamma}_n\downarrow{\gamma}$ such that $\frac{\partial P}{\partial{\gamma}}({\gamma}_n)={\langle}{\sigma}_0{\rangle}^+_{{\gamma}_n}$ for all $n$, and similarly for $-$ boundary. The result follows.
We say that there is a *unique state* at ${\gamma}$ (or at ${\lambda},{\delta},{\gamma}$) if for all finite $A{\subseteq}{\mathbb{K}}$, the limit ${\langle}{\sigma}_A{\rangle}:=\lim_{\Lambda}{\langle}{\sigma}_A{\rangle}^{b,{\alpha}}_{\Lambda}$ exists and is independent of the boundary condition $(b,{\alpha})$. Note that, by linearity, it is equivalent to require that all the limits ${\langle}\nu'_A{\rangle}:=\lim_{\Lambda}{\langle}\nu'_A{\rangle}^{b,{\alpha}}_{\Lambda}$ exist and are independent of the boundary condition. Alternatively, there is a unique state if and only if the measures ${\langle}\cdot{\rangle}^{b,{\alpha}}_{\Lambda}$ all converge weakly to the same limiting measure.
\[uniq\_lem\] There is a unique state at ${\gamma}\in{\mathbb{R}}$ if and only if $P$ is differentiable at ${\gamma}$. There is a unique state at any ${\gamma}\neq0$.
We have that $$f_A:=\sum_{x\in A}\nu'_x-\nu'_A$$ is increasing in ${\sigma}$. By the [[fkg]{}]{}-inequality, Lemma \[ising\_fkg\], we have that ${\langle}f_A{\rangle}^+_{\Lambda}\geq{\langle}f_A{\rangle}^-_{\Lambda}$. It follows on letting ${\Lambda}\uparrow{\mathbf{\Theta}}$, and using translation invariance as well as Lemma \[press\_der\_lem1\], that $$0\leq {\langle}\nu'_A{\rangle}^+-{\langle}\nu'_A{\rangle}^-\leq
\frac{1}{2}\sum_{x\in A}({\langle}{\sigma}_x{\rangle}^+-{\langle}{\sigma}_x{\rangle}^-)
=\frac{|A|}{2}\Big(\frac{\partial P}{\partial{\gamma}^+}-
\frac{\partial P}{\partial{\gamma}^-}\Big),$$ where $|A|$ is the number of elements in $A$. Hence ${\langle}\nu'_A{\rangle}^+={\langle}\nu'_A{\rangle}^-$ whenever $\frac{\partial P}{\partial{\gamma}}$ exists. Since ${\langle}\nu'_A{\rangle}^-\leq{\langle}\nu'_A{\rangle}^{b,{\alpha}}\leq{\langle}\nu'_A{\rangle}^+$ for all $(b,{\alpha})$ (a consequence of Lemma \[ising\_fkg\]), the first claim follows.
The next part makes use of the facts about convex functions stated above; this part of the argument originates in [@preston_ghs]. Let ${\gamma}>0$, and use the free boundary condition. We already know that $P$ and each $P^{\mathrm{f}}_{\Lambda}$ is convex. The [[ghs]{}]{}-inequality, which is standard for the classical Ising model and proved for the current model in Lemma \[ghs\_lem\], implies that each $\bar M^{\mathrm{f}}_{\Lambda}$ has non*positive* second derivative for ${\gamma}>0$, and hence that each $\bar M^{\mathrm{f}}_{\Lambda}$ is concave. Moreover, each $\bar M^{\mathrm{f}}_{\Lambda}$ lies between $-1$ and $1$. There therefore exists a sequence ${\Lambda}_n$ of simple regions such that the sequence $\bar M^{\mathrm{f}}_{{\Lambda}_n}$ converges pointwise to a limiting function which we denote by $M^{\mathrm{f}}_{\infty}$. If $0<{\gamma}<{\gamma}'$ then by the fundamental theorem of calculus and the bounded convergence theorem, we have that $$\begin{split}
P({\gamma}')-P({\gamma})&=\lim_{n\rightarrow\infty}
\big(P^{\mathrm{f}}_{{\Lambda}_n}({\gamma}')-P^{\mathrm{f}}_{{\Lambda}_n}({\gamma})\big)\\
&=\lim_{n\rightarrow\infty}\int_{{\gamma}}^{{\gamma}'}\bar M^{\mathrm{f}}_{{\Lambda}_n}({\gamma})\:d{\gamma}=\int_{{\gamma}}^{{\gamma}'}M^{\mathrm{f}}_{\infty}({\gamma})\:d{\gamma}.
\end{split}$$ The function $M^{\mathrm{f}}_{\infty}$ is concave, and hence continuous, in ${\gamma}>0$. It therefore follows from the above that $P$ is in fact differentiable at each ${\gamma}>0$ (with derivative $M^{\mathrm{f}}_{\infty}$). The result follows since $P(-{\gamma})=P({\gamma})$ for all ${\gamma}>0$.
Whenever there is a unique infinite-volume state at ${\gamma}$, we will denote it by ${\langle}\cdot{\rangle}={\langle}\cdot{\rangle}_{\gamma}$.
\[press\_der\_lem2\] For each ${\gamma}\neq0$ and each $(b,{\alpha})$, we have that $$M:=\frac{\partial P}{\partial {\gamma}}=
\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}} M^{b,{\alpha}}_{\Lambda}=\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}} \bar M^{b,{\alpha}}_{\Lambda}.$$
The proof of Lemma \[uniq\_lem\] shows that at each ${\gamma}\neq0$ the derivative of $P$ is $M^{\mathrm{f}}_{\infty}$. Since for all $(b,{\alpha})$ and ${\Lambda}$, the function $P^{b,{\alpha}}_{\Lambda}({\gamma})$ is convex and differentiable with $$\frac{\partial P^{b,{\alpha}}_{\Lambda}}{\partial{\gamma}}=\bar M^{b,{\alpha}}_{\Lambda}$$ it follows from the properties of convex functions that $\bar M^{b,{\alpha}}_{\Lambda}({\gamma})\rightarrow M({\gamma})$ at all ${\gamma}\neq0$. That also $M^{b,{\alpha}}_{\Lambda}\rightarrow M$ for ${\gamma}\neq0$ follows from the the fact that $M^-_{\Lambda}\leq M^{b,{\alpha}}_{\Lambda}\leq M^+_{\Lambda}$ and the fact that $\lim M^\pm_{\Lambda}=\lim \bar M^\pm_{\Lambda}$ as we saw at .
Lemma \[press\_der\_lem2\] implies in particular that $$M=\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}}{\langle}{\sigma}_0{\rangle}^\pm_{\Lambda}$$ at all ${\gamma}\neq0$. We know from Lemma \[pm\_states\] that the limits $$M_\pm:=\lim_{{\Lambda}\uparrow{\mathbf{\Theta}}}{\langle}{\sigma}_0{\rangle}^\pm_{\Lambda}$$exist also at ${\gamma}=0$. By Lemma \[press\_der\_lem1\] there is a unique state at ${\gamma}=0$ if and only if $M_+(0)=M_-(0)$. We sometimes call $M_+(0)$ the *spontaneous magnetization*.
Note that for all ${\Lambda}$ and all ${\gamma}>0$ we have $M^+_{\Lambda}(-{\gamma})=-M^-_{\Lambda}({\gamma})$, so that $\lim M^+_{\Lambda}(-{\gamma})=-M({\gamma})$. Hence $M$ is an *odd* function of ${\gamma}\neq0$. Note also that $$M_+(0)=\lim_{{\gamma}\downarrow 0} M({\gamma}).$$ Indeed, rather more is true: by repeating the argument at with ${\sigma}_A$ in place of ${\sigma}_0$, it follows that the state ${\langle}\cdot{\rangle}^+$ of Lemma \[pm\_states\] may be written as the weak limit $$\label{plus_ising}
{\langle}\cdot{\rangle}^+_{{\gamma}=0}=\lim_{{\gamma}\downarrow 0} {\langle}\cdot{\rangle}_{\gamma}$$ where ${\langle}\cdot{\rangle}_{\gamma}$ is the unique state at ${\gamma}>0$. Thus we may summarize the results of this section as follows.
\[ising\_summary\_thm\] There is a unique state at all ${\gamma}\neq0$ and there is a unique state at ${\gamma}=0$ if and only if $$M_+(0)\equiv\lim_{{\gamma}\downarrow0}M({\gamma})=0.$$
We now recall the remaining parameters ${\lambda}$, ${\delta}$ and ${\beta}$. As previously, we set ${\delta}=1$, $\rho={\lambda}/{\delta}$, and write $$M^{\beta}(\rho,{\gamma})=M^{\beta}(\rho,1,{\gamma}).$$ It follows from Lemma \[cor\_mon\_lem\] that $M^{\beta}_+(\rho,0)$ is an increasing function of $\rho$. This motivates the following definition.
\[ising\_crit\_val\_def\] We define the *critical value* $$\rho^{\beta}_{\mathrm{c}}:=\inf\{\rho>0:M^{\beta}_+(\rho,0)>0\}.$$
From Remark \[inf\_vol\_rk\] and it follows that this $\rho^{\beta}_{\mathrm{c}}$ coincides with the ‘percolation threshold’ $\rho_{\mathrm{c}}(2)$ for the $q=2$ space–time random-cluster model as defined in Definition \[crit\_def\]. More information about $\rho^{\beta}_{\mathrm{c}}$ and the behaviour of $M^{\beta}$ and related quantities near the critical point may be found in Section \[cons\_sec\].
The quantum Ising model: random-parity representation and sharpness of the phase transition {#qim_ch}
===========================================================================================
> [*Summary.*]{} We develop a ‘random-parity’ representation for the space–time Ising model; this is the space–time analog of the random-current representation. The random-parity representation is then used to derive a number of differential inequalities, from which one can deduce many important properties of the phase transition of the quantum Ising model, such as sharpness of the transition.
Classical and quantum Ising models {#sec-backg}
----------------------------------
Recall from the Introduction that the (transverse field) quantum Ising model on the finite graph $L$ is given by the Hamiltonian $$H=-\tfrac{1}{2}{\lambda}\sum_{e=uv\in E}{\sigma}_u^{(3)}{\sigma}_v^{(3)}
-{\delta}\sum_{v\in V}{\sigma}_v^{(1)},$$ acting on the Hilbert space ${\mathcal{H}}=\bigotimes_{v\in V}{\mathbb{C}}^2$. We refer to that chapter for definitions of the notation used. In the quantum Ising model the number ${\beta}>0$ is thought of as the ‘inverse temperature’. We define the positive temperature states $$\label{qi_states_eq}
{\nu}_{L,{\beta}}(Q)=\frac{1}{Z_L({\beta})}{\mathrm{tr}}(e^{-{\beta}H}Q),$$ where $Z_L({\beta})={\mathrm{tr}}(e^{-{\beta}H})$ and $Q$ is a suitable matrix. The *ground state* is defined as the limit ${\nu}_L$ of ${\nu}_{L,{\beta}}$ as ${\beta}\rightarrow\infty$. If $(L_n: n \ge 1)$ is an increasing sequence of graphs tending to the infinite graph ${\mathbb{L}}$, then we may also make use of the *infinite-volume* limits $${\nu}_{L,{\beta}}=\lim_{n\rightarrow\infty}{\nu}_{L_n,{\beta}},{\quad\quad}{\nu}_L=\lim_{n\rightarrow\infty}{\nu}_{L_n}.$$ The existence of such limits is discussed in [@akn], see also the related discussion of limits of space–time Ising measures in Section \[inf\_potts\_sec\].
The quantum Ising model is intimately related to the space–time Ising model, one manifestation of this being the following. Recall that if ${|\psi\rangle}$ denotes a vector then ${\langle\psi|}$ denotes its conjugate transpose. The state ${\nu}_{L,{\beta}}$ of gives rise to a probability measure $\mu$ on $\{-1,+1\}^V$ by $$\mu({\sigma})=\frac{{\langle}{\sigma}|e^{-{\beta}H}|{\sigma}{\rangle}}{{\mathrm{tr}}(e^{-{\beta}H})},{\quad\quad}{\sigma}\in\{-1,+1\}^V.$$ When ${\gamma}=0$, it turns out that $\mu$ is the law of the vector $({\sigma}_{(v,0)}: v\in V)$ under the space–time Ising measure of (with periodic boundary, see below). See [@akn] and the references therein. It therefore makes sense to study the phase diagram of the quantum Ising model via its representation in the space–time Ising model. Note, however, that in our analysis it is crucial to work with ${\gamma}>0$, and to take the limit ${\gamma}\downarrow 0$ later. The role played in the classical model by the external field will in our analysis be played by the ‘ghost-field’ ${\gamma}$ rather than the ‘physical’ transverse field ${\delta}$. (In fact, ${\gamma}$ corresponds to a ${\sigma}^{(3)}$-field, see [@crawford_ioffe].)
In most of this chapter we will be working with periodic boundary conditions in the ${\mathbb{R}}$-direction. That is to say, for simple regions of the form we will identify the endpoints of the the ‘time’ interval $[-{\beta}/2,{\beta}/2]$, and think of this interval as the circle of circumference ${\beta}$. We will denote this circle by ${\mathbb{S}}={\mathbb{S}}_{\beta}$ and thus our simple regions will be of the form $L\times{\mathbb{S}}$ for some finite graph $L$. We shall generally (until Section \[cons\_sec\]) keep ${\beta}>0$ fixed, and thus suppress reference to ${\beta}$. Similarly, we will generally suppress reference to the boundary condition. Thus we will write for instance ${\Sigma}(D)$ for the set of spin configurations permitted by $D$ (see the discussion before ).
General regions of the form will usually be thought of as subsets of the simple region $L\times{\mathbb{S}}$. Thus, for $v\in V$, we let $K_v\subseteq{\mathbb{S}}$ be a finite union of disjoint intervals, and we write $K_v=\bigcup_{i=1}^{m(v)} I^v_i$. As before, no assumption is made on whether the $I^v_i$ are open, closed, or half-open. With the $K_v$ given, we define $F$ and ${\Lambda}$ as in .
For simplicity of notation we replace in this chapter the functions ${\lambda},{\gamma}$ in by $2{\lambda},2{\gamma}$, respectively. Thus the space–time Ising measure on a region ${\Lambda}=(K,F)$ has partition function $$Z'=\int d\mu_{\delta}({D})\sum_{{\sigma}\in{\Sigma}({D})}\exp\left\{\int_F{\lambda}(e){\sigma}_e\, de+
\int_K{\gamma}(x){\sigma}_x\, dx\right\},
\label{o12}$$ where ${\sigma}_e={\sigma}_{(u,t)}{\sigma}_{(v,t)}$ if $e=(uv,t)$. See . As previously, we write ${\langle}f{\rangle}$ for the mean of a ${\mathcal{G}}_{\Lambda}$-measurable $f:{\Sigma}\rightarrow{\mathbb{R}}$ under this measure. Thus for example $$\label{st_Ising_eq}
{\langle}{\sigma}_A{\rangle}=
\frac{1}{Z'}\int d\mu_{\delta}({D})\sum_{{\sigma}\in{\Sigma}({D})}
{\sigma}_A\exp\left\{\int_F{\lambda}(e){\sigma}_e\, de+
\int_K{\gamma}(x){\sigma}_x\, dx\right\}.$$ Note that in this chapter we denote the partition function by $Z'$.
It is essential for our method in this chapter that we work on general regions of the form given in . The reason for this is that, in the geometrical analysis of currents, we shall at times remove from $K$ a random subset called the ‘backbone’, and the ensuing domain has the form of . Note that considering this general class of regions also allows us to revert to a ‘free’ rather than a ‘vertically periodic’ boundary condition. That is, by setting $K_v =[-{\beta}/2,{\beta}/2)$ for all $v\in V$, rather than $K_v = [-{\beta}/2,{\beta}/2]$, we effectively remove the restriction that the ‘top’ and ‘bottom’ of each $v\times{\mathbb{S}}$ have the same spin.
Whenever we wish to emphasize the roles of particular $K$, ${\lambda}$, ${\delta}$, ${\gamma}$, we include them as subscripts. For example, we may write ${\langle}{\sigma}_A{\rangle}_K$ or ${\langle}{\sigma}_A{\rangle}_{K,{\gamma}}$ or $Z'_{\gamma}$, and so on.
### Statement of the main results
Let $0$ be a given point of $V\times {\mathbb{S}}$. We will be particularly concerned with the *magnetization* and *susceptibility* of the space–time Ising model on ${\Lambda}=L\times {\mathbb{S}}$, given respectively by $$\begin{aligned}
M=M_{\Lambda}({\lambda},{\delta},{\gamma}) &:={\langle}{\sigma}_0{\rangle},\label{def-mag}\\
\chi=\chi_{\Lambda}({\lambda},{\delta},{\gamma}) &:=\frac{\partial M}{\partial {\gamma}}
=\int_{\Lambda}{\langle}{\sigma}_0;{\sigma}_x{\rangle}\,dx,
\label{def-susc}\end{aligned}$$ where we recall that the *truncated* two-point function ${\langle}{\sigma}_0;{\sigma}_x{\rangle}$ is given by $${\langle}{\sigma}_A;{\sigma}_B{\rangle}:= {\langle}{\sigma}_A{\sigma}_B{\rangle}- {\langle}{\sigma}_A{\rangle}{\langle}{\sigma}_B{\rangle}.
\label{o20}$$ Note that, for simplicity of notation, we will in most of this chapter keep $M$ and $\chi$ free from sub- and superscripts even though they refer to finite-volume quantities. Some basic properties of these quantities were discussed in Section \[ising\_uniq\_sec\].
Our main choice for $L$ is a box $[-n,n]^d$ in the $d$-dimensional cubic lattice ${\mathbb{Z}}^d$ where $d \ge 1$, with a periodic boundary condition. That is to say, apart from the usual nearest-neighbour bonds, we also think of two vertices $u$, $v$ as joined by an edge whenever there exists $i\in\{1,2,\dots,d\}$ such that $u$ and $v$ differ by exactly $2n$ in the $i$th coordinate. Subject to this boundary condition, $M$ and $\chi$ do not depend on the choice of origin $0$. We shall pass to the infinite-volume limit as $L\uparrow {\mathbb{Z}}^d$. The model is over-parametrized, and we shall, as before, normally assume ${\delta}=1$, and write $\rho={\lambda}/{\delta}$. The critical point ${\rho_{\mathrm{c}}}={\rho_{\mathrm{c}}}^{\beta}$ is given as in Definition \[ising\_crit\_val\_def\] by $${\rho_{\mathrm{c}}}^{\beta}:=\inf\{\rho: M^{\beta}_+(\rho)>0\},
\label{o1}$$ where $$M^{\beta}_+(\rho):=\lim_{{\gamma}\downarrow 0}M^{\beta}(\rho,{\gamma}),
\label{o2}$$ is the magnetization in the limiting state ${\langle}\cdot{\rangle}^{\beta}_+$ as ${\gamma}\downarrow 0$. As in Theorem \[crit\_nontriv\], we have that: $$\begin{split}
\text{if $d \ge 2$}&:\quad 0<{\rho_{\mathrm{c}}}^{\beta}<{\infty}\text{ for ${\beta}\in(0,{\infty}]$},\\
\text{if $d=1$}&:\quad {\rho_{\mathrm{c}}}^{\beta}={\infty}\text{ for ${\beta}\in(0,{\infty})$}, \ 0<{\rho_{\mathrm{c}}}^{\infty}<{\infty}.
\end{split}
\label{critvals}$$
Complete statements of our main results are deferred until Section \[cons\_sec\], but here are two examples of what can be proved.
\[ed\_thm\] Let $u,v\in {\mathbb{Z}}^d$ where $d \ge 1$, and $s,t\in{\mathbb{R}}$. For ${\beta}\in(0,{\infty}]$:
if $0 < \rho < {\rho_{\mathrm{c}}}^{\beta}$, the two-point correlation function $\langle {\sigma}_{(u,s)}{\sigma}_{(v,t)}\rangle^{\beta}_+$ of the space–time Ising model decays exponentially to $0$ as $|u-v|+|s-t|\to{\infty}$,
if $\rho\ge{\rho_{\mathrm{c}}}^{\beta}$, $\langle {\sigma}_{(u,s)}{\sigma}_{(v,t)}\rangle^{\beta}_+ \ge M^{\beta}_+(\rho)^2>0$.
Theorem \[ed\_thm\] is what is called ‘sharpness of the phase transition’: there is no intermediate regime in which correlations decay to zero slowly. (See for example [@chayes_chayes86] and [@ghm] for examples of systems where this does occur).
\[mf\_thm\] Let ${\beta}\in(0,{\infty}]$. In the notation of Theorem \[ed\_thm\], there exists $c = c(d)>0$ such that $$M^{\beta}_+(\rho) \geq c(\rho-{\rho_{\mathrm{c}}}^{\beta})^{1/2}{\quad\quad}\text{for }
\rho>{\rho_{\mathrm{c}}}^{\beta}.$$
These and other facts will be stated and proved in Section \[cons\_sec\]. Their implications for the infinite-volume quantum model will be elaborated around –.
The approach used here is to prove a family of differential inequalities for the finite-volume magnetization $M(\rho,{\gamma})$. This parallels the methods established in [@ab; @abf] for the analysis of the phase transitions in percolation and Ising models on discrete lattices, and indeed our arguments are closely related to those of [@abf]. Whereas new problems arise in the current context and require treatment, certain aspects of the analysis presented here are simpler that the corresponding steps of [@abf]. The application to the quantum model imposes a periodic boundary condition in the ${\beta}$ direction; some of our conclusions are valid for the space–time Ising model with a free boundary condition.
The following is the principal differential inequality we will derive. (Our results are in fact valid in greater generality, see the statement before Assumption \[periodic\_assump\].)
\[main\_pdi\_thm\] Let $d\ge 1$, ${\beta}<{\infty}$, and $L = [-n,n]^d$ with periodic boundary. Then $$\label{ihp18}
M\leq {\gamma}\chi+M^3+2{\lambda}M^2\frac{\partial M}{\partial {\lambda}}
-2{\delta}M^2\frac{\partial M}{\partial {\delta}}.$$
A similar inequality was derived in [@abf] for the classical Ising model, and our method of proof is closely related to that used there. Other such inequalities have been proved for percolation in [@ab] (see also [@grimmett_perc]), and for the contact model in [@aizenman_jung; @bezuidenhout_grimmett]. As observed in [@ab; @abf], the powers of $M$ on the right side of determine the bounds of Theorems \[ed\_thm\](ii) and \[mf\_thm\] on the critical exponents. The cornerstone of our proof is a ‘random-parity representation’ of the space–time Ising model.
The analysis of the differential inequalities, following [@ab; @abf], reveals a number of facts about the behaviour of the model. In particular, we will show the exponential decay of the correlations ${\langle}{\sigma}_0{\sigma}_x{\rangle}_+$ when $\rho<{\rho_{\mathrm{c}}}^{\beta}$ and ${\gamma}=0$, as asserted in Theorem \[ed\_thm\], and in addition certain bounds on two critical exponents of the model. See Section \[cons\_sec\] for further details.
We draw from [@akn; @aizenman_nacht] in the following summary of the relationship between the phase transitions of the quantum and space–time Ising models. Let $u,v\in V$, and $$\tau^{\beta}_{L}(u,v) := {\mathrm{tr}}\bigl({\nu}_{L,{\beta}}(Q_{u,v})\bigr),{\quad\quad}Q_{u,v} = {\sigma}^{(3)}_u{\sigma}^{(3)}_v.$$It is the case that $$\tau^{\beta}_{L}(u,v) = {\langle}{\sigma}_A {\rangle}^{\beta}_L
\label{o3}$$ where $A=\{(u,0),(v,0)\}$, and the role of ${\beta}$ is stressed in the superscript. Let $\tau_L^{\infty}$ denote the limit of $\tau^{\beta}_{L}$ as ${\beta}\to{\infty}$. For ${\beta}\in(0,{\infty}]$, let $\tau^{\beta}$ be the limit of $\tau^{\beta}_L$ as $L\uparrow {\mathbb{Z}}^d$. (The existence of this limit may depend on the choice of boundary condition on $L$, and we return to this at the end of Section \[cons\_sec\].) By Theorem \[ed\_thm\], $$\tau^{\beta}(u,v) \le c'e^{-c|u-v|},
\label{o4}$$ where $c'$, $c$ depend on $\rho$, and $c>0$ for $\rho<{\rho_{\mathrm{c}}}^{\beta}$ and ${\beta}\in(0,{\infty}]$. Here, $|u-v|$ denotes the $L^1$ distance from $u$ to $v$. The situation when $\rho={\rho_{\mathrm{c}}}^{\beta}$ is more obscure, but one has that $$\limsup_{|v|\to{\infty}}\tau^{\beta}(u,v) \le M^{\beta}_+(\rho),
\label{o4a}$$ so that $\tau^{\beta}(u,v) \to 0$ whenever $M^{\beta}_+(\rho)=0$. It is proved at Theorem \[crit\_val\_cor\] that ${\rho_{\mathrm{c}}}^{\infty}=2$ and $M^{\infty}_+(2)=0$ when $d=1$.
By the [[[fkg]{}]{}]{} inequality, and the uniqueness of infinite clusters in the space–time [random-cluster]{} model (see Theorem \[inf\_clust\_uniq\]), $$\tau^{\beta}(u,v) \ge M^{\beta}_+(\rho-)^2 > 0,
\label{o5}$$ when $\rho > {\rho_{\mathrm{c}}}^{\beta}$ and ${\beta}\in(0,{\infty}]$, where $f(x-):= \lim_{y\uparrow x}f(y)$. The proof is discussed at the end of Section \[cons\_sec\].
The critical value ${\rho_{\mathrm{c}}}^{\beta}$ depends of course on the number of dimensions. We shall in the next chapter use Theorem \[ed\_thm\] and planar duality to show that ${\rho_{\mathrm{c}}}^{\infty}=2$ when $d=1$, and in addition that the transition is of second order in that $M^{\infty}_+(2)=0$. See Theorem \[crit\_val\_cor\]. The critical point has been calculated by other means in the quantum case, but we believe that the current proof is valuable. Two applications to the work of [@bjo0; @GOS] are summarized in Section \[sec\_1d\].
Here is a brief outline of the contents of this chapter. Formal definitions are presented in Section \[sec-backg\]. The random-parity representation of the quantum Ising model is described in Section \[rcr\_sec\]. This representation may at first sight seem quite different from the random-current representation of the classical Ising model on a discrete lattice. It requires more work to set up than does its discrete cousin, but once in place it works in a very similar, and sometimes simpler, manner. We then state and prove, in Section \[ssec-switching\], the fundamental ‘switching lemma’. In Section \[sw\_appl\_sec\] are presented a number of important consequences of the switching lemma, including [[[ghs]{}]{}]{} and Simon–Lieb inequalities, as well as other useful inequalities and identities. In Section \[pf\_sec\], we prove the somewhat more involved differential inequality of Theorem \[main\_pdi\_thm\], which is similar to the main inequality of [@abf]. Our main results follow from Theorem \[main\_pdi\_thm\] in conjunction with the results of Section \[sw\_appl\_sec\]. Finally, in Sections \[cons\_sec\] and \[sec\_1d\], we give rigorous formulations and proofs of our main results.
This chapter forms the contents of the article [@bjogr2], which has been published in the Journal of Statistical Physics. The quantum mean-field, or Curie–Weiss, model has been studied using large-deviation techniques in [@chayes_ioffe_curie-weiss], see also [@grimmett_stp]. There is a very substantial overlap between the results reported here and those of the independent and contemporaneous article [@crawford_ioffe]. The basic differential inequalities of Theorems \[main\_pdi\_thm\] and \[three\_ineq\_lem\] appear in both places. The proofs are in essence the same despite some superficial differences. We are grateful to the authors of [@crawford_ioffe] for explaining the relationship between the random-parity representation of Section \[rcr\_sec\] and the random-current representation of [@ioffe_geom Section 2.2]. As pointed out in [@crawford_ioffe], the appendix of [@chayes_ioffe_curie-weiss] contains a type of switching argument for the mean-field model. A principal difference between that argument and those of [@crawford_ioffe; @ioffe_geom] and the current work is that it uses the classical switching lemma developed in [@aiz82], applied to a discretized version of the mean-field system.
The random-parity representation {#rcr_sec}
--------------------------------
The classical Ising model on a discrete graph $L$ is a ‘site model’, in the sense that configurations comprise spins assigned to the vertices (or ‘sites’) of $L$. As described in the Introduction, the classical random-current representation maps this into a bond-model, in which the sites no longer carry random values, but instead the *edges* $e$ (or ‘bonds’) of the graph are replaced by a random number $N_e$ of parallel edges. The bond $e$ is called *even* ([respectively]{}, *odd*) if $N_e$ is even ([respectively]{}, odd). The odd bonds may be arranged into paths and cycles. One cannot proceed in the same way in the above space–time Ising model.
There are two possible alternative approaches. The first uses the fact that, conditional on the set $D$ of deaths, ${\Lambda}$ may be viewed as a discrete structure with finitely many components, to which the random-current representation of [@aiz82] may be applied. This is explained in detail around below. Another approach is to forget about ‘bonds’, and instead to concentrate on the parity configuration associated with a current-configuration, as follows.
The circle ${\mathbb{S}}$ may be viewed as a continuous limit of a ring of equally spaced points. If we apply the random-current representation to the discretized system, but only record whether a bond is even or odd, the representation has a well-defined limit as a partition of ${\mathbb{S}}$ into even and odd sub-intervals. In the limiting picture, even and odd intervals carry different weights, and it is the properties of these weights that render the representation useful. This is the essence of the main result in this section, Theorem \[rcr\_thm\]. We will prove this result without recourse to discretization.
We now define two additional random processes associated with the space–time Ising measure on ${\Lambda}$. The first is a random colouring of $K$, and the second is a random (finite) weighted graph. These two objects will be the main components of the random-parity representation.
### Colourings {#ssec-col}
Let ${\overline}K$ be the closure of $K$. A set of *sources* is a finite set $A{\subseteq}{\overline}K$ such that: each $a\in A$ is the endpoint of at most one maximal subinterval $I^v_i$. (This last condition is for simplicity later.) Let $B\subseteq F$ and $G\subseteq K$ be finite sets. Let $S=A \cup G \cup V(B)$, where $V(B)$ is the set of endpoints of bridges of $B$, and call members of $S$ *switching points*. As usual we shall assume that $A$, $G$ and $V(B)$ are disjoint.
We shall define a colouring $\psi^A=\psi^A(B,G)$ of $K{\setminus}S$ using the two colours (or labels) ‘even’ and ‘odd’. This colouring is constrained to be ‘valid’, where a valid colouring is defined to be a mapping $\psi:K{\setminus}S \to\{{\mathrm{even}}, {\mathrm{odd}}\}$ such that:
the label is constant between two neighbouring switching points, that is, $\psi$ is constant on any sub-interval of $K$ containing no members of $S$,
the label always switches at each switching point, which is to say that, for $(u,t) \in S$, $\psi(u,t-) \ne \psi(u,t+)$, whenever these two values are defined,
for any pair $v$, $k$ such that $I^v_k\neq{\mathbb{S}}$, in the limit as we move along $v\times I^v_k$ towards either endpoint of $v\times I^v_k$, the colour converges to ‘odd’ if that endpoint lies in $S$, and to ‘even’ otherwise.
If there exists $v\in V$ and $1\leq k\leq m(v)$ such that $v\times {\overline}{I^v_k}$ contains an *odd* number of switching points, then conditions (i)–(iii) cannot be satisfied; in this case we set the colouring $\psi^A$ to a default value denoted $\#$.
Suppose that (i)–(iii) *can* be satisfied, and let $$W=W(K):=\{v\in V: K_v={\mathbb{S}}\}.$$If $W = {\varnothing}$, then there exists a unique valid colouring, denoted $\psi^A$. If $r=|W|\ge 1$, there are exactly $2^r$ valid colourings, one for each of the two possible colours assignable to the sites $(w,0)$, $w \in W$; in this case we let $\psi^A$ be chosen uniformly at random from this set, independently of all other choices.
We write $M_{B,G}$ for the probability measure (or expectation when appropriate) governing the randomization in the definition of $\psi^A$: $M_{B,G}$ is the uniform (product) measure on the set of valid colourings, and it is a point mass if and only if $W={\varnothing}$. See Figure \[colouring\_fig\].
Fix the set $A$ of sources. For (almost every) pair $B$, $G$, one may construct as above a (possibly random) colouring $\psi^A$. Conversely, it is easily seen that the pair $B$, $G$ may (almost surely) be reconstructed from knowledge of the colouring $\psi^A$. For given $A$, we may thus speak of a configuration as being either a pair $B$, $G$, or a colouring $\psi^A$. While $\psi^A(B,G)$ is a colouring of $K {\setminus}S$ only, we shall sometimes refer to it as a colouring of $K$.
![Three examples of colourings for given $B\subseteq F$, $G\subseteq K$. Points in $G$ are written $g$. Thick line segments are ‘odd’ and thin segments ‘even’. In this illustration we have taken $K_v={\mathbb{S}}$ for all $v$. *Left and middle*: two of the eight possible colourings when the sources are $a$, $c$. *Right*: one of the possible colourings when the sources are $a$, $b$, $c$.[]{data-label="colouring_fig"}](thesis.6 "fig:") ![Three examples of colourings for given $B\subseteq F$, $G\subseteq K$. Points in $G$ are written $g$. Thick line segments are ‘odd’ and thin segments ‘even’. In this illustration we have taken $K_v={\mathbb{S}}$ for all $v$. *Left and middle*: two of the eight possible colourings when the sources are $a$, $c$. *Right*: one of the possible colourings when the sources are $a$, $b$, $c$.[]{data-label="colouring_fig"}](thesis.7 "fig:") ![Three examples of colourings for given $B\subseteq F$, $G\subseteq K$. Points in $G$ are written $g$. Thick line segments are ‘odd’ and thin segments ‘even’. In this illustration we have taken $K_v={\mathbb{S}}$ for all $v$. *Left and middle*: two of the eight possible colourings when the sources are $a$, $c$. *Right*: one of the possible colourings when the sources are $a$, $b$, $c$.[]{data-label="colouring_fig"}](thesis.8 "fig:")
The next step is to assign weights ${\partial}\psi$ to colourings $\psi$. The ‘failed’ colouring $\#$ is assigned weight ${\partial}\# =0$. For every valid colouring $\psi$, let ${\mathrm{ev}}(\psi)$ ([respectively]{}, ${\mathrm{odd}}(\psi)$) denote the subset of $K$ that is labelled even ([respectively]{}, odd), and let $$\partial\psi :=\exp\bigl\{2{\delta}({\mathrm{ev}}(\psi))\bigr\},
\label{def-wt}$$where $${\delta}(U):=\int_{U}{\delta}(x)\,dx, {\quad\quad}U \subseteq K.$$ Up to a multiplicative constant depending on $K$ and ${\delta}$ only, ${\partial}\psi$ equals the square of the probability that the odd part of $\psi$ is death-free.
### Random-parity representation {#ssec-rpr}
The expectation $E({\partial}\psi^A)$ is taken over the sets $B$, $G$, and over the randomization that takes place when $W \ne {\varnothing}$, that is, $E$ denotes expectation with respect to the measure $d\mu_{\lambda}(B) d\mu_{\gamma}(G) dM_{B,G}$. The notation has been chosen to harmonize with that used in [@abf] in the discrete case: the expectation $E(\partial\psi^A)$ will play the role of the probability $P(\partial\underline n=A)$ of [@abf]. The main result of this section now follows.
\[rcr\_thm\] For any finite set $A \subseteq {\overline}K$ of sources, $${\langle}{\sigma}_A{\rangle}=\frac{E(\partial\psi^A)}{E(\partial\psi^{\varnothing})}.
\label{ihp8}$$
We introduce a second random object in advance of proving this. Let $D$ be a finite subset of $K$. The set $(v\times K_v){\setminus}{D}$ is a union of maximal death-free intervals which we write $v\times J^v_k$, and where $k=1,2,\dotsc,n$ and $n=n(v,{D})$ is the number of such intervals. We write $V({D})$ for the collection of all such intervals.
For each $e=uv\in E$, and each $1\leq k\leq n(u)$ and $1\leq l\leq n(v)$, let $$J^e_{k,l}:=J^u_k\cap J^v_l,$$and $$E(D)=\bigl\{e\times J^e_{k,l}:e\in E,\ 1\leq k\leq n(u),\ 1\leq l\leq n(v),
\,J^e_{k,l}\neq{\varnothing}\bigr\}.$$Up to a finite set of points, $E(D)$ forms a partition of the set $F$ induced by the ‘deaths’ in $D$.
![*Left*: The partition $E({D})$. We have: $K_v={\mathbb{S}}$ for $v \in V$, the lines $v\times K_v$ are drawn as solid, the lines $e\times K_e$ as dashed, and elements of ${D}$ are marked as crosses. The endpoints of the $e\times J^e_{k,l}$ are the points where the dotted lines meet the dashed lines. *Right*: The graph $G({D})$. In this illustration, the dotted lines are the $v\times K_v$, and the solid lines are the edges of $G({D})$.[]{data-label="bridge_part_fig"}](thesis.9 "fig:") ![*Left*: The partition $E({D})$. We have: $K_v={\mathbb{S}}$ for $v \in V$, the lines $v\times K_v$ are drawn as solid, the lines $e\times K_e$ as dashed, and elements of ${D}$ are marked as crosses. The endpoints of the $e\times J^e_{k,l}$ are the points where the dotted lines meet the dashed lines. *Right*: The graph $G({D})$. In this illustration, the dotted lines are the $v\times K_v$, and the solid lines are the edges of $G({D})$.[]{data-label="bridge_part_fig"}](thesis.10 "fig:")
The pair $$G({D}):=(V({D}),E({D}))$$may be viewed as a graph, illustrated in Figure \[bridge\_part\_fig\]. We will use the symbols $\bar v$ and $\bar e$ for typical elements of $V({D})$ and $E({D})$, respectively. There are natural weights on the edges and vertices of $G({D})$: for $\bar e=e\times J^e_{k,l}\in E({D})$ and $\bar v=v\times J^v_k\in V({D})$, let $$J_{\bar e}:= \int_{J^e_{k,l}}{\lambda}(e,t)\,dt,
\qquad
h_{\bar v}:= \int_{J^v_k}{\gamma}(v,t)\,dt.
\label{o10}$$ Thus the weight of a vertex or edge is its measure, calculated according to ${\lambda}$ or ${\gamma}$, respectively. By , $$\sum_{\bar e\in E({D})} J_{\bar e} + \sum_{\bar v\in V({D})}h_{\bar v}
= \int_{F}{\lambda}(e)\,de + \int_{K}{\gamma}(x)\,dx.
\label{o11}$$
With ${\Lambda}=(K,F)$ as in , we consider the partition function $Z'=Z'_K$ given in .
For each $\bar v\in V({D})$, $\bar e\in E({D})$, the spins ${\sigma}_v$ and ${\sigma}_e$ are constant for $x\in\bar v$ and $e\in\bar e$, respectively. Denoting their common values by ${\sigma}_{\bar v}$ and ${\sigma}_{\bar e}$ respectively, the summation in equals $$\begin{gathered}
\sum_{{\sigma}\in{\Sigma}({D})}\exp\left\{
\sum_{\bar e\in E({D})}{\sigma}_{\bar e}\int_{\bar e}{\lambda}(e)\, de+
\sum_{\bar v\in V({D})}{\sigma}_{\bar v}\int_{\bar v}{\gamma}(x)\, dx\right\}\\
=\sum_{{\sigma}\in{\Sigma}({D})}\exp\left\{
\sum_{\bar e\in E({D})}J_{\bar e}{\sigma}_{\bar e}+
\sum_{\bar v\in V({D})}h_{\bar v}{\sigma}_{\bar v}\right\}.
\label{ihp2}\end{gathered}$$ The right side of is the partition function of the discrete Ising model on the graph $G({D})$, with pair couplings $J_{\bar e}$ and external fields $h_{\bar v}$. We shall apply the random-current expansion of [@abf] to this model.
For convenience of exposition, we introduce the extended graph $$\begin{aligned}
{\widetilde}G({D})&=({\widetilde}V({D}),{\widetilde}E({D}))\label{ihp3}\\
&:=
\bigl(V({D})\cup\{{{\Gamma}}\},E({D})\cup\{\bar v {{\Gamma}}: \bar v \in V({D})\}\bigr)
\nonumber\end{aligned}$$ where ${{\Gamma}}$ is the ghost-site. We call members of $E({D})$ *lattice-bonds*, and those of ${\widetilde}E({D}){\setminus}E({D})$ *ghost-bonds*. Let $\Psi({D})$ be the random multigraph with vertex set ${\widetilde}V({D})$ and with each edge of ${\widetilde}E({D})$ replaced by a random number of parallel edges, these numbers being independent and having the Poisson distribution, with parameter $J_{\bar e}$ for lattice-bonds $\bar e$, and parameter $h_{\bar v}$ for ghost-bonds $\bar v{{\Gamma}}$.
Let $\{\partial\Psi({D})=A\}$ denote the event that, for each $\bar v\in V({D})$, the total degree of $\bar v$ in $\Psi({D})$ *plus* the number of elements of $A$ inside $\bar v$ (when regarded as an interval) is even. (There is $\mu_{\delta}$-probability $0$ that $A$ contains some endpoint of some $V(D)$, and thus we may overlook this possibility.) Applying the discrete random-current expansion, and in particular [@grimmett_rcm eqn (9.24)], we obtain by that $$\sum_{{\sigma}\in{\Sigma}({D})}\exp\left\{
\sum_{\bar e\in E({D})}J_{\bar e}{\sigma}_{\bar e}+
\sum_{\bar v\in V({D})}h_{\bar v}{\sigma}_{\bar v}\right\}=
c 2^{|V({D})|}P_D(\partial\Psi({D})={\varnothing}),$$ where $P_D$ is the law of the edge-counts, and $$c=\exp\left\{\int_F {\lambda}(e)\,de + \int_K{\gamma}(x)\,dx\right\}.
\label{o21}$$
By the same argument applied to the numerator in (adapted to the measure on ${\Lambda}$, see the remark after ), $$\label{rcr_step1_eq}
{\langle}{\sigma}_A{\rangle}=
\frac{E(2^{|V({D})|}{\hbox{\rm 1\kern-.27em I}}\{\partial\Psi({D})=A\})}
{E(2^{|V({D})|}{\hbox{\rm 1\kern-.27em I}}\{\partial\Psi({D})={\varnothing}\})},$$ where the expectation is with respect to $\mu_{\delta}\times P_D$. The claim of the theorem will follow by an appropriate manipulation of .
Here is another way to sample $\Psi({D})$, which allows us to couple it with the random colouring $\psi^A$. Let $B\subseteq F$ and $G\subseteq K$ be finite sets sampled from $\mu_{\lambda}$ and $\mu_{\gamma}$ respectively. The number of points of $G$ lying in the interval $\bar v=v\times J^v_k$ has the Poisson distribution with parameter $h_{\bar v}$, and similarly the number of elements of $B$ lying in $\bar e=e\times J^e_{k,l}\in E({D})$ has the Poisson distribution with parameter $J_{\bar e}$. Thus, for given ${D}$, the multigraph $\Psi(B,G,{D})$, obtained by replacing an edge of ${\widetilde}E({D})$ by parallel edges equal in number to the corresponding number of points from $B$ or $G$, respectively, has the same law as $\Psi({D})$. Using the *same* sets $B$, $G$ we may form the random colouring $\psi^A$.
The numerator of satisfies $$\begin{aligned}
&E(2^{|V({D})|}{\hbox{\rm 1\kern-.27em I}}\{\partial\Psi({D})=A\})\label{ihp5}\\
&\hskip1cm=\iint d\mu_{\lambda}(B)\, d\mu_{\gamma}(G)\,\int d\mu_{\delta}({D})\,
2^{|V({D})|}{\hbox{\rm 1\kern-.27em I}}\{\partial\Psi(B,G,{D})=A\}\nonumber\\
&\hskip1cm= \mu_{\delta}(2^{|V(D)|})
\iint d\mu_{\lambda}(B)\, d\mu_{\gamma}(G)\,{\widetilde}\mu(\partial\Psi(B,G,{D})=A),
\nonumber\end{aligned}$$ where ${\widetilde}\mu$ is the probability measure on ${\mathcal{F}}$ satisfying $$\frac{d{\widetilde}\mu}{d\mu_{\delta}}(D) \propto 2^{|V({D})|}.
\label{ihp10}$$ Therefore, by , $${\langle}{\sigma}_A{\rangle}=
\frac{{\widetilde}P(\partial\Psi(B,G,{D})=A)}
{{\widetilde}P(\partial\Psi(B,G,{D})={\varnothing})},
\label{ihp9}$$ where ${\widetilde}P$ denotes the probability under $\mu_{\lambda}\times\mu_{\gamma}\times{\widetilde}\mu$. We claim that $${\widetilde}\mu({\partial}\Psi(B,G,{D})=A) = s M_{B,G}({\partial}\psi^A(B,G)),
\label{ihp13}$$ for all $B$, $G$, where $s$ is a constant, and the expectation $M_{B,G}$ is over the uniform measure on the set of valid colourings. Claim follows from this, and the remainder of the proof is to show . The constants $s$, $s_j$ are permitted in the following to depend only on ${\Lambda}$, ${\delta}$.
Here is a special case: $${\widetilde}\mu(\partial\Psi(B,G,{D})=A)=0$$ if and only if some interval ${\overline}{I^v_k}$ contains an odd number of switching points, if and only if $\psi^A(B,G) =\#$ and $\partial\psi^A(B,G)=0$. Thus holds in this case.
Another special case arises when $K_v=[0,{\beta})$ for all $v\in V$, that is, the ‘free boundary’ case. As remarked earlier, there is a unique valid colouring $\psi^A=\psi^A(B,G)$. Moreover, $|V({D})|=|{D}|+|V|$, whence from standard properties of Poisson processes, ${\widetilde}\mu=\mu_{2{\delta}}$. It may be seen after some thought (possibly with the aid of a diagram) that, for given $B$, $G$, the events $\{\partial\Psi(B,G,{D})=A\}$ and $\{{D}\cap{\mathrm{odd}}(\psi^A)={\varnothing}\}$ differ by an event of $\mu_{2{\delta}}$-probability $0$. Therefore, $$\begin{aligned}
{\widetilde}\mu(\partial\Psi(B,G,{D})=A)&=
\mu_{2{\delta}}( {D}\cap{\mathrm{odd}}(\psi^A)={\varnothing})\label{ihp7}\\
&=\exp\{-2{\delta}({\mathrm{odd}}(\psi^A))\}\nonumber\\
&= s_1\exp\{2{\delta}({\mathrm{ev}}(\psi^A))\}=s_1 \partial\psi^A,
\nonumber\end{aligned}$$ with $s_1=e^{-2{\delta}(K)}$. In this special case, holds.
For the general case, we first note some properties of ${\widetilde}\mu$. By the above, we may assume that $B$, $G$ are such that ${\widetilde}\mu(\partial\Psi(B,G,{D})=A)>0$, which is to say that each ${\overline}{I_k^v}$ contains an even number of switching points. Let $W = \{v\in V: K_v={\mathbb{S}}\}$ and, for $v\in V$, let ${D}_v=D\cap(v\times K_v)$ and ${d}(v)=|{D}_v|$. By , $$\begin{aligned}
\frac{d{\widetilde}\mu}{d\mu_{\delta}}(D) \propto 2^{|V({D})|}&=
\prod_{w\in W}2^{1\vee {d}(w)}\prod_{v\in V\setminus W}2^{m(v)+{d}(v)}\\
&\propto 2^{|{D}|} \prod_{w\in W} 2^{{\hbox{\rm 1\kern-.27em I}}\{{d}(w)=0\}},\end{aligned}$$ where $a\vee b = {\mathrm{max}}\{a,b\}$, and we recall the number $m(v)$ of intervals $I^v_k$ that constitute $K_v$. Therefore, $$\frac{d{\widetilde}\mu}{d\mu_{2{\delta}}}(D) \propto \prod_{w\in W} 2^{{\hbox{\rm 1\kern-.27em I}}\{{d}(w)=0\}}.$$ Three facts follow.
The sets ${D}_v$, $v\in V$ are independent under ${\widetilde}\mu$.
For $v\in V\setminus W$, the law of ${D}_v$ under ${\widetilde}\mu$ is $\mu_{2{\delta}}$.
For $w\in W$, the law $\mu_w$ of ${D}_w$ is that of $\mu_{2{\delta}}$ skewed by the Radon–Nikodym factor $2^{{\hbox{\rm 1\kern-.27em I}}\{{d}(w)=0\}}$, which is to say that $$\begin{aligned}
\mu_w({D}_w \in H)
&= \frac1{{\alpha}_w}\Bigl[2\mu_{2{\delta}}({D}_w\in H,\,{d}(w)=0) \label{ihp15}\\
&\hskip3cm +
\mu_{2{\delta}}({D}_w\in H,\,{d}(w)\ge 1)\bigr],
\nonumber\end{aligned}$$ for appropriate sets $H$, where $${\alpha}_w=\mu_{2{\delta}}({d}(w)=0)+1.$$
Recall the set $S=A\cup G\cup V(B)$ of switching points. By (a) above, $$\begin{aligned}
{\widetilde}\mu(\partial\Psi(B,G,{D})=A)&=
{\widetilde}\mu(\forall v,k:\, |S\cap {\overline}{J^v_k}|\mbox{ is even})\label{ihp11}\\
&= \prod_{v\in V} {\widetilde}\mu(\forall k:\, |S\cap {\overline}{J^v_k}|\mbox{ is even}).
\nonumber\end{aligned}$$ We claim that $${\widetilde}\mu(\forall k:\, |S\cap {\overline}{J^v_k}|\mbox{ is even})=
s_2(v)M_{B,G}\Bigl(\exp\bigl\{2{\delta}\bigl({\mathrm{ev}}(\psi^A)\cap(v\times K_v)\bigr)\bigr\}\Bigr),
\label{ihp12}$$ where $M_{B,G}$ is as before. Recall that $M_{B,G}$ is a product measure. Once is proved, follows by and .
For $v\in V{\setminus}W$, the restriction of $\psi^A$ to $v\times K_v$ is determined given $B$ and $G$, whence by (b) above, and the remark prior to , $$\begin{aligned}
{\widetilde}\mu(\forall k:\, |S\cap {\overline}{J^v_k}|\mbox{ is even})&=
\mu_{2{\delta}}(\forall k:\, |S\cap {\overline}{J^v_k}|\mbox{ is even})
\label{ihp14}\\
&= \exp\bigl\{-2{\delta}\bigl({\mathrm{odd}}(\psi^A)\cap(v\times K_v)\bigr)\bigr\}.
\nonumber\end{aligned}$$ Equation follows with $s_2(v) =\exp\{-2{\delta}(v\times K_v)\}$.
For $w\in W$, by , $$\begin{aligned}
&{\widetilde}\mu(\forall k:\, |S\cap J^w_k|\mbox{ is even})\\
&\hskip1cm =\frac1{{\alpha}_w}\Bigl[2\mu_{2{\delta}}({D}_w={\varnothing})
+\mu_{2{\delta}}({D}_w\ne{\varnothing},\,\forall k:\, |S\cap J^w_k|\mbox{ is even})\Bigr]\\
&\hskip1cm=\frac1{{\alpha}_w}\Bigl[\mu_{2{\delta}}({D}_w={\varnothing})+
\mu_{2{\delta}}(\forall k:\, |S\cap J^w_k|\mbox{ is even})\Bigr].\end{aligned}$$ Let $\psi=\psi^A(B,G)$ be a valid colouring with $\psi(w,0) = {\mathrm{even}}$. The colouring ${\overline}\psi$, obtained from $\psi$ by flipping all colours on $w \times K_w$, is valid also. We take into account the periodic boundary condition, to obtain this time that $$\begin{aligned}
&\mu_{2{\delta}}(\forall k:\,|S\cap {\overline}{J^w_k}|\mbox{ is even})\\
&\quad =\mu_{2{\delta}}\bigl(\{{D}_w\cap{\mathrm{odd}}(\psi)={\varnothing}\}\cup
\{{D}_w\cap{\mathrm{ev}}(\psi)={\varnothing}\}\bigr)\\
&\quad =\mu_{2{\delta}}({D}_w\cap{\mathrm{odd}}(\psi)={\varnothing})+
\mu_{2{\delta}}({D}_w\cap{\mathrm{ev}}(\psi)={\varnothing})
-\mu_{2{\delta}}({D}_w={\varnothing}),\end{aligned}$$ whence $$\begin{aligned}
&{\alpha}_w {\widetilde}\mu(\forall k:\,|S\cap {\overline}{J^w_k}|\mbox{ is even})\\
&\hskip1cm =\mu_{2{\delta}}({D}_w\cap{\mathrm{odd}}(\psi)={\varnothing})
+\mu_{2{\delta}}({D}_w\cap{\mathrm{ev}}(\psi)={\varnothing})\nonumber\\
&\hskip1cm =2M_{B,G}\Bigl(\exp\bigl\{-2{\delta}\bigl({\mathrm{odd}}(\psi^A)\cap(w\times K_w)\bigr)\bigr\}\Bigr),
\nonumber\end{aligned}$$ since ${\mathrm{odd}}(\psi^A) = {\mathrm{odd}}(\psi)$ with $M_{B,G}$-probability $\frac12$, and equals ${\mathrm{ev}}(\psi)$ otherwise. This proves with $s_2(w) = 2\exp\{-2{\delta}(w\times K_w)\}/{\alpha}_w$.
By keeping track of the constants in the above proof, we arrive at the following statement, which will be useful later.
\[Z’\] The partition function $Z'=Z_K'$ of satisfies $$Z' = 2^N e^{{\lambda}(F)+{\gamma}(K)-{\delta}(K)} E({\partial}\psi^{\varnothing}),$$ where $N=\sum_{v\in V}m(v)$ is the total number of intervals comprising $K$.
We denote $Z_K=E(\partial\psi^{\varnothing})$, which is thus a constant multiple of $Z'$.
### The backbone
The concept of the backbone is key to the analysis of [@abf], and its definition there has a certain complexity. The corresponding definition is rather easier in the current setting, because of the fact that bridges, deaths, and sources have (almost surely) no common point.
We construct a total order on $K$ by: first ordering the vertices of $L$, and then using the natural order on $[0,{\beta})$. Let $A\subseteq {\overline}K$, $B\subseteq F$ and $G\subseteq K$ be finite. Let $\psi$ be a valid colouring. We will define a sequence of directed odd paths called the *backbone* and denoted $\xi=\xi(\psi)$. Suppose $A=(a_1,a_2,\dotsc,a_n)$ in the above ordering. Starting at $a_1$, follow the odd interval (in $\psi$) until you reach an element of $S=A\cup G \cup V(B)$. If the first such point thus encountered is the endpoint of a bridge, cross it, and continue along the odd interval; continue likewise until you first reach a point $t_1\in A\cup G$, at which point you stop. Note, by the validity of $\psi$, that $a_1\ne t_1$. The odd path thus traversed is denoted $\zeta^1$; we take $\zeta^1$ to be closed (when viewed as a subset of ${\mathbb{Z}}^d\times{\mathbb{R}}$). Repeat the same procedure with $A$ replaced by $A\setminus\{a_1,t_1\}$, and iterate until no sources remain. The resulting (unordered) set of paths $\xi=(\zeta^1,\dotsc,\zeta^k)$ is called the *backbone* of $\psi$. The backbone will also be denoted at times as $\xi=\zeta^1\circ\dotsb\circ\zeta^k$. We define $\xi(\#)={\varnothing}$. Note that, apart from the backbone, the remaining odd segments of $\psi$ form disjoint self-avoiding cycles (or ‘eddies’). Unlike the discrete setting of [@abf], there is a (a.s.) unique way of specifying the backbone from knowledge of $A$, $B$, $G$ and the valid colouring $\psi$. See Figure \[backbone\_fig\].
The backbone contains all the sources $A$ as endpoints, and the configuration outside $\xi$ may be any sourceless configuration. Moreover, since $\xi$ is entirely odd, it does not contribute to the weight $\partial\psi$ in . It follows, using properties of Poisson processes, that the conditional expectation $E(\partial\psi^A\mid\xi)$ equals the expected weight of any sourceless colouring of $K{\setminus}\xi$, which is to say that, with $\xi:= \xi(\psi^A)$, $$\label{backb_cond_eq}
E(\partial\psi^A\mid\xi)=E_{K\setminus\xi}(\partial\psi^{\varnothing})
= Z_{K\setminus\xi}.$$ Cf. and , and recall Remark \[rem-as\]. We abbreviate $Z_K$ to $Z$, and recall from Lemma \[Z’\] that the $Z_R$ differ from the partition functions $Z_R'$ by certain multiplicative constants.
![A valid colouring configuration $\psi$ with sources $A=\{a,b,c,d\}$, and its backbone $\xi=\zeta^1\circ\zeta^2$. Note that, in this illustration, bridges protruding from the sides ‘wrap around’, and that there are no ghost-bonds.[]{data-label="backbone_fig"}](thesis.11)
Let $\Xi$ be the set of all possible backbones as $A$, $B$, and $G$ vary, regarded as sequences of directed paths in $K$; these paths may, if required, be ordered by their starting points. For $A \subseteq{\overline}K$ and $\nu\in \Xi$, we write $A\sim\nu$ if there exist $B$ and $G$ such that $M_{B,G}(\xi(\psi^A)=\nu)>0$. We define the *weight* ${w}^A(\nu)$ by $${w}^A(\nu) = {w}^A_K(\nu):=
\begin{cases} \dfrac{Z_{K{\setminus}\nu}}{Z} &\text{if } A \sim \nu,\\
0 &\text{otherwise}.
\end{cases}
\label{ihp16}$$ By and Theorem \[rcr\_thm\], with $\xi=\xi(\psi^A)$, $$\label{backbone_rep_eq}
E({w}^A(\xi))=\frac{E(E({\partial}\psi^A\mid\xi))}{Z} =
\frac{E({\partial}\psi^A)}{E({\partial}\psi^{\varnothing})} ={\langle}{\sigma}_A{\rangle}.$$
For $\nu^1,\nu^2\in \Xi$ with $\nu^1\cap \nu^2={\varnothing}$ (when viewed as subsets of $K$), we write $\nu^1\circ \nu^2$ for the element of $\Xi$ comprising the union of $\nu^1$ and $\nu^2$.
Let $\nu =\zeta^1\circ\dotsb\circ\zeta^k\in\Xi$ where $k \ge 1$. If $\zeta^i$ has starting point $a_i$ and endpoint $b_i$, we write $\zeta^i:a_i\rightarrow b_i$, and also $\nu:a_1\rightarrow b_1,\dotsc,a_k\rightarrow b_k$. If $b_i \in G$, we write $\zeta^i: a_i \rightarrow {{\Gamma}}$. There is a natural way to ‘cut’ $\nu$ at points $x$ lying on $\zeta^i$, say, where $x\ne a_i, b_i$: let $\bar\nu^1=\bar\nu^1(\nu,x)=
\zeta^1\circ\cdots\circ\zeta^{i-1}\circ \zeta^i_{\le x}$ and $\bar\nu^2=\bar\nu^2(\nu,x)=\zeta^i_{\ge x}\circ\zeta^{i+1}\circ\dots\circ\zeta^k$, where $\zeta^i_{\le x}$ ([respectively]{}, $\zeta^i_{\ge x})$ is the closed sub-path of $\zeta^i$ from $a_i$ to $x$ ([respectively]{}, $x$ to $b_i$). We express this decomposition as $\nu=\bar\nu^1\circ\bar\nu^2$ where, this time, each $\bar\nu^i$ may comprise a number of disjoint paths. The notation ${\overline}\nu$ will be used only in a situation where there has been a cut.
We note two special cases. If $A=\{a\}$, then necessarily $\xi(\psi^A):a\rightarrow {{\Gamma}}$, so $${\langle}{\sigma}_a{\rangle}=E\bigl({w}^a(\xi)\cdot{\hbox{\rm 1\kern-.27em I}}\{\xi:a\rightarrow {{\Gamma}}\}\bigr).
\label{special1}$$ If $A=\{a,b\}$ where $a<b$ in the ordering of $K$, then $${\langle}{\sigma}_a{\sigma}_b{\rangle}=E\bigl({w}^{ab}(\xi)\cdot{\hbox{\rm 1\kern-.27em I}}\{\xi:a\rightarrow b\}\bigr)
+E\bigl({w}^{ab}(\xi)\cdot{\hbox{\rm 1\kern-.27em I}}\{\xi:a\rightarrow {{\Gamma}},\,b\rightarrow {{\Gamma}}\}\bigr).
\label{special2}$$ The last term equals $0$ when ${\gamma}\equiv0$.
Finally, here is a lemma for computing the weight of $\nu$ in terms of its constituent parts. The claim of the lemma is, as usual, valid only ‘almost surely’.
\[backb2\] (a) Let $\nu^1,\nu^2 \in \Xi$ be disjoint, and $\nu=\nu^1\circ\nu^2$, $A\sim\nu$. Writing $A^i=A\cap\nu^i$, we have that $${w}^A(\nu)={w}^{A^1}(\nu^1){w}^{A^2}_{K\setminus\nu^1}(\nu^2).$$ (b) Let $\nu = {\overline}\nu^1\circ {\overline}\nu^2$ be a cut of the backbone $\nu$ at the point $x$, and $A \sim \nu$. Then $${w}^A(\nu)={w}^{B^1}({\overline}\nu^1){w}^{B^2}_{K\setminus{\overline}\nu^1}({\overline}\nu^2).$$ where $B^i=A^i\cup\{x\}$.
By , the first claim is equivalent to $$\frac{Z_{K\setminus\nu}}{Z}{\hbox{\rm 1\kern-.27em I}}\{A\sim\nu\}=
\frac{Z_{K\setminus\nu^1}}{Z}{\hbox{\rm 1\kern-.27em I}}\{A^1\sim\nu^1\}
\frac{Z_{K\setminus(\nu^1\cup\nu^2)}}{Z_{K\setminus\nu^1}}{\hbox{\rm 1\kern-.27em I}}\{A^2\sim\nu^2\}.$$ The right side vanishes if and only if the left side vanishes. When both sides are non-zero, their equality follows from the fact that $Z_{K\setminus\nu}=Z_{K\setminus(\nu^1\cup\nu^2)}$. The second claim follows similarly, on adding $x$ to the set of sources.
The switching lemma {#sw_sec}
-------------------
We state and prove next the principal tool in the random-parity representation, namely the so-called ‘switching lemma’. In brief, this allows us to take two independent colourings, with different sources, and to ‘switch’ the sources from one to the other in a measure-preserving way. In so doing, the backbone will generally change. In order to preserve the measure, the *connectivities* inherent in the backbone must be retained. We begin by defining two notions of connectivity in colourings. We work throughout this section in the general set-up of Section \[ssec-col\].
### Connectivity and switching {#ssec-switching}
Let $B{\subseteq}F$, $G{\subseteq}K$ be finite sets, let $A \subseteq{\overline}K$ be a finite set of sources, and write $\psi^A=\psi^A(B,G)$ for the colouring given in the last section. In what follows we think of the ghost-bonds as bridges to the ghost-site ${{\Gamma}}$.
Let $x,y\in K^{{\Gamma}}:= K \cup\{{{\Gamma}}\}$. A *path* from $x$ to $y$ in the configuration $(B,G)$ is a self-avoiding path with endpoints $x$, $y$, traversing intervals of $K^{{\Gamma}}$, and possibly bridges in $B$ and/or ghost-bonds joining $G$ to ${{\Gamma}}$. Similarly, a *cycle* is a self-avoiding cycle in the above graph. A *route* is a path or a cycle. A route containing no ghost-bonds is called a *lattice-route*. A route is called *odd* (in the colouring $\psi^A$) if $\psi^A$, when restricted to the route, takes only the value ‘odd’. The failed colouring $\psi^A=\#$ is deemed to contain no odd paths.
Let $B_1,B_2{\subseteq}F$, $G_1,G_2{\subseteq}K$, and let $\psi_1^A=\psi_1^A(B_1,G_1)$ and $\psi_2^B=\psi_2^B(B_2,G_2)$ be the associated colourings. Let ${\Delta}$ be an auxiliary Poisson process on $K$, with intensity function $4{\delta}(\cdot)$, that is independent of all other random variables so far. We call points of ${\Delta}$ *cuts*. A route of $(B_1\cup B_2, G_1\cup G_2)$ is said to be *open* in the triple $(\psi_1^A,\psi_2^B,{\Delta})$ if it includes no sub-interval of ${\mathrm{ev}}(\psi_1^A)\cap{\mathrm{ev}}(\psi_2^B)$ containing one or more elements of ${\Delta}$. In other words, the cuts break paths, but only when they fall in intervals labelled ‘even’ in *both* colourings. See Figure \[connectivity\_fig\]. In particular, if there is an odd path $\pi$ from $x$ to $y$ in $\psi_1^A$, then $\pi$ constitutes an open path in $(\psi_1^A,\psi_2^B,{\Delta})$ irrespective of $\psi_2^B$ and ${\Delta}$. We let $$\{x{\leftrightarrow}y\mbox{ in }\psi_1^A,\psi_2^B,{\Delta}\}$$ be the event that there exists an open path from $x$ to $y$ in $(\psi_1^A,\psi_2^B,{\Delta})$. We may abbreviate this to $\{x{\leftrightarrow}y\}$ when there is no ambiguity.
![Connectivity in pairs of colourings. *Left*: $\psi_1^{ac}$. Middle: $\psi_2^{\varnothing}$. *Right*: the triple $\psi_1^{ac},\psi_2^{\varnothing},{\Delta}$. Crosses are elements of ${\Delta}$ and grey lines are where either $\psi_1^{ac}$ or $\psi_2^{\varnothing}$ is odd. In $(\psi_1^{ac},\psi_2^{\varnothing},{\Delta})$ the following connectivities hold: $a{\nleftrightarrow}b$, $a{\leftrightarrow}c$, $a{\leftrightarrow}d$, $b{\nleftrightarrow}c$, $b{\nleftrightarrow}d$, $c{\leftrightarrow}d$. The dotted line marks $\pi$, one of the open paths from $a$ to $c$.[]{data-label="connectivity_fig"}](thesis.12 "fig:") ![Connectivity in pairs of colourings. *Left*: $\psi_1^{ac}$. Middle: $\psi_2^{\varnothing}$. *Right*: the triple $\psi_1^{ac},\psi_2^{\varnothing},{\Delta}$. Crosses are elements of ${\Delta}$ and grey lines are where either $\psi_1^{ac}$ or $\psi_2^{\varnothing}$ is odd. In $(\psi_1^{ac},\psi_2^{\varnothing},{\Delta})$ the following connectivities hold: $a{\nleftrightarrow}b$, $a{\leftrightarrow}c$, $a{\leftrightarrow}d$, $b{\nleftrightarrow}c$, $b{\nleftrightarrow}d$, $c{\leftrightarrow}d$. The dotted line marks $\pi$, one of the open paths from $a$ to $c$.[]{data-label="connectivity_fig"}](thesis.13 "fig:") ![Connectivity in pairs of colourings. *Left*: $\psi_1^{ac}$. Middle: $\psi_2^{\varnothing}$. *Right*: the triple $\psi_1^{ac},\psi_2^{\varnothing},{\Delta}$. Crosses are elements of ${\Delta}$ and grey lines are where either $\psi_1^{ac}$ or $\psi_2^{\varnothing}$ is odd. In $(\psi_1^{ac},\psi_2^{\varnothing},{\Delta})$ the following connectivities hold: $a{\nleftrightarrow}b$, $a{\leftrightarrow}c$, $a{\leftrightarrow}d$, $b{\nleftrightarrow}c$, $b{\nleftrightarrow}d$, $c{\leftrightarrow}d$. The dotted line marks $\pi$, one of the open paths from $a$ to $c$.[]{data-label="connectivity_fig"}](thesis.14 "fig:")
There is an analogy between open paths in the above construction and the notion of connectivity in the random-current representation of the discrete Ising model. Points labelled ‘odd’ or ‘even’ above may be considered as collections of infinitesimal parallel edges, being odd or even in number, respectively. If a point is ‘even’, the corresponding number of edges may be $2,4,6,\dotsc$ *or* it may be 0; in the ‘union’ of $\psi_1^A$ and $\psi_2^B$, connectivity is broken at a point if and only if both the corresponding numbers equal 0. It turns out that the correct law for the set of such points is that of ${\Delta}$.
Here is some notation. For any finite sequence $(a,b,c,\dots)$ of elements in $K$, the string $abc\dotsc$ will denote the subset of elements that appear an odd number of times in the sequence. If $A\subseteq {\overline}K$ is a finite set with odd cardinality, then for any pair $(B,G)$ for which there exists a valid colouring $\psi^A(B,G)$, the number of ghost-bonds must be odd. Thinking of these as bridges to ${{\Gamma}}$, ${{\Gamma}}$ may thus be viewed as an element of $A$, and we make the following remark.
\[gGremark\] For $A \subseteq {\overline}K$ with $|A|$ odd, we shall use the expressions $\psi^A$ and $\psi^{A\cup\{{{\Gamma}}\}}$ interchangeably.
We call a function $F$, acting on $(\psi_1^A,\psi_2^B,{\Delta})$, a *connectivity function* if it depends only on the connectivity properties using open paths of $(\psi_1^A,\psi_2^B,{\Delta})$, that is, the value of $F$ depends only on the set $\{(x,y)\in (K^{{\Gamma}})^2: x {\leftrightarrow}y\}$. In the following, $E$ denotes expectation with respect to $d\mu_{\lambda}d\mu_{\gamma}dM_{B,G} dP$, where $P$ is the law of ${\Delta}$.
\[sl\] Let $F$ be a connectivity function and $A,B\subseteq {\overline}K$ finite sets. For $x,y\in K^{{\Gamma}}$, $$\begin{aligned}
\label{sw_eq_1}
&E\bigl(\partial\psi_1^A\partial\psi_2^B\cdot F(\psi_1^A,\psi_2^B,{\Delta})
\cdot {\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }\psi_1^A,\psi_2^B,{\Delta}\}\bigr)\\
&\hskip1cm =E\Big(\partial\psi_1^{A{\triangle}xy}\partial\psi_2^{B{\triangle}xy}
\cdot F(\psi_1^{A{\triangle}xy},\psi_2^{B{\triangle}xy},{\Delta})\cdot \nonumber\\
&\hskip4cm \cdot {\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }\psi_1^{A{\triangle}xy},\psi_2^{B{\triangle}xy},{\Delta}\}\Big).
\nonumber\end{aligned}$$ In particular, $$\label{sw_eq_2}
E(\partial\psi_1^{xy}\partial\psi_2^B)
=E\bigl(\partial\psi_1^{\varnothing}\partial\psi_2^{B{\triangle}xy}
\cdot{\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }
\psi_1^{\varnothing},\psi_2^{B{\triangle}xy},{\Delta}\}\bigr).$$
Equation follows from with $A=\{x,y\}$ and $F\equiv 1$, and so it suffices to prove . This is trivial if $x=y$, and we assume henceforth that $x\neq y$. Recall that $W=\{v\in V:K_v={\mathbb{S}}\}$ and $|W|=r$.
We prove first for the special case when $F\equiv 1$, that is, $$\begin{gathered}
\label{sw_eq_3}
E\bigl(\partial\psi_1^A\partial\psi_2^B
\cdot {\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }\psi_1^A,\psi_2^B,{\Delta}\}\bigr)\\
=E\bigl(\partial\psi_1^{A{\triangle}xy}\partial\psi_2^{B{\triangle}xy}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }\psi_1^{A{\triangle}xy},\psi_2^{B{\triangle}xy},{\Delta}\}\bigr),\end{gathered}$$ and this will follow by conditioning on the pair $Q=(B_1\cup B_2,G_1\cup G_2)$.
Let $Q$ be given. Conditional on $Q$, the law of $(\psi_1^A,\psi_2^B)$ is given as follows. First, we allocate each bridge and each ghost-bond to either $\psi_1^A$ or $\psi_2^B$ with equal probability (independently of one another). If $W \ne {\varnothing}$, then we must also allocate (uniform) random colours to the points $(w,0)$, $w \in W$, for each of $\psi_1^A$, $\psi_2^B$. If $(w,0)$ is itself a source, we work instead with $(w,0+)$. (Recall that the pair $(B',G')$ may be reconstructed from knowledge of a valid colouring $\psi^{A'}(B',G')$.) There are $2^{|Q|+2r}$ possible outcomes of the above choices, and each is equally likely.
The process ${\Delta}$ is independent of all random variables used above. Therefore, the conditional expectation, given $Q$, of the random variable on the left side of equals $$\label{sw_cond_eq}
\frac{1}{2^{|Q|+2r}}\sum _{{{\mathcal{Q}}^{A,B}}}\partial Q_1\partial Q_2\,
P(x{\leftrightarrow}y\mbox{ in }Q_1,Q_2,{\Delta}),$$ where the sum is over the set ${{\mathcal{Q}}^{A,B}}={{\mathcal{Q}}^{A,B}}(Q)$ of all possible pairs $(Q_1,Q_2)$ of values of $(\psi_1^A,\psi_2^B)$. The measure $P$ is that of ${\Delta}$.
We shall define an invertible (and therefore measure-preserving) map from ${{\mathcal{Q}}^{A,B}}$ to ${{\mathcal{Q}}^{A{\triangle}xy,B{\triangle}xy}}$. Let $\pi$ be a path of $Q$ with endpoints $x$ and $y$ (if such a path $\pi$ exists), and let $f_\pi:{{\mathcal{Q}}^{A,B}}\to{{\mathcal{Q}}^{A{\triangle}xy,B{\triangle}xy}}$ be given as follows. Let $(Q_1,Q_2)\in{{\mathcal{Q}}^{A,B}}$, say $Q_1=Q_1^A(B_1,G_1)$ and $Q_2=Q_2^B(B_2,G_2)$ where $Q=(B_1\cup B_2, G_1\cup G_2)$. For $i=1,2$, let $B_i'$ ([respectively]{}, $G_i'$) be the set of bridges ([respectively]{}, ghost-bonds) in $Q$ lying in exactly one of $B_i$, $\pi$ ([respectively]{}, $G_i$, $\pi$). Otherwise expressed, $(B_i',G_i')$ is obtained from $(B_i,G_i)$ by adding the bridges/ghost-bonds of $\pi$ ‘modulo 2’. Note that $(B_1'\cup B_2', G_1' \cup G_2') = Q$.
If $W = {\varnothing}$, we let $R_1=R_1^{A {\triangle}xy}$ ([respectively]{}, $R_2^{B{\triangle}xy}$) be the unique valid colouring of $(B_1',G_1')$ with sources $A{\triangle}xy$ ([respectively]{}, $(B_2',G_2')$ with sources $B{\triangle}xy$), so $R_1=\psi^{A{\triangle}xy}(B_1',G_1')$, and similarly for $R_2$. When $W\ne {\varnothing}$ and $i=1,2$, we choose the colours of the $(w,0)$, $w\in W$, in $R_i$ in such a way that $R_i \equiv Q_i$ on $K{\setminus}\pi$.
It is easily seen that the map $f_\pi:(Q_1,Q_2) \mapsto (R_1,R_2)$ is invertible, indeed its inverse is given by the same mechanism. See Figure \[connectivity\_after\_switch\_fig\].
![Switched configurations. Taking $Q_1^{ac}$, $Q_2^{\varnothing}$ and $\pi$ to be $\psi_1^{ac}$, $\psi_2^{\varnothing}$ and $\pi$ of Figure \[connectivity\_fig\], respectively, this figure illustrates the ‘switched’ configurations $R_1^{\varnothing}$ and $R_2^{ac}$ (left and right, respectively).[]{data-label="connectivity_after_switch_fig"}](thesis.15 "fig:") ![Switched configurations. Taking $Q_1^{ac}$, $Q_2^{\varnothing}$ and $\pi$ to be $\psi_1^{ac}$, $\psi_2^{\varnothing}$ and $\pi$ of Figure \[connectivity\_fig\], respectively, this figure illustrates the ‘switched’ configurations $R_1^{\varnothing}$ and $R_2^{ac}$ (left and right, respectively).[]{data-label="connectivity_after_switch_fig"}](thesis.16 "fig:")
By , $$\partial Q_1\partial Q_2=
\exp\bigl\{2{\delta}({\mathrm{ev}}(Q_1))+2{\delta}({\mathrm{ev}}(Q_2))\bigr\}.
\label{o14}$$ Now, $$\begin{aligned}
\label{o15}
{\delta}({\mathrm{ev}}(Q_i))
&={\delta}({\mathrm{ev}}(Q_i)\cap\pi)+{\delta}({\mathrm{ev}}(Q_i)\setminus\pi)\\
&={\delta}({\mathrm{ev}}(Q_i)\cap\pi)+{\delta}({\mathrm{ev}}(R_i)\setminus\pi),
\nonumber\end{aligned}$$ and $$\begin{aligned}
&{\delta}({\mathrm{ev}}(Q_1)\cap\pi)+{\delta}({\mathrm{ev}}(Q_2)\cap\pi) - 2{\delta}\bigl({\mathrm{ev}}(Q_1)\cap{\mathrm{ev}}(Q_2)\cap\pi\bigr) \\
&\hskip1cm ={\delta}\bigl({\mathrm{ev}}(Q_1)\cap{\mathrm{odd}}(Q_2)\cap\pi\bigr)+{\delta}\bigl({\mathrm{odd}}(Q_1)\cap{\mathrm{ev}}(Q_2)\cap\pi\bigr)\\
&\hskip1cm={\delta}\bigl({\mathrm{odd}}(R_1)\cap{\mathrm{ev}}(R_2)\cap\pi\bigr)+{\delta}\bigl({\mathrm{ev}}(R_1)\cap{\mathrm{odd}}(R_2)\cap\pi\bigr)\\
&\hskip1cm ={\delta}({\mathrm{ev}}(R_1)\cap\pi)+{\delta}({\mathrm{ev}}(R_2)\cap\pi) -2{\delta}\bigl({\mathrm{ev}}(R_1)\cap{\mathrm{ev}}(R_2)\cap\pi\bigr),\end{aligned}$$ whence, by –, $$\begin{aligned}
\label{switched_weights_eq}
\partial Q_1\partial Q_2=
\partial R_1\partial R_2&
\exp\bigl\{-4{\delta}\bigl({\mathrm{ev}}(R_1)\cap{\mathrm{ev}}(R_2)\cap\pi\bigr)\bigr\}\\
&\times \exp\bigl\{4{\delta}\bigl({\mathrm{ev}}(Q_1)\cap{\mathrm{ev}}(Q_2)\cap\pi\bigr)\bigr\}.
\nonumber\end{aligned}$$
The next step is to choose a suitable path $\pi$. Consider the final term in , namely $$P(x{\leftrightarrow}y\mbox{ in }Q_1,Q_2,{\Delta}).$$ There are finitely many paths in $Q$ from $x$ to $y$, let these paths be $\pi_1,\pi_2,\dotsc,\pi_n$. Let ${\mathcal{O}}_k={\mathcal{O}}_k(Q_1,Q_2,{\Delta})$ be the event that $\pi_k$ is the earliest such path that is open in $(Q_1,Q_2,{\Delta})$. Then $$\begin{aligned}
\label{path_prob_eq}
&\hskip-1cm P(x{\leftrightarrow}y\mbox{ in }Q_1,Q_2,{\Delta})\\
&=\sum_{k=1}^n P({\mathcal{O}}_k)\nonumber\\
&=\sum_{k=1}^n P\bigl({\Delta}\cap[{\mathrm{ev}}(Q_1)\cap{\mathrm{ev}}(Q_2)\cap \pi_k] = {\varnothing}\bigr)
P({\widetilde}{\mathcal{O}}_k)\nonumber\\
&=\sum_{k=1}^n
\exp\bigl\{-4{\delta}\bigl({\mathrm{ev}}(Q_1)\cap{\mathrm{ev}}(Q_2)\cap\pi_k\bigr)\bigr\}
P({\widetilde}{\mathcal{O}}_k),
\nonumber\end{aligned}$$ where ${\widetilde}{\mathcal{O}}_k = {\widetilde}{\mathcal{O}}_k(Q_1,Q_2,{\Delta})$ is the event that each of $\pi_1,\dotsc,\pi_{k-1}$ is rendered non-open in $(Q_1,Q_2,{\Delta})$ through the presence of elements of ${\Delta}$ lying in $K{\setminus}\pi_k$. In the second line of , we have used the independence of ${\Delta}\cap \pi_k$ and ${\Delta}\cap(K{\setminus}\pi_k)$.
Let $(R_1^k,R_2^k) =f_{\pi_k}(Q_1,Q_2)$. Since $R_i^k \equiv Q_i$ on $K {\setminus}\pi_k$, we have that ${\widetilde}{\mathcal{O}}_k(Q_1,Q_2,{\Delta}) = {\widetilde}{\mathcal{O}}_k(R_1^k,R_2^k,{\Delta})$. By and , the summand in equals $$\begin{aligned}
& \sum_{k=1}^n \partial Q_1\partial Q_2
\exp\bigl\{-4{\delta}\bigl({\mathrm{ev}}(Q_1)\cap{\mathrm{ev}}(Q_2)\cap\pi_k\bigr)\bigr\}P({\widetilde}{\mathcal{O}}_k)\\
&\hskip1cm=\sum_{k=1}^n \partial R_1^k\partial R_2^k
\exp\bigl\{-4{\delta}\bigl({\mathrm{ev}}(R_1^k)\cap{\mathrm{ev}}(R_2^k)\cap\pi_k\bigr)\bigr\}
P({\widetilde}{\mathcal{O}}_k)\\
&\hskip1cm=\sum_{k=1}^n \partial R_1^k
\partial R_2^k \,P({\mathcal{O}}_k(R_1^k,R_2^k,{\Delta})).\end{aligned}$$
Summing the above over ${{\mathcal{Q}}^{A,B}}$, and remembering that each $f_{\pi_k}$ is a bijection between ${{\mathcal{Q}}^{A,B}}$ and ${{\mathcal{Q}}^{A{\triangle}xy,B{\triangle}xy}}$, becomes $$\begin{aligned}
\frac1{2^{|Q|+2r}} \sum_{k=1}^n\,
& \sum_{(R_1,R_2)\in{{\mathcal{Q}}^{A{\triangle}xy,B{\triangle}xy}}}\partial R_1\partial R_2\,
P({\mathcal{O}}_k(R_1,R_2,{\Delta}))\\
&= \frac1{2^{|Q|+2r}} \sum_{{{\mathcal{Q}}^{A{\triangle}xy,B{\triangle}xy}}}\partial R_1\partial R_2\,
P(x{\leftrightarrow}y\mbox{ in }R_1,R_2,{\Delta}).\end{aligned}$$ By the argument leading to , this equals the right side of , and the claim is proved when $F\equiv 1$.
Consider now the case of general connectivity functions $F$ in . In , the factor $P(x{\leftrightarrow}y\mbox{ in }Q_1,Q_2,{\Delta})$ is replaced by $$P\bigl(F(Q_1,Q_2,{\Delta})\cdot {\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}y\mbox{ in }Q_1,Q_2,{\Delta}\}\bigr),$$ where $P$ is expectation with respect to ${\Delta}$. In the calculation , we use the fact that $$P(F\cdot {\hbox{\rm 1\kern-.27em I}}_{{\mathcal{O}}_k})=P(F\mid{\mathcal{O}}_k)P({\mathcal{O}}_k)$$ and we deal with the factor $P({\mathcal{O}}_k)$ as before. The result follows on noting that, for each $k$, $$P\bigl(F(Q_1,Q_2,{\Delta}){\,\big|\,}{\mathcal{O}}_k(Q_1,Q_2,{\Delta})\bigr)=
P\bigl(F(R_1^k,R_2^k,{\Delta}){\,\big|\,}{\mathcal{O}}_k(R_1^k,R_2^k,{\Delta})\bigr).$$ This holds because: (i) the configurations $(Q_1,Q_2,{\Delta})$ and $(R_1^k,R_2^k,{\Delta})$ are identical off $\pi_k$, and (ii) in each, all points along $\pi_k$ are connected. Thus the connectivities are identical in the two configurations.
### Applications of switching {#sw_appl_sec}
In this section are presented a number of inequalities and identities proved using the random-parity representation and the switching lemma. With some exceptions (most notably ) the proofs are adaptations of the proofs for the discrete Ising model that may be found in [@abf; @grimmett_rcm].
For $R\subseteq K$ a finite union of intervals, let $${\widetilde}R:=\{(uv,t)\in F: \mbox{either } (u,t)\in R
\mbox{ or }(v,t)\in R\mbox{ or both}\}.$$ Recall that $W=W(K)=\{v\in V: K_v={\mathbb{S}}\}$, and $N=N(K)$ is the total number of intervals constituting $K$.
\[rw\_mon\_lem\] Let $R\subseteq K$ be finite union of intervals, and let $\nu\in\Xi$ be such that $\nu\cap
R={\varnothing}$. If $A \subseteq {\overline}{K{\setminus}R}$ is finite and $A\sim\nu$, then $${w}^A(\nu)\leq 2^{r(\nu)-r'(\nu)}{w}^A_{K\setminus R}(\nu),$$ where $$\begin{aligned}
r(\nu) &= r(\nu,K) := |\{w\in W: \nu\cap (w\times K_w) \ne {\varnothing}\}|,\\
r'(\nu) &= r(\nu,K{\setminus}R).\end{aligned}$$
By and Lemma \[Z’\], $$\begin{aligned}
\label{r0}
{w}^A(\nu)&=\frac{Z_{K\setminus\nu}}{Z_K}\\
&=2^{N(K)-N(K{\setminus}\nu)}
e^{{\lambda}({\widetilde}\nu)+{\gamma}(\nu)-{\delta}(\nu)}\frac{Z'_{K{\setminus}\nu}}{Z'_K}.
\nonumber\end{aligned}$$ We claim that $$\label{r05}
\frac{Z'_{K{\setminus}\nu}}{Z'_K}\leq\frac{Z'_{K{\setminus}(R\cup\nu)}}{Z'_{K{\setminus}R}},$$ and the proof of this follows.
Recall the formula for $Z'_K$ in terms of an integral over the Poisson process $D$. The set $D$ is the union of independent Poisson processes $D'$ and $D''$, restricted respectively to $K{\setminus}\nu$ and $\nu$. We write $P'$ ([respectively]{}, $P''$) for the probability measure (and, on occasion, expectation operator) governing $D'$ ([respectively]{}, $D''$). Let ${\Sigma}(D')$ denote the set of spin configurations on $K{\setminus}\nu$ that are permitted by $D'$. By , $$\label{Z'_split_eq}
Z'_K= P'\left( \sum_{{\sigma}'\in{\Sigma}(D')}Z_\nu'({\sigma}')
\exp\left\{\int_{F{\setminus}{\widetilde}\nu}{\lambda}(e){\sigma}'_e\,de+\int_{K{\setminus}\nu}{\gamma}(x){\sigma}'_x\,dx\right\}\right),$$ where $$Z_\nu'({\sigma}') = P''\left(\sum_{{\sigma}''\in{\widetilde}{\Sigma}(D'')}
\exp\left\{\int_{{\widetilde}\nu}{\lambda}(e){\sigma}_e\,de+\int_{\nu}{\gamma}(x){\sigma}_x\,dx\right\}\cdot {\hbox{\rm 1\kern-.27em I}}_C({\sigma}')\right)$$ is the partition function on $\nu$ with boundary condition ${\sigma}'$, and where ${\sigma}$, ${\widetilde}{\Sigma}(D'')$, and $C=C({\sigma}',D'')$ are given as follows.
The set $D''$ divides $\nu$, in the usual way, into a collection $V_\nu(D'')$ of intervals. From the set of endpoints of such intervals, we distinguish the subset ${\mathcal{E}}$ that: (i) lie in $K$, and (ii) are endpoints of some interval of $K{\setminus}\nu$. For $x\in{\mathcal{E}}$, let ${\sigma}'_x =\lim_{y\to x} {\sigma}'_y$, where the limit is taken over $y\in K{\setminus}\nu$. Let ${\widetilde}V_\nu(D'')$ be the subset of $V_\nu(D'')$ containing those intervals with no endpoint in ${\mathcal{E}}$, and let ${\widetilde}{\Sigma}(D'') =\{-1,+1\}^{{\widetilde}V_\nu(D'')}$.
Let ${\sigma}'\in {\Sigma}(D')$, and let ${\mathcal{I}}$ be the set of maximal sub-intervals $I$ of $\nu$ having both endpoints in ${\mathcal{E}}$, and such that $I \cap D''={\varnothing}$. Let $C=C(D'')$ be the set of ${\sigma}'\in{\Sigma}(D')$ such that, for all $I\in{\mathcal{I}}$, the endpoints of $I$ have equal spins under ${\sigma}'$. Note that $$\label{o30}
{\hbox{\rm 1\kern-.27em I}}_C({\sigma}') = \prod_{I\in{\mathcal{I}}} \tfrac12({\sigma}'_{x(I)}{\sigma}'_{y(I)} + 1),$$ where $x(I)$, $y(I)$ denote the endpoints of $I$.
Let ${\sigma}''\in {\widetilde}{\Sigma}(D'')$. The conjunction ${\sigma}$ of ${\sigma}'$ and ${\sigma}''$ is defined except on sub-intervals of $\nu$ lying in $V_\nu(D''){\setminus}{\widetilde}V_\nu(D'')$. On any such sub-interval with exactly one endpoint $x$ in ${\mathcal{E}}$, we set ${\sigma}\equiv {\sigma}'_x$. On the event $C$, an interval of $\nu$ with both endpoints $x(I)$, $y(I)$ in ${\mathcal{E}}$ receives the spin ${\sigma}\equiv {\sigma}_{x(I)} = {\sigma}_{y(I)}$. Thus, ${\sigma}\in {\Sigma}(D'\cup D'')$ is well defined for ${\sigma}'\in C$.
By , $$\frac{Z'_K}{Z'_{K{\setminus}\nu}}={\langle}Z'_\nu({\sigma}'){\rangle}_{K{\setminus}\nu}.$$ Taking the expectation ${\langle}\cdot{\rangle}_{K{\setminus}\nu}$ inside the integral, the last expression becomes $$P''\left(\sum_{{\sigma}''\in{\widetilde}{\Sigma}(D'')}\left{\langle}\exp\left\{\int_{{\widetilde}\nu}{\lambda}(e){\sigma}_e\,de\right\}
\exp\left\{\int_{\nu}{\gamma}(x){\sigma}_x\,dx\right\}
\cdot {\hbox{\rm 1\kern-.27em I}}_C({\sigma}')\right{\rangle}_{K{\setminus}\nu}\right)$$ The inner expectation may be expressed as a sum over $k,l\geq 0$ (with non-negative coefficients) of iterated integrals of the form $$\frac1{k!}\,\frac1{l!}\,\iint\limits_{{\widetilde}\nu^k\times\nu^l}{\lambda}(\mathbf e){\gamma}(\mathbf x)
{\langle}{\sigma}_{e_1}\cdots{\sigma}_{e_k}{\sigma}_{x_1}\cdots{\sigma}_{x_l}\cdot {\hbox{\rm 1\kern-.27em I}}_C{\rangle}_{K{\setminus}\nu}
\,d\mathbf e \,d\mathbf x,
\label{o32}$$ where we have written $\mathbf e=(e_1,\dotsc,e_k)$, and ${\lambda}(\mathbf e)$ for ${\lambda}(e_1)\dotsb{\lambda}(e_k)$ (and similarly for $\mathbf x$). We may write $${\langle}{\sigma}_{e_1}\cdots{\sigma}_{e_k}{\sigma}_{x_1}\cdots{\sigma}_{x_l}\cdot {\hbox{\rm 1\kern-.27em I}}_C{\rangle}_{K{\setminus}\nu}
={\langle}{\sigma}'_S{\sigma}''_T\cdot {\hbox{\rm 1\kern-.27em I}}_C{\rangle}_{K{\setminus}\nu}={\sigma}''_T{\langle}{\sigma}_S'\cdot {\hbox{\rm 1\kern-.27em I}}_C{\rangle}_{K{\setminus}\nu},$$ for sets $S\subseteq {\overline}{K{\setminus}\nu}$, $T\subseteq \nu$ determined by $e_1,\dotsc,e_k,x_1,\dotsc,x_l$ and $D''$ only. We now bring the sum over ${\sigma}''$ inside the integral of . For $T\neq{\varnothing}$, $$\sum_{{\sigma}''\in{\widetilde}{\Sigma}(D'')}{\sigma}''_T{\langle}{\sigma}_S'\cdot {\hbox{\rm 1\kern-.27em I}}_C{\rangle}_{K{\setminus}\nu}=0,$$ so any non-zero term is of the form $$\label{r1}
{\langle}{\sigma}_S'\cdot {\hbox{\rm 1\kern-.27em I}}_C{\rangle}_{K{\setminus}\nu}.$$
By , may be expressed in the form $$\sum_{i=1}^s2^{-a_i}{\langle}{\sigma}'_{S_i}{\rangle}_{K{\setminus}\nu}
\label{o35}$$ for appropriate sets $S_i$ and integers $a_i$. By Lemma \[cor\_mon\_lem\], $${\langle}{\sigma}'_{S_i}{\rangle}_{K{\setminus}\nu}\geq{\langle}{\sigma}'_{S_i}{\rangle}_{K{\setminus}(R\cup\nu)}.$$ On working backwards, we obtain .
By –, $${w}^A(\nu)\leq 2^U{w}^A_{K\setminus R}(\nu),$$ where $$\begin{aligned}
U&=\bigl[N(K)-N(K{\setminus}\nu)\bigr]- \bigl[N(K{\setminus}R)-N(K{\setminus}(R\cup\nu) )\bigr]\\
&=r(\nu)-r'(\nu)\end{aligned}$$ as required.
For distinct $x,y,z\in K^{{\Gamma}}$, let $$\begin{aligned}
{\langle}{\sigma}_x;{\sigma}_y;{\sigma}_z{\rangle}&:=
{\langle}{\sigma}_{xyz}{\rangle}-{\langle}{\sigma}_{x}{\rangle}{\langle}{\sigma}_{yz}{\rangle}\\
&\hskip1.5cm -{\langle}{\sigma}_{y}{\rangle}{\langle}{\sigma}_{xz}{\rangle}-{\langle}{\sigma}_{z}{\rangle}{\langle}{\sigma}_{xy}{\rangle}+2{\langle}{\sigma}_{x}{\rangle}{\langle}{\sigma}_{y}{\rangle}{\langle}{\sigma}_{z}{\rangle}.\end{aligned}$$
\[ghs\_lem\] For distinct $x,y,z\in K^{{\Gamma}}$, $$\label{ghs_1_eq}
{\langle}{\sigma}_x;{\sigma}_y;{\sigma}_z{\rangle}\leq 0.$$ Moreover, ${\langle}{\sigma}_x{\rangle}$ is concave in ${\gamma}$ in the sense that, for bounded, measurable functions ${\gamma}_1,{\gamma}_2: K\to{{\mathbb{R}}_+}$ satisfying ${\gamma}_1\le{\gamma}_2$, and $\theta\in[0,1]$, $$\theta{\langle}{\sigma}_x{\rangle}_{{\gamma}_1}+(1-\theta){\langle}{\sigma}_x{\rangle}_{{\gamma}_2}\leq
{\langle}{\sigma}_x{\rangle}_{\theta{\gamma}_1+(1-\theta){\gamma}_2}.$$
The proof of this follows very closely the corresponding proof for the classical Ising model [@ghs]. We include it here because it allows us to develop the technique of ‘conditioning on clusters’, which will be useful later.
We prove via the following more general result. Let $(B_i,G_i)$, $i=1,2,3$, be independent sets of bridges/ghost-bonds, and write $\psi_i$, $i=1,2,3$, for corresponding colourings (with sources to be specified through their superscripts). We claim that, for any four points $w, x, y, z\in K^{{\Gamma}}$, $$\label{ghs_2_eq}
\begin{split}
&E\bigl(\partial\psi_1^{\varnothing}\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}\bigr)-
E\bigl(\partial\psi_1^{\varnothing}\partial\psi_2^{wz}\partial\psi_3^{xy}\bigr)
\\
&\quad\leq E(\partial\psi_1^{\varnothing}\partial\psi_2^{wx} \partial\psi_3^{yz})
+E(\partial\psi_1^{\varnothing}\partial\psi_2^{wy}\partial\psi_3^{xz})
-2E(\partial\psi_1^{wx}\partial\psi_2^{wy}\partial\psi_3^{wz}).
\end{split}$$ Inequality follows by Theorem \[rcr\_thm\] on letting $w={{\Gamma}}$.
The left side of is $$\begin{aligned}
&E(\partial\psi_1^{\varnothing})\bigl[
E(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz})-
E(\partial\psi_2^{wz}\partial\psi_3^{xy})\bigr]\\
&\hskip3cm =
Z\, E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}\bigr),\end{aligned}$$ by the switching lemma \[sl\]. When $\partial\psi_3^{wxyz}$ is non-zero, parity constraints imply that at least one of $\{w{\leftrightarrow}x\}\cap \{y{\leftrightarrow}z\}$ and $\{w{\leftrightarrow}y\}\cap
\{x{\leftrightarrow}z\}$ occurs, but that, in the presence of the indicator function they cannot both occur. Therefore, $$\begin{aligned}
\label{ghs_pf_1_eq}
&E(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\})\\
&\hskip1cm
=E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\leftrightarrow}x\}\bigr)\nonumber\\
&\hskip3cm +
E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\leftrightarrow}y\}\bigr).
\nonumber\end{aligned}$$ Consider the first term. By the switching lemma, $$E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\leftrightarrow}x\}\bigr)=
E\bigl(\partial\psi_2^{wx}\partial\psi_3^{yz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}\bigr).
\label{o17}$$
We next ‘condition on a cluster’. Let $C_z=C_z(\psi_2^{wx},\psi_3^{yz},{\Delta})$ be the set of all points of $K$ that are connected by open paths to $z$. Conditional on $C_z$, define new independent colourings $\mu_2^{\varnothing}$, $\mu_3^{yz}$ on the domain $M=C_z$. Similarly, let $\nu_2^{wx}$, $\nu_3^{\varnothing}$ be independent colourings on the domain $N=K\setminus C_z$, that are also independent of the $\mu_i$. It is not hard to see that, if $w{\nleftrightarrow}z$ in $(\psi_2^{wx},\psi_3^{yz},{\Delta})$, then, conditional on $C_z$, the law of $\psi_2^{wx}$ equals that of the superposition of $\mu_2^{\varnothing}$ and $\nu_2^{wx}$; similarly the conditional law of $\psi_3^{yz}$ is the same as that of the superposition of $\mu_3^{yz}$ and $\nu_3^{\varnothing}$. Therefore, almost surely on the event $\{w {\nleftrightarrow}z\}$, $$\begin{aligned}
E(\partial\psi_2^{wx}\partial\psi_3^{yz}\mid C_z)&=
E'({\partial}\mu_2^{\varnothing})E'({\partial}\nu_2^{wx})E'({\partial}\mu_3^{yz})E'({\partial}\nu_3^{\varnothing})\label{o19}\\
&={\langle}{\sigma}_{wx}{\rangle}_N E'({\partial}\mu_2^{\varnothing})E'({\partial}\nu_2^{\varnothing})
E'({\partial}\mu_3^{yz})E'({\partial}\nu_3^{\varnothing})\nonumber\\
&\leq
{\langle}{\sigma}_{wx}{\rangle}_KE(\partial\psi_2^{\varnothing}\partial\psi_3^{yz}\mid C_z),
\nonumber\end{aligned}$$ where $E'$ denotes expectation conditional on $C_z$, and we have used Lemma \[cor\_mon\_lem\]. Returning to –, $$\begin{aligned}
&E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{wxyz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\leftrightarrow}x\}\bigr)\\
&\hskip2cm \leq
{\langle}{\sigma}_{wx}{\rangle}E(\partial\psi_2^{\varnothing}\partial\psi_3^{yz}
\cdot{\hbox{\rm 1\kern-.27em I}}\{w{\nleftrightarrow}z\}).\end{aligned}$$ The other term in satisfies the same inequality with $x$ and $y$ interchanged. Inequality follows on applying the switching lemma to the right sides of these two last inequalities, and adding them.
The concavity of ${\langle}{\sigma}_x{\rangle}$ follows from the fact that, if $$T=\sum_{k=1}^n a_k{\hbox{\rm 1\kern-.27em I}}_{A_k}$$ is a step function on $K$ with $a_k\ge 0$ for all $k$, and ${\gamma}(\cdot)={\gamma}_1(\cdot)+{\alpha}T(\cdot)$, then $$\frac{\partial^2}{\partial {\alpha}^2}{\langle}{\sigma}_x{\rangle}=\sum_{k,l=1}^na_ka_l\iint_{A_k\times A_l} dy\,dz\, {\langle}{\sigma}_x;{\sigma}_y;{\sigma}_z{\rangle}\leq 0.$$ Thus, the claim holds whenever ${\gamma}_2-{\gamma}_1$ is a step function. The general claim follows by approximating ${\gamma}_2-{\gamma}_1$ by step functions, and applying the dominated convergence theorem.
For the next lemma we assume for simplicity that ${\gamma}\equiv 0$ (although similar results can easily be proved for ${\gamma}\not\equiv 0$). We let $\bar{\delta}\in{\mathbb{R}}$ be an upper bound for ${\delta}$, thus ${\delta}(x)\leq\bar{\delta}<{\infty}$ for all $x\in K$. Let $a,b\in K$ be two distinct points. A closed set $T\subseteq K$ is said to *separate* $a$ from $b$ if every lattice path from $a$ to $b$ (whatever the set of bridges) intersects $T$. Moreover, if ${\varepsilon}>0$ and $T$ separates $a$ from $b$, we say that $T$ is an *${\varepsilon}$-fat separating set* if every point in $T$ lies in a closed sub-interval of $T$ of length at least ${\varepsilon}$.
\[simon\_lem\] Let ${\gamma}\equiv 0$. If ${\varepsilon}>0$ and $T$ is an ${\varepsilon}$-fat separating set for $a,b\in K$, $${\langle}{\sigma}_a{\sigma}_b{\rangle}\leq\frac{1}{{\varepsilon}}\exp(8{\varepsilon}\bar{\delta})
\int_T{\langle}{\sigma}_a{\sigma}_x{\rangle}{\langle}{\sigma}_x{\sigma}_b{\rangle}\, dx.$$
By Theorems \[rcr\_thm\] and \[sl\], $${\langle}{\sigma}_a{\sigma}_x{\rangle}{\langle}{\sigma}_x{\sigma}_b{\rangle}=
\frac{1}{Z^2}E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot{\hbox{\rm 1\kern-.27em I}}\{a{\leftrightarrow}x\}),$$ and, by Fubini’s theorem, $$\int_T{\langle}{\sigma}_a{\sigma}_x{\rangle}{\langle}{\sigma}_x{\sigma}_b{\rangle}\;dx=
\frac{1}{Z^2}E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot|{\widehat}T|),$$ where ${\widehat}T=\{x\in T:a{\leftrightarrow}x\}$ and $|\cdot|$ denotes Lebesgue measure. Since ${\gamma}\equiv 0$, the backbone $\xi = \xi(\psi_2^{ab})$ consists of a single (lattice-) path from $a$ to $b$ passing through $T$. Let $U$ denote the set of points in $K$ that are separated from $b$ by $T$, and let $X$ be the point at which $\xi$ exits $U$ for the first time. Since $T$ is assumed closed, $X\in T$. See Figure \[simon\_fig\].
![The Simon inequality. The separating set $T$ is drawn with solid black lines, and the backbone $\xi$ with a grey line.[]{data-label="simon_fig"}](thesis.18)
For $x\in T$, let $A_x$ be the event that there is no element of ${\Delta}$ within the interval of length $2{\varepsilon}$ centered at $x$. Thus, $P(A_x)=\exp(-8{\varepsilon}\bar{\delta})$. On the event $A_X$, we have that $|{\widehat}T|\geq{\varepsilon}$, whence $$\begin{aligned}
\label{simon_eq1_eq}
E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot|{\widehat}T|)&\geq
E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot|{\widehat}T|\cdot{\hbox{\rm 1\kern-.27em I}}\{A_X\})\\
&\geq{\varepsilon}E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot{\hbox{\rm 1\kern-.27em I}}\{A_X\}).
\nonumber\end{aligned}$$ Conditional on $X$, the event $A_X$ is independent of $\psi_1^{\varnothing}$ and $\psi_2^{ab}$, so that $$E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot|{\widehat}T|)\geq{\varepsilon}\exp(-8{\varepsilon}\bar{\delta})
E(\partial\psi_1^{\varnothing}\partial\psi_2^{ab}),
\label{o18}$$ and the proof is complete.
Just as for the classical Ising model, only a small amount of extra work is required to deduce the following improvement of Lemma \[simon\_lem\].
\[lieb\_lem\] Under the assumptions of Lemma \[simon\_lem\], $${\langle}{\sigma}_a{\sigma}_b{\rangle}\leq\frac{1}{{\varepsilon}}\exp(8{\varepsilon}\bar{\delta})
\int_T{\langle}{\sigma}_a{\sigma}_x{\rangle}_{\overline{T}}\,{\langle}{\sigma}_x{\sigma}_b{\rangle}\;dx,$$ where ${\langle}\cdot{\rangle}_{{\overline}T}$ denotes expectation with respect to the measure restricted to ${\overline}T$.
Let $x \in T$, let ${\overline}\psi_1^{ax}$ denote a colouring on the restricted region $U$, and let $\psi_2^{xb}$ denote a colouring on the full region $K$ as before. We claim that $$\label{lieb_sw_eq}
E(\partial{\overline}\psi_1^{ax}\partial\psi_2^{xb})=
E\bigl(\partial{\overline}\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot {\hbox{\rm 1\kern-.27em I}}\{a{\leftrightarrow}x\mbox{ in }\overline{T}\}\bigr).$$ The use of the letter $E$ is an abuse of notation, since the ${\overline}\psi$ are colourings of $U$ only.
Equation may be established using a slight variation in the proof of the switching lemma. We follow the proof of that lemma, first conditioning on the set $Q$ of all bridges and ghost-bonds in the two colourings taken together, and then allocating them to the colourings $Q_1$ and $Q_2$, uniformly at random. We then order the paths $\pi$ of $Q$ from $a$ to $x$, and add the earliest open path to both $Q_1$ and $Q_2$ ‘modulo 2’. There are two differences here: firstly, any element of $Q$ that is not contained in $U$ will be allocated to $Q_2$, and secondly, we only consider paths $\pi$ that lie inside $U$. Subject to these two changes, we follow the argument of the switching lemma to arrive at .
Integrating over $x \in T$, $$\int_T{\langle}{\sigma}_a{\sigma}_x{\rangle}_{\overline{T}}\,{\langle}{\sigma}_x{\sigma}_b{\rangle}\;dx=
\frac{1}{Z_{{\overline}T}Z}E(\partial{\overline}\psi_1^{\varnothing}\partial\psi_2^{ab}
\cdot|{\widehat}T|),$$ where this time ${\widehat}T=\{x\in T:a{\leftrightarrow}x\mbox{ in }U\}$. The proof is completed as in –.
For the next lemma we specialize to the situation that is the main focus of this chapter, namely the following. Similar results are valid for other lattices and for summable translation-invariant interactions.
\[periodic\_assump\]
- The graph $L=[-n,n]^d \subseteq {\mathbb{Z}}^d$ where $d \ge 1$, with periodic boundary condition.
- The parameters ${\lambda}$, ${\delta}$, ${\gamma}$ are non-negative constants.
- The set $K_v={\mathbb{S}}$ for every $v\in V$.
Under the periodic boundary condition, two vertices of $L$ are joined by an edge whenever there exists $i\in\{1,2,\dots,d\}$ such that their $i$-coordinates differ by exactly $2n$.
Under Assumption \[periodic\_assump\], the model is invariant under automorphisms of $L$ and, furthermore, the quantity ${\langle}{\sigma}_x{\rangle}$ does not depend on the choice of $x$. Let $0$ denote some fixed but arbitrary point of $K$, and let $M=M({\lambda},{\delta},{\gamma})={\langle}{\sigma}_0{\rangle}$ denote the common value of the ${\langle}{\sigma}_x{\rangle}$.
For $x,y\in K$, we write $x\sim y$ if $x=(u,t)$ and $y=(v,t)$ for some $t \ge 0$ and $u,v$ adjacent in $L$. We write $\{x{\overset{z}{{\leftrightarrow}}} y\}$ for the complement of the event that there exists an open path from $x$ to $y$ not containing $z$. Thus, $x{\overset{z}{{\leftrightarrow}}} y$ if: either $x {\nleftrightarrow}y$, or $x {\leftrightarrow}y$ and every open path from $x$ to $y$ passes through $z$.
\[three\_ineq\_lem\] Under Assumption \[periodic\_assump\], the following hold. $$\begin{aligned}
\label{dg_bound_eq}
\frac{\partial M}{\partial{\gamma}}&=\frac{1}{Z^2}\int_K dx\;
E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr)
\leq \frac{M}{{\gamma}}.\\
\label{dl_bound_eq}
\frac{\partial M}{\partial{\lambda}}&=\frac{1}{2Z^2}\int_K dx
\sum_{y\sim x} E\bigl(\partial\psi_1^{0xy{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr)
\leq 2dM\frac{\partial M}{\partial{\gamma}}.\\
\label{dd_bound_eq}
-\frac{\partial M}{\partial{\delta}}&=
\frac{2}{Z^2}\int_K dx \: E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)
\leq \frac{2M}{1-M^2}\frac{\partial M}{\partial{\gamma}}.\end{aligned}$$
With the exception of , the proofs mimic those of [@abf] for the classical Ising model. For the equality in , note that $$\frac{\partial M}{\partial {\gamma}}=\int_K {\langle}{\sigma}_0;{\sigma}_x{\rangle}\,dx.$$ Now $${\langle}{\sigma}_0;{\sigma}_x{\rangle}={\langle}{\sigma}_0{\sigma}_x{\rangle}- {\langle}{\sigma}_0{\rangle}{\langle}{\sigma}_x{\rangle}=
\frac{1}{Z^2}(E(\partial\psi_1^{0x}\partial\psi_2^\varnothing)-
E(\partial\psi_1^{0{\Gamma}}\partial\psi_2^{x{\Gamma}}))$$ and the difference $E(\partial\psi_1^{0x}\partial\psi_2^\varnothing)-
E(\partial\psi_1^{0{\Gamma}}\partial\psi_2^{x{\Gamma}})$ on the right hand side equals $$E(\partial\psi_1^{0x}\partial\psi_2^\varnothing)-
E(\partial\psi_1^{0x}\partial\psi_2^\varnothing\cdot
{\hbox{\rm 1\kern-.27em I}}\{0\leftrightarrow {\Gamma}\})
=E(\partial\psi_1^{0x}\partial\psi_2^\varnothing
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\not\leftrightarrow {\Gamma}\})$$ by the switching lemma. For the inequality in , the concavity of $M$ in ${\gamma}$ means that for all ${\gamma}_2\geq{\gamma}_1>0$, $$\frac{\partial M}{\partial{\gamma}}\leq
\frac{M({\lambda},{\delta},{\gamma}_2)-M({\lambda},{\delta},{\gamma}_1)}{{\gamma}_2-{\gamma}_1}.$$ Letting ${\gamma}_1\rightarrow0$ and using the continuity of $M$ and the fact that $M({\lambda},{\delta},0)=0$ for all ${\lambda},{\delta}>0$, the result follows.
Similarly, for the equality in we note that $$\frac{\partial M}{\partial {\lambda}}=\int_F {\langle}{\sigma}_0;{\sigma}_e{\rangle}\,de
=\frac{1}{2}\int_K dx\sum_{y\sim x}
({\langle}{\sigma}_0{\sigma}_x{\sigma}_y{\rangle}- {\langle}{\sigma}_0{\rangle}{\langle}{\sigma}_x{\sigma}_y{\rangle}).$$ Again $$\begin{split}
{\langle}{\sigma}_0{\sigma}_x{\sigma}_y{\rangle}- {\langle}{\sigma}_0{\rangle}{\langle}{\sigma}_x{\sigma}_y{\rangle}&=
\frac{1}{Z^2}(E(\partial\psi_1^{0xy{\Gamma}}\partial\psi_2^\varnothing)-
E(\partial\psi_1^{0{\Gamma}}\partial\psi_2^{xy}))\\
&=E(\partial\psi_1^{0xy{\Gamma}}\partial\psi_2^\varnothing
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\not\leftrightarrow {\Gamma}\})
\end{split}$$ by the switching lemma. For the inequality, $$\begin{split}
\frac{\partial M}{\partial {\lambda}} &=
\frac{1}{2}\int_K dx\sum_{y\sim x}
\big({\langle}{\sigma}_0{\sigma}_x{\sigma}_y{\rangle}-{\langle}{\sigma}_0{\rangle}{\langle}{\sigma}_x{\sigma}_y{\rangle}\big)\\
&\leq\frac{1}{2}\int_K dx\sum_{y\sim x}
\big({\langle}{\sigma}_x{\rangle}{\langle}{\sigma}_0{\sigma}_y{\rangle}+{\langle}{\sigma}_y{\rangle}{\langle}{\sigma}_0{\sigma}_x{\rangle}-2{\langle}{\sigma}_0{\rangle}{\langle}{\sigma}_x{\rangle}{\langle}{\sigma}_y{\rangle}\big)\\
&=\int_K dx\:{\langle}{\sigma}_0;{\sigma}_x{\rangle}\sum_{y\sim x}
{\langle}{\sigma}_y{\rangle}\\
&=2dM\int_K dx\:{\langle}{\sigma}_0;{\sigma}_x{\rangle}=
2dM\frac{\partial M}{\partial {\gamma}},
\end{split}$$ where we have used the [[ghs]{}]{}-inequality and translation invariance.
Here is the proof of . Let $|\cdot|$ denote Lebesgue measure. By differentiating $$M=\frac{E(\partial\psi^{0{{\Gamma}}})}{E(\partial\psi^{\varnothing})}=
\frac{E(\exp(2{\delta}|{\mathrm{ev}}(\psi^{0{{\Gamma}}})|))}{E(\exp(2{\delta}|{\mathrm{ev}}(\psi^{\varnothing})|))},$$ with respect to ${\delta}$, we obtain that $$\begin{aligned}
\frac{\partial M}{\partial {\delta}}&=
\frac{2}{Z^2}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
\bigl[|{\mathrm{ev}}(\psi_1^{0{{\Gamma}}})|-|{\mathrm{ev}}(\psi_2^{\varnothing})|\bigr]\bigr)\label{ihp17}\\
&=\frac{2}{Z^2}\int dx\,
E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
\bigl[{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_2^{\varnothing})\}-{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_1^{0{{\Gamma}}})\}\bigr]\bigr).
\nonumber\end{aligned}$$
Consider the integrand in . Since $\psi_2^{\varnothing}$ has no sources, all odd routes in $\psi_2^{\varnothing}$ are necessarily cycles. If $x\in{\mathrm{odd}}(\psi_2^{\varnothing})$, then $x$ lies in an odd cycle. We shall assume that $x$ is not the endpoint of a bridge, since this event has probability 0. It follows that, on the event $\{0{\leftrightarrow}{{\Gamma}}\}$, there exists an open path from $0$ to ${{\Gamma}}$ that avoids $x$ (since any path can be re-routed around the odd cycle of $\psi_2^{\varnothing}$ containing $x$). Therefore, the event $\{0{\overset{x}{{\leftrightarrow}}} {{\Gamma}}\}$ does not occur, and hence $$\begin{aligned}
&E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_2^{\varnothing})\}\bigr)\label{dd_pf_1_eq}\\
&\hskip2cm =E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_2^{\varnothing})\}\cdot
{\hbox{\rm 1\kern-.27em I}}\{0{\overset{x}{{\leftrightarrow}}} {{\Gamma}}\}^{\mathrm{c}}\bigr).
\nonumber\end{aligned}$$
We note next that, if $\partial\psi_1^{0{{\Gamma}}}\ne 0$ and $0{\overset{x}{{\leftrightarrow}}}{{\Gamma}}$, then necessarily $x\in{\mathrm{odd}}(\psi_1^{0{{\Gamma}}})$. Hence, $$\begin{aligned}
\label{dd_pf_2_eq}
&E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_1^{0{{\Gamma}}})\}\bigr)\\
&\hskip1cm =E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_1^{0{{\Gamma}}})\}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}^{\mathrm{c}}\bigr)\nonumber\\
&\hskip5cm +E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr).
\nonumber\end{aligned}$$ We wish to switch the sources $0{{\Gamma}}$ from $\psi_1$ to $\psi_2$ in the right side of . For this we need to adapt some details of the proof of the switching lemma to this situation. The first step in the proof of that lemma was to condition on the union $Q$ of the bridges and ghost-bonds of the two colourings; then, the paths from $0$ to ${{\Gamma}}$ in $Q$ were listed in a fixed *but arbitrary* order. We are free to choose this ordering in such a way that paths not containing $x$ have precedence, and we assume henceforth that the ordering is thus chosen. The next step is to find the earliest open path $\pi$, and ‘add $\pi$ modulo 2’ to both $\psi_1^{0{{\Gamma}}}$ and $\psi_2^{\varnothing}$. On the event $\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}^{\mathrm{c}}$, this earliest path $\pi$ does not contain $x$, by our choice of ordering. Hence, in the new colouring $\psi_1^{\varnothing}$, $x$ continues to lie in an ‘odd’ interval (recall that, outside $\pi$, the colourings are unchanged by the switching procedure). Therefore, $$\begin{aligned}
&E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_1^{0{{\Gamma}}})\}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}^{\mathrm{c}}\bigr)\\
&\hskip 2cm =E\bigl(\partial\psi_1^{\varnothing}\partial\psi_2^{0{{\Gamma}}}\cdot
{\hbox{\rm 1\kern-.27em I}}\{x\in{\mathrm{odd}}(\psi_1^{\varnothing})\}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}^{\mathrm{c}}\bigr).
\nonumber\end{aligned}$$ Relabelling, putting the last expression into , and subtracting from , we obtain $$\label{dd_pf_3_eq}
\frac{\partial M}{\partial {\delta}}=
-\frac{2}{Z^2}\int dx\:
E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)$$ as required.
Turning to the inequality, let $C^x_z$ denote the set of points that can be reached from $z$ along open paths *not containing $x$*. When conditioning $E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)$ on $C^x_0$ as in the proof of the [[ghs]{}]{} inequality, we find that $\psi_1^{0{{\Gamma}}}$ is a combination of two independent colourings, one inside $C^x_0$ with sources $0x$, and one outside $C^x_0$ with sources $x{{\Gamma}}$. As in , using Lemma \[cor\_mon\_lem\] as there, $$\begin{aligned}
\label{dd_pf_4_eq}
E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)&=
E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}{\langle}{\sigma}_x{\rangle}_{K\setminus C^x_0}
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)\\
&\leq M\cdot E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr).
\nonumber\end{aligned}$$ We split the expectation on the right side according to whether or not $x{\leftrightarrow}{{\Gamma}}$. Clearly, $$\label{dd_pf_5_eq}
E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}
\cdot {\hbox{\rm 1\kern-.27em I}}\{x{\nleftrightarrow}{{\Gamma}}\}\bigr)\leq
E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{x{\nleftrightarrow}{{\Gamma}}\}\bigr).$$ By the switching lemma \[sl\], the other term satisfies $$E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}
\cdot {\hbox{\rm 1\kern-.27em I}}\{x{\leftrightarrow}{{\Gamma}}\}\bigr)=
E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{x{{\Gamma}}}
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr).$$ We again condition on a cluster, this time $C^x_{{\Gamma}}$, to obtain as in that $$\label{dd_pf_6_eq}
E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{x{{\Gamma}}}
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)\leq
M\cdot E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr).$$ Combining , , with , we obtain by that $$-\frac{\partial M}{\partial {\delta}}\leq 2M\frac{\partial M}{\partial {\gamma}}+
M^2\Big(-\frac{\partial M}{\partial {\delta}}\Big),$$ as required.
Proof of the main differential inequality {#pf_sec}
-----------------------------------------
In this section we will prove Theorem \[main\_pdi\_thm\], the differential inequality which, in combination with the inequalities of the previous section, will yield information about the critical behaviour of the space–time Ising model. The proof proceeds roughly as follows. In the random-parity representation of $M={\langle}{\sigma}_0{\rangle}$, there is a backbone from $0$ to ${{\Gamma}}$ (that is, to some point $g \in G$). We introduce two new sourceless configurations; depending on how the backbone interacts with these configurations, the switching lemma allows a decomposition into a combination of other configurations which, via Theorem \[three\_ineq\_lem\], may be transformed into derivatives of the magnetization.
Throughout this section we work under Assumption \[periodic\_assump\], that is, *we work with a translation-invariant model on a cube in the $d$-dimensional lattice*, while noting that our conclusions are valid for more general interactions with similar symmetries. The arguments in this section borrow heavily from [@abf]. As in Theorem \[three\_ineq\_lem\], the main novelty in the proof concerns connectivity in the ‘vertical’ direction (the term $R_v$ in – below).
By Theorem \[rcr\_thm\], $$M=\frac{1}{Z}E(\partial\psi_1^{0{{\Gamma}}})
=\frac{1}{Z^3}E(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}).$$ We shall consider the backbone $\xi=\xi(\psi_1^{0{{\Gamma}}})$ and the open cluster $C_{{\Gamma}}$ of ${{\Gamma}}$ in $(\psi_2^{\varnothing},\psi_3^{\varnothing},{\Delta})$. All connectivities will refer to the triple $(\psi_2^{\varnothing},\psi_3^{\varnothing},{\Delta})$. Note that $\xi$ consists of a single path with endpoints $0$ and ${{\Gamma}}$. There are four possibilities, illustrated in Figure \[decomp\_fig\], for the way in which $\xi$, viewed as a directed path from $0$ to ${{\Gamma}}$, interacts with $C_{{\Gamma}}$:
$\xi\cap C_{{\Gamma}}$ is empty,
$0 \in \xi\cap C_{{\Gamma}}$,
$0 \notin \xi\cap C_{{\Gamma}}$, and $\xi$ first meets $C_{{\Gamma}}$ immediately after a bridge,
$0 \notin \xi\cap C_{{\Gamma}}$, and $\xi$ first meets $C_{{\Gamma}}$ at a cut, which necessarily belongs to ${\mathrm{ev}}(\psi_2^{\varnothing})\cap{\mathrm{ev}}(\psi_3^{\varnothing})$.
Thus, $$M=T+R_0+R_h+R_v,
\label{ihp19}$$ where $$\label{ihp20}
\begin{split}
T&=\frac{1}{Z^3}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}\bigr),\\
R_0&=\frac{1}{Z^3}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\leftrightarrow}{{\Gamma}}\}\bigr),\\
R_h&=\frac{1}{Z^3}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{\mbox{first point on $\xi\cap C_{{\Gamma}}$ is a bridge of $\xi$}\}\bigr),\\
R_v&=\frac{1}{Z^3}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{\mbox{first point on $\xi\cap C_{{\Gamma}}$ is a cut}\}\bigr).
\end{split}$$ We will bound each of these terms separately.
![Illustrations of the four possibilities for $\xi\cap C_{{\Gamma}}$. Ghost-bonds in $\psi^{0{{\Gamma}}}$ are labelled $g$. The backbone $\xi$ is drawn as a solid black line, and $C_{{\Gamma}}$ as a grey rectangle.[]{data-label="decomp_fig"}](thesis.19 "fig:") ![Illustrations of the four possibilities for $\xi\cap C_{{\Gamma}}$. Ghost-bonds in $\psi^{0{{\Gamma}}}$ are labelled $g$. The backbone $\xi$ is drawn as a solid black line, and $C_{{\Gamma}}$ as a grey rectangle.[]{data-label="decomp_fig"}](thesis.20 "fig:") ![Illustrations of the four possibilities for $\xi\cap C_{{\Gamma}}$. Ghost-bonds in $\psi^{0{{\Gamma}}}$ are labelled $g$. The backbone $\xi$ is drawn as a solid black line, and $C_{{\Gamma}}$ as a grey rectangle.[]{data-label="decomp_fig"}](thesis.21 "fig:") ![Illustrations of the four possibilities for $\xi\cap C_{{\Gamma}}$. Ghost-bonds in $\psi^{0{{\Gamma}}}$ are labelled $g$. The backbone $\xi$ is drawn as a solid black line, and $C_{{\Gamma}}$ as a grey rectangle.[]{data-label="decomp_fig"}](thesis.22 "fig:")
By the switching lemma, $$\begin{aligned}
R_0&=\frac{1}{Z^3}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\leftrightarrow}{{\Gamma}}\}\bigr)\label{ihp21}\\
&=\frac{1}{Z^3}E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{0{{\Gamma}}}
\partial\psi_3^{0{{\Gamma}}}\bigr)=M^3.
\nonumber\end{aligned}$$
Next, we bound $T$. The letter $\xi$ will always denote the backbone of the first colouring $\psi_1$, with corresponding sources. Let $X$ denote the location of the ghost-bond that ends $\xi$. By conditioning on $X$, $$\begin{split}
T&=\frac{1}{Z^3}\int P(X\in dx)\,E\bigl(\partial\psi_1^{0{{\Gamma}}}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}{\,\big|\,}X=x\bigr)\\
&\leq \frac{{\gamma}}{Z^3}\int dx\, E\bigl(\partial\psi_1^{0x}\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}\bigr).
\end{split}
\label{o51}$$ We study the last expectation by conditioning on $C_{{\Gamma}}$ and bringing one of the factors $1/Z$ inside. By – and conditional expectation, $$\begin{aligned}
&\frac{1}{Z}E\bigl(\partial\psi_1^{0x}\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}{\,\big|\,}C_{{\Gamma}}\bigr)\label{o23}\\
&\hskip3cm =E\Bigl( Z^{-1}E({\partial}\psi_1^{0x} \mid \xi, C_{{\Gamma}}){\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\} {\,\Big|\,}C_{{\Gamma}}\Bigr)
\nonumber\\
&\hskip3cm =E\bigl({w}^{0x}(\xi)\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}{\,\big|\,}C_{{\Gamma}}\bigr).
\nonumber\end{aligned}$$ By Lemma \[rw\_mon\_lem\], $${w}^{0x}(\xi) \le 2^{r(\xi)-r'(\xi)}
{w}_{K\setminus C_{{\Gamma}}}^{0x}(\xi)\quad\mbox{on}\quad \{\xi\cap C_{{\Gamma}}= {\varnothing}\},
\label{o22}$$ where $$r(\xi) = r(\xi,K),\qquad
r'(\xi) = r(\xi,K{\setminus}C_{{\Gamma}}).$$ Using and , we have $$\begin{aligned}
&E\bigl({w}^{0x}(\xi)
\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}{\,\big|\,}C_{{\Gamma}}\bigr)\label{o24} \\
&\hskip3cm \leq E\bigl( 2^{r(\xi)-r'(\xi)}{w}_{K\setminus C_{{\Gamma}}}^{0x} (\xi)\cdot
{\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}{\,\big|\,}C_{{\Gamma}}\bigr)\nonumber \\
&\hskip3cm \le{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}.
\nonumber\end{aligned}$$
The last step merits explanation. Recall that $\xi=\xi(\psi_1^{0x})$, and assume $\xi\cap C_{{\Gamma}}={\varnothing}$. Apart from the randomization that takes place when $\psi_1^{0x}$ is one of several valid colourings, the law of $\xi$, $P(\xi\in d\nu)$, is a function of the positions of bridges and ghost-bonds along $\nu$ only, that is, the existence of bridges where needed, and the non-existence of ghost-bonds along $\nu$. By and Lemma \[rw\_mon\_lem\], with $\Xi_{K{\setminus}C} := \{\nu\in \Xi: \nu\cap C = {\varnothing}\}$ and $P$ the law of $\xi$, $$\begin{aligned}
&E\bigl({w}^{0x}(\xi)\cdot {\hbox{\rm 1\kern-.27em I}}\{\xi\cap C_{{\Gamma}}={\varnothing}\}{\,\big|\,}C_{{\Gamma}}\bigr)\\
&\hskip3cm =\int_{\Xi_{K{\setminus}C_{{\Gamma}}}} w^{0x}(\nu)\, P(d\nu)\\
&\hskip3cm \le \int_{\Xi_{K{\setminus}C_{{\Gamma}}}}2^{r(\nu)-r'(\nu)} w^{0x}_{K{\setminus}C_{{\Gamma}}}(\nu)
\left(\tfrac12\right)^{r(\nu)} \mu(d\nu) \end{aligned}$$ for some measure $\mu$, where the factor $(\frac12)^{r(\nu)}$ arises from the possible existence of more than one valid colouring. Now, $\mu$ is a measure on paths which by the remark above depends only locally on $\nu$, in the sense that $\mu(d\nu)$ depends only on the bridge- and ghost-bond configurations along $\nu$. In particular, the same measure $\mu$ governs also the law of the backbone in the *smaller* region $K\setminus C_{{\Gamma}}$. More explicitly, by with $P_{K{\setminus}C_{{\Gamma}}}$ the law of the backbone of the colouring $\psi_{K{\setminus}C_{{\Gamma}}}^{0x}$ defined on $K{\setminus}C_{{\Gamma}}$, we have $$\begin{aligned}
{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}
&=\int_{\Xi_{K{\setminus}C_{{\Gamma}}}} w^{0x}_{K{\setminus}C_{{\Gamma}}}(\nu)\, P_{K{\setminus}C_{{\Gamma}}}(d\nu)\\
&= \int_{\Xi_{K{\setminus}C_{{\Gamma}}}} w^{0x}_{K{\setminus}C_{{\Gamma}}}(\nu)
\left(\tfrac12\right)^{r'(\nu)}\, \mu(d\nu).\end{aligned}$$ Thus follows.
Therefore, by –, $$\begin{aligned}
T&\leq \frac{{\gamma}}{Z^2}\int dx\: E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}\cdot
{\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr)\\
&={\gamma}\int dx\: \frac{1}{Z^2}E\bigl(\partial\psi_2^{0x}
\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr)\nonumber\\
&={\gamma}\frac{\partial M}{\partial {\gamma}},
\nonumber\end{aligned}$$ by ‘conditioning on the cluster’ $C_{{\Gamma}}$ and Theorem \[three\_ineq\_lem\].
Next, we bound $R_h$. Suppose that the bridge bringing $\xi$ into $C_{{\Gamma}}$ has endpoints $X$ and $Y$, where we take $X$ to be the endpoint not in $C_{{\Gamma}}$. When the bridge $XY$ is removed, the backbone $\xi$ consists of two paths: $\zeta^1:0\rightarrow X$ and $\zeta^2:Y\rightarrow {{\Gamma}}$. Therefore, $$\begin{aligned}
R_h&= \frac{1}{Z^3}\int P(X\in dx)\,E\bigl(\partial\psi_1^{0{{\Gamma}}}
\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}{\,\big|\,}X=x\bigr)\\
&\le\frac{{\lambda}}{Z^3}\int dx \, \sum_{y\sim x}E\bigl({\partial}\psi_1^{0xy{{\Gamma}}}
\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot{\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}},\,y{\leftrightarrow}{{\Gamma}}\}\cdot {\hbox{\rm 1\kern-.27em I}}\{J_\xi\}\bigr),
\nonumber\end{aligned}$$ where $\xi=\xi(\psi_1^{0xy{{\Gamma}}})$ and $$J_\xi=\bigl\{\xi=\zeta^1\circ\zeta^2,
\,\zeta^1:0\rightarrow x,\,\zeta^2:y\rightarrow {{\Gamma}},\,
\zeta^1\cap C_{{\Gamma}}={\varnothing}\bigr\}.$$ As in , $$R_h
\leq \frac{{\lambda}}{Z^2}\int dx\:\sum_{y\sim x}
E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}},\, y{\leftrightarrow}{{\Gamma}}\}\cdot
{w}^{0xy{{\Gamma}}}(\xi)\cdot{\hbox{\rm 1\kern-.27em I}}\{J_\xi\}\bigr).
\label{ihp23}$$
By Lemmas \[backb2\](a) and \[rw\_mon\_lem\], on the event $J_\xi$, $$\begin{aligned}
{w}^{0xy{{\Gamma}}}(\xi) &= {w}^{0x}(\zeta^1) {w}^{y{{\Gamma}}}_{K{\setminus}\zeta^1}(\zeta^2)\\
&\le 2^{r-r'}{w}^{0x}_{K{\setminus}C_{{\Gamma}}}(\zeta^1){w}^{y{{\Gamma}}}_{K{\setminus}\zeta^1}(\zeta^2),\end{aligned}$$ where $r = r(\zeta^1,K)$ and $r'= r(\zeta^1, K{\setminus}C_{{\Gamma}})$. By Lemma \[cor\_mon\_lem\] and the reasoning after , $$\begin{aligned}
E\bigl({w}^{0xy{{\Gamma}}}(\xi) \cdot {\hbox{\rm 1\kern-.27em I}}\{J_\xi\}{\,\big|\,}\zeta^1, C_{{\Gamma}}\bigr)
&\leq 2^{r-r'}
{w}_{K\setminus C_{{\Gamma}}}^{0x}(\zeta^1)\cdot {\langle}{\sigma}_y{\rangle}_{K{\setminus}\zeta^1}\\
&\leq M\cdot 2^{r-r'} {w}_{K\setminus C_{{\Gamma}}}^{0x}(\zeta^1),\end{aligned}$$ so that, similarly, $$E\bigl({w}^{0xy{{\Gamma}}}(\xi)\cdot {\hbox{\rm 1\kern-.27em I}}\{J_\xi\}{\,\big|\,}C_{{\Gamma}}\bigr) \le
M \cdot {\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}.
\label{o25}$$ We substitute into the summand in , using the switching lemma, conditioning on the cluster $C_{{\Gamma}}$, and the bound ${\langle}{\sigma}_y{\rangle}_{C_{{\Gamma}}}\leq M$, to obtain the upper bound $$\begin{aligned}
&M\cdot E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot
{\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}},\, y{\leftrightarrow}{{\Gamma}}\}\cdot
{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}\bigr)\\
&\hskip2.5cm =M\cdot E\bigl(\partial\psi_2^{y{{\Gamma}}}\partial\psi_3^{y{{\Gamma}}}
\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\cdot {\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}\bigr)\nonumber\\
&\hskip2.5cm =M\cdot E\bigl(\partial\psi_2^{0xy{{\Gamma}}}\partial\psi_3^{\varnothing}{\langle}{\sigma}_y{\rangle}_{C_{{\Gamma}}} \cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr)\nonumber\\
&\hskip2.5cm \leq M^2\cdot E\bigl(\partial\psi_2^{0xy{{\Gamma}}}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr).
\nonumber\end{aligned}$$ Hence, by , $$\begin{aligned}
R_h &\leq {\lambda}M^2\frac{1}{Z^2}\int dx\,\sum_{y\sim x}
E\bigl(\partial\psi_2^{0xy{{\Gamma}}}\partial\psi_3^{\varnothing}{\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}}\}\bigr)\\
&=2{\lambda}M^2 \frac{\partial M}{\partial {\lambda}}.\end{aligned}$$
Finally, we bound $R_v$. Let $X\in{\Delta}\cap{\mathrm{ev}}(\psi_2^{\varnothing})\cap{\mathrm{ev}}(\psi_3^{\varnothing})$ be the first point of $\xi$ in $C_{{\Gamma}}$. In a manner similar to that used for $R_h$ at above, and by cutting the backbone $\xi$ at the point $x$, $$R_v\le\frac{1}{Z^2}\int P(X\in dx)\,
E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}},\,x{\leftrightarrow}{{\Gamma}}\}\cdot
{w}^{0{{\Gamma}}}(\xi)\cdot {\hbox{\rm 1\kern-.27em I}}\{J_\xi\}\bigr),
\label{ihp22}$$ where $$J_\xi= 1\bigl\{\xi={\overline}\zeta^1\circ{\overline}\zeta^2,\, {\overline}\zeta^1:0\rightarrow x,\,{\overline}\zeta^2:x\rightarrow {{\Gamma}},
\, \zeta^1\cap C_{{\Gamma}}={\varnothing}\bigr\}.$$ As in , $$\begin{aligned}
E({w}^{0{{\Gamma}}}(\xi)\cdot {\hbox{\rm 1\kern-.27em I}}\{J_\xi\} \mid C_{{\Gamma}})
&= E\bigl(E({w}^{0{{\Gamma}}}(\xi)\cdot {\hbox{\rm 1\kern-.27em I}}\{J_\xi\}\mid {\overline}\zeta^1,C_{{\Gamma}}){\,\big|\,}C_{{\Gamma}}\bigr)\\
&\leq E\bigl({\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}\cdot {\langle}{\sigma}_x{\rangle}_{K\setminus\zeta^1}{\,\big|\,}C_{{\Gamma}}\bigr)\\
&\leq {\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}\cdot M.\end{aligned}$$ By therefore, $$R_v\leq M\frac{1}{Z^2}\int P(X\in dx)\,
E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0{\nleftrightarrow}{{\Gamma}},\,x{\leftrightarrow}{{\Gamma}}\}
{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C_{{\Gamma}}}\bigr).$$ By removing the cut at $x$, the origin $0$ becomes connected to ${{\Gamma}}$, but only via $x$. Thus, $$R_v\leq 4{\delta}M\frac{1}{Z^2}\int dx\:
E\bigl(\partial\psi_2^{\varnothing}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}},\,x{\leftrightarrow}{{\Gamma}}\}
{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C^x_{{\Gamma}}}\bigr),$$ where $C^x_{{\Gamma}}$ is the set of points reached from ${{\Gamma}}$ along open paths not containing $x$. By the switching lemma, and conditioning twice on the cluster $C_{{\Gamma}}^x$, $$\begin{aligned}
R_v&\leq4{\delta}M\frac{1}{Z^2}\int dx\:
E\bigl(\partial\psi_2^{x{{\Gamma}}}\partial\psi_3^{x{{\Gamma}}}
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}
{\langle}{\sigma}_0{\sigma}_x{\rangle}_{K\setminus C^x_{{\Gamma}}}\bigr)\\
&=4{\delta}M\frac{1}{Z^2}\int dx\, E\bigl(\partial\psi_2^{0{{\Gamma}}}\partial\psi_3^{x{{\Gamma}}}
\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)\\
&=4{\delta}M\frac{1}{Z^2}\int dx\, E\bigl(\partial\psi_2^{0{{\Gamma}}}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}{\langle}{\sigma}_x{\rangle}_{C^x_{{\Gamma}}}\bigr)\\
&\leq 4{\delta}M^2\frac{1}{Z^2}\int dx\,
E\bigl(\partial\psi_2^{0{{\Gamma}}}\partial\psi_3^{\varnothing}\cdot {\hbox{\rm 1\kern-.27em I}}\{0\overset{x}{{\leftrightarrow}} {{\Gamma}}\}\bigr)\\
&=-2{\delta}M^2 \frac{\partial M}{\partial {\delta}},\end{aligned}$$ by , as required.
Consequences of the inequalities {#cons_sec}
--------------------------------
In this section we formulate the principal results of this chapter, and show how the differential inequalities of Theorems \[main\_pdi\_thm\] and \[three\_ineq\_lem\] may be used to prove them. We will rely in this section on the results in Section \[inf\_potts\_sec\], and we work under Assumption \[periodic\_assump\], unless otherwise stated. It is sometimes inconvenient to use periodic boundary conditions, and we revert to the free condition where necessary.
We shall consider the infinite-volume limit as $L \uparrow {\mathbb{Z}}^d$; the ground state is obtained by letting ${\beta}\to{\infty}$ also. Let $n$ be a positive integer, and set $L_n = [-n,n]^d$ with periodic boundary condition. Let ${\Lambda}_n^{\beta}:= [-n,n]^d \times[-\frac12 {\beta}, \frac12{\beta}]$. The symbol ${\beta}$ will appear as superscript in the following; the superscript ${\infty}$ is to be interpreted as the ground state. Let $0=(0,0)$ and $$M^{\beta}_{n}({\lambda},{\delta},{\gamma}) ={\langle}{\sigma}_0{\rangle}_{L_{n}}^{\beta}={\langle}{\sigma}_0{\rangle}_{{\Lambda}^{\beta}_{n}}$$ be the magnetization in ${\Lambda}_{n}^{\beta}$, noting that $M_n^{\beta}\equiv 0$ when ${\gamma}=0$.
We have from the results in Section \[press\_sec\] that the limits $$\label{m_lim_eq}
M^{\beta}:= \lim_{n\to{\infty}} M^{\beta}_n,\quad
M^{\infty}:= \lim_{n,{\beta}\to{\infty}} M^{\beta}_n,$$ exist for all ${\gamma}\in{\mathbb{R}}$ (where, in the second limit, ${\beta}={\beta}_n$ is comparable to $n$ in the sense that Assumption \[vanhove\] holds). Note that $M^{\beta}({\lambda},{\delta},0)=0$ for ${\beta}\in(0,{\infty}]$. Recall that we set ${\delta}=1$, $\rho={\lambda}/{\delta}$, and write $$M^{\beta}(\rho,{\gamma})= M^{\beta}(\rho,1,{\gamma}),\qquad {\beta}\in(0,{\infty}],$$ with a similar notation for other functions.
Recall the following facts. From Theorem \[ising\_summary\_thm\] there is a unique infinite-volume state ${\langle}\cdot{\rangle}^{\beta}$ at every ${\gamma}>0$. Letting ${\langle}\cdot {\rangle}^{\beta}_+$ be the limiting state as ${\gamma}\downarrow 0$, there is a unique state at $(\rho,0)$ if and only if $$M^{\beta}_+(0):={\langle}{\sigma}_0{\rangle}^{\beta}_+=0.$$ From the state ${\langle}\cdot{\rangle}^{\beta}_+$ may alternatively be obtained as the infinite volume limit of the $+$ boundary states taken with ${\gamma}=0$. The critical value $${\rho_{\mathrm{c}}}^{\beta}:=\inf\{\rho>0:M^{\beta}_+(\rho)>0\},
\label{crit_val_defs_eq}$$ see also and . We shall have need later for the infinite-volume limit ${\langle}\cdot{\rangle}^{{\mathrm{f}},{\beta}}$, as $n\to{\infty}$, with *free* boundary condition in the ${\mathbb{Z}}^d$ direction (or in both directions, if ${\beta}\rightarrow{\infty}$). This limit exists by Theorem \[potts\_lim\_thm\]. Note from Theorem \[ising\_summary\_thm\] that $${\langle}\cdot{\rangle}^{{\mathrm{f}},{\beta}}_{{\gamma}=0} = {\langle}\cdot{\rangle}^{\beta}_{{\gamma}=0}={\langle}\cdot {\rangle}^{\beta}_+
\quad \text{if} \quad M^{\beta}_+(\rho)=0.
\label{o40}$$ The superscript ‘f’ shall always indicate the free boundary condition.
For ${\beta}\in(0,{\infty}]$, let $\phi_\rho^{b,{\beta}}$, $b\in\{{\mathrm{f}},{\mathrm{w}}\}$, be the $q=2$ [random-cluster]{} measures of Theorem \[conv\_lem\], with ${\gamma}=0$. By Theorem \[correlation\], these measures are non-decreasing in $\rho$, and, as we saw in , $$\label{mel61}
{\phi_{\rho}}^{{\mathrm{w}},{\beta}} \le {\phi_{\rho'}}^{{\mathrm{f}},{\beta}}, \qquad\text{when }0\leq\rho<\rho'.$$ As in Remark \[inf\_vol\_rk\], for ${\beta}\in(0,{\infty}]$, $${\phi_{\rho}}^{{\mathrm{w}},{\beta}}(x{\leftrightarrow}y) = {\langle}{\sigma}_x{\sigma}_y{\rangle}^{\beta}_+,
\quad {\phi_{\rho}}^{{\mathrm{w}},{\beta}}(0{\leftrightarrow}{\infty}) =M_+(\rho).
\label{mel60}$$ By , the [[[fkg]{}]{}]{} inequality (Theorem \[rc\_fkg\]), and the uniqueness of the unbounded cluster (Theorem \[inf\_clust\_uniq\]), $${\langle}{\sigma}_x {\sigma}_y{\rangle}^{\beta}_+ \ge {\phi_{\rho}}^{{\mathrm{w}},{\beta}}(x{\leftrightarrow}{\infty})
{\phi_{\rho}}^{{\mathrm{w}},{\beta}}(y{\leftrightarrow}{\infty})= M^{\beta}_+(\rho)^2.
\label{o50}$$
Let ${\beta}\in(0,{\infty})$. Using the concavity of $M^{\beta}$ implied by Lemma \[ghs\_lem\], as well as the properties of convex functions in Proposition \[conv\_prop\], the derivative ${\partial}M^{\beta}/{\partial}{\gamma}$ exists for all ${\gamma}\in{\mathcal{C}}\subseteq(0,\infty)$, where ${\mathcal{C}}$ is a set whose complement has measure zero. When ${\gamma}\in{\mathcal{C}}$, $$\label{chi_lim_eq}
\chi^{\beta}_n(\rho,{\gamma}):=\frac{\partial M^{\beta}_n}{\partial{\gamma}}\rightarrow
\chi^{\beta}(\rho,{\gamma}):=\frac{\partial M^{\beta}}{\partial{\gamma}}< \infty.$$The corresponding conclusion holds also as $n,{\beta}\to{\infty}$. Furthermore, by the [[ghs]{}]{}-inequality, Lemma \[ghs\_lem\], $\chi^{\beta}$ is decreasing in ${\gamma}\in{\mathcal{C}}$, which implies that the limits $$\chi^{\beta}_+(\rho) := \lim_{{\gamma}\downarrow0} \chi^{\beta}(\rho,{\gamma}), \qquad {\beta}\in(0,{\infty}].$$ exist when taken along sequences in ${\mathcal{C}}$.
The limit $$\begin{aligned}
\chi^{{\mathrm{f}},{\beta}}(\rho,0) &:=
\lim_{n\to{\infty}}\left(\left.\frac{{\partial}M^{{\mathrm{f}},{\beta}}_n}{{\partial}\gamma}\right|_{{\gamma}=0}\right)
\label{mel65}\\
&=\lim_{n\to{\infty}}\int_{{\Lambda}_n^{\beta}} {\langle}{\sigma}_0{\sigma}_x{\rangle}_{n,{\gamma}=0}^{{\mathrm{f}},{\beta}}\,dx
=\int{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}}_{{\gamma}=0}\,dx
\nonumber\end{aligned}$$ exists by monotone convergence, see Lemma \[cor\_mon\_lem\]. Let $${\rho_{\mathrm{s}}}^{\beta}:=\inf\{\rho>0:\chi^{{\mathrm{f}},{\beta}}(\rho,0)=\infty\},\qquad {\beta}\in(0,{\infty}].
\label{o27}$$ We shall see in Theorem \[0mass\] that $\chi^{{\mathrm{f}},{\beta}}({\rho_{\mathrm{s}}}^{\beta},0)={\infty}$.
It will be useful later to note that $$\label{i2}
\chi_+^{\beta}(\rho)\geq\chi^{{\mathrm{f}},{\beta}}(\rho,0)\quad\text{whenever }
M^{\beta}_+(\rho)=0,\qquad {\beta}\in(0,{\infty}].$$ To see this, let ${\gamma}\in{\mathcal{C}}$ and first note from Fatou’s lemma that $$\chi^{\beta}(\rho,{\gamma})\geq\int {\langle}{\sigma}_0;{\sigma}_x{\rangle}^{\beta}_{\gamma}\, dx,$$ where we have written ${\langle}\cdot{\rangle}^{\beta}_{\gamma}$ for the unique state at ${\gamma}$. Hence, using also the monotone convergence theorem and the [[ghs]{}]{}-inequality, $$\chi^{\beta}_+(\rho)=\lim_{\substack{{\gamma}\downarrow 0\\{\gamma}\in{\mathcal{C}}}}
\chi^{\beta}(\rho,{\gamma})\geq\lim_{\substack{{\gamma}\downarrow 0\\{\gamma}\in{\mathcal{C}}}}
\int{\langle}{\sigma}_0;{\sigma}_x{\rangle}^{\beta}_{\gamma}\, dx=\int{\langle}{\sigma}_0;{\sigma}_x{\rangle}^{\beta}_+\, dx.$$ When $M_+(0)=0$ there is a unique state at ${\gamma}=0$, so that ${\langle}{\sigma}_0;{\sigma}_x{\rangle}^{\beta}_+={\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}}_{{\gamma}=0}$ which by gives . It will follow in particular from Theorem \[0mass\] that $\chi^{\beta}_+(\rho^{\beta}_{\mathrm{s}})={\infty}$. Of course, similar arguments are valid for the limit $n,{\beta}\rightarrow{\infty}$.
By and Lemma \[cor\_mon\_lem\] we have that $\chi^{{\mathrm{f}},{\beta}}(\rho,0)$ is increasing in $\rho$. We claim that $${\rho_{\mathrm{s}}}^{\beta}\le {\rho_{\mathrm{c}}}^{\beta};
\label{mel64}$$ it will follow that there is a unique equilibrium state when ${\gamma}=0$ and $\rho<{\rho_{\mathrm{s}}}^{\beta}$. First note that, by and , if $\rho<\rho'<{\rho_{\mathrm{s}}}^{\beta}$ then $$M_+(\rho)=\phi^{{\mathrm{w}},{\beta}}_\rho(0\leftrightarrow{\infty})
\leq\phi^{{\mathrm{f}},{\beta}}_{\rho'}(0\leftrightarrow{\infty}),$$ so it suffices to show that $\phi^{{\mathrm{f}},{\beta}}_{\rho}(0\leftrightarrow{\infty})=0$ if $\rho<{\rho_{\mathrm{s}}}^{\beta}$. To see this, note that if $\phi^{{\mathrm{f}},{\beta}}_{\rho}(0\leftrightarrow{\infty})>0$ then certainly $$\chi^{{\mathrm{f}},{\beta}}(\rho,0)=
\int_{{\mathbb{Z}}^d\times[-\frac12{\beta},\frac12{\beta}]}
{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}} \;dx=
\phi^{\mathrm{f}}_\rho(|C_0|)=\infty,$$ where $C_0$ denotes the cluster at the origin, and $|\cdot|$ denotes Lebesgue measure.
For $x\in{\mathbb{Z}}^d\times{\mathbb{R}}$, let $\|x\|$ denote the supremum norm of $x$.
\[exp\_decay\_cor\] Let ${\beta}\in(0,{\infty}]$ and $\rho <{\rho_{\mathrm{s}}}^{\beta}$. There exists ${\alpha}={\alpha}(\rho)>0$ such that $${\langle}{\sigma}_0{\sigma}_x{\rangle}^{\beta}\leq e^{-{\alpha}\|x\|},\qquad x\in{\mathbb{Z}}^d\times{\mathbb{R}}.$$
Fix ${\beta}\in(0,{\infty})$ and ${\gamma}=0$, and let $\rho<{\rho_{\mathrm{s}}}^{\beta}$, so that applies. By the uniqueness of the equilibrium state, we have that $$\chi^{{\mathrm{f}},{\beta}}(\rho,0)=
\int_{{\mathbb{Z}}^d\times[-\frac12{\beta},\frac12{\beta}]}
{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\beta}} \;dx=
\sum_{k\geq1}\int_{C_k^{\beta}}{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\beta}} \;dx,$$ where $C_k^{\beta}:={\Lambda}^{\beta}_k\setminus{\Lambda}^{\beta}_{k-1}$. Since $\rho<{\rho_{\mathrm{s}}}^{\beta}$, the last summation converges, whence, for sufficiently large $k$, $$\label{exp_cond_eq}
\int_{C_k^{\beta}}{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\beta}} \, dx<e^{-8}.$$ The result follows from the the Simon inequality, Lemma \[simon\_lem\], with the 1-fat separating sets $C_k^{\beta}$ using standard arguments (see [@grimmett_rcm Corollary 9.38] for more details on the method). A similar argument holds when ${\beta}={\infty}$.
Let ${\beta}\in(0,{\infty}]$, ${\gamma}=0$ and define the *mass* $$m^{\beta}(\rho):=\liminf_{\|x\|\rightarrow\infty}
\left(-\frac{1}{\|x\|}\log{\langle}{\sigma}_0{\sigma}_x{\rangle}^{\beta}_\rho\right)$$ By Theorem \[exp\_decay\_cor\] and , $$m^{\beta}(\rho)
\begin{cases} >0 &\text{if }\rho<{\rho_{\mathrm{s}}}^{\beta},\\
=0 &\text{if } \rho>{\rho_{\mathrm{c}}}^{\beta}.
\end{cases}
\label{mel66}$$
\[0mass\] Except when $d=1$ and ${\beta}<{\infty}$, $m^{\beta}({\rho_{\mathrm{s}}}^{\beta})=0$ and $\chi^{{\mathrm{f}},{\beta}}({\rho_{\mathrm{s}}}^{\beta},0)={\infty}$.
Let $d \ge 2$, ${\gamma}=0$, and fix ${\beta}\in(0,{\infty})$. We use the Lieb inequality, Lemma \[lieb\_lem\], and the argument of [@lieb80; @simon80], see also [@grimmett_rcm Corollary 9.46]. It is necessary and sufficient for $m^{\beta}(\rho)>0$ that $$\label{exp_cond_eq_2}
\int_{C_n^{\beta}}{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}}_{n,\rho}\, dx<e^{-8}\quad\mbox{for some }n.$$ Necessity holds because the integrand is no greater than ${\langle}{\sigma}_0{\sigma}_x{\rangle}^{\beta}$. Sufficiency follows from Lemma \[lieb\_lem\], as in the proof of Theorem \[exp\_decay\_cor\].
By , $$\begin{aligned}
\frac{\partial}{\partial\rho}{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}}_{n,\rho}
&=
\frac{1}{2} \int_{{\Lambda}_n^{\beta}}dy\,\sum_{z\sim y}{\langle}{\sigma}_0{\sigma}_x;{\sigma}_y{\sigma}_z{\rangle}^{{\mathrm{f}},{\beta}}_{n,\rho}\\
&\leq d{\beta}(2n+1)^d.\end{aligned}$$ Therefore, if $\rho'>\rho$, $$\int_{C_n^{\beta}}{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}}_{n,\rho'}\,dx\leq d[{\beta}(2n+1)^d]^2(\rho'-\rho)
+\int_{C_n^{\beta}}{\langle}{\sigma}_0{\sigma}_x{\rangle}^{{\mathrm{f}},{\beta}}_{n,\rho}\,dx.$$ Hence, if holds for some $\rho$, then it holds for $\rho'$ when $\rho'-\rho>0$ is sufficiently small.
Suppose $m^{\beta}({\rho_{\mathrm{s}}}^{\beta})>0$. Then $m^{\beta}(\rho')>0$ for some $\rho'>{\rho_{\mathrm{s}}}^{\beta}$, which contradicts $\chi^{{\mathrm{f}},{\beta}}(\rho',0)={\infty}$, and the first claim of the theorem follows. A similar argument holds when $d=1$ and ${\beta}={\infty}$. The second claim follows similarly: if $\chi^{{\mathrm{f}},{\beta}}({\rho_{\mathrm{s}}}^{\beta},0)<{\infty}$, then holds with $\rho={\rho_{\mathrm{s}}}^{\beta}$, whence $m^{\beta}(\rho')>0$ and $\chi^{{\mathrm{f}},{\beta}}(\rho',0)<{\infty}$ for some $\rho'>{\rho_{\mathrm{s}}}^{\beta}$, a contradiction. (See also [@aizenman_tree-graph].)
We are now ready to state the main results. We will adapt the arguments of [@ab Lemmas 4.1, 5.1] (see also [@abf; @grimmett_perc]) to prove the following.
\[ab\_thm\] There are constants $c_1$, $c_2>0$ such that, for ${\beta}\in(0,{\infty}]$, $$\begin{aligned}
\label{ab_1_eq}
M^{\beta}({\rho_{\mathrm{s}}},{\gamma})&\geq c_1{\gamma}^{1/3},\\
\label{ab_2_eq}
M^{\beta}_+(\rho,0)&\geq c_2(\rho-{\rho_{\mathrm{s}}}^{\beta})^{1/2},\end{aligned}$$ for small, positive ${\gamma}$ and $\rho-{\rho_{\mathrm{s}}}^{\beta}$, [respectively]{}.
This is vacuous when $d=1$ and ${\beta}<{\infty}$; see . The exponents in the above inequalities are presumably sharp in the corresponding mean-field model (see [@abf; @af] and Remark \[remark\_mf\]). It is standard that a number of important results follow from Theorem \[ab\_thm\], of which we state the following here.
\[eq\_cor\] For $d \ge 1$ and ${\beta}\in(0,{\infty}]$, we have that ${\rho_{\mathrm{c}}}^{\beta}={\rho_{\mathrm{s}}}^{\beta}$.
Except when $d=1$ and ${\beta}<{\infty}$, this is immediate from and . In the remaining case, ${\rho_{\mathrm{c}}}^{\beta}={\rho_{\mathrm{s}}}^{\beta}={\infty}$.
We will describe the case when ${\beta}<{\infty}$ is fixed; the ground state case is proved by a similar method. The argument is based on [@ab].
We start by proving . If $M^{\beta}_+(\rho_{\mathrm{s}},0)>0$ there is nothing to prove, so we assume that $M^{\beta}_+(\rho_{\mathrm{s}},0)=0$. The inequalities of Theorems \[three\_ineq\_lem\] and \[main\_pdi\_thm\] may be combined to obtain $$\label{i1}
M^{\beta}_n\leq (M^{\beta}_n)^3+\chi^{\beta}_n\cdot
\left({\gamma}+4d{\lambda}(M^{\beta}_n)^3+4{\delta}\frac{(M^{\beta}_n)^3}{1-(M^{\beta}_n)^2}\right).$$ Set ${\delta}=1$ and $\rho=\rho^{\beta}_{\mathrm{s}}$, and write $f_n({\gamma})=2M^{\beta}_n(\rho^{\beta}_{\mathrm{s}},{\gamma})$. Recall that the sequence $f_n({\gamma})$ converges as $n\rightarrow{\infty}$ to some $f({\gamma})$ for all ${\gamma}\geq0$, and that the derivatives $f_n'=2\chi^{\beta}_n$ converge for ${\gamma}\in{\mathcal{C}}$ to some $g({\gamma})$ which is decreasing in ${\gamma}$. Moreover, from the discussion around and the assumption that $M^{\beta}_+(\rho_{\mathrm{s}},0)=0$ it follows that $$\lim_{\substack{{\gamma}\downarrow 0\\{\gamma}\in{\mathcal{C}}}} g({\gamma})={\infty}.$$
Multiplying through by $1-(M^{\beta}_n)^2$ and discarding non-positive terms on the right hand side, we may deduce from that the functions $f_n$ satisfy the inequality $$f_n({\gamma})\leq{\gamma}\cdot f'_n({\gamma})+a\cdot f'_n({\gamma})f_n({\gamma})^3
+f_n({\gamma})^3,\qquad {\gamma}\geq0,$$ where $a>0$ is an appropriate constant depending on ${\lambda}$ and $d$ only. For ${\gamma}>0$ we may rewrite this as $$\frac{1}{f'_n({\gamma})}\frac{d}{d{\gamma}}\Big[\frac{{\gamma}}{f_n({\gamma})}\Big]\leq
f'_n({\gamma})\Big(a+\frac{1}{f'_n({\gamma})}\Big).$$ Letting ${\gamma}>{\varepsilon}>0$ and integrating from ${\varepsilon}$ to ${\gamma}$ it follows that $$\frac{{\gamma}}{f_n({\gamma})}-\frac{{\varepsilon}}{f_n({\varepsilon})}\leq
\int_{\varepsilon}^{\gamma}f'_n(x)f_n(x)\Big(a+\frac{1}{f'_n(x)}\Big)\,dx.$$ Using of Theorem \[three\_ineq\_lem\], it follows on letting ${\varepsilon}\downarrow0$ that $$\label{i3}
\frac{{\gamma}}{f_n({\gamma})}-\frac{1}{f'_n(0)}\leq
\int_0^{\gamma}f'_n(x)f_n(x)\Big(a+\frac{1}{f'_n(x)}\Big)\,dx.$$
Now suppose that ${\gamma}>0$ lies in ${\mathcal{C}}$. If ${\gamma}$ is sufficiently small then $g({\gamma})\geq1.1$, and for such a ${\gamma}$ fixed we have for sufficiently large $n$ that $f'_n({\gamma})\geq1$. Since $f'_n$ is decreasing in ${\gamma}$ we may deduce from that $$\frac{{\gamma}}{f_n({\gamma})}-\frac{1}{f'_n(0)}\leq(a+1)
\int_0^{\gamma}f'_n(x)f_n(x)\,dx=\frac{a+1}{2}f_n({\gamma})^2$$ Letting $n\rightarrow{\infty}$ it follows that $$\frac{{\gamma}}{f({\gamma})}\leq\frac{a+1}{2}f({\gamma})^2$$ as required.
Let us now turn to . Note first that if $\rho={\lambda}/{\delta}$ then $$\frac{\partial M^{\beta}_n}{\partial{\lambda}}=\frac{1}{{\delta}}
\frac{\partial M^{\beta}_n}{\partial\rho}\quad\text{and}\quad
\frac{\partial M^{\beta}_n}{\partial{\delta}}=-\frac{{\lambda}}{{\delta}^2}
\frac{\partial M^{\beta}_n}{\partial\rho},$$ so that the inequality of Theorem \[main\_pdi\_thm\] may be rewritten as $$M^{\beta}_n\leq{\gamma}\frac{\partial M^{\beta}_n}{\partial{\gamma}}+(M^{\beta}_n)^3
+2\rho (M^{\beta}_n)^2\frac{\partial M^{\beta}_n}{\partial\rho}.$$ This may in turn be rewritten as $$\frac{\partial}{\partial{\gamma}}(\log M^{\beta}_n)+
\frac{1}{{\gamma}}\frac{\partial}{\partial\rho}
(\rho (M^{\beta}_n)^2-\rho)\geq0.$$ We wish to integrate this over the rectangle $[\rho^{\beta}_{\mathrm{s}},\rho']\times[{\gamma}_0,{\gamma}_1]$ for $\rho'>\rho^{\beta}_{\mathrm{s}}$ and ${\gamma}_1>{\gamma}_0>0$. Since $M^{\beta}_n$ is increasing in $\rho$ and in ${\gamma}$ we deduce, after discarding a term $-\rho_{\mathrm{s}}^{\beta}M^{\beta}_n(\rho^{\beta}_{\mathrm{s}},{\gamma})^2$, that $$\label{i4}
(\rho'-\rho^{\beta}_{\mathrm{s}})\log\Big(
\frac{M^{\beta}_n(\rho',{\gamma}_1)}{M^{\beta}_n(\rho^{\beta}_{\mathrm{s}},{\gamma}_0)}\Big)+
(\rho' M^{\beta}_n(\rho',{\gamma}_1)^2-\rho'+\rho^{\beta}_{\mathrm{s}})
\log\frac{{\gamma}_1}{{\gamma}_0}\geq0.$$ We may let $n\rightarrow{\infty}$ in , to deduce that the same inequality is valid with $M^{\beta}_n$ replaced by $M^{\beta}$. It follows from that $$\liminf_{{\gamma}_0\downarrow 0}
\frac{\log\Big(
\frac{M^{\beta}_n(\rho',{\gamma}_1)}{M^{\beta}_n(\rho^{\beta}_{\mathrm{s}},{\gamma}_0)}\Big)}{\log({\gamma}_1/{\gamma}_0)}
\leq \frac{1}{3}.$$ It follows that $$\frac{1}{3}(\rho'-\rho^{\beta}_{\mathrm{s}})+\rho' M^{\beta}(\rho',{\gamma}_1)
-(\rho'-\rho^{\beta}_{\mathrm{s}})\geq0,$$ which on letting ${\gamma}_1\downarrow0$ gives the result.
\[remark\_mf\] Let ${\beta}\in(0,{\infty}]$. Except when $d=1$ and ${\beta}<{\infty}$, one may conjecture the existence of exponents $a=a(d,{\beta})$, $b=b(d,{\beta})$ such that $$\begin{aligned}
{2}
M^{\beta}_+(\rho)&=(\rho-{\rho_{\mathrm{c}}}^{\beta})^{(1+{\mathrm{o}}(1))a }
\qquad&&\mbox{as }\rho\downarrow{\rho_{\mathrm{c}}}^{\beta},\\
M^{\beta}({\rho_{\mathrm{c}}}^{\beta},{\gamma})&={\gamma}^{(1+{\mathrm{o}}(1))/b}\qquad&&\mbox{as }{\gamma}\downarrow 0.\end{aligned}$$ Theorem \[ab\_thm\] would then imply that $a\leq \frac12$ and $b \geq 3$. In [@chayes_ioffe_curie-weiss Theorem 3.2] it is proved for the ground-state quantum Curie–Weiss, or mean-field, model that the corresponding $a =\frac12$. It may be conjectured that the values $a=\frac12$ and $b=3$ are attained for the space–time Ising model on ${\mathbb{Z}}^d\times [-\frac12 {\beta},\frac12{\beta}]$ for $d$ sufficiently large, as proved for the classical Ising model in [@af]. See also Section \[af\_sec\].
Finally, a note about . The [random-cluster]{} measure corresponding to the quantum Ising model is *periodic* in both ${\mathbb{Z}}^d$ and ${\beta}$ directions, and this complicates the infinite-volume limit. Since the periodic [random-cluster]{} measure dominates the free [random-cluster]{} measure, for ${\beta}\in(0,{\infty})$, as in and , $$\begin{aligned}
{2}
\liminf_{n\to{\infty}} \tau^{\beta}_{L_n}(u,v) &\ge
{\langle}{\sigma}_{(u,0)}{\sigma}_{(v,0)}{\rangle}_{+,\rho'}^{\beta}\qquad&&\text{for } \rho'<\rho\\
&\to M_+(\rho-)^2 \qquad &&\text{as } \rho'\uparrow \rho,\end{aligned}$$ and a similar argument holds in the ground state also.
Applications and extensions {#appl_ch}
===========================
> [*Summary.*]{} First we prove that the critical ratio for the ground state quantum Ising model on ${\mathbb{Z}}$ is $\rho^{\infty}_{\mathrm{c}}=2$; we then extend this result to more complicated (‘star-like’) graphs. Next we discuss the possible applications of ‘reflection positivity’ to strengthen the results of Chapter \[qim\_ch\] when $d\geq 3$, and conclude with a discussion of versions of the random-parity representation of the Potts model.
In one dimension {#sec_1d}
----------------
The quantum Ising model on ${\mathbb{Z}}$ has been thoroughly studied in the mathematical physics literature. It is an example of what is called an ‘exactly solvable model’: using transfer matrices and related techniques, the critical ratio and other important quantities have been computed, see for example [@pfeuty70] or [@sachdev99] and references therein. In this section we prove by graphical methods that the critical value coincides with the self-dual value of Section \[duality\_sec\]. The graphical method is valuable in that it extends to more complicated geometries, as in the next section. In the light of , we shall study only the ground state, and we shall suppress the superscript ${\infty}$.
\[crit\_val\_cor\] Let ${\mathbb{L}}={\mathbb{Z}}$. Then $\rho_{\mathrm{c}}=2$, and the transition is of second order in that $M_+(2)=0$.
We mention an application of this theorem. In an account [@GOS] of so-called ‘entanglement’ in the quantum Ising model on the subset $[-m,m]$ of ${\mathbb{Z}}$, it was shown that the reduced density matrix $\nu_{m}^L$ of the block $[-L,L]$ satisfies $$\|\nu_m^L - \nu_n^L\| \le \min\{2, C L^{{\alpha}}e^{-cm}\},\qquad 2\le m<n<{\infty},$$ where $C$ and ${\alpha}$ are constants depending on $\rho={\lambda}/{\delta}$, and $c=c(\rho)>0$ whenever $\rho < 1$. Using Theorems \[exp\_decay\_cor\] and \[crit\_val\_cor\], we have that $c(\rho)>0$ for $\rho<2$.
We adapt the well-known methods [@grimmett_rcm Chapter 6] for the discrete random-cluster model. Write $\phi_\rho^{\mathrm{f}}$ and $\phi_\rho^{\mathrm{w}}$ for the free and wired $q=2$ [random-cluster]{} measures, respectively. By Theorem \[potts\_lim\_thm\] and Remark \[inf\_vol\_rk\], and the representation of the state ${\langle}\cdot{\rangle}_+$, we have that $${\langle}{\sigma}_x{\sigma}_y{\rangle}_+={\phi_{\rho}}^{\mathrm{w}}(x{\leftrightarrow}y),
\qquad{\langle}{\sigma}_x{\rangle}_+={\phi_{\rho}}^{\mathrm{w}}(x{\leftrightarrow}\infty).
\label{o26}$$ Recall from Theorem \[duality\_thm\] that the measures $\phi_\rho^{\mathrm{f}}$ and $\phi_{4/\rho}^{\mathrm{w}}$ are mutually dual. By Zhang’s argument, Theorem \[zhang\_thm\], we know of the self-dual point $\rho=2$ that $$\label{o52}
\phi_2^{\mathrm{f}}(0{\leftrightarrow}{\infty}) = 0$$ and hence that $\rho_{\mathrm{c}}\geq 2$.
We show next that $\rho_{\mathrm{c}}\le 2$, following the method developed for percolation to be found in [@grimmett_perc; @grimmett_rcm]. Suppose that ${\rho_{\mathrm{c}}}>2$. Consider the ‘lozenge’ $D_n$ of side length $n$, as illustrated in Figure \[zhang\_fig\] on p. , and its ‘dual’ $D_n^{\mathrm{d}}$. Let $A_n$ denote the event that there is an open path from the bottom left to the top right of $D_n$ in ${\omega}$, and let $A_n^{\mathrm{d}}$ be the ‘dual’ event that there is in ${\omega}_{\mathrm{d}}$ an open path from the top left to the bottom right of $D_n^{\mathrm{d}}$. The events $A_n$ and $A_n^{\mathrm{d}}$ are complementary, so we have by duality and symmetry under reflection that $$1=\phi_2^{\mathrm{f}}(A_n)+\phi_2^{\mathrm{f}}(A_n^{\mathrm{d}})=
\phi_2^{\mathrm{f}}(A_n)+\phi_2^{\mathrm{w}}(A_n)\leq 2\phi_2^{\mathrm{w}}(A_n).$$ However, if $2<\rho_{\mathrm{c}}$ then we have by and Theorem \[exp\_decay\_cor\] that $\phi_2^{\mathrm{w}}(A_n)$ decays to zero in the manner of $C n^2 e^{-{\alpha}n}$ as $n\to{\infty}$, a contradiction.
We show that $M_+(2)=0$ by adapting a simple argument developed by Werner in [@WW08] for the classical Ising model on ${\mathbb{Z}}^2$. Certain geometrical details are omitted. Let $\pi^{\mathrm{f}}$ be the Ising state obtained with free boundary condition, as in Theorem \[potts\_lim\_thm\]. Recall that $\pi^{\mathrm{f}}$ may be obtained from the random-cluster measures $\phi_2^{\mathrm{f}}$ by assigning to the clusters spin $\pm 1$ independently at random, with probability $1/2$ each. By Lemma \[ergod\_lem\], $\pi^{\mathrm{f}}$ is ergodic.
The binary relations ${\overset{\pm}{{\leftrightarrow}}}$ are defined as follows. A *path* of ${\mathbb{Z}}\times{\mathbb{R}}$ is a path of ${\mathbb{R}}^2$ that: traverses a finite number of line-segments of ${\mathbb{Z}}\times{\mathbb{R}}$, and is permitted to connect them by passing between any two points of the form $(u,t)$, $(u\pm 1,t)$. For $x,y\in{\mathbb{Z}}\times{\mathbb{R}}$, we write $x{\overset{+}{{\leftrightarrow}}} y$ ([respectively]{}, $x{\overset{-}{{\leftrightarrow}}} y$) if there exists a path with endpoints $x$, $y$ all of whose elements are labelled $+1$ ([respectively]{}, $-1$). (In particular, for any $x$ we have that $x{\overset{+}{{\leftrightarrow}}} x$ and $x{\overset{-}{{\leftrightarrow}}} x$.) Let $N^+$ ([respectively]{}, $N^-$) be the number of unbounded $+$ ([respectively]{}, $-$) Ising clusters with connectivity relation ${\overset{+}{{\leftrightarrow}}}$ ([respectively]{}, ${\overset{-}{{\leftrightarrow}}}$). By the Burton–Keane argument, as in Theorem \[inf\_clust\_uniq\], one may show that either $\pi^{\mathrm{f}}(N^+=1) = 1$ or $\pi^{\mathrm{f}}(N^+=0)=1$. The former would entail also that $\pi^{\mathrm{f}}(N^-=1)=1$, by the $\pm$ symmetry in the coupling with the random-cluster measure. With an application of Zhang’s argument as in Theorem \[zhang\_thm\], however, one can show that this is impossible. Therefore, $$\pi^{\mathrm{f}}(N^\pm = 0) = 1.
\label{o31}$$
Recall that ${\langle}\cdot{\rangle}^+=\pi^{\mathrm{w}}$. There is a standard argument for deriving $\pi^{\mathrm{f}}={\langle}\cdot{\rangle}^+$ from , of which the idea is roughly as follows. (See [@ACCN] or [@grimmett_rcm Thm 5.33] for examples of similar arguments applied to the [random-cluster]{} model.) Let ${\Lambda}_n=[-n,n]^2 \subseteq {\mathbb{Z}}\times {\mathbb{R}}$, and let $m<n$. We call a set $S\subseteq{\Lambda}_n$ a *separating set* if any path from ${\Lambda}_m$ to $\partial{\Lambda}_n$ contains an element of $S$. We adopt the harmless convention that, for any spin-configuration ${\sigma}$, the subset of ${\Lambda}_n$ labelled $+1$ is closed, compare Remark \[rem-as\]. By , for given $m$, and for ${\varepsilon}>0$ and large $n$, the event $A_{m,n}= \{{\Lambda}_m {\overset{-}{{\leftrightarrow}}}{\partial}{\Lambda}_n\}^{\mathrm{c}}$ satisfies $\pi^{\mathrm{f}}(A_{m,n}) > 1-{\varepsilon}$. On $A_{m,n}$, there is a separating set labelled entirely $+$; let us call any such separating set a $+$-separating set. Let $U$ denote the set of all points in ${\Lambda}_n$ which are $-$-connected to $\partial{\Lambda}_n$ (note that this includes $\partial{\Lambda}_n$ itself). Write $S=S({\sigma})$ for $\partial ({\Lambda}_n{\setminus}U)$. Note that $S\subseteq {\Lambda}_n{\setminus}{\Lambda}_m$ is a $+$-separating set. See Figure \[2nd\_order\_fig\].
![Sketch of an Ising configuration ${\sigma}$, with the set $S({\sigma})$ drawn bold; $S$ is a $+$-separating set.[]{data-label="2nd_order_fig"}](thesis.30)
For any closed separating set $S_1$, define $\hat S_1$ to be the union of $S_1$ with the unbounded component of $({\mathbb{Z}}\times{\mathbb{R}}){\setminus}S_1$. Also let $\tilde S_1$ be the set of points in ${\Lambda}_n$ that are separated from $\partial{\Lambda}_n$ by $S_1$. The event $\{S({\sigma})=S_1\}$ is ${\mathcal{G}}_{\hat S_1}$-measurable, i.e. it depends only on the restriction of ${\sigma}$ to $\hat S_1$. Let $B{\subseteq}{\Lambda}_m$ be a finite set, and recall the notation $\nu'_B$ at . By the [[dlr]{}]{}-property of Lemma \[potts\_cond\] (the natural extension of which holds also for infinite-volume measures) we deduce that $$\pi^{\mathrm{f}}(\nu'_B\mid A_{m,n},S)=\pi^{\mathrm{w}}_{\tilde S}(\nu'_B\mid A_{m,n}).$$ Let $n\rightarrow{\infty}$ to deduce, using also the [[fkg]{}]{}-inequality of Lemma \[ising\_fkg\], that $$\pi^{\mathrm{f}}(\nu'_B\mid S)=\pi^{\mathrm{w}}_{\tilde S}(\nu'_B)\geq\pi^{\mathrm{w}}(\nu'_B).$$ By integrating, and letting $m\rightarrow{\infty}$, we obtain that $\pi^{\mathrm{f}}(\nu'_B)\geq\pi^{\mathrm{w}}(\nu'_B)$ for all finite sets $B{\subseteq}{\mathbb{Z}}\times{\mathbb{R}}$. Since the reverse inequality $\pi^{\mathrm{f}}(\nu'_B)\le \pi^{\mathrm{w}}(\nu'_B)$ always holds (by Lemma \[potts\_cond\] and Lemma \[ising\_fkg\] again), we deduce that $\pi^{\mathrm{f}}=\pi^{\mathrm{w}}$ as claimed.
One way to conclude that $M_+(2)=0$ is to use the [random-cluster]{} representation again. By and the above, $$\phi_2^{\mathrm{f}}(0{\leftrightarrow}{\infty}) = \phi_2^{\mathrm{w}}(0{\leftrightarrow}{\infty})=0,$$ whence $M_+(2) = \phi_2^{\mathrm{w}}(0{\leftrightarrow}{\infty}) = 0$.
On star-like graphs {#starlike_sec}
-------------------
We now extend Theorem \[crit\_val\_cor\] of the previous section, to show that the critical ratio $\rho_{\mathrm{c}}(2)=2$ for a larger class of graphs than just ${\mathbb{Z}}$. This section forms the contents of the article [@bjo0].
The class of graphs for which we prove that the critical ratio is $2$ includes for example the *star graph*, which is the junction of several copies of ${\mathbb{Z}}$ at a single point. See Figure \[star\_fig\]. It also includes many other planar graphs (see Definition \[starlike\_def\]).
![The star graph has a central vertex of degree $k\geq3$ and $k$ infinite arms, on which each vertex has degree 2. In this illustration, $k=4$.[]{data-label="star_fig"}](thesis.23)
The result for the star is perhaps not unexpected, since the star is only ‘locally’ different from ${\mathbb{Z}}$: if you go far enough out on one of the ‘arms’ then the star ‘looks like’ ${\mathbb{Z}}$. However, as pointed out before, the quantum Ising model on the star, unlike on ${\mathbb{Z}}$, is not exactly solvable, and graphical methods are the only known way to prove this result.
The Ising model on the star-graph has recently arisen in the study of boundary effects in the two-dimensional classical Ising model, see for example [@trombettoni; @martino_etal]. Similar geometries have also arisen in different problems in quantum theory, such as transport properties of quantum wire systems, see [@chamon03; @hou08; @lal02].
Throughout this section we consider the ground-state only, that is to say we let ${\beta}={\infty}$; reference to ${\beta}$ will be suppressed. We also let ${\lambda},{\delta}>0$ be constant and ${\gamma}=0$. Let ${\mathbb{L}}=({\mathbb{V}},{\mathbb{E}})$ be a fixed *star-like graph*:
\[starlike\_def\] A star-like graph is a countably infinite connected planar graph, in which all vertices have finite degree and only finitely many vertices have degree larger than two.
Such a graph is illustrated in Figure \[graph\_fig\]; note that the star graph of Figure \[star\_fig\] is an example in which exactly one vertex has degree at least three.
![A star-like graph ${\mathbb{L}}$ (left) and its line-hypergraph ${\mathbb{H}}$ (right). Any vertex of degree $\geq3$ in ${\mathbb{L}}$ is associated with a “polygonal” (hyper)edge in ${\mathbb{H}}$.[]{data-label="graph_fig"}](thesis.24 "fig:") ![A star-like graph ${\mathbb{L}}$ (left) and its line-hypergraph ${\mathbb{H}}$ (right). Any vertex of degree $\geq3$ in ${\mathbb{L}}$ is associated with a “polygonal” (hyper)edge in ${\mathbb{H}}$.[]{data-label="graph_fig"}](thesis.25 "fig:")
The following is the main result of this section.
\[starlike\_thm\] Let ${\mathbb{L}}$ be any star-like graph. Then the critical ratio of the ground state quantum Ising model on ${\mathbb{L}}$ is $\rho_{\mathrm{c}}(2)=2$.
Simpler arguments than those presented here can be used to establish the analogous result when $q=1$, namely that $\rho_{\mathrm{c}}(1)=1$. Also, the same arguments can be used to calculate the critical probability of the discrete graphs ${\mathbb{L}}\times{\mathbb{Z}}$ when $q=1,2$. As in the case ${\mathbb{L}}={\mathbb{Z}}$, an essential ingredient of the proof is the exponential decay of correlations below $\rho_{\mathrm{c}}$.
Recall that a *hypergraph* is a set ${\mathbb{W}}$ together with a collection ${\mathbb{B}}$ of subsets of ${\mathbb{W}}$, called *edges* (or hyperedges). A graph is a hypergraph in which all edges contain two elements. In our analysis we will use a suitably defined hypergraph ‘dual’ of ${\mathbb{L}}$. To be precise, let ${\mathbb{H}}=({\mathbb{W}},{\mathbb{B}})$ be the *line-hypergraph* of ${\mathbb{L}}$, given by letting ${\mathbb{W}}={\mathbb{E}}$ and letting the set $\{e_1,\dotsc,e_n\}\subseteq {\mathbb{E}}={\mathbb{W}}$ be an edge (that is, an element of ${\mathbb{B}}$) if and only if $e_1,\dotsc,e_n$ are all the edges of ${\mathbb{L}}$ adjacent to some particular vertex of ${\mathbb{L}}$. Note that only finitely many edges of ${\mathbb{H}}$ have size larger than two, since ${\mathbb{L}}$ is star-like.
Fix an arbitrary planar embedding of ${\mathbb{L}}$ into ${\mathbb{R}}^2$; we will typically identify ${\mathbb{L}}$ with its embedding. We let ${\mathcal{O}}$ denote an arbitrary but fixed vertex of ${\mathbb{L}}$ which has degree at least two; we think of ${\mathcal{O}}$ as the ‘origin’. There is a natural planar embedding of ${\mathbb{H}}$ defined via the embedding ${\mathbb{L}}$, in which an edge of size more than two is represented as a polygon. See Figure \[graph\_fig\]. In this section we will use the symbol ${\mathbb{X}}$ in place of ${\mathbf{\Theta}}$ for ${\mathbb{L}}\times{\mathbb{R}}$, and will identify ${\mathbb{X}}$ with the corresponding subset of ${\mathbb{R}}^3$. Similarly, we write ${\mathbb{Y}}={\mathbb{H}}\times{\mathbb{R}}$ for the ‘dual’ of ${\mathbb{X}}$, also thought of as a subset of ${\mathbb{R}}^3$. We will often identify ${\omega}=(B,D)\in{\Omega}$ with its embedding, ${\omega}\equiv({\mathbb{X}}\setminus D)\cup B$. We let ${\Lambda}_n$ be the simple region corresponding to ${\beta}=n$ and $L$ the subgraph of ${\mathbb{L}}$ induced by the vertices at graph distance at most $n$ from ${\mathcal{O}}$, see . Note that ${\Lambda}_n\uparrow{\mathbb{X}}$. In this section we let uppercase $\Phi^b_n$ denote the random-cluster measure on ${\Lambda}_n$ with parameters ${\lambda},{\delta}>0$, ${\gamma}=0$, $q=2$ and boundary condition $b\in\{0,1\}$, where, as in Section \[duality\_sec\], we let $0$ and $1$ denote the free and wired boundary conditions, respectively.
Given any configuration ${\omega}\in{\Omega}$, one may as in the case ${\mathbb{L}}={\mathbb{Z}}$ associate with it a *dual* configuration on ${\mathbb{Y}}$ by placing a death wherever ${\omega}$ has a bridge, and a (hyper)bridge wherever ${\omega}$ has a death. Recall Figure \[duality\_fig\] on p. . More precisely, we let ${\Omega}_{\mathrm{d}}$ be the set of pairs of locally finite subsets of ${\mathbb{B}}\times{\mathbb{R}}$ and ${\mathbb{W}}\times{\mathbb{R}}$, and for each ${\omega}=(B,D)\in{\Omega}$ we define its dual to be ${\omega}_{\mathrm{d}}:=(D,B)$. As before, we may identify ${\omega}_{\mathrm{d}}$ with its embedding in ${\mathbb{Y}}$, noting that some bridges may be embedded as polygons. We let $\Psi^b_n$ and $\Psi^b$ denote the laws of ${\omega}_{\mathrm{d}}$ under $\Phi_n^{1-b}$ and $\Phi^{1-b}$ respectively.
We will frequently be comparing the random-cluster measures on ${\mathbb{X}}$ and ${\mathbb{Y}}$ with the random-cluster measures on ${\mathbb{Z}}\times{\mathbb{R}}$; the latter may be regarded as a subset of both ${\mathbb{X}}$ and ${\mathbb{Y}}$ (in a sense made more precise below). We will reserve the lower-case symbols $\phi^b_n,\phi^b$ for the random-cluster measures on ${\mathbb{Z}}\times{\mathbb{R}}$ with the same parameters as $\Phi^b_n$ (where $\phi^b_n$ lives on the simple region given by ${\beta}=n$ and $L=[-n,n]$). We will write $\psi_n^{1-b},\psi^{1-b}$ for the dual measures of $\phi^b_n,\phi^b$ on ${\mathbb{Z}}\times{\mathbb{R}}$; thus by Theorem \[duality\_thm\], the measures $\psi_n^{1-b},\psi^{1-b}$ are random cluster measures with parameters $q'=q$, ${\lambda}'=q{\delta}$ and ${\delta}'={\lambda}/q$, and boundary condition $1-b$.
Here is a brief outline of the proof of Theorem \[starlike\_thm\]. First we make the straightforward observation that $\rho_{\mathrm{c}}(2)\leq 2$. Next, we use exponential decay to establish the existence of certain infinite paths in the dual model on ${\mathbb{Y}}$ when ${\lambda}/{\delta}<2$. Finally, we show how to put these paths together to form ‘blocking circuits’ in ${\mathbb{Y}}$, which prevent the existence of infinite paths in ${\mathbb{X}}$ when ${\lambda}/{\delta}<2$. Parts of the argument are inspired by [@gkr].
\[upperbound\_lem\] For ${\mathbb{L}}$ any star-like graph, $\rho_{\mathrm{c}}(2)\leq 2$.
Since ${\mathbb{L}}$ is star-like, it contains an isomorphic copy of ${\mathbb{Z}}$ as a subgraph. Let $Z$ be such a subgraph; we may assume that ${\mathcal{O}}\in Z$. We may identify $\phi^b_n,\phi^b$ with the random-cluster measures on $Z\times{\mathbb{R}}$. For each $n\geq 1$, let $C_n$ be the event that no two points in ${\Lambda}_n\cap(Z\times{\mathbb{R}})$ are connected by a path which leaves $Z\times{\mathbb{R}}$. Each $C_n$ is a decreasing event. It follows from the [[dlr]{}]{}-property, Lemma \[del\_contr\], that $\Phi^b_n(\cdot\mid C_n)=\phi^b_n(\cdot)$. If $A$ is an increasing local event defined on $Z\times{\mathbb{R}}$, this means that $$\phi^b_n(A)=\Phi^b_n(A\mid C_n)\leq\Phi^b_n(A),$$ i.e. $\phi^b_n\leq\Phi^b_n$ for all $n$. Letting $n\rightarrow\infty$ it follows that $\phi^b\leq\Phi^b$. If ${\lambda}/{\delta}>2$ then $\phi^b(({\mathcal{O}},0)\leftrightarrow\infty)>0$ and it follows that also $$\Phi^b(({\mathcal{O}},0)\leftrightarrow\infty)>0,$$ which is to say that $\rho_{\mathrm{c}}(2)\leq2$.
### Infinite paths in the half-plane
Let us now establish some facts about the random-cluster model on the ‘half-plane’ ${\mathbb{Z}}_+\times{\mathbb{R}}$ which will be useful later. Our notation is as follows: for $n\geq 1$, let $$\begin{split}
S_n&=\{(a,t)\in{\mathbb{Z}}\times{\mathbb{R}}: -n\leq a\leq n, |t|\leq n\}\\
S_n(m,s)&=S_n+(m,s)=\{(a+m,t+s)\in{\mathbb{Z}}\times{\mathbb{R}}: (a,t)\in S_n\}.
\end{split}$$For brevity write $T_n=S_n(n,0)$. For $b\in\{0,1\}$ and ${\Delta}$ one of $S_n,T_n$, we let $\phi^b_{\Delta}$ denote the $q=2$ random-cluster measure on the simple region in ${\mathbb{X}}$ with $K={\Delta}$ with boundary condition $b$ and parameters ${\lambda},{\delta}$. Note that $$\phi^b=\lim_{n\rightarrow\infty}\phi^b_{S_n},\qquad
\psi^b=\lim_{n\rightarrow\infty}\psi^b_{S_n}.$$ We will also be using the limits $$\phi^{\mathrm{sw}}=\lim_{n\rightarrow\infty} \phi^1_{T_n},\qquad
\psi^{\mathrm{sf}}=\lim_{n\rightarrow\infty} \psi^0_{T_n},$$which exist by similar arguments to Theorem \[conv\_lem\]. (The notation ‘${\mathrm{sw}}$’ and ‘${\mathrm{sf}}$’ is short for ‘side wired’ and ‘side free’, respectively.) These are measures on configurations ${\omega}$ on ${\mathbb{Z}}_+\times{\mathbb{R}}$; standard arguments let us deduce all the properties of $\phi^{\mathrm{sw}}$ and $\psi^{\mathrm{sf}}$ that we need. In particular $\psi^{\mathrm{sf}}$ and $\phi^{\mathrm{sw}}$ are mutually dual (with the obvious interpretation of duality) and they enjoy the positive association property of Theorem \[rc\_fkg\] and the finite energy property of Lemma \[fin\_en\].
Let $W$ be the ‘wedge’ $$W=\{(a,t)\in{\mathbb{Z}}_+\times{\mathbb{R}}: 0\leq t\leq a/2+1\},$$ and write $0$ for the origin $(0,0)$.
\[wedge\_lem\] Let ${\lambda}/{\delta}<2$. Then $$\psi^{\mathrm{sf}}(0\leftrightarrow\infty\mbox{ in } W)>0.$$
Here is some intuition behind the proof of Lemma \[wedge\_lem\]. The claim is well-known with $\psi^0$ in place of $\psi^{\mathrm{sf}}$, by standard arguments using duality and exponential decay. However, $\psi^{\mathrm{sf}}$ is stochastically smaller than $\psi^0$, so we cannot deduce the result immediately. Instead we pass to the dual $\phi^{\mathrm{sw}}$ and establish directly a lack of blocking paths. The problem is the presence of the infinite ‘wired side’; we get the required fast decay of two-point functions by using the following result.
\[exp\_prop\] Let ${\lambda}/{\delta}<2$. There is ${\alpha}>0$ such that for all $n$, $$\phi^1_{S_n}(0\leftrightarrow \partial S_n)\leq e^{-{\alpha}n}.$$
In words, correlations decay exponentially under finite volume measures if they do so under infinite volume measures. Results of this type for the classical Ising and random-cluster models appear in many places. In [@campanino_ioffe_velenik_08] and [@cerf_messikh_06] it is proved for general $q\geq1$ random-cluster models in two dimensions, and more general results about the two-dimensional case appear in [@alexander04]. A proof of general results of this type for the classical Ising model in any dimension appears in [@higuchi93_ii]. Below we adapt the argument in [@higuchi93_ii] to the current setting, with the difference that we shorten the proof by using the Lieb inequality, Lemma \[lieb\_lem\], in place of the [[ghs]{}]{}-inequality; use of the Lieb-inequality was suggested by Grimmett (personal communication). Note that the same argument works on ${\mathbb{Z}}^d$ for any $d\geq1$.
Let $\hat S_n\supseteq S_n$ denote the ‘tall’ box $$\hat S_n=\{(a,t)\in{\mathbb{Z}}\times{\mathbb{R}}: -n\leq a\leq n, |t|\leq n+1\}.$$ We will use a random-cluster measure on $\hat S_n$ which has non-constant ${\lambda},{\delta}$, and nonzero ${\gamma}$. The particular intensities we use are these. Fix $n$, and fix $m\geq0$, which we think of as large. Let ${\lambda}(\cdot)$, ${\delta}(\cdot)$ and ${\gamma}_m(\cdot)$ be given by $$\begin{split}
{\delta}(a,t)&=\left\{
\begin{array}{ll}
{\delta}, & \mbox{if } (a,t)\in S_n \\
0, & \mbox{otherwise},
\end{array}
\right.\\
{\lambda}(a+1/2,t)&=\left\{
\begin{array}{ll}
{\lambda}, & \mbox{if } (a,t)\in S_n \mbox{ and } (a+1,t)\in S_n\\
0, & \mbox{otherwise},
\end{array}
\right.\\
{\gamma}_m(a,t)&=\left\{
\begin{array}{ll}
{\lambda}, & \mbox{if exactly one of }
(a,t) \mbox{ and } (a+1,t) \mbox{ is in } S_n\\
m, & \mbox{if } (a,t)\in \hat S_n\setminus S_n\\
0, & \mbox{otherwise}.
\end{array}
\right.
\end{split}$$ In words, the intensities are as usual ‘inside’ $S_n$ and in particular there is no external field in the interior; on the left and right sides of $S_n$, the external field simulates the wired boundary condition; and on top and bottom, the external field simulates an approximate wired boundary (as $m\rightarrow\infty$). We let $\tilde \phi^b_{m,n}$ denote the random-cluster measure on $\hat S_n$ with intensities ${\lambda}(\cdot),{\delta}(\cdot),{\gamma}_m(\cdot)$ and boundary condition $b\in\{0,1\}$. Note that $\tilde\phi^0_{m,n}$ and $\phi^0_{S_n}$ agree on events defined on $S_n$, for any $m$.
Let $X$ denote $\hat S_n\setminus S_n$ together with the left and right sides of $S_n$. By the Lieb inequality, Lemma \[lieb\_lem\], we have that $$\begin{split}
\tilde\phi^1_{m,n}(0\leftrightarrow {\Gamma})&\leq
e^{8{\delta}}\int_X dx\;\tilde\phi^0_{m,n}(0\leftrightarrow x)
\tilde\phi^1_{m,n}(x\leftrightarrow {\Gamma}) \\
&\leq e^{8{\delta}}\int_X dx\;\tilde\phi^0_{m,n}(0\leftrightarrow x),
\end{split}$$ since (with these intensities) $X$ separates $0$ from ${\Gamma}$. Therefore, by stochastic domination by the infinite-volume measure, $$\label{o76}
\tilde\phi^1_{m,n}(0\leftrightarrow {\Gamma})\leq
e^{8{\delta}}\int_X dx\;\phi^0(0\leftrightarrow x).$$ All the points $x\in X$ are at distance at least $n$ from the origin. By exponential decay in the infinite volume, Theorem \[exp\_decay\_cor\], it follows from that there is an absolute constant $\tilde{\alpha}>0$ such that $$\tilde\phi^1_{m,n}(0\leftrightarrow {\Gamma})\leq e^{8{\delta}}|X|e^{-\tilde{\alpha}n}=
e^{8{\delta}}(8n+2)e^{-\tilde{\alpha}n}.$$ Now let $C$ be the event that all of $\hat S_n\setminus S_n$ belongs to the connected component of ${\Gamma}$, which is to say that all points on $\hat S_n\setminus S_n$ are linked to ${\Gamma}$. Then by the [[dlr]{}]{}-property of random-cluster measures the conditional measure $\tilde\phi^1_{m,n}(\cdot\mid C)$ agrees with $\phi^1_{S_n}(\cdot)$ on events defined on $S_n$. Therefore $$\begin{split}
\phi^1_{S_n}(0\leftrightarrow \partial S_n)&=
\tilde\phi^1_{m,n}(0\leftrightarrow \partial S_n\mid C)
=\tilde\phi^1_{m,n}(0\leftrightarrow {\Gamma}\mid C)\\
&\leq\frac{\tilde\phi^1_{m,n}(0\leftrightarrow {\Gamma})}{\tilde\phi^1_{m,n}(C)}
\leq\frac{e^{8{\delta}}}{\tilde\phi^1_{m,n}(C)}\cdot (8n+2)e^{-\tilde{\alpha}n}.
\end{split}$$ Since $\tilde\phi^1_{m,n}(C)\rightarrow 1$ as $m\rightarrow\infty$ we conclude that $$\phi^1_{S_n}(0\leftrightarrow \partial S_n)
\leq e^{8{\delta}}(8n+2)e^{-\tilde{\alpha}n}.$$ Since each $\phi^1_{S_n}(0\leftrightarrow \partial S_n)<1$ it is a simple matter to tidy this up to get the result claimed.
Let $T=\{(a,a/2+1): a\in{\mathbb{Z}}_+\}$ be the ‘top’ of the wedge $W$. We claim that $$\sum_{n\geq 1}\phi^{\mathrm{sw}}((n,0)\leftrightarrow T \mbox{ in } W)<\infty.$$ Once this is proved, it follows from the Borel–Cantelli lemma that with probability one under $\phi^{\mathrm{sw}}$, at most finitely many of the points $(n,0)$ are connected to $T$ inside $W$. Hence under the dual measure $\psi^{\mathrm{sf}}$ there is an infinite path inside $W$ with probability one, and by the [[dlr]{}]{}- and positive association properties it follows that $$\psi^{\mathrm{sf}}(0\leftrightarrow\infty\mbox{ in } W)>0,$$ as required.
To prove the claim we note that, if $n$ is larger than some constant, then the event ‘$(n,0)\leftrightarrow T \mbox{ in } W$’ implies the event ‘$(n,0)\leftrightarrow \partial S_{n/3}(n,0)$’. The latter event, being increasing, is more likely under the measure $\phi^1_{S_{n/3}(n,0)}$ than under $\phi^{\mathrm{sw}}$. But by Proposition \[exp\_prop\], $$\phi^1_{S_{n/3}(n,0)}((n,0)\leftrightarrow \partial S_{n/3}(n,0))
=\phi^1_{S_{n/3}}(0\leftrightarrow \partial S_{n/3})\leq e^{-{\alpha}n/3},$$ which is clearly summable.
The next lemma uses a variant of standard blocking arguments.
\[circuits\_lem\] Let ${\lambda}/{\delta}<2$. There exists ${\varepsilon}>0$ such that for each $n$, $$\psi^{\mathrm{sf}}((0,2n+1)\leftrightarrow (0,-2n-1)\mbox{ off } T_n)\geq {\varepsilon}.$$
Let $L_n=\{(a,n):a\geq0)\}$ be the horizontal line at height $n$, and let ${\varepsilon}>0$ be such that $\psi^{\mathrm{sf}}(0\leftrightarrow\infty\mbox{ in } W)\geq\sqrt{\varepsilon}$. We claim that $$\psi^{\mathrm{sf}}((0,-2n-1)\leftrightarrow L_{2n+1}\mbox{ off } T_n)\geq \sqrt{\varepsilon}.$$ Clearly $\psi^{\mathrm{sf}}$ is invariant under reflection in the $x$-axis and under vertical translation, see Lemma \[invar\_lem\]. Thus once the claim is proved we get that $$\begin{split}
\psi^{\mathrm{sf}}((0,2n+1)&\leftrightarrow (0,-2n-1)\mbox{ off } T_n)\\
&\geq\psi^{\mathrm{sf}}((0,-2n-1)\leftrightarrow L_{2n+1}\mbox{ off } T_n\\
&\qquad\qquad\mbox{ and } (0,2n+1)\leftrightarrow L_{-2n-1}\mbox{ off } T_n)\\
&\geq (\sqrt{\varepsilon})^2,
\end{split}$$ as required. See Figure \[paths\_fig\].
![Construction of a ‘half-circuit’ in ${\mathbb{Z}}_+\times{\mathbb{R}}$. With probability one, any infinite path in the lower wedge must reach the line $L_{2n+1}$, and similarly for any infinite path in the upside-down wedge. Any pair of such paths starting on the horizontal axis must cross.[]{data-label="paths_fig"}](thesis.28)
The claim follows if we prove that $$\label{strip_eqn}
\psi^{\mathrm{sf}}(0\leftrightarrow\infty\mbox{ in } R)=0,$$ where $R$ is the strip $$R=\{(a,t):a\geq 0,-2n-1\leq t\leq 2n+1\}.$$ However, follows from the [[dlr]{}]{}-property, Lemma \[del\_contr\], the stochastic domination of Theorem \[comp\_perc\], and the Borel–Cantelli lemma; these combine to show that the event ‘no bridges between $\{k\}\times[-2n-1,2n+1]$ and $\{k+1\}\times[-2n-1,2n+1]$’ must happen for infinitely many $k$ with $\psi^{\mathrm{sf}}$-probability one. In more detail: we have that $\psi^{\mathrm{sf}}\leq\mu$, where $\mu$ is the percolation measure with parameters ${\lambda},{\delta}$; under $\mu$ the events above are independent, so $$\psi^{\mathrm{sf}}(0\leftrightarrow\infty\mbox{ in } R)\leq
\mu(0\leftrightarrow\infty\mbox{ in } R)=0.$$
### Proof of Theorem \[starlike\_thm\]
We may assume that ${\mathbb{L}}\neq{\mathbb{Z}}$, since the case ${\mathbb{L}}={\mathbb{Z}}$ is known. Let ${\lambda}/{\delta}<2$, and recall that ${\mathbb{L}}$ consists of finitely many infinite ‘arms’, where each vertex has degree two, together with a ‘central’ collection of other vertices. On each of the arms, let us fix one arbitrary vertex (of degree two) and call it an *exit point*. Let $U$ denote the set of exit points of ${\mathbb{L}}$.
Given an exit point $u\in U$, call its two neighbours $v$ and $w$; we may assume that they are labelled so that only $v$ is connected to the origin ${\mathcal{O}}$ by a path not including $u$. If the edge $uv$ were removed from ${\mathbb{L}}$, the resulting graph would consist of two components, where we denote by $J_u$ the component containing $w$. Let $\hat \Phi^b_n,\hat \Phi^b$ denote the marginals of $\Phi^b_n,\Phi^b$ on $X_u:=J_u\times{\mathbb{R}}$; similarly let $\hat\Psi^b_n,\hat\Psi^b$ denote the marginals of the dual measures. Of course $X_u$ is isomorphic to the half-plane graph considered in the previous subsection. By positive association and the [[dlr]{}]{}-property of random-cluster measures, $\hat\Phi_n^0\leq\phi^1_{T_n(u)}$, so letting $n\rightarrow\infty$ also $\hat\Phi^0\leq\phi^{\mathrm{sw}}$. Passing to the dual, it follows that $\hat\Psi^1\geq\psi^{\mathrm{sf}}$. The (primal) edge $uv$ is a *vertex* in the line-hypergraph; denoting it still by $uv$ we therefore have by Lemma \[circuits\_lem\] that there is an ${\varepsilon}>0$ such that for all $n$, $$\Psi^1((uv,-2n-1)\leftrightarrow(uv,2n+1)
\mbox{ off }T_n(u)\mbox{ in }X_u)\geq{\varepsilon}.$$ Here $T_n(u)$ denotes the copy of the box $T_n$ contained in $X_u$. Letting $A$ denote the intersection of the events above over all exit points $u$, and letting $A_1=A_1(n)$ be the dual event $A_1=\{{\omega}_{\mathrm{d}}:{\omega}\in A\}$, it follows from positive association that $\Phi^0(A_1)\geq{\varepsilon}^k$, where $k=|U|$ is the number of exit points. Note that $A_1$ is a decreasing event in the primal model.
On $A_1$, no point in $T_n(u)$ can reach $\infty$ without passing the line $\{u\}\times[-2n-1,2n+1]$, since there is a dual blocking path in $X_u$. Let $I$ denote the (finite) subgraph of ${\mathbb{L}}$ spanned by the complement of all the $J_u$ for $u\in U$, and let $A_2=A_2(n)$ denote the event that for all vertices $v\in I$, the intervals $\{v\}\times[2n+1,2n+2]$ and $\{v\}\times[-2n-1,-2n-2]$ all contain at least one death and the endpoints of no bridges (in the primal model). There is $\eta>0$ independent of $n$ such that $\Phi^0(A_2)\geq\eta$. So by positive association $\Phi^0(A_1\cap A_2)\geq\eta{\varepsilon}^k>0$. We have that $A_1\cap A_2{\subseteq}A_3$, where $A_3$ is the event that no point inside the union of $I\times[-n,n]$ with $\cup_{u\in U}T_n(u)$ lies in an unbounded connected component. See Figure \[blocking\_fig\].
![The dashed lines indicate dual paths that block any primal connection from the interior to $\infty$. Note that this figure illustrates only the simplest case when ${\mathbb{L}}$ is a junction of lines at a single point.[]{data-label="blocking_fig"}](thesis.29)
Taking the intersection of the $A_3=A_3(n)$ over all $n$, it follows that $$\Phi^0(\mbox{there is no unbounded connected component})\geq\eta{\varepsilon}^k.$$ The event that there is no unbounded connected component is a tail event. By tail-triviality, Proposition \[tail\_triv\], it follows that whenever ${\lambda}/{\delta}<2$ then $$\Phi^0(0\not\leftrightarrow\infty)=1.$$ In other words, $\rho_{\mathrm{c}}(2)\geq 2$. Combined with the opposite bound in Lemma \[upperbound\_lem\], this gives the result.
One may ask if, as in the case ${\mathbb{L}}={\mathbb{Z}}^d$, the phase transition on star-like graphs is of second order, and if there is exponential decay of correlations below the critical point. We do not know how to prove such results: Zhang’s argument (Theorem \[zhang\_thm\]) fails on star-like graphs, and so do the arguments for Theorem \[main\_pdi\_thm\], due to the lack of symmetry.
Reflection positivity {#af_sec}
---------------------
The theory of reflection positivity was first developed in [@frohlich_lieb78; @froehlich_etal78; @froehlich_etal80], originally as a way to prove the existence of discontinuous phase transitions in a wide range of models in statistical physics. A model which is reflection positive (see definitions below) will satisfy what are called ‘Gaussian domination bounds’ and ‘chessboard estimates’. The latter will not be touched upon here, see the review [@biskup_rp] and references therein. One may think of the Gaussian domination bounds, and the related ‘infrared bound’, as a way of bounding certain quantities in the model by corresponding quantities in another, simpler, model, namely what is called the ‘Gaussian free field’. Very roughly, existence of a phase transition in the Gaussian free field therefore implies existence of a phase transition in your reflection positive model.
In [@af], it was shown that Gaussian domination bounds could also be used in another way for the Ising model. By relating the bounds to quantities that appear naturally in the random-current representation of the Ising model, Aizenman and Fernández were able to establish that the behaviour of the classical Ising model on ${\mathbb{Z}}^d$ resembles that of the ‘mean field’ Ising model when $d$ is large, in fact already when $d\geq4$. In this section we will state more precisely the sense in which ‘large $d$ resembles mean field’, and give a very brief sketch of the arguments involved. We will also indicate how one might extend the results of [@af] to the quantum Ising model; this is currently work in progress.
In this section we will only be considering the case when ${\mathbb{L}}={\mathbb{Z}}^d$ for some $d\geq1$, and $L=(V,E)=[-n,n]^d$ for some $n$, with periodic boundary (see Assumption \[periodic\_assump\]). For $j=1,\dotsc,d$, write $e_j$ for the element of $V$ whose $j$th coordinate is $1$ and whose other coordinates are zero. For ${\sigma}\in\{-1,+1\}^V$, we write its classical Ising weight in this section as $$\label{q1}
\exp\Big({\beta}\sum_{xy\in E}J_{xy}{\sigma}_x{\sigma}_y+{\gamma}\sum_{x\in V}{\sigma}_x\Big),$$ where ${\beta},{\gamma},J_e\geq0$. We assume that the model is translation invariant in that $J_{xy}\equiv J_{y-x}$, where, for $z\in V$, $J_z\geq0$ and $J_z=0$ unless $z=e_j$ for some $j$. We also assume that $J_{e_j}=J_{-e_j}$ for all $j=1,\dotsc,d$.
The classical Ising model displays a phase-transition in ${\beta}$ when ${\gamma}=0$, at the critical value ${\beta}_{\mathrm{c}}$. As in the quantum Ising model (Theorem \[ab\_thm\]), the infinite-volume magnetization $M=M({\beta},{\gamma})$ satisfies the inequalities $$\begin{aligned}
\label{q0}
M&\geq c_2({\beta}-{\beta}_{\mathrm{c}})^{1/2},
&\text{for ${\gamma}=0$ and ${\beta}\downarrow{\beta}_{\mathrm{c}}$}, \\
M&\geq c_1{\gamma}^{1/3}, &\text{for ${\beta}={\beta}_{\mathrm{c}}$ and ${\gamma}\downarrow 0$},
\nonumber\end{aligned}$$ for some constants $c_1,c_2$ (this was first proved in [@abf]). As mentioned in Remark \[remark\_mf\], it is conjectured that the limits $$\label{q-1}
a=\lim_{{\beta}\downarrow{\beta}_{\mathrm{c}}}\frac{\log M({\beta},0)}{\log({\beta}-{\beta}_{\mathrm{c}})},\qquad
\frac{1}{b}=\lim_{{\gamma}\downarrow 0}\frac{\log M({\beta}_{\mathrm{c}},{\gamma})}{\log{\gamma}}$$ exist. Using the random-current representation coupled with results from reflection positivity, [@af] shows that these limits do indeed exist when $d\geq 4$, and that is sharp in that $a=1/2$ and $b=3$. The values $a=1/2$ and $b=3$ are called the ‘mean field’ values because they are known to be the correct critical exponents for the Ising model on the complete graph (this result is ‘well-known’, but see [@fisher64; @fisher67] for reviews). Intuitively, complete graphs are infinite-dimensional, so the higher $d$ is the closer one may expect the behaviour to be to that on the complete graph. The results of [@af] confirm this, and show that the ‘critical dimension’ is at most $d=4$. Their method is roughly as follows.
For $j=1,\dotsc,d$ we let $P_i=\{x=(x_1,\dotsc,x_d)\in V: x_j=0\}$, and we let $P_j^+=\{x\in V:x_j>0\}$ and $P_j^-=\{x\in V:x_j<0\}$. The symbol $\theta_i$ will denote reflection in $P_i$, thus $\theta_j(x_1,\dotsc,x_j,\dotsc,x_d)=(x_1,\dotsc,-x_j,\dotsc,x_d)$. Write ${\mathcal{F}}_{P_j^+}$ and ${\mathcal{F}}_{P^-_j}$ for the ${\sigma}$-algebras of events defined on $P_j^+$ and $P_j^-$, respectively.
Although we will be using the concept of reflection positivity only for the Ising measure , the definition makes sense in greater generality, as follows. Let $S\subseteq{\mathbb{R}}$ be a compact set, and endow $S^V$ with the product ${\sigma}$-algebra. Fix $j\in\{1,\dotsc,d\}$, and let $\psi$ denote a probability measure on $S^V$ which is invariant under $\theta_j$. For $s=(s_x:x\in V)\in S^V$, write $\theta_j(s)=(s_{\theta_j(x)}:x\in V)$, and for $f:S^V\rightarrow{\mathbb{R}}$ define $\theta_j f(s)=f(\theta_j (s))$.
The probability measure $\psi$ is *reflection positive* with respect to $\theta_j$ if for all ${\mathcal{F}}_{P_j^+}$-measurable $f:S^V\rightarrow{\mathbb{R}}$, we have that $$\psi(f\cdot\theta_j f)\geq 0.$$
\[rp\_lem\]$\;$
- Any product measure on $S^V$ invariant under $\theta_j$ is reflection positive with respect to $\theta_j$,
- The Ising measure is reflection positive with respect to all the $\theta_j$.
For a proof of this standard fact, see for example [@biskup_rp]. It follows from Lemma \[rp\_lem\] that the Ising model satisfies the following ‘Gaussian domination’ bounds. For $p\in[-\pi,\pi]^d$, let $$G(p):=\sum_{x\in V}{\langle}{\sigma}_o{\sigma}_x{\rangle}_{{\gamma}=0}e^{ip\cdot x}$$ be the Fourier transform of ${\langle}{\sigma}_0{\sigma}_x{\rangle}_{{\gamma}=0}$, where $i=\sqrt{-1}$ and $p \cdot x$ denotes the usual dot product. Due to our symmetry assumptions we see that the complex conjugate ${\overline}{G(p)}=G(-p)=G(p)$ so that $G(p)\in {\mathbb{R}}$. Also define $$E(p):=\frac{1}{2}\sum_{x\in V}(1-e^{ip\cdot x})J_x;$$ similarly we see that $E(p)\in{\mathbb{R}}$.
\[gd\_prop\] $$G(p)\leq \frac{1}{2{\beta}E(p)}.$$
Before we describe how this relates to the random-current representation, we note that a simple calculation shows that $E(p)\geq c\sum_{j=1}^dp_j^2$, which at least gives some indication of why Gaussian domination may be particularly useful for large $d$.
The link to the random-current representation is roughly as follows. Define the *bubble diagram* $$B_0=\sum_{x\in V}{\langle}{\sigma}_0{\sigma}_x{\rangle}_{{\gamma}=0}^2.$$ Recall that $M={\langle}{\sigma}_0{\rangle}$ and that we write $\chi=\partial M/\partial{\gamma}$. We saw in Section \[sw\_appl\_sec\] that random-current arguments imply the [[ghs]{}]{}-inequality, namely that $\partial \chi/\partial{\gamma}\leq 0$. In [@af], elaborations of such arguments (for the discrete model) show that in fact $$\label{q2}
\frac{\partial\chi}{\partial{\gamma}}\leq
-\frac{|1-\tanh({\gamma})B_0/M|_+^2}{96B_0(1+2{\beta}B_0)^2}
\tanh({\gamma})\chi^4,$$ where $|x|_+=x\vee 0$. The bubble diagram appears here as it becomes necessary to consider the existence of two independent currents between sites $0$ and $x$. Inequality is an improvement on the [[ghs]{}]{}-inequality if $B_0$ is finite; thus the first task is to obtain bounds on $B_0$. Such bounds are provided primarily by Gaussian domination. The link is provided via Parseval’s identity: $$B_0=\frac{1}{(2\pi)^d}\int_{[-\pi,\pi]^d} G(p)^2\,dp.$$ By careful use of Gaussian domination and other bounds, one may establish bounds on $B_0$ for ${\beta}$ close to the critical value ${\beta}_{\mathrm{c}}$. More precisely, one may show that there are constants $0<c_1,c_2<{\infty}$ such that $$\begin{aligned}
B_0&\leq c_1, &\text{if } d>4,\\
B_0&\leq c_2 |\log({\beta}_{\mathrm{c}}-{\beta})|, &\text{if } d=4,\end{aligned}$$ as ${\beta}\uparrow{\beta}_c$. Careful manipulation and integration of then gives that there are constants $c_1',c_2',c_1'',c_2''$ such that the infinite-volume magnetization $M$ satisfies the following. First, as ${\beta}\downarrow{\beta}_{\mathrm{c}}$ for ${\gamma}=0$, $$\begin{aligned}
M&\leq c_1'({\beta}-{\beta}_{\mathrm{c}})^{1/2}, &\text{if } d>4,\\
M&\leq c_2'({\beta}-{\beta}_{\mathrm{c}})^{1/2} |\log({\beta}-{\beta}_{\mathrm{c}})|^{3/2}, &\text{if } d=4,\end{aligned}$$ and second, for ${\beta}={\beta}_{\mathrm{c}}$ and ${\gamma}\downarrow0$, $$\begin{aligned}
M&\leq c_1''{\gamma}^{1/3}, &\text{if } d>4,\\
M&\leq c_2'' {\gamma}^{1/3}|\log {\gamma}|, &\text{if } d=4.\end{aligned}$$ These are the complementary bounds to needed to show that the limits exist and take the values $a=1/2$ and $b=3$.
There are two main steps to extending the results of [@af] to the quantum (or space–time) Ising model: first, to establish reflection positivity and the related Gaussian domination bound, and second, to verify that the random-parity representation can produce an inequality of the form . There is essentially only one known way of showing that a measure is reflection positive, which is to show that it has a density against a product measure which is of a prescribed form [@biskup_rp Lemma 4.4]. Preliminary calculations suggest that this method works also for the space–time Ising model. Although the random-current manipulations in [@af] leading up to are considerably more delicate than those presented in Chapter \[qim\_ch\] of this work and involve some new ideas such as ‘dilution’, preliminary calculations again suggest that it should be possible to extend them as required.
Random currents in the Potts model
----------------------------------
The main results of this work have relied on the random-parity representation for the space–time Ising model. It is natural to ask if there is a similar representation for the $q\geq 3$ Potts model. Here we will discuss this question, to start with in the context of the *classical* (discrete) Potts model on a finite graph $L=(V,E)$. For simplicity we will assume free boundary condition and zero external field; it is easy to adapt the results here to positive fields.
It is shown in [@grimmett_rcm Chapter 9] (see also [@essam_tsallis; @magalhaes_essam]) that the $q$-state Potts model with $q\geq3$ possesses a *flow representation*, which is akin to the random-current representation, in that the two-point correlation function may be written as the ratio of two expected values. This representation is as follows.
Let the integer $q\geq 2$ be fixed. For ${\underline}n=(n_e:e\in E)$ a vector of non-negative integers, define the graph $L_{{\underline}n}=(V,E_{{\underline}n})$ by replacing each edge $e$ of $L$ by $n_e$ parallel edges. If $P=(P_e:e\in E)$ is a collection of finite sets with $|P_e|=n_e$, we identify $L_P$ with $L_{{\underline}n}$, and interpret $P_e$ as the set of edges replacing $e$. We assign to the elements of $E_{{\underline}n}$ arbitrary directions and write $\vec{e}$ for directed elements of $E_{{\underline}n}$; if $\vec{e}$ is adjacent to a vertex $x\in V$ and is directed into $x$ we write $\vec{e}\mapsto x$, and if $\vec{e}$ is directed out of $x$ we write $\vec{e}\mapsfrom x$. We say that a function $f:E_{{\underline}n}\rightarrow\{1,\dotsc,q-1\}$ is a (nonzero) *mod $q$ flow on $L_{{\underline}n}$* (or $q$-flow for short) if for all $x\in V$ we have that $$\sum_{\substack{\vec{e}\in E_{{\underline}n}: \\e\mapsfrom x}} f(\vec{e})-
\sum_{\substack{\vec{e}\in E_{{\underline}n}: \\e\mapsto x}} f(\vec{e})
\equiv 0 \quad\text{(mod $q$)}.$$ Let $C(L_{{\underline}n};q)$ denote the number of mod $q$ flows on $L_{{\underline}n}$ (this is called the flow polynomial of $L_{{\underline}n}$). It is easy to see that this number does not depend on the directions chosen on the edges (if the direction of an edge $\vec{e}$ is reversed we can replace $f(\vec{e})$ by $q-f(\vec{e})$).
For each $e\in E$, let ${\beta}'_e\geq0$, and recall that the Potts weight of an element $\nu\in\{1,\dotsc,q\}^V={\mathcal{N}}$ is $$\label{p1}
\exp\Big(\sum_{e=xy\in E}{\beta}'_e{\delta}_{\nu_x,\nu_y}\Big),$$ so that the partition function is $$\label{p2}
Z=\sum_{\nu\in{\mathcal{N}}} \exp\Big(\sum_{e=xy\in E}{\beta}'_e{\delta}_{\nu_x,\nu_y}\Big).$$ Let ${\beta}_e={\beta}_e'/q$ and let the collection $P=(P_e:e\in E)$ of finite sets be given by letting the $|P_e|$ be independent Poisson random variables, each with parameter ${\beta}_e$. Write ${\mathbb{P}}_{\beta}$ for the probability measure governing the $P_e$ and ${\mathbb{E}}_{\beta}$ for the corresponding expectation operator.
The flow representation of $Z$ is $$\label{fl_z}
Z=\exp\Big(2\sum_{e\in E}{\beta}_e\Big)q^{|V|}{\mathbb{E}}_{\beta}[C(L_P;q)].$$ In fact, more is true. For $x,y\in V$, let $L_{{\underline}n}^{xy}=(V,E_{{\underline}n}\cup\{xy\})$ denote the graph $L_{{\underline}n}$ with an edge added from $x$ to $y$. Write ${\langle}\cdot{\rangle}$ for the expected value under the $q$-state Potts measure defined by –. Then for any $x,y\in V$ we have that $$\label{fl_2p}
q{\langle}{\hbox{\rm 1\kern-.27em I}}\{\nu_x=\nu_y\}{\rangle}-1=
\frac{{\mathbb{E}}_{\beta}[C(L_P^{xy};q)]}{{\mathbb{E}}_{\beta}[C(L_P;q)]}.$$
Here is a simple observation that changes the expected value in into a probability. For ${\underline}n\in{\mathbb{Z}}_+^E$, let $F_{q}({\underline}n)$ denote the set of functions $f:V\rightarrow\{1,\dotsc,q-1\}$. Then $$\label{fl_prob}
\begin{split}
{\mathbb{E}}_{\beta}[C(L_P;q)]&=\sum_{{\underline}n\in{\mathbb{Z}}_+^E}\prod_{e\in E}
\frac{{\beta}_e^{n_e}}{n_e!}e^{-{\beta}_e}\sum_{f\in F_{q}({\underline}n)}
{\hbox{\rm 1\kern-.27em I}}\{f\text{ is $q$-flow}\}\\
&=\exp\Big((q-2)\sum_{e\in E}{\beta}_e\Big)
\sum_{{\underline}n\in{\mathbb{Z}}_+^E}\prod_{e\in E}
\frac{((q-1){\beta}_e)^{n_e}}{n_e!}e^{-(q-1){\beta}_e}\\
&\qquad\cdot\frac{1}{(q-1)^{\sum_{e\in E}n_e}}
\sum_{f\in F_{q}({\underline}n)}{\hbox{\rm 1\kern-.27em I}}\{f\text{ is $q$-flow}\}\\
&=\exp\Big((q-2)\sum_{e\in E}{\beta}_e\Big)
{\mathbb{P}}(\psi\text{ is $q$-flow on $L_{P'}$}),
\end{split}$$ where, under ${\mathbb{P}}$, the collection $P'=(P'_e:e\in E)$ is given by letting the $|P'_e|$ be independent Poisson random variables with parameters $(q-1){\beta}_e$ respectively, and $\psi$ is, given $P'$, a uniformly chosen element of $F_q(P')$. (As before, arbitrary directions are assigned to the elements of $E_{P'}$, but the probability that $\psi$ is a $q$-flow does not depend on the choice of directions.)
We now show that a similar representation to holds for the two-point correlation functions , and indeed for more general correlation functions. As in Section \[gks\_sec\] we will use the variables $${\sigma}_x=\exp\Big(\frac{2\pi i\nu_x}{q}\Big), \qquad\nu_x=1,\dotsc,q.$$ We write $Q\subseteq{\mathbb{C}}$ for the set of $q$th roots of unity, and ${\Sigma}=Q^V$. For ${\underline}r\in{\mathbb{Z}}^V$ and ${\sigma}\in{\Sigma}$ we let $${\sigma}^{{\underline}r}=\prod_{x\in V}{\sigma}_x^{r_x}.$$ Note that it is equivalent to regard $r_x$ as an element of ${\mathbb{Z}}/(q{\mathbb{Z}})$, the integers modulo $q$. Let ${\mathbb{P}}$, $P'$ and $\psi$ be as in , and write $\{\psi\equiv 0\}$ for the event that $\psi$ is a $q$-flow. More generally, write $\{\psi+{\underline}r\equiv 0\}$ for the event that for each $x\in V$, $$\sum_{\substack{\vec{e}\in E_{P'}: \\e\mapsfrom x}}
\psi(\vec{e})-
\sum_{\substack{\vec{e}\in E_{P'}: \\e\mapsto x}}
\psi(\vec{e})
\equiv -r_x \quad\text{(mod $q$)}.$$ (Recall that we have assigned arbitrary directions to the elements of $E_{P'}$.)
\[potts\_flow\_thm\] In the discrete Potts model with zero field and coupling constants ${\beta}'_e$, $${\langle}{\sigma}^{{\underline}r}{\rangle}=
\frac{{\mathbb{P}}(\psi+{\underline}r\equiv 0)}{{\mathbb{P}}(\psi\equiv 0)}.$$
Before proving this, note that if ${\sigma}\in{\mathcal{N}}$ and $x,y\in V$, then $\tau_{xy}:={\sigma}_x{\sigma}_y^{-1}$ has the property that $\tau_{xy}=1$ if and only if ${\sigma}_x={\sigma}_y$, and in fact $$\frac{1}{q}\sum_{r=0}^{q-1}\tau_{xy}^r={\delta}_{{\sigma}_x,{\sigma}_y}.$$ Thus the partition function may be written $$\label{p3}
\begin{split}
Z&=\sum_{{\sigma}\in{\Sigma}}\exp\Big(\sum_{e=xy\in E}{\beta}'_e{\delta}_{{\sigma}_x,{\sigma}_y}\Big)\\
&=\sum_{{\sigma}\in{\Sigma}}\exp\Big(\frac{1}{2}\sum_{x,y\in V}
{\beta}_{xy}\sum_{r=1}^{q-1}\tau_{xy}^r\Big)
\cdot\exp\Big(\sum_{e\in E}{\beta}_e\Big),
\end{split}$$ where the first sum inside the exponential is over all ordered pairs $x,y\in V$, and we set ${\beta}_{xy}={\beta}_e$ if $e\in E$ is an edge between $x$ and $y$, and ${\beta}_{xy}=0$ otherwise. Note finally that $\tau_{xy}\neq\tau_{yx}$ in general.
We perform a calculation on the factor $$\sum_{{\sigma}\in{\Sigma}}\exp\Big(\frac{1}{2}\sum_{x,y\in V}
{\beta}_{xy}\sum_{r=1}^{q-1}\tau_{xy}^r\Big)$$ which appears on the right-hand-side of ; this will only re-prove the relation , but it will be clear that a simple extension of the calculation will give the result.
Let us write $\tilde{\beta}_{xy}={\beta}_{xy}/2$. We have that $$\label{rcr1}
\begin{split}
\sum_{{\sigma}\in{\Sigma}}\exp\Big(\frac{1}{2}\sum_{x,y\in V}
{\beta}_{xy}\sum_{r=1}^{q-1}\tau_{xy}^r\Big)
&=\sum_{{\sigma}\in{\Sigma}}\prod_{x,y\in V}\prod_{r=1}^{q-1}
\sum_{m\geq 0}\frac{1}{m!}(\tilde{\beta}_{xy}\tau_{xy}^r)^m\\
&=\sum_{{\sigma}\in{\Sigma}}\sum_{{\underline}m} w({\underline}m)
\prod_{x,y\in V}\prod_{r=1}^{q-1}(\tau_{xy}^r)^{m_{x,y,r}},
\end{split}$$ where the vector ${\underline}m=(m_{x,y,r}:x,y\in V,r=1,\dotsc,q-1)$ consists of non-negative integers and $$w({\underline}m)=\prod_{x,y\in V}\prod_{r=1}^{q-1}
\frac{\tilde{\beta}_{xy}^{m_{x,y,r}}}{m_{x,y,r}!}$$ is an un-normalized Poisson weight on ${\underline}m$. Reordering we obtain $$\label{rcr2}
\sum_{{\sigma}\in{\Sigma}}\exp\Big(\frac{1}{2}\sum_{x,y\in V}
{\beta}_{xy}\sum_{r=1}^{q-1}\tau_{xy}^r\Big)=
\sum_{{\underline}m}w({\underline}m)\sum_{{\sigma}\in{\Sigma}}\prod_{x,y\in V}
\tau_{xy}^{M_{xy}}$$ where $$M_{xy}=\sum_{r=1}^{q-1} r\cdot m_{x,y,r}.$$
We may interpret $m_{x,y,r}$ as a random number of edges, each of which is directed from $x$ to $y$ and receives flow value $r$. Then $M_{xy}$ is the total flow from $x$ to $y$. Up to the constant multiple $\exp\big((q-1)\sum_e{\beta}_e\big)$, the quantity equals the expected value of the quantity $$\label{rcr3}
\sum_{{\sigma}\in{\Sigma}}\prod_{x,y\in V}
\tau_{xy}^{M_{xy}}$$ when the $m_{x,y,r}$ have the Poisson distribution with parameter $\tilde{\beta}_{xy}$ and are chosen independently.
The quantity simplifies, as follows. Let $a\in V$ be fixed, and let $L_a=(V_a,E_a)$ denote $L$ with $a$ removed. Then $$\begin{split}
\sum_{{\sigma}\in{\Sigma}}\prod_{x,y\in V}\tau_{xy}^{M_{xy}}&=
\sum_{{\sigma}\in{\Sigma}}\Big(\prod_{b\sim a}\tau_{ab}^{M_{ab}}
\tau_{ba}^{M_{ba}}\Big)
\prod_{x,y\in V_a}\tau_{xy}^{M_{xy}}\\
&=\sum_{{\sigma}\in{\Sigma}}\Big(\prod_{b\sim a}{\sigma}_a^{M_{ab}-M_{ba}}
{\sigma}_b^{M_{ba}-M_{ab}}\Big)
\prod_{x,y\in V_a}\tau_{xy}^{M_{xy}}.
\end{split}$$ Write $M_a=\sum_{b\sim a}(M_{ab}-M_{ba})$. We may now take out the factor $$\label{p4}
\sum_{{\sigma}_a\in Q}{\sigma}_a^{M_a}=q\cdot {\hbox{\rm 1\kern-.27em I}}_{\{M_a\equiv 0\text{ (mod $q$)}\}}.$$ Proceeding as above with the remaining vertices of $L$ we obtain that $$\sum_{{\sigma}\in{\Sigma}}\prod_{x,y\in V}\tau_{xy}^{M_{xy}}=
q^{|V|}\cdot {\hbox{\rm 1\kern-.27em I}}\{M_a\equiv 0\text{ (mod $q$) for all }a\in V\}.$$ Thus $$Z=q^{|V|}\exp\Big(q\sum_{e\in E}{\beta}_e\Big)
\Pr(M_a\equiv 0\;\forall a\in V)$$
It remains to show that the distribution of $M$ coincides with that of $\psi$. This is easy: given $P'$, do the following. First, assign for all $e\in E$ each of the $|P'_e|$ edges replacing $e$ a direction uniformly a random; the number of edges directed from $x$ to $y$ then has the Poisson distribution with parameter $(q-1){\beta}_e/2$. Next, assign each directed edge a value $1,\dotsc,q-1$ uniformly at random; the number of edges directed from $x$ to $y$ with value $r$ then has the Poisson distribution with parameter $\tilde{\beta}_e$. The corresponding element of $F_q(P')$ is uniformly chosen given the edge numbers and directions, and since the probability of obtaining a $q$-flow does not depend on the choice of directions, we are done.
To obtain the full result in the theorem, repeat the above steps with the numerator of ${\langle}{\sigma}^{{\underline}r}{\rangle}$. The quantity $M_a$ in must then be replaced by $M_a+r_a$, but the rest of the calculation is as before. It follows that $$\label{p5}
{\langle}{\sigma}^{{\underline}r}{\rangle}=
\frac{q^{|V|}\exp\Big(q\sum_{e\in E}{\beta}_e\Big){\mathbb{P}}(\psi+{\underline}r\equiv 0)}
{q^{|V|}\exp\Big(q\sum_{e\in E}{\beta}_e\Big){\mathbb{P}}(\psi\equiv 0)}
=\frac{{\mathbb{P}}(\psi+{\underline}r\equiv 0)}{{\mathbb{P}}(\psi\equiv 0)}$$
It is straightforward to extend Theorem \[potts\_flow\_thm\] to an analogous representation for the space–time model, and we sketch this here. First, by conditioning on the set $D$, one obtains (as in ) a discrete graph $G(D)=(V(D),E(D))$. By applying the formulas in the numerator and denominator of on the graph $G(D)$, one obtains a representation of the form . One may then repeat the procedure in the proof of Theorem \[rcr\_thm\] to obtain a formula in terms of weighted labellings; these labellings are defined as follows.
Let ${\Lambda}=(K,F)$ and ${\beta}$ be as in Chapter \[qim\_ch\]. Fix an arbitrary ordering of the vertices $V$ of $L$. Let $B\subseteq F$ be a Poisson process with rate $(q-1){\lambda}$. We assign directions to the elements of $B$ by letting a bridge between $(u,t)\in K$ and $(v,t)\in K$ be directed from $u$ to $v$ if $u$ comes before $v$ in the ordering of $V$. We then assign to each element of $B$ a weight from $\{1,\dotsc,q-1\}$ uniformly at random, these choices being independent.
Let $A\subseteq K^\circ$ be a finite set (which lies in the interior of $K$ only for convenience of exposition). Let ${\underline}r=(r_x:x\in A)$ be a vector of integers, indexed by $A$, and let $S{\subseteq}K$ denote the union of $A$ with the set of endpoints of bridges in $B$. Given the above, a labelling $\psi^{{\underline}r}$ is a map $K\rightarrow{\mathbb{Z}}/(q{\mathbb{Z}})$, which is constrained to be ‘valid’ in that:
1. on each subinterval of each $K_v$, the label is constant between elements of $S$,
2. as we move along a subinterval of $K_v$ ($v\in V$) in the increasing ${\beta}$ direction, the label changes at elements of $S$; if the label is $t$ before reaching $x\in S$, then the label just after $x$ is
- $t+r$ if $x$ is the endpoint of a bridge directed into $x$ and which has weight $r$,
- $t-r$ if $x$ is the endpoint of a bridge directed out of $x$ with weight $r$,
- $t-r_x$ if $x\in A$,
3. as one moves towards an endpoint of an interval $I^v_k\neq{\mathbb{S}}$ (in either direction) the label converges to $0$.
As for the random-parity representation of the space–time Ising model, these conditions do not uniquely define $\psi^{{\underline}r}$ if there is a $v\in V$ such that $K_v={\mathbb{S}}$. If this is the case, the label at $0$ is chosen uniformly at random for each such $v$, these choices being independent.
A valid labelling is given the weight $$\partial\psi^{{\underline}r}:=\exp(q{\delta}({\mathcal{L}}_0(\psi^{{\underline}r}))),$$ where ${\mathcal{L}}_0(\psi^{{\underline}r})$ is the set labelled $0$ in $\psi^{{\underline}r}$. In the following, ${\underline}r=0$ denotes the vector which takes the value $0$ at all $x\in A$; we let $E(\cdot)$ denote the expectation over $B$ as well as the weights assigned to the elements of $B$, and the randomization which takes place when there are several valid labellings.
\[st\_potts\_flow\_thm\] In the space–time Potts model, $${\langle}{\sigma}^{{\underline}r}{\rangle}=\frac{E(\partial\psi^{{\underline}r})}
{E(\partial\psi^0)}.$$
The usefulness of Theorems \[potts\_flow\_thm\] and \[st\_potts\_flow\_thm\] when $q\geq3$ is questionable. Mod $q$ flows with $q\geq3$ are considerable more complicated than mod $2$ flows, and there does not seem to be a useful switching lemma (along the lines of Theorem \[sl\] or its discrete version [@abf]) for general $q$.
The Skorokhod metric and tightness {#skor_app}
==================================
In this appendix we define carefully the Skorokhod metric on ${\Omega}$ and show that the sequence $\phi^b_n$ of random-cluster measures in Section \[rc\_wl\_sec\] is tight, proving Lemma \[tight\_lem\]. We will rely partly on the notation and results in [@ethier_kurtz Chapter 3]; see also [@lindvall Appendix 1].
A function $f:{\mathbb{R}}\rightarrow{\mathbb{R}}$ is called *càdlàg* if it is right-continuous and has left limits. We let ${\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}})$ denote the set of increasing càdlàg step functions on ${\mathbb{R}}$ with values in ${\mathbb{Z}}$, and which take the value $0$ at $0$. It is straightforward to modify the definitions and results of [@ethier_kurtz Chapter 3], which concern càdlàg functions on $[0,{\infty})$ with values in some metric space $E$, to apply to the set ${\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}})$. Specifically, we define the Skorokhod metric on ${\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}})$ as follows. Let $U$ denote the set of strictly increasing bijections $u:{\mathbb{R}}\rightarrow{\mathbb{R}}$ which are Lipschitz continuous and for which the quantity $${\alpha}(u):=\sup_{t>s}\;\log\Big| \frac{u(t)-u(s)}{t-s}\Big|$$ is finite. For $a,b\in{\mathbb{Z}}$ let $r(a,b)={\delta}_{a,b}$, and note that $r$ is a metric on ${\mathbb{Z}}$. The Skorokhod metric on ${\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}})$ is by definition given by $$d'(f,g)=\inf_{u\in U}
\Big[{\alpha}(u)\wedge \int_{-y}^y e^{-|y|}d'(f,g,u,y)\,dy\Big],$$ where $$d'(f,g,u,y)=\sup_{t\in{\mathbb{R}}}
r(f((t\wedge y)\vee -y),g((u(t)\wedge y)\vee -y)).$$ It may be checked, as in [@ethier_kurtz pp. 117], that $d'$ is indeed a metric, and that the metric space $({\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}}),d')$ is complete and separable.
Recall that we are given a countable graph ${\mathbb{L}}=({\mathbb{V}},{\mathbb{E}})$. Let ${\mathbb{T}}$ denote the countable set $${\mathbb{T}}=({\mathbb{V}}\times\{\mathrm{d}\})\cup({\mathbb{V}}\times\{\mathrm{g}\})\cup{\mathbb{E}},$$ and let $\upsilon:{\mathbb{T}}\rightarrow\{1,2,\dotsc\}$ denote an arbitrary bijection. Then we formally define the set ${\Omega}$ to be the product space ${\Omega}={\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}})^{\mathbb{T}}$. For ${\omega}\in{\Omega}$ and $x\in{\mathbb{T}}$, the restriction ${\omega}_x$ of ${\omega}$ to $x\times{\mathbb{R}}$ (not to be confused with the ${\omega}_x$ of Section \[rcm\_si\]) is to be interpreted as: the process of deaths on $x\times{\mathbb{R}}$ if $x\in{\mathbb{V}}\times\{\mathrm{d}\}$, or the process of ghost-bonds on $x\times{\mathbb{R}}$ if $x\in{\mathbb{V}}\times\{\mathrm{g}\}$, or the process of bridges on $x\times{\mathbb{R}}$ if $x\in{\mathbb{E}}$. In this section we do *not* overlook events of probability zero, that is Remark \[rem-as\] does not apply.
We define the Skorokhod metric $d$ on ${\Omega}$ by $$d({\omega},{\omega}')=\sum_{x\in{\mathbb{T}}}e^{-\upsilon(x)}d'({\omega}_x,{\omega}_x').$$
Note that the sum is absolutely convergent since $d'$ is bounded, and in fact also $d$ is bounded. It is straightforward to check that $d$ is indeed a metric on ${\Omega}$, and (using the dominated convergence theorem) that $({\Omega},d)$ is a complete metric space. It is also separable, hence Polish. The ${\sigma}$-algebra ${\mathcal{F}}$ on ${\Omega}$ generated by $d$ agrees with that generated by all the coordinate functions $\pi_{x,t}:{\omega}\mapsto{\omega}_x(t)$ for $x\in{\mathbb{T}}$ and $t\in{\mathbb{R}}$, see [@ethier_kurtz Proposition 3.7.1]. The fact that all finite tuples of such coordinate functions forms a convergence determining class (a fact used in Theorem \[conv\_lem\]) follows as in [@ethier_kurtz Theorem 3.7.8].
In order to establish tightness of the sequence $\phi^{b_n}_n$ we must find compact sets in ${\Omega}$. Since $({\Omega},d)$ is a metric space, compactness is equivalent to sequential compactness. If for each $x\in{\mathbb{T}}$, the set $A_x$ is (sequentially) compact in $({\mathcal{D}}^0_{\mathbb{Z}}({\mathbb{R}}),d')$, then by a straightforward diagonal argument the set $A=\bigotimes_{x\in{\mathbb{T}}}A_x$ is a compact subset of $({\Omega},d)$.
As a witness for the tightness of $\{\phi^{b_n}_n:n\geq 1\}$ we will use the product $A$ of the following compact sets $A_x$. For each $x\in{\mathbb{T}}$, let $\xi_x:[0,{\infty})\rightarrow(0,{\infty})$ be a strictly positive function, to be specified later. Let $A_x$ be the set of ${\omega}\in{\Omega}$ such that for all $t>0$, all jumps of ${\omega}_x$ in the interval $[-t,t]$ are separated from each other by at least $\xi_x(t)$. It follows from the characterization in [@ethier_kurtz Theorem 3.6.3] that $A_x$ is compact (alternatively, it is not hard to deduce the sequential compactness of $A_x$ using a diagonal argument).
It remains to show that we can choose the functions $\xi_x$ so as to get a uniform lower bound on $\phi^{b_n}_n(A)$ which is arbitrarily close to 1. We can use stochastic domination, Corollary \[comp\_perc\], to reduce this to checking the tightness of *a single percolation measure*, as follows. If $x\in{\mathbb{V}}\times\{\mathrm{d}\}$ then the event $A_x$ is increasing, otherwise it is decreasing. Thus $A=\bigcap_{x\in{\mathbb{T}}}A_x=A^+\cap A^-$ where $$A^+=\bigcap_{x\in{\mathbb{V}}\times\{\mathrm{d}\}}A_x\qquad\text{and}\qquad
A^-=\bigcap_{x\in({\mathbb{V}}\times\{\mathrm{g}\})\cup{\mathbb{E}}}A_x$$ are increasing and decreasing events, respectively. We have that $$\label{ek1}
\phi^{b_n}_n(A)\geq\phi^{b_n}_n(A^+)+\phi^{b_n}_n(A^-)-1.$$ The events $A^+,A^-$ are not local events, but by writing them as decreasing limits of local events it is easy to justify the following application of Corollary \[comp\_perc\] to . For suitable choices of the parameters ${\lambda}_i,{\delta}_i,{\gamma}_i$ ($i=1,2$) which are multiples of the original parameters ${\lambda},{\delta},{\gamma}$ we have that $$\label{ek2}
\phi^{b_n}_n(A)\geq\mu_{{\lambda}_1,{\delta}_1,{\gamma}_1}(A^+)+\mu_{{\lambda}_2,{\delta}_2,{\gamma}_2}(A^-)-1.$$ Clearly, any lower bound on the right-hand-side of is a uniform lower bound on the $\phi^{b_n}_n(A)$.
Let us focus on $A^+$, since $A^-$ is similar. Suppose we can, for any ${\varepsilon}>0$, choose $\xi_x$ so that $$\mu_{{\lambda}_1,{\delta}_1,{\gamma}_1}(A_x)\geq e^{-{\varepsilon}/\upsilon(x)^2}.$$ Then, since the $A_x$ are independent under $\mu_{{\lambda}_1,{\delta}_1,{\gamma}_1}$, we will have that $$\mu_{{\lambda}_1,{\delta}_1,{\gamma}_1}(A^+)\geq\exp\Big(-{\varepsilon}\frac{\pi^2}{6}\Big),$$ which is enough. The event $A_x$ concerns only the process $D$ of deaths on $x\times{\mathbb{R}}$. We may replace ${\delta}_1$ by a constant upper bound. By adjusting parameters it follows that we are done if we prove the following: for any ${\varepsilon}>0$ we have that $$\label{ek3}
P(N\in A_x)\geq 1-{\varepsilon},$$ where $P$ is the measure governing the Poisson process $N$ of rate 1 on ${\mathbb{R}}$. The proof of is a straightforward exercise on Poisson processes, but we include it for completeness.
For $I\subseteq{\mathbb{R}}$ and $a\in{\mathbb{R}}$ we write $aI=\{at:t\in I\}$. Define $I_1^+=I_1^-=[-1,1]$ and for $k\geq2$ let $I^+_k$ be the closed interval of length $1/k$ with left endpoint $1+1/2+1/3+\dotsb+1/(k-1)$; let $I^-_k=-I^+_k$. Since the series $\sum\tfrac{1}{k}$ diverges, the $I^\pm_k$ ($k\geq1$) cover ${\mathbb{R}}$. Next let $J^+_k$ ($k\geq1$) be the closed interval whose left and right endpoints are at the midpoints of $I^+_k$ and $I^+_{k+1}$ respectively; let $J^-_k=-J^+_k$. Note that $|J^\pm_k|=(|I^\pm_k|+|I^\pm_{k+1}|)/2\geq \tfrac{1}{k+1}$. Let ${\varepsilon}>0$ and let $A'$ be the event that each ${\varepsilon}I^\pm_k$ and each ${\varepsilon}J^\pm_k$ ($k\geq1$) contains at most one element of $N$.
We claim that $A'\subseteq A_x$ for $\xi_x(t)={\varepsilon}e^{-t/{\varepsilon}}/4$. Suppose $A'$ happens and $s\in N$. We may assume $s\in{\varepsilon}I^+_k$ with $k\geq2$ (the other cases are similar). Then $s$ also lies in either ${\varepsilon}J^+_{k-1}$ or ${\varepsilon}J^+_k$. Hence the closest possible other point of $N$ is a distance at least $\tfrac{{\varepsilon}}{2(k+1)}$ from $s$. Let $t>0$ and suppose $s\in N\cap[0,t]$. Let $k$ be maximal with $I^+_k\cap[0,t]\neq\varnothing$. Then $$t\geq{\varepsilon}\sum_{i=1}^{k-1}\frac{1}{i}\geq{\varepsilon}\log k,$$ and the closest point to $s$ in $N$ is a distance at least $$\frac{{\varepsilon}}{2(k+1)}\geq\frac{{\varepsilon}}{2(e^{t/{\varepsilon}}+1)}\geq
\frac{{\varepsilon}}{4}e^{-t/{\varepsilon}}.$$ Similarly if $s<0$. Hence $A'\subseteq A_x$ as claimed.
It is well-known that there is an absolute constant $C$ such that for $\eta>0$ small and $I$ a fixed interval of length at most $\eta$, we have that $P(|N\cap I|\geq 2)\leq C\eta^2$. Clearly we have $$\begin{split}
P(N\in A')&\geq 1-2\sum_{k\geq 2} P(|N\cap {\varepsilon}I^+_k|\geq 2)-\\
&\qquad-2\sum_{k\geq 1} P(|N\cap {\varepsilon}J^+_k|\geq 2)
- P(|N\cap {\varepsilon}I_1|\geq 2)\\
&\geq 1-{\varepsilon}^2 C\cdot 2\pi^2/3.
\end{split}$$ This proves the result.
Proof of Proposition \[cond\_meas\_rcm\] {#pfs_app}
========================================
This is essentially straightforward, but notationally intricate. We write $(\eta,{\omega},\tau)_{{\Lambda},{\Delta}}$ for the configuration which equals $\eta$ inside the smallest set ${\Lambda}$, equals ${\omega}$ in the intermediate region ${\Delta}\setminus{\Lambda}$, and equals $\tau$ outside ${\Delta}$. For readability, let us write $k_{\Lambda}(\cdot;\tau)$ in place of $k_{\Lambda}^\tau(\cdot)$ in what follows.
Let $A'\in{\mathcal{F}}_{{\Delta}\setminus{\Lambda}}$. Then $$\begin{gathered}
\label{abc0}
\phi^\tau_{\Delta}({\hbox{\rm 1\kern-.27em I}}_{A'}(\cdot)\phi^{(\cdot,\tau)_{\Delta}}_{\Lambda}(A))=
\iint{\hbox{\rm 1\kern-.27em I}}_{A'}({\omega}){\hbox{\rm 1\kern-.27em I}}_A((\eta,{\omega},\tau)_{{\Lambda},{\Delta}})
\:d\phi^{({\omega},\tau)_{\Delta}}_{\Lambda}(\eta)d\phi^\tau_{\Delta}({\omega})
\\=\iint{\hbox{\rm 1\kern-.27em I}}_{A\cap {A'}}((\eta,{\omega},\tau)_{{\Lambda},{\Delta}})
\frac{q^{k_{\Lambda}(\eta;({\omega},\tau)_{\Delta})}}{Z^{({\omega},\tau)_{\Delta}}_{\Lambda}}
\frac{q^{k_{\Delta}({\omega};\tau)}}{Z^{\tau}_{\Delta}}\:d\mu(\eta)d\mu({\omega}).\end{gathered}$$ Note that if $({\alpha},{\beta})_{\Lambda}\in{\Omega}$ then $$\label{cpt_count_eq}
k_{\Delta}(({\alpha},{\beta})_{\Lambda};\tau)=k_{\Lambda}(\alpha;(\beta,\tau)_{\Delta})+\tilde k_{\Delta}(\beta;\tau),$$ where $\tilde k_{\Delta}$ counts the number of components in ${\Delta}$ which do not intersect ${\Lambda}$. Let ${\alpha},{\beta}$ be independent with law $\mu$; then ${\omega}$ has the law of $({\alpha},{\beta})_{\Lambda}$. Use on each power of $q$ in to see that $$\begin{aligned}
\phi^\tau_{\Delta}\big(&{\hbox{\rm 1\kern-.27em I}}_{A'}(\cdot)\phi^{(\cdot,\tau)}_{\Lambda}(A)\big)\\
&=\iiint{\hbox{\rm 1\kern-.27em I}}_{A\cap {A'}}((\eta,\beta,\tau)_{{\Lambda},{\Delta}})
\frac{q^{k_{\Lambda}(\alpha;(\beta,\tau)_{\Delta})}q^{k_{\Delta}((\eta,\beta)_{\Lambda};\tau)}}
{Z_{\Lambda}^{(\beta,\tau)_{\Delta}}Z_{\Delta}^\tau}
\:d\mu(\eta)d\mu(\alpha)d\mu(\beta)\\
&=\int{\hbox{\rm 1\kern-.27em I}}_{A\cap {A'}}(({\omega}',\tau)_{\Delta})
\frac{q^{k_{\Delta}({\omega}';\tau)}}{Z^\tau_{\Delta}}
\bigg(\int\frac{q^{k_{\Delta}(\alpha;({\omega}',\tau))}}{Z^{({\omega}',\tau)}_{\Lambda}}
\:d\mu(\alpha)\bigg)\:d\mu({\omega}')\\
&=\phi_{\Delta}^\tau(A\cap {A'}),\end{aligned}$$ where ${\omega}'=(\eta,\beta)_{\Lambda}$. This proves the claim.
|
Introduction
============
A broad suite of astrophysical and cosmological observations provides compelling evidence for the existence of dark matter. Although its ultimate nature is unknown, the large-scale dynamics of dark matter is essentially that of a self-gravitating collisionless fluid. In an expanding universe, gravitational instability leads to the formation and growth of structure in the dark matter distribution. The existence of localized, highly overdense dark matter clumps, or halos, is a key prediction of cosmological nonlinear gravitational collapse. The distribution of dark matter halo masses is termed the halo mass function and constitutes one of the most important probes of cosmology. At low redshifts, $z\leq 2$, the mass function at the high-mass end (cluster scales) is very sensitive to variations in cosmological parameters, such as the matter content of the Universe $\Omega_{\rm m}$, the dark energy content along with its equation-of-state parameter, $w$ [@Holder01], and the normalization of the primordial fluctuation power spectrum, $\sigma_8$. At higher redshifts, the halo mass function is important in probing quasar abundance and formation sites [@Haiman01], as well as the reionization history of the Universe [@Furl06].
Many recently suggested reionization scenarios are based on the assumption that the mass function is given reliably by modified Press-Schechter type fits (Press & Schechter 1974, hereafter PS; Bond et al. 1991). However, the theoretical basis of this approach is at best heuristic and careful numerical studies are required in order to obtain accurate results. Two examples serve to illustrate this statement. Reed et al. (2003) report a discrepancy with the Sheth-Tormen fit (Sheth & Tormen 1999, hereafter ST) of $\sim$50% at a redshift of $z=15$ (we explain the different fitting formulae and their origin in §\[massf\]). Heitmann et al. (2006a) show that the Press-Schechter form can be severely incorrect at high redshifts: at $z\ge 10$, the predicted mass function sinks below the numerical results by an order of magnitude at the upper end of the relevant mass scale. Consequently, incorrect, or at best imprecise, predictions for the reionization history can result from the failure of fitting formulae.
Since halo formation is a complicated nonlinear gravitational process, the current theoretical understanding of the mass, spatial distribution, and inner profiles of halos remains at a relatively crude level. Numerical simulations are therefore crucial as drivers of theoretical progress, having been instrumental in obtaining important results such as the Navarro-Frenk-White (NFW) profile [@NFW97] for dark matter halos and an (approximate) universal form for the mass function (Jenkins et al. 2001, hereafter Jenkins). In order to better understand the evolution of the mass function at high redshifts, a number of numerical studies have been carried out. High–redshift simulations, however, suffer from their own set of systematic issues, and simulation results can be at considerable variance with each other, differing on occasion by as much as an order of magnitude!
Motivated by all of these reasons we have carried out a numerical investigation of the evolution of the mass function with the aim of attaining good control over both statistical and, more importantly, possible systematic errors in $N$-body simulations. Our first results have been reported in condensed form in Heitmann et al. (2006a). Here we provide a more detailed and complete exposition of our work, including several new results.
We first pay attention to simulation criteria for obtaining accurate mass functions with the aim of reducing systematic effects. Our two most significant points are that simulations must be started early enough to obtain accurate results and that the box sizes must be large enough to suppress finite-volume artifacts. As in most recent work following that of Jenkins, we define halo masses using a friends-of-friends (FOF) halo finder with linking length $b=0.2$. This choice introduces systematic issues of its own (e.g., connection to spherical overdensity mass as a function of redshift), which we touch on as relevant below. As it is not quantitatively significant in the context of this paper, we leave a detailed discussion to later work (Z. Lukić et al., in preparation; see also @Reed07).
The more detailed results in this paper enable us to study the mass function at statistical and systematic accuracies reaching a few percent over most of our redshift range, a substantial improvement over most previous work. At this level we find discrepancies with the “universal” fit of Jenkins at low redshifts ($z<5$), but it must be kept in mind that the universality of the original fit was only meant to be at the $\pm
20\%$ level. Recently, [@Reed07] have found violation of universality at high redshifts (up to $z=30$). To fit the mass function they have incorporated an additional free parameter, the effective spectral index $n_{\rm eff}$, with the aim of understanding and taking into account the extra redshift dependence missing from conventional mass–function–fitting formulae. Our simulation results are consistent with the trends found by [@Reed07] at low redshifts ($z\leq 5$), but at higher redshifts we do not observe a statistically significant violation of the universal form of the mass function.
Results from some previous simulations have reported good agreement with the Press-Schechter mass function at high redshifts. Since the Press-Schechter fit has been found significantly discrepant with low–redshift results ($z<5$), this would imply a strong disagreement with extending the well-validated low–redshift notion of (approximate) mass function universality to high $z$. Our conclusion is that the simulations on which these findings were based violated one or more of the criteria to be discussed below.
As simulations are perforce restricted to finite volumes, the obtained mass function clearly cannot represent that of an infinite box. Not only is sampling a key issue, but also the fact that simulations with periodic boundary conditions have no fluctuations on scales larger than the box size. To minimize and test for these effects we were conservative in our choices of box size and the mass range probed in each individual box. We also used nested-volume simulations to directly test for finite-volume effects. Because we used multiple boxes and averaged mass function results over the box ensemble, extended Press-Schechter theory can be used to correct for residual finite volume–effects (@Mo96; @Barkana04); this approach is different from the individual box corrections applied by [@Reed07]. Details are given in §\[simvol\].
The paper is organized as follows. In §\[massf\] we give a brief overview of the mass function and popular fitting formulae, discussing as well previous numerical work on the halo mass function at high redshifts. In §\[code\] we give a short description of the $N$-body code MC$^2$ ([**M**]{}esh-based [**C**]{}osmology [ **C**]{}ode) and a summary of the performed simulations. In §\[icevol\] we derive and discuss some simple criteria for the starting redshift and consider systematic errors related to the numerical evolution such as mass and force resolution and time stepping. These considerations in turn specify the input parameters for the simulations in order to span the desired mass and redshift range for our investigation. In §\[resint\] we present results for the mass function at different redshifts as well as the halo growth function. Here we also discuss the importance of post-processing corrections such as FOF particle sampling compensation and finite-volume effects. We discuss our results and conclude in §\[conclusion\].
Definitions and Previous Work {#massf}
=============================
The mass function describes the number density of halos of a given mass. In order to determine the mass function in simulations one has to first [*identify*]{} the halos and then [*define*]{} their mass. No precise theoretical basis exists for these operations. Nevertheless, depending on the situation at hand, the observational and numerical communities have adopted a few “standard” ways of defining halos and their associated masses. For a recent review of these issues with regard to observations, see, e.g., [@Voit05], but for a more theoretically oriented review, see, e.g., [@White01].
Halo Mass {#halomass}
---------
There are basically two ways to find halos in a simulation. One, the overdensity method, is based on identifying overdense regions above a certain threshold. The threshold can be set with respect to the critical density $\rho_{\rm c}=3H^2/8\pi G$ (or the background density $\rho_{\rm b}=\Omega_{\rm m}\rho_{\rm c}$, where $\Omega_{\rm m}$ is the matter density of the Universe including dark matter and baryons). The mass $M_\Delta$ of a halo identified this way is defined as the mass enclosed in a sphere of radius $r_\Delta$ whose mean density is $\Delta\rho_{\rm c}$. Common values for $\Delta$ range from 100 to 500 (or even higher). As explained in [@Voit05], cluster observers prefer higher values for $\Delta$. Properties of clusters are easier to observe in higher density regions and these regions are more relaxed than the outer parts which are subject to the effects of inflow and incomplete mixing. The disadvantage of defining a halo in this manner is that sphericity of halos is implied, an assumption which may be easily violated, e.g., in the case of halos that formed in a recent merger event or halos at high redshifts. At higher redshifts, the nonlinear mass scale $M_*$ decreases rapidly, and the ratio of the considered halo mass $M_{\rm halo}$ to $M_*$ can become large. This translates into producing large-scale structures roughly analogous to supercluster structures today. While these structures are gravitationally bound, they are often not virialized, nor spherical. Even the much smaller structures (which are considered in this paper) are not virialized at high redshifts, and therefore, assumptions about sphericity are most likely violated. Hence the spherical overdensity method does not suggest itself as an obvious way to identify halos at high redshift.
The other method, the FOF algorithm, is based on finding neighbors of particles and neighbors of neighbors as defined by a given separation distance (see, e.g., Einasto et al. 1984; Davis et al. 1985). The FOF algorithm leads to halos with arbitrary shapes since no prior symmetry assumptions have been made. The halo mass is defined simply as the sum of particles which are members of the halo. While this definition is easy to apply to simulations, the connection to observations is difficult to establish directly. (For an investigation of connections between different definitions of halos masses and approximate conversions between them, see White 2001).
It is important to keep in mind that the definition of a halo is essentially the adoption of some sort of convention for the halo boundary. In reality, a sharp distinction between the particles in a halo and particles in the simulation “field” does not exist. Jenkins showed that the choice of a FOF finder with a linking length $b=0.2$ to define halo masses provides the best fit for a universal form of the mass function. This choice has since been adopted by many numerical practitioners as a standard convention. A useful discussion of the various halo definitions can be found in [@White02].
In this paper we use the FOF algorithm to identify halos and their masses. It was recently pointed out by Warren et al. (2006, hereafter Warren) that FOF masses suffer from a systematic problem when halos are sampled by relatively small numbers of particles. Although halos can be robustly identified with as few as 20 particles, if a given halo has too few particles, its FOF mass turns out to be systematically too high. We describe how we compensate for this effect in §\[masscorr\]. In the current paper, all results for the mass function are displayed at a fixed FOF linking length of $b=0.2$, using the Warren correction.
Defining the Mass Function {#massdef}
--------------------------
The exact definition of the mass function, e.g., integrated versus differential form or count versus number density, varies widely in the literature. To characterize different fits, Jenkins introduced the scaled differential mass function $f(\sigma, z)$ as a fraction of the total mass per $\ln\sigma^{-1}$ that belongs to halos: $$\label{fsigma}
f(\sigma, z) \equiv \frac{d\rho/\rho_b}{d\ln\sigma^{-1}}
=\frac{M}{\rho_{\rm b}(z)} \frac{dn(M,z)}
{d\ln[\sigma^{-1}(M,z)]}.$$ Here $n(M,z)$ is the number density of halos with mass $M$, $\rho_{\rm b}(z)$ is the background density at redshift $z$, and $\sigma(M,z)$ is the variance of the linear density field. As pointed out by Jenkins, this definition of the mass function has the advantage that to a good accuracy it does not explicitly depend on redshift, power spectrum, or cosmology; all of these are encapsulated in $\sigma(M,z)$. For the most part, we will display the mass function $$\label{FM}
{F}(M,z) \equiv {dn \over d\log M}$$ as a function of $\log M$ itself. \[In §\[resint\] we include results for $f(\sigma,z)$.\]
To compute $\sigma(M,z)$, the power spectrum $P(k)$ is smoothed with a spherical top-hat filter function of radius $R$, which on average encloses a mass $M$ ($R =
[3M/4\pi \rho_{\rm b}(z)]^{1/3}$): $$\sigma^2(M,z) =
\frac{d^2(z)}{2\pi^2}\int^{\infty}_{0}k^2P(k)W^2(k,M)dk,
\label{sig}$$ where $W(k,M)$ is the top-hat filter: $$\begin{aligned}
W(r) & = & \left\{ \begin{array}{rl}
\frac{3}{4 \pi R^3}, & r<R \\
0, & r>R
\end{array} \right.\\
W(k) & = & \frac{3}{(kR)^3} \left[ \sin (kR) - kR \cos (kR) \right].\end{aligned}$$ The redshift dependence enters only through the growth factor $d(z)$, normalized so that $d(0)=1$: $$\sigma(M,z) = \sigma(M,0) d(z).$$ In the approximation of negligible difference in the CDM and baryon peculiar velocities, the growth function in a $\Lambda$CDM universe is given by (Peebles 1980) $$d(a) = \frac{D^{+}(a)}{D^{+}(a=1)},$$ where we consider $d$ as a function of the cosmological scale factor $a=1/(1+z)$, and $$D^+(a)=\frac{5\Omega_{\rm m}}{2}\,\frac{H(a)}{H_0}
\int_0^a\frac{da'}{[a'H(a')/H_0]^3}
\label{D_no_baryons}$$ with $H(a)/H_0= \left[\Omega_{\rm m}/a^3+(1-\Omega_{\rm
m})\right]^{1/2}$. In particular, for $z\gg1$, when matter dominates the cosmological constant, $D^+(a)\simeq a$.
Even in linear theory, equation (\[D\_no\_baryons\]) is only an approximation because baryons began their gravitational collapse with velocities different from those of CDM particles. Until recombination at $z\sim 1100$, well into the matter era with non-negligible growth of CDM inhomogeneities, the baryons were held against collapse by the pressure of the CMB photons (see, e.g. @husug96). While thereafter the relative baryon-CDM velocity decayed as $1/a$, the residual velocity difference was sufficient to affect the growth function $d(z)$ at $z=50$ by more than $1\%$ and at $z=10$ by about $0.2\%$ (@Yoshida03bar; @Naoz06).
Fitting Functions {#fits}
-----------------
Over the last three decades several different fitting forms for the mass function have been suggested. The mass function is not only a sensitive measure of cosmological parameters by itself but also a key ingredient in analytic and semianalytic modeling of the dark matter distribution, as well as of several aspects of the formation, evolution, and distribution of galaxies. Therefore, if a reliable and accurate fit for the mass function applicable to a wide range of cosmologies and redshifts were to exist, it would be of obvious utility. In this section we briefly review the common fitting functions and compare them at different redshifts.
The first analytic model for the mass function was developed by PS. Their theory accounts for a spherical overdense region in an otherwise smooth background density field, which then evolves as a Friedmann universe with a positive curvature. Initially, the overdensity expands, but at a slower rate than the background universe (thus enhancing the density contrast), until it reaches the ‘turnaround’ density, after which collapse begins. Although from a purely gravitational standpoint this collapse ends with a singularity, it is assumed that in reality – due to the spherical symmetry not being exact – the overdense region will virialize. For an Einstein-de Sitter universe, the density of such an overdense region at the virialization redshift is $z\approx 180
\rho_{\rm c}(z)$. At this point, the density contrast from the linear theory of perturbation growth \[$\delta(\vec{x},z) = d(z)
\delta(\vec{x},0)$\] would be $\delta_{\rm c}(z) \approx 1.686$ in an Einstein-de Sitter cosmology. For $\Omega_{\rm m} < 1$, the value of the threshold parameter $\delta_{\rm c}$ can vary (see Lacey & Cole 1993), but the dependence on cosmology has little quantitative significance (see, e.g., Jenkins). Thus, throughout this paper we adopt $\delta_{\rm c} = 1.686$.
[llcc]{} Reference & Fitting Function $f(\sigma)$& Mass Range & Redshift range\
ST, Sheth & Tormen (2001) & $0.3222 \sqrt{\frac{2a}{\pi}}
\frac{\delta_{\rm c}}{\sigma} \exp \left[ - \frac{a \delta^2_{\rm
c}}{2\sigma^2} \right] \left[ 1 + \left( \frac{\sigma^2}{a
\delta^2_{\rm c}} \right) ^ p \right]$ & unspecified & unspecified\
Jenkins &$ 0.315 \exp
\left[ -| \ln \sigma^{-1} + 0.61 |^{3.8} \right]$ & $-1.2\le\ln\sigma^{-1}\le1.05$ & $z=0-5$\
Reed et al. (2003) &$ f_{\rm ST}(\sigma)\,
\exp\left\{-0.7/\left[\sigma(\cosh(2\sigma))^5\right]\right\}$& $-1.7\le
\ln\sigma^{-1}\le0.9$ & $z=0-15$\
Warren & $0.7234 \left( \sigma^{-1.625} + 0.2538 \right)
\exp \left[ -\frac{1.1982}{\sigma^2}\right]$ & $(10^{10}-10^{15}) h^{-1}M_{\odot} $ & $z=0$\
Reed et al. (2007) & $A \sqrt{\frac{2a}{\pi}} \left[1 +
\left(\frac{\sigma^2}{a\delta_c^2}\right)^p+0.6G_1(\sigma)+0.4G_2(\sigma)\right]$ & $-0.5\le\ln\sigma^{-1}\le1.2$ & $z=0-30$\
& $ \times\frac{\delta_c}{\sigma}
\exp\left[-\frac{ca\delta_c^2}{2\sigma^2}
-\frac{0.03}{(n_{\rm eff}+3)^2}
\left(\frac{\delta_c}{\sigma}\right)^{0.6}\right]$ &\
Following the above reasoning and with the assumption that the initial density perturbations are described by a homogeneous and isotropic Gaussian random field, the PS mass function is specified by $$f_{\rm PS}(\sigma) = \sqrt{\frac{2}{\pi}} \frac{\delta_{\rm c}}{\sigma}
\exp \left( - \frac{\delta^2_{\rm c}}{2\sigma^2} \right).$$ The PS approach assumes that all mass is inside halos, as enforced by the constraint $$\int^{+\infty}_{-\infty} f_{\rm PS}(\sigma)\, d\ln \sigma^{-1} = 1.$$ While as a first rough approximation the PS mass function agrees with simulations at $z=0$ reasonably well, it overpredicts the number of low–mass halos and underpredicts the number of massive halos at the current epoch. Furthermore, it is significantly in error at high redshifts (see, e.g., Springel et al. 2005; Heitmann et al. 2006a; §\[timeevo\]).
After PS, several suggestions were made in order to improve the mass function fit. These suggestions were based on more refined dynamical modeling, direct fitting to simulations, or a combination of the two.
Using empirical arguments ST proposed an improved mass function fit of the form: $$f_{\rm ST}(\sigma) = 0.3222 \sqrt{\frac{2a}{\pi}} \frac{\delta_{\rm
c}}{\sigma} \exp \left( - \frac{a \delta^2_{\rm c}}{2\sigma^2} \right)
\left[ 1 + \left( \frac{\sigma^2}{a \delta^2_{\rm c}} \right) ^ p \right],$$ with $a=0.707$ and $p=0.3$. (Sheth & Tormen 2002 suggest $a=0.75$ as an improved value.) Sheth et al. (2001) rederived this fit theoretically by extending the PS approach to an elliptical collapse model. In this model, the collapse of a region depends not only on its initial overdensity but also on the surrounding shear field. The dependence is chosen such that it recovers the Zel’dovich approximation [@Zeldovich70] in the linear regime. A halo is considered virialized when the third axis collapses (see also Lee & Shandarin (1998) for an earlier, different approach to the same idea).
Jenkins combined high resolution simulations for four different CDM cosmologies ($\tau$CDM, SCDM, $\Lambda$CDM, and OCDM) spanning a mass range of over 3 orders of magnitude ($\sim (10^{12}-10^{15}) \,h^{-1}M_{\sun}$), and including several redshifts between $z=5$ and 0. Independent of the underlying cosmology, the following fit provided a good representation of their numerical results (within $\pm 20\%$): $$f_{\rm Jenkins}(\sigma) = 0.315 \exp
\left( -| \ln \sigma^{-1} + 0.61 |^{3.8} \right).$$ The above formula is very close to the Sheth-Tormen fit, leading to some improvement at the high-mass end. The disadvantage is that it cannot be simply extrapolated beyond the range of the fit, since it was tuned to a specific set of simulations.
By performing 16 nested-volume dark matter simulations, Warren was able to obtain significant halo statistics spanning a mass range of 5 orders of magnitude ($\sim
(10^{10}-10^{15})\,h^{-1} M_{\sun}$). Because this represents by far the largest uniform set of simulations–based on multiple boxes with the same cosmology run with the same code–we use it as a reference standard throughout this paper. Using a functional form similar to ST, Warren determined the best mass function fit to be $$f_{\rm Warren}(\sigma) = 0.7234 \left( \sigma^{-1.625} + 0.2538 \right)
\exp \left( -\frac{1.1982}{\sigma^2}\right).$$ For a quantitative comparison of the different fits at different redshifts, we show the ratio of the PS, Jenkins, and ST fits with respect to the Warren fit in Figure \[plotone\]. We do not show the Jenkins fit below $10^{11}\,h^{-1}M_\odot$ at $z=0$ since it diverges in this regime. The original ST fit, the Jenkins fit, and the Warren fit all give similar predictions. The discrepancy between PS and the other fits becomes more severe for higher masses at high redshifts. PS dramatically underpredicts halos in the high-mass range at high redshifts (assuming that the other fits lead to reasonable results in this regime). For low-mass halos the disagreement becomes less severe. For $z=0$ the Warren fit agrees, especially in the low-mass range below $10^{13}\,h^{-1} M_\odot$, to better than 5% with the ST fit. At the high-mass end the difference increases up to 20%. The Jenkins fit leads to similar results over the considered mass range. At higher redshifts and intermediate-mass ranges around $10^{9}\,h^{-1}M_\odot$, the Warren and ST fit disagree by roughly a factor of 2.
Several other groups have suggested modifications of the ST fit. In §\[resint\] we compare our results with two of them. [@Reed03] suggest an empirical adjustment to the ST fit by multiplying it with an exponential function, leading to $$f_{\rm Reed03}(\sigma)=f_{\rm ST}(\sigma)\,
\exp\left\{-0.7/\left[\sigma(\cosh(2\sigma))^5\right]\right\},$$ valid over the range $-1.7\le \ln \sigma^{-1} \le 0.9$. This adjustment leads to a suppression of the ST fit at large $\sigma^{-1}$. In [@Reed07] the adjustment to the ST fit is slightly modified again, leading to the following new fit: $$\begin{aligned}
f_{\rm Reed07}(\sigma)&=& A \sqrt{\frac{2a}{\pi}} \left[1 +
\left(\frac{\sigma^2}{a\delta_c^2}\right)^p+0.6G_1+0.4G_2\right]\nonumber\\
&&\times\frac{\delta_c}{\sigma}
\exp\left[-\frac{ca\delta_c^2}{2\sigma^2}
-\frac{0.03}{(n_{\rm eff}+3)^2}
\left(\frac{\delta_c}{\sigma}\right)^{0.6}\right],\\
G_1&=&\exp\left[-\frac{\ln(\sigma^{-1}-0.4)^2}{2(0.6)^2}\right],
\label{g1_def}\\
G_2&=&\exp\left[-\frac{\ln(\sigma^{-1}-0.75)^2}{2(0.2)^2}\right],
\label{g2_def}\end{aligned}$$ with $c=1.08$, $ca=0.764$, and $A=0.3222$. The adjustment has very similar effects to that of [@Reed03], as we show in §\[resint\]. [@Reed07] note that the (small) suppression of the mass function relative to ST as a function of redshift seen in simulations (see also Heitmann et al. 2006a) can be treated by adding an extra parameter, the power spectral slope at the scale of the halo radius, $n_{\rm eff}$ (formally defined by equation (\[n\_eff\_def\]) below). We return to this issue when we discuss our numerical results in §5. We summarize the described, most commonly used fitting functions in Table \[tabone\].
Although fitting functions may be a useful way to approximately encapsulate results from simulations, meaningful comparisons to observations require overcoming many hurdles, e.g., an operational understanding of the definition of halo mass (see, e.g., White 2001), how it relates to various observations, and error control in $N$-body codes (see, e.g., O’Shea et al. 2005; Heitmann et al. 2005). In this paper, our focus is first on identifying possible systematic problems in the $N$-body simulations themselves and how they can be avoided and controlled.
Halo Growth Function {#halog}
--------------------
A useful way to study the statistical evolution of halo masses in simulations is to transform the mass function into the halo growth function, $n(M_1,M_2,z)\equiv \int_{M_1}^{M_2}F\,d\log M$ [@Heitmann06], which measures the mass-binned number density of halos as a function of redshift. The halo growth function, plotted versus redshift in Figure \[plottwo\], shows at a glance how many halos in a particular mass bin and box volume are expected to exist at a certain redshift. This helps set the required mass and force resolution in a simulation which aims to capture halos at high redshifts. For a given simulation volume, the halo growth function directly predicts the formation time of the first halos in a given mass range.
In order to derive this quantity approximately, we first convert an accurate mass function fit (we use the Warren fit here) into a function of redshift $z$. It has been shown recently by us (Heitmann et al. 2006a) that mass function fits work reliably enough out to at least $z=20$, and can therefore be used to estimate the halo growth function. Figure \[plottwo\] shows the evolution of eight different mass bins, covering the mass range investigated in this paper, as a function of redshift $z$. As expected from the paradigm of hierarchical structure formation in a $\Lambda$CDM cosmology, small halos form much earlier than larger ones. An interesting feature in the lower mass bins is that they have a maximum at different redshifts. The number of the smallest halos grows until a redshift of $~z=2$ and then declines when halos start merging and forming much more massive halos. This feature is reflected in a crossing of the mass functions at different redshifts for small halos.
Mass Function at High Redshift: Previous Work {#evolreview}
---------------------------------------------
![Summary of recent work on the mass function at high redshift. The mass function fits are shown at $z=10$ (top) and $z=20$ (bottom) for the cosmology used throughout this paper (the other groups used slightly different parameters). At $z=10$, Jang-Condell & Hernquist (2001) (gray shaded region) cover the very low mass range using a very small box, as do Cen et al. (2004) (green shaded region). The larger boxes of [@Reed07] and Springel et al. (2005) (red shaded region) lead to results at higher halo masses. Note that in this regime the PS fit deviates substantially from the other fits, while at the very low mass end all fits tend to merge. Our suite of variable box sizes covers a mass range of 10$^7$ to 10$^{13.5}\,h^{-1}M_\odot$ between $z=0$ and 20, a much larger range than previously covered by any group with a uniform set of simulations. At $z=20$ Yoshida et al. (2003a, 2003b, 2003c, 2003d) cover the very low mass end of the mass function, while Zahn et al. (2007) investigate larger mass halos. Our simulations overlap with both of them at the edges. By combining a heterogeneous set of simulations, [@Reed07] cover a wide range in mass and redshift. Figure quality reduced for the arXiv version of the paper.[]{data-label="plotthree"}](f3a.ps){width="7.5cm"}
![Summary of recent work on the mass function at high redshift. The mass function fits are shown at $z=10$ (top) and $z=20$ (bottom) for the cosmology used throughout this paper (the other groups used slightly different parameters). At $z=10$, Jang-Condell & Hernquist (2001) (gray shaded region) cover the very low mass range using a very small box, as do Cen et al. (2004) (green shaded region). The larger boxes of [@Reed07] and Springel et al. (2005) (red shaded region) lead to results at higher halo masses. Note that in this regime the PS fit deviates substantially from the other fits, while at the very low mass end all fits tend to merge. Our suite of variable box sizes covers a mass range of 10$^7$ to 10$^{13.5}\,h^{-1}M_\odot$ between $z=0$ and 20, a much larger range than previously covered by any group with a uniform set of simulations. At $z=20$ Yoshida et al. (2003a, 2003b, 2003c, 2003d) cover the very low mass end of the mass function, while Zahn et al. (2007) investigate larger mass halos. Our simulations overlap with both of them at the edges. By combining a heterogeneous set of simulations, [@Reed07] cover a wide range in mass and redshift. Figure quality reduced for the arXiv version of the paper.[]{data-label="plotthree"}](f3b.ps){width="7.5cm"}
[cccccccc]{} & Box Size & Resolution & & & Particle Mass & Smallest Halo &\
\[0pt\][Mesh]{} & ($h^{-1}$Mpc) & ($h^{-1}$kpc) & \[0pt\][$z_{\rm in}$]{} & \[0pt\][$z_{\rm final}$]{} & ($h^{-1}M_\odot$) & ($h^{-1}M_\odot$) & \[0pt\][No. of Realizations]{}\
1024$^3$ & 256 & 250 & 100 & 0 & $8.35\times 10^{10}$ & $3.34 \times 10^{12}$ & 5\
1024$^3$ & 128 & 125 & 200 & 0 & $1.04\times 10^{10}$ & $4.18 \times 10^{11}$ & 5\
1024$^3$ & 64 & 62.5 & 200 & 0 & $1.31\times 10^9$ & $5.22 \times 10^{10}$ & 5\
1024$^3$ & 32 & 31.25 & 150 & 5 & $1.63\times 10^8$ & $6.52 \times 10^{9}$ & 5\
1024$^3$ & 16 & 15.63 & 200 & 5 &$2.04\times 10^7$ & $8.16\times 10^{8}$ & 5\
1024$^3$ & 8 & 7.81 & 250 & 10 & $2.55\times 10^6$ & $1.02 \times 10^{8}$ & 20\
1024$^3$ & 4 & 3.91 & 500 & 10 & $3.19\times 10^5$ & $1.27 \times 10^{7}$ & 15\
Most of the effort to characterize, fit, and evaluate the mass function from simulations has been focused on or near the current cosmological epoch, $z\sim 0$. This is mainly for two reasons: (1) so far most observational constraints have been derived from low-redshift objects ($z<1$); (2) the accurate numerical evaluation of the mass function at high redshifts is a nontrivial task.
The increasing reach of telescopes on the ground and in space, such as the upcoming James Webb Space Telescope, allows us to study the Universe at higher and higher redshifts. Recent discoveries include 970 galaxies at redshifts between $z=1.5$ and $z=5$ from the VIMOS VLT Deep Survey [@Lefevre05], and the recent observation of a galaxy at $z=6.5$ [@Mobasher05]. The epoch of reionization (EOR) is of central importance to the formation of cosmic structure. Although our current observational knowledge of the EOR is rather limited, future 21 cm experiments have the potential for revolutionizing the field. Proposed low-frequency radio telescopes include LOFAR (Low Frequency Array) [^1], the Mileura Wide Field Array (MWA) [@Bowman06][^2], and the next-generation SKA (Square Kilometer Array) [^3]. The observational progress is an important driver for high-redshift mass function studies.
Theoretical studies of the mass function at high redshifts are challenging due to the small masses of the halos at early times. In order to capture these small-mass halos, high mass and force resolution are both required. For the large simulation volumes typical in cosmological studies, this necessitates a very large number of particles, as well as very high force resolution. Such simulations are very costly, and only a very limited number can be performed, disallowing exploration of a wide range of possible simulation parameters. Alternatively, many smaller volume simulation boxes, each with moderate particle loading, can be employed. This leads automatically to high force and mass resolution in grid codes (such as particle-mesh \[PM\]) and also reduces the costs for achieving sufficient resolution for particle codes (such as tree codes) or hybrid codes (such as TreePM). The disadvantages of this strategy are the limited statistics in individual realizations (because fewer halos form in a smaller box) and the unreliability of simulations below an intermediate redshift at which the largest mode in the box is still (accurately) linear. In addition, results from small boxes may be biased, since they only focus on a small region and volume. Therefore, one must show that the simulations are free from finite-volume artifacts, e.g. missing tidal forces, and run a sufficient number of statistically independent simulations to reduce the sample variance. Both strategies, employing large volume or multiple small-volume simulations, have been followed in the past in order to obtain results at high redshifts. The different mass ranges investigated by different groups are shown in Figure \[plotthree\]. The fits are shown for redshifts $z=10$ and 20. In the Appendix we provide a very detailed discussion on previous findings as organized by simulation volume.
In summary, there is considerable variation in the high-redshift ($z>10$) mass function as found by different groups, independent of box size and simulation algorithm. Broadly speaking, the results fall into two classes: either consistent with linear theory scaling of a universal form (Jenkins, Reed, ST, or Warren) at low redshift (Reed et al. 2003, 2007; Springel et al. 2005; Heitmann et al. 2006a; Maio et al. 2006; Zahn et al. 2007) or more consistent with the PS fit (Jang-Condell & Hernquist 2001; Yoshida et al. 2003a, 2003b, 2003c; Cen et al. 2004; Iliev et al. 2006; Trac & Cen 2006).
Our aim here is to determine the evolution of the mass function accurately, at the few percent level, and at the same time characterize many of the numerical and physical factors that control the error in the mass function (details below). We follow up on our previous work [@Heitmann06] and analyze a large suite of $N$-body simulations with varying box sizes between 4 and $256\,h^{-1}$Mpc, including many realizations of the small boxes, to study the mass function at redshifts up to $z=20$ and to cover a large mass range between $10^7$ and $10^{13.5}\,h^{-1}
M_\odot$. With respect to our previous work, the number of small-box realizations has been increased to improve the statistics at high redshifts. Our results categorically rule out the PS fit as being more accurate than any of the more modern forms at [*any*]{} redshift up to $z=20$, the discrepancy increasing with redshift.
The Code and the Simulations {#code}
============================
All simulations in this paper are carried out with the parallel PM code MC$^2$. This code solves the Vlasov-Poisson equations for an expanding universe. It uses standard mass deposition and force interpolation methods allowing periodic or open boundary conditions with second-order (global) symplectic time stepping and fast fourier transform based Poisson solves. Particles are deposited on the grid using the cloud-in-cell method. The overall computational scheme has proven to be accurate and efficient: relatively large time steps are possible with exceptional energy conservation being achieved. MC$^2$ has been extensively tested against state-of-the-art cosmological simulation codes (Heitmann et al. 2005, 2007).
We use the following cosmology for all simulations: $$\begin{aligned}
&&\Omega_{\rm tot}=1.0,~~~\Omega_{\rm CDM}=0.253,~~~
\Omega_{\rm baryon}=0.048, \nonumber\\
&&\sigma_8=0.9,~~~ H_0=70\ {\rm km\ s^{-1}\,Mpc^{-1}},~~~n=1,\end{aligned}$$ in concordance with cosmic microwave background and large scale structure observations [@Mactavish05] (the third-year Wilkinson Microwave Anisotropy Probe observations suggest a lower value of $\sigma_8$; @Spergel06). The transfer functions are generated with CMBFAST [@Seljak96]. We summarize the different runs, including their force and mass resolution, in Table \[tabtwo\]. As mentioned earlier, we identify halos with a standard FOF halo finder with a linking length of $b=0.2$. Despite several shortcomings of the FOF halo finder, e.g., the tendency to link up two halos which are close to each other (see, e.g., Gelb & Bertschinger 1994, Summers et al. 1995) or statistical biases (Warren), the FOF algorithm itself is well defined and very fast. As discussed in §\[halomass\], we adopt the correction for sampling bias given by Warren when presenting our results.
Initial Conditions and Time Evolution {#icevol}
=====================================
In a near-ideal simulation with very high mass and force resolution, the first halos would form very early. By $z=50$, a redshift commonly used to start cosmological simulations, a large number of small halos would already be present (see, e.g., Reed et al. \[2005\] for a discussion of the first generation of star-forming halos). In a more realistic situation, however, the initial conditions at $z=50$ have of course no halos, the particles having moved only the relatively small distance assigned by the initial Zel’dovich step. Only after the particles have traveled a sufficient distance and come close together can they interact locally to form the first halos. In the following we estimate the redshift when the Zel’dovich grid distortion equals the interparticle spacing, leading to the most conservative estimate for the redshift of possible first halo formation. From this estimate, we derive the necessary criterion for the starting redshift for a given box size and particle number.
![Probability distribution of $|\nabla{\bf\phi}|$ in units of the interparticle spacing $\Delta_{\rm p}$. All curves shown are drawn from $256^3$ particle simulations from an initial density grid of $256^3$ zones. The physical box sizes are $126\,h^{-1}$Mpc (black line), $32\,h^{-1}$Mpc (red line), and $8\,h^{-1}$Mpc (green line). As expected, $\langle|\nabla{\bf\phi}|\rangle$ increases with decreasing box size (which is equivalent to increasing force resolution). Therefore, $z_{\rm in}$ and $z_{\rm cross}$ are higher for the smaller boxes.[]{data-label="plotfour"}](f4.eps){width="80mm"}
Initial Redshift
----------------
In order to capture halos at high redshifts, we have found that it is very important to start the simulation sufficiently early. We consider two criteria for setting the starting redshift: (1) ensuring the linearity of all the modes in the box used to sample the initial matter power spectrum, and (2) restricting the initial particle move to prevent interparticle crossing and to keep the particle grid distortion relatively small. The first criterion is commonly used to identify the starting redshift in simulations. However, as shown below, it fails to provide sufficient accuracy of the mass functions, accuracy which can be obtained when a second (much more restrictive) control is applied. Furthermore, it is important to allow a sufficient number of expansion factors between the starting redshift $z_{\rm in}$ and the highest redshift of physical significance. This is needed to make sure that artifacts from the Zel’dovich approximation are negligible and that the memory of the artificial particle distribution imposed at $z_{\rm in}$ (grid or glass) is lost by the time any halo physics is to be extracted from the simulation results.
Although not studied here, it is important to note that high-redshift starts do require the correct treatment of baryons as noted in §\[massdef\]. In addition, redshift starts that are too high can lead to force errors for a variety of reasons, e.g., interpolation systematics, round-off, and correlated errors in tree codes.
### Initial Perturbation Amplitude
[cccc]{} Box Size & $k_{\rm Ny}$ & &\
($h^{-1}$Mpc) & ($h$Mpc$^{-1}$) & \[0pt\][$T(z=0,\, k_{\rm Ny})$]{} & \[0pt\][$z_{\rm in}$]{}\
126 & 6.3 & 0.0002 & 33\
32 & 25 & 1.7$\cdot 10^{-5}$ & 45\
16 & 50 & 4.8$\cdot 10^{-6}$ & 50\
8 & 100 & 1.3$\cdot 10^{-6}$ & 55\
The initial redshift in simulations is often determined from the requirement that all mode amplitudes in the box below the particle Nyquist wavenumber characterized by $k_{\rm Ny}/2$ with $k_{\rm
Ny}=2\pi/\Delta_{\rm p}$, where $\Delta_{\rm p}$ is the mean interparticle spacing, be sufficiently linear. The smaller the box size chosen (keeping the number of particles fixed), the larger the largest $k$-value. Therefore, in order to ensure that the smallest initial mode in the box is well in the linear regime, the starting redshift must increase as the box size decreases. In the following we give an estimate based on this criterion for the initial redshift for different simulation boxes. We (conservatively) require the dimensionless power spectrum $\Delta^2=k^3P(k)/2\pi^2$ to be smaller than $0.01$ at the initial redshift. The initial power spectrum is given by $$\Delta^2(k_{\rm Ny},z_{\rm in})
=\frac{k^3 P(k_{\rm Ny},z_{\rm in})}{2\pi^2}
\sim\frac{B\ k^{n+3} T^2(k_{\rm Ny},z=0)}{2\pi^2 (z_{\rm in}+1)^2},$$ where $B$ is the normalization of the primordial power spectrum (see, e.g., Bunn & White \[1997\] for a fitting function for $B$ including COBE results) and $T(k)$ is the transfer function. We assume the spectral index to be $n=1$, which is sufficient to obtain an estimate for the initial redshift. For a $\Lambda$CDM universe the normalization is roughly $B\sim 3.4 \times 10^6 (h^{-1}{\rm Mpc})^4$. Therefore, $z_{\rm in}$ is simply determined by $$z_{\rm in}\simeq 4150 \ k_{\rm Ny}^2T(z=0,k_{\rm Ny}).$$ We present some estimates for different box sizes in Table \[tabthree\]. For the smaller boxes ($<8~h^{-1}$Mpc), the estimates for the initial redshifts are at around $z_{\rm in}=50$.
It is clear that this criterion simply sets a minimal requirement for $z_{\rm in}$ and neglects the fact that the initial particle move should be small enough to maintain the dynamical accuracy of perturbation theory (linear or higher order) used to set the initial conditions. Also, this criterion certainly does not tell us that if, e.g., $z_{\rm
in}=50$, then we may already trust the mass function at, say, $z=30$. An example of this is provided by the results of [@Reed03], who find that their high-redshift results between $z=7$ and 15 have not converge if they start their simulations at $z_{\rm in}=69$. (A value of $z_{\rm in}=139$ was claimed to be sufficient in their case.)
We now consider another criterion – ostensibly similar in spirit – that particles should not move more than a certain fraction of the interparticle spacing in the initialization step. This second criterion demands much higher redshift starts.
### First Crossing Time {#firstcross}
![Average redshift of first crossing (top) and highest redshift of first crossing (bottom) as a function of box size. The initial conditions (five different realizations) are shown for boxes between 1 and 512$\,h^{-1}$Mpc with 128$^3$ and 256$^3$ particles. For each initial condition, $z_{\rm cross}^{\rm
first}$ and $z_{\rm cross}^{\rm rms}$ are shown by the crosses. The solid lines show the average from the five realizations. As expected, scatter from the different realizations is larger for smaller boxes. These plots provide estimates of the required initial redshift for a simulation since $|\nabla\phi|/\Delta_{\rm p}$ is $z$-independent in the Zel’dovich approximation (see text).[]{data-label="plotfive"}](f5.eps){width="80mm"}
In cosmological simulations, initial conditions are most often generated using the Zel’dovich approximation [@Zeldovich70]. Initially each particle is placed on a uniform grid or in a glass configuration and is then given a displacement determined by the relation $$\label{zeldo}
{\bf x} = {\bf q} -d(z)\nabla{\bf\phi},$$ where ${\bf q}$ is the Lagrangian coordinate of each particle. The gradient of the potential $\bf{\phi}$ is independent of the redshift $z$. The Zel’dovich approximation holds in the mildly nonlinear regime, as long as particle trajectories do not cross each other (no caustics have formed). Studying the magnitude of $|{\nabla\bf \phi}|$ allows us to estimate two important redshift values: first, the initial redshift $z_{\rm in}$ at which the particles should not have moved on average more than a fraction of the interparticle spacing $\Delta_{\rm p}=L_{\rm box}/n_{\rm p}$, where $L_{\rm box}$ is the physical box size and $n_{\rm p}$ the number of particles in the simulation; second, the redshift at which particles first move more than the interparticle spacing, $z_{\rm cross}$, i.e., at which they have traveled on average a distance greater than $\Delta_{\rm p}$.
{width="180mm"}
For a given realization of the power spectrum, the magnitude of $|{\nabla\bf \phi}|$ depends on two parameters: the physical box size and the interparticle spacing. Together these parameters determine the range of scales under consideration. The smaller the box, the smaller the scales; therefore, $|{\nabla\bf \phi}|$ increases and both $z_{\rm in}$ and $z_{\rm cross}$ increase. Increasing the resolution has the same effect. In Figure \[plotfour\] we show the probability distribution function for $|{\nabla\phi}|$ for three different box sizes, $8$, $32$, and $126\,h^{-1}$Mpc, representing values studied by other groups, as well as in this paper. To make the comparison between the different box sizes more straightforward, we have scaled $|{\nabla\bf \phi}|$ with respect to the interparticle spacing $\Delta_{\rm p}$. All curves are drawn from simulations with 256$^3$ particles on a 256$^3$ grid, in accordance with the set up of our initial conditions. The behavior of the probability function follows our expectations: the smaller the box, or the higher the force resolution, the larger the initial displacements of the particles on average. From the mean and maximum values of such a distribution we can determine appropriate values for $z_{\rm in}$ and $z_{\rm cross}$. For our estimates we assume $d(z)\simeq 1/(1+z)$, which is valid for high redshifts. The maximum and rms initial displacements of the particles can then be easily calculated: $$\begin{aligned}
\delta^{\rm max}_{\rm in}&\simeq&
\frac{{\rm max}(|\nabla\phi|/\Delta_{\rm p})}{1+z_{\rm in}},\\
\delta^{\rm rms}_{\rm in}&\simeq&
\frac{{\rm rms}(|\nabla\phi|/\Delta_{\rm p})}{1+z_{\rm in}}.\end{aligned}$$ The very first “grid crossing” of a particle occurs when $\delta^{\rm
max}_{\rm in}=1$; on average the particles have moved more than one particle spacing when $\delta^{\rm rms}_{\rm in}=1$. This leads to the following estimates: $$\begin{aligned}
z^{\rm first}_{\rm cross}&\simeq&
{\rm max}(\nabla\phi/\Delta_{\rm p})-1,\\
z^{\rm rms}_{\rm cross}&\simeq&
{\rm rms}(\nabla\phi/\Delta_{\rm p})-1.\end{aligned}$$ We show these two redshifts in Figure \[plotfive\] for 10 different box sizes ranging from 1 to 512$\,h^{-1}$Mpc and for 256$^3$ and 128$^3$ particles. The top panel shows the average redshift of the first crossing as a function of box size (which corresponds to the maximum in Fig. \[plotfour\]). The bottom panel shows the redshift where the first “grid crossing” occurs (corresponding to the right tail in Fig. \[plotfour\]). To estimate the scatter in the results, we have generated five different realizations for each box. As expected, the small boxes show much more scatter. The average redshift of the first crossing in the 1$\,h^{-1}$Mpc box varies between $z=63$ and 83, while there is almost no scatter in the 512$\,h^{-1}$Mpc box. Since $|\nabla\phi|/\Delta_{\rm p}$ is independent of redshift in the Zel’dovich approximation, a simple scaling determines the appropriate initial redshift from these plots. For example, if a particle should not have moved more than 0.3$\Delta_{\rm p}$ on average at the initial redshift, the average redshift of first crossing has to be multiplied by a factor $1/0.3=3.\bar{3}$. For an $8\,h^{-1}$Mpc box this leads to a minimum starting redshift of $z=230$, while for a $126\,h^{-1}$Mpc box this suggests a starting redshift of $z_{\rm in}=50.$ The 128$^3$ particle curve can be scaled to the 256$^3$ particle curve by multiplying by a factor of 2. Curves for different particle loadings can be obtained similarly.
Transients and Mixing
---------------------
The Zel’dovich approximation matches the exact density and velocity fields to linear order in Lagrangian perturbation theory. Therefore, there is in principle an error arising from the resulting discrepancy with the density and velocity fields given by the exact growing mode initialized in the far past.
This error is linear in the number of expansion factors between $z_{\rm in}$ and the redshift of interest $z_{\rm phys}$. It has been explored in the context of simulation error by [@Valageas02] and by [@Crocce06]. Depending on the quantity being calculated, the number of expansion factors between $z_{\rm in}$ and $z_{\rm phys}$ required to limit the error to some given value may or may not be easy to estimate. For example, unlike quantities such as the skewness of the density field, there is no analytical result for how this error impacts the determination of the mass function. Neither does there exist any independent means of validating the result aside from convergence studies. Nevertheless, it is clear that to be conservative, one should aim for a factor of $\sim 20$ in expansion factor in order to anticipate errors at the several percent level, a rule of thumb that has been followed by many $N$-body practitioners (and often violated by others!). This rule of thumb gives redshift starts that are roughly in agreement with the estimates in the previous subsection. Convergence tests done for our simulations show that the suppression in the mass function is very small (less than 1%) for simulations whose evolution covers a factor of 15 in the expansion factor and can be up to 20% for simulations that evolved by only 5 expansion factors. However, due to modest particle loads, we were unable to distinguish between the error induced by too few expansion factors and the breakdown of the Zel’dovich approximation.
Another possible problem, independent of the accuracy of the Zel’dovich approximation, is the initial particle distribution itself. Whether based on a grid or a glass, the small-distance ($k>k_{\rm Ny}$) mass distribution is clearly not sampled at all by the initial condition. Therefore, unlike the situation that would arise if a fully dynamically correct initial condition were given, some time must elapse before the correct small-separation statistics can be established in the simulation. Thus, all other things being equal, for the correct mass function to exist in the box, one must run the simulation forward by an amount sufficiently greater than the time taken to establish the correct small-scale power on first-halo scales while erasing memory on these scales of the initial conditions. If this is not done, structure formation will be suppressed, leading to a lowering of the halo mass function.
Because there is no fully satisfactory way to calculate $z_{\rm in}$ in order to compute the mass function at a given accuracy, we subjected [ *every*]{} simulation box to convergence tests in the mass function while varying $z_{\rm in}$. The results shown in this paper are all converged to the sub-percent level in the mass function. We give an example of one such convergence test below.
### Initial Redshift Convergence Study
As mentioned above, we have tested and validated our estimates for the initial redshift for all the boxes used in the simulation suite via convergence studies. Here, we show results for an 8$\,h^{-1}$Mpc box with initial redshifts $z_{\rm in}=50$, 150, and 250 in Figure \[plotsix\], where the mass functions at $z=10$ are displayed. For the lowest initial redshift, $z_{\rm in}=50$, the average initial particle movement is 1.87$\Delta_{\rm p}$, while some particles travel as much as 5.03$\Delta_{\rm p}$. This clearly violates the requirement that the initial particle grid distortion be kept sufficiently below 1 grid cell. The starting redshift $z_{\rm
in}=150$ leads to an average displacement of 0.63$\Delta_{\rm p}$ and a maximum displacement of 1.71$\Delta_{\rm p}$, and therefore just barely fulfills the requirements. For $z_{\rm in}=250$ we find an average displacement in this particular realization of 0.37$\Delta_{\rm p}$ and a maximum displacement of 1.00$\Delta_{\rm p}$.
The bottom plot in each of the three panels of Figure \[plotsix\] shows the ratio of the mass functions with respect to the Warren fit. In the middle and right panels the ratio for the largest halo is outside the displayed range. The mass function from the simulation started at $z_{\rm in}=50$ (left panel) is noticeably lower, $\sim 15\%$, than for the other two simulations. The mass functions from the two higher redshift starts are in good agreement, showing that the choice for average grid distortion of approximately 0.3$\Delta_{\rm p}$ is conservative, and that one can safely use (0.5–0.6)$\Delta_{\rm p}$. The general conclusion illustrated by Figure \[plotsix\] is that if a simulation is started too late, halos are found to be missing over the entire mass range. With the late start, there is less time to form bound objects. Also, some particles that are still streaming towards a halo do not have enough time to join it. Both of these artifacts lead to an overall downshift of the mass function.
To summarize, requiring a limit on initial displacements sets the starting redshift much higher than simply demanding that all modes in the box stay linear. Indeed, the commonly used latter criterion (with $\delta^{\rm rms}\sim 0.1$) is not adequate for computing the halo mass function at high redshifts. One must verify that the chosen $z_{\rm in}$ sets an early enough start as shown here. We comment on previous results from other groups with respect to this finding below in §\[conclusion\].
Force and Mass Resolution
-------------------------
We now take up an investigation of the mass and force resolution requirements. The first useful piece of information is the size of the simulation box: from Figure \[plottwo\] we can easily translate the number density into when the first halo is expected to appear in a box of volume $V$. For example, a horizontal line at $n=10^{-6}$ would tell us at what redshift we would expect on average to find 1 halo of a certain mass in a $(100\,h^{-1}$Mpc)$^3$ box. The first halo of mass $~10^{11}-10^{12}\,h^{-1}M_\odot$ will appear at $z\simeq 15.5$, and the first cluster-like object of mass $~10^{14}-10^{15}\,h^{-1}M_\odot$ at $z\simeq 2$. Of course, these statements only hold if the mass and force resolution are sufficient to resolve these halos. The mass of a particle in a simulation, and hence the halo mass, is determined by three parameters: the matter content of the Universe $\Omega_{\rm m}$, including baryons and dark matter, the physical box size $L_{\rm box}$, and the number of simulation particles $n_{\rm p}^3$: $$m_{\rm particle}= 2.775 \times 10^{11}\Omega_{\rm m}
\left(\frac{L_{\rm box}}{n_{\rm p}\,h^{-1}{\rm Mpc}}\right)^3 \,h^{-1} {M}_\odot.
\label{mpart}$$
{width="180mm"}
The required force resolution to resolve the chosen smallest halos can be estimated very simply. Suppose we aim to resolve a virialized halo with comoving radius $r_{\Delta}$ at a given redshift $z$, where $\Delta$ is the overdensity parameter with respect to the critical density $\rho_{\rm c}$. The comoving radius $r_\Delta$ is given by $$r_\Delta=9.51 \times 10^{-5}
\left[\frac{\Omega(z)}{\Omega_{\rm m}}\right]^{1/3}\left(
\frac{1}{\Delta}\frac{M_{\Delta {\rm c}}}{h^{-1}{\rm M_\odot}}\right)^{1/3}
\,h^{-1}{\rm Mpc},
\label{rdelta}$$ where $\Omega(z)=\Omega_{\rm m}(1+z)^3/[\Omega_{\rm
m}(1+z)^3+\Omega_\Lambda]$ and the halo mass $M_{\Delta {\rm c}}=
m_{\rm part} n_{\rm h}$, where $n_{\rm h}$ is the number of particles in the halo. We measure the force resolution in terms of $$\delta_{\rm f}=\frac{L_{\rm box}}{n_{\rm g}}.$$ In the case of a grid code, $n_{\rm g}$ is literally the number of grid points per linear dimension; for any other code, $n_{\rm g}$ stands for the number of “effective softening lengths” per linear dimension. To resolve halos of mass $M_{\Delta{\rm c}}$, a minimal requirement is that the code resolution be smaller than the radius of the halo we wish to resolve: $$\delta_{\rm f} < r_\Delta.
\label{resol}$$ Note that this minimal resolution requirement is aimed only at capturing halos of a certain mass, not at resolving their interior profile. Next, inserting the expression for the particle mass (eq. \[\[mpart\]\]) and the comoving radius (eq. \[\[rdelta\]\]) into the requirement (eq \[\[resol\]\]) and employing the relation between the interparticle spacing $\Delta_{\rm p}$ and the box size $\Delta_{\rm
p}=L_{\rm box}/n_{\rm p}$, the resolution requirement reads $$\frac{\delta_{\rm f}}{\Delta_{\rm p}}<
0.62\left[\frac{n_h\Omega(z)}{\Delta}\right]^{1/3}.
\label{resreq}$$ We now illustrate the use of this simple relation with an example. Let $\Delta=200$ and consider a $\Lambda$CDM cosmology with $\Omega_{\rm
m}=0.3$. Then for PM codes for which $\delta_{\rm
f}/\Delta_{\rm p}=n_{\rm p}/n_{\rm g}$, we have the following conclusions. If the number of mesh points is the same as the number of particles ($n_{\rm p}=n_{\rm g}$), halos with less than 2500 particles cannot be accurately resolved. If the number of mesh points is increased to 8 times the particle number ($n_{\rm p}=1/2 n_{\rm
g}$), commonly used for cosmological simulations with PM codes, the smallest halo reliably resolved has roughly 300 particles, and if the resolution is increased to a ratio of 1 particle per 64 grid cells, which we use in the main PM simulations in this paper, halos with roughly 40 particles can be resolved. It has been shown in [@Heitmann05] that this ratio (1:64) does not cause collisional effects and that it leads to consistent results in comparison to high-resolution codes. Note that increasing the resolution beyond this point will not help, since it is unreliable to sample halos with too few particles. Note also that a similar conclusion holds for any simulation algorithm and not just for PM codes.
In Figure \[plotseven\] we show results from a resolution convergence test at $z=0$. We run 256$^3$ particles in a 126$\,h^{-1}$Mpc box with three different resolutions: 0.5, 0.25, and 0.125$\,h^{-1}$Mpc. The vertical line in each figure shows the mass below which the resolution is insufficient to capture all halos following condition (\[resreq\]). In all three cases, the agreement with the theoretical prediction is excellent.
![Top: One of the 32$\,h^{-1}$Mpc box realizations run with 250, 125, 50 and 5 time steps between $z_{\rm in}=150$ and $z_{\rm final}=5$. The mass function is shown at the final redshift $z=5$. Data points for all runs except the one with five time steps are so close that they are difficult to distinguish. Bottom: A 126$\,h^{-1}$Mpc box with 300, 100, 8, and 5 time steps between $z_{\rm
in}=50$ and $z_{\rm final}=0$. The agreement for the very large halos for 100 and 300 time steps is essentially perfect. Poisson error bars are shown.[]{data-label="ploteight"}](f8.eps){width="80mm"}
Time Stepping
-------------
Next, we consider the question of time-step size and estimate the minimal number of time steps required to resolve the halos of interest. We begin with a rough estimate of the characteristic particle velocities in halos. For massive halos, the halo mass $M_{200}$ and its velocity dispersion are connected by the approximate relation (Evrard 2004): $$M_{200}\simeq\frac{10^{15}\,h^{-1}M_\odot}{H/H_0}
\left(\frac{\sigma_{v}}{1080
{\rm \,km/s}}\right)^3.$$ A more accurate expression can be found in [@Evrard07], but the above is more than sufficient for our purposes. At high redshift, $\Omega_\Lambda$ can be neglected, and we can express the velocity dispersion as a function of redshift: $$\sigma_{\rm v}\simeq
10^{-2}\sqrt{1+z}\left(\frac{M_{200}}{\,h^{-1}M_\odot}\right)^{1/3}{\rm km s^{-1}}.$$ In a time $\delta t$, the characteristic scale length $\delta l$ is given by $\delta l\simeq \sigma_{\rm v}\delta t$ or $$\label{deltat}
\delta t\simeq\frac{\delta l}{\sigma_{v}}=\frac{100\,\delta l/{\rm km}}{\sqrt{1+z}}
\left(\frac{M_{200}}{\,h^{-1}M_\odot}\right)^{-1/3} {\rm s}.$$ The scale factor $a$ is a convenient time variable for codes working in comoving units, such as ours. Expressed in terms of the scale factor, equation (\[deltat\]) reads: $$\delta a \simeq 10^4\frac{\delta l}{h^{-1}{\rm Mpc}}\left( \frac{M_{200}}
{h^{-1}M_\odot}\right)^{-1/3}.$$ We are interested in the situation where $\delta l$ is actually the force resolution, $\delta_{\rm f}$. In a single time step, the distance moved should be small compared to $\delta_{\rm f}$; i.e., the actual time step should be smaller than $\delta a$ estimated from the above equation when $\delta l$ is replaced on the right–hand side with $\delta_{\rm f}$. Let us consider a concrete example for the case of a PM code where $\delta_{\rm f}=L_{\rm box}/n_{\rm g}$ as explained earlier. For a “medium” box size of $L_{\rm box}=256\,h^{-1}{\rm Mpc}$ and a grid size of $n_{\rm g}=1024$, $\delta_{\rm f}=0.25\,h^{-1}{\rm
Mpc}$. For a given box, the highest mass halos present have the largest $\sigma_{\rm v}$ and give the tightest constraints on the time step. For the chosen box size, a good candidate halo mass scale is $M_{200}\sim 10^{15}\,h^{-1} M_\odot$ (this could easily be less, but it does not change the result much). In this case, $$\delta a \simeq 0.025.$$ If, for illustration, we start a simulation at $z=50$ and evolve it down to $z=0$, this translates to roughly 40 time steps. We stress that this estimate is aimed only at avoiding disruption of the halos themselves, and is certainly not sufficient to resolve the [*inner*]{} structure of the halo.
In Figure \[ploteight\] we show two tests of the time step criterion. The top panel shows the result from a 32$\,h^{-1}$Mpc box at redshift $z=5$. The simulation starts at $z_{\rm in}=150$ and is evolved with 50, 125, and 250 time steps down to $z=5$. Following the argument above for this box size, one would expect all three choices to be acceptable, and the excellent agreement across these runs testifies that this is indeed the case. We also carried out a run with only five time steps, which yields a clearly lower ($\sim 20 \%$) mass function than the others, but not as much as one would probably expect from such an imprecise simulation.
The bottom panel shows the results from a 126$\,h^{-1}$Mpc box at $z=0$. This simulation was started at $z{\rm in}=50$ and run to $z=0$ with 5, 8, 100, and 300 time steps. Again, as we would predict, the agreement is very good for the last two simulations, and the convergence is very fast, confirming our estimate that only ${\cal{O}}(10)$ time steps is enough to get the correct halo mass function. Overall, the halo mass function appears to be a very robust measure, not very sensitive to the number of time steps. Nevertheless, we used a conservatively large number of time steps, e.g., 500 for the simulations stopping at $z=0$ and 300 for those stopping at $z=10$.
In the previous subsections we have discussed and tested different error control criteria for obtaining the correct simulated mass function at all redshifts. These criteria are (1) a sufficiently early starting redshift to guarantee the accuracy of the Zel’dovich approximation at that redshift and provide enough time for the halos to form; (2) sufficient force and mass resolution to resolve the halos of interest at any given redshift; and (3) sufficient numbers of time steps. Violating any of these criteria [*always*]{} leads to a suppression of the mass function. Most significantly, our tests show that a late start (i.e., starting redshift too low) leads to a suppression over the entire mass range under consideration, and is a likely explanation of the low mass function results in the literature. As intuitively expected, insufficient force resolution leads to a suppression of the mass function at the low-mass end, while errors associated with time stepping are clearly subdominant and should not be an issue in the vast majority of simulations.
Results and Interpretation {#resint}
==========================
In this section we present the results from our simulation suite. We describe how the data are obtained as well as the post-processing corrections applied. The latter include compensation for FOF halo mass bias induced by finite (particle number) sampling, and the (small) systematic suppression of the mass function induced by the finite volume of the simulation boxes.
Binning of Simulation Data
--------------------------
Before venturing into the simulation results, we first describe how they were obtained and reported from individual simulations. We used narrow mass bins while conservatively keeping the statistical shot noise of the binned points no worse than some given value. Bin widths $\Delta \log M$ were chosen such that the bins contain an equal number of halos $N_{\rm h}$. The worst-case situation occurs at $z=20$ for the 8$\,h^{-1}$Mpc box, which has $N_{\rm h}=80$; the 4$\,h^{-1}$Mpc box at the same redshift has $N_{\rm h}=400$. At $z=15$ we have $N_{\rm h}=150$, 1600, and 3000 for box sizes 16, 8, and 4$\,h^{-1}$Mpc, respectively. At $z=10$ the smallest value $N_{\rm
h}=450$ is for the 32$\,h^{-1}$Mpc box, while at $z=5$ and 0 we essentially always have $N_{\rm h}>10000$.
With a mass function decreasing monotonically with $M$, this binning strategy results in bin widths increasing monotonically with $M$. The increasing bin size may cause a systematic deviation – growing towards larger masses – from an underlying “true” continuous mass function. The data points for the binned mass function give the average number of halos per volume in a bin, $$\bar{{F}}\equiv N_{\rm h}/(V\Delta\log M),
\label{fbar}$$ plotted versus an average halo mass, averaged by the [*number of halos*]{} in the bin: $$\bar{M}\equiv \sum_{\rm bin}M/N_{\rm h}.
\label{mbar}$$ Assuming that the true mass function $dn/d\log M$ has some analytic form ${F}(M)$, a systematic deviation due to the binning prescription $$\epsilon_{\rm{bin}}\equiv\frac{\bar{{F}}-{F}(\bar{M})}{{F}(\bar{M})}$$ can be evaluated by computing $\bar{{F}}$ and $\bar{M}$ as $$\bar{{F}}=\frac{\int_{\Delta M} dn}{\Delta\log M},~~~\bar{M}
=\frac{\int_{\Delta M} Mdn}{\int_{\Delta M}dn},$$ where $dn\equiv F(M)\,d\log M$ and the integrations are over a mass range $[M,M+\Delta M]$. For the leading-order term of the Taylor expansion of $\epsilon_{\rm bin}(\Delta M)$, we find $$\epsilon_{\rm{bin}}\simeq\frac{F^{\prime\prime}-2(F^{\prime})^2/F}{24F}(\Delta M)^2,
\label{epsb}$$ where the primes denote $\partial/\partial M$. A characteristic magnitude of this $\epsilon_{\rm bin}$ for a general ${F}(M)$ is $(\Delta M/M)^2/24$. However, in our case, where the relevant scales $k\gg k_{\rm eq}\sim0.01 h$Mpc$^{-1}$, $\epsilon_{\rm bin}$ has a much stronger suppression, as explained below.
We know that the mass function is close to the universal form, $${F}(M)= \frac{\rho_{\rm b}}{M} f(\sigma)\frac{d\ln\sigma^{-1}}{d\log M}
\label{univ2}$$ (see, eq. \[\[fsigma\]\]). Note that for $k\gg k_{\rm eq}$, $\sigma^{-1}(M)$ is a slowly varying function, i.e., $$\frac{d\log \sigma^{-1}}{d\log M} \equiv \frac{n_{\rm eff}+3}{6}
\label{n_eff_def}$$ is much smaller than unity, and the derivative $d\log
\sigma^{-1}/d\log M$ also changes slowly with $M$. Then, despite the steepness of ${F}(\sigma)$ at small $\sigma$, the factor $f(\sigma)\,d\ln\sigma^{-1}/d\log M$ in equation (\[univ2\]) depends weakly on $M$. Therefore, the mass function ${F}(M)$ is close to being inversely proportional to $M$. In the limit of exact inverse proportionality, ${F}\propto M^{-1}$, equation (\[epsb\]) tells us that $\epsilon_{\rm{bin}}\rightarrow 0$. This effective cancellation of the two terms on the right-hand side of equation (\[epsb\]) makes the binning error negligible to the accuracy of our ${F}(M)$ reconstruction whenever a bin width $\Delta \log M$ does not exceed $0.5$. To confirm the absence of any systematic offsets due to the binning, we binned the data into $\log M$ intervals 5 times narrower and wider, with no apparent change in the inferred ${F}(M)$ dependence.
We remark that the situation could be quite different with another binning choice. For example, if the binned masses $\bar{M}$ were chosen at the centers of the corresponding $\log M$ intervals, $\log
\bar M=[\log M+\log(M+\Delta M)]/2$, the systematic binning deviation $$\epsilon_{\rm{bin}}^{({\rm center})}\simeq
\frac{F^{\prime\prime}+F^{\prime}/M}{24F}(\Delta M)^2$$ would have no special cancellation for the studied type of mass function. A corresponding binning error would be about 2 orders of magnitude larger than that of equations (\[fbar\]) and (\[mbar\]).
The statistical error bars used are Poisson errors, following the improved definition of Heinrich (2003): $$\label{heinrich}
\sigma_{\pm}=\sqrt{N_{\rm h}+\frac{1}{4}}\pm \frac{1}{2}.$$ At large values of $N_{h}$, these error bars asymptote to the familiar form $\sqrt N_{\rm h}$. At smaller values of $N_{\rm h}$ – which are of minor concern here – equation \[heinrich\] has several advantages over the standard Poisson error definition, some being (1) it is nonzero for $N_{\rm
h}=0$; (2) the lower edge of the error bar does not go all the way to zero when $N_{\rm h}=1$; (3) the asymmetry of the error bars reflects the asymmetry of the Poisson distribution.
Finally, as noted earlier and discussed in the next section, all the results shown in the following include a correction for the sampling bias of FOF halos according to equation (\[halocorr\]). This mass correction brings down the low-mass end of the mass function.
FOF Mass Correction {#masscorr}
-------------------
![FOF mass correction for halos in 4 (dark blue), 8 (black), 16 (light blue), and 32 (yellow) $h^{-1}$Mpc boxes. To show the effect clearly, we plot the ratio of our data to the Warren fit. Crosses show the uncorrected mass function and squares the mass function after correction, following eq. (\[halocorr\]). Note the smooth behavior of the corrected mass function as opposed to the mass-function jumps across box sizes for the uncorrected data.[]{data-label="plotnine"}](f9.eps){width="7.5cm"}
The mass of a halo as determined by the FOF algorithm displays a systematic bias with the number of particles used to sample the halo. Too few particles lead to an increase in the estimated halo mass. By systematically subsampling a large halo population from N-body simulations (at $z=0$), Warren determined an empirical correction for this undersampling bias. For a halo with $n_{\rm h}$ particles, his correction factor for the FOF mass is given by $$\label{halocorr}
n_{\rm h}^{\rm corr}=n_{\rm h}\left(1-n_{\rm h}^{-0.6}\right).$$ We have carried out an independent exercise to check the systematic bias of the FOF halo mass as a function of particle number based on Monte Carlo sampling of an NFW halo mass profile with varying concentration and particle number, as well as by direct checks against simulations (e.g., Fig. \[plotnine\]); our results are broadly consistent with equation (\[halocorr\]). Details will be presented elsewhere (Z. Lukić et al., in preparation). In this associated work we also address how overdensity masses connect to FOF masses, how this relation depends on the different linking length used for the FOF finder, and the properties of the halo itself, such as the concentration.
The effect of the FOF sampling correction can be quickly gauged by considering a few examples: for a halo with 50 particles, the mass reduction is almost 10%, for a halo with 500 particles, it is $\sim
2.4\%$, and for a well-sampled halo with 5000 particles, it is only 0.6%. As a cautionary remark, this correction formula does not represent a general recipe but can depend on variables such as the halo concentration. Since the conditions under which different simulations are carried out can differ widely, corrections of this type should be checked for applicability on a case-by-case basis. Note also that the correction for the mass function itself depends on how halos move across mass bins once the FOF correction is taken into account.
The choice of the mass function range in a given simulation box always involves a compromise: too wide a dynamic range leads to poor statistics at the high-mass end and possible volume-dependent systematic errors, and too narrow a range leads to possible undersampling biases. Our choice here reflects the desire to keep good statistical control over each mass bin at the expense of wide mass coverage, compensating for this by using multiple box sizes. Therefore, in our case it is important to demonstrate control over the FOF mass bias. An example of this is shown in Figure \[plotnine\], where results from four box sizes demonstrate the successful application of the Warren correction to simulation results at $z=10$.
Simulation Mass and Growth Function {#simvol}
-----------------------------------
The complete set of simulations, summarized in Table \[tabtwo\], allows us to study the mass function spanning the redshift range from $z=20$ to 0. The mass range covers dwarf to massive galaxy halos at $z=0$ (cluster scales are best covered by much bigger boxes as in Warren and @Reed07), and at higher redshifts goes down to $10^7\,h^{-1}M_\odot$, the mass scale above which gas in halos can cool via atomic line cooling (@Tegmark97).
Time Evolution of the Mass Function {#timeevo}
-----------------------------------
{width="160mm"}
Halo mass functions from the multiple-box simulations are shown in Figure \[plotten\], with results being reported at five different redshifts with no volume corrections applied. The combination of box sizes is necessary because larger boxes do not have the mass resolution to resolve very small halos at early redshifts, while smaller boxes cannot be run to low redshifts. The bottom plot of each panel shows the ratio of the numerically obtained mass function, and various other fits, to the Warren fit as scaled by linear theory (for volume-corrected results, see Fig. \[plottwelve\]). Displaying the ratio has the advantage over showing relative residuals that large discrepancies (more than 100%) appear more clearly. For all redshifts, the agreement with the Warren fit is at the 20% level. The ST fit matches the simulations for small masses very well but overpredicts the number of halos at large masses. This overprediction becomes worse at higher redshifts. For example, at $z=15$ ST overpredicts halos of 10$^9 \,h^{-1}M_\odot$ by a factor of 2. Reed et al. (2003) found a similar result: the ST fit at $z=15$ for halos with mass larger than $10^{10} \,h^{-1}M_\odot$ disagrees with their simulation by 50%. Agreement with the Reed et al. (2003, 2007) fits is also good, within the 10% level. (For a further discussion focused around the question of universality, see Section 5.7.) The PS fit in general is not satisfactory over a larger mass range at any redshift. It crosses the other fits at different redshifts for different masses. Away from this crossing region, however, the disagreement can be as large as an order of magnitude, e.g. for $z=20$ over the entire mass range we consider here.
Halo Growth Function {#halogres}
--------------------
As discussed in §\[halog\] the halo growth function (the number density of halos in mass bins as a function of redshift) offers an alternative avenue to study the time evolution of the mass function. Figure \[ploteleven\] shows the halo growth function for an 8$\,h^{-1}$Mpc box for three different starting redshifts, $z_{\rm
in}=50$, 150, and 250 (these are the same simulations as in Fig. \[plotsix\]). The results are displayed at three redshifts, $z=20$, 15, and 10 and for three mass bins, $10^8 -
10^9\,h^{-1}M_\odot$, $10^9 - 10^{10}\,h^{-1}M_\odot$, and $10^{10} -
10^{11}\,h^{-1}M_\odot$.
Assuming that the Warren fit scales at least approximately to high redshifts, the first halos in the lowest mass bin are predicted to form at $z_{\rm form}\sim 25$ (see Fig. \[plotsix\]). We have found that if $z_{\rm form}$ is not sufficiently far removed from $z_{\rm in}$, formation of the first halos is significantly delayed/suppressed. In turn, this leads to suppressions of the halo growth function and the mass function at high redshifts. As shown in Figure \[ploteleven\], the suppression can be quite severe at high redshifts: the simulation result at $z=20$ from the late start at $z_{\rm in}=50$ is an [*order of magnitude*]{} lower than that from $z_{\rm in}=250$. At lower redshifts, the discrepancy decreases, and results from late-start simulations begin to catch up with the results from earlier starts. Coincidentally, the suppression due to the late start at $z_{\rm in}=50$ is rather close to the PS prediction which is very significantly below the Warren fit in the mass and redshift range of interest (see Fig. \[ploteleven\]). We take up this point further below.
![Halo growth function for an 8$\,h^{-1}$Mpc box started from three different redshifts. The blue data points results from the $z=50$ start, the turquoise data points from the $z=150$ start, and orange from the $z=250$ start, which is the redshift satisfying our starting criteria. The two fits shown are the Warren fit (solid line) and the PS fit (dashed line). Three different mass bins are shown. It is interesting to note that the late start seems to follow the PS fit at high redshift.[]{data-label="ploteleven"}](f11.eps){width="80mm"}
Finite-Volume Corrections
-------------------------
The finite size of simulation boxes can compromise results for the mass function in multiple ways. It is important to keep in mind that finite-volume boxes cannot be run to lower than some redshift, $z_{\rm
final}$, the stopping point being determined by when nonlinear scales approach close enough to the box size. Approaching too near this point delays the ride-up of nonlinear power towards the low-$k$ end, with a possible suppression of the mass function.
As a consequence of this delay, the evolution (incorrectly) appears more linear at large scales than it actually should, as compared to the $P(k)$ obtained in a much bigger box. Therefore, verifying linear evolution of the lowest $k$-mode is by itself [*not*]{} sufficient to establish that the box volume chosen was sufficiently large. For all of our overlapping-volume simulations we have checked that the power spectra were consistent across boxes up to the lowest redshift from which results have been reported (Table 1 lists the stopping redshifts).
Aside from testing for numerical convergence, it is important to show that finite-volume effects are also under control, especially any suppression of the mass function with decreasing box size (due to lack of large-scale power on scales greater than the box size). Several heuristic analyses of this effect have appeared in the literature. Rather than rely solely on the unknown accuracy of these results, however, here we also numerically investigate possible systematic differences in the mass function with box size.
Over the redshifts and mass ranges probed in each of our simulation boxes, we find no direct evidence for an error caused by finite volume (at more than the $\sim 20\%$ level), as already emphasized previously in [@Heitmann06]. (Overlapping box-size results over different mass ranges are shown in Fig. 3 of @Heitmann06.) Figure \[plotten\] shows the corresponding results in the present work. This is not to say that there are no finite-volume effects (the very high-mass tail in a given box must be biased low simply from sampling considerations) but that their relative amplitude is small. Below we discuss how to correct the mass function for finite box size.
### Volume Corrections from Universality {#sec_boxvolume_univ}
Let us first assume that mass function universality holds strictly, in other words, that for any initial condition the number of halos can be described by a certain scaled mass function (eq. \[\[fsigma\]\]) in which $\sigma(M)$ is the variance of the top-hat-smoothed linear density field. In the case of infinite simulation volume, $\sigma(M)$ is determined by equation (\[sig\]), and the mass function ${F}(M)$ of equation (\[FM\]) is $${F}(M) \equiv {dn \over d\log M}=\frac{\rho_b}{M}f(\sigma)
\frac{d\ln\sigma^{-1}}{d\log M}\ .
\label{eq_F_infinite}$$ In an ensemble of finite-volume boxes, however, one necessarily measures a different quantity: $${F}'(M') \equiv {dn' \over d\log M'}=\frac{\rho_b}{M'}f(\sigma')
\frac{d\ln\sigma'^{-1}}{d\log M'}\ .
\label{eq_F_finite}$$ Here $\sigma'(M')$ is determined by the (discrete) power spectrum of the simulation ensemble, although if universality holds as assumed, $f$ in equations (\[eq\_F\_infinite\]) and (\[eq\_F\_finite\]) is the same function.
Since we are, in general, interested in the mass function which corresponds to an infinite volume, we can then correct the data obtained from our simulations as follows: for each box size we can define a function $M'(M)$ such that $$\sigma(M) \equiv \sigma'(M'(M)).
\label{sprime}$$ Using equations (\[eq\_F\_infinite\]) – (\[sprime\]), we determine ${F}(M)$ as $${F}(M)={F}'(M')\,\frac{dM'(M)}{dM}.$$ Thus, the corrected number of halos in each bin is calculated as $$dn = dn' \frac{M'}{M} \ .
\label{volcorr}$$
The universality must eventually break down for sufficiently small boxes or high accuracy because the nonlinear coupling of modes is more complicated than that described by the smoothed variance. This violation can be partly corrected for by modifying the functional form of $\sigma'(M')$. Therefore, we also explore other choices of $\sigma'(M')$ which may better represent the mass function in the box. To address this question we provide a short summary of the Press-Schechter approach.
### Motivation from Isotropic Collapse
We first consider the idealized case of a random isotropic perturbation of pressureless matter and assume that the primordial overdensity at the center of this perturbation has a Gaussian probability distribution. The probability of local matter collapse at the center is then fully determined by the local variance of the primordial overdensity $\sigma^2$. Consequently, for the isotropic case the contribution of Fourier modes of various scales to the collapse probability is fully quantified by their contribution to $\sigma^2$.
To see this, consider the evolution of matter density $\rho_{\rm loc}$ at the center of the spherically symmetric density perturbation. For transparency of argument, let us focus on the evolution during the matter-dominated era; it is straightforward to generalize the argument to include a dark energy component $\rho_{\rm de}(z)$, homogeneous on the length scales of interest, by a substitution $\rho_{\rm loc} \to
\rho_{\rm m,\,\rm loc}+\rho_{\rm de}$ in equations (\[Freed\_eq\]) and (\[q\_collapse\_def\]). By Birkhoff’s law, the evolution of $\rho_{\rm loc}$ and the central Hubble flow $H_{\rm loc}\equiv
\frac{1}{3}\bm{\nabla}\cdot\bm{v}_{\rm loc}$ are governed by the closed set of the Friedmann and conservation equations, $$\begin{aligned}
H_{\rm loc}^2&=&\frac{8\pi G \rho_{\rm loc}}{3}
-\frac{\kappa}{a_{\rm loc}^2},
\label{Freed_eq}\\
a_{\rm loc}&\equiv&\left(\frac{\rho_0}{\rho_{\rm loc}}\right)^{1/3},~~
\frac{da_{\rm loc}}{dt}=H_{\rm loc}a_{\rm loc},
\label{a_loc}\end{aligned}$$ where $\kappa$ is a constant determined by the initial conditions, $\rho_0$ is arbitrary (e.g., $\rho_0=\left.\rho_b\right|_{z=0}$), and $t$ is the proper time.
The degree of nonlinear collapse at the center can be quantified by a dimensionless parameter $$q\equiv 1 -\frac{3H_{\rm loc}^2}{8\pi G\rho_{\rm loc}}.
\label{q_collapse_def}$$ First consider early times, when the evolution is linear, and let $\rho_{\rm loc}=\rho_b(1+\delta)$. Then for the growing perturbation modes during matter domination $H_{\rm loc}=\bar H(1-\delta/3)$. Given these initial conditions, which set the initial $\rho_{\rm loc}$ and the constant $\kappa$ in equation (\[Freed\_eq\]), the subsequent evolutions of $\rho_{\rm loc}$, $H_{\rm loc}$, and therefore $q$ are determined unambiguously.
During the linear evolution in the matter era $q=5\delta/3$ is small and grows proportionally to the cosmological scale factor $a$. For positive overdensity, nonlinear collapse begins when $q$ becomes of order unity, reaching its maximal value $q=1$ when $H_{\rm loc}=0$, and decreasing rapidly afterwards. (We can observe the latter by rewriting eq. \[\[q\_collapse\_def\]\] as $$q=\frac{3\kappa}{8\pi G a_{\rm loc}^2\rho_{\rm loc}}
\propto\rho_{\rm loc}^{-1/3},$$ having applied eqs. \[\[Freed\_eq\]\] and \[\[a\_loc\]\].) Nonlinear collapse of matter at the center of the considered region can be said to occur either when $q\rightarrow 0$ or when $q$ reaches a critical “virialization” value $q_c$.
Now it is easy to argue that in the isotropic case the Press-Schechter approach gives the true probability of the collapse, $P(q>q_c,z)$, for a redshift $z$. Indeed, the evolution of $q$ is set deterministically by the primordial density perturbation at the center; for adiabatic initial conditions specifically, it is set by the curvature perturbation $\zeta$ at the center. Since higher values of $\zeta$ lead to earlier collapse, $$P(q>q_c,z)=P(\zeta>\zeta_c(z))=
\frac{1}{2}\mathop{\rm erfc}\left[\frac{\zeta_c(z)}{\sqrt2\,\sigma}\right],
\label{collapse_prob_isotrop}$$ where the last equality uses the explicit form of $P(\zeta)$ as a Gaussian distribution with a variance $\sigma^2$.
![Mass function data corrected for finite box volume by the extended Press-Schechter prescription of §\[sec\_boxvolume\_num\] (squares). We show the results as a ratio with respect to the Warren fit and follow the conventions of Fig. \[plotten\]. We also display the volume-uncorrected data (crosses). Note that the volume-corrected data join smoothly across the box-size boundaries. This box correction brings the results very close to universal behavior at high redshifts (see Fig.\[plotfifteen\]).[]{data-label="plottwelve"}](f12.eps){width="80mm"}
If the considered isotropic distribution is confined by a (spherical) boundary and $\sigma$ at the center is reduced by removal of large-scale power, then equation (\[collapse\_prob\_isotrop\]) should accurately describe the corresponding change of the collapse probability. In numerical simulations, due to the imposition of periodic boundary conditions, there is no power on scales larger than the box size. In this case the variance $\sigma$ should be specified by the analogue of equation (\[sig\]) with the integral replaced by a sum over discrete modes.
For the mass function (eq. \[\[eq\_F\_infinite\]\]), a constant reduction of the variance $\sigma^2(M)$ due to the removal of large-scale power leads to a suppression of the mass function at the high-mass end and, counterintuitively, a boost at the low-mass end. The latter is easily understood as follows: The $\sigma$-dependent terms of equation (\[eq\_F\_infinite\]), $$f(\sigma)\frac{d\ln\sigma^{-1}}{d\log M}=\frac{d\rho(M)/\rho_b}{d\log M},$$ give the fraction of the total matter density that belongs to the halos of mass $M$. When the variance is decreased by the box boundaries, this fraction is boosted at low masses due to a shift of halo formation to an earlier stage, where a larger fraction of matter is bound into low-mass objects.
### Numerical Results and Comparisons {#sec_boxvolume_num}
Following the above intuition, we employ the extended Press-Schechter formalism (@Bond91) to correct for the missing fluctuation variance on box scales. This formalism, while clearly inadequate at various levels in describing halo formation in realistic simulations (@Bond91; @Katz93; @White96), has nevertheless been very successful as a central engine in describing the statistics of cosmological structure formation. As shown by [@Mo96] using $N$-body simulations, the biasing of halos in a spherical region with respect to the average mass overdensity in that region is very well described by the extended Press-Schechter approach. [@Barkana04] discussed the suppression of the halo mass function in terms of this bias, and suggested a prescription for adjusting large-volume mass function fits such as Warren or ST to small boxes. Here we do not follow this path but directly work with the numerical data by correcting the number of halos in each bin as in equation (\[volcorr\]).
In the extended Press-Schechter scenario of halo formation, $\sigma'$ on the right-hand side of equation (\[eq\_F\_finite\]) would be approximately connected with $\sigma$ via $\sigma'^2 = \sigma^2 -
\sigma_{R({\rm box})}^2$ [@Bond91], where $\sigma_{R({\rm
box})}^2$ is the variance of fluctuations in spheres that contain the simulation volume. Since extended Press-Schechter theory is derived for spherical regions, while our simulation boxes are cubes, we define $R({\rm box})$ as the radius of a sphere enclosing the same volume as in the simulations.
The action of this correction is shown in Figure \[plottwelve\]. Finite-volume corrections are subdominant to statistical error at $z=0$ and 5. At higher redshifts, the corrections produce results that are consistent across box sizes, i.e., that have no systematic shape changes or “jumps” across box boundaries. Moreover, the action of the corrections is to bring the simulation results closer to a universal behavior. We discuss this aspect further below.
{width="80mm"}
Mass function corrected for a finite box using the assumption of strict universality, as described in §\[sec\_boxvolume\_univ\] (squares). Again, we show uncorrected data as well (crosses), and follow the conventions of Fig. \[plotten\]. This correction produces a clear systematic shift in the results across box boundaries. \[plotthirteen\]
For completeness, we mention two other approaches aimed at box-adjusting the mass function. The first (@Yoshida03c; @Bagla06) simply replaces the original mass variance (eq. (\[sig\])) with $$\sigma^2_{\rm box}(M,z) =
\frac{d^2(z)}{2\pi^2}\int^{\infty}_{2\pi/L}k^2P(k)W^2(k,M)dk\ ,
\label{sigbox}$$ the lower cut-off arising from imposing periodic boundary conditions ($L$ is the box-size). (For enhanced fidelity with simulations, the integral in eq. \[\[sigbox\]\] goes to a sum over the simulation box modes.) This approach basically assumes that $\sigma$ defined via an infrared cutoff is the appropriate replacement for the infinite-volume mass variance. Figure \[plotthirteen\] shows the effect of this suggested correction: At $z=0$ and 5 it is not noticeable, but at higher redshifts the correction is significant relative to the accuracy with which the binned mass function is determined. Furthermore, it exhibits systematic shape changes and offsets across boxes, in contrast to the results shown in Figure \[plottwelve\]. For example, at $z=10$ the corrected data at the crossover point between the 4 and 8$\,h^{-1}$Mpc boxes ($\sim 10^8\,h^{-1}M_{\odot}$) have an offset of $5\%$. We conclude that this approach is disfavored by our simulation results.
An alternative strategy is to estimate the mass variance from each realization of $P(k)$ in the individual simulation boxes and to treat every box individually, as done in [@Reed07]. This has in fact two purposes: to compensate for the realization-to-realization variation in density fluctuations (which could be a problem for small boxes) and also to compensate for an overall suppression in the mass function as discussed above. The disadvantage is that each of many realizations now has a different $\sigma(M)$ for a given value of $M$.
Mass Function Universality
--------------------------
Finally, we investigate the universality of the mass function found by Jenkins. Approximate universality is expected from the analytic arguments of PS and the extended, excursion-set formulation of Bond et al. (1991). The universal behavior of halo formation persists even in the model of ellipsoidal collapse of ST, in which the predicted mass function is no longer of the PS form. On the other hand, the universality cannot be exact if the nonlinear interactions of different scales are fully accounted for: The nonlinear evolution that leads to the formation of halos of mass $M$ must involve multiple degrees of freedom that are described by more parameters than the overall variance of the primordial overdensity smoothed by a top-hat filter $W(r,M)$. The universality is expected to be violated at sufficiently high resolution of the mass function even in the PS-type spherical collapse model: It is more reasonable to represent the probability of the collapse not by a fraction of particles at the center of spheres enclosing a mass $M$ but by any fraction of particles belonging to such spheres [@beta106]. The improved mass-function derived from this argument deviates somewhat from a universal form [@beta206].
To investigate the extent our numerical simulations are consistent with universality, we combine our results for $f(\sigma,z)$ as a function of the variance $\sigma^{-1}$ from the entire simulation set in one single curve at various redshifts. This curve is expected to be independent of redshift if universality holds. We display the results in Figure \[plotfourteen\] for the raw data and in Figure \[plotfifteen\] for the same data after applying the volume corrections discussed earlier.
In the raw data of Figure \[plotfourteen\], the agreement with the various fits is quite tight (except for PS) until $\ln\sigma^{-1}>0.3$. Beyond this point, the multiple-redshift simulation results do not lie on top of each other; in the absence of any possible systematic deviation, this would denote a failure of the universality of the FOF, $b=0.2$ mass function at small $\sigma$. Note also that beyond this point the ST and Jenkins fits have a steeply rising asymptotic behavior (relative to the Warren fit). The Reed et al. (2003) fit, meant to be valid over the range $-1.7\le
\ln \sigma^{-1} \le 0.9$, is in better agreement with our results, to the extent that a single fit can be overlaid on the data.
The ostensible violation of universality seen above is small, however, and subject to a systematic correction due to the finite simulation volume(s). On applying the correction discussed in Section 5.6.3, we obtain the results shown in Figure \[plotfifteen\], the key difference being that beyond $\ln\sigma^{-1}>0.3$ the multiple-redshift simulation results now lie on top of each other and, within the statistical resolution of our simulations, are consistent with universal behavior. Specifically, we do not observe the sort of violation reported by [@Reed07] at high redshifts. This could be due to several factors. The finite-sampling FOF mass correction and the finite-volume corrections we employ are different from those of [@Reed07] and the boxes we use at high redshifts are significantly larger. We note also that the difference between the Warren fit and the $z$-dependent fit of [@Reed07] does not appear to be statistically very significant given their data.
Conclusions and Discussion {#conclusion}
==========================
![Scaled differential mass function from all simulations, prior to applying finite-volume corrections. Fits shown are Warren (red), PS (dark blue), ST (black), Jenkins (light blue), and Reed et al. (2003) (yellow). Dashed lines denote an extrapolation beyond the original fitting range. The bottom panel shows the ratio relative to the Warren fit. The failure of the different redshift results to lie on top of each other at small values of $\sigma$ indicate a possible violation of universality.[]{data-label="plotfourteen"}](f14.eps){width="7.5cm"}
We have investigated the halo mass function from $N$-body simulations over a large mass and redshift range. A suite of 60 overlapping-volume simulations with box sizes ranging from 4 to 256$\,h^{-1}$Mpc allowed us to cover the halo mass range from 10$^7$ to 10$^{13.5}\,h^{-1}M_\odot$ and an effective redshift range from $z=0$ to 20.
In order to reconcile conflicting results for the mass function at high redshifts, as well as to investigate the reality of the breakdown of the universality of the mass function, we have studied various sources of error in $N$-body computations of the mass function. A set of error control criteria need to be satisfied in order to obtain accurate mass functions. These simple criteria include an estimate for the necessary starting redshift, for the required mass and force resolution to resolve the halos of interest at a certain mass and redshift, and for the number of time steps.
The criteria for the initial redshift appear to be particularly restrictive. For small boxes, commonly used in the study of the formation of the first objects in the Universe, significantly higher initial redshifts are required than is the normal practice. A violation of this criterion leads to a strong suppression of the mass function, most severe at high redshifts. Recent results by other groups may be contaminated due to a violation of this requirement; a careful re-analysis of small-box simulations is apparently indicated.
The force resolution criterion is especially useful for grid codes, PM as well as adaptive mesh. The mass function can be obtained reliably from PM codes down to small-mass and up to high-mass halos provided the halos are adequately resolved. The resolution criterion is also very useful in setting refinement levels for adaptive mesh refinement (AMR) codes. As shown recently (O’Shea et al. 2005; Heitmann et al. 2005, 2007) the mass function from AMR codes is suppressed at the low-mass end if the base refinement level is too coarse. A more detailed analysis on how to incorporate our results to improve the efficiency of AMR codes is underway.
The results for the required number of time steps to resolve the mass functions is somewhat surprising. The halo mass function appears to be very robust with respect to the number of time steps chosen to follow the evolution, even though the inner structure of the halos will certainly not be correct. Even a small number of time steps is sufficient to obtain a close-to-correct mass function at $z=0$. This considerably simplifies the study of the mass function and its evolution.
![Volume-corrected scaled differential mass function following Fig.\[plotfourteen\]. Note the significantly improved agreement with universal behavior (overlapping results beyond $\ln\sigma^{-1}\sim0.3$).[]{data-label="plotfifteen"}](f15.eps){width="7.5cm"}
Since finite-volume effects can also lead to a suppression of the mass function, we have tried to minimize the importance of these effects by avoiding too-small box sizes, by using overlapping boxes, and by restricting the mass range investigated in a given box size. In addition, we have found that a box-size correction motivated by the extended Press-Schechter formalism for the mass variance appears to give consistent results when applied to our multiple-box simulation ensembles.
We now briefly comment on results found previously by other groups. Jang-Condell & Hernquist (2001) find good agreement with the PS fit at $z=10$ for a mass range $4 \times 10^{5} - 4 \times 10^{8}
\,h^{-1}M_\odot$. The crossover of PS with the more accurate fits at $z=10$ takes place in exactly this region (see Figs. \[plotone\] and \[plotten\]). Therefore, all fits are very close, and the mass function from a single 1 Mpc box at a single redshift as shown in Jang-Condell & Hernquist (2001) cannot distinguish between them.
As mentioned earlier in §\[evolreview\], good agreement with the PS result has been reported at high redshifts (some results being even lower than PS) by several other groups (Yoshida et al. 2003a, 2003b, 2003c; @Cen04; @Trac06). The simulations of [@Cen04] and [@Trac06] were started at $z_{\rm in}\sim 50$, substantially below the starting redshift that would be suggested by our work. The very large number of particles in the [@Trac06] simulation requires a high starting redshift (Fig. \[plotfive\]). Therefore, the depressed mass function results of these simulations are very consistent with a too-low initial redshift. ([@Trac06] have recently rerun their simulations with a much higher initial redshift \[$z=300$\], and now find results consistent with ours.) The initial particle density of the [@Iliev05] simulations is very close to that of our 16$\,h^{-1}$Mpc box, in which case also a high redshift start is indicated (we used $z_{\rm in}=200$). Finally, the initial redshift of the Yoshida et al. papers, $z_{\rm in}=100$, for boxes of size $\sim 1\,h^{-1}$Mpc, also appears to be significantly on the low side.
We have compared our simulation results for the mass function with various fitting functions commonly used in the literature. The recently introduced ($z=0$) fit of Warren leads to good agreement (at the 20% level with no volume correction, and at the 5% level with volume correction) at all masses and all redshifts we considered. Other modern fits, such as Reed et al. (2003, 2007), also lie within this range. These fits do not suffer from the overprediction of large halos at high redshifts observed for the ST fit. The PS fit performs poorly over almost all the considered mass and redshift ranges, at certain points falling below the simulations by as much as an order of magnitude.
The evolution of the mass function can be used to test the (approximate) universality of the FOF, $b=0.2$ mass function. At low redshifts our data are in good agreement with those of [@Reed07] (at $z=5$), finding a (possible) mild redshift dependence (at the 10% level). At higher redshifts, however, we find that volume corrections are important to the extent that little statistically significant evidence for breakdown of universality remains in our mass function data. A full theoretical understanding of this very interesting result remains to be elucidated.
We have made no attempt to provide a fitting function for our data due to several reasons. First, the current simulation state of the art has not reached the point that one can be confident of percent-level agreement between results from different simulations even in regimes that are not statistics-dominated [@Heitmann07]. Second, simulations have not sufficiently explored the extent to which universal forms for the mass function are indeed applicable as cosmological parameters are systematically varied. Third, absent even a compelling phenomenological motivation for the choice of fitting functions, there is an inherent arbitrariness in the entire procedure. Finally, it is not clear how to connect the FOF mass function to observations. In general, tying together mass-observable relations requires close coupling of simulations and observational strategies. In studies of cosmological parameter estimation, we support working directly with simulations rather than with derived quantities, which would add another layer of possible systematic error. Because observations already significantly constrain the parametric range, and are a smooth function of the parameters, this approach is quite viable in practice (@coscal1; @coscal2).
We thank Kevork Abazajian, Nick Gnedin, Daniel Holz, Lam Hui, Gerard Jungman, Savvas Koushiappas, Andrey Kravtsov, Steve Myers, Ken Nagamine, Brian O’Shea, Sergei Shandarin, Ravi Sheth, Mike Warren, and Simon White for helpful discussions. We are particularly grateful to Adam Lidz and Darren Reed for their help, many discussions, and sharing their results with us. We are indebted to Mike Warren for use of his parallel FOF halo finder. The authors acknowledge support from the Institute of Geophysics and Planetary Physics at Los Alamos National Laboratory. S. B., S. H., and K. H. acknowledge support from the Department of Energy via the Laboratory-Directed Research and Development program of Los Alamos. P. M. R. acknowledges support from the University of Illinois at Urbana-Champaign and the National Center for Supercomputing Applications. S. H., K. H., and P. M. R. acknowledge the hospitality of the Aspen Center for Physics, where part of this work was carried out. The calculations described herein were performed using the computational resources of Los Alamos National Laboratory. A special acknowledgement is due to supercomputing time awarded to us under the LANL Institutional Computing Initiative. P. M. R. and Z. L. also acknowledge support under a Presidential Early Career Award from the US Department of Energy, Lawrence Livermore National Laboratory (contract B532720).
\[append\]
In this Appendix we discuss in detail previous results on the mass function at high redshift. As explained in the main paper, these results are often contradictory. We structure our discussion with respect to the physical volume simulated.
Small-Volume Simulations
------------------------
Small-box simulations of side $\sim 1\,h^{-1}$Mpc have been performed by several groups. Using a treecode with softening length $0.4\,h^{-1}$kpc, and a 1$\,h^{-1}$Mpc box with $128^3$ particles, [@Jangcondell01] evolved their simulation from $z_{\rm in}=100$ to $z=10$. With a halo finder that combined overdensity criteria with an FOF algorithm, the mass function was determined over the range $10^{5.5} - 10^{8.1}\,h^{-1}M_\odot$, keeping halos with as few as eight particles. At $z=10$ they found “remarkably close agreement” with the PS fit but did not quantify the agreement explicitly.
In a series of papers, Yoshida et al. ran simulations with similar box sizes as above, most including the effects of gas dynamics. The simulations were performed with the TreePM/smoothed particle hydrodynamics code GADGET-II [@Springel05b] and followed the evolution of 2$\times$ 324$^3$ particles (324$^3$ in the case of dark matter only), covering a halo mass range of 10$^5$-10$^{7.5}M_\odot$. All simulations were started at $z_{\rm in}=100$ from “glass” initial conditions (Baugh et al. 1995; White 1996), in contrast to the grid-based initial conditions used here. The focus of [@Yoshida03a] was the origin of primordial star-forming clouds. As part of that investigation, a dark-matter-only simulation in a 1.6$\,h^{-1}$Mpc box was carried out. The halo density results for $z=20$ to 32 lay systematically below the PS prediction, with the discrepancy being worse at high redshifts. The authors argued that this low abundance of halos was (possibly) due to finite-box-size effects. In [@Yoshida03b], the mass function at $z=20$ for a warm dark matter model was compared with CDM, with the simulation set up being very similar to that of [@Yoshida03a], a 1 Mpc box started at $z=100$. The results obtained were also similar; at $z=20$ the CDM mass function was in good agreement with the PS fit. In a third paper, [@Yoshida03c], a running spectral index was considered. Here results for a standard CDM mass function for a 1 Mpc box were given, this time at $z=17$ and 22. Consistent with their previous results, they found good agreement with PS at these redshifts. (The FOF linking length used in the last paper was $b=0.2$, while in the first two papers $b=0.164$ was chosen. This did not appear to make much of a difference, however.) These papers do not quantitatively compare the numerical mass function to the PS fit. (In contrast to these findings, a recent 1 Mpc box GADGET-II simulation with $z_{\rm
in}\sim 120$ has been performed by [@Maio06] who find good agreement with the Warren fit as extrapolated by linear theory – in clear disagreement with PS.)
A similar strategy was followed in [@Cen04] who investigated dark matter halos in a mass range of $10^{6.5}$ to $10^9\,h^{-1} M_\odot$, using a TreePM code (Xu 1995; Bode et al. 2000). The box size was taken to be $4\,h^{-1}$Mpc, the softening length was set at $0.14\,h^{-1}$kpc, $512^3$ particles were used, and the simulations had a starting redshift of $z_{\rm in}=53$. Halos were identified using the overdensity scheme DENMAX [@Bertschinger91]. Among other quantities, they studied the mass function between $z=11$ and 6 and found that the PS function “provides a good fit” but without explicit quantification.
Overall, these small-box simulations, run with different codes and different halo finders, all found a “depressed” mass function (see Fig. \[plotone\]), consistent with PS and deviating very significantly from the predictions of the more modern fitting forms. In contrast, other simulations also using small boxes have come to quite different conclusions. For example, in [@Reed07], a large suite of different box sizes and simulations was used to cover the mass range between 10$^5$ and 10$^{11.5}
\,h^{-1}M_\odot$ at high redshift. The smallest boxes considered in this study were 1 $\,h^{-1}$Mpc on a side. The authors studied the halo mass function at redshifts out to $z=30$, implementing a correction scheme to account for finite-box effects, as discussed in more detail below. Overall, [@Reed07] confirmed previous results as found by [@Reed03] and [@Heitmann06]: PS underestimates the mass function considerably (by at least a factor of 5 at high redshift and high masses), and ST overpredicts the halo abundance at high redshift.
Large-Volume Simulations
-------------------------
The large-box strategy is exemplified by a recent dark matter simulation with the GADGET-II code [@Springel05a]. The evolution of 2160$^3$ particles in a $500 \,h^{-1}$Mpc box was followed from $z_{\rm in}=127$ until $z=0$. The softening length was $5\,h^{-1}$kpc. The high mass and force resolution was sufficient to study the mass function reliably down to a redshift of $z=10$, covering a mass range of $10^{10}$ to $10^{16} \,h^{-1}M_\odot$, with halos being identified by a standard FOF algorithm with $b=0.2$. The results are consistent with the Jenkins fit, even though the mass function points at redshifts $z=1.5$, 3.06, and 5.72 are slightly higher than the Jenkins fit and slightly lower for $z=10$. No residuals were shown nor quantitative statements made.
In two recent papers, [@Iliev05] and [@Zahn06] investigated cosmic reionization, providing mass function results at high redshift as part of this work. [@Iliev05] ran a PM simulation with PMFAST (Merz et al. 2005) in a 100$\,h^{-1}$Mpc box with 1624$^3$ particles on a 3248$^3$ mesh. They present results for the mass function at redshifts between $z=6$ and 18.5, using a spherical overdensity halo finder. At lower redshifts they find good agreement with ST, and at high redshift ($z>10$) the results are closer to PS (because of their limited mass range, a more quantitative statement is difficult to make). [@Zahn06] ran a 1024$^3$ particle simulation (dark matter only) in a 65.6$\,h^{-1}$Mpc box with GADGET-II and analyzed the FOF, $b=0.2$ mass function out to $z=20$. Between $z=6$ and 14 they found good agreement with ST in the mass range of 10$^9$ to $10^{12}M_\odot$. At $z=20$ they found that the simulation results were below ST but above PS, in relatively good agreement with the recent findings of [@Heitmann06] and [@Reed07].
Medium Volume Simulations
--------------------------
[@Reed03] chose a compromise between the large- and small-box strategies by picking a 50$\,h^{-1}$Mpc box sampled with 432$^3$ particles. The tree code PKDGRAV was used to evolve the simulation from different starting redshifts between $z_{\rm in}=139$ and 69 until $z=0$. The smallest halo contained 75 particles, leading to a mass range of roughly 10$^{10}$ to $10^{14.5}\,h^{-1}M_\odot$. Good agreement (better than 10%) was found with the ST fit up to $z\simeq 10$. For higher redshifts, the ST fit overpredicted the number of halos, up to 50% at $z=15$. At this high redshift, statistics were lacking, and the resolution was not sufficient to resolve very small halos. A more recent 50$\,h^{-1}$Mpc simulation with PMFAST with $z_{\rm in}=60$ has been carried out by [@Trac06] using a spherical overdensity definition of halo mass. In this work, the mass function, in the redshift range $6<z<15$, is found to be in very good agreement with PS, in gross contradiction with the results of most of the other simulations mentioned above. (This contradiction has recently been resolved by rerunning their simulation with $z_{in}=300$ and identifying halos with a $b=0.2$ FOF finder.)
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[^1]: See http://www.lofar.org
[^2]: See http://haystack.mit.edu/arrays/MWA/
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|
---
abstract: 'Anisotropic cosmological models are constructed in $f(R,T)$ gravity theory to investigate the dynamics of universe concerning the late time cosmic acceleration. Using a more general and simple approach, the effect of the coupling constant and anisotropy on the cosmic dynamics have been investigated. In the present study it is found that cosmic anisotropy affects substantially the cosmic dynamics.'
author:
- 'B. Mishra [^1],Sankarsan Tarai [^2], S. K. Tripathy[^3]'
title: '**Anisotropic cosmological reconstruction in $f(R,T)$ gravity**'
---
**PACS number**: 04.50kd.\
**Keywords**: $f(R,T)$ cosmology, Bianchi Type $VI_h$, Anisotropic Universe, cosmic strings
Introduction
============
Cosmological models are constructed in recent times to account for the predicted late time cosmic acceleration usually by incorporating possible dark energy candidates in the field equations or by modifying the geometrical part of the action. Amidst the debate that, whether dark energy exists or whether there really occurs a substantial cosmic acceleration [@Riess98; @Perlm99; @Nielsen16], researchers have devoted a lot of time in proposing different dark energy models. These models are also tested against the observational data accumulated over a long period of time. Some vector-tensor models are also proposed to explain the cosmic speed up phenomena without adopting these approaches. In these vector-tensor models, the presence of a vector field such as the electromagnetic field provides the necessary acceleration [@Fer07; @Jim08; @Jim09; @Jim09a; @Dale12]. Usually in General Relativity (GR), it is not possible to explain the late time cosmic acceleration without the assumption of additional dynamical degrees of freedom besides the tensor modes. Some scalar fields are considered as a solution to this. These scalar fields are usually ghost fields having negative kinetic energy, at least around flat, cosmological or spherically symmetric backgrounds e.g. Bulware-Desser mode in massive gravity [@Bul72], bending mode in the self-accelerating branch of Dvali-Gabadadze-Porrati model [@Koyama07; @Sbisa15; @Gumru16]. Among all the constructed models to understand the cosmic speed up phenomena, geometrically modified gravity theories have attracted substantial research attention. In geometrically modified theories, instead of incorporating some additional matter fields (may be ghost scalar fields), the Einstein Hilbert action is modified considering some extra geometrical objects. These models thereby provide a ghost free and stable alternative to GR. In this context, Harko et al. have proposed $f(R,T)$ gravity theory in which, the geometry part of the action has been modified in such a manner that, the usual Ricci Scalar $R$ in the action is replaced by a function $f(R,T)$ of Ricci Scalar $R$ and the trace of the energy momentum tensor $T$ [@Harko11]. In that work, Harko et al. have suggested three different possible forms of the functional $f(R,T)$ such as $f(R,T)= R+2f(T)$, $f(R,T)= f_1(R)+f_2(T)$ and $f(R,T)= g_1(R)+g_2(R)g_3(T)$, where $f(T), f_1(R), f_2(T), g_1(R), g_2(R)$ and $g_3(T)$ are some arbitrary plausible functions or $R$ and $T$. In $f(R,T)$ gravity theory, the cosmic acceleration is achieved from the geometrical modification and a bit of matter content coupled to the geometrical part of the action. Many workers have used different forms of these functionals to address the issue of mysterious dark energy and the late time cosmic phenomena [@Shami15; @Myrza12; @Hound12; @Alvar13; @Barri14; @Vacar14; @Shari12; @Shari14; @Yous16a; @Yous16b; @Sahoo16; @Aktas17]. Alvarenga et al. studied the scalar perturbations [@Alvar13], Shabani and Ziaie studied the stability of the model [@Shab17; @Shab17a], Sharif and Zubair investigated the energy conditions and stability of power law solutions [@Sharif13] in this modified gravity theory. Sharif and Zubair [@Sharif12a] and Jamil et al. [@Jamil12] have studied thermodynamic aspects of $f(R,T)$ theory. There are some good works available in literature in the context of astrophysical applications of this theory [@Moraes17; @Zubair15; @Alha16].
With the advent of recent observations regarding the cosmic anisotropy [@Anton10; @Planck14; @Javan15; @Lin16; @Bengaly15; @Zhou17; @Andrade18], there has been an increase in the interest to investigate on the breakdown of the standard cosmology based on cosmic isotropy [@Campa06; @Grup10; @SKT14; @Saadeh16; @SKT17; @Deng18]. In view of this, anisotropic cosmological models that bear a similarity to Bianchi morphology have gained much importance [@Koivisto08; @SKT15; @Campa07; @Campa09]. In the context of geometry modification to explain the late time cosmic dynamics and to take into account the cosmic anisotropy, many workers have constructed some Bianchi type cosmological models in $f(R,T)$ gravity [@Mishr14; @Mishr16; @Sharif17; @Mishra18]. However, a lot remain to be explored in this modified gravity theory in the context of different unanswered issues concerning the late time cosmic acceleration and cosmic anisotropy.\
In the discussion of cosmological models, space-times admit a three-parameter group of automorphisms are important. When the group is simply transitive over three-dimension, constant-time subspace is useful. Bianchi has shown that there are only nine distinct sets of structure constants for groups of this type. So, Bianchi type space-times admit a three parameter group of motions with a manageable number of degrees of freedom. Kramer et al.[@Kramer80] provided a complete list of Bianchi types I–IX space-times. In this work, we have constructed some anisotropic cosmological models in $f(R,T)$ gravity. We have adopted a simple approach to the cosmic anisotropy to investigate the effect of anisotropy on cosmic anisotropy. In order to provide some anisotropic directional pressure, we have considered an anisotropic source along x-direction such as the presence of one dimensional cosmic strings. The effect of the coupling constant in the determination of the cosmic evolution has been investigated. We organise the work as follows: In Sect-II, some basic equations concerning different properties of the universe are derived for Bianchi $VI_h$ model in the framework of the modified $f(R,T)$ gravity. The dynamical features of the models are discussed in Sect-III. Considering the dominance of quark matter that have not yielded to the hadronization process, we have derived the quark energy density and pressure and their evolutionary behaviour in Sect-IV. We conclude in Sect-V.
Basic Equations
===============
The field equation in $f(R,T)$ gravity for the choice of the functional $f(R,T)=f(R)+f(T)$ is given by [@Harko11; @Mishra16a] $$\label{eq:2}
f_R (R) R_{ij}-\frac{1}{2}f(R)g_{ij}-\left(\nabla_i \nabla_j-g_{ij}\Box\right)f_R(R)=\left[8\pi +f_T(T),\right]T_{ij}+\left[f_T(T)p+\frac{1}{2}f(T)\right]g_{ij}$$ where $f_R=\frac{\partial f(R)}{\partial R}$ and $f_T=\frac{\partial f(T)}{\partial T}$. We wish to consider a functional form of $f(R,T)$ so that the field equations in the modified gravity theory can be reduced to the usual field equations in GR under suitable substitution of model parameters. In this context, we have a popular choice, $f(R,T)=R+2\beta T$ [@Das17; @Mishr14; @Mishr16; @Moraes15; @Shamir15]. However, we consider a time independent cosmological constant $\Lambda_0$ in the functional so that $f(R,T)= R+2\Lambda_0+2\beta T$. Here $\beta$ is a coupling constant. For this particular choice of the functional $f(R,T)$, the field equation in the modified theory of gravity becomes,
$$\label{eq:3}
R_{ij}-\frac{1}{2}Rg_{ij}=\left[8\pi+2\beta\right]T_{ij} + \left[\left(2p+T\right)\beta+\Lambda_0\right] g_{ij}$$
which can also be written as $$\label{eq:3}
R_{ij}-\frac{1}{2}Rg_{ij}=\left[8\pi+2\beta\right]T_{ij} + \Lambda(T) g_{ij}.$$
Here $\Lambda(T)=\left(2p+T\right)\beta+\Lambda_0$ can be identified as the effective time dependent cosmological constant. If $\beta=0$, the above modified field equation reduces to the Einstein field equation in GR with a cosmological constant $\Lambda_0$. One can note that, the effective cosmological constant $\Lambda(T)$ picks up its time dependence through the matter field. For a given matter field described through an energy momentum tensor, the effective cosmological constant can be expressed in terms of the matter components. In the present work, we consider the energy momentum tensor as $T_{ij}=(p+\rho)u_iu_j - pg_{ij}-\xi x_ix_j$, where $u^{i}u_{i}=-x^{i}x_{i}=1$ and $u^{i}x_{i}=0$. In a co moving coordinate system, $u^{i}$ is the four velocity vector and $p$ is the proper isotropic pressure of the fluid. $\rho$ is the energy density and $\xi$ is the string tension density. The strings are considered to be one dimensional and thereby contribute to the anisotropic nature of the cosmic fluid. The direction of the cosmic strings is represented through $x^{i}$ that are orthogonal to $u^{i}$.
The field equations of the modified $f(R,T)$ gravity theory, for Bianchi type $VI_h$ space-time described through the metric $ds^2 = dt^2 - A^2dx^2- B^2e^{2x}dy^2 - C^2e^{2hx}dz^2$ now have the explicit forms
$$\label{eq:6}
\frac{\ddot{B}}{B}+\frac{\ddot{C}}{C}+\frac{\dot{B}\dot{C}}{BC}- \frac{h}{A^2}= -\alpha(p-\xi) +\rho \beta+\Lambda_0$$
$$\label{eq:7}
\frac{\ddot{A}}{A}+\frac{\ddot{C}}{C}+\frac{\dot{A}\dot{C}}{AC}- \frac{h^2}{A^2}=-\alpha p +(\rho+\xi)\beta+\Lambda_0$$
$$\label{eq:8}
\frac{\ddot{A}}{A}+\frac{\ddot{B}}{B}+\frac{\dot{A}\dot{B}}{AB}- \frac{1}{A^2}=-\alpha p +(\rho+\xi)\beta+\Lambda_0$$
$$\label{eq:9}
\frac{\dot{A}\dot{B}}{AB}+\frac{\dot{B}\dot{C}}{BC}+\frac{\dot{C}\dot{A}}{CA}-\frac{1+h+h^2}{A^2}=
\alpha \rho -\left(p-\xi\right)\beta +\Lambda_0$$
$$\label{eq:10}
\frac{\dot{B}}{B}+ h\frac{\dot{C}}{C}- (1+h)\frac{\dot{A}}{A}=0.$$
An over dot over a field variable denotes ordinary differentiation with respect to the cosmic time. Here $\alpha = 8\pi+3\beta$ and $A=A(t),B=B(t), C=C(t)$. An interesting component in this space time is the constant exponent $h$, which takes integral values $-1,0,1$. These three integral values decide the behaviour of the model. However, Tripathy et al. [@skt15] and Mishra et al. [@skt16] have shown from the calculation of the energy and momentum of diagonal Bianchi type universes that, the value $h=-1$ is favoured compared to other values. Moreover, only in this value of the exponent $h$, the total energy of an isolated universe vanishes. In view of this, in the present work, we assume this value of $h$ i.e. $h=-1$ and study the dynamics of the anisotropic universe in presence of anisotropic energy sources. The directional Hubble rates may be considered as $H_x=\frac{\dot{A}}{A}$, $H_y=\frac{\dot{B}}{B}$ and $H_z=\frac{\dot{C}}{C}$. With $h=-1$, it is straightforward to get $H_y=H_z$ from and consequently the mean Hubble parameter becomes, $H=\frac{1}{3}(H_x+2H_z)$. The set of field equations can be reduced to
$$\begin{aligned}
2\dot{H_z}+3H^2_z+\frac{1}{A^2} &=& -\alpha(p-\xi)+\rho\beta+\Lambda_0,\label{eq:16}\\
\dot{H_x}+\dot{H_z}+H^2_x+H^2_z+H_xH_z-\frac{1}{A^2} &=& -\alpha p+\left(\rho+\xi\right)\beta+\Lambda_0,\label{eq:17}\\
2H_xH_z+H_z^2-\frac{1}{A^2} &=& \alpha \rho-\left(p-\xi\right)\beta+\Lambda_0. \label{eq:18}\end{aligned}$$
From the above field equations -, we obtain the expressions for pressure, energy density and the string tension density as
$$\begin{aligned}
p &=& \frac{1}{\alpha^2-\beta^2}\left[\left(s_1-s_2+s_3\right)\beta-s_2\alpha+\left(\alpha-\beta\right)\Lambda_0\right],\label{eq:19}\\
\rho &=& \frac{1}{\alpha^2-\beta^2}\left[s_3\alpha-s_1\beta-\left(\alpha-\beta\right)\Lambda_0\right], \label{eq:20}\\
\xi &=& \frac{s_1-s_2}{\alpha-\beta}. \label{eq:21}\end{aligned}$$
Consequently, the equation of state parameter $\omega$ and the effective cosmological constant $\Lambda$ can be expressed as $$\begin{aligned}
\omega &=& -1+\left(\alpha+\beta\right)\frac{s_2-s_3}{s_1\beta-s_3\alpha+\left(\alpha-\beta\right)\Lambda_0},\label{eq:22}\\
\Lambda &=& \frac{\beta}{\alpha^2-\beta^2}\left[(s_2+s_3)\alpha-(2s_1-s_2+s_3)\beta-(\alpha+\beta)(s_2-s_1)-2(\alpha-\beta)\Lambda_0\right]+\Lambda_0.\label{eq:23}\end{aligned}$$
In the above equations, $s_1, s_2$ and $s_3$ are functions of the directional Hubble parameters and scale factor: $s_1=2\dot{H_z}+3H^2_z+\frac{1}{A^2}$, $s_2=\dot{H_x}+\dot{H_z}+H^2_x+H^2_z+H_xH_z-\frac{1}{A^2}$ and $2H_xH_z+H_z^2-\frac{1}{A^2}$. Eqns - describe the dynamical behaviour of the model. Once the evolutionary behaviour of the functions $s_1, s_2$ and $s_3$ are obtained from some assumed dynamics, the dynamical nature of the model can be studied easily and the modified gravity model can be reconstructed accordingly.
Dynamical Parameters
====================
We intend to investigate the cosmic history through the assumption of an assumed dynamics concerning the late time cosmic acceleration. In view of this, we assume the scalar expansion be governed by an inverse function of cosmic time i.e. $\theta=(H_x+2H_z)=\frac{m}{t}$ and also we assume that $\theta$ be proportional to the shear scalar $\sigma^2=\frac{1}{2}\left(\sum H_i^2-\frac{1}{3}\theta^2\right); i=x, y,z$. Consequently, $H_x=\left(\frac{km}{k+2}\right)\frac{1}{t}$, $H_y=H_z=\left(\frac{m}{k+2}\right)\frac{1}{t}$. The directional scale factors can be expressed as $A=t^{km/(k+2)}$, $B=C=t^{m/(k+2)}$.
For such an assumption, the functions $s_1, s_2$ and $s_3$ reduce to $$\begin{aligned}
s_1 &=& \left[\frac{3m^2-2(k+2)m}{(k+2)^2}\right]\frac{1}{t^2}+\frac{1}{t^{\frac{2km}{k+2}}}, \\
s_2 &=& \left[\frac{(k^2+k+1)m^2-(k+1)(k+2)m}{(k+2)^2}\right]\frac{1}{t^2}-\frac{1}{t^{\frac{2km}{k+2}}}, \\
s_3 &=& \left[\frac{(2k+1)m^2}{(k+2)^2}\right]\frac{1}{t^2}-\frac{1}{t^{\frac{2km}{k+2}}}.\end{aligned}$$
From the field eqns. -, the pressure, energy density and string tension density can be obtained as:
$$\begin{aligned}
p &=& \frac{1}{(\alpha^2-\beta^2)}\left[\left(\frac{\phi_1}{(k+2)^2}\right)\frac{1}{t^2}+\frac{(\alpha+\beta)}{t^{\frac{2km}{k+2}}}+(\alpha-\beta)\Lambda_0\right], \\
\rho &=& \frac{1}{(\alpha^2-\beta^2)}\left[\left(\frac{\phi_2}{(k+2)^2}\right)\frac{1}{t^2}+\frac{(\beta-\alpha)}{t^{\frac{2km}{k+2}}}-(\alpha-\beta)\Lambda_0\right],\\
\xi &=& \frac{1}{(\alpha-\beta)}\left[\frac{(k-1)(m^2-m)}{(k+2)^2t^2}-\frac{2}{t^{\frac{2km}{k+2}}}\right],\end{aligned}$$
where $\phi_1=m\{(k^2+k-2)\beta+(k^2+3k+2)\alpha\}-m^2\{(k^2-k-3)\beta-(k^2+k+1)\alpha\}$ and $\phi_2=(2k+1)m^2\alpha-(3m^2-2km-4m)\beta$ are redefined constants. These physical quantities evolve with the cosmic expansion. Their evolution is governed by two time dependent factors: one behaving like $t^{-2}$ and the other behaving as $t^{-\frac{2km}{k+2}}$. Since $m$ and $k$ are positive quantities, the magnitude of the physical quantities (neglecting their sign) decrease monotonically with cosmic time. It is interesting to note that, $\xi$ also decreases from a large value in the initial epoch to small values at late phase of cosmic evolution. This behaviour of $\xi$ implies that, at the initial phase, more anisotropic components are required than at late phase.
{width="80.00000%"}
From eqs. - , we obtain the equation of state parameter $\omega=\frac{p}{\rho}$ and the effective cosmological constant $\Lambda$ respectively as
$$\begin{aligned}
\omega &=& -1+(\alpha+\beta)\left[\frac{\phi_3}{\phi_4+(\alpha-\beta)(k+2)^2 \left\{\Lambda_0 t^2-t^{2\left(\frac{k-km+2}{k+2}\right)}\right\}}\right],\\
\Lambda &=&\frac{\beta}{(\alpha^2-\beta^2)} \left[ \frac{\phi_5}{(k+2)^2t^2}-\frac{2(\alpha+\beta)}{t^{\frac{2km}{k+2}}}-2(\alpha-\beta)\Lambda_0 \right]-\frac{\phi_6}{(k+2)^2t^2}+\frac{\beta}{(\alpha-\beta)t^{\frac{2km}{k+2}}}+\Lambda_0,\end{aligned}$$
where $\phi_3=(k^2-2k)m^2-(k^2+2k+3)m$, $\phi_4=(3m^2-2km-4)\beta-(2k+1)m^2 \alpha$, $\phi_5=\{(k+1)\alpha+(k-3)\beta\}(m^2-m)$ and $\phi_6=\frac{\beta(k-1)(m^2-m)}{(\alpha-\beta)}$ are some constants.
{width="80.00000%"}
{width="80.00000%"}
The dynamical nature of the model can be assessed through the evolution of the equation of state parameter $\omega$. In Figure 1, $\omega$ is plotted as function of redshift for four different values of the coupling constant $\beta$ namely $\beta =0, 0.5, 1.0$ and 2.0. $\beta=0$ refers to the case in GR. The anisotropic parameter is considered to be $k=0.7$ and $m$ is fixed from the observationally constrained value of deceleration parameter $q=-0.598$ [@Montiel14]. For all the cases considered here, $\omega$ becomes a negative quantity and remains in the quintessence region through out the period of evolution considered in the work. It decreases from some higher value at the beginning to low values at late times. However, at late phase of cosmic evolution, $\omega$ grows up a little bit which may be due to the anisotropic effect of cosmic strings.
The coupling constant $\beta$ affects the dynamical behaviour of the equation of state parameter. In order to understand the effect of the $\beta$ on $\omega$, the equation of state at the present epoch is plotted as a function of $\beta$ in Figure 2 for three different values of $k$. One can note that, $\omega$ increases with the increase in the value of the coupling constant. In view of the recent observations predicting an accelerating universe, the value of coupling constant $\beta$ should have a lower value i.e. $\beta \leq 1$.
In Figure 3, we have shown the effect of anisotropy on the equation of state parameter. In the figure, we assume three representative values of the anisotropy i.e $k=0.7, 0.8$ and 0.9 for a given coupling constant $\beta=0.5$. Anisotropy brings a substantial change in the magnitude as well as the behaviour of the equation of state parameter. There occurs a flipping behaviour of $\omega$ at a redshift $z_f \simeq 4$. At a cosmic time earlier to $z_f$, with the increase in the anisotropy of the model, $\omega$ assumes a higher value. In other words, prior to $z_f$, higher the value of $k$, higher is the $\omega$. It displays an opposite behaviour at cosmic times later to $z_f$. Also, at the redshift $z_f$, curves corresponding to all $k$ considered here cross each other. In general, the rate of evolution of the equation of state parameter increases with the increase in the value of the anisotropic parameter.
Anisotropic universe with quark matter
======================================
One can believe that, quarks and gluons did not yield to hadronization and resisted as a perfect fluid that spread over the universe and may contribute to the accelerated expansion. Here we will reconstruct an anisotropic cosmological model with non interacting quarks that may well be dealt as a Fermi gas with an equation of state given by [@Kapusta94; @Aktas07]
$$p_q=\frac{\rho_q}{3}-B_c,$$
where $p_q$ is the quark pressure, $\rho_q$ is the quark energy density and $B_c$ is the bag constant. We assume that quarks exist along with one dimensional cosmic strings without any interaction. The quark energy density can then be expressed as $\rho_q=\rho-\xi-B_c$. Going in the same manner as described in the previous section, we can have the expressions for the quark pressure and quark energy density as $$\begin{aligned}
\rho_q &=& \frac{1}{\alpha^2-\beta^2}\left[(\alpha+\beta)s_2+s_3\alpha-(\alpha+2\beta)s_1-\left(\alpha-\beta\right)\Lambda_0\right]-B_c, \label{eq:26}\\
p_q &=& \frac{1}{3\left(\alpha^2-\beta^2\right)}\left[(\alpha+\beta)s_2+s_3\alpha-(\alpha+2\beta)s_1-\left(\alpha-\beta\right)\Lambda_0\right]-\frac{4B_c}{3} \label{eq:27}\end{aligned}$$
If we put $\beta=0$, the model reduces to that in GR with a cosmological constant. In that case, the above equations reduce to $$\begin{aligned}
\rho_q &=& \frac{1}{8\pi}\left[s_2+s_3-s_1-\Lambda_0\right]-B_c, \label{eq:28}\\
p_q &=& \frac{1}{24\pi}\left[s_2+s_3-s_1-\Lambda_0\right]-\frac{4B_c}{3} \label{eq:29}\end{aligned}$$
Substituting the expressions for $s_1, s_2$ and $s_3$ in eqs. and , the quark matter energy density and quark pressure are obtained as
$$\begin{aligned}
\rho_q &=& \frac{1}{\alpha^2-\beta^2}\left[\frac{\phi_7}{(k+2)^2}\frac{1}{t^2}+\frac{(\alpha+3\beta)}{t^{\frac{2km}{k+2}}}-\left(\alpha-\beta\right)\Lambda_0\right]-B_c,\\
p_q &=& \frac{1}{3(\alpha^2-\beta^2)}\left[\frac{\phi_7}{(k+2)^2}\frac{1}{t^2}+\frac{(\alpha+3\beta)}{t^{\frac{2km}{k+2}}}-\left(\alpha-\beta\right)\Lambda_0\right]-\frac{4B_c}{3},\end{aligned}$$
where $\phi_7=\phi_2-(k-1)(m^2-m)(\alpha+\beta)$. For some reasonable value of the coupling parameter $\beta$ and the anisotropic parameter $k$, the quark energy density and quark pressure decrease smoothly with the cosmic evolution. Bag constant certainly has a role to play at late times when the value of $\rho_q$ and $p_q$ are mostly dominated by this quantity.
Conclusion
==========
This paper reports the investigation of the dynamical behaviour of an anisotropic Bianchi type $VI_h$ universe in the presence of one dimensional cosmic strings and quark matter. Anisotropic cosmological models are reconstructed for a power assumption of the scale factor in the frame work of $f(R,T)$ gravity. In the process of reconstruction and study of dynamical features of the model, we chose the functional $f(R,T)$ as $f(R,T)=R+2\Lambda_0+2\beta T$. From some general expressions of the physical quantities, we derived the expression of the equation of state parameter and the effective cosmological constant. The effects of anisotropy $k$ and the coupling constant $\beta$ are investigated. It is observed that, with an increase in the coupling constant the equation of state parameter assumes a higher value. Anisotropy is observed to affect largely to the dynamics of the model. The equation of state parameter undergoes an increased rate of growth with an increase in the anisotropy. We hope, the present study will definitely put some light in the context of the uncertainty prevailing in the studies of the late time cosmic phenomena.
Acknowledgment
==============
BM and SKT thank IUCAA, Pune (India) for hospitality and support during an academic visit where a part of this work is accomplished.ST thanks University Grants Commission (UGC), New Delhi, India, for the financial support to carry out the research work. The authors are very thankful to the anonymous reviewer for his useful comments that helped us to improvise the manuscript.
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[^1]: Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India, E-mail:[email protected]
[^2]: Department of Mathematics, Birla Institute of Technology and Science-Pilani, Hyderabad Campus, Hyderabad-500078, India, E-mail:[email protected]
[^3]: Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India, E-mail:tripathy\_ [email protected]
|
---
abstract: 'We compute the effect of Markovian bulk dephasing noise on the staggered magnetization of the XXZ Heisenberg chain, as the system evolves after a Néel quench. For sufficiently weak system-bath coupling, the unitary dynamics are found to be preserved up to a single exponential damping factor. This is a consequence of the interplay between $\mathbb{PT}$ symmetry and weak symmetries, which strengthens previous predictions for $\mathbb{PT}$-symmetric Liouvillian dynamics. Requirements are a non-degenerate $\mathbb{PT}$-symmetric generator of time evolution $\hat{\mathcal{L}}$, a weak parity symmetry and an observable that is anti-symmetric under this parity transformation. The spectrum of $\hat{\mathcal{L}}$ then splits up into symmetry sectors, yielding the same decay rate for all modes that contribute to the observable’s time evolution. This phenomenon may be realized in trapped ion experiments and has possible implications for the control of decoherence in out-of-equilibrium many-body systems.'
author:
- Moos van Caspel$^1$
- 'Vladimir Gritsev$^{1,2}$'
bibliography:
- 'xxz\_dephasing.bib'
title: 'Symmetry-protected coherent relaxation of open quantum systems'
---
Introduction {#section:introduction}
============
The theory of open quantum systems has a long history, finding countless applications in quantum optics, nanotechnology, quantum information and other fields of physics [@Breuer]. Particularly in the past decade, there has been a drive to apply this formalism to the realm of many-body quantum systems. Rapid developments in the fields of cold atoms and quantum computation form major incentives to improve our understanding of the interaction between a quantum system and its environment. Whether one is interested in shielding a system from decoherence or driving it towards a specific steady state, the theoretical challenges are largely the same and they are formidable.
Analytical methods to tackle dissipative many-body systems are few and far between. Most efforts are focused on Markovian baths, allowing a formulation in terms of a Lindblad master equation. Exact solutions have been given for quadratic fermionic systems [@Eisler2011; @Prosen2010; @Prosen2010a; @Torres2014; @Znidaric2010], but this excludes most bulk dissipation in spin systems. Various algebraic methods have been used to solve Lindblad equations [@Ringel2012; @Bolanos2015; @Mesterhazy2017], requiring the unitary and dissipative parts to form a closed algebra. Finally, very specific models have been mapped to integrable closed systems, solvable by Bethe ansatz [@Essler_prosen]. On the other hand, numerical approaches are typically restricted to either very short time scales or to the infinite-time limit. Properties of the full relaxation process are surprisingly difficult to probe, but the presence of symmetries can simplify the problem.
Symmetry structures in the context of open quantum systems have been studied mostly in relation to stationary states and conserved quantities [@AlbertLiang; @Nicacio2016; @Wilming2017; @Manzano2014]. They are closely tied to the theory of decoherence-free subspaces and subsystems, which are considered promising candidates for quantum memory [@Zanardi1997]. However, symmetries can also play an important role in the dynamics at shorter time scales. These dynamics satisfy the Lindblad equation and are generated by the Liouvillian superoperator $\hat{\mathcal{L}}$, acting on the space of linear operators. Symmetries allow for a separation of operator sectors, which split up the spectrum of $\hat{\mathcal{L}}$. For an observable, this means that large parts of the spectrum may not contribute toward the time evolution of its expectation value, depending on the symmetry properties of the observable. We use this phenomenon, combined with the spectral structure of a $\mathbb{PT}$-symmetric Liouvillian [@Prosen2012], to study a scenario where dissipation affects the dynamics of observables in a predictable and coherent manner.
In general, adding a non-unitary part to a system’s time evolution introduces many new time scales, corresponding to the decay rates of the different modes of the time evolution’s generator $\hat{\mathcal{L}}$. For a generic system, all of these modes will contribute at intermediate times, affecting the dynamics in a highly nontrivial way. In the system we study — a XXZ Heisenberg chain affected by bulk dephasing noise — some observables are protected by symmetry from all but one of the system’s decay rates. The result is an overall damping factor such that the unitary dynamics are preserved for weak system-bath coupling. This surprising effect should be experimentally measurable and may be relevant for the control of decoherence in many-body quantum gates.
The structure of the paper is as follows: in section \[section:lindblad\], we review the Lindblad master equation and its spectral properties. Section \[section:symmetries\] details the different types of symmetries and their interplay in Liouvillian dynamics. Section \[section:xxz\] describes how these symmetries apply to the XXZ Heisenberg chain with bulk dephasing. Finally, we study the staggered magnetization after a Néel quench in section \[section:neel\], as an example of symmetry-protected coherent relaxation. The appendix shows a detailed perturbation theory calculation of corrections to the staggered magnetization.
Lindbladian time evolution {#section:lindblad}
==========================
Markovian dynamics can always be described in terms of a Lindblad master equation of the form $$\begin{aligned}
\begin{split}
\frac{\d \rho}{\d t} &= -i \left[ H, \rho \right]\\ &+ \gamma \sum_i \left( L_i \rho L^\+_i - \frac{1}{2} L^\+_i L_i \rho - \frac{1}{2} \rho L^\+_i L_i \right),
\end{split}\end{aligned}$$ where $\gamma>0$ is the system-bath coupling strength and $L_i$ are the so-called jump operators, which encode the interaction between the system and the bath. This form can typically be derived from a microscopic theory by integrating out the bath and applying several approximations, such as the Born-Markov and the Rotating Wave approximation [@Breuer]. It is often convenient to write the Lindblad equation in superoperator form: $$\begin{aligned}
\frac{\d \rho}{\d t} = \hat{\mathcal{L}}\rho ~~~ \Rightarrow ~~~ \rho(t) = e^{t \hat{\mathcal{L}}} \rho(0).\end{aligned}$$ where the *Liouvillian superoperator* $\hat{\mathcal{L}}$ is a trace, hermiticity and positivity-preserving linear map, such that it maps one density matrix to another. Superoperators act on the space $\mathcal{B}(\mathcal{H})$, consisting of all linear operators acting on the Hilbert space of quantum states $\mathcal{H}$. In turn, $\mathcal{B}(\mathcal{H})$ itself can be treated as a Hilbert space with the Hilbert-Schmidt inner product: $(A,B) \equiv \operatorname{tr}(A^\+ B)$. In what follows, we will be particularly concerned with non-degenerate Liouvillians, which are diagonalizable[^1] and can therefore be written as a spectral decomposition: $$\begin{aligned}
\begin{split}
\hat{\mathcal{L}}\rho &= \sum_m \lambda_m \operatorname{tr}(v_m^\+ \rho)\, u_m\\ &\Rightarrow ~~~ \rho(t) = \sum_m e^{t \lambda_m} \operatorname{tr}(v_m^\+ \rho(0))\, u_m,
\end{split}\end{aligned}$$ where $\hat{\mathcal{L}} u_m = \lambda_m u_m$ and $\hat{\mathcal{L}}^\+ v_m = \lambda_m^* v_m$ such that $\operatorname{tr}(v_m^\+ u_m) = 1$. Since $\hat{\mathcal{L}}$ is not Hermitian, its left and right eigenmodes are not equal and its eigenvalues $\lambda_m$ are generally complex. Furthermore, $\operatorname{Re}(\lambda_m) \leq 0$ or else $\rho(t)$ would blow up in the infinite time limit. Thanks to Brouwer’s fixed point theorem, there must be at least one zero eigenvalue $\lambda_0 = 0$. The corresponding eigenmode $u_0$ is known as a *steady state* of the time evolution. Symmetries can result in multiple steady states, as we will show in section \[section:symmetries\]. One can also have persistent oscillations with $\lambda \neq 0$ on the imaginary axis, but these are rare and will not be discussed further in this work. All other modes $u_m$ with $\operatorname{Re}(\lambda_m) \neq 0$ are known as *decay modes*, and they must be traceless operators.
Studying the spectrum of the Liouvillian can tell you much about the non-unitary dynamics. One particular quantity of interest is the *dissipative gap*, defined as $\Gamma = \operatorname{min}_{\text{decay modes}}\{|\operatorname{Re}(\lambda_m)|\}$. The gap determines the longest timescale in the system. At long times, generic observables decay exponentially at rate $\Gamma$. Expanding the time evolution of the expectation values of observables yields: $$\begin{aligned}
\la O(t) \ra = \operatorname{tr}(O \rho(t)) = \sum_m e^{t \lambda_m} \operatorname{tr}(v_m^\+ \rho(0)) \operatorname{tr}(O u_m). \label{eq:obs_spectral}\end{aligned}$$ At sufficiently long times, the dissipative gap dominates all higher decay modes and determines the rate at which the steady state is approached. For some systems the gap may close in the thermodynamic limit, leading to algebraic decay [@CaiBarthel]. But as we will see, the presence of symmetries can throw a wrench into this simplified picture. Each symmetry sector has its own gap and the decay rates can be vastly different between observables.
Lastly, it is illuminating to consider the spectrum of the Liouvillian for a closed system, i.e. $\gamma = 0$. The eigenvalues are purely imaginary and given by $\lambda = i(\epsilon_i - \epsilon_j)$, corresponding to the eigenmodes $|\psi_i\ra \la\psi_j|$ with $H|\psi_i\ra = \epsilon_i |\psi_i\ra$. There is a degeneracy at $\lambda = 0$, the size of the Hilbert space, as projectors onto energy eigenstates are naturally stationary. If we then turn on a weak dissipation, degenerate perturbation theory shows that these diagonal modes $|\psi_i\ra \la\psi_i|$ will hybridize and their eigenvalues will spread out. In the case of Hermitian jump operators $L_i^\+ = L_i$ or in the presence of $\mathbb{PT}$ symmetry (see section \[section:symmetries\]), they will stay on the real axis.
Symmetries in Hilbert space, Liouville space and beyond {#section:symmetries}
=======================================================
In the context of unitary time evolution, discrete symmetries are relatively straightforward. They are typically generated by a Hermitian operator $O$, acting on the Hilbert space $\mathcal{H}$, such that $[H, O] = 0$. As a result, energy eigenstates are simultaneously eigenstates of $O$. The Hilbert space can therefore be separated into blocks, one for each eigenvalue of $O$, which are preserved under unitary time evolution. If there are multiple, mutually commuting symmetries, then there will be subblocks within each symmetry block.
When adding a dissipative, non-unitary part to the time evolution, this story becomes slightly more complicated [@AlbertLiang]. Symmetry on the level of the Hilbert space $\mathcal{H}$, as described above, still exists and we will call this a *strong symmetry*, following Ref. \[\]. In the case of Lindbladian evolution, the operator $O$ should not only commute with $H$, but also with each jump operator individually: $[L_i,O] = 0 ~ \forall i$. Once again the Hilbert space separates into blocks. Of course non-unitarity will produce mixed states, but it only mixes states within the same symmetry block.
This block structure of $\mathcal{H}$ can be lifted to the space of linear operators $\mathcal{B}(\mathcal{H})$, which we will call Liouville space. To make this more precise, consider $n$ symmetry blocks $\mathcal{U}_i$ that form a partition of $\mathcal{H}$. We can then partition $\mathcal{B}(\mathcal{H})$ into $n^2$ blocks $\hat{\mathcal{U}}_{i,j}$ spanned by operators of the form $|\psi\ra \la \phi|$ with the states $|\psi\ra \in \mathcal{U}_i$ and $|\phi\ra \in \mathcal{U}_j$. Because of the strong symmetry, the $n$ ‘diagonal’ blocks[^2] $\hat{\mathcal{U}}_{i,i}$ must each have their own steady state. In rare cases, some ‘off-diagonal’ blocks may also contain fixed points of the Liouvillian, yielding what is known as a decoherence-free subspace [@Lidar2003].
However, one can have a block structure in Liouville space without the strict conditions of a strong symmetry. The only requirement for such a structure is a unitary superoperator that commutes with the Liouvillian: $[\hat{\mathcal{L}}, \hat{O}] = 0$, where unitarity is defined as preserving the Hilbert-Schmidt inner product. This is known as a *weak symmetry* (or a dynamical symmetry in some literature [@Baumgartner2008]). Note that this requirement is immediately satisfied in case of a strong symmetry by defining $\hat{O}\rho = O \rho O^\+$. But a weak symmetry by itself does not imply the presence of multiple steady states. In general, only one block will contain the steady state, while the others are spanned by traceless decay modes. In section \[section:xxz\], examples of both weak and strong symmetries will be discussed in detail.
It is necessary to understand the symmetry structures of Liouville space when studying the time evolution of observables. Each of the Liouvillian’s decay modes is confined to one symmetry block. If an observable has no components in a given symmetry block, then it is clear from Eq. (\[eq:obs\_spectral\]) that any decay modes in this block will not contribute towards the observable’s time evolution. This can severely impact which parts of the spectrum are relevant, depending on the observables of interest. An extreme example is the staggered magnetization in the XXZ chain with dephasing, as we will see in section \[section:neel\]. One more symmetry is needed to produce such a case, of a special type that acts on the Liouvillian superoperator itself. Table \[tab:symmetries\] shows an overview of the three different types of symmetries.
$\mathbb{PT}$ symmetry in Lindbladian time evolution was first described in Ref. \[\]. Since it features prominently in the rest of this work, we will summarize its properties here but refer to the original paper for details. A Liouvillian is $\mathbb{PT}$-symmetric when it satisfies the condition: $$\begin{aligned}
&\hat{\mathcal{P}}\hat{\mathcal{L}}'\hat{\mathcal{P}} = - (\hat{\mathcal{L}}')^\+ \label{eq:ptsymmetry} \\
&\hat{\mathcal{L}}' = \hat{\mathcal{L}} + \gamma \delta \hat{\id},~~~~~ \delta \equiv -\frac{\operatorname{tr} \hat{\mathcal{L}}}{\gamma \operatorname{tr}\hat{\id}} \label{eq:tracelessL}\end{aligned}$$ where $\mathcal{L}'$ is the traceless part of the Liouvillian, $\hat{\mathcal{P}}$ is some (unitary) parity superoperator (with $\hat{\mathcal{P}}^2 = \id$) and the Hermitian adjoint is again defined using the Hilbert-Schmidt inner product. Since the unitary part of $\hat{\mathcal{L}}$ is traceless, $\operatorname{tr} \hat{\mathcal{L}}$ is proportional to $\gamma$, such that the scaling factor $\delta$ is dimensionless and does not depend on the coupling strength. In words, this is an antisymmetry relating the adjoint of the traceless part of the Liouvillian to a parity transformation of the same. While this seems highly specific and not very physical, it can be considered a generalization of $\mathcal{PT}$-symmetric quantum mechanics [@Bender]. $\mathbb{PT}$-symmetric Liouvillians have some very nice properties and turn out to be surprisingly prevalent in spin systems [@Prosen2012a].
*Symmetry * *Acts on* *Condition* *Ex. XXZ*
--------------- ----------------------------------------- ------------------------------------------------------------------------------------ -----------------------------------
Strong $\mathcal{H}$ $ [H, O] = [L_i, O]$ = 0 $\sum_i S^z_i$
Weak $\mathcal{B}(\mathcal{H})$ $[\hat{\mathcal{L}}, \hat{O}] = 0$ $\hat{R}, \hat{F}$
$\mathbb{PT}$ $\mathcal{B}(\mathcal{B}(\mathcal{H}))$ $\hat{\mathcal{P}}\hat{\mathcal{L}}'\hat{\mathcal{P}} = - (\hat{\mathcal{L}}')^\+$ $\hat{\mathcal{P}} \rho = F \rho$
: Overview of different types of symmetries, acting on the hierarchy of Hilbert spaces. The last column shows the examples from the XXZ Heisenberg spin chain, as described in section \[section:xxz\]. $\mathcal{B}(A)$ refers to the vector space of linear operators acting on space A.[]{data-label="tab:symmetries"}
The spectrum of a $\mathbb{PT}$-symmetric Liouvillian shows a second reflection symmetry axis in the complex plane, at $\operatorname{Re} \lambda = -\gamma \delta$. This is in addition to the reflection symmetry across the real axis, which is guaranteed by hermiticity conservation. In the absence of degeneracies and for sufficiently weak system-bath coupling, *all* eigenvalues lie on these two axes. This can be seen by applying perturbation theory to the $\gamma = 0$ case, as mentioned at the end of section \[section:lindblad\]. $\mathbb{PT}$ symmetry guarantees that the diagonal operators (in the energy eigenbasis) stay on the real axis when the dissipation is turned on [@Prosen2012]. Meanwhile, the off-diagonal coherences are confined to move along $\operatorname{Re} \lambda = -\gamma \delta$ as $\gamma$ is increased. Only when two eigenvalues collide (thereby creating a degeneracy), they might shoot off into the complex plane. This can be described as a spontaneous breaking of the $\mathbb{PT}$ symmetry and at these points the Liouvillian becomes non-diagonalizable [@Kanki2016].
While the decay modes with eigenvalues on the real axis originate from purely diagonal operators, the perturbation does yield non-zero off-diagonal elements to first order in $\gamma$. Likewise, those on the vertical axis may have non-zero diagonal elements under the dissipative perturbation[^3]. The claim that decoherence is purely determined by modes with decay rate $\gamma \delta$ is therefore only true for the asymptotic limit $\gamma \rightarrow 0$. But beyond this limit, the presence of weak symmetries can divide the spectrum in such a way that all decay modes on the real axis are confined to one symmetry sector.
One can understand this as follows: consider a weak symmetry $[\hat{\mathcal{L}}, \hat{O}] = 0$ where $\hat{O}$ has $n$ distinct eigenvalues, which label the different blocks that partition the space of operators. Unless the dissipation is fine-tuned in a very particular way, the Hamiltonian and dissipative parts of $\hat{\mathcal{L}}$ should separately commute with $\hat{O}$. This implies that $\hat{O}H = H$, such that the Hamiltonian is found in the sector corresponding to eigenvalue 1, i.e. the invariant subspace of $\hat{O}$. Writing $H = \sum_i \epsilon_i |\psi_i\ra \la \psi_i|$, the individual projectors onto the energy eigenstates must also be part of that sector, assuming that $\hat{O}$ is not specifically constructed to permute these different projectors (in which case it would be unlikely to commute with the dissipator). In other words, all operators that are diagonal in the energy eigenbasis are invariant under $\hat{O}$ and must belong to the same symmetry block. In the presence of $\mathbb{PT}$-symmetry, these are precisely the operators responsible for the eigenvalues on the real axis! As the dissipation is turned on perturbatively and the eigenvalues spread along the axis, these diagonal decay modes will be mixed with others (introducing off-diagonal components), but only those within the same symmetry sector. Due to the weak symmetry, the block structure is preserved.
We have shown that, for $\mathbb{PT}$-symmetric Liouvillian dynamics with a weak symmetry, all decay modes on the real axis belong to the same symmetry sector. As mentioned before, this becomes relevant when studying the time-evolution of the expectation value of observables. For observables outside out of this sector, with no components invariant under $\hat{O}$, the decay modes on the real axis do not contribute. In case of sufficiently weak system-bath coupling $\gamma$, all other decay modes lie on the vertical access and decay at the same rate. The result is an overall exponential damping factor, on top of the unitary dynamics of the closed system. The interplay between weak and $\mathbb{PT}$-symmetry, and its effect on the dynamics of observables, constitutes our main result. The rest of the paper is dedicated to a concrete example of the phenomenon.
XXZ Heisenberg spin chain with bulk dephasing {#section:xxz}
=============================================
As an example, we consider the spin-$\frac{1}{2}$ XXZ anisotropic Heisenberg chain, given by the Hamiltonian $$\begin{aligned}
H = J \sum_{i=1}^{N-1} \left(S_i^x S_{i+1}^x + S_i^y S_{i+1}^y + \Delta S_i^z S_{i+1}^z \right)\end{aligned}$$ with coupling strength $J$, anisotropy $\Delta$, zero magnetic field and open boundary conditions. For the dissipative part, we consider bulk dephasing noise, defined by the $N$ jump operators $L_i = S_i^z$. This open quantum system cannot be solved by any known analytical methods, except in the $\Delta = 0$ limit where it can be mapped to a Hubbard model and is solvable by Bethe ansatz [@Essler_prosen]. Nonetheless there have been some good numerical studies on the system, in particular on the scaling of its dissipative gap [@Znidaric].
Since the total magnetization $M = \sum_i S^z_i$ commutes with the Hamiltonian and with all jump operators $L_i$, it serves as the generator of a strong symmetry. This means that there are $2N-1$ magnetization blocks in $\mathcal{H}$ and the same number of diagonal blocks in $\mathcal{B(H)}$, each of which has its own steady state. These steady states are the maximally mixed states within each sector, as is easily checked by insertion into the Lindblad master equation. Thanks to the block structure, we can safely restrict ourselves to the zero-magnetization sector, which contains a lot of interesting physics. Note that the energy spectrum within this sector is non-degenerate, except at specific values of $\Delta$ corresponding to the XXZ model’s roots of unity [@Deguchi2001].
![The spectrum of the Liouvillian for the dissipative XXZ Heisenberg chain with $N = 6$, $\Delta = 0.3$ and three different values of the system-bath coupling $\gamma$. The plot axes, as well as $\gamma$, are in units of the nearest-neighbor coupling $J$. Eigenvalues are labeled according to their symmetry sector $\hat{\mathcal{U}}_{p,q}$ with $p, q \in \{\pm 1\}$. At the top, $\gamma = 0.003 < \gamma_{PT} \approx 0.013$ shows all eigenvalues located on the two axes of reflection. Those on the real axis all belong to sector $\hat{\mathcal{U}}_{+,+}$. As $\gamma$ is increased, the $\mathbb{PT}$ symmetry is spontaneously broken and eigenvalues of all sectors move away from the vertical axes, into the complex plane.[]{data-label="fig:spectrum"}](spectrum5){width="\columnwidth"}
The zero-magnetization sector contains two additional weak symmetries, corresponding to spatial reflection and spin inversion. $$\begin{aligned}
{2}
R &= \prod_{i=1}^{N/2}\Big(S^+_i S^-_{N+1-i} + &&S^-_i S^+_{N+1-i} + 2S^z_i S^z_{N+1-i} + \frac{1}{2}\id \Big) \nonumber \\
& &&\Rightarrow ~~~ R S^z_i R = S^z_{N+1-i} \\[6pt]
F &= \prod_{i=1}^N \left(S^+_i + S^-_i\right) &&\Rightarrow ~~~ F S^z_i F = - S^z_i.\end{aligned}$$ Both are parity operators, i.e. $R^2 = F^2 = \id$ with eigenvalues $\pm 1$. $R$ and $F$ commute with the Hamiltonian and which each other, but not with the individual jump operators. However, it is easy to check that the superoperators $\hat{R} \rho \equiv R \rho R$ and $\hat{F}\rho \equiv F \rho F$ do commute with the Liouvillian. Therefore the zero-magnetization sector of the Liouville space is split into four blocks $\hat{\mathcal{U}}_{p,q}$ labeled by the eigenvalues $p, q \in \{\pm 1\}$ of $\hat{R}$ and $\hat{F}$. The steady state, being proportional to the identity matrix, naturally is found in $\hat{\mathcal{U}}_{+,+}$. In fact, any decay mode that is purely diagonal in the energy eigenbasis will belong to this symmetry block. This can be seen as follows: since any energy eigenstate $|\psi\ra$ is also an eigenstate of $R$ and $F$ with eigenvalue $\pm 1$, we conclude that $|\psi\ra\la \psi|$ must be invariant under the superoperators $\hat{R}$ and $\hat{F}$. This is relevant, considering that the system is also $\mathbb{PT}$-symmetric.
The traceless part of the Liouvillian, as defined in (\[eq:tracelessL\]), is given by $$\begin{aligned}
\hat{\mathcal{L}}'\rho = -i \left[H, \rho \right] + \gamma \sum_i S^z_i \rho S^z_i.\end{aligned}$$ The parity superoperator $\hat{\mathcal{P}}$ is given by left-multiplication of the spin inversion $F$, such that $\hat{\mathcal{P}} \rho = F \rho$. It is now simple to check that the condition (\[eq:ptsymmetry\]) for a $\mathbb{PT}$-symmetric Liouvillian is satisfied: $$\begin{aligned}
\begin{split}
\mathcal{P}\mathcal{L'}\mathcal{P} \rho &= -i F \left[ H, F \rho \right] + \gamma \sum_i F S^z_i F \rho S^z_i\\ &= -i \left[ H, \rho \right] - \gamma \sum_i S^z_i \rho S^z_i = -(\mathcal{L}')^\+ \rho,
\end{split}\end{aligned}$$ Figure \[fig:spectrum\] shows the Liouvillian spectrum for three values of $\gamma$. For sufficiently weak coupling, all eigenvalues are located along the two axes of reflection. Those along the real axis all correspond to decay modes in the $\hat{\mathcal{U}}_{+,+}$ symmetry block, which can be understood as follows: in the limit $\gamma \rightarrow 0$, these decay modes are purely diagonal in the energy eigenbasis and therefore even under $\hat{R}$ and $\hat{F}$. Because the dissipation preserves the symmetry structure in Liouville space, the modes must remain in the $\hat{\mathcal{U}}_{+,+}$ sector as the perturbation is turned on, even though they are no longer purely diagonal. In section \[section:neel\], we will see how this affects observables such as the staggered magnetization.
As $\gamma$ is further increased, the dynamics undergoes a transition where the $\mathbb{PT}$ symmetry is spontaneously broken and some of the eigenvalues leave the two axes. In Ref. \[\], an estimate is given for the critical coupling strength $\gamma_{PT}$ at which this happens. By computing the operator norm of the dissipator[^4] and estimating, in turn, the average density of states, we find the following expression for our model: $$\begin{aligned}
\gamma_{PT} \approx J \, \frac{(N-1)^2}{N}\, \binom{N}{N/2}^{-2}.\end{aligned}$$ Unfortunately this quantity decays exponentially as $N$ becomes large. However, even for coupling strengths well above $\gamma_{PT}$, the effects of the $\mathbb{PT}$ symmetry remain visible. Figure \[fig:spectrum\_variance\] shows the spread in the real part of eigenvalues, both for all eigenvalues and for only those in the double-odd sector $\hat{\mathcal{U}}_{-,-}$. As can be seen, the variance within the odd sector is far below that of the total variance for a significant region of parameter space, even after the sharp jump at $\gamma = \gamma_{PT}$. This also ties into the results of Ref. \[\], where a critical coupling $\gamma_c$ is described, at which the global dissipative gap switches from the even to the odd symmetry sector. This coupling $\gamma_c$ scales as $~ N^{-2}$, rather than exponentially.
![The variance of the real part of the Liouvillian spectrum as a function of the coupling strength $\gamma$, computed in units of $J$ for the dissipative XXZ Heisenberg chain with $N = 8$ and $\Delta = 0.3$. The (blue) circles indicate the variance over all eigenvalues, while the (red) diamonds include only those in the double-odd symmetry sector $\hat{\mathcal{U}}_{-,-}$. Values are rescaled by a factor $\gamma^{-2}$ to account for a uniform linear dependence on $\gamma$. A discontinuity around $\gamma_{PT} \approx 0.0003$ is clearly visible.[]{data-label="fig:spectrum_variance"}](variance_spectrum_n8_d03_v3){width="\columnwidth"}
Staggered magnetization after a Néel quench {#section:neel}
===========================================
The interplay of $\mathbb{PT}$ symmetry and weak parity symmetries results in an interesting structure within the Liouvillian spectrum of the XXZ chain with dephasing noise. To find out whether this is more than just a mathematical oddity, let us consider one of the natural observables for this system. The staggered magnetization is defined as $$\begin{aligned}
M_s = \frac{1}{N} \sum_{i=1}^N (-1)^i S_i^z.\end{aligned}$$ Its expectation value is maximized in the Néel state, defined as $|\text{N\'eel}\ra = |\!\downarrow \uparrow \downarrow \uparrow\! \ldots \ra$ in the local spin basis. We can imagine preparing the system in the Néel state and looking at the evolution of the staggered magnetization after the state is released. This can be described as a quantum quench from the Ising antiferromagnet ($\Delta \rightarrow \infty$) to the XXZ model, which was studied numerically (in the absence of dissipation) in Refs. \[\]. Since the Néel state has non-zero overlaps with all energy eigenstates in the zero-magnetization sector, the unitary dynamics at short times is extremely complex and impossible to study analytically, even using the tools of integrability. The numerics show that the staggered magnetization $M_s$ decays exponentially, modulated by an oscillation in the gapless regime. In the non-interacting limit ($\Delta = 0$), the decay becomes algebraic and is described exactly by a Bessel function.
Because the Néel quench provides such a rich unitary dynamics, it is well-suited to see the extreme effects of the symmetry structure within Liouville space. Naively, one would expect the dissipation to introduce many new timescales into the system, effectively destroying the characteristic behavior of the closed system. Looking back to Eq. (\[eq:obs\_spectral\]), the factor $\operatorname{tr}(v_m^\+ \rho(0))$ is non-zero for all decay modes, due to the nature of the Néel state. As it is an eigenstate of neither $R$ nor $F$, the density matrix $\rho(0)$ has components in all four symmetry blocks. The factor $\operatorname{tr}(O u_m)$, on the other hand, depends on the symmetry properties of the observable.
Assuming that $N$ is even, the staggered magnetization is antisymmetric under both of the parity symmetries: $$\begin{aligned}
R M_s R = F M_s F = -M_s \label{eq:MsSym}\end{aligned}$$ and is therefore located within the $\hat{\mathcal{U}}_{-,-}$ symmetry block of Liouville space. It will be orthogonal under the Hilbert-Schmidt inner product to any decay modes within other sectors. As a result, only the decay modes in $\hat{\mathcal{U}}_{-,-}$ will contribute toward the time evolution of $\la M_s \ra$, regardless of the initial state. And thanks to the $\mathbb{PT}$ symmetry, for $\gamma < \gamma_{PT}$ all those modes have eigenvalues on the vertical symmetry axis and hence decay with the same rate $\delta$. The weak dephasing noise only introduces one new timescale after the Néel quench, yielding an overall exponential damping factor on top of the existing (unitary) exponential decay of the staggered magnetization.
This can be made more explicit by applying perturbation theory in $\gamma$ to Eq. (\[eq:obs\_spectral\]), expanding $\lambda_m$, $u_m$ and $v_m$. Since the perturbation does not mix modes from different symmetry sectors, the expansion only involves off-diagonal coherences and there are no degeneracies. The calculation is done in the appendix. In addition to the overall factor $e^{-\gamma \delta t}$, we see a $\gamma^2$ correction to the expectation value, due the shift of the decay modes along the vertical axis.
![The staggered magnetization after a Néel quench of the XXZ spin chain with $N = 12$ and $\Delta = 0.4$, for various values of the bulk dephasing strength $\gamma$ (in units of $J$). For the closed system and for weak dephasing, the expectation value shows an exponentially damped oscillation. The oscillatory behavior is largely unchanged for $\gamma < 0.1$. Dashed lines show exponential fits of the envelopes. The bottom panel shows the resulting decay rates due to the dephasing $1/\tau_{diss} = 1/\tau(\gamma) - 1/\tau(0)$, with error bars acquired from the exponential fit. It is clear that $1/\tau_{diss}$ is proportional to $\gamma$. Parameters for this computation were chosen to minimize finite-size effects.[]{data-label="fig:ms_rates"}](ms_rates_n12_d04){width="\columnwidth"}
We have numerically confirmed the above using a master equation solver [@qutip] within the relevant symmetry sector [@Sandvik] and the results are shown in figure \[fig:ms\_rates\]. Even for $\gamma$ much higher than $\gamma_{PT} \approx 10^{-5}$, the exponentially damped oscillations are preserved. The only effect of the bulk dephasing is an increase of the decay rate, proportional to $\gamma$, as predicted.
Discussion
==========
We have shown that the effect of weak bulk dephasing on the staggered magnetization of the XXZ spin chain consists of a single exponential damping factor $e^{-\gamma \delta}$. This stems from the combination of various symmetries, acting on the different levels of a hierarchy of Hilbert spaces. On the level of quantum states, the conserved magnetization generates a strong symmetry, allowing a restriction to the zero-magnetization sector. On the level of operators, there are two weak symmetries in the form of reflection $\hat{R}$ and spin inversion $\hat{F}$, which divide the Liouvillian spectrum into four blocks $\hat{\mathcal{U}}_{p,q}$ with $p,q \in \{\pm 1\}$. And on the superoperator level, the $\mathbb{PT}$ symmetry of the Liouvillian forces its spectrum into a unique shape. The result is a spectral separation of the symmetry sectors, where all modes contributing to the staggered magnetization $M_s$ will decay at the same rate.
It is now interesting to define a general recipe, that can be applied to look for similar behavior in other systems. The required ingredients are a non-degenerate, $\mathbb{PT}$-symmetric Liouvillian and an observable of interest that is anti-symmetric under an additional weak parity symmetry. Since such anti-symmetries are built into the algebra of fermionic and spin systems, we suspect the phenomenon to be quite prevalent in such many-body models. Unfortunately it may be more difficult to find those properties in the simple bosonic systems that serve as popular models in quantum optics. Whether a $\mathbb{PT}$-symmetric Liouvillian is even possible in a purely bosonic system is an interesting open question.
In Ref. \[\], a boundary driven XXZ chain is given as an example of $\mathbb{PT}$ symmetry. There, one relevant observable is the spin current $J = i \sum_{i=1}^{N-1}\left(S_i^+ S_{i+1}^- - S_i^- S_{i+1}^+\right)$ which has vanishing diagonal elements in the energy eigenbasis, just like the staggered magnetization in our example above. The reason for this is that $J$ also is odd under the parity symmetries $R$ and $F$. However, this is not enough to ensure that the spin current relaxes with a uniform rate, except in the limit of $\gamma \rightarrow 0$. As we have seen, the decay modes on the real axis do have non-zero off-diagonal elements. Unlike the staggered magnetization under bulk dephasing, $J$ is not protected from these modes by a weak symmetry. That is because the driving of the spin chain is no longer symmetric under the superoperators $\hat{R}$ and $\hat{F}$ individually, but only under their product [@Buca2012]: $[\hat{\mathcal{L}}, \hat{R}\hat{F}] = 0$. The spin current is found in the even sector of this weak symmetry, and so are the decay modes on the real axis. It can easily be checked numerically that the contribution of these modes to the expectation value is small but non-zero. Particularly at long times, they may have a noticeable effect due to the slower decay rates. Another observable that *is* confined to the odd symmetry sector is the total magnetization, which is not conserved by the boundary driving.
Also worth noting is that the addition of long-range interactions does not break any of the symmetries described for the XXZ spin chain. Going beyond nearest-neighbor coupling will affect $\gamma_{PT}$, but the structure of the symmetry sectors and the Liouvillian spectrum will remain the same. This is experimentally relevant in the context of trapped ions, which allow quantum simulation of spin chains with highly tunable long-range interactions [@Neyenhuis2017; @Gras2014; @Bermudez2017]. For such systems, bulk dephasing corresponds to local magnetic fluctuations within the trap, although there are also methods to control the dissipation [@Mueller2011]. It is our hope that the phenomenon of symmetry-protected coherent relaxation may be detectable in these kind of experiments.
Acknowledgements
================
This work is part of the Delta-ITP consortium, a program of the Netherlands Organization for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW). The authors want to thank Enej Ilievski, Jean-Sébastien Caux and Dirk Schuricht for fruitful discussions and suggestions.
Perturbation theory of [$\la M_s \ra$]{}
========================================
In this extra material, we will derive corrections to the staggered magnetization in the XXZ Heisenberg chain, resulting from weak bulk dephasing. We will draw heavily on the symmetry arguments from sections \[section:xxz\] and \[section:neel\]. As a starting point, consider Eq. (\[eq:obs\_spectral\]) in the $\gamma = 0$ case. Assuming non-degenerate energy eigenstate $H|\mu\ra = \epsilon_{\mu} |\mu \ra$, we find that $u_m = v_m = |\mu\ra \la \nu|$ with $\lambda_m = i(\epsilon_\mu - \epsilon_\nu)$. Since $M_s$ is confined to the double-odd symmetry sector $\hat{\mathcal{U}}_{-,-}$, only modes with $\mu \neq \nu$ need to be considered.
Now we can turn on the dissipation and apply perturbation theory to these off-diagonal modes. Writing the perturbation as $$\begin{aligned}
\hat{\mathcal{D}} \rho = \hat{\mathcal{D}}^\+ \rho = -\delta \rho + \sum_i S^z_i \rho S^z_i,\end{aligned}$$ we find $$\begin{aligned}
\lambda_m &= \lambda_m^{(0)} + \gamma \operatorname{tr}(u_m^\+ \hat{\mathcal{D}} u_m) + \mathcal{O}(\gamma^2) \nonumber\\[5pt]
&= i(\epsilon_\mu - \epsilon_\nu) - \gamma\delta + \gamma \sum_{i,j} \la \nu | S^z_i | \nu \ra \la \mu | S^z_j | \mu \ra + \mathcal{O}(\gamma^2) \nonumber\\
&= i(\epsilon_\mu - \epsilon_\nu) - \gamma\delta + \mathcal{O}(\gamma^2),\end{aligned}$$ where we have used that $\la \nu | S^z_i | \nu \ra = 0$. Similarly, the first-order correction to the decay modes becomes: $$\begin{aligned}
u_m \approx |\mu\ra \la \nu| - i \gamma \sum_{\substack{\mu', \nu' \\ \neq \mu, \nu}} \sum_{i,j} \frac{\la \mu'|S^z_i|\mu\ra \la \nu|S^z_j|\nu' \ra}{\epsilon_\mu - \epsilon_\nu - \epsilon_{\mu'} + \epsilon_{\nu'}} \, |\mu'\ra \la \nu'|. \nonumber\end{aligned}$$ Note that the XXZ Hamiltonian is real and symmetric (most easily seen in the Pauli-representation of the local spin basis), which means that the matrix elements of $S^z_i$ are also real: $\la \mu | S^z_i | \nu \ra = \la \nu | S^z_i | \mu \ra$. Therefore, the first-order correction is purely imaginary.
The operators $M_s$ and $\rho_0 = |\text{N\'eel}\ra \la \text{N\'eel}|$ likewise have only real matrix elements. Plugging the results above into Eq. (\[eq:obs\_spectral\]), we find: $$\begin{aligned}
&\la M_s(t) \ra = e^{-\gamma \delta t} \sum_{\mu, \nu \neq \mu} e^{it(\epsilon_\mu - \epsilon_\nu) + \mathcal{O}(\gamma^2)} \\
\times &\Big(\la\mu|\rho_0|\nu\ra + i\gamma \sum_{\substack{\mu', \nu' \\ \neq \mu, \nu}} \sum_{i,j} \frac{\la \mu'|S^z_i|\mu\ra \la \nu|S^z_j|\nu' \ra}{\epsilon_\mu - \epsilon_\nu - \epsilon_{\mu'} + \epsilon_{\nu'}} \la \mu'|\rho_0|\nu'\ra \Big) \nonumber \\
\times &\Big(\la\mu|M_s|\nu\ra - i\gamma \sum_{\substack{\mu', \nu' \\ \neq \mu, \nu}} \sum_{i,j} \frac{\la \mu'|S^z_i|\mu\ra \la \nu|S^z_j|\nu' \ra}{\epsilon_\mu - \epsilon_\nu - \epsilon_{\mu'} + \epsilon_{\nu'}} \la \mu'|M_s|\nu'\ra \Big) \nonumber\end{aligned}$$ The cross terms, representing the corrections of order $\mathcal{O}(\gamma)$, are purely imaginary and cancel out when completing the sum over $\mu$ and $\nu$. As a result, the leading order correction due to the shifting decay modes is proportional to $\gamma^2$: $$\begin{aligned}
\la M_s(t) \ra = e^{-\gamma \delta t} \left(\la M_s(t) \ra_0 + \mathcal{O}(\gamma^2) \right),\end{aligned}$$ where $\la M_s(t) \ra_0$ is the time-evolution for the closed system, as described in [@Barmettler].
[^1]: This is not generally true for Liouvillians. However, a non-diagonalizable superoperator can be expressed in a comparable form using a Jordan decomposition. Most of the following qualitative statements will still hold true in this situation, although one can get power-law contributions to the expansion of $\rho(t)$. See e.g. [@Prosen2010; @Pletyukhov2010]
[^2]: These should *not* be thought of in the sense of a block-diagonal matrix. For example, one can block-diagonalize the Liouvillian superoperator $\hat{\mathcal{L}}$, in which case *all* blocks $\hat{\mathcal{U}}_{i,j}$ will be on the diagonal. Instead, these ‘diagonal’ blocks relate to the diagonal matrix elements of operators.
[^3]: Unless the unitary and dissipative parts of the Liouvillian commute with one another. In that case, diagonal and off-diagonal modes will remain separated. This would make the dynamics largely trivial, though.
[^4]: As we are concerned with pure dephasing, the dissipator is diagonal in the local spin basis. This makes it straightforward to find the largest eigenvalue.
|
---
abstract: 'Because of unboundedness of the general relativity action, Euclidean version of the path integral in general relativity requires definition. Area tensor Regge calculus is considered in the representation with independent area tensor and finite rotation matrices. Being integrated over rotation matrices the path integral measure in area tensor Regge calculus is rewritten by moving integration contours to complex plain so that it looks as that one with effective action in the exponential with positive real part. We speculate that positivity of the measure can be expected in the most part of range of variation of area tensors.'
author:
- |
V.M. Khatsymovsky\
[*Budker Institute of Nuclear Physics*]{}\
[ *Novosibirsk, 630090, Russia*]{}\
[*E-mail address: [email protected]*]{}
title: On positivity of quantum measure and of effective action in area tensor Regge calculus
---
PACS numbers: 04.60.-m Quantum gravity
The formal nonrenormalisability of quantum version of general relativity (GR) may cause us to try to find alternatives to the continuum description of underlying spacetime structure. An example of such the alternative description may be given by Regge calculus (RC) suggested in 1961 [@Regge]. It is the exact GR developed in the piecewise flat spacetime which is a particular case of general Riemannian spacetime [@Fried]. In turn, the general Riemannian spacetime can be considered as limiting case of the piecewise flat spacetime [@Fein]. Any piecewise flat spacetime is simlicial one: it can be represented as collection of a (countable) number of the flat 4-dimensional [*simplices*]{}(tetrahedrons), and its geometry is completely specified by the countable number of the freely chosen lengths of all edges (or 1-simplices). Thus, RC implies a [*discrete*]{} description [*alternative*]{} to the usual continuum one. For a review of RC and alternative discrete gravity approaches see, e. g., [@RegWil].
The discrete nature of the simplicial description presents a difficulty in the (canonical) quantization of such the theory due to the absence of a regular continuous coordinate playing the role of time. Therefore one cannot immediately develop Hamiltonian formalism and canonical (Dirac) quantization. To do this we need to return to the partially continuum description, namely, with respect to only one direction shrinking sizes of all the simplices along this direction to those infinitely close to zero. The linklengths and other geometrical quantities become functions of the continuous coordinate taken along this direction. We can call this coordinate time $t$ and develop quantization procedure with respect to this time. The result of this procedure can be formulated as some path integral measure. It is quite natural to consider this measure as a (appropriately defined) limiting continuous time form of a measure on the set of the original completely discrete simplicial spacetimes. This last completely discrete measure is just the object of interest to be found. The requirement for this measure to have the known limiting continuous time form can be considered as a starting postulate in our construction. The issuing principles are of course not unique, and another approaches to defining quantum measure in RC based on another physical principles do exist [@HamWil1; @HamWil2].
The above condition for the completely discrete measure to possess required continuous time limit does not defines it uniquely as long as only one fixed direction which defines $t$ is considered. However, different coordinate directions should be equivalent and we have a right to require for the measure to result in the canonical quantization measure in the continuous time limit [*whatever*]{} coordinate direction is chosen to define a time. These requirements are on the contrary a priori too stringent, and it is important that on some configuration superspace (extended in comparison with superspace of the genuine simplicial geometries) such the measure turns out to exist.
Briefly speaking, we should, first, find continuous time limit for Regge action, recast it in the canonical Hamiltonian form and write out the Hamiltonian path integral, the measure in the latter being called for a moment the continuous time measure; second, we should check for existence and (if exists) find the measure obeying the property to tend in the continuous time limit (with concept “to tend” being properly defined) to the found continuous time measure irrespectively of the choice of the time coordinate direction. When passing to the continuous time RC we are faced with the difficulty that the description of the infinitely flattened in some direction simplex purely in terms of the lengths is singular.
The way to avoid singularities in the continuous time limit is to extend the set of variables via adding the new ones having the sense of angles and considered as independent variables. Such the variables are the finite rotation matrices which are the discrete analogs of the connections in the continuum GR. The situation considered is analogous to that one occurred when recasting the Einstein action in the Hilbert-Palatini form, $$\label{S-HilPal} {1\over 2}\int{R\sqrt{g}{\rm d}^4x} \Leftarrow {1\over
8}\int{\epsilon_{abcd}\epsilon^{\lambda\mu\nu\rho}e^a_{\lambda}e^b_{\mu}
[\partial_{\nu}+\omega_{\nu},\partial_{\rho}+\omega_{\rho}]^{cd}{\rm d}^4x},$$
where the tetrad $e^a_{\lambda}$ and connection $\omega^{ab}_{\lambda}$ = $-\omega^{ba}_{\lambda}$ are independent variables, the RHS being reduced to LHS in terms of $g_{\lambda\mu}$ = $e^a_{\lambda}e_{a\mu}$ if we substitute for $\omega^{ab}_{\lambda}$ solution of the equations of motion for these variables in terms of $e^a_{\lambda}$. The Latin indices $a$, $b$, $c$, ... are the vector ones with respect to the local Euclidean frames which are introduced at each point $x$.
Now in RC the Einstein action in the LHS of (\[S-HilPal\]) becomes the Regge action, $$\label{S-Regge} \sum_{\sigma^2}{\alpha_{\sigma^2}|\sigma^2|},$$
where $|\sigma^2|$ is the area of a triangle (the 2-simplex) $\sigma^2$, $\alpha_{\sigma^2}$ is the angle defect on this triangle, and summation run over all the 2-simplices $\sigma^2$. The discrete analogs of the tetrad and connection, edge vectors and finite rotation matrices, were first considered in [@Fro]. The local Euclidean frames live in the 4-simplices now, and the analogs of the connection are defined on the 3-simplices $\sigma^3$ and are the matrices $\Omega_{\sigma^3}$ connecting the frames of the pairs of the 4-simplices $\sigma^4$ sharing the 3-faces $\sigma^3$. These matrices are the finite SO(4) rotations in the Euclidean case (or SO(3,1) rotations in the Lorentzian case) in contrast with the continuum connections $\omega^{ab}_{\lambda}$ which are the elements of the Lee algebra so(4)(so(3,1)) of this group. This definition includes pointing out the direction in which the connection $\Omega_{\sigma^3}$ acts (and, correspondingly, the opposite direction, in which the $\Omega^{-1}_{\sigma^3}$ = $\bar{\Omega}_{\sigma^3}$ acts), that is, the connections $\Omega$ are defined on the [*oriented*]{} 3-simplices $\sigma^3$. Instead of RHS of (\[S-HilPal\]) we use exact representation which we suggest in our work [@Kha1], $$\label{S-RegCon}S(v,\Omega) = \sum_{\sigma^2}{|v_{\sigma^2}|\arcsin{v_{\sigma^2}\circ
R_{\sigma^2}(\Omega)\over |v_{\sigma^2}|}}$$
where we have defined $A\circ B$ = ${1\over 2}A^{ab}B_{ab}$, $|A|$ = $(A\circ A)^{1/2}$ for the two tensors $A$, $B$; $v_{\sigma^2}$ is the dual bivector of the triangle $\sigma^2$ in terms of the vectors of its edges $l^a_1$, $l^a_2$, $$\label{v=ll} v_{\sigma^2ab} = {1\over 2}\epsilon_{abcd}l^c_1l^d_2$$
(in some 4-simplex frame containing $\sigma^2$). The curvature matrix $R_{\sigma^2}$ on the 2-simplex $\sigma^2$ is the path ordered product of the connections $\Omega^{\pm 1}_{\sigma^3}$ on the 3-simplices $\sigma^3$ sharing $\sigma^2$ along the contour enclosing $\sigma^2$ once and contained in the 4-simplices sharing $\sigma^2$, $$\label{R-Omega} R_{\sigma^2} = \prod_{\sigma^3\supset\sigma^2}{\Omega^{\pm
1}_{\sigma^3}}.$$
As we can show, when substituting as $\Omega_{\sigma^3}$ the genuine rotations connecting the neighbouring local frames as functions of the genuine Regge lengths into the equations of motion for $\Omega_{\sigma^3}$ with the action (\[S-RegCon\]) we get exactly the closure condition for the surface of the 3-simplex $\sigma^3$ (vanishing the sum of the bivectors of its 2-faces) written in the frame of one of the 4-simplices containing $\sigma^3$, that is, the identity. This means that (\[S-RegCon\]) is the exact representation for (\[S-Regge\]). At the same time, general solution to the equations of motion is wider than that leading to $R_{\sigma^2}(\Omega)$ rotating around $\sigma^2$ by the defect angle $\alpha_{\sigma^2}$.
We can pass to the continuous time limit in (\[S-RegCon\]) in a nonsingular manner and recast it to the canonical (Hamiltonian) form [@Kha2]. This allows us to write out Hamiltonian path integral. The above problem of finding the measure which results in the Hamiltonian path integral measure in the continuous time limit whatever coordinate is chosen as time has solution in 3 dimensions [@Kha3]. A specific feature of the 3D case important for that is commutativity of the dynamical constraints leading to a simple form of the functional integral. The 3D action looks like (\[S-RegCon\]) with area tensors $v_{\sigma^2}$ substituted by the egde vectors ${\mbox{\boldmath$l$}}_{\sigma^1}$ independent of each other. In 4 dimensions, the variables $v_{\sigma^2}$ are not independent but obey a set of (bilinear) [*intersection relations*]{}. For example, tensors of the two triangles $\sigma^2_1$, $\sigma^2_2$ sharing an edge satisfy the relation $$\label{v*v}\epsilon_{abcd}v^{ab}_{\sigma^2_1}v^{cd}_{\sigma^2_2} = 0.$$
These purely geometrical relations can be called kinematical constraints. The idea is to construct quantum measure first for the system with formally independent area tensors. That is, originally we concentrate on quantization of the dynamics while kinematical relations of the type (\[v\*v\]) are taken into account at the second stage. Note that the RC with formally independent (scalar) areas have been considered in the literature [@RegWil; @BarRocWil].
The theory with formally independent area tensors can be called area tensor RC. Consider the Euclidean case. The Einstein action is not bounded from below, therefore the Euclidean path integral itself requires careful definition. Our result for the constructed in the above way completely discrete quantum measure [@Kha4] can be written as a result for vacuum expectations of the functions of the field variables $v$, $\Omega$. Upon passing to integration over imaginary areas with the help of the formal replacement of the tensors of a certain subset of areas $\pi$ over which integration in the path integral is to be performed, $$\pi \rightarrow -i\pi,$$ the result reads $$\begin{aligned}
\label{VEV2}<\Psi (\{\pi\},\{\Omega\})> & = & \int{\Psi (-i\{\pi\}, \{\Omega\})\exp{\left (-\!
\sum_{\stackrel{t-{\rm like}}{\sigma^2}}{\tau _{\sigma^2}\circ
R_{\sigma^2}(\Omega)}\right )}}\nonumber\\
& & \hspace{-20mm} \cdot \exp{\left (i
\!\sum_{\stackrel{\stackrel{\rm not}{t-{\rm like}}}{\sigma^2}} {\pi_{\sigma^2}\circ
R_{\sigma^2}(\Omega)}\right )}\prod_{\stackrel{\stackrel{\rm
not}{t-{\rm like}}}{\sigma^2}}{\rm d}^6
\pi_{\sigma^2}\prod_{\sigma^3}{{\cal D}\Omega_{\sigma^3}} \nonumber\\ & \equiv &
\int{\Psi (-i\{\pi\},\{\Omega\}){\rm d} \mu_{\rm area}(-i\{\pi\},\{\Omega\})},\end{aligned}$$
where ${\cal D}\Omega_{\sigma^3}$ is the Haar measure on the group SO(4) of connection matrices $\Omega_{\sigma^3}$. Appearance of some set ${\cal F}$ of triangles $\sigma^2$ integration over area tensors of which is omitted (denoted as “$t$-like” in (\[VEV2\]))is connected with that integration over [*all*]{} area tensors is generally infinite, in particular, when normalizing measure (finding $<1>$). Indeed, different $R_{\sigma^2}$ for $\sigma^2$ meeting at a given link $\sigma^1$ are connected by Bianchi identities [@Regge]. Therefore for the spacetime of Minkowsky signature (when exponent is oscillating over all the area tensors) the product of $\delta^6(R_{\sigma^2} - {\bar{R}}_{\sigma^2})$ for all these $\sigma^2$ which follow upon integration over area tensors for these $\sigma^2$ contains singularity of the type of $\delta$-function squared. To avoid this singularity we should confine ourselves by only integration over area tensors on those $\sigma^2$ on which $R_{\sigma^2}$ are independent, and complement ${\cal F}$ to this set of $\sigma^2$ are those $\sigma^2$ on which $R_{\sigma^2}$ are by means of the Bianchi identities functions of these independent $R_{\sigma^2}$. Let us adopt regular way of constructing 4D simplicial structure of the 3D simplicial geometries (leaves) of the same structure. Denote by $A$, $B$, $C$, ... vertices of the 4D simplicial complex while $n$-simplex $\sigma^n$ is denoted by the set of its $n + 1$ vertices in round brackets (unordered sequence), $(A_1A_2...)$. The $i$, $k$, $l$, ... are vertices of the current leaf, $i^+$, $k^+$, $l^+$, ... and $i^-$, $k^-$, $l^-$, ... are corresponding vertices of the nearest future and past in $t$ leaves. Or, dealing with Euclidean time, we shall speak of the “upper” and “lower” leaves, respectively. Each vertex is connected by links (edges) with its $\pm$-images. These links (of the type of $(ii^+)$, $(ii^-)$) will be called $t$-[*like*]{} ones (do not mix with the term “timelike” which is reserved for the local frame components). The [*leaf*]{} links $(ik)$ are completely contained in the 3D leaf. There may be [*diagonal*]{} links $(ik^+)$, $(ik^-)$ connecting a vertex with the $\pm$-images of its neighbors. We call arbitrary simplex $t$-[*like*]{} one if it has $t$-like edge, the [*leaf*]{} one if it is completely contained in the 3D leaf and [*diagonal*]{} one in other cases. It can be seen that the set of the $t$-like triangles is fit for the role of the above set ${\cal F}$. In the case of general 4D simplicial structure we can deduce that the set ${\cal F}$ of the triangles with the Bianchi-dependent curvatures pick out some one-dimensional field of links, and we can simply take it as definition of the coordinate $t$ direction so that ${\cal F}$ be just the set of the $t$-like triangles. Also existence of the set ${\cal F}$ naturally fits our initial requirement that limiting form of the full discrete measure when any one of the coordinates (not necessarily $t$!) is made continuous by flattening the 4-simplices in the corresponding direction should coincide with Hamiltonian path integral (with that coordinate playing the role of time). Namely, in the Hamiltonian formalism absence of integration over area tensors of triangles which pick out some coordinate $t$ ($t$-like ones) corresponds to some gauge fixing.
There is the invariant (Haar) measure ${\cal D}\Omega$ in (\[VEV2\]) which looks natural from symmetry considerations. From the formal point of view, in the Hamiltonian formalism (when one of the coordinates is made continuous) this arises when we write out standard Hamiltonian path integral for the Lagrangian with the kinetic term $\pi_{\sigma^2}\circ{\bar{\Omega}}_{\sigma^2}\dot{\Omega}_{\sigma^2}$ [@Kha3; @Kha4]. To this end, one might pass to the variables $\Omega_{\sigma^2}
\pi_{\sigma^2}\!$ = $\!P_{\sigma^2}$ and $\Omega_{\sigma^2}$ (in 3D case used in [@Wael; @Kha3]). The kinetic term $P\dot{\Omega}$ with arbitrary matrices $P$, $\Omega$ leads to the standard measure ${\rm d}^{16}P{\rm d}^{16}\Omega$, but there are also $\delta$-functions taking into account II class constraints to which $P$, $\Omega$ are subject, $\delta^{10}({\bar{\Omega}}\Omega - 1)\!$ $\!\delta^{10}({\bar{\Omega}}P +
\bar{P}\Omega)$. Integrating out these just gives ${\rm d}^6\pi{\cal D}^6\Omega$. Following our strategy of recovering full discrete measure from requirement that it reduces to the Hamiltonian path integral whatever coordinate is made continuous, the same Haar measure should be present also in the full discrete measure.
One else specific feature of the quantum measure is the absence of the inverse trigonometric function ’arcsin’ in the exponential, whereas the Regge action ($\ref{S-RegCon}$) contains such functions. This is connected with using the canonical quantization at the intermediate stage of derivation: in gravity this quantization is completely defined by the constraints, the latter being equivalent to those ones without $\arcsin$ (in some sense on-shell).
In what follows, it is convenient to split antisymmetric matrices ($\pi$ and generator of $R$) into self- and antiselfdual parts, then the measure (\[VEV2\]) splits into two factors, in the self- and antiselfdual sectors, $$\begin{aligned}
\pi_{ab} & \equiv & {1\over 2}{\,^+\!\pi}_k{\,^+\!\Sigma}^k_{ab} + {1\over 2}{\,^-\!\pi}_k{\,^-\!\Sigma}^k_{ab} \\{\,^{\pm}\!R}& = & \exp ({\,^{\pm}\!\mbox{\boldmath$\phi$}}{\,^{\pm}\!\mbox{\boldmath$\Sigma$}}) = \cos \!{\,^{\pm}\!\phi}+ {\,^{\pm}\!\mbox{\boldmath$\Sigma$}}{\,^{\pm}\!\mbox{\boldmath$n$}}\sin \!{\,^{\pm}\!\phi}\\ {{\rm d}}\mu_{\rm area} & = & {{\rm d}}{\,^+\!\mu}_{\rm area} {{\rm d}}{\,^-\!\mu}_{\rm area}. $$
Here ${\,^{\pm}\!\mbox{\boldmath$n$}}$ = ${\,^{\pm}\!\mbox{\boldmath$\phi$}}/ {\,^{\pm}\!\phi}$ is unit vector and the basis of self- and antiselfdual matrices $i{\,^{\pm}\!\Sigma}^k_{ab}$ obeys the Pauli matrix algebra.
Since as pointed out below the eqs. (\[S-RegCon\]) - (\[R-Omega\]) the classical equations of motion for $\Omega$ do not restrict the resulting $R_{\sigma^2}(\Omega )$ be exactly the rotation around $\sigma^2$ by the defect angle $\alpha_{\sigma^2}$, the sense of $\Omega$, $R(\Omega)$ as physical observables is restricted. Consider averaging functions of only area tensors $\pi_{\sigma^2}$. By the properties of invariant measure, integrations over $\prod{\cal D}\Omega_{\sigma^3}$ in (\[VEV2\]) reduce to integrations over $\prod{\cal D}R_{\sigma^2}$ with independent $R_{\sigma^2}$ (i. e. $\sigma^2$ are just not $t$-like) and some number of connections $\prod{\cal D}\Omega_{\sigma^3}$ which we can call gauge ones. The expectation value of any field monomial, $<\pi^{a_1b_1}_{\sigma^2_1}...\pi^{a_nb_n}_{\sigma^2_n}>$ reduces to the (derivatives of) $\delta$-functions $\delta (R^{a_ib_i}_{\sigma^2_i} -
R^{b_ia_i}_{\sigma^2_i} )$ which are then integrated out over ${\cal D}R_{\sigma^2_i}$ giving finite nonzero answer. This is consequence of i) the underlying Dirac-Hamilton principle of quantization (leading to ${{\rm d}}^6\pi_{\sigma^2}{\cal D}R_{\sigma^2}$ in the measure) and of ii) conception of independent area tensors (integrations over ${{\rm d}}^6\pi_{\sigma^2}$ are independent leading to $\delta$-functions). This holds in the Minkowsky spacetime as well (and in the first instance since oscillating exponent is present there from the very beginning). The Euclidean expectations values correspond to the Minkowskian ones in the spacelike region. The formal passing to the Euclidean version by simply writing $\exp (-\pi_{\sigma^2}\circ R_{\sigma^2})$ in the measure (not with additional substitution $\pi_{\sigma^2}\!$ $\rightarrow$ $\!-i\pi_{\sigma^2}$ in the integration variables as in (\[VEV2\])) might result, upon integrating over ${\cal D}R_{\sigma^2}$, in appearance of the terms with both factors, $\exp (+|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|)$ and $\exp (-|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|)$. This is consequence of that iii) $R_{\sigma^2}$ are [*finite*]{} SO(4) rotations, not elements of Lee group so(4) - therefore the stationary phase points in the integrals over ${\cal D}R_{\sigma^2_i}$ correspond just to ${\,^{\pm}\!\pi}_{\sigma^2}\circ{\,^{\pm}\!R}_{\sigma^2}\!$ = $\!+|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|$ and ${\,^{\pm}\!\pi}_{\sigma^2}\circ{\,^{\pm}\!R}_{\sigma^2}\!$ = $\!-|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|$. Due to the above mentioned finiteness of area monomial VEVs the growing exponents $\exp (+|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|)$ should be excluded. Thus, the measure upon integration over connections should exponentially decrease with areas as $\exp (-|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|)$. Once again, collect the reasons for that,\
i) Dirac-Hamiltonian canonical quantization;\
ii) conception of independent area tensors;\
iii) connection matrices being finite SO(4) rotations, not elements of the Lee group so(4).
The definition of the Euclidean version (\[VEV2\]) via $\pi_{\sigma^2}\!$ $\rightarrow$ $\!-i\pi_{\sigma^2}$, as well as of Minkowskian one, contains oscillating exponent. It is possible to reproduce the results of above considered calculation of area monomial VEVs through $\delta$-functions of (antisymmetric part of) the curvature by integrating monomials with monotonic exponent in terms of genuine $\pi_{\sigma^2}$ by moving integration contour over curvature to complex plane [@Kha5]. This contour should start at ${\,^{\pm}\!\pi}_{\sigma^2}\circ{\,^{\pm}\!R}_{\sigma^2}\!$ = $\!+|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|$, not at ${\,^{\pm}\!\pi}_{\sigma^2}\circ{\,^{\pm}\!R}_{\sigma^2}\!$ = $\!-|{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}|$. If ${\,^{\pm}\!R}$ appears in the exponential in the form ${\,^{\pm}\!\pi}\circ{\,^{\pm}\!R}$, then appropriate complex change of variable ${\,^{\pm}\!\mbox{\boldmath$\phi$}}$ parameterizing ${\,^{\pm}\!R}$ corresponds to $$\begin{aligned}
\label{phi-i-eta} {\,^{\pm}\!\phi}& = & {\pi \over 2} + i{\,^{\pm}\!\eta}, ~~~ -\infty < {\,^{\pm}\!\eta}< +\infty, \\\label{theta-i-zeta} {\,^{\pm}\!\theta}& = & i{\,^{\pm}\!\zeta}, ~~~~~~~~~~~~~ 0 \leq {\,^{\pm}\!\zeta}< +\infty\end{aligned}$$
where ${\,^{\pm}\!\theta}$ is the azimuthal angle of ${\,^{\pm}\!\mbox{\boldmath$\phi$}}$ w.r.t. ${\,^{\pm}\!\mbox{\boldmath$\pi$}}$, the polar angle ${\,^{\pm}\!\chi}$ remaining the same.
Now generalize (\[phi-i-eta\]), (\[theta-i-zeta\]) to the case when ${\,^{\pm}\!R}$ enters in the form ${\,^{\pm}\!m}\circ {\,^{\pm}\!R}$ where ${\,^{\pm}\!m}$ has not only antisymmetric, but also scalar part, $${\,^{\pm}\!m}= {1 \over 2}{\,^{\pm}\!\mbox{\boldmath$m$}}\cdot {\,^{\pm}\!\mbox{\boldmath$\Sigma$}}+ {1 \over 2}{\,^{\pm}\!m}_0 \cdot 1.$$
Of course, in this case ${\,^{\pm}\!m}$ can not be (anti)selfdual part of anything nor (anti)selfdual matrix itself. Here index $\pm$ means simply that it is sum of products of (anti)selfdual matrices. The latter arise when we express curvatures on $t$-like triangles in terms of independent ones with the help of Bianchi identities. These curvatures can depend on the given ${R_{\sigma^2}}^{\pm 1}$ linearly or not depend at all. Therefore ${\,^{\pm}\!m_{\sigma^2}}\circ {\,^{\pm}\!R_{\sigma^2}}$ is the general form of dependence on the given ${\,^{\pm}\!R_{\sigma^2}}$ in the exponential of (\[VEV2\]). General form of integral over given curvature matrix is $$\int \exp (-{\,^{\pm}\!\mbox{\boldmath$m$}}{\,^{\pm}\!\mbox{\boldmath$n$}}\sin \!{\,^{\pm}\!\phi}- {\,^{\pm}\!m}_0 \cos \!{\,^{\pm}\!\phi}){\sin^2 \!{\,^{\pm}\!\phi}\over {\,^{\pm}\!\phi}^2} {{\rm d}}^3 {\,^{\pm}\!\mbox{\boldmath$\phi$}},$$
where, remind, ${\,^{\pm}\!\mbox{\boldmath$n$}}$ = ${\,^{\pm}\!\mbox{\boldmath$\phi$}}/{\,^{\pm}\!\phi}$, and azimuthal angle of ${\,^{\pm}\!\mbox{\boldmath$\phi$}}$ w.r.t. ${\,^{\pm}\!\mbox{\boldmath$m$}}$ is ${\,^{\pm}\!\theta}$. Apply (\[phi-i-eta\]) and then ($\ref{theta-i-zeta}$) to the [*shifted*]{} ${\,^{\pm}\!\phi}$, $$\begin{aligned}
& & {\,^{\pm}\!\theta}= i{\,^{\pm}\!\zeta}, ~~~ {\,^{\pm}\!\phi}+ {\,^{\pm}\!\alpha}= {\pi \over 2} +
i{\,^{\pm}\!\eta}, \\ & & \cos \!{\,^{\pm}\!\alpha}= {\sqrt{{\,^{\pm}\!\mbox{\boldmath$m$}}^2} \cosh
\!{\,^{\pm}\!\zeta}\over \sqrt{{\,^{\pm}\!\mbox{\boldmath$m$}}^2\cosh^2 \!{\,^{\pm}\!\zeta}+ {\,^{\pm}\!m}^2_0}},
~~~ \sin \!{\,^{\pm}\!\alpha}= {{\,^{\pm}\!m}_0 \over \sqrt{{\,^{\pm}\!\mbox{\boldmath$m$}}^2\cosh^2 \!{\,^{\pm}\!\zeta}+ {\,^{\pm}\!m}^2_0}}. $$
The general case of complex ${\,^{\pm}\!\mbox{\boldmath$m$}}$, ${\,^{\pm}\!m}_0$ is implied. Important is that ${\,^{\pm}\!\Sigma}_k$ are real-valued so that orthogonal conjugation operation is commuting with analytic continuation. The branch of the function $\sqrt{z}$ is chosen in the complex plane of $z$ with cut along negative real half-axis such that $\sqrt{1}$ = 1. (In particular, this means that $\Re \sqrt{z}$ $\!\geq\!$ 0.) The integral over ${\,^{\pm}\!\eta}$, ${\,^{\pm}\!\zeta}$ transforms to give $$\label{K1m} \int \exp (-{\,^{\pm}\!m}\circ {\,^{\pm}\!R}) {\sin^2 \!{\,^{\pm}\!\phi}\over {\,^{\pm}\!\phi}^2} {{\rm d}}^3
{\,^{\pm}\!\mbox{\boldmath$\phi$}}= {4\pi \over \sqrt{{\rm tr}{\,^{\pm}\!\bar{m}}{\,^{\pm}\!m}}}K_1\left
(\sqrt{{\rm tr}{\,^{\pm}\!\bar{m}}{\,^{\pm}\!m}}\right ).$$
The $K_1$ is the modified Bessel function.
The idea is to try to find some set of the 2-simplices ${\cal M}$ so that exponential in (\[VEV2\]) be representable in the form $$\label{-mR-piR} - \sum_{\sigma^2 \in {\cal M}} {m_{\sigma^2}}\circ {R_{\sigma^2}}- \sum_{\sigma^2 \not\in {\cal M}}
{\pi_{\sigma^2}}\circ {R_{\sigma^2}}$$
where ${m_{\sigma^2}}$ = ${\pi_{\sigma^2}}$ + (linear in $\{ {\tau_{\sigma^2}}\}$ terms). The notation $\{ \dots \}$ means “the set of …”. The set $\{ {m_{\sigma^2}}\}$ depend on $\{ {\tau_{\sigma^2}}\}$ and on $\{ {R_{\sigma^2}}| \sigma^2 \not\in {\cal M} \}$, but not on $\{ {R_{\sigma^2}}| \sigma^2
\in {\cal M} \}$. Then integrations over $\{ {R_{\sigma^2}}| \sigma^2 \in {\cal M} \}$ can be explicitly performed according to eq. (\[K1m\]) giving $$\begin{aligned}
\label{dN1} {{\rm d}}{\,^{\pm}\!\mu}_{\rm area} & \equiv & {{\rm d}}{\,^{\pm}\!{\cal N}}\prod_{\stackrel{\stackrel{\rm not} {t-{\rm
like}}}{\sigma^2}}{\rm d}^3 {\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}, \nonumber\\ & & \hspace{-28mm} {{\rm d}}{\,^{\pm}\!{\cal N}}\Longrightarrow \left [ \prod_{\sigma^2 \in {\cal M}}{K_1\left (\sqrt{{\rm
tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}\right ) \over \sqrt{{\rm tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}} \right ] \exp
\left ( -\sum_{\stackrel{\stackrel{\rm not} {t-{\rm like}}}{\sigma^2 \not \in {\cal
M}}} |{\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}|\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}\right ) \nonumber
\\ & & \hspace{-10mm} \cdot \prod_{\stackrel{\stackrel{\rm not} {t-{\rm
like}}}{\sigma^2 \not \in {\cal M}}} \cosh^2 \!{\,^{\pm}\!\eta_{\sigma^2}}{{\rm d}}\cosh
\!{\,^{\pm}\!\eta_{\sigma^2}}{{\rm d}}\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}{{\rm d}}{\,^{\pm}\!\chi_{\sigma^2}}\end{aligned}$$
where $\{ {m_{\sigma^2}}| \sigma^2 \in {\cal M} \}$ depend on $\{ {\,^{\pm}\!\eta_{\sigma^2}},
{\,^{\pm}\!\zeta_{\sigma^2}}, {\,^{\pm}\!\chi_{\sigma^2}}| \sigma^2 \not\in {\cal M} \}$ through ${R_{\sigma^2}}$ parameterized by these, $$\begin{aligned}
{\,^{\pm}\!R_{\sigma^2}}& = & -i\sinh \!{\,^{\pm}\!\eta_{\sigma^2}}+ {\,^{\pm}\!\mbox{\boldmath$\Sigma$}}\cdot \!{\,^{\pm}\!\mbox{\boldmath$n$}_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}},
\nonumber\\ {\,^{\pm}\!\mbox{\boldmath$n$}_{\sigma^2}}& = & {{\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}\over | {\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}|} \cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}+
i(\sinh \!{\,^{\pm}\!\zeta_{\sigma^2}})({\,^{\pm}\!\mbox{\boldmath$e$}_{\!1\sigma^2}}\cos \!{\,^{\pm}\!\chi_{\sigma^2}}+ {\,^{\pm}\!\mbox{\boldmath$e$}_{2\sigma^2}}\sin \!{\,^{\pm}\!\chi_{\sigma^2}})\end{aligned}$$
where ${\,^{\pm}\!\mbox{\boldmath$e$}_{\!1\sigma^2}}$, ${\,^{\pm}\!\mbox{\boldmath$e$}_{2\sigma^2}}$ together with ${\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}/ | {\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}|$ form orthonormal triple.
Rewrite (\[dN1\]) as $$\begin{aligned}
\label{dN2} {{\rm d}}{\,^{\pm}\!{\cal N}}& \Longrightarrow & \exp \left ( -\sum_{\sigma^2 \in {\cal M}} \sqrt{{\rm
tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}\right. \nonumber\\ & &
\hspace{-28mm} \left. -\sum_{\stackrel{\stackrel{\rm not} {t-{\rm like}}}{\sigma^2
\not \in {\cal M}}} |{\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}|\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}\right )
\prod_{\stackrel{\stackrel{\rm not} {t-{\rm like}}}{\sigma^2}} \cosh^2 \!{\,^{\pm}\!\eta_{\sigma^2}}{{\rm d}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}{{\rm d}}\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}{{\rm d}}{\,^{\pm}\!\chi_{\sigma^2}}\end{aligned}$$
where abstract dummy variables $\{ {\,^{\pm}\!\eta_{\sigma^2}}, {\,^{\pm}\!\zeta_{\sigma^2}}, {\,^{\pm}\!\chi_{\sigma^2}}|
\sigma^2 \in {\cal M} \}$ and integrations over them are introduced to represent $K_1$ differently from what is given by equation (\[K1m\]) read from right to left. Remarkable is that it looks as path integral measure with [*positive*]{} (real part of) effective action whereas general relativity action remains unbounded from below upon formal Wick rotation. The price is that exponential in (\[dN2\]) has imaginary part, and positivity of the Euclidean measure (upon integrating out curvature matrices) does not follow automatically as in the case of the usual field theory with bounded action since explicitly real form of (\[dN2\]) reads $$\begin{aligned}
\label{dN2cos} {{\rm d}}{\,^{\pm}\!{\cal N}}& \Longrightarrow & \exp \left ( -\sum_{\sigma^2 \in {\cal M}} \Re
\sqrt{{\rm tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}\right.
\nonumber\\ & & \hspace{-28mm} \left. -\sum_{\stackrel{\stackrel{\rm not} {t-{\rm
like}}}{\sigma^2 \not \in {\cal M}}} |{\,^{\pm}\!\mbox{\boldmath$\pi$}_{\sigma^2}}|\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}\right ) \cos \left( \sum_{\sigma^2 \in {\cal M}} \Im \sqrt{{\rm
tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}\right) \nonumber\\ & &
\hspace{-28mm} \cdot \prod_{\stackrel{\stackrel{\rm not} {t-{\rm like}}}{\sigma^2}}
\cosh^2 \!{\,^{\pm}\!\eta_{\sigma^2}}{{\rm d}}\cosh \!{\,^{\pm}\!\eta_{\sigma^2}}{{\rm d}}\cosh \!{\,^{\pm}\!\zeta_{\sigma^2}}{{\rm d}}{\,^{\pm}\!\chi_{\sigma^2}}\end{aligned}$$
that is nonconstant in sign due to cosine. Below we speculate that positivity should be expected in the most part of (if not in the whole) range of variation of area tensors ${\pi_{\sigma^2}}$ if ${\tau_{\sigma^2}}$ are sufficiently small.
To construct the set ${\cal M}$, note that due to the Bianchi identities dependence on the matrix ${R_{\sigma^2}}$ on the given leaf/diagonal triangle $\sigma^2$ in the exponential of (\[VEV2\]) comes from all the triangles constituting together with this $\sigma^2$ a closed surface. This is surface of the $t$-like 3-prism, one base of which is just the given $\sigma^2$, the lateral surface consists of $t$-like triangles and goes to infinity. In practice, replace this infinity by some lowest (initial) leaf where another base $\sigma^2_0$ is located the tensor of which $\pi_{\sigma^2_0}$ is taken as boundary value. Consider a variety of such prisms with upper bases $\sigma^2$ placed in the uppest (final) leaf such that any link in this leaf belongs to one and only one of these bases. That is, lateral surfaces of different prisms do not have common triangles. Then the terms ${m_{\sigma^2}}\circ {R_{\sigma^2}}$ in (\[-mR-piR\]) represent contribution from these prisms, ${\cal M}$ being the set of their bases in the uppest leaf.
To really reduce the measure to such form, we should express the curvature matrices on the $t$-like triangles in terms of those on the leaf/diagonal ones. The curvature on a leaf/diagonal triangle $\sigma^2$ as product of $\Omega$s includes the two matrices $\Omega$ on the $t$-like tetrahedrons $\sigma^3$ adjacent to $\sigma^2$ from above and from below. Knowing curvatures on the set of leaf/diagonal triangles inside any t-like 3-prism allows to successively express matrix $\Omega$ on any $t$-like tetrahedron inside the prism in terms of matrix $\Omega$ on the uppest $t$-like tetrahedron in this prism taken as boundary value. Expressions for the considered curvatures look like (fig.\[3prism\])
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$$\begin{aligned}
& & \hspace{-18mm} \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots
\nonumber\\ R_{(ikl)} & = & \dots {\bar{\Omega}}_{(i^-ikl)} \dots \Omega_{(ik^+kl)} \dots
\nonumber\\ R_{(ik^+)} & = & \dots {\bar{\Omega}}_{(ik^+kl)} \dots \Omega_{(ik^+l^+l)} \dots \\
R_{(ik^+l^+)} & = & \dots {\bar{\Omega}}_{(ik^+l^+l)} \dots \Omega_{(i^+ik^+l^+)} \dots
\nonumber\\ & & \hspace{-18mm} \dots \dots \dots \dots \dots \dots \dots \dots \dots
\dots \dots \nonumber\end{aligned}$$
The dots in expressions for $R$ mean matrices $\Omega$ on the leaf/diagonal tetrahedrons which can be considered as gauge ones. We can step-by-step express $\Omega_{(i^-ikl)}$ $\rightarrow$ $\Omega_{(ik^+kl)}$ $\rightarrow$ $\Omega_{(ik^+l^+l)}$ $\rightarrow$ $\Omega_{(i^+ik^+l^+)}$ $\rightarrow$ …where the arrow means “in terms of”. Knowing $\Omega$s on $t$-like tetrahedrons we can find the curvatures on $t$-like triangles, the products of these $\Omega$s, $$R_{(i^+ikl)} = \Omega^{\epsilon_{(ikl_n)l_{n-1}}}_{(i^+ikl_n)} \dots
\Omega^{\epsilon_{(ikl_1)l_n}}_{(i^+ikl_1)}.$$
Here $\epsilon_{(ikl)m}$ = $\pm 1$ is some sign function. Thereby we find contribution of the $t$-like triangles in terms of independent curvature matrices (on the leaf/diagonal triangles).
In the continuum path integral formalism, one usually imposes boundary (initial/final) conditions to unambiguously define the measure. Consideration of the two previous paragraphs says that in our case fixing the initial leaf area tensors $\pi_{\sigma^2_0}$ and final connections on the $t$-like tetrahedrons is appropriate. Thereby, in particular, nontrivial integrations reduce to those over matching each other sets ${{\rm d}}^6 {\pi_{\sigma^2}}$ and ${\cal D}^6 {R_{\sigma^2}}$ on the leaf/diagonal $\sigma^2$.
Note an important particular case when integrations in (\[dN1\]) are made over the whole sets $\{ {\pi_{\sigma^2}}| \sigma^2 \not\in {\cal M} \}$ and $\{ {R_{\sigma^2}}| \sigma^2 {\rm ~
is ~ not ~} t{\rm -like} \}$. The resulting measure factorizes over the 3-prisms with upper bases constituting ${\cal M}$, $$\begin{aligned}
\label{factoriz} {{\rm d}}{\,^{\pm}\!\mu}_{\rm area} \Longrightarrow \prod_{\sigma^2 \in {\cal M}}{K_1\left (\sqrt{{\rm
tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}\right ) \over \sqrt{{\rm tr}{\,^{\pm}\!\bar{m}_{\sigma^2}}{\,^{\pm}\!m_{\sigma^2}}}} {{\rm d}}^3
{\,^{\pm}\!\mbox{\boldmath$\pi$}}_{\sigma^2}.\end{aligned}$$
Here ${m_{\sigma^2}}$ is taken at $\{ {R_{\sigma^2}}= 0 | {\rm ~ not ~} t{\rm -like ~ }
\sigma^2 \not\in {\cal M} \}$ and differs from ${\pi_{\sigma^2}}$ by a constant, ${m_{\sigma^2}}= {\pi_{\sigma^2}}- {\pi_{\sigma^2}}^{(0)}$. In turn, ${\pi_{\sigma^2}}^{(0)}$ differs from the initial area tensor $\pi_{\sigma^2_0}$ by the lateral 3-prism surface tensor: ${\pi_{\sigma^2}}^{(0)} -
\pi_{\sigma^2_0}$ is algebraic sum of tensors $\tau_{\sigma^2}$ for $\sigma^2$ constituting the lateral surface. The ${\pi_{\sigma^2}}^{(0)}$ has geometrical meaning of [*expected*]{} value of area tensor ${\pi_{\sigma^2}}$ when the surface of the 3-prism closes due to the (classical) equations of motion. The measure (\[factoriz\]) describes quantum fluctuation of ${\pi_{\sigma^2}}$ around ${\pi_{\sigma^2}}^{(0)}$. The (\[factoriz\]) is explicitly positive.
Thus, to represent exponential in (\[VEV2\]) in the form (\[-mR-piR\]) sufficient is to divide the whole set of links in the uppest 3D leaf into triples forming the triangles and take this set of triangles as ${\cal M}$ in (\[-mR-piR\]). It is clear that such set ${\cal M}$ does exist not for an arbitrary 3D leaf (at least the number of links should be multiple of 3). In fig.\[M-simplex\] probably the simplest periodic cell of simplicial lattice is shown where the set ${\cal M}$ (shaded triangles) is also periodic.
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(-5,0)(9,2)[100]{}[.]{}(-5,0)(4,7)[100]{}[.]{} (1200,0)[(2,1)[400]{}]{}(1200,500)[(2,1)[400]{}]{}(1700,0)[(2,1)[400]{}]{}(1700,500)[(2,1)[400]{}]{}(1200,0)[(1,0)[500]{}]{}(1200,500)[(1,0)[500]{}]{}(1600,200)[(1,0)[500]{}]{}(1600,700)[(1,0)[500]{}]{}(1200,0)[(0,1)[500]{}]{}(1700,0)[(0,1)[500]{}]{}(1600,200)[(0,1)[500]{}]{}(2100,200)[(0,1)[500]{}]{} (1195,500)(18,4)[50]{}[.]{}(1695,0)(8,14)[50]{}[.]{}(1695,0)(-7,14)[14]{}[.]{}(1595,200)(-16,12)[25]{}[.]{}(1195,500)(11.5,-11.5)[44]{}[.]{}(1595,200)(11.5,11.5)[44]{}[.]{} (1020,350)[+]{} (400,-150)[C0]{}(1600,-150)[C1]{}
Genuine simplicial decomposition possesses quite complex combinatorics, so let us demonstrate main features of the result of above calculation by using as example the cubic decomposition. The latter can be viewed as [*sub*]{}-minisuperspace of simplicial system if one starts from the simplest periodic simplicial complex with elementary 4-cubic cell divided by diagonals emitted from one of its vertices into 24 4-simplices [@RocWil]. Each 3-cube face built on three coordinate directions is divided into 6 tetrahedrons, and we simply put $\Omega$s on these tetrahedrons to be the same on the whole 3-cube. There are also the 3-cube faces built on two coordinate and one diagonal direction, and we put $\Omega$s on the tetrahedrons forming these faces to be 1. Each 2-face (square) is divided into two triangles, and the curvature matrices on these triangles resulting from our choice of connections turn out to be the same on this square and, besides, these differs from 1 only on the square built on two coordinate directions, not on diagonal(s).
Introduce some cubic notations and definitions. By $\lambda$ we denote link in the coordinate direction $\lambda$; $\lambda$, $\mu$, $\nu$, $\rho$, …= 1, 2, 3, 4. Let the coordinate 4 be $t$. By ${{\cal S}q}$ denote a square. In particular, ${{\cal S}q}$ = $|\lambda\mu|$ means the square built on the coordinate directions $\lambda$, $\mu$. The connection matrix $\Omega_{\lambda}$ is that one on the 3-cube built on the coordinates $\mu$, $\nu$, $\rho$ (and also denoted as $|\mu\nu\rho|$) complement to $\lambda$. The set ${\cal M}$ for cubic decomposition of 3D leaf has periodic cell consisting of 2 $\!\times\!$ 2 $\!\times\!$ 2 elementary cubes, see fig.\[cube\] corresponding to0.03mm
(1600,1200)
(0,800)[(0,-1)[800]{}]{}(0,800)[(3,1)[600]{}]{}(0,800)[(1,0)[800]{}]{}
(300,900)[(0,-1)[400]{}]{}(600,1000)[(0,-1)[400]{}]{}(0,400)[(3,1)[600]{}]{}(0,400)[(1,0)[800]{}]{}(300,500)[(1,0)[610]{}]{}(960,500)[(1,0)[140]{}]{}(400,400)[(3,1)[465]{}]{}(0,0)[(3,1)[300]{}]{}(0,0)[(1,0)[800]{}]{}(300,100)[(1,0)[800]{}]{}(400,0)[(3,1)[300]{}]{}(1000,1000)[(0,-1)[400]{}]{}(700,900)[(0,-1)[800]{}]{}(1100,500)[(0,-1)[400]{}]{}(400,400)[(0,-1)[400]{}]{}(700,900)[(3,1)[300]{}]{}(800,400)[(0,-1)[400]{}]{}
(300,500)[(0,-1)[400]{}]{}(300,900)[(1,0)[800]{}]{}(600,600)[(0,-1)[400]{}]{}(600,1000)[(1,0)[800]{}]{}(400,800)[(0,-1)[400]{}]{}(400,800)[(3,1)[300]{}]{}(800,800)[(0,-1)[400]{}]{}(800,800)[(3,1)[600]{}]{}(600,600)[(1,0)[800]{}]{}(800,400)[(3,1)[600]{}]{}(300,100)[(3,1)[300]{}]{}(600,200)[(1,0)[800]{}]{}(700,100)[(3,1)[300]{}]{}(800,0)[(3,1)[600]{}]{}(1400,1000)[(0,-1)[800]{}]{}(1000,600)[(0,-1)[400]{}]{}(1100,900)[(0,-1)[400]{}]{}
(900,500)(100,0)[3]{}
(-2,-1)
(1100,450)(0,-50)[4]{}[(-2,-1)[400]{}]{} (800,100)(100,0)[2]{}
(2,1)
(550,400)(100,0)[3]{}
(-2,-1)
(800,375)(0,-50)[4]{}[(-2,-1)[400]{}]{}
(450,0)(100,0)[3]{}
(2,1)
(50,400)(50,0)[7]{}[(3,1)[300]{}]{} (50,0)(50,0)[7]{}[(3,1)[300]{}]{}
(300,900)(0,-60)[5]{}
(1,-1)
(600,660)(0,60)[4]{}
(-1,1)
(700,900)(0,-60)[5]{}
(1,-1)
(1000,660)(0,60)[4]{}
(-1,1)
(855,525)[2]{}(905,570)(9,3)[10]{}[.]{}(920,475)[1]{}(640,660)[3]{}(615,515)[O]{}
$${\cal M} = \sum_{k_1, k_2, k_3} T^{2k_1}_1 T^{2k_2}_2 T^{2k_3}_3 \left (|23| + {\bar{T}}_1
|23| + {\bar{T}}_3 |31| + {\bar{T}}_{12} |12| + {\bar{T}}_{23} |31| + {\bar{T}}_{123} |12| \right )$$
in the uppest leaf. Here $k_1$, $k_2$, $k_3$ are integers, $T_{\lambda}$ is translation to the neighboring vertex in the direction $\lambda$. Expressions for the measure follow from those for the simplicial case (\[dN1\]) – (\[dN2\]) by replacing $\sigma^2$ $\rightarrow$ ${{\cal S}q}$. There are several choices of the 4-cube containing a given square in the frame of which tensor of this square is defined. If the two area tensors are defined in the same frame, the result of integrating the measure over connections will depend on the scalar constructed of these two tensors. Therefore it seems to be a good idea to define area tensors in possibly different frames, as in fig.\[frames\], case (b).0.04mm
(1800,1200)
(568,580)[$\odot$]{}(468,680)[$\odot$]{}(668,680)[$\odot$]{} (1168,580)[$\odot$]{} (1268,680)[$\odot$]{}(1068,680)[$\odot$]{} (825,50)[(-1,2)[310]{}]{}(975,50)[(1,2)[310]{}]{} (840,0)[(b)]{} (625,600)[(1,0)[550]{}]{}(600,575)[(0,-1)[550]{}]{}(1200,575)[(0,-1)[550]{}]{}(25,600)[(1,0)[550]{}]{}(1225,600)[(1,0)[550]{}]{}(600,635)[(0,1)[440]{}]{}(1200,635)[(0,1)[440]{}]{} (800,700)[(0,-1)[200]{}]{}(1000,500)[(0,1)[200]{}]{} (680,500)[$\Omega_1$]{}(1020,490)[${\bar{\Omega}}_1$]{}(885,520)[3]{} (860,1020)[(-1,-2)[140]{}]{}(940,1020)[(1,-2)[140]{}]{}(845,1050)[(a)]{} (300,800)[(4,-1)[365]{}]{}(300,800)[(2,-1)[165]{}]{} (1500,800)[(-4,-1)[365]{}]{}(1500,800)[(-2,-1)[165]{}]{} (500,950)[(1,0)[200]{}]{}(1100,950)[(1,0)[200]{}]{}(750,350)[(-1,0)[250]{}]{}(1300,350)[(-1,0)[250]{}]{}(300,500)[(0,1)[200]{}]{}(1500,700)[(0,-1)[200]{}]{} (560,805)[1]{}(1215,810)[$T_31$]{}(150,800)[$\tau_{|42|}$]{}(1515,800)[$T_3\tau_{|42|}$]{}(490,975)[${\bar{\Omega}}_3$]{}(1220,975)[$T_3{\bar{\Omega}}_3$]{}
(600,700)(0,50)[6]{}
(2,1)
(1200,950)[(-2,-1)[470]{}]{}(1200,900)[(-2,-1)[570]{}]{}(1200,850)[(-2,-1)[500]{}]{}(1200,800)[(-2,-1)[400]{}]{}(1200,700)[(-2,-1)[200]{}]{}(1200,650)[(-2,-1)[100]{}]{}(900,600)[(2,1)[170]{}]{} (380,250)[${\bar{T}}_1\Omega_3$]{}(1220,250)[${\bar{T}}_1T_3\Omega_3$]{} (80,490)[${\bar{T}}_3{\bar{\Omega}}_1$]{}(1520,500)[$T_3\Omega_1$]{} (500,500)[2]{}(1250,500)[$T_32$]{}
Of course, corresponding curvature matrices should be defined in the same frames. With this rule of definition the curvature matrices on the lateral squares of, say, the 3-prism with base $|23|$ take the form ${\bar{T}}^n_4R_{{{\cal S}q}}$, $n$ = 1, 2, 3, …, ${{\cal S}q}$ = $|42|$, $|43|$, $T_3|42|$, $T_2|43|$, $$\begin{aligned}
R_{|42|} & = & \left({\bar{T}}_3{\bar{\Omega}}_1\right)\left({\bar{T}}_1\Omega_3\right)\Omega_1{\bar{\Omega}}_3,
\nonumber\\ T_3R_{|42|} & = & \left( T_3{\bar{\Omega}}_3 \right) {\bar{\Omega}}_1 \left( T_3{\bar{T}}_1\Omega_3
\right) \left( T_3\Omega_1 \right), \nonumber\\ R_{|43|} & = & \Omega_2 {\bar{\Omega}}_1 \left(
{\bar{T}}_1 {\bar{\Omega}}_2 \right)\left( {\bar{T}}_2 \Omega_1 \right), \\ T_2R_{|43|} & = & \left( T_2 {\bar{\Omega}}_1
\right) \left( T_2{\bar{T}}_1{\bar{\Omega}}_2 \right) \Omega_1 \left( T_2\Omega_2 \right). \nonumber\end{aligned}$$
(By default, notations $\Omega_{\lambda}$, $R_{|\lambda\mu|}$ are referred to the uppest leaf.) Denote $\Omega_4$ $\equiv$ $U$. The matrices ${\bar{T}}^n_4\Omega_{\alpha}$, $n$ = 1, 2, 3, …, $\alpha$, $\beta$, $\gamma$, …= 1, 2, 3 can be found in terms of $\Omega_{\alpha}$, ${\bar{T}}^k_4R_{|\alpha\beta|}$, ${\bar{T}}^k_4U$, $k$ = 0, 1, …, $n-1$ from $$\begin{aligned}
R_{|23|} & = & {\bar{U}}\left( {\bar{T}}_4 {\bar{\Omega}}_1 \right) \left( {\bar{T}}_1 U \right) \Omega_1,
\nonumber\\ & & \hspace{-15mm} \dots {\rm cycle ~ perm ~ 1, 2, 3} \dots .\end{aligned}$$
Thereby we find contribution of the $t$-like squares in terms of independent curvature matrices (on the leaf squares) and eventually matrices $m_{{{\cal S}q}}$.
For $m_{|23|}$ the result reads $$\begin{aligned}
\label{m23} & & \hspace{-20mm} m_{|23|} = \pi_{|23|} + \sum^N_{n=1} \left( \prod^{k=1}_{n-1}
{\bar{T}}^k_4 {\bar{R}}_{|23|} \right) \left\{ \left( \prod^{k=0}_{n-1} {\bar{T}}^k_4 T_3 {\bar{R}}_{|12|} \right)
\left( {\bar{T}}^n_4 T_3 \tau_{|42|} \right) \left( \prod^{n-1}_{k=0} {\bar{T}}^k_4 T_3 R_{|23|}
\right) \right. \nonumber \\ & & \hspace{-20mm} \cdot \left( \prod^{n-1}_{k=0} {\bar{T}}^k_4
T_3 {\bar{T}}_1 R_{|12|} \right) - \left( \prod^{n-1}_{k=0} {\bar{T}}^k_4 R_{|12|} \right) \left(
{\bar{T}}^n_4 \tau_{|42|} \right) \left( \prod^{n-1}_{k=0} {\bar{T}}^k_4 {\bar{T}}_3 R_{|23|} \right)
\left( \prod^{k=0}_{n-1} {\bar{T}}^k_4 {\bar{T}}_1 {\bar{R}}_{|12|} \right) \nonumber \\ & & + \left(
\prod^{n-1}_{k=0} {\bar{T}}^k_4 R_{|31|} \right) \left( {\bar{T}}^n_4 \tau_{|43|} \right) \left(
\prod^{n-1}_{k=0} {\bar{T}}^k_4 {\bar{T}}_2 R_{|23|} \right) \left( \prod^{k=0}_{n-1} {\bar{T}}^k_4 {\bar{T}}_1
{\bar{R}}_{|31|} \right) \\ & & \left. - \left( \prod^{k=0}_{n-1} {\bar{T}}^k_4 T_2 {\bar{R}}_{|31|}
\right) \left( {\bar{T}}^n_4 T_2 \tau_{|43|} \right) \left( \prod^{n-1}_{k=0} {\bar{T}}^k_4 T_2
R_{|23|} \right) \left( \prod^{n-1}_{k=0} {\bar{T}}^k_4 T_2 {\bar{T}}_1 R_{|31|} \right) \right\}
\nonumber\end{aligned}$$
where $N$ + 1 is the number of leaves, and the products of matrices are ordered according to the rule $$\prod^n_{k=0} A_k = A_n A_{n-1} \dots A_0.$$
For other $m_{{{\cal S}q}}$ we cyclically permute 1, 2, 3 and translate in the directions 1, 2, 3. For simplicity, here we have put equal to 1 the boundary values $\Omega_{\alpha}$ on the uppest leaf and to zero boundary values $\pi_{{{\cal S}q}_0}$ on the lowest leaf. Besides that, gauge matrices ${\bar{T}}^n_4U$, $n$ = 0, 1, 2, …are set to be 1. This corresponds to extending the local frame from any 4-cube to the whole t-like 4-prism containing this 4-cube.
At the point $\{ {\,^{\pm}\!\zeta}_{{{\cal S}q}} = 0, {\,^{\pm}\!\eta}_{{{\cal S}q}} = 0 | {{\cal S}q}\not\in {\cal M} \}$ where the factor in the measure corresponding to contribution from the squares ${{\cal S}q}$ $\not\in$ ${\cal M}$ reaches its maximum, and for uniform orthogonal lattice take ${\,^{\pm}\!\pi}_{|23|}$ = $A {\,^{\pm}\!\Sigma}_1/4$, ${\,^{\pm}\!\tau}_{|41|}$ = $\pm \varepsilon {\,^{\pm}\!\Sigma}_1/4$, …cycle permutations of 1, 2, 3 …, then ${\,^{\pm}\!R}_{|23|}$ = ${\,^{\pm}\!\Sigma}_1$, …. In (\[m23\]) we find sum of sign-altered terms so that $m_{|23|} = \pi_{|23|} +
O(\varepsilon)$. For estimate, let $A$, $\varepsilon$ be typical areas of the leaf and $t$-like squares, respectively. In the explicitly real expression for the measure (\[dN2cos\]) to be integrated, the cosine may become negative if for some variable $\zeta$ or $\eta$ we have $\sinh \zeta
= O(A/\varepsilon)$ or $\sinh \eta = O(A/\varepsilon)$. However, contribution from negative half-wave of cosine to the entire integral over $\zeta, \eta$-variables is dumped by the factor $\exp (-O(A^2/\varepsilon))$ in this case. Therefore at $A \geq
A_0 = O(\sqrt{\varepsilon})$ contribution of the negative half-waves of cosine is dominated by positive ones, and resulting ${\,^{\pm}\!{\cal N}}$ is positive.
This is quite rough, sufficient estimate. In reality, the region of positivity of ${\,^{\pm}\!{\cal N}}$ well may be larger then this or even coincide with the whole range of varying the area tensors. The one-dimensional example is inequality $\int^{\infty}_0
f(x) \cos x {{\rm d}}x > 0$ which can be easily proved to hold for [*any*]{} concave function ($f''(x)\!$ $\!>\!$ 0; in particular, for $f(x)\!$ = $\!\exp (-kx)$ at [*any*]{} $k > 0$). And even the inequality ${\,^{\pm}\!{\cal N}}\!$ $\!>\!$ 0 is, generally speaking, redundant for positivity means only ${\,^{+}\!{\cal N}}{\,^{-}\!{\cal N}}$ $>$ 0. Especially this circumstance is expected to promote the measure be positive when ${\,^{+}\!{\cal N}}$ and ${\,^{-}\!{\cal N}}$ are dependent. This takes place on the physical hypersurface singled out by the relations on area tensors of the type (\[v\*v\]) which connect ${\,^{+}\!v_{\sigma^2}}$ and ${\,^{-}\!v_{\sigma^2}}$.
Thus, completely discrete version of path integral in simplicial gravity can be naturally formulated with some boundary (initial/final) conditions. Representation of simplicial general relativity action in terms of area tensors and finite rotation matrices (connection and curvature) is used. Discrete connection and curvature on classical solutions of the equations of motion are not, strictly speaking, genuine connection and curvature, but more general quantities and, therefore, these do not appear as arguments of a function to be averaged, but are the integration (dummy) variables. Despite of unboundedness of general relativity action, path integral can be written in the form resembling that with positive (real part of) effective action by moving integration contours over curvature to complex plane. This effective action is not purely real, but arguments are given that the resulting path integral measure is expected to be positively defined upon integrating over connection matrices. Up to some integrable factor, this measure is dominated by the product of exponentially (in area) falling off factors on separate areas.
It is interesting that our arguments use simplicial structure although built in a simple regular way of similar 3-dimensional leaves, but with rather complex structure of these leaves themselves; simplest leaf will not do. The work to extend the results to arbitrary structure is in order.
The present work was supported in part by the Russian Foundation for Basic Research through Grant No. 05-02-16627-a.
[99]{} T. Regge, General relativity theory without coordinates. - Nuovo Cimento [**19**]{}, 568 (1961). R. Friedberg, T. D. Lee, Derivation of Regge’s action from Einstein’s theory of general relativity. - Nucl. Phys. B [**242**]{}, 145 (1984). G. Feinberg, R. Friedberg, T. D. Lee, M. C. Ren, Lattice gravity near the continuum limit. - Nucl. Phys. B [**245**]{}, 343 (1984). T. Regge and R.M. Williams, Discrete structures in gravity. - Journ. Math. Phys., [**41**]{}, 3964 (2000), gr-qc/0012035. H. Hamber and R.M. Williams, Newtonian Potential in Quantum Regge Gravity. - Nucl.Phys. B, [**435**]{}, 361 (1995), hep-th/9406163. H. Hamber and R.M. Williams, On the Measure in Simplicial Gravity. - Phys. Rev. D, [**59**]{}, 064014 (1999), hep-th/9708019. J. Fröhlich, Regge Calculus and Discretized Gravitational Functional Integrals, I. H. E. S. preprint 1981 (unpublished); Non-Perturbative Quantum Field Theory: Mathematical Aspects and Applications, Selected Papers (World Scientific, Singapore, 1992), pp. 523-545. V. M. Khatsymovsky, Tetrad and self-dual formulations of Regge calculus. - Class. Quantum Grav. [**6**]{}, L249 (1989). V. M. Khatsymovsky, Regge calculus in the canonical form. - Gen. Rel. Grav. [**27**]{}, 583 (1995), gr-qc/9310004. V. M. Khatsymovsky, A version of quantum measure in Regge calculus in three dimensions. - Class. Quantum Grav. [**11**]{}, 2443 (1994), gr-qc/9310040. J.W. Barrett, M. Roček and R.M. Williams, A note on area variables in Regge calculus. - Class. Quantum Grav., [**16**]{}, 1373 (1999), gr-qc/9710056. V. M. Khatsymovsky, Area expectation values in quantum area Regge calculus. - Phys. Lett. [**560B**]{}, 245 (2003), gr-qc/0212110. V. M. Khatsymovsky, Length expectation values in quantum Regge calculus. - Phys. Lett. [**586B**]{}, 411 (2004), gr-qc/0401053. H. Waelbroeck, 2+1 lattice gravity. - Class. Quantum Grav. [**7**]{}, 751 (1990). V. M. Khatsymovsky, Path integral in area tensor Regge calculus and complex connections. - Phys. Lett. [**637B**]{}, 350 (2006), gr-qc/0602116. M. Rocek and R.M. Williams, Quantum Regge calculus. - Phys. Lett. [**104B**]{}, 31 (1981).
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abstract: 'We analyze the entanglement evolution of two cavity photons being affected by the dissipation of two individual reservoirs. Under an arbitrary local unitary operation on the initial state, it is shown that there is only one parameter which changes the entanglement dynamics. For the bipartite subsystems, we show that the entanglement of the cavity photons is correlated with that of the reservoirs, although the local operation can delay the time at which the photon entanglement disappears and advance the time at which the reservoir entanglement appears. Furthermore, via a new defined four-qubit entanglement measure and two three-qubit entanglement measures, we study the multipartite entanglement evolution in the composite system, which allows us to analyze quantitatively both bipartite and multipartite entanglement within a unified framework. In addition, we also discuss the entanglement evolution with an arbitrary initial state.'
author:
- Wei Wen$^1$
- 'Yan-Kui Bai$^{2,3}$'
- Heng Fan$^3$
title: 'Entanglement evolution in multipartite cavity-reservoir systems under local unitary operations'
---
Introduction
============
As one of the most subtle phenomena in many-body systems, quantum entanglement has now been an important physical resource widely used in quantum communication and quantum computation [@hor09rev; @ple07qic]. Therefore it is fundamental to characterize the entanglement nature in quantum systems, especially at a quantitative level. Till now, although bipartite entanglement is well understood in many aspects, the entanglement in multipartite systems is far from clear and thus deserve further exploration.
Entanglement dynamical behavior is an important property in practical quantum information processing. This is because entanglement is fragile and always decays due to unwanted interactions between the system and its environment. A theoretical study of two-atom spontaneous emission shows that entanglement does not always decay in an asymptotic way and it can be corrupted in a finite time [@tyu04prl], which is referred to as entanglement sudden death (ESD). Some earlier studies also pointed out this fact that even a very weakly dissipative environment can disentangle the quantum system in a finite time [@zyc01pra; @akr01pra; @daf03pra; @dod04pra]. The ESD phenomenon has recently received a lot of attentions [@ban06jpa; @san06pra; @der06pra; @sun07pra; @lfw09pra; @fan09pra; @ajh10pra; @ysw10pra] (see also a review paper [@tyu09sci] and references therein), and, experimentally, it has been detected in photon [@alm07sci] and atom systems [@lau07prl].
A deep understanding on the ESD phenomenon concerns the problem where the lost entanglement goes. To answer the question, it is proper to enlarge the system to include its environment. Recently, López *et al* analyzed the entanglement evolution in a composite system consisting of entangled cavity photons with individual reservoirs [@clo08prl], and show that the entanglement sudden birth (ESB) of reservoir-reservoir subsystem must happen whenever the ESD of cavity-cavity subsystem occurs. Moreover, in Ref. [@byw09pra], Bai *et al* presented a entanglement monogamy relation in multipartite systems and analyzed quantitatively the bipartite entanglement transfer in the multipartite cavity-reservoir system.
However, in the above analysis, the multipartite entanglement in the composite cavity-reservoir system is not well characterized, although the residual entanglement [@byw09pra] can indicate its existence. Moreover, the authors only consider the symmetric initial state like ${|\phi\rangle}=\alpha{|00\rangle}+\beta{|11\rangle}$. When the initial state is asymmetric, the entanglement evolution can be very different. For example, a $\sigma_{x}$ operation acting on the symmetric state can change the evolution of the entangled cavity photons from the ESD route to the asymptotic decay route, although the two kinds of initial states have the equal entanglement. Therefore, it is desirable to consider the entanglement dynamical behavior for the asymmetric case and, particularly, find a good entanglement measure to characterize the genuine multipartite entanglement evolution.
In this paper, for the asymmetric initial state modulated by an arbitrary local unitary (LU) operation, we analyze its entanglement evolution in the multipartite cavity-reservoir system. In Sec. II, we derive the effective output state under the LU operation, in which there is only one parameter affecting the entanglement dynamics. In Sec. III, we analyze the bipartite entanglement transfer in the composite system, and point out the cavity photon entanglement is still correlated with the reservoir entanglement although the local operation can delay the ESD time and advance the ESB time. In Sec. IV, the multipartite entanglement evolution is studied via a new defined four-qubit entanglement measure and two three-qubit entanglement measures. In Sec. V, within a unified framework, we investigate the relation between bipartite entanglement transfer and multipartite entanglement transition in the composite system. Finally, we discuss the entanglement evolution with an arbitrary initial state and give a brief conclusion in Sec. VI.
The effective output state under the LU operation
=================================================
Before the derivation of the effective output state under the LU operation, we first recall the multipartite cavity-reservoir system. In Ref. [@clo08prl], López *et al* considered two entangled cavity photons being affected by the dissipation of two individual $N$-mode reservoirs where the interaction of a single cavity-reservoir system is described by the Hamiltonian $$\label{1}
\hat{H}=\hbar \omega
\hat{a}^{\dagger}\hat{a}+\hbar\sum_{k=1}^{N}\omega_{k}
\hat{b}_k^{\dagger}\hat{b}_k+\hbar\sum_{k=1}^{N}g_{k}(\hat{a}
\hat{b}_{k}^{\dagger}+\hat{b}_{k}\hat{a}^{\dagger}).$$ The authors analyzed the entanglement evolution with the symmetric initial state $$\label{2}
{|\Phi_{0}\rangle}=(\alpha{|00\rangle}+\beta{|11\rangle})_{c_1c_2}{|00\rangle}_{r_1r_2},$$ in which the reservoirs are in the vacuum state and the quantum state of cavity photons is invariant under the permutation of the qubits $c_1$ and $c_2$. They show that, along the time evolution, the ESD of two photons can happen when the initial state amplitudes satisfy the condition $\alpha<\beta$, and this procedure is *necessarily* related to the ESB of two reservoirs.
Now, we consider the asymmetric initial state modulated by an arbitrary single-qubit LU operation. Without loss of generality, we assume that the operation acts on the first cavity, and then the initial state can be written as $$\label{3}
{|\Phi_{0}^a\rangle}=U_{c_1}{|\Phi_{0}\rangle}.$$ For an arbitrary single qubit LU operation, one can decompose it as [@nie00book] $$\label{4}
U(\zeta,\eta,\gamma,
\delta)=e^{i\zeta}R_{z}(\eta)R_{y}(\gamma)R_{z}(\delta),$$ where the $e^{i\zeta}$ is a global phase shift and $R_{k}(\theta)=\mbox{exp}(-i\theta \sigma_k/2)$ is the rotation along the $k(=y,z)$ axis with the $\sigma_k$ being the Pauli matrix. In this case, the output state under the time evolution is $$\begin{aligned}
\label{5}
{|\Phi_t\rangle}&=&U_{c_1r_1}(\hat{H},t)\otimes
U_{c_2r_2}(\hat{H},t){|\Phi_0^a\rangle}\nonumber\\
&\simeq&U_{c_1r_1}(\hat{H}^{\prime},t) \otimes
U_{c_2r_2}(\hat{H}^{\prime\prime},t)[R_y(\gamma)_{c_1}{|\Phi_0\rangle}],\end{aligned}$$ where $\hat{H}^{\prime}=R_z^{\dagger}(\eta)_{c_1}\hat{H}R_z(\eta)_{c_1}$, $\hat{H}^{\prime\prime}=R_z^{\dagger}(\delta)_{c_2}\hat{H}R_z(\delta)_{c_2}$, and the $\simeq$ means the states on two sides are equivalent up to some LU operations (for a detail derivation, see the appendix). After considering the effect of the evolution $U_{c_1r_1}(\hat{H}^{\prime},t)$ on the entanglement dynamics, we find that it is equivalent to that of the evolution $U_{c_1r_1}(\hat{H},t)$ (in the appendix, we give the proof). The case for the evolution $U_{c_2r_2}(\hat{H}^{\prime\prime},t)$ is similar. Then Eq. (5) can be rewritten as $$\label{6}
{|\Phi_t\rangle} \simeq U_{c_1r_1}(\hat{H},t)\otimes U_{c_2r_2}(\hat{H},t)
[R_y(\gamma)_{c_1}{|\Phi_0\rangle}],$$ which means that, under an arbitrary LU operation $U_{c_1}(\zeta,\eta,\gamma,\delta)$, the entanglement evolution is only sensitive to the rotation $R_y(\gamma)_{c_1}$.
Therefore, the effective initial state for the entanglement evolution is $$\label{7}
{|\Psi_0\rangle}=R_y(\gamma)_{c_1}{|\Phi_0\rangle}=(\alpha{|\tilde{0}0\rangle}
+\beta{|\tilde{1}1\rangle})_{c_1c_2}{|00\rangle}_{r_1r_2},$$ in which the new basic vectors are ${|\tilde{0}\rangle}=\mbox{cos}(\gamma/2){|0\rangle}+\mbox{sin}(\gamma/2){|1\rangle}$ and ${|\tilde{1}\rangle}=-\mbox{sin}(\gamma/2){|0\rangle}+\mbox{cos}(\gamma/2){|1\rangle}$. For the output state, we use the approximation [@clo08prl] $$\label{8}
U(\hat{H},t)_{cr}{|10\rangle}=\xi{|10\rangle}+\chi{|01\rangle},$$ where the amplitudes are $\xi(t)=\mbox{exp}(-\kappa t/2)$ and $\chi(t)=[1-\mbox{exp}(-\kappa t)]^{1/2}$ in the limit of $N\rightarrow \infty$ for a reservoir with a flat spectrum. Then the effective output state has the form $$\begin{aligned}
\label{9}
{|\Psi_t\rangle}&=&\alpha(\mbox{cos}\frac{\gamma}{2}{|00\rangle}
+\mbox{sin}\frac{\gamma}{2}{|\phi_t\rangle})_{c_1r_1}{|00\rangle}_{c_2r_2}\nonumber\\
&&-\beta(\mbox{sin}\frac{\gamma}{2}{|00\rangle}
-\mbox{cos}\frac{\gamma}{2}{|\phi_t\rangle})_{c_1r_1}{|\phi_t\rangle}_{c_2r_2},\end{aligned}$$ where ${|\phi_t\rangle}=\xi(t){|10\rangle}_{cr}+\chi(t){|01\rangle}_{cr}$ and the parameter $\gamma$ being chosen in the range $[0,\pi]$.
Two-qubit entanglement evolution under the LU operation
=======================================================
According to the effective output state ${|\Psi_t\rangle}$ in Eq. (9), we can derive the density matrices of different subsystems and analyze their entanglement dynamical behaviors. We first consider the subsystem of two cavity photons, for which its density matrix is $$\label{10}
\rho_{c_1c_2}(t)=\psi_1+\psi_2+\psi_3+\psi_4,$$ where $\psi_i={{|\psi_i\rangle}{\langle \psi_i|}}$ and the four non-normalized pure state components are ${|\psi_1\rangle}=\alpha\mbox{cos}(\gamma/2){|00\rangle}+\alpha\mbox{sin}(\gamma/2)
\xi{|10\rangle}-\beta\mbox{sin}(\gamma/2)\xi{|01\rangle}+
\beta\mbox{cos}(\gamma/2)\xi^2{|11\rangle}$, ${|\psi_2\rangle}=\beta\mbox{sin}(\gamma/2)\chi{|00\rangle}-\beta\mbox{cos}(\gamma/2)
\xi\chi{|10\rangle}$, ${|\psi_3\rangle}=\alpha\mbox{sin}(\gamma/2)\chi{|00\rangle}+\beta\mbox{cos}(\gamma/2)
\xi\chi{|01\rangle}$, and ${|\psi_4\rangle}=\beta\mbox{cos}(\gamma/2)\chi^2{|00\rangle}$, respectively. For the two reservoirs, its density matrix is similar to that of the cavity photons and the following relation holds $$\label{11}
\rho_{r_1r_2}(t)=S_{\xi\leftrightarrow\chi}[\rho_{c_1c_2}(t)],$$ where $S_{\xi\leftrightarrow\chi}$ exchanges the parameters $\xi$ and $\chi$ (*i.e.*, $\xi\rightarrow\chi$ and $\chi\rightarrow\xi$).
Based on the previous analysis in Ref. [@byw09pra], we choose the square of the concurrence to characterize the two-qubit entanglement evolution. The concurrence is defined as [@woo98prl] $C(\rho_{ij})=\mbox{max}(0,
\sqrt{\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4})$ with the decreasing nonnegative real numbers $\lambda_{i}$ being the eigenvalues of the matrix $R_{ij}=\rho_{ij}(\sigma_y\otimes\sigma_y)\rho_{ij}^{\ast}
(\sigma_y\otimes\sigma_y)$. After computing the eigenvalues of the matrices $R_{c_1c_2}$ and $R_{r_1r_2}$ [@expl2], we can obtain $$\begin{aligned}
\label{12}
C_{c_1c_2}^2(t)&=&4[\mbox{max}(|\alpha\beta\xi^2|-|\beta\xi\chi|^2\mbox{cos}^2
(\gamma/2),0)]^2,\nonumber\\
C_{r_1r_2}^2(t)&=&4[\mbox{max}(|\alpha\beta\chi^2|-|\beta\xi\chi|^2\mbox{cos}^2
(\gamma/2),0)]^2.\end{aligned}$$ Combining the concurrences and the expressions of $\xi(t)$ and $\chi(t)$ in Eq. (8), we know that, for a given asymmetric initial state (given the parameters $\alpha$, $\beta$ and $\gamma$), the cavity photons entanglement decreases and the reservoir entanglement increases along with the time evolution.
It is still an unsolved problem whether or not the ESD of cavity photons and the ESB of the reservoirs are correlated in the asymmetric case. With the conditions $C_{c_1c_2}(t)=0$ and $C_{r_1r_2}(t)=0$, we can deduce the times of the ESD and the ESB, which have the forms $$\begin{aligned}
\label{13}
t_{ESD}(\rho_{c_1c_2})&=&-\frac{1}{\kappa}\mbox{ln}
\left(1-\frac{\alpha}{\beta\cdot\mbox{cos}^2
(\gamma/2)}\right),\nonumber\\
t_{ESB}(\rho_{r_1r_2})&=&\frac{1}{\kappa}\mbox{ln}
\frac{\beta\cdot\mbox{cos}^2(\gamma/2)}{\alpha},\end{aligned}$$ where the parameter $\kappa$ is the dissipative constant (note that $\xi=\mbox{exp}(-\kappa t/2)$ in the entanglement evolution). According to the two times, we can derive that the ESD of two photons occurs when $\beta\cdot\mbox{cos}^2(\gamma/2)>\alpha$, as is the case for the ESB of two reservoirs. This means that the correlation between the ESD and the ESB *still* holds for the asymmetric initial states, i.e., the ESB of the reservoirs must happen when the ESD of cavity photons occurs.
As an example, we choose the initial state parameters as $\alpha=1/\sqrt{10}$ and $\beta=3/\sqrt{10}$. In Fig. 1(a), the concurrence $C_{c_1c_2}^2$ is plotted as a function of the time $\kappa t$ and the rotation parameter $\gamma$. For a fixed value of $\kappa t$, the photon entanglement increases with the parameter $\gamma$. When the $\gamma$ is given, the $C_{c_1c_2}^2$ decreases along the time $\kappa t$. The ESD line (the purple line) is also plotted in the figure, where the $\gamma$ can delay the ESD time. It is interesting that the entanglement evolution changes to the asymptotical decay route before the $\gamma$ attains to the value $\pi$, and the critical value is $\gamma=2\mbox{arccos}\sqrt{1/3}\approx 1.91063$. In Fig. 1(b), the entanglement evolution of $C_{r_1r_2}^2$ is plotted, where the parameter $\gamma$ can increase the reservoir entanglement and advance the ESB time (the purple line). The critical value for the route transition is also $\gamma=2\mbox{arccos}\sqrt{1/3}$. Moreover, depending on the value of the $\gamma$, the ESB can manifest before, simultaneously and after the ESD.
The two-qubit entanglement of subsystem $c_1r_1$ has the form $$\label{14}
C_{c_1r_1}^2(t)=\xi^2\chi^2[1+(\beta^2-\alpha^2)\mbox{cos}(\gamma)]^2.$$ In Fig. 1(c), the concurrence is plotted as a function of the parameters $\kappa t$ and $\gamma$. The maximum of $C_{c_1r_1}^2(t)$ appears at the time $\kappa t=\mbox{ln}2$, and the entanglement decreases with the $\gamma$. However, the $\gamma$ does not change the entanglement of subsystem $c_2r_2$, because the rotation $R_y(\gamma)$ acts on the first cavity. For the subsystems $c_1r_2$ and $c_2r_1$, we can get that they have the equal entanglement, which can be expressed as $$\label{15}
C_{c_1r_2}^2(t)=4[\mbox{max}(|\alpha\beta\xi\chi|-|\beta\xi\chi|^2
\mbox{cos}^2(\gamma/2),0)]^2.$$ In Fig. 1(d), the concurrence is plotted. When $\gamma=0$, the entanglement evolution experiences the ESD at the time $\kappa t=\mbox{ln}[3(3-\sqrt{5})/2]$ and the ESB at the time $\kappa t=\mbox{ln}[3(3+\sqrt{5})/2]$ (the two intersections between the purple line and the $\kappa t$ axis), then the entanglement changes asymptotically. Along with the increase of the parameter $\gamma$, the time window between the ESD and ESB decreases, and the window become a point when $\gamma=2\mbox{arccos}\sqrt{2/3}\approx 1.23096$. After this value, both the ESD and the ESB phenomena disappear.
Multipartite entanglement evolution under the LU operation
==========================================================
Before analyzing the entanglement evolution, we first consider how to characterize the multipartite entanglement in the composite system. In Ref. [@clo08prl], the multipartite concurrence $C_N$ [@car04prl] can not characterize completely the genuine multipartite entanglement, due to its nonzero value for two Bell states. In Ref. [@byw09pra], it is shown that the genuine multipartite entanglement can be indicated by the two-qubit residual entanglement $$\label{16}
M_{c_1r_1}(\Phi_t)=C_{c_1r_1|c_2r_2}^2(t)-\sum
C_{i^{\prime}j^{\prime}}^2(t),$$ where the sum subscripts $i^{\prime}\in\{c_1, r_1\}$ and $j^{\prime}\in\{c_2, r_2\}$, respectively. However, the entanglement monotone property of $M_{c_1r_1}$ is not clear, even for the case of symmetric initial states.
The average multipartite entanglement may quantify the genuine multiqubit entanglement based on much numerical analysis, which is defined as [@byw07pra] $$\begin{aligned}
\label{17}
E_{ms}(\Psi_4)=\frac{\sum_i \tau_i(\rho_i)-2\sum_{i>j}
C_{ij}^2(\rho_{ij})}{4},\end{aligned}$$ where the $\tau_i=2(1-\mbox{tr}\rho_i^2)$ is the linear entropy and the $C_{ij}$ is the concurrence. For the four-qubit cluster-class states, an analytical proof of the entanglement monotone property for the $E_{ms}$ was given in Refs. [@baw08pra; @ren08pra].
For the effective output state ${|\Psi_t\rangle}$ in Eq. (9), we can compute the average multipartite entanglement $E_{ms}$ and the residual entanglement $M_{c_1r_1}$. After comparing the two measures, we can obtain that they are equivalent up to a constant factor $2$. Therefore, we can define entanglement measure $$\label{18}
E_{BB}(\Psi_t)=M_{c_1r_1}(\Psi_t)=2E_{ms}(\Psi_t),$$ which quantifies the genuine multipartite entanglement between the blocks $c_1r_1$ and $c_2r_2$ and its entanglement monotone property is based on the numerical analysis on the average multipartite entanglement $E_{ms}$. In Fig. 2, the $E_{BB}$ is plotted as a function of the parameters $\kappa t$ and $\gamma$, where the initial state parameters are chosen as $\alpha=1/\sqrt{10}$ and $\beta=3/\sqrt{10}$. When $\gamma=0$, the $E_{BB}(\kappa t)$ increases from $0$ to $0.36$ in the region $\kappa t\in\{0,\mbox{ln}(3/2)\}$, then it keeps invariant until the time $\kappa t=\mbox{ln}3$, finally, the $E_{BB}(\kappa t)$ decreases asymptotically. With the increase of the $\gamma$, the width of the plateau decreases and can be expressed as $$\label{19}
t_w=\mbox{ln}[3\mbox{cos}^2(\gamma/2)-1].$$ When $\gamma=2\mbox{arccos}(\sqrt{2/3})$, the width changes to zero and the evolution time is $\kappa t=\mbox{ln}2$. After this value, the block-block entanglement decreases along with the $\gamma$, and vanishes when $\gamma=\pi$.
The genuine tripartite entanglement in the composite system can be quantified by the mixed state three-tangle [@won01pra] $$\label{20}
\tau_3(\rho_{ijk})=\mbox{min}\sum_{\{p_x,\varphi^{(x)}_{ijk}\}}
p_x\tau(\varphi_{ijk}^{(x)}),$$ where $\tau(\varphi^{(x)}_{ijk})=\tau_i-C_{ij}^2-C_{ik}^2$ [@ckw00pra] is the pure state three-tangle and the minimum runs over all the pure state decompositions of $\rho_{ijk}$. The reduced density matrix of subsystem $c_1r_1c_2$ can be written as $$\label{21}
\rho_{c_1r_1c_2}(t)=\varphi_1(t)+\varphi_2(t),$$ where the non-normalized pure state components are ${|\varphi_1(t)\rangle}=\alpha\mbox{cos}(\gamma/2){|000\rangle}-\beta\mbox{sin}(\gamma/2)
\xi{|001\rangle}+\alpha\mbox{sin}(\gamma/2)\chi{|010\rangle}+\alpha\mbox{sin}(\gamma/2)
\xi{|100\rangle}+\beta\mbox{cos}(\gamma/2)\xi^2{|101\rangle}+\beta\mbox{cos}\xi\chi{|011\rangle}$ and ${|\varphi_2(t)\rangle}=\beta\mbox{sin}(\gamma/2)\chi{|000\rangle}-\beta\mbox{cos}
(\gamma/2)\chi^2{|010\rangle}-\beta\mbox{cos}(\gamma/2)\xi\chi{|100\rangle}$, respectively. It is obvious that the ${|\varphi_2\rangle}$ is a separable state and its three-tangle is zero. Moreover, for the component ${|\varphi_1\rangle}$, we can derive $\tau(\varphi_1)=0$. So, the decomposition in Eq. (21) is the optimal and the mixed state three-tangle $\tau_3(\rho_{c_1r_1c_2})$ is zero. Similarly, we can obtain that all the other mixed state three-tangles $\tau_3(\rho_{ijk})$ are zero.
Although all the $\tau_3(\rho_{ijk})$ are zero in the entanglement evolution, the three-qubit states are still entangled in the qubit-block form [@byw08pra; @loh06prl], which is not equivalent to the mixed state three-tangle and can not be accounted for the two-qubit entanglement. The qubit-block entanglement characterizes the genuine three-qubit entanglement under bipartite cut between a qubit and a block of qubits, and can be defined as $E_{q-B}(\rho_{i|jk})=C_{i|jk}^2-C_{ij}^2-C_{ik}^2$, in which the $C_{i|jk}$ quantifies bipartite entanglement between the qubits $i$ and $jk$. For the subsystems $c_1c_2r_2$ and $r_1c_2r_2$, their qubit-block entanglement are $$\begin{aligned}
\label{22}
E_{q-B}(\rho_{c_1|c_2r_2})&=&C_{c_1|c_2r_2}^2-C_{c_1c_2}^2-C_{c_1r_2}^2
\nonumber\\
E_{q-B}(\rho_{r_1|c_2r_2})&=&C_{r_1|c_2r_2}^2-C_{r_1r_2}^2-C_{c_2r_1}^2,\end{aligned}$$ where $C_{c_1|c_2r_2}^2=4\alpha^2\beta^2\xi^2$, $C_{r_1|c_2r_2}^2=4\alpha^2\beta^2\chi^2$, and the expressions of two-qubit concurrences $C_{ij}^2$ are given in Eqs. (12), (14) and (15).
In Fig. 3, we plot the qubit-block entanglement as a function of the parameters $\kappa t$ and $\gamma$, where the initial state parameters are chosen as $\alpha=1/\sqrt{10}$ and $\beta=3/\sqrt{10}$. For a given value of the $\gamma$, the qubit-block entanglement $E_{c_1|c_2r_2}$ (in Fig. 3(a)) increases first with the time $\kappa t$, and then decreases with the $\kappa
t$ after attaining to its maximal value. Along with the increase of the $\gamma$, the maximal value of $E_{c_1|c_2r_2}$ decreases. For the qubit-block entanglement $E_{r_1|c_2r_2}$ (in Fig. 3(b)), the trend of entanglement evolution is similar. In Refs. [@byw08pra; @byw09pra], it is pointed out that the qubit-block entanglement comes from the genuine multipartite entanglement in the enlarged pure state system. Here, for the multipartite cavity-reservoir system, we can derive the following relation $$\label{23}
E_{BB}(\Psi_t)=E_{q-B}(\rho_{c_1|c_2r_2}(t))
+E_{q-B}(\rho_{r_1|c_2r_2}(t)),$$ which means that the qubit-block entanglement comes from the genuine block-block entanglement in the composite system.
Entanglement transfer and entanglement transition under the LU operation
========================================================================
Because the evolution $U_{c_1r_1}(\hat{H},t)\otimes
U_{c_2r_2}(\hat{H},t)$ are two local unitary operations under the partition $c_1r_1|c_2r_2$, the bipartite entanglement $C_{c_1r_1|c_2r_2}^2$ is invariant, and the following relation holds $$\label{24}
C_{c_1r_1|c_2r_2}^2(\Psi_t)=E_{BB}(t)+\sum
C_{i^{\prime}j^{\prime}}^2(t)=4\alpha^2\beta^2,$$ where $i^{\prime}\in\{c_1, r_1\}$ and $j^{\prime}\in\{c_2, r_2\}$, respectively. Therefore, in the multipartite cavity-reservoir system, we can characterize the entanglement evolution under a unified framework, where the two qubit entanglement transfer is quantified by the concurrence and the multipartite entanglement transition is quantified by the block-block entanglement $E_{BB}$. In Fig.4, the entanglement evolution modulated by different value of $\gamma$ is plotted, where $\gamma_1=2\mbox{arccos}(\sqrt{2/3})$, $\gamma_2=2\mbox{arccos}(\sqrt{1/3})$, and the initial state parameters are $\alpha=1/\sqrt{10}$ and $\beta=3/\sqrt{10}$.
In Fig. 4(a), the parameter is chosen as $\gamma=0$, which corresponds to the symmetric initial state. In the time interval $\kappa t\in(0,\mbox{ln}(3/2))$, a part of the initial photon-photon entanglement $C_{c_1c_2}^2$ first transfers to the subsystems $c_1r_2$ and $c_2r_1$ (the brown dot-dashed line where a factor $5$ is multiplied), then the remaining photon-photon entanglement and the cavity-reservoir entanglement transition completely to the genuine block-block entanglement $E_{BB}$ (the blue solid line). Along with the time evolution, the block-block entanglement keeps invariant and is immune to the cavity-reservoir interaction in the time interval $\kappa t\in [\mbox{ln}(3/2),\mbox{ln}3]$. Finally, when $\kappa t \in(\mbox{ln}3,6]$, the multipartite entanglement $E_{BB}$ transitions to the two-qubit reservoir-reservoir entanglement (the red dotted line) and the cavity-reservoir entanglement. When the parameter $\gamma\in (0,\gamma_1]$, the trends of bipartite and multipartite entanglement evolutions are similar to those when $\gamma=0$, but the immune region of the block-block entanglement decreases with the parameter and the other evolution regions extend. In Fig. 4(b), the parameter is chosen as $\gamma=\gamma_1$, where the plateau region of the $E_{BB}$ changes to a point ($\kappa t=\mbox{ln}2$) and the procedures of entanglement transfer and entanglement transition need more time.
When the parameter $\gamma\in (\gamma_1,\gamma_2]$, the initial photon entanglement transfers to not only the subsystems $c_1r_2$ and $c_2r_1$ but also the subsystem $r_1r_2$, and the two-qubit entanglement can not transition completely to the genuine block-block entanglement. As shown in Fig. 4(c), the entanglement transfer and entanglement transition are plotted when $\gamma=\gamma_2$. Along with the increase of the $\gamma$, the decay of the photon entanglement slows down and the transfer ratio of the two-qubit entanglement increases. At the same time, the transition ratio of the block-block entanglement decreases. In Fig. 4(d), the parameter is chosen as $\gamma=\pi$, in which the transition between the two-qubit entanglement and the multipartite entanglement disappears, and the entanglement evolution consists of only two-qubit entanglement transfer.
It should be pointed out that, in the unified framework of entanglement evolution, the entanglement monotone property of $E_{BB}$ is based on the numerical analysis on the average multipartite entanglement $E_{ms}$ [@byw07pra]. The analytic proof is still an open problem.
Discussion and conclusion
=========================
In the more general case, an arbitrary initial state has the form ${|\Psi_0^g\rangle}=(\alpha_1{|00\rangle}+\alpha_2{|01\rangle}+\alpha_3{|10\rangle}
+\alpha_4{|11\rangle})_{c_1c_2}{|00\rangle}_{r_1r_2}$, which corresponds to two LU operations $U_{c_1}(\zeta_1,\eta_1,\gamma_1,\delta_1)\otimes
U_{c_2}(\zeta_2,\eta_2,\gamma_2,\delta_2)$ acting on the symmetric initial state. In this case, the analytical characterization for the entanglement evolution is not available so far. However, the correlation between the ESD of cavity photons and the ESB of reservoirs still holds. This is because we can deduce the relation $$\label{25}
\rho_{c_1c_2}^g(\xi,\chi)=S_{\xi\leftrightarrow\chi}
[\rho_{r_1r_2}^g(\xi,\chi)],$$ where the evolution $U_{cr}(\hat{H},t){|10\rangle}=\xi{|10\rangle}+\chi{|01\rangle}$ is used. Based on this relation, we can obtain that when the ESD of cavity photons occurs at the time $t_{ESD}=t_0$, the ESB of reservoirs will necessarily happen at the time $t_{ESB}=-(1/\kappa)\mbox{ln}[1-\mbox{exp}(-\kappa t_0)]$. Moreover, the entanglement evolution is restricted by the monogamy relation $$\begin{aligned}
\label{26}
C_{c_1r_1|c_2r_2}^2({|\Psi_0^g\rangle})&\geq&
C_{c_1c_2}^2(t)+C_{r_1r_2}^2(t)+C_{c_1r_2}^2(t),\nonumber\\
&&+C_{c_2r_1}^2(t)\end{aligned}$$ and the multipartite entanglement can be indicated by the two-qubit residual entanglement $M_{c_1r_1}$ [@byw09pra; @osb06prl].
The entanglement evolution with the asymmetric initial state is worth to consider for other physical systems, for example, the atoms systems [@tyu04prl], quantum dots and spin chains etc. [@dlo98pra; @ssl01pns; @xgw06pra]. Moreover, the dissipative entanglement evolution has close relation with the type of the noise environment. Therefore, the non-Markovian environment, the correlated noises, and some operator channels [@bel07prl; @nov08pra; @kon08nat; @wan10epr] are also worth to study in future.
In conclusion, we have investigated the entanglement evolution of multipartite cavity-reservoir systems with the asymmetric initial state. It is shown that there is only one parameter in the LU operation affecting the entanglement dynamics, which can delay the ESD of the photons, advance the ESB of the reservoirs, change the evolution route of bipartite entanglement, and suppress the multipartite entanglement. However, the correlation between the ESD and the ESB still holds. Furthermore, by defining the block-block entanglement, we analyze the multipartite entanglement evolution in the composite system, which allows us to study quantitatively both the entanglement transfer and the entanglement transition within a unified framework. Finally, the entanglement evolution with an arbitrary initial state is discussed.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Prof. Z. D. Wang for many useful discussions and suggestions. This work was supported by the National Basic Research Program of China (973 Program) grant Nos. 2009CB929300 and 2010CB922904. Y.K.B. was also supported by the fund of Hebei Normal University and NSF-China Grant No. 10905016.
Appendix {#appendix .unnumbered}
========
We first prove Eq. (5). The output state under the time evolution is $$\begin{aligned}
\label{27}
{|\Phi_t\rangle}&=&U_{c_1r_1}(\hat{H},t)\otimes U_{c_2r_2}(\hat{H},t)
{|\Phi_0^a\rangle}\nonumber\\
&=&U_{c_1}U_{c_1}^{\dagger}U_{c_1r_1}(\hat{H},t)\otimes U_{c_2r_2}(\hat{H},t)
[U_{c_1}{|\Phi_0\rangle}],\end{aligned}$$ where, in the second equation, the identity operator $I=U_{c_1}U_{c_1}^{\dagger}$ is inserted. Substituting the $U_{c_1}$ with the expression $R_{z}(\eta)R_{y}(\gamma)R_{z}(\delta)$ (we neglect the global phase $e^{i\zeta}$), we can obtain $$\begin{aligned}
\label{28}
{|\Phi_t\rangle}&=&U_{l}R_{z}^{\dagger}(\eta)U_{c_1r_1}(\hat{H},t)R_{z}(\eta)_{c_1}
\otimes U_{c_2r_2}(\hat{H},t)\nonumber\\
&&[R_y(\gamma)_{c_1}R_z(\delta)_{c_1}{|\Phi_0\rangle}]\\
&\simeq & U_{c_1r_1}(\hat{H}^{\prime},t)\otimes U_{c_2r_2}(\hat{H},t)
[R_y(\gamma)_{c_1}R_z(\delta)_{c_1}{|\Phi_0\rangle}]\nonumber\end{aligned}$$ where the symbol $\simeq$ means the quantum states on the two sides are equivalent up to the local unitary operation $U_l=U_{c_1}R_z^{\dagger}
(\delta)_{c_1}R_y^{\dagger}(\gamma)_{c_1}$ (note that entanglement is invariant under local unitary transformation), and we use the relation $R_z^{\dagger}(\eta)_{c_1}U_{c_1r_1}(\hat{H},t)R_z(\eta)_{c_1}=
U_{c_1r_1}(\hat{H}^{\prime},t)$ with $\hat{H}^{\prime}=R_z^{\dagger}(\eta)_{c_1}
\hat{H}R_z(\eta)_{c_1}$. Due to the symmetric property of initial state ${|\Phi_0\rangle}$, we have the relation $R_z(\delta)_{c_1}
{|\Phi_0\rangle}=R_z(\delta)_{c_2}{|\Phi_0\rangle}$. Then the output state can be expressed further as $$\begin{aligned}
\label{29}
{|\Phi_t\rangle}&=&U_{c_1r_1}(\hat{H}^{\prime},t)\otimes R_z(\delta)_{c_2}
R_z^{\dagger}(\delta)_{c_2}U_{c_2r_2}(\hat{H},t)R_z(\delta)_{c_2}\nonumber\\
&&[R_y(\gamma)_{c_1}{|\Phi_0\rangle}]\nonumber\\
&\simeq&U_{c_1r_1}(\hat{H}^{\prime},t)\otimes U_{c_2r_2}(\hat{H}^{\prime\prime},t)
[R_y(\gamma)_{c_1}{|\Phi_0\rangle}],\end{aligned}$$ where we insert the identity operator $R_z(\delta)_{c_2}
R_z^{\dagger}(\delta)_{c_2}=I$ in the first equation and use the relation $\hat{H}^{\prime\prime}=R_z^{\dagger}(\delta)_{c_2}
\hat{H}R_z(\delta)_{c_2}$ in the second equation.
Next, we will prove the effects of $\hat{H}^{\prime}$ and $\hat{H}^{\prime\prime}$ are equivalent to that of $\hat{H}$ in the entanglement evolution. Because the Hilbert space of subsystem $c_1r_1$ is spanned by ${|00\rangle},{|01\rangle},{|10\rangle}$, the creation and annihilation operators are $$\label{30}
\hat{a}^{\dagger}=\left(
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}
\right)
~\mbox{and}~ \hat{a}=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right),$$ respectively. With these expressions, the rotation operator $R_{z}(\eta)$ can be rewritten as $$\label{31}
R_z(\eta)=A\cdot \hat{a}^{\dagger}\hat{a}+B,$$ where the coefficients $A=\mbox{exp}(i\eta/2)-\mbox{exp}(-i\eta/2)$ and $B=\mbox{exp}(-i\eta/2)$, respectively. After substituting the expression of $R_{z}(\eta)$ into the Hamiltonian $\hat{H}^{\prime}$, we can derive $$\begin{aligned}
\label{32}
\hat{H}^{\prime}&=&R_z^{\dagger}(\eta)_{c_1}\hat{H}R_z(\eta)_{c_1}\nonumber\\
&=&(A \cdot
\hat{a}^{\dagger}\hat{a}+B)^{\dagger}[\hbar\omega\hat{a}^{\dagger}\hat{a}
+\hbar\sum_{k=1}^{N}\omega_{k}
\hat{b}_k^{\dagger}\hat{b}_k\nonumber\\
&&+\hbar\sum_{k=1}^{N}g_{k}(\hat{a}
\hat{b}_{k}^{\dagger}+\hat{b}_{k}\hat{a}^{\dagger})](A \cdot
\hat{a}^{\dagger}\hat{a}+B)\nonumber\\
&=&\hbar\omega\hat{a}^{\dagger}\hat{a}
+\hbar\sum_{k=1}^{N}\omega_{k}
\hat{b}_k^{\dagger}\hat{b}_k+\hbar\sum_{k=1}^{N}g_{k}(\hat{a}
\hat{b}_{k}^{\dagger}\cdot
e^{i\eta}\nonumber\\
&&+\hat{b}_{k}\hat{a}^{\dagger}\cdot e^{-i\eta})\nonumber\\
&=&V_{r_1}^{\dagger}(\eta)\hat{H}V_{r_1}(\eta),\end{aligned}$$ where $V_r(\eta)=\mbox{diag}\{1,\mbox{exp}(-i\eta)\}$, and we used the relations $\hat{a}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{a}=
\hat{a}^{\dagger}\hat{a}=\hat{N}$ and $\hat{a}^{\dagger}\hat{N}\hat{b}_k=0$ [@expl1]. Similarly, for the Hamiltonian $\hat{H}^{\prime\prime}$, we can obtain $$\label{33}
\hat{H}^{\prime\prime}=R_z^{\dagger}(\delta)_{c_2}\hat{H}R_z(\delta)_{c_2}
=V_{r_2}^{\dagger}(\delta)\hat{H}V_{r_2}(\delta)$$ with $V_{r_2}(\delta)=\mbox{diag}\{1,\mbox{exp}(-i\delta)\}$. Therefore, the output state in Eq. (29) can be written as $$\begin{aligned}
\label{34}
{|\Phi_t\rangle}
&=&V_{r_1}^{\dagger}(\eta)V_{r_2}^{\dagger}(\delta)U_{c_1r_1}(\hat{H},t)
\otimes U_{c_2r_2}(\hat{H},t)\nonumber\\
&&[R_y(\gamma)_{c_1}V_{r_1}(\eta)V_{r_2}(\delta){|\Phi_0\rangle}].\end{aligned}$$ In the above equation, the local unitary operation $V_{r_1}^{\dagger}(\eta)V_{r_2}^{\dagger}(\delta)$ does not change the entanglement evolution. Moreover, due to the reservoirs being in the vacuum state, we have $V_{r_1}(\eta)V_{r_2}
(\delta){|\Phi_0\rangle}={|\Phi_0\rangle}$. Therefore, the effective output state in the entanglement evolution has the form $$\label{35}
{|\Psi_t\rangle}=U_{c_1r_1}(\hat{H},t)\otimes U_{c_2r_2}(\hat{H},t)
[R_y(\gamma)_{c_1}{|\Phi_0\rangle}].$$ This means that, for the asymmetric initial state modulated by an arbitrary LU operation $U_{c_1}(\zeta,\eta,\gamma,\delta)$, the entanglement evolution is only sensitive to the rotation $R_y(\gamma)$.
[99]{}
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+\sqrt{\beta^2\xi^4(\alpha^2 + \beta^2cr^4\chi^4)}$, and $\sqrt{\lambda_3}=\sqrt{\lambda_4}=\beta^2\xi^2\chi^2cr^2$ with $cr=\mbox{cos}(\gamma/2)$. The square root of the eigenvalues of matrix $R_{r_1r_2}$ are same to those of matrix $R_{c_1c_2}$ after exchanging the parameters $\xi$ and $\chi$. A. R. R. Carvalho, F. Mintert, and A. Buchleitner, Phys. Rev. Lett. **93**, 230501 (2004). Y.-K. Bai, D. Yang, and Z. D. Wang, Phys. Rev. A **76**, 022336 (2007). Y.-K. Bai and Z. D. Wang, Phys. Rev. A **77**, 032313 (2008). X.-J. Ren, W. Jiang, X. Zhou, Z. W. Zhou, and G. C. Guo, Phys. Rev. A **78**, 012343 (2008). A. Wong and N. Christensen, Phys. Rev. A **63**, 044301 (2001). V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A **61**, 052306 (2000). R. Lohmayer, A. Osterloh, J. Siewert, and A. Uhlmann, Phys. Rev. Lett. **97**, 260502 (2006). Y.-K. Bai, M.-Y. Ye, and Z. D. Wang, Phys. Rev. A **78**, 062325 (2008). T. J. Osborne and F. Verstraete, Phys. Rev. Lett. **96**, 220503 (2006). D. Loss and D. V. DiVincenzo, Phys. Rev. A **57**, 120 (1998). S.-S. Li, G.-L. Long, F.-S. Bai, S.-L. Feng, and H.-Z. Zheng, Proc. Natl. Acad. Sci. U.S.A. **98**, 11847 (2001). X. Wang and Z. D. Wang, Phys. Rev. A **73**, 064302 (2006). B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. Lett. **99**, 160502 (2007). E. Novais, E. R. Mucciolo, and H. U. Baranger, Phys. Rev. A **78**, 012314 (2008). T. Konrad, F. De Melo, M Tiersch, C. Kasztelan, A. Aragao, and A. Buchleitner, Nature Physics **4**, 99 (2008). X.-B. Wang, Z.-W. Yu and J.-Z. Hu, arXiv:1001.0156.
Due to the space of system $c_1r_1$ is spanned by $\{{|00\rangle},{|01\rangle},{|10\rangle}\}$, we have $(\hat{a}^{\dagger}\hat{a}-1)\hat{a}^{\dagger}\hat{a}
=0\Rightarrow \hat{N}^2=\hat{N}$. Similarly, according to the space of system $c_1r_1$, we have $\hat{a}^{\dagger}
\hat{b}\hat{b}^{\dagger}\hat{b}\hat{N}=0$. Combining it with the commutation relation of $\hat{b}$ and $\hat{b}^{\dagger}$, we can derive $\hat{a}^{\dagger}\hat{N}\hat{b}_k=0$.
|
hep-th/9612119
CERN-TH/96-351
USC-96/HEP-B7
LPT ENS 96/70
[A NEW SUPERSYMMETRY]{} [ ]{}\
[Itzhak Bars ]{}[$^a$]{} [and Costas Kounnas ]{}[$^b$]{} [ \
]{}
[TH Division, CERN, CH-1211 Geneva 23, Switzerland]{}
[**ABSTRACT**]{}
[We propose a new supersymmetry in field theory that generalizes standard supersymmetry and we construct field theoretic models that provide some of its representations. This symmetry combines a finite number of ordinary four dimensional supersymmetry multiplets into a single multiplet with a new type of Kaluza-Klein embedding in higher dimensions. We suggest that this mechanism may have phenomenological applications in understanding family unification. The algebraic structure, which has a flavor of W-algebras, is directly motivated by S-theory and its application in black holes. We show connections to previous proposals in the literature for 12 dimensional supergravity, Yang-Mills, (2,1) heterotic superstrings and Matrix models that attempt to capture part of the secret theory behind string theory. ]{}
CERN-TH/96-351
[December 1996]{}
------------------------------------------------------------------------
width 6.7cm
[$^a$ [[On sabbatical leave from the Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA.]{}]{}]{}
[$^b$ [[On leave from Ecole Normale Supérieure, 24 rue Lhomond, F-75231, Paris, Cedex 05, FRANCE. ]{}]{}]{}
1
New supersymmetry
=================
The usual $N=1$ superalgebra in four dimensions is $$\left\{ Q_\alpha ,\bar{Q}_{\dot{\beta}}\right\} =\sigma _{\alpha \dot{\beta}
}^\mu \,p_\mu \,\,,$$ where $\alpha ,\dot{\beta}$ are Weyl spinor indices corresponding to $\left(
1/2,0\right) $ and $\left( 0,1/2\right) $ representations of the Lorentz group $SO(3,1)$. The simplest example of the new superalgebras that we will discuss in this paper is the $N=1$ case $$\left\{ Q_\alpha ,\bar{Q}_{\dot{\beta}}\right\} =\sigma _{\alpha \dot{\beta}
}^\mu \,p_\mu v_{+}\,\,. \label{newsusy}$$ Here $v_{+}=v_{0^{\prime }}+v_{1^{\prime }}$ is the light-cone component of a new operator $v_m=i\tilde{\partial}_m$ acting as a momentum in an additional $1+1$ dimensional space $y^m=(y^{0^{\prime }},y^{1^{\prime }})$ beyond the usual four dimensions $x^\mu $. We will see in section-3 that it is possible to interpret these as the 12th and 11th dimensions respectively. Note that there are two time coordinates $x^0$ and $y^{0^{\prime }}$; we will show that in our approach no problems arise due to this fact and that four dimensional physics looks quite conventional. This superalgebra has an isometry group $SO(3,1)\otimes SO(1,1)$ which is the direct product of the Lorentz groups in the $x^\mu $ and $y^m$ spaces. The group $SO(1,1)$ consists of a single parameter corresponding to a boost that mixes the $y^m$ . Its action on the Weyl spinors is an overall scale transformation with weight $1/2$, while its action on the vector $v_{+}$ is an overall scale transformation with weight $1$.
In this paper we construct field theory models that provide representations of this $N=1$ superalgebra. The fields $\Phi (x,y)$ depend on the 4D $x^\mu $ and on the 2D $y^m=y^{\pm }=y^{0^{\prime }}\pm y^{1^{\prime }}.$ We take derivatives with respect to them $\partial _\mu $ and $\tilde{\partial}_{\pm
}$ such that $p_\mu v_{+}=-\partial _\mu \tilde{\partial}_{+}.$ On this space the form of the superalgebra is $$\left\{ Q_\alpha ,\bar{Q}_{\dot{\beta}}\right\} =-\sigma _{\alpha \dot{\beta}
}^\mu \,\partial _\mu \tilde{\partial}_{+}\,\,.$$
If one considers an expansion of the fields in 2D momentum modes $$\Phi (x,y)=\sum_{k_{\pm }}\Phi _k(x)\,\,e^{-i\left(
k_{+}y^{+}+k_{-}y^{-}\right) }$$ one sees that for each mode the new superalgebra reduces to the standard $
N=1 $ superalgebra, except for a rescaling of the momentum $p^\mu k_{+}$ by a different amount for each mode. Hence the effect of this type of “Kaluza-Klein expansion” on the 4D mass spectrum is very different than the usual one. We will suggest that the extra space $y^m$ may be related to family structure (of quarks and leptons) through this new type of expansion.
As explained in section-3 our original motivation for considering this superalgebra comes from recent developments in S-theory. Some sectors of S-theory may be described as sectors in which the 12D superalgebra simplifies to $$\left\{ Q_\alpha ,Q_\beta \right\} =\gamma _{\alpha \beta }^{MN}\,\,p_M\,v_N
\label{12pv}$$ There is a related version that applies to 13D as well as to compactifications to lower dimensions ( see [@ibstheory][@ibsentropy] and section-3). The compactified form was recently used to explain the presence of up to 12 (or 13) hidden dimensions in supersymmetric black holes [@ibsentropy]. This development provided a strong motivation since the relevance of the hidden dimensions and of the superalgebra was demonstrated in a potentially physical system. There were additional hints that such new superalgebras provide a framework for understanding some deeper structures. In particular, the 12D version in (\[12pv\]) with all possible eigenvalues of the operators $p_M=i\partial /\partial x^M,\,v_N=i\partial /\partial y^N$ that satisfy certain BPS constraints was first suggested in [@ibstheory] as a basis for extending 11D supergravity to 12D supergravity. It was later used with a fixed eigenvalue of $v_N$ (that breaks $SO(10,2)$ ) in a super Yang-Mills theory [@sezgin], and in a related matrix model [@periwal] that recasts a matrix version [@susskind] of 11D M-theory [@wittenetc] to a 12D version. The same algebra was also understood to be present in a (2,1) heterotic string [@martinec]. From the point of view of the superalgebra and S-theory the models in [@sezgin][@periwal] [@martinec] are incomplete because of the fixed $v_N$. As explained in section-3, fully $SO(10,2)$ covariant generalizations of these proposals must exist by allowing all eigenvalues of $v_N\,\,$(as in section-2). Furthermore such models may be regarded as intermediate steps toward the construction of the secret theory behind string theory. These provide additional motivations for studying this type of superalgebra.
In section-3 we will show how the superalgebra (\[newsusy\]) is embedded in S-theory and how it is generalized in all possible ways up to 13 higher dimensions. But at a simpler level one may also consider the $N=1$ superalgebra (\[newsusy\]) on its own merit as a symmetry structure in field theory. Thus, a study of its representations in field theory is undertaken in section-2. In this context we have discovered a new Kaluza-Klein mechanism for embedding families in higher dimensions. We find that several families belong together in the same supermultiplet. In section-2 we provide first examples of representations and family generation mechanisms that may be generalized in several ways. The fact that this approach may have phenomenological implications is both surprizing and welcomed. Connections to M-,F-,S-theories are also given in section-3. In Section-4 we present the first steps of a superfield formalism and conclude with some observations.
Field theoretic representations
===============================
The simplest multiplet of standard $N=1$ supersymmetry is the scalar multiplet that contains the fields ($\phi ,\psi _\alpha ,F)(x)$ that appear in the Wess-Zumino (WZ) model. For the new $N=1$ superalgebra we will present four different representations that connect to the WZ representation upon Kaluza-Klein reduction (there may also be others). We will refer to them as the scalar-vector multiplets with the fields $$(\phi ,\psi _\alpha ,V_{+})(x,y)\quad or\quad (\phi ^{\prime },\psi _\alpha
^{\prime },V_{-}^{\prime })(x,y)$$ and the scalar-scalar multiplets with the fields $$(\varphi ,\chi _\alpha ,f)(x,y)\quad or\quad (\varphi ^{\prime },\chi
_\alpha ^{\prime },f^{\prime })(x,y).$$
The new $N=1$ superalgebra has the isometry $SO(3,1)\times SO(1,1),$ therefore every field must correspond to a representation of this group. Since we wish to connect to the WZ fields we take a triplet of fields $\,$that are (scalar, chiral spinor, scalar) under $SO(3,1)$, and next we choose their $
SO(1,1)$ properties as follows. The group $SO(1,1)$ contains a single parameter corresponding to boosts that mix the ($y^{0^{\prime
}},y^{1^{\prime }}),$ but rescales the light-cone components $y_{\pm }$ with opposite factors $y_{\pm }\rightarrow \Lambda ^{\pm 1}y_{\pm }$. Under the boosts the fields $(\phi ,\psi _\alpha ,V_{+})$ undergo scale transformations with the scale factor raised to the powers $\left(
0,1/2,1\right) $ respectively. The complex conjugates of these fields $(\bar{
\phi},\bar{\psi}_{\dot{\alpha}},\bar{V}_{+})$ have the same $SO(1,1)$ scales. Similarly $(\phi ^{\prime },\psi _\alpha ^{\prime },V_{-}^{\prime })$ and their complex conjugates are assigned $SO(1,1)$ weights $\left(
0,-1/2,-1\right) $, also $(\varphi ,\chi _\alpha ,f)$ have weights $\left(
0,1/2,0\right) $ and $(\varphi ^{\prime },\chi _\alpha ^{\prime },f^{\prime
})$ have weights $\left( 0,-1/2,0\right) .$ The name of the multiplets “scalar-vector” refers to the properties of $V_{+},V_{-}^{\prime }$ that are scalars under $SO(3,1)$ and vectors under $SO(1,1)$. Similarly “scalar-scalar” refers to $f,f^{\prime }$ that are scalars under both $
SO(3,1)$ and $SO(1,1)$ .
Scalar-vector multiplets
------------------------
We guess the following supersymmetry transformation rules for $(\phi ,\psi
_\alpha ,V_{+})$ by imitating the old WZ transformation rules and by requiring consistency with the isometries $SO(3,1)\times SO(1,1)$
$$\begin{aligned}
\delta \phi =-\varepsilon ^{\prime \alpha }\psi _\alpha ,\,\,\delta \psi
_\alpha =\partial _\mu \tilde{\partial}_{+}\phi \,\sigma _{\alpha \dot{\beta}
}^\mu \bar{\varepsilon}^{\prime \dot{\beta}}+\tilde{\partial}_{+}\tilde{
\partial}_{-}V_{+}\varepsilon _\alpha ,\,\,\delta V_{+}=\bar{\varepsilon}_{
\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}\alpha }\partial ^\mu \tilde{
\partial}_{+}\psi _\alpha \label{trone} \\
\delta \bar{\phi} = -\bar{\psi}_{\dot{\alpha}}\bar{\varepsilon}^{\prime
\dot{\alpha}},\,\,\delta \bar{\psi}_{\dot{\alpha}}=\varepsilon ^{\prime
\beta }\sigma _{\beta \dot{\alpha}}^\mu \partial _\mu \tilde{\partial}_{+}
\bar{\phi}\,+\tilde{\partial}_{+}\tilde{\partial}_{-}\bar{V}_{+}\bar{
\varepsilon}_{\dot{\alpha}},\,\,\delta \bar{V}_{+}=\partial ^\mu \tilde{
\partial}_{+}\bar{\psi}_{\dot{\alpha}}\,\bar{\sigma}_\mu ^{\dot{\alpha}\beta
}\varepsilon _\beta \nonumber\end{aligned}$$
The fermionic parameters $\varepsilon ^{\prime },\varepsilon $ mix the pairs ($\phi ,\psi )$ and ($\psi ,V_{+})$ respectively. We will build Lagrangians that are invariant under this transformation for arbitrary global fermionic parameters $\varepsilon ^{\prime },\varepsilon .$ However for the closure of the algebra we find that $\varepsilon ^{\prime },\varepsilon $ must be related as given below.
Taking hermitian conjugation (bar) turns the index $\alpha $ into a dotted index $\dot{\alpha}$, and also interchanges the order of anticommuting variables. Applying $C=i\sigma _2$ raises or lowers the index. We have used it in the following definitions $$\begin{aligned}
\sigma _{\alpha \dot{\beta}}^\mu &\equiv &(1,\vec{\sigma}),\quad \left( \bar{
\sigma}^\mu {}\right) ^{\dot{\alpha}\beta }\equiv C(1,\vec{\sigma}
^{*})C^T=\left( 1,-\vec{\sigma}\right) ,\quad \\
\left( \sigma ^\mu \right) ^{\dagger } &=&\sigma ^\mu ,\quad \left( \bar{
\sigma}^\mu \right) ^{\dagger }=\bar{\sigma}^\mu ,\quad \sigma ^\mu \bar{
\sigma}^\nu +\sigma ^\nu \bar{\sigma}^\mu =2\eta ^{\mu \nu }.\end{aligned}$$ Since we have anticommuting variables, the following rules apply $$\varepsilon ^{\prime \alpha }\psi _\alpha =\varepsilon _\beta ^{\prime
}C^{\beta \alpha }\psi _\alpha =\psi _\alpha C^{\alpha \beta }\varepsilon
_\beta ^{\prime }=\psi ^\beta \varepsilon _\beta ^{\prime }=-\psi _\alpha
\varepsilon ^{\prime \alpha }.$$ Then the transformation rules are consistent with hermitian conjugation.
By applying two infinitesimal transformations (\[trone\]) and antisymmetrizing $\left[ \delta _1,\delta _2\right] \phi $, $\left[ \delta
_1,\delta _2\right] \psi _\alpha ,$ $\left[ \delta _1,\delta _2\right] V_{+}$ we find the closure of the algebra by demanding consistency with eq.(\[newsusy\]) $$\begin{aligned}
\left[ \delta _1,\delta _2\right] \phi &=&-\varepsilon _2^{\prime \alpha
}\left( \delta _1\psi _\alpha \right) -\left( 1\longleftrightarrow 2\right)
\nonumber \\
&=&-\varepsilon _2^{\prime \alpha }\left( \partial _\mu \tilde{\partial}
_{+}\phi \,\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}_1^{\prime \dot{
\beta}}+(\tilde{\partial}_{+}\tilde{\partial}_{-}V_{+})\varepsilon _{1\alpha
}\right) -\left( 1\longleftrightarrow 2\right) \nonumber \\
&=&\left( \varepsilon _1^{\prime \alpha }\sigma _{\alpha \dot{\beta}}^\mu
\bar{\varepsilon}_2^{\prime \dot{\beta}}-\varepsilon _2^{\prime \alpha
}\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}_1^{\prime \dot{\beta}
}\right) \partial _\mu \tilde{\partial}_{+}\phi \, \label{phi}\end{aligned}$$ The $V_{+}$ term drops out provided we take $$\varepsilon _\alpha ^{\prime }=A\varepsilon _\alpha \label{phase}$$ where $A$ is any complex number. Similarly $$\begin{aligned}
\left[ \delta _1,\delta _2\right] V_{+} &=&\bar{\varepsilon}_{2\dot{\beta}}
\bar{\sigma}_\mu ^{\dot{\beta}\alpha }\partial ^\mu \tilde{\partial}
_{+}\left( \partial _\nu \tilde{\partial}_{+}\phi \,\sigma _{\alpha \dot{
\gamma}}^\nu \bar{\varepsilon}_1^{\prime \dot{\gamma}}+\tilde{\partial}_{+}
\tilde{\partial}_{-}V_{+}\varepsilon _{1\alpha }\right) -\left(
1\longleftrightarrow 2\right) \nonumber \\
&=&\bar{\varepsilon}_{2\dot{\beta}}\bar{\varepsilon}_1^{\prime \dot{\beta}
}\partial ^\mu \partial _\mu \tilde{\partial}_{+}^2\phi +\bar{\varepsilon}_{2
\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}\alpha }\varepsilon _{1\alpha }\,
\tilde{\partial}_{-}\tilde{\partial}_{+}^2\partial ^\mu V_{+}-\left(
1\longleftrightarrow 2\right) \nonumber \\
&=&\left( \bar{\varepsilon}_{2\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}
\alpha }\varepsilon _{1\alpha }\,-\bar{\varepsilon}_{1\dot{\beta}}\bar{\sigma
}_\mu ^{\dot{\beta}\alpha }\varepsilon _{2\alpha }\,\right) \tilde{\partial}
_{-}\tilde{\partial}_{+}^2\partial ^\mu V_{+}\end{aligned}$$ Notice that $$\begin{aligned}
\varepsilon _2^{\prime \alpha }\sigma _{\alpha \dot{\beta}}^\mu \bar{
\varepsilon}_1^{\prime \dot{\beta}} &=&-\bar{\varepsilon}_1^{\prime \dot{
\beta}}\sigma _{\dot{\beta}\alpha }^{T\mu }\varepsilon _2^{\prime \alpha }=-
\bar{\varepsilon}_{1\dot{\gamma}}^{\prime }C^{\dot{\gamma}\dot{\beta}}\sigma
_{\dot{\beta}\alpha }^{T\mu }\varepsilon _{2\delta }^{\prime }C^{\delta
\alpha } \nonumber \\
&=&-\bar{\varepsilon}_{1\dot{\gamma}}^{\prime }\left( C\sigma ^{T\mu
}C^T\right) ^{\dot{\gamma}\delta }\varepsilon _{2\delta }^{\prime }=-\bar{
\varepsilon}_{1\dot{\gamma}}^{\prime }\left( \bar{\sigma}^\mu \right) ^{\dot{
\gamma}\delta }\varepsilon _{2\delta }^{\prime }\end{aligned}$$ Therefore, if we use (\[phase\]) we obtain $$\varepsilon _2^{\prime \alpha }\sigma _{\alpha \dot{\beta}}^\mu \bar{
\varepsilon}_1^{\prime \dot{\beta}}=-\bar{\varepsilon}_{1\dot{\gamma}}\left(
\bar{\sigma}^\mu \right) ^{\dot{\gamma}\delta }\varepsilon _{2\delta
}\,\,\left| A\right| ^2$$ which gives the form $$\left[ \delta _1,\delta _2\right] V_{+}=\left( \varepsilon _1^{\prime \beta
}\sigma _{\beta \dot{\alpha}}^\mu \bar{\varepsilon}_2^{\prime \dot{\alpha}
}-\varepsilon _2^{\prime \beta }\sigma _{\beta \dot{\alpha}}^\mu \bar{
\varepsilon}_1^{\prime \dot{\alpha}}\right) \left( -\frac 1{\left| A\right|
^2}\tilde{\partial}_{+}\tilde{\partial}_{-}\right) \partial ^\mu \tilde {
\partial}_{+}V_{+}$$ In order to have the same closure property as in (\[phi\]) we must require $$\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right)
V_{+}=0.$$ The field is on shell in the extra dimensions but is[* not*]{} on shell from the point of view of 4D (for the scalar-scalar representation presented below the field is fully off-shell). We will see that this restriction follows from a Lagrangian that is fully invariant without using any mass shell conditions. Next consider the spinor $$\begin{aligned}
\left[ \delta _1,\delta _2\right] \psi _\alpha &=&\partial _\mu \tilde{
\partial}_{+}\left( \delta _1\phi \right) \,\sigma _{\alpha \dot{\beta}}^\mu
\bar{\varepsilon}_2^{\prime \dot{\beta}}+\tilde{\partial}_{+}\tilde{\partial}
_{-}\left( \delta _1V_{+}\right) \varepsilon _{2\alpha }-\left(
1\longleftrightarrow 2\right) \nonumber \\
&=&-\partial _\mu \tilde{\partial}_{+}\left( \varepsilon _1^{\prime \gamma
}\psi _\gamma \right) \,\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}
_2^{\prime \dot{\beta}}+\tilde{\partial}_{+}\tilde{\partial}_{-}\left( \bar{
\varepsilon}_{1\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}\gamma }\partial
^\mu \tilde{\partial}_{+}\psi _\gamma \right) \varepsilon _{2\alpha }-\left(
1\longleftrightarrow 2\right) \nonumber \\
&=&\frac 12\varepsilon _1^{\prime }\sigma _\nu \bar{\varepsilon}_2^{\prime
}\,\,\partial _\mu \tilde{\partial}_{+}\left( \sigma ^\mu \bar{\sigma}^\nu
\right) \psi -\frac 12\left( \bar{\varepsilon}_1\bar{\sigma}_\nu \varepsilon
_2\right) \,\tilde{\partial}_{+}^2\tilde{\partial}_{-}\partial _\mu \left(
\sigma ^\nu \bar{\sigma}^\mu \right) \psi -\left( 1\longleftrightarrow
2\right) \nonumber \\
&=&\frac 12\left( \varepsilon _1^{\prime }\sigma _\nu \bar{\varepsilon}
_2^{\prime }\,-\varepsilon _2^{\prime }\sigma _\nu \bar{\varepsilon}
_1^{\prime }\right) \left( \sigma ^\mu \bar{\sigma}^\nu +\sigma ^\nu \bar{
\sigma}^\mu \left( -\frac 1{\left| A\right| ^2}\tilde{\partial}_{+}\tilde{
\partial}_{-}\right) \right) \partial _\mu \tilde{\partial}_{+}\psi\end{aligned}$$ Note that we got an extra minus sign from the interchange of fermions from line 2 to line 3. The two terms combine to give the desired closure $$\left[ \delta _1,\delta _2\right] \psi _\alpha =\left( \varepsilon
_2^{\prime }\sigma _\nu \bar{\varepsilon}_1^{\prime }\,-\varepsilon
_1^{\prime }\sigma _\nu \bar{\varepsilon}_2^{\prime }\right) \partial ^\nu
\tilde{\partial}_{+}\psi \,_\alpha \,\,\,,$$ provided the spinor satisfies $$\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right)
\psi _\alpha =0.$$ Since both $\psi _\alpha $ and $V_{+}$ are restricted, $\phi $ must also be restricted by $$\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right)
\phi =0,$$ in order to have consistent transformation properties. As we will see in section-3, solutions of a generalized BPS condition involves a time-like condition on $v^2$. The condition $\,v^2=-\tilde{
\partial}_{+}\tilde{\partial}_{-}=\left| A\right| ^2$ is consistent with this case, and it is interesting that it followed from the closure of the superalgebra. As we will see, this is just what we need in order to connect correctly to standard physics.
So far there is no mass shell condition or equation of motion required in 4D. Therefore these are the properties of the scalar-vector representation $
(\phi ,\psi _\alpha ,V_{+})$ independent of any dynamics. Note that $\left|
A\right| $ plays the role of a label for the representation (like a Casimir eigenvalue). Next we consider a Lagrangian.
### Free supersymmetric Lagrangian
The [*free*]{} Lagrangian we propose is $\pounds _0+\pounds _0^{\prime }$ $$\begin{aligned}
\pounds _0 &=&\partial _\mu \tilde{\partial}_{+}\bar{\phi}\,\,\partial ^\mu
\tilde{\partial}_{-}\phi +\bar{\psi}_{\dot{\alpha}}\bar{\sigma}_\mu ^{\dot{
\alpha}\beta }{}\partial _\mu \tilde{\partial}_{-}\psi _\beta +\tilde{
\partial}_{-}\bar{V}_{+}\,\tilde{\partial}_{-}V_{+}, \nonumber \\
\pounds _0^{\prime } &=&\phi ^{\prime }\left( \tilde{\partial}_{+}\tilde{
\partial}_{-}+\left| A\right| ^2\right) \phi +\psi ^{\prime \alpha }\left(
\tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right) \psi
_\alpha \\
&&+V_{-}^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left|
A\right| ^2\right) V_{+}+h.c. \nonumber\end{aligned}$$ $\pounds _0$ contains the original fields $(\phi ,\psi _\alpha ,V_{+})$ and is hermitian up to total derivatives. $\pounds _0^{\prime }$ is included to impose the constraints through the equations of motion of the Lagrange multipliers $(\phi ^{\prime },\psi ^{\prime \alpha },V_{-}^{\prime })$. Their $SO(1,1)$ weights have to be the opposite of the original fields because of invariance under $SO(1,1).$ This Lagrangian is unconventional because of the number of derivatives applied on the fields, and because of the two time coordinates. According to old wisdom one should expect problems with ghosts. However one should note that in either the $x$-space or the $y$ -space there are at the most two derivatives. Also the $y$-space will be compactified in a Kaluza-Klein approach. We will see below that this structure leads to conventional physics in four dimensions without any problems, i.e. there are no ghosts in the spectrum of this model.
Each term of the Lagrangian is separately invariant under the transformations of $(\phi ,\psi _\alpha ,V_{+})$ and of $(\phi ^{\prime
},\psi ^{\prime \alpha },V_{-}^{\prime })$ (given below) for arbitrary global parameters $\varepsilon ^{\prime },\varepsilon ,$ [*without using any constraints*]{}. The constraints that are needed to close the algebra follow as an equation of motion of the auxiliary fields. Thus, before using the constraints there is an even larger supersymmetry. We demonstrate the larger symmetry by applying the supersymmetry transformations on $\pounds _0$ $$\begin{aligned}
\delta \pounds _0 &=&\left( \partial _\mu \tilde{\partial}_{+}\delta \bar{
\phi}\right) \,\partial ^\mu \tilde{\partial}_{-}\phi +\bar{\psi}_{\dot{
\alpha}}\bar{\sigma}_\mu ^{\dot{\alpha}\beta }{}\partial ^\mu \tilde{\partial
}_{-}\left( \delta \psi _\beta \right) +\left( \tilde{\partial}_{-}\delta
\bar{V}_{+}\right) \,\tilde{\partial}_{-}V_{+} \nonumber \\
&&+\partial _\mu \tilde{\partial}_{+}\bar{\phi}\,\,\left( \partial ^\mu
\tilde{\partial}_{-}\delta \phi \right) +\delta \bar{\psi}_{\dot{\alpha}}
\bar{\sigma}_\mu ^{\dot{\alpha}\beta }\partial ^\mu \tilde{\partial}_{-}\psi
_\beta +\tilde{\partial}_{-}\bar{V}_{+}\tilde{\partial}_{-}\left( \delta
V_{+}\right)\end{aligned}$$ Substituting from (\[trone\]) we have for the first line $$\begin{aligned}
&&-\partial _\mu \tilde{\partial}_{+}\bar{\psi}_{\dot{\alpha}}\bar{
\varepsilon}^{\prime \dot{\alpha}}\,\,\partial ^\mu \tilde{\partial}_{-}\phi
+\bar{\psi}_{\dot{\alpha}}\bar{\sigma}_\mu ^{\dot{\alpha}\beta }{}\partial
^\mu \tilde{\partial}_{-}\left( \partial _\nu \tilde{\partial}_{+}\phi
\,\sigma _{\beta \gamma }^\nu \bar{\varepsilon}^{\prime \dot{\gamma}}+\tilde{
\partial}_{+}\tilde{\partial}_{-}V_{+}\varepsilon _\beta \right) \nonumber
\\
&&+\tilde{\partial}_{-}\left( \partial ^\mu \tilde{\partial}_{+}\bar{\psi}_{
\dot{\alpha}}\,\bar{\sigma}_\mu ^{\dot{\alpha}\beta }\varepsilon _\beta
\right) \tilde{\partial}_{-}V_{+}\,\,,\end{aligned}$$ which is a total derivative (without using the constraints) $$\begin{aligned}
&&\partial ^\mu \left( \bar{\psi}_{\dot{\alpha}}\bar{\varepsilon}^{\prime
\dot{\alpha}}\,\tilde{\partial}_{+}\tilde{\partial}_{-}\partial _\mu \phi
\,\,+\bar{\psi}_{\dot{\alpha}}\,\bar{\sigma}_\mu ^{\dot{\alpha}\beta
}\varepsilon _\beta \,\,\tilde{\partial}_{+}\tilde{\partial}
_{-}^2V_{+}\,\right) \nonumber \\
&&-\tilde{\partial}_{+}\left( \partial _\mu \bar{\psi}_{\dot{\alpha}}\bar{
\varepsilon}^{\prime \dot{\alpha}}\,\partial ^\mu \tilde{\partial}_{-}\phi
+\partial ^\mu \bar{\psi}_{\dot{\alpha}}\,\bar{\sigma}_\mu ^{\dot{\alpha}
\beta }\varepsilon _\beta \,\,\tilde{\partial}_{-}^2V_{+}\right) \\
&&+\tilde{\partial}_{-}\left( \tilde{\partial}_{-}V_{+}\,\,\tilde{\partial}
_{+}\partial ^\mu \bar{\psi}_{\dot{\alpha}}\,\bar{\sigma}_\mu ^{\dot{\alpha}
\beta }\varepsilon _\beta \right) . \nonumber\end{aligned}$$ Similarly, the second line gives $$\begin{aligned}
&&\partial _\mu \tilde{\partial}_{+}\bar{\phi}\,\,\partial ^\mu \tilde{
\partial}_{-}\left( -\varepsilon ^{^{\prime }\alpha }\psi _\alpha \right) +
\tilde{\partial}_{-}\bar{V}_{+}\,\left( \bar{\varepsilon}_{\dot{\beta}}\bar{
\sigma}_\mu ^{\dot{\beta}\alpha }\partial ^\mu \tilde{\partial}_{+}\tilde{
\partial}_{-}\psi _\alpha \right) \nonumber \\
&&\,\,+\left( \varepsilon ^{\prime \gamma }\sigma _{\gamma \dot{\alpha}}^\nu
\partial _\nu \tilde{\partial}_{+}\bar{\phi}+\tilde{\partial}_{+}\tilde{
\partial}_{-}\bar{V}_{+}\,\bar{\varepsilon}_{\dot{\alpha}}\right) \,\bar{
\sigma}_\mu ^{\dot{\alpha}\beta }\partial ^\mu \tilde{\partial}_{-}\psi
_\beta\end{aligned}$$ which is also a total derivative $$\partial ^\mu \left( \partial ^\nu \tilde{\partial}_{+}\bar{\phi}
\,\,\varepsilon ^{\prime }\sigma _{\nu \mu }\tilde{\partial}_{-}\psi
\,\,\right) +\tilde{\partial}_{+}\left( \tilde{\partial}_{-}\bar{V}_{+}\,
\bar{\varepsilon}_{\dot{\alpha}}\bar{\sigma}_\mu ^{\dot{\alpha}\beta
}\partial ^\mu \tilde{\partial}_{-}\psi _\beta \right) .$$
Now, we turn to $\pounds _0^{\prime }.$ Its variation under the supertransformation gives $$\begin{aligned}
\delta \pounds _0^{\prime } &=&\delta \phi ^{\prime }\left( \tilde{\partial}
_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right) \phi +\delta \psi
^{\prime \alpha }\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left|
A\right| ^2\right) \psi _\alpha +\delta V_{-}^{\prime }\left( \tilde{\partial
}_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right) V_{+} \nonumber \\
&&+\phi ^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left|
A\right| ^2\right) \left( -\varepsilon ^{\prime \alpha }\psi _\alpha \right)
+V_{-}^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left|
A\right| ^2\right) \left( \bar{\varepsilon}_{\dot{\beta}}\bar{\sigma}_\mu ^{
\dot{\beta}\alpha }\partial ^\mu \tilde{\partial}_{+}\psi _\alpha \right)
\nonumber \\
&&+\psi ^{\prime \alpha }\left( \tilde{\partial}_{+}\tilde{\partial}
_{-}+\left| A\right| ^2\right) \left( \partial _\mu \tilde{\partial}_{+}\phi
\,\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}^{\prime \dot{\beta}}+
\tilde{\partial}_{+}\tilde{\partial}_{-}V_{+}\varepsilon _\alpha \right)
+h.c.\end{aligned}$$ We get $\delta \pounds _0^{\prime }=$total derivative, without using the constraints, provided the Lagrange multipliers transform under supersymmetry as follows $$\delta \phi ^{\prime }=-\partial _\mu \tilde{\partial}_{+}\psi ^{\prime
\alpha }\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}^{\prime \dot{\beta
}},\,\,\delta \psi ^{\prime \alpha }=\phi ^{\prime }\varepsilon ^{^{\prime
}\alpha }+\partial ^\mu \tilde{\partial}_{+}V_{-}^{\prime }\bar{\varepsilon}
_{\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}\alpha },\,\,\delta
V_{-}^{\prime }=\tilde{\partial}_{+}\tilde{\partial}_{-}\psi ^{\prime \alpha
}\varepsilon _\alpha$$
So, the total free Lagrangian is supersymmetric for arbitrary $\varepsilon
,\varepsilon ^{\prime }\,$without using any constraints. This larger symmetry closes into a larger set of bosonic operators included in S-theory. We will not discuss the larger symmetry in any detail but one can see its general structure in section-3. The smaller superalgebra (\[newsusy\]) is represented correctly only after we use the constraints $$\varepsilon ^{\prime }=A\varepsilon ,\quad \tilde{\partial}_{+}\tilde{
\partial}_{-}+\left| A\right| ^2=0.$$
It is possible to write an additional piece in the free supersymmetric Lagrangian involving only the primed fields. In that case the primed fields become propagating fields instead of being Lagrange multipliers. This Lagrangian is invariant without using the constraints provided the number of derivatives $\tilde{\partial}_m\,$on the fermion $\psi ^{\prime }$ is cubic. Because of the high derivatives, and because we are interested in interpreting the primed fields as non-propagating fields, we refrain from adding this additional term in the present model.
### On mass shell degrees of freedom
We can now analyze the equations of motion and determine the content of the degrees of freedom on mass shell $$\begin{aligned}
\bar{\sigma}_\mu ^{\dot{\alpha}\beta }{}\partial _\mu \tilde{\partial}
_{-}\psi _\beta &=&-\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left|
A\right| ^2\right) \psi ^{\prime \dot{\alpha}},\quad \tilde{\partial}
_{-}^2V_{+}=-\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right|
^2\right) V_{-}^{\prime }\,\quad \\
\tilde{\partial}_{+}\tilde{\partial}_{-}\partial ^\mu \partial _\mu \phi
&=&-\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right|
^2\right) \phi ^{\prime },\quad \left( \tilde{\partial}_{+}\tilde{\partial}
_{-}+\left| A\right| ^2\right) \left[ \phi ,\psi _\alpha ,V_{+}\right] =0,\end{aligned}$$ The only solutions of the last equation are expressed as a linear combination of the following complete basis $$e^{-i\left( k_{+}y^{+}+k_{-}y^{-}\right) }\left[ \phi _k\left( x\right)
,\psi _{k\alpha }\left( x\right) ,V_{+k}\left( x\right) \right] ,\quad with
{ \thinspace \thinspace \thinspace \thinspace }k_{+}k_{-}=\left|
A\right| ^2.$$ In the other equations the primed fields must be in the same basis (to match the $y^{\pm }$ dependence). However, the operator $\tilde{\partial}_{+}
\tilde{\partial}_{-}+\left| A\right| ^2$ applied on the primed fields vanishes on this basis. Therefore, the only solutions are of the form $$\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right| ^2\right)
\left[ \phi ^{\prime },\psi _\alpha ^{\prime },V_{-}^{\prime }\right]
=0,\,\quad \left( \tilde{\partial}_{+}\tilde{\partial}_{-}+\left| A\right|
^2\right) \left[ \phi ,\psi _\alpha ,V_{+}\right] =0$$ which reduce the original equations of motion to massless field equations in 4D since $\tilde{\partial}_{+}\tilde{\partial}_{-}$ or $\tilde{\partial}_{-}$ cannot vanish on these fields, $$\partial ^\mu \partial _\mu \phi =0,\quad \bar{\sigma}_\mu ^{\dot{\alpha}
\beta }{}\partial _\mu \psi _\beta =0,\quad V_{+}=0.$$ So the modes $\phi _k\left( x\right) ,\psi _{k\alpha }\left( x\right) $ are ordinary massless bosonic and fermionic fields in 4D, while $V_{+k}\left(
x\right) =0.$ For each $k^m$ one has the degrees of freedom of a scalar multiplet of ordinary $N=1$ supersymmetry. The Hilbert space constructed with these degrees of freedom has no ghosts.
### Families and Kaluza-Klein compactification
The massless modes are labeled by the Kaluza-Klein momenta $k^m$ in 11th and 12th dimensions$.$ Let us assume that these dimensions are compactified so that the momenta are quantized as follows $$k_{\pm }=\frac{n_{\pm }}{R_{\pm }}$$ where $n_{\pm }$ are integers. Then we must choose $$\left| A\right| ^2=\frac n{R_{+}R_{-}}$$ where $n$ is a positive fixed integer, and the integers $n_{\pm }$ must take all possible values such that $$n_{+}n_{-}=n. \label{int}$$ $n$ is a label of the representation. For fixed $n$ the solutions of (\[int\]) are given as follows
$$\begin{aligned}
n &=&1:\quad \left( n_{+},n_{-}\right) =\left( 1,1\right) \nonumber \\
n &=&2:\quad \left( n_{+},n_{-}\right) =\left( 2,1\right) ,\left( 1,2\right)
\nonumber \\
n &=&3:\quad \left( n_{+},n_{-}\right) =\left( 3,1\right) ,\left( 1,3\right)
\label{fam} \\
n &=&4:\quad \left( n_{+},n_{-}\right) =\left( 4,1\right) ,\left( 1,4\right)
,\left( 2,2\right) \nonumber \\
&&etc. \nonumber\end{aligned}$$
where we have listed only the positive values, assuming a positivity condition for both $k_{\pm }.$
We see that the on mass shell physical modes correspond to free fields that satisfy the operator conditions $$p^2=0,\,\,\,\quad v^2=\frac n{R_{+}R_{-}}. \label{BPS}$$ where $n/R_{+}R_{-}$ characterizes the fixed geometry in the compactified 11th and 12th dimensions (see section-3 for an interpretation of these conditions as generalized BPS constraints). For fixed geometry there are only a finite number of solutions as determined by $n$. This could be interpreted as a “family quantum number”. We have obtained a finite number of families for fixed $n$ because $k_{\pm }$ are [*both*]{} quantized. Without such a quantization the number of solutions of $k_{+}k_{-}=\left|
A\right| ^2$ is infinite. Furthermore if $n_{\pm }$ are integers but $n$ is not an integer there are no solutions at all. With the quantization of all three integers $n, n_{\pm }$ number theory comes to the rescue to give a finite number of solutions.
The Kaluza-Klein momenta in the new SUSY multiply the usual momenta $
k_{+}p_\mu $, not add. Therefore their effect is similar to the slope parameter $\alpha ^{\prime }\,\,$of strings ( but unlike $\alpha ^{\prime },$ the factor $k_{+}$ is not necessarily a constant on all fields). For massless particles their presence does not change the mass, therefore we just get repetitions of massless particles, i.e. families.
So far the model is non-interacting. We expect that in the presence of interactions, for example gauge or gravitational interactions, the operators $p_\mu ,v_m$ would be replaced by covariant derivatives in the closure of the superalgebra. The analog of the BPS conditions (\[BPS\]) would then become Laplacians and Dirac operators in the presence of interactions. Then one may consider their solutions in the presence of non-trivial geometries in $y^m$ space (analogs of Calabi-Yau, etc., but now in a space with Minkowski signature). Also, as in sections 3 and 4, one may add the other $c$ compactified dimensions and consider $SO(c+1,1)$ instead of $SO(1,1).$ Evidently the geometry is bound to modify the number of permitted solutions interpreted as families. While there are some similarities between this embedding of families in the geometries of higher dimensions, the equations and the mechanisms are different as compared to the more familiar Kaluza-Klein mechanism. We have seen already from (\[fam\]) that there are new possibilities that had not emerged before. This unexpected wealth of possibilities may have fruitful phenomenogical applications.
Scalar-scalar representations
-----------------------------
Instead of the $SO(1,1)$ vector $V_{+}$ of the previous subsection we now take an $SO(1,1)$ scalar $f$ and consider the supersymmetry transformation rules of the fields $(\varphi ,\chi _\alpha ,f)$ that are consistent with the isometries $SO(3,1)\times SO(1,1)$
$$\begin{aligned}
\,\delta \varphi &=&-\varepsilon ^{\prime \alpha }\chi _\alpha ,\,\,\,\delta
\chi _\alpha =\partial _\mu \tilde{\partial}_{+}\varphi \sigma _{\alpha \dot{
\beta}}^\mu \bar{\varepsilon}^{\prime \dot{\beta}}+s\tilde{\partial}
_{+}f\varepsilon _\alpha ,\,\,\,\delta f=\bar{\varepsilon}_{\dot{\beta}}\bar{
\sigma}_\mu ^{\dot{\beta}\alpha }\partial ^\mu \chi _\alpha \\
\delta \bar{\varphi} &=&-\bar{\chi}_{\dot{\alpha}}\bar{\varepsilon}^{\prime
\dot{\alpha}},\,\,\,\delta \bar{\chi}_{\dot{\alpha}}=\varepsilon ^{\prime
\beta }\sigma _{\beta \dot{\alpha}}^\mu \partial _\mu \tilde{\partial}_{+}
\bar{\varphi}+s^{*}\tilde{\partial}_{+}\bar{f}\bar{\varepsilon}_{\dot{\alpha}
},\,\,\delta \bar{f}=\partial ^\mu \bar{\chi}_{\dot{\alpha}}\,\bar{\sigma}
_\mu ^{\dot{\alpha}\beta }\varepsilon _\beta \nonumber\end{aligned}$$
where $s$ is some complex number to be determined. In this case the $SO(1,1)$ weight of $f$ is $0,$ which is to be contrasted to the previous case. Note again that we have two independent parameters $\varepsilon ^{\prime
},\varepsilon .$ The Lagrangian presented below is invariant under arbitrary $\varepsilon ^{\prime },\varepsilon ,s.$ Closure of the algebra will require a relation between these parameters, but it will not require any mass shell conditions as shown below.
By applying two infinitesimal transformations (\[trone\]) and antisymmetrizing $\left[ \delta _1,\delta _2\right] \varphi $, $\left[
\delta _1,\delta _2\right] \chi _\alpha ,$ $\left[ \delta _1,\delta
_2\right] f$ we find the closure of the algebra consistent with eq.(\[newsusy\]) $$\begin{aligned}
\left[ \delta _1,\delta _2\right] \varphi &=&-\varepsilon _2^{\prime \alpha
}\left( \delta _1\chi _\alpha \right) -\left( 1\longleftrightarrow 2\right)
\nonumber \\
&=&-\varepsilon _2^{\prime \alpha }\left( \partial _\mu \tilde{\partial}
_{+}\varphi \,\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}_1^{\prime
\dot{\beta}}+s\tilde{\partial}_{+}f\varepsilon _{1\alpha }\right) -\left(
1\longleftrightarrow 2\right) \\
&=&\left( \varepsilon _1^{\prime \alpha }\sigma _{\alpha \dot{\beta}}^\mu
\bar{\varepsilon}_2^{\prime \dot{\beta}}-\varepsilon _2^{\prime \alpha
}\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}_1^{\prime \dot{\beta}
}\right) \partial _\mu \tilde{\partial}_{+}\varphi \, \nonumber\end{aligned}$$ The $f$ term drops out provided we take $$\varepsilon _\alpha ^{\prime }=A\varepsilon _\alpha$$ where $A$ is any complex number. Similarly $$\begin{aligned}
\left[ \delta _1,\delta _2\right] f &=&\bar{\varepsilon}_{2\dot{\beta}}\bar{
\sigma}_\mu ^{\dot{\beta}\alpha }\partial ^\mu \left( \partial _\nu \tilde{
\partial}_{+}\varphi \,\sigma _{\alpha \dot{\gamma}}^\nu \bar{\varepsilon}
_1^{\prime \dot{\gamma}}+s\tilde{\partial}_{+}f\varepsilon _{1\alpha
}\right) -\left( 1\longleftrightarrow 2\right) \nonumber \\
&=&\bar{\varepsilon}_{2\dot{\beta}}\bar{\varepsilon}_1^{\prime \dot{\beta}
}\partial ^\mu \partial _\mu \tilde{\partial}_{+}\varphi +s\bar{\varepsilon}
_{2\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}\alpha }\varepsilon _{1\alpha
}\,\tilde{\partial}_{+}\partial ^\mu f-\left( 1\longleftrightarrow 2\right)
\nonumber \\
&=&s\left( \bar{\varepsilon}_{2\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}
\alpha }\varepsilon _{1\alpha }\,-\bar{\varepsilon}_{1\dot{\beta}}\bar{\sigma
}_\mu ^{\dot{\beta}\alpha }\varepsilon _{2\alpha }\,\right) \tilde{\partial}
_{+}\partial ^\mu f \nonumber \\
&=&-\frac s{\left| A\right| ^2}\left( \varepsilon _1^{\prime \beta }\sigma
_{\beta \dot{\alpha}}^\mu \bar{\varepsilon}_2^{\prime \dot{\alpha}
}-\varepsilon _2^{\prime \beta }\sigma _{\beta \dot{\alpha}}^\mu \bar{
\varepsilon}_1^{\prime \dot{\alpha}}\right) \partial ^\mu \tilde{\partial}
_{+}f\end{aligned}$$ In order to have the same closure property as in (\[phi\]) we must require $$s=-\left| A\right| ^2.$$ Next consider the spinor $$\begin{aligned}
\left[ \delta _1,\delta _2\right] \chi _\alpha &=&\partial _\mu \tilde{
\partial}_{+}\left( \delta _1\varphi \right) \,\sigma _{\alpha \dot{\beta}
}^\mu \bar{\varepsilon}_2^{\prime \dot{\beta}}+s\tilde{\partial}_{+}\left(
\delta _1f\right) \varepsilon _{2\alpha }-\left( 1\longleftrightarrow
2\right) \nonumber \\
&=&-\partial _\mu \tilde{\partial}_{+}\left( \varepsilon _1^{\prime \gamma
}\chi _\gamma \right) \,\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}
_2^{\prime \dot{\beta}}+s\left( \bar{\varepsilon}_{1\dot{\beta}}\bar{\sigma}
_\mu ^{\dot{\beta}\gamma }\partial ^\mu \tilde{\partial}_{+}\chi _\gamma
\right) \varepsilon _{2\alpha }-\left( 1\longleftrightarrow 2\right)
\nonumber \\
&=&\frac 12\varepsilon _1^{\prime }\sigma _\nu \bar{\varepsilon}_2^{\prime
}\,\,\partial _\mu \tilde{\partial}_{+}\left( \sigma ^\mu \bar{\sigma}^\nu
\right) \chi -\frac s2\left( \bar{\varepsilon}_1\bar{\sigma}_\nu \varepsilon
_2\right) \,\tilde{\partial}_{+}\partial _\mu \left( \sigma ^\nu \bar{\sigma}
^\mu \right) \chi -\left( 1\longleftrightarrow 2\right) \nonumber \\
&=&\frac 12\left( \varepsilon _1^{\prime }\sigma _\nu \bar{\varepsilon}
_2^{\prime }\,-\varepsilon _2^{\prime }\sigma _\nu \bar{\varepsilon}
_1^{\prime }\right) \left( \sigma ^\mu \bar{\sigma}^\nu +\sigma ^\nu \bar{
\sigma}^\mu \left( -\frac s{\left| A\right| ^2}\right) \right) \partial _\mu
\tilde{\partial}_{+}\chi\end{aligned}$$ The two terms combine to give the desired closure provided we use again $
s=-\left| A\right| ^2.$ $$\left[ \delta _1,\delta _2\right] \chi _\alpha =\left( \varepsilon
_2^{\prime }\sigma _\nu \bar{\varepsilon}_1^{\prime }\,-\varepsilon
_1^{\prime }\sigma _\nu \bar{\varepsilon}_2^{\prime }\right) \partial ^\nu
\tilde{\partial}_{+}\chi \,_\alpha \,\,\,.$$ In this version we did not need to impose any mass shell constraints in order to close the algebra.
### Free supersymmetric Lagrangian
The [*free*]{} Lagrangian we start with is $\pounds _1$ $$\pounds _1=\partial _\mu \tilde{\partial}_{+}\bar{\varphi}\,\,\partial ^\mu
\tilde{\partial}_{-}\varphi +\bar{\chi}_{\dot{\alpha}}\bar{\sigma}_\mu ^{
\dot{\alpha}\beta }{}\partial _\mu \tilde{\partial}_{-}\chi _\beta -s\tilde{
\partial}_{+}\bar{f}\,\tilde{\partial}_{-}f$$ Applying the supersymmetry transformations on $\pounds _0$ we have $$\delta \pounds _1=total\,\,\,derivative, \nonumber$$ for any $\varepsilon ^{\prime },\varepsilon ,s.$ The equations of motion that follow from this free Lagrangian require $v^2f\equiv -\tilde{\partial}
_{+}\tilde{\partial}_{-}f=0$ and $p^2v^2=0$ on both $\varphi ,\chi .$ There are two classes of solutions $$\begin{aligned}
(i) &:&p^2=0,\quad \quad v^2\neq 0, \\
(ii) &:&p^2\neq 0,\quad \quad v^2=0. \nonumber\end{aligned}$$ Neither class is physically satisfactory. In class (i) the solution for $f$ is zero while the other fields are massless. This seems fine, but since $v^2$ is not determined there are an infinite number of massless families from the point of view of 4D. In class (ii) $p^2$ is not required to be on shell by the equations of motion. To avoid these problems we add an additional part to the free Lagrangian to enforce the constraint $v^2=M^2\geq 0$ via Lagrange multipliers $$\pounds _{12}=\varphi ^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}
_{-}+M^2\right) \varphi +\chi ^{\prime \alpha }\left( \tilde{\partial}_{+}
\tilde{\partial}_{-}+M^2\right) \chi _\alpha +f^{\prime }\left( \tilde{
\partial}_{+}\tilde{\partial}_{-}+M^2\right) f+h.c$$ Now, we turn to check the invariance of $\pounds _{12}^{\prime }.$ Its variation under supertransformation gives $$\begin{aligned}
\delta \pounds _{12} &=&\delta \varphi ^{\prime }\left( \tilde{\partial}_{+}
\tilde{\partial}_{-}+M^2\right) \varphi +\delta \chi ^{\prime \alpha }\left(
\tilde{\partial}_{+}\tilde{\partial}_{-}+M^2\right) \chi _\alpha +\delta
f^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+M^2\right) f
\nonumber \\
&&+\varphi ^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}
_{-}+M^2\right) \left( -\varepsilon ^{\prime \alpha }\chi _\alpha \right)
+f^{\prime }\left( \tilde{\partial}_{+}\tilde{\partial}_{-}+M^2\right)
\left( \bar{\varepsilon}_{\dot{\beta}}\bar{\sigma}_\mu ^{\dot{\beta}\alpha
}\partial ^\mu \chi _\alpha \right) \nonumber \\
&&+\chi ^{\prime \alpha }\left( \tilde{\partial}_{+}\tilde{\partial}
_{-}+M^2\right) \left( \partial _\mu \tilde{\partial}_{+}\varphi \,\sigma
_{\alpha \dot{\beta}}^\mu \bar{\varepsilon}^{\prime \dot{\beta}}+s\tilde{
\partial}_{+}f\varepsilon _\alpha \right) +h.c.\end{aligned}$$ We get $\delta \pounds _{12}^{\prime }=$total derivative, without using the constraints, provided the primed fields transform under supersymmetry as follows $$\delta \varphi ^{\prime }=-\partial _\mu \tilde{\partial}_{+}\chi ^{\prime
\alpha }\,\sigma _{\alpha \dot{\beta}}^\mu \bar{\varepsilon}^{\prime \dot{
\beta}},\quad \delta \chi ^{\prime \alpha }=\varepsilon ^{\prime \alpha
}\varphi ^{\prime }+\bar{\varepsilon}_{\dot{\beta}}\bar{\sigma}_\mu ^{\dot{
\beta}\alpha }\partial ^\mu f^{\prime },\quad \delta f^{\prime }=s\tilde{
\partial}_{+}\chi ^{\prime \alpha }\varepsilon _\alpha$$ Remarkably, this transformation closes $\left[ \delta _1,\delta _2\right]
(\varphi ^{\prime },\chi ^{\prime },f^{\prime })$ as desired without requiring any mass shell constraints. So, the total free Lagrangian is supersymmetric, and the supersymmetry algebra closes as desired, provided $
\varepsilon ^{\prime }=A\varepsilon ,$ and $s=-M^2$ as before. The free model with $$\pounds _0^{\left( 1\right) }=\pounds _1+\pounds _{12}$$ has a physically satisfactory 4D mass spectrum. Class (ii) is completely eliminated while in class (i) there are a finite number of families as in the scalar-vector model of the previous subsection provided $$M^2=\frac n{R_{+}R_{-}},$$ and $n$ is a positive integer.
### scalar-scalar hyper-multiplet
We could have stopped here, but we also wish to build more models by exploring the possibility of adding a supersymmetric free Lagrangian involving only the primed fields. There is one that satisfies $\delta
\pounds _2=$total derivative: $$\pounds _2=\varphi ^{\prime }\bar{\varphi}^{\prime }+\chi ^{\prime \alpha
}\sigma _{\alpha \dot{\beta}}^\mu \partial _\mu \tilde{\partial}_{+}\bar{\chi
}^{\prime \dot{\beta}}-\frac 1s\partial _\mu f^{\prime }\partial ^\mu \bar{f}
^{\prime }.$$ By itself the spectrum of this Lagrangian has some of the problems of $
\pounds _1.$ However, when added to the previous terms it becomes interesting. Each term in the following total Lagrangian is supersymmetric separately $$\pounds _0^{(2)}=\pounds _1+\gamma \pounds _{12}+\pounds _2.$$ Note that there is an additional parameter $\gamma $ that cannot be absorbed into normalizations of the fields. In this model there are two scalar-scalar representations that are coupled to each other in a supersymmetric invariant way. The significance of this coupling is that now there are two propagating fermions $\chi ^\alpha ,\chi ^{\prime \alpha }$ both of which are left handed $SO(3,1)$ spinors $\left( 1/2,0\right) $ but they have opposite $
SO(1,1)$ chiralities (or weights $\pm 1/2$). Together they are equivalent to a full Dirac spinor of $SO(1,1)$ as well as of $SO(3,1)$, and their mixing term in $\gamma \pounds _{12}$ is the analog of a fermion mass term with mass $m\sim \gamma (\tilde{\partial}_{+}\tilde{\partial}_{-}+M^2)$ from the point of view of 4D. This interpretation is better understood by analyzing the coupled equations. One now finds that $\varphi ^{\prime },f$ are completely solved in terms of the other fields and the remaining two complex bosons and two chiral fermions form a massive hyper-multiplet of ordinary supersymmetry in 4D. These remaining fields can be expanded in Kaluza-Klein modes, where the modes have quantized momenta labeled by $\left(
n_{+},n_{-}\right) $. Each mode satisfies the following mass shell condition $$\begin{aligned}
p^2 &=&\frac{\gamma ^2}{k^2}\left( k^2-M^2\right) ^2 \\
&=&\frac{\gamma ^2}{R_{+}R_{-}}\frac{\left( n_{+}n_{-}-n\right) ^2}{
n_{+}n_{-}}\end{aligned}$$ where we have used $k^2=\frac{n_{+}n_{-}}{R_{+}R_{-}}$ and $M^2=\frac
n{R_{+}R_{-}}.$ Unlike our previous examples, here $k^2$ or the product $
n_{+}n_{-}$ is not fixed. A plot of $p^2$ versus $n_{+}n_{-}$ shows the following effects. For $n_{+}n_{-}=n$ there are massless modes, $p^2=0,$ provided $n$ is an integer, and this fixed integer (which is a label of the representation) determines the number of massless families as in (\[fam\] ). In addition, there are an infinite number of massive Kaluza-Klein modes for $n_{+}n_{-}\neq n$ such that their 4D mass gets bigger for $n_{+}n_{-}$ increasing toward +infinity as well as for $n_{+}n_{-}$ decreasing toward zero. There is a mass gap away from from zero mass, $n_{+}n_{-}=n,$ since $
p^2$ has quantized values in units of $\frac{\gamma ^2}{R_{+}R_{-}}$. These features are compatible with a physical interpretation of the spectrum in 4D. However this model is not entirely satisfactory because the spectrum of $
p^2$ contains tachyons when $n_{+}n_{-}$ is negative ( $k^2\sim $ space-like). Additional input is needed to prevent $n_{+}n_{-}$ from being negative. Perhaps interactions, or an appropriate interpretation of the extra $y$-space in terms of $p$-branes, will suggest how to impose $
k^2\geq 0.$ This issue does not arise in the other models presented in this paper because for them $k^2$ is fixed and positive.
S-theory origins and generalizations
====================================
In this section we describe the algebra (\[newsusy\]) in the context of a more general framework in order to display its connections to a secret theory behind string theory, and to provide a basis for generalizations.
The goal in S-theory [@ibstheory] is to extract information about the secret theory behind string theory by combining the representation structure of a generalized superalgebra with other information that may be available about the secret theory through some of its limits such as string theory, p-brane theory, D-branes and the likes. This strategy is similar to the one used in the 1960’s, with symmetries and current algebras on the one hand and experimental input on the other, which eventually led to the discovery of the Standard Model.
S-theory has two types of superalgebras with 32 real supergenerators and 528 real bosonic generators: the $SO(10,2)$ covariant type-A in 12 dimensions and the $SO(9,1)\times SO(2,1)$ covariant type-B in 13 dimensions. By a change of basis the same superalgebras may be rewritten in bases that display other symmetry structures. From the point of view of 10 dimensions the 32$_{A,B}$ spinors correspond to two 16-component spinors, such that for the type-A the 10D-chiralities are opposite while for type-B the 10D-chiralities are the same, as in type-A and type-B string theories. The two types may be embedded in a 13D superalgebra by considering the 64-component spinor space of $SO(11,2).$ Then two different $A,B$ projections reduce the 64-component spinor into distinguishable 32$_{A,B}$ fermions, and those pick out the sets 528$_{A,B}$ out of the $\frac
1264\times 65=78+286+1716$ bosons classified as antisymmetric tensors with 2,3,6 indices under $SO(11,2)$. The A and B types are T-dual to each other such that T-duality mixes the 13th dimension with the others[^1]. Therefore, even though there is no $SO(11,2)$ covariant formalism, thanks to T-duality of string theory we already know that there is a sense in which all 13 dimensions are connected to each other in the complete secret theory.
We remind the reader that one cannot consider more than 32 real supercharges [*in the flat limit*]{} of the secret theory. If there were more than 32, they would show up in 4D as more than $N=8$ supersymmetries, and this is not permitted by the absence of massless interacting particles with helicities higher than 2, in the flat limit. Similarly, with 32$_{A,B}$ supercharges there cannot be more than 528$_{A,B}$ bosonic generators since 528 is the number of independent components of a $32\times 32$ symmetric matrix. Special forms of these superalgebras are obtained in representations in which some of the 528$_{A,B}$ bosons or some of the 32$_{A,B}$ fermions vanish. The basic hypothesis of S-theory is that in the complete secret theory all of these operators are realized non-trivially when all of its sectors are taken into account. In the curved version of the secret theory more supercharges may exist, but they should vanish as the curvature vanishes. In considering curved spaces one is interested in what happens to the superalgebra of the 32$_{A,B}$ supercharges that survive in the flat limit. Those can close only on the same set of 528$_{A,B}$ bosons, but some of the latter, as well as some of the 32$_{A,B}$ fermions, could satisfy non Abelian commutation relations depending on the nature of the curved space. As suggested in [@ibstheory] various curved space models may be described as contractions of supergroups such as $OSp(1/32),\,OSp(1/64)$ and other non-Abelian supergroups.
All sectors of the secret theory would fall into some representation of S-theory. Such sectors include well known theories such as super Yang-Mills, supergravity and superstring theory. Furthermore M- and F- theories can be viewed in the same light. This is because the type-A superalgebra contains the superalgebra of 11D M-theory [@wittenetc], while the type-B superalgebra contains the superalgebra of 12D F-theory [@vafa], so these theories could be embedded in a larger theory in 12D and 13D respectively. Various new compactifications [@vafanew] of the secret theory also seem to be consistent with the overall 12D or 13D algebraic structure of S-theory (Abelian and non-Abelian). Constructing simple explicit models that provide representations of the generalized superalgebra of S-theory is likely to shed more light on the dynamical structure of the secret theory behind string theory. Section-2 is a small step in this direction and it should provide an example of the idea expressed in this paragraph.
Type-A
------
Starting with the type-A superalgebra that contains a 2-brane and a self-dual 6-brane in 12D [^2] $$\begin{aligned}
&&\left\{ {Q_\alpha ,Q_\beta }\right\} =\left( S_A\right) _{\alpha \beta }
\nonumber \\
&&S_A=\frac{1+\gamma _{13}}2C\left( \gamma ^{M_1M_2}\,\,Z_{M_1M_2}\,+\gamma
^{M_1\cdots M_6}\,\,\,Z_{M_1\cdots M_6}^{+}\right) \label{typea}\end{aligned}$$ and then reducing to 4 dimensions, one obtains the generalized $N=8$ superalgebra in 4D in a particular basis, extended with all possible 528 bosonic generators [@ibsentropy] $$\begin{aligned}
\left\{ Q_{\alpha a},Q_{\beta b}\right\} &=&\left( i\sigma _2\right)
_{\alpha \beta }\,z_{ab}+\left( i\sigma _2\vec{\sigma}\right) _{\alpha \beta
}\,\cdot \vec{F}_{ab} \nonumber \\
\left\{ \bar{Q}_{\dot{\alpha}\dot{a}},\bar{Q}_{\dot{\beta}\dot{b}}\right\}
&=&\left( i\sigma _2\right) _{\dot{\alpha}\dot{\beta}}\,z_{\dot{a}\dot{b}
}^{*}+\left( i\sigma _2\vec{\sigma}\right) _{\dot{\alpha}\dot{\beta}}\,\cdot
\vec{F}_{\dot{a}\dot{b}}^{*} \label{4d} \\
\left\{ Q_{\alpha a},\bar{Q}_{\dot{\beta}\dot{b}}\right\} &=&\sigma
_{\alpha \dot{\beta}}^\mu \,\left( \gamma _{a\dot{b}}^m\,P_{\mu m}+\gamma _{a
\dot{b}}^{m_1m_2m_3}A_{\mu m_1m_2m_3}\right) \nonumber\end{aligned}$$ where $\mu ,m$ are Lorentz indices for $SO(3,1),\,SO(c+1,1)$ respectively, $
c=6$ is the number of compactified string dimensions, and the extra $(1,1)$ correspond to the 11th and 12th dimensions. The pair of spinor indices $
\alpha a,\dot{\alpha}\dot{a}$ are Weyl spinor indices for the spacetime $
SO(3,1)\,$and internal $SO(c+1,1)$ groups, such that the Weyl projection is simultaneously left-handed or simultaneously right handed for both indices (because of the 12D Weyl projection $1+\gamma _{13}$). The ordinary $N=8$ supersymmetry with all of its Lorentz scalar central extensions is obtained for $c=6$ by keeping only $z_{ab},z_{\dot{a}\dot{b}}^{*},P_{\mu 0^{\prime }},
$ and setting the remaining Lorentz non-scalar operators to zero. In that sector one may transform to a basis with an $SU(8)$ symmetry, such that the momentum $P_{\mu 0^{\prime }}$ is a singlet under the $SU(8)$. This $SU(8)$ is the maximal compact group of $E_{7,7}$ of U-duality [@julia]. The isometry group $
SO(c+1,1)=SO(7,1)$ is not in this $SU(8)$ or even in the $E_{7,7}$ because $
P_{\mu 0^{\prime }}$ is not a singlet under $SO(c+1,1).$ The web of these symmetries is described further in [@ibjapan][@ibstheory] and it has been used to explain how the black hole entropy in 4D and 5D contains information up to 12 (or 13) hidden dimensions [@ibsentropy].
The form of the superalgebra given above may be taken with other values of $
c $, as we will do below in order to study simpler systems with fewer supersymmetries. In particular $c=0,1$ contains $N=1,2$ supersymmetry.
some sectors
------------
S-theory suggests that the other operators beyond $z_{ab},z_{\dot{a}\dot{b}
}^{*},P_{\mu 0}$ (i.e. the Lorentz non-scalars) also play a role in the secret theory. Hence we are interested in exploring models that provide representations of the more general algebra even if they correspond to a simplified sector of the algebra in which some of the operators vanish, as long as some of the novel features that relate to the Lorentz non-scalars are included. With this in mind, a greatly truncated version of the 12D type-A superalgebra (\[typea\]) was first suggested in [@ibstheory] by taking $Z_{M_1M_2}=\frac 12(p_{M_1}v_{M_2}-p_{M_2}v_{M_1})$ and $
Z_{M_1\cdots M_6}^{+}=0$ $$\left\{ Q_\alpha ,Q_\beta \right\} =\gamma _{\alpha \beta }^{MN}\,\,p_Mv_N
{\ .}\, \label{newsuper12}$$ The generic representations of this superalgebra are long supermultiplets of minimum dimension $2^{32/2}$ with $2^{15}$ bosons and $2^{15}$ fermions. Shorter multiplets also exist provided one imposes the generalized BPS constraint $$\det \left( \gamma _{\alpha \beta }^{MN}\,\,p_Mv_N\right) =\left[
p^2v^2-\left( p\cdot v\right) ^2\right] ^{16}=0.$$ There are three classes of 12D [*covariant*]{} solutions of the BPS constraint: (i) neither $p^2,v^2$ is zero, (ii) one of them is zero, (iii) both of them are zero. When $p^2$ or $v^2$ are non-zero their signs classify different representations. The physical signs and solutions must be imposed through the details of a physical theory. Some such input is the interpretation of the $Z_{M_1M_2},\,$ $Z_{M_1\cdots M_6}^{+}$ in terms of p-brane boundaries. Each solution is distinct in the sense that $SO(10,2)$ transformations cannot relate them, but obviously, solution (i) contains (ii) and (iii), and solution (ii) contains (iii) as special cases. Examples of physical representations and the issue of signs were discussed in section-2 (for the $N=1$ case).
For cases (i) and (ii) there are 16 zero and 16 non-zero supercharges and the minimal supermultiplet of dimension $2^{16/2}$ contains 128 bosons plus 128 fermions. This is the same set of massless states of 11D membrane theory [@bsp], 10D string theory, or 11D supergravity, but in the present case they are part of the spectrum of a 12D secret theory that contains these theories. Considering the low energy limit in a field theory context, this supergravity multiplet would be realized on bi-local fields $\Phi (x^M,y^M)$ on which $p_M,v_M$ act as derivatives $i\partial /\partial x^M,i\partial
/\partial y^M$ respectively. When the BPS constraints are satisfied with $$v^2=time-like,\quad v\cdot p=0 \label{timelike}$$ and $p^2=0$ on shell, these fields are directly connected to 11D supergravity fields by a Kaluza-Klein reduction in the $y$-space and keeping only one eigenvalue of $v_M$. Hence the unreduced theory must be the long sought $SO(10,2)$ supergravity, as suggested in [@ibstheory]. The non-locality is a remnant of the extended objects that are needed to realize this type of superalgebra.
Similarly, for case (iii) there are 24 zero and 8 non-zero supercharges and the minimal supermultiplet has dimension $2^{8/2}.$ So the Yang-Mills super multiplet provides a basis for realizing the superalgebra as the $(10,2)$ extension of 10D super Yang-Mills theory. However, such a field theory may be realized co-covariantlyvariantly provided one uses bi-local fields that satisfy the constraints $$v^2=0,\quad p\cdot v=0 \label{lightlike}$$ and take $p^2=0$ on shell.
The superalgebra (\[newsuper12\]) suggested in [@ibstheory] later found realizations as the supersymmetry algebra for a series of intriguing models: a 12D Yang-Mills theory [@sezgin], a 12D heterotic $(2,1)$ -string [@martinec], a covariant matrix model [@periwal] for a possible large-N matrix description of M-theory [@susskind]. These models are incomplete from the point of view of representations of the superalgebra (\[newsuper12\]) and S-theory. As suggested in [@ibsentropy] to complete the representation space one must take all eigenvalues of $p_M,v_M$ that are consistent with a given solution of the constraints (i,ii,iii), rather than taking one of them as a [*constant*]{} light-like vector. This requires bi-local fields, as in section-2. Bi-local fields, that include all the Kaluza-Klein modes consistent with a solution of the BPS constraints, contain a finite or an infinite number of eigenvalues of $v_M$ (as in section-2). Only if all such Kaluza-Klein modes are included can one maintain the 12D covariance. As currently known, the models in [@sezgin][@martinec][@periwal] correspond to keeping one Kaluza-Klein mode (the constant vector) in an expansion of another complete theory.
Another sector of the superalgebra (\[4d\]) was shown to be relevant for supersymmetric extremal black holes [@ibsentropy]. In this sector one sets to zero all bosonic operators except for $z_{ab},z_{\dot{a}\dot{b}}^{*}$ and take the special factorized form $P_{\mu m}=p_\mu v_m$. The resulting superalgebra is covariant under $SO(3,1)\times SO(c+1,1)$ which keeps track of all 12 dimensions. It was shown that the black hole entropy is invariant under this isometry and that it contains information about the hidden 12th or 13th dimension. To do so all eigenvalues of $v_m$ had to be allowed. This is the first instance in which all eigenvalues of $v_m$ showed up in a physical system, thus providing encouragement for pursuing this approach further.
In sections-1,2 of this paper we have considered a sector along the lines of the last paragraph. In this sector the discussion is simpler, and perhaps more relevant for possible physical applications. We have also specialized to the sector of zero central charges since they may be included in later investigations. Then one has $$\left\{ Q_{\alpha a},\bar{Q}_{\dot{\beta}\dot{b}}\right\} =\sigma _{\alpha
\dot{\beta}}^\mu \,\gamma _{a\dot{b}}^m\,p_\mu v_m \label{newsuper}$$ where $\mu ,m$ are Lorentz indices for $SO(3,1),\,SO(c+1,1)$ respectively as in (\[4d\]). In the Weyl sector the gamma matrices may be represented by hermitian matrices as follows $$\sigma ^\mu \,\,p_\mu =p_0+\vec{\sigma}\cdot \vec{p},\quad \gamma
^mv_m=v_{0^{\prime }}+\vec{\gamma}\cdot \vec{v}\,\,, \label{gammas}$$ where in the time directions $\mu =0,m=0^{\prime }$ the Weyl projected gamma matrices are proportional to unity. The resulting superalgebra may also be viewed as a reduction of the12D superalgebra of (\[newsuper12\]) in which the BPS constraints are satisfied in a sector [^3] . In this way with a minimal set of operators one can still probe some novel sector of S-theory that respects the isometry $SO(3,1)\times \,SO(c+1,1)$.
Type-B sectors
--------------
Similarly, one may start from the type-B superalgebra that is covariant under $SO(9,1)\times SO(2,1)$ [@ibstheory] and rewrite it in a 4D basis by using an explicit covariance $SO(3,1)\times SO(c)\times SO(2,1),$ with $
c=6$. Then the indices on the 32 spinors are $Q_{\alpha Aa},$ with $\alpha
=1,2$, denoting an $SO(3,1)$ spinor, $A=1,2,3,4$ denoting an $SO(6)$ spinor and $a=1,2$ denoting an $SO(2,1)$ spinor. $\bar{Q}_{\dot{\alpha}\dot{A}a}$ is the hermitian conjugate of the 16 complex $Q_{\alpha Aa}.$ The 10D vector index is split into a 4D index $\mu $ and a 6D index $j$. Then the 4D, N=8 superalgebra, with 528 real bosonic generators, can be put into the following form that keeps track of all 13 dimensions labeled by $\mu
=0,1,2,3;\,\,j=1,\cdots ,6;\,m=0^{\prime },1^{\prime },2^{\prime }$ $$\begin{aligned}
\left\{ Q_{\alpha Aa},Q_{\beta Bb}\right\} &=&\left( i\sigma _2\right)
_{\alpha \beta }\,\left[ \gamma _{AB}^j\left( i\tau _2\tau ^m\right)
_{ab}\left( P_{jm}+iX_{jm}\right) +\gamma _{AB}^{ijk}\left( i\tau _2\right)
_{ab}\,Y_{ijk}\right] \nonumber \\
&&+\left( i\sigma _2\vec{\sigma}\right) _{\alpha \beta }\left[ \gamma
_{AB}^i\left( i\tau _2\right) _{ab}\vec{Y}_i+\gamma _{AB}^{ijk}\left( i\tau
_2\tau _m\right) _{ab}\vec{X}_{ijk}^m\right] \nonumber \\
\left\{ \bar{Q}_{\dot{\alpha}\dot{A}\dot{a}},\bar{Q}_{\dot{\beta}\dot{B}\dot{
b}}\right\} &=&\left( i\sigma _2\right) _{\dot{\alpha}\dot{\beta}}\,\left[
\gamma _{\dot{A}\dot{B}}^j\left( i\tau _2\tau ^m\right) _{ab}\left(
P_{jm}-iX_{jm}\right) +\gamma _{\dot{A}\dot{B}}^{ijk}\left( i\tau _2\right)
_{ab}\,Y_{ijk}^{*}\right] \nonumber \\
&&+\left( i\sigma _2\vec{\sigma}\right) _{\alpha \beta }\left[ \gamma
_{AB}^i\left( i\tau _2\right) _{ab}\vec{Y}_i+\gamma _{AB}^{ijk}\left( i\tau
_2\tau _m\right) _{ab}\vec{X}_{ijk}^m\right] \\
\left\{ Q_{\alpha Aa},\bar{Q}_{\dot{\beta}\dot{B}\dot{b}}\right\} &=&\sigma
_{\alpha \dot{\beta}}^\mu \,\left(
\begin{array}{c}
\delta _{A\dot{B}}\,\left( i\tau _2\tau ^m\right) _{ab}P_{\mu m}+\delta _{A
\dot{B}}\,\left( i\tau _2\right) _{ab}\,y_\mu \\
+\gamma _{A\dot{B}}^{ij}\,\left( i\tau _2\right) _{ab}Y_{\mu ij}+\gamma _{A
\dot{B}}^{ij}\,\,\left( i\tau _2\tau _m\right) _{ab}\,X_{\mu ij}^m
\end{array}
\right) \nonumber\end{aligned}$$ The 528$_B$ bosons labeled by the letters $P,Y,X$ come from a relabeling of the 528$_B$ bosons that were denoted by the same letters in the $
SO(9,1)\times SO(3,1)$ covariant basis of [@ibstheory]. Following the route of reasoning that led to eq.(\[newsuper\]), a truncation of this superalgebra gives the form $$\left\{ Q_{\alpha Aa},\bar{Q}_{\dot{\beta}\dot{B}b}\right\} =\sigma _{\alpha
\dot{\beta}}^\mu \,\delta _{A\dot{B}}\left( i\tau _2\tau ^m\right)
_{ab}\,p_\mu v_m \label{newsuper3}$$ where we may use the $SO(2,1)$ gamma matrices $\tau ^m=\left( -i\tau _2,\tau
_3,-\tau _1\right) $ to get $$i\tau _2\tau ^mv_m=v_{0^{\prime }}+\tau _1v_{1^{\prime }}+\tau
_3v_{2^{\prime }}$$ which is consistent with the notation in (\[gammas\]).
Reduction to N=1
----------------
Both forms (\[newsuper\],\[newsuper3\]) become the standard $N=8$ supersymmetry if one keeps only one eigenvalue of a time-like $v^m,$ since then it is possible to use the isometry to rotate $v^m$ to the form $\gamma
^mv_m=1.$ Similarly, they reduce to standard $N=4$ supersymmetry if $v^m$ is light-like and fixed. However, if one allows all possible eigenvalues of a time-like or light-like $v^m,$ just as all possible eigenvalues of $p^\mu $ are allowed, then the representation space is much richer and includes novel sectors of $S$-theory. The presence of 12 or 13 dimensions manifests itself through the two distinct forms (\[newsuper\],\[newsuper3\]) in the corresponding representation spaces. Our purpose is to construct some simple models that provide explicit representations of this new type of supersymmetry, with all possible eigenvalues of $v_m,$ as in section-2, with the hope that such models will shed some light on the dynamics of S-theory.
To begin with, one may start the analysis by neglecting the $c$ compactified dimensions altogether, and keep only the 11th and 12th dimensions in (\[newsuper\]) or the 11th, 12th and 13th dimensions in (\[newsuper3\]). This corresponds to setting $c=0$ in eq.(\[newsuper\],\[newsuper3\]) and neglecting the spinor indices that correspond to the $c=6$ dimensions. In doing so we are really studying a much simpler system with fewer supersymmetries (as if the supersymmetry has been broken down from $N=8$ to $
N=1)$. With $N=8$ supersymmetry we must have supergravity. To avoid such complicated systems in the initial stages of this program, we prefer to start with $N=1$ supersymmetry and slowly work our way to higher $N$. Thus, in the present paper our goal is to provide some examples of representations for $N=1$. For higher $N$ one will need to deal with a more complicated set of auxiliary fields in building up the representations.
In the simplified $c=0$ case, $\gamma _{a\dot{b}}^mv_m$ in (\[newsuper\]) becomes a $1\times 1$ matrix $$\gamma _{a\dot{b}}^mv_m=v_{1^{\prime }}+v_{0^{\prime }}\equiv v_{+}$$ where $1^{\prime },0^{\prime }$ represent the 11th and 12th dimensions respectively. The indices $a,\dot{b}=1$ will be suppressed from now on, and the new superalgebra will be written in the form given in (\[newsusy\]). This connects to the standard $N=1$ supersymmetry if only one Kaluza-Klein mode of $v_m$ is kept. Similarly in the type-B superalgebra (\[newsuper3\] ) setting $c=0$ corresponds to neglecting the indices $A,\dot{B}$ and writing $$\left\{ Q_{\alpha a},\bar{Q}_{\dot{\beta}b}\right\} =\sigma _{\alpha \dot{
\beta}}^\mu \,\,p_\mu \,\,\left( v_{0^{\prime }}+\sigma _1v_{1^{\prime
}}+\sigma _3v_{2^{\prime }}\right) _{ab}$$ This connects to standard $N=2$ supersymmetry if only one Kaluza-Klein mode of $v_m$ is kept. The same form follows from the type-A superalgebra (\[newsuper\]) if one takes $c=1.$ Then $v_{2^{\prime }}$ represents one of compactified string dimensions $c$ rather than the 13th dimension. The map between these two equivalent cases corresponds to T-duality that mixes the 13th dimension with the compactified string dimensions.
superspace
==========
One may wonder whether the results in section-2 can be recast in a superfield formalism. The hope is that this would provide the calculus of representation theory for our superalgebra and help in constructing and analyzing interacting theories. Here we give a brief summary of such an attempt, which is an imitation of the ordinary superfield formulation with some new twists. However our formulation is only partially successful because the calculus of the representations turns out to be more tricky than just the superfield formulation. It also requires a non-trivial product rule for superfields which remains to be constructed.
Consider the superalgebra (\[newsuper\]) with $SO(3,1)\times SO(c+1,1)$ isometry. Introduce fermionic coordinates $\theta ^{\alpha a}$ and their hermitian conjugates $\bar{\theta}^{\dot{\alpha}\dot{a}}$classified by the isometry in one to one correspondence to the supercharges. Then the following representation of supercharges satisfy the algebra $$Q_{\alpha a}=\frac \partial {\partial \theta ^{\alpha a}}-\frac 12\sigma
_{\alpha \dot{\beta}}^\mu \gamma _{a\dot{b}}^m\bar{\theta}^{\dot{\beta}\dot{b
}}\partial _\mu \tilde{\partial}_m,\quad \bar{Q}_{\dot{\beta}\dot{b}}=\frac
\partial {\partial \bar{\theta}^{\dot{\beta}\dot{b}}}-\frac 12\theta
^{\alpha a}\sigma _{\alpha \dot{\beta}}^\mu \gamma _{a\dot{b}}^m\partial
_\mu \tilde{\partial}_m \label{charges}$$ Furthermore we may introduce covariant derivatives that antianticommute with both of these charges $$D_{\alpha a}=\frac \partial {\partial \theta ^{\alpha a}}+\frac 12\sigma
_{\alpha \dot{\beta}}^\mu \gamma _{a\dot{b}}^m\bar{\theta}^{\dot{\beta}\dot{b
}}\partial _\mu \tilde{\partial}_m,\quad \bar{D}_{\dot{\beta}\dot{b}}=\frac
\partial {\partial \bar{\theta}^{\dot{\beta}\dot{b}}}\_+\frac 12\theta
^{\alpha a}\sigma _{\alpha \dot{\beta}}^\mu \gamma _{a\dot{b}}^m\partial
_\mu \tilde{\partial}_m$$ A general superfield $\Phi (x,y,\theta ,\bar{\theta})$ is a double polynomial in powers of $\theta ,\bar{\theta}$ with coefficients that are ordinary fields that have consistent $SO(3,1)\times SO(c+1,1)$ assignments. A supersymmetry transformation of all the fields is given as $$\delta \Phi (x,y,\theta ,\bar{\theta})=\left( \varepsilon ^{\alpha
a}Q_{\alpha a}+\bar{\varepsilon}^{\alpha a}\bar{Q}_{\alpha a}\right) \Phi
(x,y,\theta ,\bar{\theta}) \label{transf}$$ The supersymmetry transformation of the components is read off by comparing the powers of $\theta ,\bar{\theta}$ on both sides.
As in usual supersymmetry, we introduce the concept of a chiral superfield defined by $$\bar{D}_{\dot{\beta}\dot{b}}\Phi ^{\left( chiral)\right) }(x,y,\theta ,\bar{
\theta})=0$$ The solution of this equation is $$\Phi ^{\left( chiral)\right) }(x,y,\theta ,\bar{\theta})=\exp \left( \frac
12\theta ^{\alpha a}\sigma _{\alpha \dot{\beta}}^\mu \gamma _{a\dot{b}}^m
\bar{\theta}^{\dot{\beta}\dot{b}}\partial _\mu \tilde{\partial}_m\right)
F\left( x,y,\theta \right)$$ where $F\left( x,y,\theta \right) $ is the general polynomial involving only $\theta .$ Note that, because of the double derivative, the exponential factor is not the translation operator on the $x,y$ coordinates.
Now, let’s specialize to the $N=1$ case, for which $a,\dot{b}=1$ and therefore this index is suppressed. The chiral superfield can have at the most two powers of $\theta .$ If one respects the $SO(1,1)$ assignments one finds, for example, the scalar-scalar chiral supermultiplet $$\Phi ^{\left( chiral)\right) }=\exp \left( \frac 12\theta ^\alpha \sigma
_{\alpha \dot{\beta}}^\mu \bar{\theta}^{\dot{\beta}}\partial _\mu \tilde{
\partial}_{+}\right) \left( \phi +\theta ^\alpha \chi _\alpha +\theta
^\alpha \theta _\alpha \tilde{\partial}_{+}f\right) (x,y).$$ The supersymmetry transformation applied as a differential operator in the form of eq.(\[transf\]) gives the transformation rules displayed in section-2 for the components $(\phi ,\chi _\alpha ,f).$ Closure of the superalgebra is guaranteed by the construction of eq.(\[charges\]). Evidently this property is automatically generalized to higher dimensions by the superfield formalism. For the purpose of defining (at least some) representations, as above, the superfield formalism given here is clearly useful.
One may think that this formulation supplies the technique for writing interactions. Unfortunately this does not work in a straightforward manner. If one takes a function of the superfield $W(F)$, e.g. a polynomial, and constructs a new chiral superfield $$\exp \left( \frac 12\theta ^{\alpha a}\sigma _{\alpha \dot{\beta}}^\mu
\gamma _{a\dot{b}}^m\bar{\theta}^{\dot{\beta}\dot{b}}\partial _\mu \tilde{
\partial}_m\right) W(F)$$ then the transformation law of the original superfields $F$ are not compatible with the transformation applied on the superfield $W(F),$ if super transformations are applied naively by $\delta =\varepsilon ^{\alpha
a}Q_{\alpha a}+\bar{\varepsilon}^{\alpha a}\bar{Q}_{\alpha a}$ as a differential operator. This is because $Q_{\alpha a}$ contains the double derivative structure $\partial _\mu \tilde{\partial}_m$ which is not distributive as a single derivative structure. That is, it does not satisfy the Leibnitz rule on naive products of the superfield. Hence representations are not combined into new irreducible ones by naive superfield manipulations. The correct combination rules remain to be discovered. This probably requires the construction of a “star product” of superfields that is compatible with the Leibnitz rule. At this stage it is not clear whether component methods or superfield methods will be more efficient in providing the techniques for constructing interactions.
In this paper we concentrated on the simplest $N=1$ superalgebra (\[newsuper\]) with only two new dimensions $y^m$ and constructed some of its representations in the context of field theory. Our purpose was to provide some concrete examples of representations and to show that they can be connected to familiar 4D physics. Surprizingly, a new mechanism for embedding a few families in higher compactified dimensions emerged. The possibility of phenomenological applications is intriguing and encouraging for pursuing further the ideas in this paper. The construction of representations of the higher dimensional cases is aided by the superfield formalism suggested here.
Acknowledgments
===============
We would like to thank I. Bakas, S. Ferrara and A. Schwimmer for discussions. The work of I. Bars was partially supported by the U.S. Department of Energy under grant number DE-FG03-84ER40168. The work of C. Kounnas is partially supported by EEC grant ERB-FMRX-CT 96/0045.
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[^1]: In the complete theory probably only the self T-dual subset are actually the same operators while the remainder are T-dual without being identical operators.
[^2]: The 12D momentum operator $P_M\gamma _{\alpha \beta }^M$ cannot appear, and $
Z_{M_1M_2}$ is not the 12D Lorentz generator. So, this algebra is not the extension of the conformal superalgebra in 12D. The $Z$’s have to do with p-brane open boundaries in flat and curved dimensions, or with wrappings of p-branes in dimensions with non-trivial topologies. The embedding of 11D in 12D with this interpretation was presented in 1995 in a conference [@ibjapan] as the first suggestion of 12 dimensions as a step beyond the 11D M-theory. Since this superalgebra is type-A, not type-B, this 12D is distinguishable than the one suggested later in F-theory [@vafa]. Also, in the type-B superalgebra, one must distinguish the explicit isometry $SO(2,1)$ that acts on 3D (including the 13th dimension) from the $SL(2)$ of U-duality that is not an explicit isometry of the superalgebra, but is used in describing a 12D F-theory.
[^3]: More generally (\[newsuper12\]) allows also the components $p_m$ and $
v_\mu .$ In their presence we must have bi-local fields, but we have specialized to ordinary local fields in 12D by working in the sector in which $p_m$ and $v_\mu $ are zero. In this way we have lost the full $
SO(10,2)$ but still have an isometry that keeps track of all 12 dimensions. Case (i) cannot be realized in this sector, but cases (ii) and (iii) remain.
|
---
abstract: 'We present reflectance measurements in the infrared region on a single crystal the rare earth scandate DyScO$_3$. Measurements performed between room temperature and 10 K allow to determine the frequency of the infrared-active phonons, never investigated experimentally, and to get information on their temperature dependence. A comparison with the phonon peak frequency resulting from *ab initio* computations is also provided. We finally report detailed data on the frequency dependence of the complex refractive index of DyScO$_3$ in the terahertz region, which is important in the analysis of terahertz measurements on thin films deposited on DyScO$_3$.'
author:
- 'L. Baldassarre, A. Perucchi'
- 'S. Lupi'
- 'P. Dore'
title: 'Far infrared properties of the rare-earth scandate DyScO$_3$'
---
epsf
Introduction
============
DyScO$_3$, together with other rare-earth scandates ($R$ScO$_3$), has recently received considerable attention, since it is considered to be among the best substrates for the epitaxial growth of high-quality ABO$_3$ perovskite-type thin films [@thin_films]. On such thin films it is possible to induce ferroelectric and multiferroic properties tailoring their lattice constants by changing $R$ in $R$ScO$_3$. For example, SrTiO$_3$ exhibits strain-induced ferroelectricity if grown on a RScO$_3$ substrate[@haeni]. Furthermore, scandates are considered to be some of the most promising candidates to substitute SiO$_2$ as gate dielectric in MOSFET, thanks to the high value of their static dielectric constant $\epsilon_0$ [@haeni; @delugas]. SrTiO$_3$/DyScO$_3$ heterostructures are also widely used for applications in the terahertz (THz) range [@kuzel].
We remark that scandates are increasingly used as substrates for film growth and that the optical investigation of a film often allows basic studies which can be difficult when only small size single crystals are available. In particular, the large and flat surface of a film permits accurate optical measurements in the far-IR and THz regions, which have an important role in studying superconducting films [@Tink]. When the radiation penetration depth is larger than the film thickness the optical response of the substrate affects the measured far-IR/THz spectrum, and the complex dielectric function $\tilde{\epsilon}(\omega)=\epsilon_{1}(\omega)+i\epsilon_{2}(\omega)$ of the film material cannot be obtained through the Kramers-Kronig (KK) transformations. In this case more elaborate procedures [@Dress; @paolod-old] must be used to extract the $\tilde{\epsilon}(\omega)$ of the film from the reflectance or transmittance data, which require the knowledge of the substrate complex refractive index $\tilde{n}(\omega)=n(\omega)+ik(\omega)$. While the far-infrared properties of common perovskites-like substrates as SrTiO$_3$ and LaAlO$_3$ are well known [@calvani; @zhang; @nkSTO], no far-IR data are available in the DyScO$_3$ case. For this system, theoretical calculations of the phonon modes have been reported [@delugas], which could be compared only with data from recent Raman investigations[@kreisel; @kreisel2].
Moreover, both infrared (IR) and Raman spectroscopy are of interest in investigating the structural properties of oxide materials with the perovskite structure, since the study of the optical phonons can provide direct information on even subtle structural distortions of the ideal perovskite structure.
We have performed reflectance measurements in the IR region on a DyScO$_3$ single crystal, at a number of temperatures in the 10-300 K range. The infrared-active phonons of DyScO$_3$ have been investigated for the first time, allowing a direct comparison with the results of the $ab$ $initio$ calculations [@delugas]. Moreover the frequency and temperature dependence of the complex refractive index $\tilde{n}(\omega)$ in the far-IR/THz region has been obtained.
Experimental details and results
================================
Reflectance measurements were performed at near-normal incidence on the 110 surface of a DyScO$_3$ (DSO) single crystal. The sample was glued with silver paint, to ensure thermal contact, on an optically black cone [@conetto] mounted on the end of a Helitran cryostat’s coldfinger. Such cone was aligned, with the use of three tilting screws, so to have the sample surface perpendicular to the incident radiation. By employing a home-built reflectivity unit, measurements were performed over a broad energy range (50$\div$12000 cm$^{-1}$) with a Michelson interferometer at a spectral resolution of 2 cm$^{-1}$.
To obtain the absolute value of the reflectivity R($\omega$) we employed the overfilling technique [@conetto]. A metallic film of Au is deposited $in$ $situ$ on the sample surface and used as reference. This allows not only to prevent thermal disalignment but also to take into account any effect of diffraction due to the sample size or diffusion if the sample surface is not *mirror-like*.
The optical reflectivity R$(\omega)$ is plotted up to 800 cm$^{-1}$ in Fig. \[rifle\] at various temperatures. Several phonon lines, which show a weak temperature dependence, can be observed in this spectral region. At higher frequencies R($\omega$), in agreement with the insulating properties of the DyScO$_3$ crystal, is nearly flat and constant at a value of about 0.1 up to the highest measured frequency (see Fig.\[rifle2\]).
=8.5cm
=8.5cm
Analysis and discussion
=======================
DyScO$_3$ has an orthorhombically-distorted perovskite structure with space group $Pnma$ with a 20-atom primitive cell. Therefore, one expects, as in the isostructural LaMnO$_3$ manganite [@iliev; @fedorov; @smirnova; @paolone], 60 phonons, among which 24 are Raman active (corresponding to the irriducible representations $7A_g$, $5B_{1g}$, $7B_{2g}$, $5B_{3g}$), 25 are infrared active ($9B_{1u}$, $7B_{2u}$, $9B_{3u}$), 8 silent ($8A_u$), and 3 acoustic ($B_{1u}$, $B_{2u}$, $B_{3u}$). By measuring the reflectance on the 110 surface, we should detect all the $B_{1u}$ modes [@kreisel; @iliev; @fedorov; @smirnova] corresponding to the dipole moment oscillating along $z$, where the orthorhombic axis $z$ is defined as $z=[001]$ (see Ref.[@kreisel] for more details), and a linear combination of the $B_{2u}$ and $B_{3u}$ corresponding to the dipole oscillating along $y$ and $x$, respectively [@fedorov].
In order to perform the KK analysis [@Dress], we first performed a Lorentz fit on R$(\omega)$ so to extrapolate its behavior as $\omega \rightarrow 0$. We have fitted the low-frequency part of R$(\omega)$ (*i.e.* below 400 cm$^{-1}$) in order to obtain a careful description in the THz region. This procedure resulted in a convincing extrapolation of the experimental data down to zero frequency and in reliable results in the far-IR once the KK transformations are performed. We remark that data have been collected up to 12000 cm$^{-1}$, so to avoid problems in the KK procedure due to the high-frequency extrapolation of the spectra. The resulting $\epsilon_{2}(\omega)$ are reported in Fig. 2 for frequencies below 600 cm$^{-1}$ at selected temperatures.
At 10 K, 19 of the expected 25 IR-active modes are clearly visible in the spectrum. As in the case of the isostructural LaMno$_3$ system, the three phonon modes (external, bending, and stretching modes) proper of the cubic perovskite structure [@Burns] are split in several phonons because of the orthorombic distortion. The low frequency phonon modes originate from the external mode, i.e. are due to vibrations of the Dy-cation sublattice with respect to the network of ScO$_6$ octahedra. The high frequency phonons originate from the stretching mode of oxygens in the ScO$_6$ octhaedra, the phonons at intermediate frequencies from the bending mode. However, due to the strong orthorhombic lattice distortions, none of the modes can be classified as purely bending or as purely stretching as these modes imply considerable changes of both the bond angles and bond lengths in the ScO$_6$ octahedra [@fedorov]. In the vibrational spectra we distinguish one phonon mode around 100 cm$^{-1}$ and one around 115 cm$^{-1}$. These frequencies are close to those predicted by *ab initio* calculations for a $B_{3u}$ and a B$_{1u}$ modes, respectively. Their dipole moment is due to small displacements of Sc-O ions $vs$ Dy that do not compensate along the $x$ direction[@delugas] while the larger vertical ($z$) displacements produce no dipole moment. The intensity of these modes can in principle be enhanced by disorder, which can also increase the static dielectric constant $\epsilon_0$.
To determine the central frequency of the phonon modes and their temperature dependence, we have fitted $\epsilon_2(\omega)$ within the Lorentz model by using: $$\tilde{\epsilon}=\epsilon_1(\omega) + i\epsilon_2(\omega) = \epsilon_\infty+\sum_j \frac{A_j{\omega_j}^2}{{\omega_j}^2-\omega^2-i\omega\gamma_j} \label{lorentz}$$ where $A_j$, $\omega_j$ and $\gamma_j$ are respectively the mode strength, the central frequency and the damping of each phonon mode $j$. The resulting curves obtained with $\epsilon_\infty\sim 4$ are in good agreement with the data, as shown for T= 10 K and 300 K in Fig.\[fig3\]. The phonon frequencies resulting from the fits are reported in Table \[table1\] and tentatively compared with the calculations of Ref.[@delugas] As our measurements were performed without polarizing the incoming beam no information about the polarization of the phonon peaks can be extracted from the experimental data. We made a tentative assignment by simply comparing the phonon frequencies obtained from the fit with the computed ones. Taking into account the complexity of this class of materials, we believe that the overall agreement between theory and experiment should be considered as reasonable.
Most of the phonons harden their central frequency with decreasing T and reduce their half-height half-width $\gamma$, as expected and in good agreement with the behavior found in Ref.[@kreisel2] at higher temperatures. A closer inspection of data shows that the broad peak, visible at 300 K just below 200 cm$^{-1}$ is due to two different phonon modes at 190 cm$^{-1}$ and at 199 cm$^{-1}$ (see Table \[table1\]). The apparent splitting of such peak is due to the softening of the more intense mode (at 190 cm$^{-1}$) to 185 cm$^{-1}$ and the simultaneous hardening of the second mode. Some other weak modes also show a light softening by decreasing T as reported in Table.\[table1\]. This softening might correspond to subtle lattice modifications (still within the $Pnma$ crystal structure) as T decreases. To the best of our knowledge, the T-behavior of the DyScO$_3$ lattice parameters has been investigated between 298 - 1273 K, that is the temperature range of interest for perovskite thin film growth[@lattice], while no data exist at lower temperatures. Therefore one cannot link unambiguously the modifications in the IR-active phonon response to a structural rearrangement.
=8.5cm
[ p[.08]{}p[.08]{}p[.08]{}p[0.08]{} p[0.095]{} ]{} $\omega_{exp}$ & $\omega_{exp}$ &$A_j$ & $\omega_{theo}$ & Mode\
(cm$^{-1}$) & (cm$^{-1}$)& & (cm$^{-1})$ & Symmetry\
300 K & 10 K & 10 K & Ref.&Ref.\
101 & 104 & 0.52& 97.9 & B$_3u$\
113& 117 & 0.96&111.0&B$_1u$\
131&135& 15.35 &129.4&B$_2u$\
160& 159 & 1.12& 173.7&B$_1u$\
190& 185 & 0.37& 193.3&B$_2u$\
199& 205&1.14 &197.0&B$_3u$\
- & - &-& 231.4&B$_3u$\
256 & 257 & 0.85 & 278.8&B$_1u$\
- & - & - &283.2&B$_2u$\
- & - & -& 285.3&B$_3u$\
295&298&2.53&293.9&B$_1u$\
306& 309 & 0.69&328.8&B$_1u$\
344&332& 0.04& 334.4&B$_3u$\
346&344& 0.14& 336.4&B$_2u$\
354&358&0.23&356.7&B$_3u$\
371&367&0.44&368.6&B$_2u$\
381&385&1.51&369.5&B$_1u$\
- & - & -& 412.9&B$_1u$\
- & - & -&419.2&B$_3u$\
434&436&0.35&435.5&B$_2u$\
- & - & -&445.3&B$_2u$\
- & - & -&478.1&B$_3u$\
484&482&0.03&484.8&B$_1u$\
501&503&0.12&509.7&B$_1u$\
539&544&0.07&532.6&B$_3u$\
570&570&0.01&-&\
\
It is important to notice that the low-frequency modes, due to vibrations of the $R$-cation sublattice with respect to the ScO$_6$ octahedral network, are those with higher intensity. This finding supports the theoretical prediction[@delugas] that in DyScO$_3$ the high value of the static dielectric constant $\epsilon_0$, i.e. $\tilde{\epsilon}(\omega\rightarrow0)$, is associated with low-frequency ionic vibrations. On the basis of the employed procedure, $\epsilon_0$ is given by $\epsilon_\infty+\sum_jA_j$ (see Eq.\[lorentz\]). Since the fitting procedure provides the $A_j$ value for each mode (as reported in Table \[table1\]), we find that the phononic contribution to $\epsilon_0$, i.e. $\sum_jA_j$ is large, of about 21, mainly due to the phonon mode at 135 cm$^{-1}$. Moreover we find a total static dielectric constant $\epsilon_0\approx 25$, as the sum of $\epsilon_\infty\sim4$ and the phononic contribution, in excellent agreement with theory[@delugas] and experiments on single crystals[@haeni].
Finally we report in Fig.\[fig4\] the frequency dependance of the complex refractive index $\tilde{n}=n+ik$ in the THz region as directly obtained from $\tilde{\epsilon}=\epsilon_{1}+i\epsilon_{2}$. Below 100 cm$^{-1}$ $n$ is nearly flat reaching a value of about 5 for $\omega \rightarrow 0$, while the vanishingly small $k$ value indicates the absence of an appreciable absorption in the same spectral region.
=8.5cm
.
The $n$ and $k$ data at 10 K have been recently employed in the analysis of the THz measurements performed on a film[@perucchi], grown on a DSO substrate[@JJ], of the BaFe$_{1.84}$Co$_{0.16}$As$_2$ compound, that belongs to the class of the new Fe-based superconductors which attracted strong attention since their recent discovery [@pnict1].
Conclusions
===========
We have presented here the first experimental data on the IR phonon spectrum of the rare earth scandate DyScO$_3$. In the temperature range between 10 and 300 K no dramatic changes occur in the phonon response indicating that the $Pnma$ structure is stable down to the lowest measured temperature. The overall agreement with the recent *ab initio* calculations is to be considered satisfactory. We finally note that the present work can provide reference data and extrinsic peak selection for future infrared investigations of thin films grown on DyScO$_3$ substrates.
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---
abstract: 'The exact nuclear time-dependent potential energy surface arises from the exact decomposition of electronic and nuclear motion, recently presented in \[A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)\]. Such time-dependent potential drives nuclear motion and fully accounts for the coupling to the electronic subsystem. We investigate the features of the potential in the context of electronic non-adiabatic processes and employ it to study the performance of the classical approximation on nuclear dynamics. We observe that the potential, after the nuclear wave-packet splits at an avoided crossing, develops dynamical steps connecting different regions, along the nuclear coordinate, in which it has the same slope as one or the other adiabatic surface. A detailed analysis of these steps is presented for systems with different non-adiabatic coupling strength. The exact factorization of the electron-nuclear wave-function is at the basis of the decomposition. In particular, the nuclear part is the true nuclear wave-function, solution of a time-dependent Schroedinger euqation and leading to the exact many-body density and current density. As a consequence, the Ehrenfest theorem can be extended to the nuclear subsystem and Hamiltonian, as discussed here with an analytical derivation and numerical results.'
author:
- Federica Agostini
- Ali Abedi
- Yasumitsu Suzuki
- 'E. K. U. Gross'
title: 'Mixed quantum-classical dynamics on the exact time-dependent potential energy surface: A fresh look at non-adiabatic processes'
---
Introduction {#sec: intro}
============
The Born-Oppenheimer (BO) [@BO], or adiabatic, treatment of the coupled motion of electrons and nuclei is among the most fundamental approximations in modern condensed-matter theory and forms the basis of our understanding of dynamical processes in molecules and solids. It offers a practical way to visualize a molecule or solid as a set of nuclei moving on a single potential energy surface (PES) generated by the electrons in a given eigenstate. However, it is based on the assumption that the electrons adjust instantaneously to adiabatic changes of the nuclear positions, and a variety of interesting phenomena in physics, chemistry and biology take place in the regime where this approximation breaks down. Prominent examples are the process of vision [@cerulloN2010; @schultenBJ2009; @ishidaJPCB2012], photo-synthesis [@tapaviczaPCCP2011; @flemingN2005], photo-voltaic processes [@rozziNC2013; @silvaNM2013; @jailaubekovNM2013], proton-transfer/hydrogen storage [@sobolewski; @varella; @hammes-schiffer; @marx] as well as phonon-induced superconductivity.
Non-adiabatic molecular processes are usually explained in terms of BOPESs and transitions between the BO electronic states. In this context, the solution of the time-dependent Schrödinger equation (TDSE) is expanded in the complete system of BO electronic states, leading to a nuclear wave-packet with contributions on several BOPESs that undergo transitions in the regions of strong non-adiabatic coupling. This approach provides a formally exact description of the complete system if all the electronic states are taken into account. However, practical applications are limited to a small number of degrees of freedom. For large systems, the only feasible way of dealing with non-adiabatic processes is the introduction of classical or semi-classical approximations for the nuclear motion, coupled, non-adiabatically, to the (quantum mechanical) electrons. Although widely investigated [@pechukas; @ehrenfest; @TSH; @kapral-ciccotti], the nature of the force driving the classical nuclei in this mixed quantum-classical treatment has not yet been fully identified.
Recently [@steps], this problem has been addressed from a novel perspective by referring to the exact representation of the full molecular wave-function [@AMG; @AMG2] as a single product of a purely nuclear wave-function and an electronic factor that parametrically depends on the nuclear coordinates. In this framework, a TDSE for the nuclear wave-function is derived, where a time-dependent potential energy surface (TDPES) and a time-dependent vector potential arise as exact concepts and provide the *driving force* for the nuclear evolution.
The present paper discusses situations where the vector potential can be set to zero by an appropriate choice of gauge, thus leaving the TDPES as the only potential responsible for the nuclear dynamics. In this case, the force on the nuclei, in a classical sense, can be obtained as the gradient of the TDPES. But, is this the true classical force on the nuclei? We will try to address this issue by employing the exact TDPES, that is known for the simple system studied here, for the propagation of classical trajectories in order to (i) examine the quality of the classical approximation for the nuclear motion and (ii) get insight into the properties of approximated classical forces for an eventual mixed quantum-classical treatment of non-adiabatic processes. Moreover, we will discuss the connections [@steps] between such novel approach, based on a *single* TDPES, and the well-established description in terms of several static (coupled) BOPESs.
The paper is organized as follows. In Section \[sec: background\], the exact factorization of the time-dependent electron-nuclear wave-function is presented and the equations that govern the evolution of the electronic and nuclear subsystems are discussed. The TDPES is investigated and analyzed in detail in Section \[sec: pes\] for systems showing different degree of non-adiabaticity. Section \[sec: dynamics\] presents some results obtained by performing classical dynamics on the exact surface and in Section \[sec: ehrenfest\] we discuss the Ehrenfest theorem in the exact factorization representation of the full wave-function. In Section \[sec: conclusion\] some concluding words are given.
Exact decomposition of the electronic and nuclear motion {#sec: background}
========================================================
In the absence of a time-dependent external field, a system of interacting electrons and nuclei is described, non-relativistically, by the Hamiltonian $$\label{eqn: hamiltonian}
\hat H = \hat T_n+\hat H_{BO},$$ where $\hat T_n$ is the nuclear kinetic energy operator and $$\label{eqn: boe}
\hat{H}_{BO}(\dulr,\dulR) = \hat{T}_e(\dulr) + \hat{W}_{ee}(\dulr) + \hat{V}_{en}(\dulr,\dulR) + \hat{W}_{nn}(\dulR),$$ is the standard BO electronic Hamiltonian. The symbols $\dulr$ and $\dulR$ are used to collectively indicate the coordinates of $N_{e}$ electrons and $N_n$ nuclei, respectively. It has been proved in [@AMG; @AMG2] that the full time-dependent electron-nuclear wave function, $\Psi(\dulr,\dulR,t)$, that is the solution of the TDSE, $$\label{eqn: tdse}
\hat H\Psi(\dulr,\dulR,t)=i\hbar\partial_t \Psi(\dulr,\dulR,t),$$ can be exactly factorized to the correlated product, $$\label{eqn: factorization}
\Psi(\dulr,\dulR,t)=\chi(\dulR,t)\Phi_\dulR(\dulr,t),$$ of the nuclear wave-function, $\chi(\dulR,t)$, and the electronic wave-function, $\Phi_\dulR(\dulr,t)$, that parametrically depends on the nuclear configuration and satisfies the partial normalization condition (PNC), $$\int d\dulr \left|\Phi_\dulR(\dulr,t)\right|^2 = 1, \quad\forall\,\,\dulR,t.$$ The PNC is an essential element of this representation. Without imposing the PNC, the full wave-function can be factorized in many different (unphysical) ways. It is the PNC that makes the factorization (\[eqn: factorization\]) unique up to within a $(\dulR,t)$-dependent gauge transformation, $$\label{eqn:
gauge}
\begin{array}{rcl}
\chi(\dulR,t)\rightarrow\tilde\chi(\dulR,t)&=&e^{-\frac{i}{\hbar}\theta(\dulR,t)}\chi(\dulR,t) \\
\Phi_\dulR(\dulr,t)\rightarrow\tilde\Phi_\dulR(\dulr,t)&=&e^{\frac{i}{\hbar}\theta(\dulR,t)}\Phi_\dulR(\dulr,t).
\end{array}$$ Another important implication of imposing the PNC is that the diagonal of the $N$-body nuclear density matrix of the complete system is equal to $|\chi(\dulR,t)|^2$.
The stationary variations [@frenkel] of the quantum mechanical action[^1] w.r.t. $\Phi_\dulR(\dulr,t)$ and $\chi(\dulR,t)$ lead to the derivation of the equations of motion $$\begin{aligned}
\left(\hat{H}_{BO}(\dulr,\dulR)+\hat U_{en}^{coup}[\Phi_\dulR,\chi]-\epsilon(\dulR,t)\right)
\Phi_{\dulR}(\dulr,t)&=&i\hbar\partial_t \Phi_{\dulR}(\dulr,t)\label{eqn: exact electronic eqn} \\
\left(\sum_{\nu=1}^{N_n} \frac{\left[-i\hbar\nabla_\nu+\bA_\nu(\dulR,t)\right]^2}{2M_\nu} + \epsilon(\dulR,t)
\right)\chi(\dulR,t)&=&i\hbar\partial_t \chi(\dulR,t). \label{eqn: exact nuclear eqn}\end{aligned}$$ Here, $\epsilon(\dulR,t)$ is the TDPES, defined as $$\label{eqn: tdpes}
\epsilon(\dulR,t)=\left\langle\Phi_\dulR(t)\right|\hat{H}_{BO}+\hat U_{en}^{coup}-i\hbar\partial_t\left|
\Phi_\dulR(t)\right\rangle_\dulr,$$ $\hat U_{en}^{coup}[\Phi_\dulR,\chi]$ is what we name “electron-nuclear coupling operator”, defined as $$\begin{aligned}
\hat U_{en}^{coup}[\Phi_\dulR,\chi]=&\sum_{\nu=1}^{N_n}\frac{1}{M_\nu}\left[
\frac{\left[-i\hbar\nabla_\nu-\bA_\nu(\dulR,t)\right]^2}{2}\right.\label{eqn: enco} \\
&\left.+\left(\frac{-i\hbar\nabla_\nu\chi}{\chi}+\bA_\nu(\dulR,t)\right)
\left(-i\hbar\nabla_\nu-\bA_{\nu}(\dulR,t)\right)\right],\nonumber\end{aligned}$$ and $\bA_{\nu}\left(\dulR,t\right)$ is the time-dependent vector potential potential, $$\label{eqn: vector potential}
\bA_{\nu}\left(\dulR,t\right) = \left\langle\Phi_\dulR(t)\right|-i\hbar\nabla_\nu\left.\Phi_\dulR(t)
\right\rangle_\dulr.$$ The symbol $\left\langle\,\,\cdot\,\,\right\rangle_\dulr$ indicates an integration over electronic coordinates only.
In Eqs. (\[eqn: exact electronic eqn\]) and (\[eqn: exact nuclear eqn\]), $\hat
U_{en}^{coup}[\Phi_\dulR,\chi]$, $\epsilon(\dulR,t)$ and $\bA_{\nu}\left(\dulR,t\right)$ mediate the coupling between the electronic and nuclear motions in a formally exact way. The electron-nuclear coupling operator, $\hat
U_{en}^{coup}[\Phi_\dulR,\chi]$, in the electronic equation (\[eqn: exact electronic eqn\]), depends on the nuclear wave-function and the first and second derivatives of the electronic wave-function with respect to the nuclear coordinates. This operator includes the coupling to the nuclear subsystem beyond the parametric dependence in the BO Hamiltonian $\hat H_{BO}(\dulr,\dulR)$. The nuclear equation (\[eqn: exact nuclear eqn\]), on the other hand, has a particularly appealing form of a Schrödinger equation that contains a time-dependent vector potential (\[eqn: vector potential\]) and a time-dependent scalar potential (\[eqn: tdpes\]) that uniquely [^2] govern the nuclear dynamics and yield the nuclear wave-function. $\chi(\dulR,t)$ is interpreted as the nuclear wave-function since it leads to an $N$-body nuclear density, $\Gamma(\dulR,t)=\vert\chi(\dulR,t)\vert^2$, and an $N$-body current density, ${\bf J}_\nu(\dulR,t)=Im(\chi^*\nabla_\nu\chi)+ \Gamma(\dulR,t){\bf A}_\nu$, which reproduce the true nuclear $N$-body density and current density obtained from the full wave-function $\Psi(\dulr,\dulR,t)$ [@AMG2]. The uniqueness of $\epsilon(\dulR,t)$ and $\bA_{\nu}(\dulR,t)$ can be straightforwardly proved by following the steps of the current-density version [@Ghosh-Dhara] of the Runge-Gross theorem [@RGT]. The scalar potential and the vector potential transform as $$\begin{aligned}
\tilde{\epsilon}(\dulR,t) &=& \epsilon(\dulR,t)+\partial_t\theta(\dulR,t)\label{eqn: transformation of epsilon} \\
\tilde{\bf A}_{\nu}(\dulR,t) &=& {\bf A}_{\nu}(\dulR,t)+\nabla_\nu\theta(\dulR,t),\label{eqn: transformation of A}\end{aligned}$$ under the gauge transformation (\[eqn: gauge\]).
Time-dependent potential energy surface {#sec: pes}
=======================================
In this work, we present a detailed study of the TDPES for strongly coupled electronic and nuclear motions. In order to obtain the TDPES, the full electron-nuclear wave-function has to be calculated. Therefore, we need to choose a system that is simple enough to allow for a numerically exact treatment and that nevertheless exhibits characteristic features associated with non-adiabatic dynamics. Here, we employ the model of Shin and Metiu [@MM], consisting of three ions and a single electron, as depicted in Fig. \[fig: metiu model\].
![Schematic representation of the model system described by the Hamiltonian (\[eqn: metiu-hamiltonian\]). $R$ and $r$ indicate the coordinates of the moving ion and electron, respectively, in one dimension. $L$ is the distance between the fixed ions.[]{data-label="fig: metiu model"}](./Figure1.pdf)
Two ions are fixed at a distance of $L=19.0$ $a_0$, the third ion and the electron are free to move in one dimension along the line joining the two fixed ions. The Hamiltonian of this system reads $$\begin{aligned}
\label{eqn: metiu-hamiltonian}
\hat{H}(r,R)= &-\frac{1}{2}\frac{\partial^2}{\partial r^2}-\frac{1}{2M}\frac{\partial^2}{\partial R^2} +
\frac{1}{\left|\frac{L}{2}
-R\right|}+\frac{1}{\left|\frac{L}{2} + R\right|}-\frac{\mathrm{erf}\left(\frac{\left|R-r\right|}{R_f}\right)}
{\left|R - r\right|}\nonumber\\
&-\frac{\mathrm{erf}\left(\frac{\left|r-\frac{L}{2}
\right|}{R_r}\right)}{\left|r-\frac{L}{2}\right|}-\frac{\mathrm{erf}\left(\frac{\left|r+\frac{L}{2}\right|}
{R_l}\right)}{\left|r+\frac{L}{2}\right|}.\end{aligned}$$ Here, the symbols $\dulr$ and $\dulR$ are replaced by $r$ and $R$, the coordinates of the electron and the movable ion measured from the center of the two fixed ions and $M=1836$ is the mass of the movable ion. The parameters $R_f$, $R_l$ and $R_r$ specify the interactions between the charged particles and can be tuned to have different couplings between the electronic and nuclear motions.
To obtain the TDPES, we first solve the TDSE (\[eqn: tdse\]) for the complete system and obtain the full wave-function, $\Psi(r,R,t)$. This is done by the numerical integration of the TDSE using SPO-technique [@spo], with the time-steps of $2.4\times10^{-3}$ $fs$ (or $0.1$ $a.u.$). The nuclear density is calculated, at each time, as the marginal probability of the configuration $\dulR$ [^3] from the full wave-function $$\left|\chi(\dulR,t)\right|^2=\int d\dulr \left|\Psi(\dulr,\dulR,t)\right|^2.$$ The phase $S(\dulR,t)$ of $\chi(\dulR,t)$ is determined by the choice of the gauge. We use the exact equality $$\label{eqn: vector-exact}
\bA_\nu(\dulR,t) = \left|\chi(\dulR,t)\right|^{-2}\mbox{Im}\int d\dulr\,\Psi^*(\dulr,\dulR,t)\nabla_\nu
\Psi(\dulr,\dulR,t) - \nabla_\nu S(\dulR,t)$$ which follows immediately from the definition (\[eqn: vector potential\]) of the vector potential by inserting the factorization (\[eqn: factorization\]). The gauge is chosen by setting the vector potential to zero $A(R,t)\equiv0$ in Eq. (\[eqn: vector-exact\]), which is possible in our specific example because we are dealing with a one-dimensional system. Obviously, the choice of the gauge does not affect any physical observable. $S(R,t)$ is thus determined from the expression $$S(R,t)=\int^R dR'
\left|\chi(R',t)\right|^{-2}\mbox{Im}\int dr\,\Psi^*(r,R',t)\nabla_{R'}\Psi(r,R',t).$$ From the calculated exact nuclear wave-function $\chi(\dulR,t)=e^{-\frac{i}{\hbar}S(\dulR,t)}|\chi(\dulR,t)|$, we obtain the TDPES $\epsilon(\dulR,t)$ from Eq. (\[eqn: tdpes\]) by explicitly calculating the electronic wave-function $\Phi_\dulR(\dulr,t)=\Psi(\dulr,\dulR,t)/\chi(\dulR,t)$. Alternatively, we may invert the nuclear equation (\[eqn: exact nuclear eqn\]). In the gauge we have implemented to perform the calculations, the TDPES alone determines the time evolution of $\chi(\dulR,t)$. In order to investigate the TDPES in detail, we study its gauge-invariant (GI) and gauge-dependent (GD) constituents separately (it can be easily proven that $\tilde{\epsilon}_{GI}(\dulR,t)=\epsilon_{GI}(\dulR,t)$ and $\tilde{\epsilon}_{GD}(\dulR,t)=\epsilon_{GD}(\dulR,t)+\partial_t\theta(\dulR,t)$ under the transformations in Eqs. (\[eqn: gauge\])), $$\epsilon(\dulR,t)=\epsilon_{GI}(\dulR,t)+\epsilon_{GD}(\dulR,t),$$ where $$\label{eqn: gi tdpes}
\epsilon_{GI}(\dulR,t)=\left\langle\Phi_\dulR(t)\right|\hat{H}_{BO}\left|\Phi_\dulR(t)\right\rangle_\dulr
+\sum_{\nu=1}^{N_n}\bigg(\frac{\hbar^2}{2M_\nu}\left\langle\nabla_\nu\Phi_\dulR(t)|\nabla_\nu\Phi_\dulR(t)
\right\rangle_\dulr-\frac{\bA^2_\nu(\dulR,t)}{2M_\nu}\bigg),$$ with the second term on the RHS obtained from the action of the electron-nuclear coupling operator in Eq. (\[eqn: enco\]) on the electronic wave-function, and $$\label{eqn: gd tdpes}
\epsilon_{GD}(\dulR,t)=\left\langle\Phi_\dulR(t)\right|-i\hbar\partial_t\left|\Phi_\dulR(t)\right\rangle_\dulr.$$ The GI part of the TDPES, $\epsilon_{GI}$, is not affected by the gauge transformation (\[eqn: gauge\]). The GD part, on the other hand, depends on the choice of the gauge. They both have important features [@steps] that will be discussed and analyzed in the following section. For this analysis, we will use a representation in terms of the BO electronic states, $\varphi_{\dulR}^{(l)}(\dulr)$, and BOPESs, $\epsilon_{BO}^{(l)}(\dulR)$, which are the eigenstates and eigenvalues of the BO electronic Hamiltonian (\[eqn: boe\]), respectively. If the full wave-function is expanded in this basis, $$\label{eqn: expansion of Psi}
\Psi(\dulr,\dulR,t)=\sum_l F_l(\dulR,t)\varphi_\dulR^{(l)}(\dulr),$$ then the nuclear density may be written as $$\label{eqn: chi and Fl}
\left|\chi(\dulR,t)\right| = \sqrt{\sum_{l}\left|F_l(\dulR,t)\right|^2}.$$ This identity is obtained by integrating the squared modulus of Eq. (\[eqn: expansion of Psi\]) over the electronic coordinates with normalized adiabatic states. The exact electronic wave-function may also be expanded in terms of the BO states, $$\label{eqn: expansion of Phi}
\Phi_\dulR(\dulr,t)=\sum_l C_l(\dulR,t)\varphi_\dulR^{(l)}(\dulr).$$ The expansion coefficients of Eqs. (\[eqn: expansion of Psi\]) and (\[eqn: expansion of Phi\]) are related, $$\label{eqn: relation coefficients}
F_l(\dulR,t)= C_l(\dulR,t)\chi(\dulR,t),$$ by virtue of the factorization (\[eqn: factorization\]). The PNC then reads $$\label{eqn: PNC on BO}
\sum_l\left|C_l(\dulR,t)\right|^2=1,\quad\forall\,\,\dulR,t.$$ In the cases studied in the following sections, the initial wave-function is the product of a real-valued normalized Gaussian wave-packet, centered at $R_c=-4.0$ $a_0$ with variance $\sigma=1/\sqrt{2.85}$ $a_0$ (black line in Fig. \[fig: BO-data\]), and the second BO electronic state, $\varphi_{R}^{(2)}(r)$.
Steps in the TDPES in strong non-adiabatic regime {#sec: strong coupling}
-------------------------------------------------
We first study a case in which the electronic and nuclear motions are strongly coupled. In order to produce that situation, we choose the parameters of the Hamiltonian (\[eqn: metiu-hamiltonian\]) as $R_f=5.0$ $a_0$, $R_l=3.1$ $a_0$ and $R_r=4.0$ $a_0$ such that the first BOPES, $\epsilon^{(1)}_{BO}$, is strongly coupled to the second BOPES, $\epsilon^{(2)}_{BO}$, around the avoided crossing at $R_{ac}=-1.90~a_0$ and there is a weak coupling to the rest of the surfaces. The four lowest BOPESs for this set of parameters are shown in Fig. \[fig: BO-data\] (left panel), along with the initial nuclear density. Energies are given in atomic (Hartree) units $\epsilon_h$. The same figure (right panel) presents the time-evolution of the populations of the BO states, $$\label{eqn: population BO}
\rho_{l}(t) = \int d\dulR \left|F_l(\dulR,t)\right|^2,$$ and underlines the strong non-adiabatic character of the system with the intense population exchange taking place at the passage through the avoided crossing ($t\simeq12$ $fs$).
![Left: lowest four BO surfaces, as functions of the nuclear coordinate. The first (red line) and second (green line) surfaces will be considered in the actual calculations that follow, the third and forth (dashed black lines) are shown for reference. The squared modulus (reduced by ten times and rigidly shifted in order to superimpose it on the energy curves) of the initial nuclear wave-packet is also shown (black line). Right: populations of the BO states along the time evolution. The strong non-adiabatic nature of the model is underlined by the population exchange at the crossing of the coupling region.[]{data-label="fig: BO-data"}](./Figure2.pdf)
As recently discussed [@steps], the GI part of the TDPES (\[eqn: gi tdpes\]) shows, in general, two distinct features: (i) in the vicinity of the avoided crossing, as the nuclear wave-packet passes through the region of non-adiabatic coupling between different BOPESs, $\epsilon_{GI}(R,t)$ resembles the *diabatic* surface that smoothly connects the two adiabatic surfaces; (ii) a bit further away from the avoided crossing, it shows *dynamical steps* between regions in $R$-space where it is on top of one or the other BOPES. The GD part of the TDPES (\[eqn: gd tdpes\]), on the other hand, is a piecewise constant function of the nuclear coordinate. This is illustrated in detail in Fig. \[fig: snapshots strong\] that contains the GI part of the TDPES (upper panel), the GD part of the TDPES (middle panel) and the nuclear density together with $|F_1|^2$ and $|F_2|^2$ (lower panel) for three different snapshots of time. In all the plots, the regions highlighted within the boxes are the regions which we refer to in the following discussion. Outside such regions, the value of the nuclear density drops under the numerical accuracy and the resulting potentials are not meaningful. That is why the TDPES are trimmed. The left panels show, at the initial time-step, (top) the GI part of the TDPES (black dots), with the two lowest BOPESs ($\epsilon_{BO}^{(1)}(R)$, dashed red line, and $\epsilon_{BO}^{(2)}(R)$, dashed green line) as reference, (center) the GD part of the exact potential (dark-green dots) and (bottom) the nuclear density (dashed black line) and its components from on the BO states (see Eq. (\[eqn: chi and Fl\])), $|F_1(R,t)|^2$ (red line) and $|F_2(R,t)|^2$ (green line).
![TDPES and nuclear densities at different time-steps, namely $t=0$ $fs$, $t=10.88$ $fs$ and $t=26.61$ $fs$. The different panels show: (top) GI part of the TDPES (black dots) and the two lowest BOPESs (first, dashed red line, and second, dashed green line) as reference; (center) the GD part of the TDPES (green dots); (bottom) nuclear density (dashed black line) and $|F_l(R,t)|^2$ ($l=1$ red line and $l=2$ green line). The gray boxes define the regions in $R$-space where the energies have been calculated, since the nuclear density is (numerically) not zero.[]{data-label="fig: snapshots strong"}](./Figure3.pdf)
At time $t=0$ $fs$, the electronic wave-function, $\Phi_R(r,t)$, coincides with the second adiabatic state $\varphi_R^{(2)}(r)$, therefore the GI component of the TDPES is identical with $\epsilon_{BO}^{(2)}(R)$, apart from a slight deviation due to the second term in Eq. (\[eqn: gi tdpes\]). This is easily confirmed by the expression of $\epsilon_{GI}(R,t)$ in terms of the BO states and energies $$\begin{aligned}
\epsilon_{GI}(R,t)&=\sum_{l}\left|C_l(R,t)\right|^2\epsilon_{BO}^{(l)}(R)+\frac{\hbar^2}{2M}
\left[\sum_{l,k}C_l^*(R,t)C_k(R,t)d_{lk}^{(2)}(R)\right.
\label{eqn: gi tdpes on BO}\\
&\left.\sum_{l,k}\left({C_l^*}'(R,t)C_k(R,t)-C_l^*(R,t)C_k'(R,t)\right)d_{lk}^{(1)}(R)+
\sum_l\left|C_l'(R,t)\right|^2\right],\nonumber\end{aligned}$$ where we use the prime to indicate the spatial derivative of the coefficients and we introduced the non-adiabatic couplings $$\begin{aligned}
d_{lk}^{(1)}(R)=&\left\langle\varphi_{R}^{(l)}\right.\left|\nabla_{R}\varphi_{R}^{(k)}\right\rangle_r &
=-{d_{kl}^{(1)}}^*(R) \\
d_{lk}^{(2)}(R)=&\left\langle\nabla_{R}\varphi_{R}^{(l)}\right.\left|\nabla_{R}\varphi_{R}^{(k)}\right\rangle_r &
= {d_{kl}^{(2)}}^*(R).\end{aligned}$$ The leading term in Eq. (\[eqn: gi tdpes on BO\]) is the average of the BOPESs weighted by $\left|C_l(R,t)
\right|^2$, since the second term is $\mathcal O(M^{-1})$. The GD component of the TDPES in Eq. (\[eqn: gd tdpes\]), in terms of the BO states, becomes $$\label{eqn: gd tdpes on BO}
\epsilon_{GD}(R,t) = \sum_{l}\left|C_l(R,t)\right|^2\dot\gamma_{l}(R,t)$$ where $\dot\gamma_{l}(R,t)$ is the time-derivative of the phase of the coefficients $C_l(R,t)=
e^{\frac{i}{\hbar}\gamma_l(R,t)}|C_l(R,t)|$. The nuclear density, along with its components on the BO states from Eq. (\[eqn: chi and Fl\]), is presented in the bottom panels of Fig. \[fig: snapshots strong\]. At the initial time, $\left|\chi(R,t)\right|^2=\left|F_2(R,t)\right|^2$.
At $t=10.88$ $fs$ in Fig. \[fig: snapshots strong\] (central panels), (top) the GI part of the TDPES resembles the diabatic surface [@MM] that smoothly passes through the avoided crossing. This behavior allows the nuclear density moving on the upper BOPES to be partially “transferred” to the lower state, as the consistent increase of the population of state $\varphi_R^{(1)}(r)$ (red curve in the bottom plot in Fig. \[fig: snapshots strong\]) confirms. In the region highlighted by the dashed box, the GD part of the exact potential is constant, therefore, it does not affect nuclear dynamics.
At later times ($t=26.61$ $fs$ shown in the right panels of Fig. \[fig: snapshots strong\]), when the nuclear wave-packet has split at the avoided crossing, both components of the TDPES present a pronounced stepwise behavior: the GI part follows one or the other BOPES in different regions of $R$-space that are connected by a step, whereas the GD part is stepwise constant, with steps appearing in the same region.
The overall shape of the TDPES, at initial times, is determined by the GI part, as the effect of the GD part is no more than a constant shift. Hence, the TDPES, that drives the nuclear dynamics, behaves like a diabatic surface and “opens” in the direction of the wave-packet’s motion in order to facilitate the population exchange between the adiabatic states. After the wave-packet splits at the avoided crossing, in different regions in $R$-space, the TDPES is parallel to one or the other BOPES and a step forms in the transition region. Therefore, the motion of the components $F_l(R,t)$ of the nuclear wave-packet is driven by single adiabatic surfaces and not (like, e.g., in Ehrenfest dynamics) by an average electronic potential. This feature is reminiscent of the way the well-known *trajectory surface hopping* (TSH) scheme [@TSH] deals with the non-adiabatic dynamics. In this approach, the components (in our case identified by the symbol $|F_l(R,t)|^2$) of the nuclear density on different BO states are represented by *bundles* of classical trajectories evolving, independently from one another, on different BO surfaces. The ratio of the total number of trajectories occupying, at each time, the surfaces approximates the population $\rho_l$ of the corresponding BO state. The success of this method in reproducing non-adiabatic processes becomes clear in the light of the fact that the exact TDPES itself is parallel to different BOPESs in different regions along the nuclear coordinate. The usually abrupt transitions between the adiabatic surfaces, i.e., the steps in the exact treatment, are reminiscent to the stochastic jumps between BO surfaces in TSH.
Analysis of the steps
---------------------
The behavior of the GI part of the TDPES is mainly determined by the first term in Eq. (\[eqn: gi tdpes on BO\]). The steps appear in the region around $R_0$, the cross-over of $|F_1(R,t))|^2$ and $|F_2(R,t))|^2$. In particular, at this point $|F_1(R_0,t)|^2=|F_2(R_0,t)|^2=|X(t)|$ and, irrespective of this value, the expansion coefficients in the electronic wave-function ([\[eqn: expansion of Phi\]]{}) have the value $|C_1(R_0,t)|^2=|C_2(R_0,t)|^2=1/2$. This relation holds as consequence of Eq. (\[eqn: relation coefficients\]), which can be written as $$\left|C_l(R_0,t)\right|^2 = \frac{\left|F_l(R_0,t)\right|^2}{\left|F_1(R_0,t)\right|^2+\left|F_2(R_0,t)
\right|^2}=\frac{1}{2}\quad \mbox{with}\quad l=1,2,$$ and is clearly shown in Fig. \[fig: steps analysis\].
![Top: GI part (black line) and the GD part (blue line, rigidly shifted along the energy axis) of the exact potential at time $t=26.61$ $fs$. The first (dashed red) and second (dashed green) BOPESs are shown as reference. Bottom: coefficients $|F_l(R,t)|^2$ of the expansion of the full wave-function (Eq. (\[eqn: expansion of Psi\])) on the BO states ($l=1$ dashed red line, $l=2$ dashed green line) and coefficients $|C_l(R,t)|^2$ of the expansion of the electronic wave-function ($l=1$ continuous red line, $l=2$ continuous green line); the black line represents the nuclear density. $R_0$ is the position where the coefficients $|F_1(R,t)|^2$ and $|F_2(R,t)|^2$ have the same value and the dashed box highlights the region of the step.[]{data-label="fig: steps analysis"}](./Figure4.pdf)
Here we present, in the upper panel, the GI part (black line) and the GD part (blue line, rigidly shifted along the energy axis) of the exact potential at time $t=26.62$ $fs$. The BO surfaces (dashed red and green lines) are also plotted as reference. In the lower panel, we plot the coefficients of the expansions in Eq. (\[eqn: expansion of Psi\]) (dashed red and green lines) and in Eq. (\[eqn: expansion of Phi\]) (continuous red and green lines). The continuous black line represents the nuclear density.
The expression of the GI component of the TDPES for a two-state system, from Eq. (\[eqn: gi tdpes on BO\]), is $$\epsilon_{GI}(R,t) \simeq \left|C_1(R,t)\right|^2
\epsilon_{BO}^{(1)}(R)+\left|C_2(R,t)\right|^2\epsilon_{BO}^{(2)}(R),$$ neglecting terms $\mathcal O(M^{-1})$. If $\left|C_l(R,t)\right|^2$ is Taylor-expanded around $R_0$, up to within the linear deviations, $$\begin{aligned}
\left|C_{\mathop{}_2^1}(R,t)\right|^2&\simeq&\left.\frac{\left|F_{\mathop{}_2^1}(R,t)\right|^2}{\left|\chi(R,t)
\right|^2}\right|_{R_0}+ \left.\nabla_{R}\frac{\left|F_{\mathop{}_2^1}(R,t)\right|^2}{\left|\chi(R,t)\right|^2}
\right|_{R_0} (R-R_0)\nonumber \\
&=&\frac{1}{2}\pm \frac{\alpha(t)}{2}\left(R-R_0\right),\end{aligned}$$ one can identify the parameter $\alpha(t)$, defined as $$\label{eqn: definition of alpha}
\alpha(t) = \frac{\left(\nabla_R\left|F_1(R,t)\right|\right)_{R_0}-\left(\nabla_R\left|F_2(R,t)
\right|\right)_{R_0}}{\left|X(t)\right|},$$ where $\alpha(t)$ is the slope of the coefficients in the step region from which the width of the region can be determined. Using the relation $0\leq\left|C_1(R,t)\right|^2\leq 1$, we get $$0\leq \frac{1}{2}+\frac{\alpha(t)}{2}\left(R-R_0\right)\leq 1\quad\mbox{with}\quad\frac{\Delta R}{2}=
\left|R-R_0\right|\leq \frac{1}{\alpha(t)}.$$ Therefore, $\Delta R$ is small because the step is steep, as consequence of a large $\alpha(t)$. $\alpha(t)$ can be large either because $|X(t)|$ is small, i.e., the cross-over is located in a region of small nuclear density, or because the terms in the numerator of Eq. (\[eqn: definition of alpha\]) have opposite slopes at $R_0$ (this is the case depicted in Fig. \[fig: steps analysis\]). Outside the region $\Delta R$, one or the other coefficients $|C_l(R,t)|^2$ dominates, thus leading to $$\label{eqn: gi tdpes outside step
region}
\epsilon_{GI}(R,t)=\left\lbrace
\begin{array}{cc}
\epsilon_{BO}^{(2)}(R), & R< R_0 \\
& \\
\epsilon_{BO}^{(1)}(R), & R> R_0.
\end{array}
\right.$$ The GD part of the TDPES can be analyzed similarly: $\epsilon_{GD}(R,t)$ from Eq. (\[eqn: gd tdpes on BO\]) may be written, in terms of the two BO states, as $$\epsilon_{GD}(R,t) = \left|C_1(R,t)\right|^2 \dot\gamma_1(R,t)+\left|C_2(R,t)\right|^2\dot\gamma_2(R,t)$$ and we recall that $\gamma_l(R,t)$ is the phase of the coefficient $C_l(R,t)$. As in Eq. (\[eqn: gi tdpes outside step region\]), outside the step region, this part of the potential becomes $$\label{eqn: gd tdpes outside step region}
\epsilon_{GD}(R,t)=\left\lbrace
\begin{array}{cc}
\dot\gamma_2(R,t), & R< R_0 \\
& \\
\dot\gamma_1(R,t), & R> R_0.
\end{array}
\right.$$ Moreover, Fig. \[fig: steps analysis\] shows that in these regions $\dot\gamma_1(R,t)$ and $\dot\gamma_2(R,t)$ are constant functions of $R$. This is a consequence of the gauge we chose. The gauge condition, $A(R,t)=\langle\Phi_R(t)|-i\hbar\nabla_R\Phi_R(t)\rangle_r =0$, in terms of the two BO states involved in the dynamics, reads $$\begin{aligned}
0=\sum_{l=1,2}\left|C_l(R,t)\right|^2\nabla_R\gamma_l(R,t)-\frac{i\hbar}{2}\nabla_R\sum_{l=1,2}\left|C_l(R,t)
\right|^2 \nonumber \\
-i\hbar\sum_{l,k=1,2}C_l^*(R,t)C_k(R,t)d_{lk}^{(1)}(R).\end{aligned}$$ However, the second term of the RHS is identically zero, due to the PNC in Eq. (\[eqn: PNC on BO\]), and the third term can be neglected, due to the presence of the non-adiabatic couplings, $d_{lk}^{(1)}(R)$, that are small far from the avoided crossing. The gauge condition then states $$\left|C_1(R,t)\right|^2\nabla_R\gamma_1(R,t) = -\left|C_2(R,t)\right|^2\nabla_R\gamma_2(R,t),$$ or equivalently $$\begin{aligned}
\nabla_R\gamma_2(R,t) = 0 &\quad\mbox{for}\quad R<R_0\quad&\mbox{where}\quad\left|C_1(R,t)\right|^2=0 \\
\nabla_R\gamma_1(R,t) = 0 &\quad\mbox{for}\quad R>R_0\quad&\mbox{where}\quad\left|C_2(R,t)\right|^2=0.\end{aligned}$$ We obtain $\gamma_l(R,t)=\Gamma_l(t)$, namely the phase of the coefficient $C_l(R,t)$ is only a function of time (constant in space) in the region where the squared modulus of the corresponding coefficient is equal to unity. Similarly, $\dot\gamma_l(R,t)=\dot\Gamma_l(t)$, as shown in Fig. \[fig: steps analysis\].
In the step region, around $R_0$, the expression of the TDPES can be approximated as $$\begin{aligned}
\epsilon(R,t) =& \frac{\epsilon_{BO}^{(1)}(R)+\epsilon_{BO}^{(2)}(R)}{2}+\frac{\dot\gamma_1(R,t)+
\dot\gamma_2(R,t)}{2}\nonumber \\
&+\alpha(t)\left[\frac{\epsilon_{BO}^{(1)}(R)-\epsilon_{BO}^{(2)}(R)}{2}+\frac{\dot\gamma_1(R,t)-
\dot\gamma_2(R,t)}{2}\right](R-R_0). \label{eqn: full potential at R0}\end{aligned}$$ The first two terms on the RHS are the average of the BO energies plus the average value of the time-derivative of the phases $\gamma_1(R,t)$ and $\gamma_2(R,t)$; the terms in square brackets are the energy gaps between the BO surfaces and between the time-derivative of the phases, which give the contribution proportional to the parameter $\alpha(t)$. From Fig. \[fig: steps analysis\], we notice that, around $R_0$, the slope of $\epsilon_{GD}$ is opposite to the slope of $\epsilon_{GI}$ and this is a general feature in the studied system (in the absence of a time-dependent external field). Therefore, the GD part reduces the height of the steps in the GI part. We will see the effect of this contribution on (classical) nuclear dynamics in the section \[sec: dynamics\].
Steps in the TDPES in weak non-adiabatic regime
-----------------------------------------------
In this section, we study a case of weaker non-adiabatic coupling between the two lowest BO states. In order to make the coupling weaker, we choose the parameters in the Hamiltonian (\[eqn: metiu-hamiltonian\]) as $L=19.0$ $a_0$, $R_f=3.8$ $a_0$, $R_l=2.0$ $a_0$ and $R_r=5.5$ $a_0$. The BO surfaces, along with the evolution of the populations of the BO states, are shown in Fig. \[fig: BO-data weak\].
![Same as Fig. \[fig: BO-data\] but for weaker non-adiabatic coupling between the two lowest BO states.[]{data-label="fig: BO-data weak"}](./Figure5.pdf)
The initial conditions for the dynamical evolution of this system are the same as in the previous example, however the coupling between the two lowest electronic states is weaker, thus leading to a reduced population exchange, clearly shown in Fig. \[fig: BO-data weak\] (right panel). Nonetheless, the process described here shows similarities to the previous case, as can be seen from Fig. \[fig: snapshots weak\].
![Same as Fig. \[fig: snapshots strong\] but for a weaker non-adiabatic coupling between the two lowest BO states, at time-steps $9.68$ $fs$, $27.33$ $fs$ and $32.65$ $fs$.[]{data-label="fig: snapshots weak"}](./Figure6.pdf)
The GI part of the TDPES presents again two main features, (i) the diabatization at the avoided crossing, when the nuclear wave-packet crosses the region of relatively strong non-adiabatic coupling and (ii) the steps at the cross-over of $|F_1(R,t)|^2$ and $|F_2(R,t)|^2$, signature of the splitting of the nuclear density. The GD part is either constant, before the splitting at the avoided crossing, or stepwise constant, with steps appearing in the same region as the steps in the GI term, but with opposite slope. At different snapshots of time, i.e., $9.68$ $fs$, $27.33$ $fs$ and $32.65$ $fs$, these properties are shown in Fig. \[fig: snapshots weak\], along with the nuclear density and its components on the BO states. The notation used in the figures is the same as in Fig. \[fig: snapshots strong\].
A slightly different behavior from the situation of strong non-adiabatic coupling can be identified in $\epsilon_{GI}(R,t)$ before the passage through the avoided crossing. As the nuclear wave-packet approaches the avoided crossing, the GI part of the TDPES “opens” towards the direction of motion, resembling the diabatic surface that connects the BO surfaces through the avoided crossing. This is clearly shown in Fig. \[fig: diabatization\] (left) at time $t=9.68$ $fs$ for the strongly coupled system. In the case of weaker non-adiabatic coupling, $\epsilon_{GI}(R,t)$, at the avoided crossing, lies between the BO surfaces, as shown in Fig. \[fig: diabatization\] (right).
![Diabatization feature of $\epsilon_{GI}(R,t)$ (blue dots) for the two model systems (left panel, strong coupling at $t=9.68$ $fs$, and right panel, weak coupling at $t=6.29$ $fs$) presented here. The dashed lines represent the BO surfaces ($\epsilon_{BO}^{(1)}(R)$ red line and $\epsilon_{BO}^{(2)}(R)$ green line) and the continuous black line represents the nuclear density (reduced by a factor 10 and rigidly shifted along the $y$-axis).[]{data-label="fig: diabatization"}](./Figure7.pdf)
Therefore, the diabatization feature strictly depends on the strength of the non-adiabatic coupling and, in general, can be viewed as a transient configuration of the GI part of the TDPES before the formation of the steps.
Classical dynamics on PESs {#sec: dynamics}
==========================
In section \[sec: pes\], we have addressed some of the generic features of the TDPES that governs the nuclear dynamics in the presence of non-adiabatic electronic transitions. As discussed before, some of these features, in particular the step that bridges between the two parts of the TDPES that are parallel to the BOPESs, are reminiscent of the jumping between the BOPESs in TSH methods [@TSH]. These algorithms are based on the mixed quantum-classical treatment of the electronic and nuclear dynamics using stochastic jumps between BO surfaces. Therefore, an ensemble of classical trajectories with different initial conditions is needed to achieve statistically reasonable outcomes. On the other hand, the TDPES is the exact time-dependent potential that governs the nuclear dynamics (in general together with the vector potential) and contains the back-reaction resulting from the exact coupling to the electronic subsystem. This brings us to investigate how the TDPES drives the classical dynamics of point-like nuclei.
In order to understand how the generic features of the TDPES affect the classical nuclear dynamics, we have employed the surfaces presented in section (\[sec: pes\]) to calculate the forces acting on the nuclear degree of freedom. We compare the resulting dynamics using the forces that are calculated from the gradient of the TDPES and from the gradient of its GI part. The classical propagation starts at the initial position $R_c=-4.0$ $a_0$ with zero initial momentum. Here, we use the velocity-Verlet algorithm to integrate Hamilton’s equations, $$\label{eq: hamilton-eom}
\left\lbrace
\begin{array}{ccl}
\dot R &=& \dfrac{P}{M} \\
&& \\
\dot P &=& -\nabla_R\epsilon(R)\,\,\mbox{ or }\,\,-\nabla_R\epsilon_{GI}(R),
\end{array}
\right.$$ using the same time-steps as in the quantum propagation ($\delta t = 2.4\times10^{-3}$ $fs$). In Fig. \[fig: position and velocity\] (upper panels) we present the evolution of the classical position compared to the average nuclear position from the quantum calculation, for strong and weak coupling. In both cases, a single trajectory, evolving on the exact surface (blue lines in Fig. \[fig: position and velocity\]), is able to reproduce the mean nuclear path (dashed black lines) fairly well. A slight deviation from the quantum results happens only towards the end of the simulated trajectories. When the classical forces are calculated from the GI part of the TDPES, the corresponding classical trajectory in the strong coupling case, does not show a large deviation from the exact calculation. However, in the weak coupling case, after $20$ $fs$, the classical trajectory deviates considerably from the quantum mean path. This behavior is also confirmed by the pronounced increase of the velocity of the classical particle moving on $\epsilon_{GI}$, shown in Fig. \[fig: position and velocity\] (lower panels).
![Classical position (upper panels) and velocity (lower panels) and mean nuclear position and velocity as functions of time for the systems in the presence of strong non-adiabatic coupling (left) and of weak non-adiabatic coupling (right). The dashed black line represents the average nuclear values from quantum calculation, the blue and orange lines are the positions and velocities of the classical particle when it evolves on the exact potential and on the GI part of the potential, respectively.[]{data-label="fig: position and velocity"}](./Figure8.pdf)
We now have a closer look at the classical dynamics and try to find out the source of the deviations, especially in the weaker coupling case. Fig. \[fig: evolution\] shows the classical positions calculated from the full TDPES (blue dots) and the GI part of it (orange dots) together with the corresponding potentials and the exact nuclear densities at the times indicated in the plots. It can be seen in the figure that the classical particle evolving on the GI part of the potential, in the case of weaker coupling, at the moment of the step formation feels an intense force, as its position is exactly in the region of the step (see $t=23.71$ $fs$ in Fig. \[fig: evolution\]). This happens also in the case of the strong coupling (see the blue line referring to the velocity in Fig. \[fig: position and velocity\], left plot), to a lesser extent and the velocity of the classical particle does not show a strong peak. The evolution of the classical particle on the GI part, in the case of the strong coupling, shows that the step forms in the direction of larger nuclear density (see plot at $t=22.25$ $fs$), hence, the classical particle correctly follows the step and its position is approximately the mean nuclear position. However, in the case of weaker coupling, the step forms in the direction of smaller nuclear density and the classical particle can not move “up the hill” to follow the nuclear mean path, leading to a large deviation of the classical position from the quantum mean value. The intense force felt by the classical particle drives it to an unphysical region, where the nuclear density is very small. The presence of the GD part of the TDPES is responsible for the decrease (or even the inversion) of the “energy gap” in the GI part, thus producing a better agreement between classical and quantum results.
![Upper panels: strong coupling results. Lower panels: weak coupling results. The figure shows classical positions (dots) at different times, as indicated in the plots, with the corresponding potentials, $\epsilon_{GI}(R,t)$ (orange lines) and $\epsilon(R,t)$ (blue lines). The nuclear density (dashed black line) is plotted as reference, along with the mean position (black arrows).[]{data-label="fig: evolution"}](./Figure9.pdf){width="\textwidth"}
From comparing the classical and quantum dynamics shown in Fig. \[fig: evolution\], we observe that in the strong coupling case (upper panel), at $t=4.84$ $fs$ and at $t=11.37$ $fs$, the nuclear wave-packet has not yet crossed the avoided crossing, thus the GD part of the TDPES is a constant. Therefore, the classical force calculated from the TDPES is identical with the one calculated from its GI part. At these times, the classical positions of the nuclei evolving on the GI part of the potential (orange dots in the figure) and on the full TDPES (blue dots) coincide with the mean position of the nuclear wave-packet (black arrows). On the other hand, in the weaker coupling case (lower panels), a similar behavior is seen only before the wave-packet splitting, at $t=7.26$ $fs$ and $t=12.09$ $fs$. At later times, namely $t=22.25$ $fs$ for the strong coupling case and $t=23.71$ $fs$ for the weaker coupling case, the steps develop in $\epsilon_{GI}$ and the classical particle evolving on this potential follows the direction in which the step is forming: in the case of strong coupling, this region coincides with the region associated with larger nuclear density, whereas this is not the case for the weaker coupling case. As discussed above, this feature explains why the positions of the particles on $\epsilon$ and on $\epsilon_{GI}$, for the system in the presence of strong non-adiabatic coupling, are close to each other also at later times ($t=29.03$ $fs$ in Fig. \[fig: evolution\]), whereas they deviate in the weaker coupling regime as clearly shown in the figure at time $t=31.45$ $fs$.
The results presented in this section offer interesting insights into possible ways of modeling non-adiabatic processes, within a mixed quantum-classical treatment. On one hand, the gradient of the GI part of the exact potential is the force that drives the classical nuclear motion and we have shown that such force is “adiabatic” in the sense that, far from the step, it is produced by a single BOPES. On the other hand, the GD part does not affect such force, but contributes in diminishing the energy separation between the two sides of the step. This energy barrier almost disappears in the full TDPES, but the difference in slopes indeed persists. If a gauge is chosen such that $\epsilon_{GD}(R,t)\equiv 0$, the non-zero vector potential compensates the effect of the energy step in the GI part of the TDPES by adding a kinetic energy contribution (the vector potential appears in the kinetic term of the nuclear Hamiltonian in Eq. (\[eqn: exact nuclear eqn\])). Such contribution would energetically favor the transfer of classical point-particles from one side of the step to the other. Once again, the comparison with TSH is inevitable: in the latter, different adiabatic surfaces are energetically accessible by the classical nuclei because of the *stochastic* jumps and the subsequent momentum rescaling (in order to impose energy conservation); in the scheme based on the exact TDPES, depending on the gauge, either the GD part of the potential is responsible for bringing “energetically closer” different BOPES or the vector potential gives the necessary kinetic energy contribution. So far, we have described where the steps appear, how they form and how they affect nuclear motion. From these observations, we expect that rigorous mixed quantum-classical schemes for dealing with non-adiabatic processes can be deduced in a systematic way from the classical forces associated with the exact TDPES and the exact vector potential.
Ehrenfest theorem for the nuclear wave-function {#sec: ehrenfest}
===============================================
In section \[sec: dynamics\], we studied the classical nuclear dynamics on the TDPES. However, we did not provide any argument on how that study can be associated with a classical limit of the nuclear motion that is able to, approximately, reproduce the expectation values of the nuclear position and momentum of the complete electron-nuclear system. Here, using the Ehrenfest theorem, we show how the nuclear position and momentum calculated from Eq. (\[eq: hamilton-eom\]) can be linked to the expectation values of the nuclear position and momentum of the complete electron-nuclear system.
The Ehrenfest theorem [@ehrenfest] relates the time-derivative of the expectation value of a quantum-mechanical operator $\hat
O$ to the expectation value of the commutator of that operator with the Hamiltonian, i.e. $$\frac{d}{dt}\langle\hat O(t)\rangle = \frac{1}{i\hbar}\left\langle\left[\hat O(t),\hat H\right]\right\rangle+
\langle\partial_t\hat O(t)\rangle.$$ The second term on the RHS refers to the explicit time-dependence of $\hat O$. In particular, the theorem leads to the classical-like equations of motion for the mean value of position and momentum operators. For a system of electrons and nuclei, described by the Hamiltonian in Eq. (\[eqn: hamiltonian\]) and the wave-function $\Psi(\dulr,\dulR,t)$, the mean values of the $\nu$-th nuclear position $\hat{\bf R}_{\nu}$ and momentum $\hat{\bf
P}_{\nu}$ operators evolve according to the classical Hamilton’s equations $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf R}_{\nu}\rangle_{\Psi}=\frac{1}{i\hbar} \left\langle\left[\hat{\bf R}_\nu,
\hat H(\dulr,\dulR)\right]\right\rangle_\Psi&=&\frac{\langle\hat{\bf P}_{\nu}\rangle_{\Psi}}{M_{\nu}}
\label{eqn: general ehrenfest 1}\\
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=\frac{1}{i\hbar} \left\langle\left[\hat{\bf P}_\nu,
\hat H(\dulr,\dulR)\right]\right\rangle_\Psi&=& \langle-\nabla_{\nu}\big(\hat{V}_{en}(\dulr,\dulR)+\hat{W}_{nn}(\dulR)\big)
\rangle_{{\Psi}}.
\label{eqn: general ehrenfest 2}\end{aligned}$$ Here, the operators do not depend explicitly on time and we indicate the integration over the full wave-function (electronic and nuclear coordinates) by $\langle\,\cdot\,\rangle_{{\Psi}}$. On the other hand, the nuclear equation (\[eqn: exact nuclear eqn\]) is a Schrödinger equation that contains a time-dependent vector potential and a time-dependent scalar potential. Therefore, the Ehrenfest theorem for the nuclear subsystem reads $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf R}_{\nu}\rangle_\chi&=&\frac{1}{i\hbar}\left\langle\left[\hat{\bf R}_\nu,
\hat H_n(\dulR)\right]\right\rangle_\chi\label{eqn: ehrenfest 1}\\
\frac{d}{dt}\langle\hat{\widetilde{\bf P}}_{\nu}\rangle_\chi&=&\frac{1}{i\hbar}\left\langle\left[
\hat{\widetilde{\bf P}}_\nu,\hat H_n(\dulR)\right]\right\rangle_\chi+
\left\langle\partial_t{\bf A}_\nu(\dulR,t)\right\rangle_\chi \label{eqn: ehrenfest 2}\end{aligned}$$ where [@AMG2] $$\hat{\widetilde{\bf P}}_{\nu} = -i\hbar\nabla_\nu+{\bf A}_{\nu}(\dulR,t)$$ is the expression of the nuclear canonical momentum operator in position representation, and $$\hat H_n(\dulR) = \sum_{\nu=1}^{N_n} \frac{\left[-i\hbar\nabla_\nu+\bA_\nu(\dulR,t)\right]^2}{2M_\nu} +
\epsilon(\dulR,t) \label{eqn: nuclear-Hamiltonian}$$ is the nuclear Hamiltonian from Eq. (\[eqn: exact nuclear eqn\]). Note that the average operation is performed only on the nuclear wave-function as indicated by $\langle\,\cdot\,\rangle_{\chi}$. An explicit time-dependence appears in the expression of the momentum operator, due to the presence of the vector potential. This dependence is accounted for in the second term on the RHS of Eq. (\[eqn: ehrenfest 2\]). While Eq. (\[eqn: ehrenfest 1\]) is easily obtained from Eq. (\[eqn: general ehrenfest 1\]) by performing the integration over the electronic part of full wave-function, Eq. (\[eqn: ehrenfest 2\]) is more involved and will be proved as follows. We rewrite LHS of Eq. (\[eqn: general ehrenfest 2\]) as $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=&\int d\dulr d\dulR\,
\left[\Phi_\dulR^*(\dulr,t)\partial_t\chi^*(\dulR,t)+\chi^*(\dulR,t)\partial_t\Phi_\dulR^*(\dulr,t)\right]
\hat{\bf P}_\nu\chi(\dulR,t)\Phi_\dulR(\dulr,t) \nonumber\\
&+\int d\dulr d\dulR\,
\chi^*(\dulR,t)\Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu
\left[\Phi_\dulR(\dulr,t)\partial_t\chi(\dulR,t)+\chi(\dulR,t)\partial_t\Phi_\dulR(\dulr,t)\right].\end{aligned}$$ $\hat{\bf P}_\nu$ being a differential operator in position representation, its action on the factorized wave-function is $$\hat{\bf P}_\nu\chi(\dulR,t)\Phi_\dulR(\dulr,t)= \left(\hat{\bf P}_\nu\chi(\dulR,t)\right)\Phi_\dulR(\dulr,t)+
\chi(\dulR,t)\left(\hat{\bf P}_\nu\Phi_\dulR(\dulr,t)\right).$$ Then we use the nuclear equation (\[eqn: exact nuclear eqn\]) for $$\partial_t\chi(\dulR,t)=\frac{1}{i\hbar}\hat H_n(\dulR) \chi(\dulR,t)$$ and its complex-conjugated ($\hat H_n(\dulR)$ is hermitian), the definition of the (real) vector potential $${\bf A}_\nu(\dulR,t) = \int d\dulr \,\Phi_\dulR^*(\dulr,t) \hat{\bf P}_\nu\Phi_\dulR(\dulr,t)$$ and the PNC, to derive $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=\frac{1}{i\hbar}\int d\dulR\,\chi^*(\dulR,t)
&\left(\hat{\widetilde{\bf P}}_\nu\hat H_n(\dulR)-\hat H_n(\dulR)\hat{\widetilde{\bf P}}_\nu\right)\chi(\dulR,t)
\nonumber \\
+\int d\dulR\left|\chi(\dulR,t)\right|^2 \int d\dulr &\left[\left(\partial_t\Phi_\dulR^*(\dulr,t)\right)
\hat{\bf P}_\nu\Phi_\dulR(\dulr,t) + \Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu\partial_t\Phi_\dulR(\dulr,t)\right]\end{aligned}$$ with $\hat{\widetilde{\bf P}}_\nu=\hat{\bf P}_\nu+{\bf A}_\nu(\dulR,t)$. Using the relation $$\left(\partial_t\Phi_\dulR^*(\dulr,t)\right)\hat{\bf P}_\nu\Phi_\dulR(\dulr,t) =
\partial_t\left(\Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu\Phi_\dulR(\dulr,t)\right) -
\Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu\partial_t\Phi_\dulR(\dulr,t),$$ for the term in the square brackets, leads to $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=\int d\dulR\,\chi^*(\dulR,t)\left(
\frac{1}{i\hbar}\left[\hat{\widetilde{\bf P}}_\nu,\hat H_n(\dulR)\right]+\partial_t{\bf A}_\nu(\dulR,t)\right)
\chi(\dulR,t),\end{aligned}$$ recovering the term on the RHS of Eq. (\[eqn: ehrenfest 2\]). A similar procedure [@AMG2] yields the relation $$\begin{aligned}
\langle\hat{\bf P}_\nu\rangle_\Psi &= \int d\dulr d\dulR \,\Phi_\dulR^*(\dulr,t)\chi^*(\dulR,t)
\left[\left(\hat{\bf P}_\nu\chi(\dulR,t)\right)\Phi_\dulR(\dulr,t)+\chi(\dulR,t)\hat{\bf
P}_\nu\Phi_\dulR(\dulr,t)\right] \nonumber \\
&=\int d\dulR \,\chi^*(\dulR,t)\left[\hat{\bf P}_\nu+{\bf A}_\nu(\dulR,t)\right]\chi(\dulR,t)=\langle\hat{\widetilde{\bf
P}}_\nu\rangle_\chi,\end{aligned}$$ which proves the identity of the LHSs of Eqs. (\[eqn: general ehrenfest 2\]) and (\[eqn: ehrenfest 2\]).
We have proved the Ehrenfest theorem for the nuclear wave-function and nuclear Hamiltonian, deriving exact relations for the evolution of the mean values of nuclear position and momentum operators over the complete system. This outcome is consistent with the interpretation of $\chi(\dulR,t)$ as the proper nuclear wave-function that reproduces the nuclear density and current density of the complete system (see the discussion in section \[sec: background\]).
In the one-dimensional system studied here, the gauge is chosen such that $A(R,t)=0$, therefore, the Ehrenfest equations become $$\begin{aligned}
\frac{d}{dt}\langle\hat R\rangle_\chi=\frac{1}{i\hbar}\left\langle\left[\hat R,\hat H_n\right]\right\rangle_\chi
&=&\frac{\langle\hat P\rangle_\chi}{M}\label{eqn: ehrenfest 1 one-d}\\
\frac{d}{dt}\langle\hat P\rangle_\chi=\frac{1}{i\hbar}\left\langle\left[\hat P,\hat H_n\right]\right\rangle_\chi
&=&\langle-\nabla_R\epsilon(R,t)\rangle_\chi, \label{eqn: ehrenfest 2 one-d}\end{aligned}$$ where the mean force generating the classical-like evolution is determined as the expectation value, on the nuclear wave-function, of the gradient of the TDPES. If we replace the nuclear wave-function in Eqs. (\[eqn: ehrenfest 1 one-d\]) and (\[eqn: ehrenfest 2 one-d\]) by a delta-function centered at the classical position, we get Eqs. (\[eq: hamilton-eom\]) that was used in section \[sec: dynamics\] to generate classical dynamics on the exact PES. That is why the classical nuclear dynamics on the TDPES could actually approximate the mean nuclear position and momentum.
We have numerically simulated classical dynamics under the following equations of motion $$\left\lbrace
\begin{array}{ccl}
\dot R &=&\dfrac{P}{M}\\
&& \\
\dot P&=&\langle-\nabla_R\epsilon(R,t)\rangle_\chi,
\end{array}
\right.$$ where $\epsilon(R,t)$ is obtained from the solution of the TDSE with Hamiltonian (\[eqn: metiu-hamiltonian\]), for both sets of parameters producing strong and weak non-adiabatic coupling between the two lowest BO surfaces. The initial conditions for the classical evolution are exactly the initial mean position and mean velocity of the quantum particle.
![Left: nuclear position as a function of time. Right: nuclear velocity as a function of time. The average position and velocity calculated from the quantum-mechanical (QM) propagation are shown as dotted red (strong coupling) and dotted green (weak coupling) lines. The long-dashed (strong coupling) and short-dashed (weak coupling) black lines are the results of classical propagation driven by the average force (AV) as in Eqs. (\[eqn: ehrenfest 1 one-d\]) and (\[eqn: ehrenfest 2 one-d\]).[]{data-label="fig: ehrenfest"}](./Figure10.pdf)
The results are shown in Fig. \[fig: ehrenfest\], where we plot the mean position (left) and velocity (right) as functions of time from quantum-mechanical calculations, compared to the values of position and velocity of a classical particle moving according to the average force $\langle-\nabla_R\epsilon(R,t)\rangle_\chi$. As expected by the proof of the Ehrenfest theorem involving the nuclear wave-function $\chi(\dulR,t)$ and the nuclear Hamiltonian $\hat H_n$ presented in this section, the classical trajectory perfectly follows the evolution of the quantum mean values. In section \[sec: dynamics\], we studied the classical nuclear dynamics on the TDPES. However, we did not provide any argument on how that study can be associated with a classical limit of the nuclear motion that is able to, approximately, reproduce the expectation values of the nuclear position and momentum of the complete electron-nuclear system. Here, using the Ehrenfest theorem, we show how the nuclear position and momentum calculated from Eq. (\[eq: hamilton-eom\]) can be linked to the expectation values of the nuclear position and momentum of the complete electron-nuclear system.
The Ehrenfest theorem [@ehrenfest] relates the time-derivative of the expectation value of a quantum-mechanical operator $\hat
O$ to the expectation value of the commutator of that operator with the Hamiltonian, i.e. $$\frac{d}{dt}\langle\hat O(t)\rangle = \frac{1}{i\hbar}\left\langle\left[\hat O(t),\hat H\right]\right\rangle+
\langle\partial_t\hat O(t)\rangle.$$ The second term on the RHS refers to the explicit time-dependence of $\hat O$. In particular, the theorem leads to the classical-like equations of motion for the mean value of position and momentum operators. For a system of electrons and nuclei, described by the Hamiltonian in Eq. (\[eqn: hamiltonian\]) and the wave-function $\Psi(\dulr,\dulR,t)$, the mean values of the $\nu$-th nuclear position $\hat{\bf R}_{\nu}$ and momentum $\hat{\bf
P}_{\nu}$ operators evolve according to the classical Hamilton’s equations $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf R}_{\nu}\rangle_{\Psi}=\frac{1}{i\hbar} \left\langle\left[\hat{\bf R}_\nu,
\hat H(\dulr,\dulR)\right]\right\rangle_\Psi&=&\frac{\langle\hat{\bf P}_{\nu}\rangle_{\Psi}}{M_{\nu}}
\label{eqn: general ehrenfest 1}\\
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=\frac{1}{i\hbar} \left\langle\left[\hat{\bf P}_\nu,
\hat H(\dulr,\dulR)\right]\right\rangle_\Psi&=& \langle-\nabla_{\nu}\big(\hat{V}_{en}(\dulr,\dulR)+\hat{W}_{nn}(\dulR)\big)
\rangle_{{\Psi}}.
\label{eqn: general ehrenfest 2}\end{aligned}$$ Here, the operators do not depend explicitly on time and we indicate the integration over the full wave-function (electronic and nuclear coordinates) by $\langle\,\cdot\,\rangle_{{\Psi}}$. On the other hand, the nuclear equation (\[eqn: exact nuclear eqn\]) is a Schrödinger equation that contains a time-dependent vector potential and a time-dependent scalar potential. Therefore, the Ehrenfest theorem for the nuclear subsystem reads $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf R}_{\nu}\rangle_\chi&=&\frac{1}{i\hbar}\left\langle\left[\hat{\bf R}_\nu,
\hat H_n(\dulR)\right]\right\rangle_\chi\label{eqn: ehrenfest 1}\\
\frac{d}{dt}\langle\hat{\widetilde{\bf P}}_{\nu}\rangle_\chi&=&\frac{1}{i\hbar}\left\langle\left[
\hat{\widetilde{\bf P}}_\nu,\hat H_n(\dulR)\right]\right\rangle_\chi+
\left\langle\partial_t{\bf A}_\nu(\dulR,t)\right\rangle_\chi \label{eqn: ehrenfest 2}\end{aligned}$$ where [@AMG2] $$\hat{\widetilde{\bf P}}_{\nu} = -i\hbar\nabla_\nu+{\bf A}_{\nu}(\dulR,t)$$ is the expression of the nuclear canonical momentum operator in position representation, and $$\hat H_n(\dulR) = \sum_{\nu=1}^{N_n} \frac{\left[-i\hbar\nabla_\nu+\bA_\nu(\dulR,t)\right]^2}{2M_\nu} +
\epsilon(\dulR,t) \label{eqn: nuclear-Hamiltonian}$$ is the nuclear Hamiltonian from Eq. (\[eqn: exact nuclear eqn\]). Note that the average operation is performed only on the nuclear wave-function as indicated by $\langle\,\cdot\,\rangle_{\chi}$. An explicit time-dependence appears in the expression of the momentum operator, due to the presence of the vector potential. This dependence is accounted for in the second term on the RHS of Eq. (\[eqn: ehrenfest 2\]). While Eq. (\[eqn: ehrenfest 1\]) is easily obtained from Eq. (\[eqn: general ehrenfest 1\]) by performing the integration over the electronic part of full wave-function, Eq. (\[eqn: ehrenfest 2\]) is more involved and will be proved as follows. We rewrite LHS of Eq. (\[eqn: general ehrenfest 2\]) as $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=&\int d\dulr d\dulR\,
\left[\Phi_\dulR^*(\dulr,t)\partial_t\chi^*(\dulR,t)+\chi^*(\dulR,t)\partial_t\Phi_\dulR^*(\dulr,t)\right]
\hat{\bf P}_\nu\chi(\dulR,t)\Phi_\dulR(\dulr,t) \nonumber\\
&+\int d\dulr d\dulR\,
\chi^*(\dulR,t)\Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu
\left[\Phi_\dulR(\dulr,t)\partial_t\chi(\dulR,t)+\chi(\dulR,t)\partial_t\Phi_\dulR(\dulr,t)\right].\end{aligned}$$ $\hat{\bf P}_\nu$ being a differential operator in position representation, its action on the factorized wave-function is $$\hat{\bf P}_\nu\chi(\dulR,t)\Phi_\dulR(\dulr,t)= \left(\hat{\bf P}_\nu\chi(\dulR,t)\right)\Phi_\dulR(\dulr,t)+
\chi(\dulR,t)\left(\hat{\bf P}_\nu\Phi_\dulR(\dulr,t)\right).$$ Then we use the nuclear equation (\[eqn: exact nuclear eqn\]) for $$\partial_t\chi(\dulR,t)=\frac{1}{i\hbar}\hat H_n(\dulR) \chi(\dulR,t)$$ and its complex-conjugated ($\hat H_n(\dulR)$ is hermitian), the definition of the (real) vector potential $${\bf A}_\nu(\dulR,t) = \int d\dulr \,\Phi_\dulR^*(\dulr,t) \hat{\bf P}_\nu\Phi_\dulR(\dulr,t)$$ and the PNC, to derive $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=\frac{1}{i\hbar}\int d\dulR\,\chi^*(\dulR,t)
&\left(\hat{\widetilde{\bf P}}_\nu\hat H_n(\dulR)-\hat H_n(\dulR)\hat{\widetilde{\bf P}}_\nu\right)\chi(\dulR,t)
\nonumber \\
+\int d\dulR\left|\chi(\dulR,t)\right|^2 \int d\dulr &\left[\left(\partial_t\Phi_\dulR^*(\dulr,t)\right)
\hat{\bf P}_\nu\Phi_\dulR(\dulr,t) + \Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu\partial_t\Phi_\dulR(\dulr,t)\right]\end{aligned}$$ with $\hat{\widetilde{\bf P}}_\nu=\hat{\bf P}_\nu+{\bf A}_\nu(\dulR,t)$. Using the relation $$\left(\partial_t\Phi_\dulR^*(\dulr,t)\right)\hat{\bf P}_\nu\Phi_\dulR(\dulr,t) =
\partial_t\left(\Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu\Phi_\dulR(\dulr,t)\right) -
\Phi_\dulR^*(\dulr,t)\hat{\bf P}_\nu\partial_t\Phi_\dulR(\dulr,t),$$ for the term in the square brackets, leads to $$\begin{aligned}
\frac{d}{dt}\langle\hat{\bf P}_{\nu}\rangle_{\Psi}=\int d\dulR\,\chi^*(\dulR,t)\left(
\frac{1}{i\hbar}\left[\hat{\widetilde{\bf P}}_\nu,\hat H_n(\dulR)\right]+\partial_t{\bf A}_\nu(\dulR,t)\right)
\chi(\dulR,t),\end{aligned}$$ recovering the term on the RHS of Eq. (\[eqn: ehrenfest 2\]). A similar procedure [@AMG2] yields the relation $$\begin{aligned}
\langle\hat{\bf P}_\nu\rangle_\Psi &= \int d\dulr d\dulR \,\Phi_\dulR^*(\dulr,t)\chi^*(\dulR,t)
\left[\left(\hat{\bf P}_\nu\chi(\dulR,t)\right)\Phi_\dulR(\dulr,t)+\chi(\dulR,t)\hat{\bf
P}_\nu\Phi_\dulR(\dulr,t)\right] \nonumber \\
&=\int d\dulR \,\chi^*(\dulR,t)\left[\hat{\bf P}_\nu+{\bf A}_\nu(\dulR,t)\right]\chi(\dulR,t)=\langle\hat{\widetilde{\bf
P}}_\nu\rangle_\chi,\end{aligned}$$ which proves the identity of the LHSs of Eqs. (\[eqn: general ehrenfest 2\]) and (\[eqn: ehrenfest 2\]).
We have proved the Ehrenfest theorem for the nuclear wave-function and nuclear Hamiltonian, deriving exact relations for the evolution of the mean values of nuclear position and momentum operators over the complete system. This outcome is consistent with the interpretation of $\chi(\dulR,t)$ as the proper nuclear wave-function that reproduces the nuclear density and current density of the complete system (see the discussion in section \[sec: background\]).
In the one-dimensional system studied here, the gauge is chosen such that $A(R,t)=0$, therefore, the Ehrenfest equations become $$\begin{aligned}
\frac{d}{dt}\langle\hat R\rangle_\chi=\frac{1}{i\hbar}\left\langle\left[\hat R,\hat H_n\right]\right\rangle_\chi
&=&\frac{\langle\hat P\rangle_\chi}{M}\label{eqn: ehrenfest 1 one-d}\\
\frac{d}{dt}\langle\hat P\rangle_\chi=\frac{1}{i\hbar}\left\langle\left[\hat P,\hat H_n\right]\right\rangle_\chi
&=&\langle-\nabla_R\epsilon(R,t)\rangle_\chi, \label{eqn: ehrenfest 2 one-d}\end{aligned}$$ where the mean force generating the classical-like evolution is determined as the expectation value, on the nuclear wave-function, of the gradient of the TDPES. If we replace the nuclear wave-function in Eqs. (\[eqn: ehrenfest 1 one-d\]) and (\[eqn: ehrenfest 2 one-d\]) by a delta-function centered at the classical position, we get Eqs. (\[eq: hamilton-eom\]) that was used in section \[sec: dynamics\] to generate classical dynamics on the exact PES. That is why the classical nuclear dynamics on the TDPES could actually approximate the mean nuclear position and momentum.
We have numerically simulated classical dynamics under the following equations of motion $$\left\lbrace
\begin{array}{ccl}
\dot R &=&\dfrac{P}{M}\\
&& \\
\dot P&=&\langle-\nabla_R\epsilon(R,t)\rangle_\chi,
\end{array}
\right.$$ where $\epsilon(R,t)$ is obtained from the solution of the TDSE with Hamiltonian (\[eqn: metiu-hamiltonian\]), for both sets of parameters producing strong and weak non-adiabatic coupling between the two lowest BO surfaces. The initial conditions for the classical evolution are exactly the initial mean position and mean velocity of the quantum particle.
![Left: nuclear position as a function of time. Right: nuclear velocity as a function of time. The average position and velocity calculated from the quantum-mechanical (QM) propagation are shown as dotted red (strong coupling) and dotted green (weak coupling) lines. The long-dashed (strong coupling) and short-dashed (weak coupling) black lines are the results of classical propagation driven by the average force (AV) as in Eqs. (\[eqn: ehrenfest 1 one-d\]) and (\[eqn: ehrenfest 2 one-d\]).[]{data-label="fig: ehrenfest"}](./Figure10.pdf)
The results are shown in Fig. \[fig: ehrenfest\], where we plot the mean position (left) and velocity (right) as functions of time from quantum-mechanical calculations, compared to the values of position and velocity of a classical particle moving according to the average force $\langle-\nabla_R\epsilon(R,t)\rangle_\chi$. As expected by the proof of the Ehrenfest theorem involving the nuclear wave-function $\chi(\dulR,t)$ and the nuclear Hamiltonian $\hat H_n$ presented in this section, the classical trajectory perfectly follows the evolution of the quantum mean values.
Conclusion {#sec: conclusion}
==========
In a system of interacting electrons and nuclei, the nuclear dynamics is fully determined by the TDPES and the time-dependent vector potential defined in the framework of the exact decomposition of the electronic and nuclear motions, as presented in this paper. We investigated some situations in which the vector potential can be gauged away, thus making the TDPES responsible for the nuclear evolution. This time-dependent scalar potential presents distinct and general features that can be analyzed in terms of its GI and GD components. The former, (i) in the region of an avoided crossing has a pronounced *diabatic* character, smoothly connecting different BOPESs along the direction of the nuclear wave-packet’s motion, and, (ii) further away from the avoided crossing, *dynamical steps* appear between regions in which the (GI part of the) exact potential coincides with one or the other BOPES. The latter is either constant, if the nuclear wave-packet does not split, or stepwise constant, with the step at the same position, and with opposite slope, as in the GI part of the TDPES. We have analyzed in detail these features and discussed the connections with a classical picture of the nuclear evolution. To this end, we calculated the classical forces from the TDPES and from its GI component and performed classical nuclear dynamics driven by those forces. The importance of the GD part of the potential is evident as it improves the agreement of classical results with the quantum-mechanical calculations. We conclude that, if the exact TDPES is available, a single classical trajectory is able to reproduce quantum results fairly well, as long as quantum nuclear effects, such as tunneling or splitting of the nuclear wave-packet, are negligible. We have seen, in the example presented in the paper, that the splitting of the nuclear wave-function at the avoided crossing, that cannot be captured in the classical study, is responsible for the deviation of the classical results from the expected quantum behavior. Further analysis involving the propagation of multiple independent trajectories on the exact TDPES are envisaged. Such a multi-trajectory approach should be able to reproduce non-adiabatic effects, as those described above.
The development of mixed quantum-classical schemes to treat the non-adiabatic coupled electron-nuclear dynamics is still a challenging topic in physics and chemistry. Investigating the properties of the exact potential, that incorporates the effects of the electronic quantum dynamics on the nuclei, is a first step towards understanding the key features of approximated potentials and algorithms. We did not consider here cases where the vector potential cannot be gauged away. This will be the subject of future investigations.
In the final part of the paper, we have shown that the Ehrenfest theorem applied to calculate the mean nuclear position and momentum based on the nuclear equation alone reproduces the mean values calculated from the complete electron-nuclear system.
Acknowledgements {#acknowledgements .unnumbered}
================
Partial support from the Deutsche Forschungsgemeinschaft (SFB 762) and from the European Commission (FP7-NMP-CRONOS) is gratefully acknowledged.
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[^1]: The PNC is inserted in the calculation of the stationary variations of the quantum mechanical action by means of Lagrange multipliers.
[^2]: The scalar and vector potentials are uniquely determined up to within a gauge transformation, given in Eqs. (\[eqn: transformation of epsilon\]) and (\[eqn: transformation of A\]). However, as expected, the nuclear Hamiltonian in Eq. (\[eqn: exact nuclear eqn\]) is form-invariant under such transformations.
[^3]: We reintroduce the bold-double underlined symbols for electronic and nuclear positions whenever the statements have general validity.
|
---
abstract: 'An atom structure of type $\mathcal{T}$ is said to be strongly representable if all atomic algebras (of the same type $\mathcal{T}$) with that atom structure are representable. We show that for any finite $n\geq 3$ and any signature $\mathcal{T}$ between $Df_{n}$ and $QEA_{n}$, the class of strongly representable atom structures of type $\mathcal{T}$ is not elementary. We extensively use graphs and games as introduced in algebraic logic by Hirsch and Hodkinson.'
author:
- |
Mohamed Khaled and Tarek Sayed Ahmed\
Department of Mathematics, Faculty of Science,\
Cairo University, Giza, Egypt.
title: Strongly representable atom structures
---
Introduction
============
In [@hirsh], Hirsch and Hodkinson proved that for finite $n\geq 3$, the class of strongly representable cylindric-type atom structures of dimension $n$ is not definable by any set of first-order sentences: it is not elementary class. Their method depends on that $RCA_n$ is a variety, an atomic algebra $\A$ will be in $RCA_n$ if all the equations defining $RCA_n$ are valid in $\A$. From the point of view of $At\A$, each equation corresponds to a certain universal monadic second-order statement, where the universal quantifiers are restricted to ranging over the sets of atoms that are defined by elements of $\A$. Such a statement will fail in $\A$ if $At\A$ can be partitioned into finitely many $\A$-definable sets with certain properties - they call this a bad partition. This idea can be used to show that $RCA_n$ (for $n\geq 3$) is not finitely axiomatizable, by finding a sequence of atom structures, each having some sets that form a bad partition, but with the minimal number of sets in a bad partition increasing as we go along the sequence. This can yield algebras not in $RCA_n$ but with an ultraproduct that is in $RCA_n$. In this article we extend the result of Hirsch and Hodkinson to any class of strongly representable atom structure having signature between the diagonal free atom structures and the quasi polyadic equality atom structures (recall the definitions of such algebras from [@tarski] and [@thompson]). As in [@hirsh] we deal only with finite dimensional algebras. Fix a finite dimension $n <\omega$, with $n\geq 3$.
Atom structures
===============
The action of the non-boolean operators in a completely additive atomic $BAO$ is determined by their behavior over the atoms, and this in turn is encoded by the atom structure of the algebra.
(**Atom Structure**)\
Let $\mathcal{A}=\langle A, +, -, 0, 1, \Omega_{i}:i\in I\rangle$ be an atomic boolean algebra with operators $\Omega_{i}:i\in I$. Let the rank of $\Omega_{i}$ be $\rho_{i}$. The *atom structure* $At\mathcal{A}$ of $\mathcal{A}$ is a relational structure
$\langle At\mathcal{A}, R_{\Omega_{i}}:i\in I\rangle$
where $At\mathcal{A}$ is the set of atoms of $\mathcal{A}$ as before, and $R_{\Omega_{i}}$ is a $(\rho(i)+1)$-ary relation over $At\mathcal{A}$ defined by$$R_{\Omega_{i}}(a_{0},
\cdots, a_{\rho(i)})\Longleftrightarrow\Omega_{i}(a_{1}, \cdots,
a_{\rho(i)})\geq a_{0}.$$
Similar ’dual’ structure arise in other ways, too. For any not necessarily atomic $BAO$ $\mathcal{A}$ as above, its *ultrafilter frame* is the structure$$\mathcal{A}_{+}=\langle Uf(\mathcal{A}),
R_{\Omega_{i}}:i\in I\rangle,$$ where $Uf(\mathcal{A})$ is the set of all ultrafilters of (the boolean reduct of) $\mathcal{A}$, and for $\mu_{0}, \cdots, \mu_{\rho(i)}\in
Uf(\mathcal{A})$, we put $R_{\Omega_{i}}(\mu_{0}, \cdots,
\mu_{\rho(i)})$ iff $\{\Omega(a_{1}, \cdots,
a_{\rho(i)}):a_{j}\in\mu_{j}$ for $0<j\leq\rho(i)\}\subseteq\mu_{0}$.
(**Complex algebra**)\
Conversely, if we are given an arbitrary structure $\mathcal{S}=\langle S, r_{i}:i\in I\rangle$ where $r_{i}$ is a $(\rho(i)+1)$-ary relation over $S$, we can define its *complex algebra*$$\mathfrak{Cm}(\mathcal{S})=\langle \wp(S),
\cup, \setminus, \phi, S, \Omega_{i}\rangle_{i\in
I},$$where $\wp(S)$ is the power set of $S$, and $\Omega_{i}$ is the $\rho(i)$-ary operator defined by$$\Omega_{i}(X_{1}, \cdots, X_{\rho(i)})=\{s\in
S:\exists s_{1}\in X_{1}\cdots\exists s_{\rho(i)}\in X_{\rho(i)},
r_{i}(s, s_{1}, \cdots, s_{\rho(i)})\},$$ for each $X_{1}, \cdots, X_{\rho(i)}\in\wp(S)$.
It is easy to check that, up to isomorphism, $At(\mathfrak{Cm}(\mathcal{S}))\cong\mathcal{S}$ always, and $\mathcal{A}\subseteq\mathfrak{Cm}(At\mathcal{A})$ for any completely additive atomic $BAO$ $\mathcal{A}$. If $\mathcal{A}$ is finite then of course $\mathcal{A}\cong\mathfrak{Cm}(At\mathcal{A})$.\
- [Atom structure of diagonal free-type algebra is $\mathcal{S}=\langle S, R_{c_{i}}:i<n\rangle$, where the $R_{c_{i}}$ is binary relation on $S$.]{}
- [Atom structure of cylindric-type algebra is $\mathcal{S}=\langle S, R_{c_{i}}, R_{d_{ij}}:i, j<n\rangle$, where the $R_{d_{ij}}$, $R_{c_{i}}$ are unary and binary relations on $S$. The reduct $\mathfrak{Rd}_{df}\mathcal{S}=\langle S,
R_{c_{i}}:i<n\rangle$ is an atom structure of diagonal free-type.]{}
- [Atom structure of substitution-type algebra is $\mathcal{S}=\langle S, R_{c_{i}}, R_{s^{i}_{j}}:i, j<n\rangle$, where the $R_{d_{ij}}$, $R_{s^{i}_{j}}$ are unary and binary relations on $S$, respectively. The reduct $\mathfrak{Rd}_{df}\mathcal{S}=\langle S,
R_{c_{i}}:i<n\rangle$ is an atom structure of diagonal free-type.]{}
- [Atom structure of quasi polyadic-type algebra is $\mathcal{S}=\langle S, R_{c_{i}}, R_{s^{i}_{j}}$, $R_{s_{ij}}:i, j<n\rangle$, where the $R_{c_{i}}$, $R_{s^{i}_{j}}$ and $R_{s_{ij}}$ are binary relations on $S$. The reducts $\mathfrak{Rd}_{df}\mathcal{S}=\langle S,
R_{c_{i}}:i<n\rangle$ and $\mathfrak{Rd}_{Sc}\mathcal{S}=\langle S,
R_{c_{i}}, R_{s^{i}_{j}}:i, j<n\rangle$ are atom structures of diagonal free and substitution types, respectively.]{}
- Atom structure of quasi polyadic equality-type algebra is $\mathcal{S}=\langle S, R_{c_{i}}, R_{d_{ij}}, R_{s^{i}_{j}}, R_{s_{ij}}:i, j<n\rangle$, where the $R_{d_{ij}}$ is unary relation on $S$, and $R_{c_{i}}$, $R_{s^{i}_{j}}$ and $R_{s_{ij}}$ are binary relations on $S$.
- [The reduct $\mathfrak{Rd}_{df}\mathcal{S}=\langle S, R_{c_{i}}:i\in
I\rangle$ is an atom structure of diagonal free-type.]{}
- [The reduct $\mathfrak{Rd}_{ca}\mathcal{S}=\langle S, R_{c_{i}},
R_{d_{ij}}:i, j\in I\rangle$ is an atom structure of cylindric-type.]{}
- [The reduct $\mathfrak{Rd}_{Sc}\mathcal{S}=\langle S, R_{c_{i}},
R_{s^{i}_{j}}:i, j\in I\rangle$ is an atom structure of substitution-type.]{}
- [The reduct $\mathfrak{Rd}_{qa}\mathcal{S}=\langle S, R_{c_{i}},
R_{s^{i}_{j}}, R_{s_{ij}}:i, j\in I\rangle$ is an atom structure of quasi polyadic-type.]{}
An algebra is said to be representable if and only if it is isomorphic to a subalgebra of a direct product of set algebras of the same type.
\[strong rep\]Let $\mathcal{S}$ be an $n$-dimensional algebra atom structure. $\mathcal{S}$ is *strongly representable* if every atomic $n$-dimensional algebra $\mathcal{A}$ with $At\mathcal{A}=\mathcal{S}$ is representable. We write $SDfS_{n}$, $SCS_{n}$, $SSCS_{n}$, $SQS_{n}$ and $SQES_{n}$ for the classes of strongly representable ($n$-dimensional) diagonal free, cylindric, substitution, quasi polyadic and quasi polyadic equality algebra atom structures, respectively.
Note that for any $n$-dimensional algebra $\mathcal{A}$ and atom structure $\mathcal{S}$, if $At\mathcal{A}=\mathcal{S}$ then $\mathcal{A}$ embeds into $\mathfrak{Cm}\mathcal{S}$, and hence $\mathcal{S}$ is strongly representable iff $\mathfrak{Cm}\mathcal{S}$ is representable.
Graphs and Strong representability
==================================
In this section, by a graph we will mean a pair $\Gamma=(G, E)$, where $G\not=\phi$ and $E\subseteq G\times G$ is a reflexive and symmetric binary relation on $G$. We will often use the same notation for $\Gamma$ and for its set of nodes ($G$ above). A pair $(x, y)\in E$ will be called an edge of $\Gamma$. See [@graph] for basic information (and a lot more) about graphs.
Let $\Gamma=(G, E)$ be a graph.
1. [A set $X\subset G$ is said to be *independent* if $E\cap(X\times X)=\phi$.]{}
2. [The *chromatic number* $\chi(\Gamma)$ of $\Gamma$ is the smallest $\kappa<\omega$ such that $G$ can be partitioned into $\kappa$ independent sets, and $\infty$ if there is no such $\kappa$.]{}
- For an equivalence relation $\sim$ on a set $X$, and $Y\subseteq
X$, we write $\sim\upharpoonright Y$ for $\sim\cap(Y\times Y)$. For a partial map $K:n\rightarrow\Gamma\times n$ and $i, j<n$, we write $K(i)=K(j)$ to mean that either $K(i)$, $K(j)$ are both undefined, or they are both defined and are equal.
- For any two relations $\sim$ and $\approx$. The composition of $\sim$ and $\approx$ is the set $$\sim\circ\approx=\{(a, b):\exists c(a\sim c\wedge c\approx b)\}.$$
Let $\Gamma$ be a graph. We define an atom structure $\eta(\Gamma)=\langle H, D_{ij},
\equiv_{i}, \equiv_{ij}:i, j<n\rangle$ as follows:
1. $H$ is the set of all pairs $(K, \sim)$ where $K:n\rightarrow \Gamma\times n$ is a partial map and $\sim$ is an equivalent relation on $n$ satisfying the following conditions
1. [If $|n\diagup\sim|=n$, then $dom(K)=n$ and $rng(K)$ is not independent subset of $n$.]{}
2. [If $|n\diagup\sim|=n-1$, then $K$ is defined only on the unique $\sim$ class $\{i, j\}$ say of size $2$ and $K(i)=K(j)$.]{}
3. [If $|n\diagup\sim|\leq n-2$, then $K$ is nowhere defined.]{}
2. [$D_{ij}=\{(K, \sim)\in H : i\sim j\}$.]{}
3. [$(K, \sim)\equiv_{i}(K', \sim')$ iff $K(i)=K'(i)$ and $\sim\upharpoonright(n\setminus\{i\})=\sim'\upharpoonright(n\setminus\{i\})$.]{}
4. [$(K, \sim)\equiv_{ij}(K', \sim')$ iff $K(i)=K'(j)$, $K(j)=K'(i)$, and $K(\kappa)=K'(\kappa) (\forall\kappa\in n\setminus\{i, j\})$ and if $i\sim j$ then $\sim=\sim'$, if not, then $\sim'=\sim\circ[i,
j]$.]{}
It may help to think of $K(i)$ as assigning the nodes $K(i)$ of $\Gamma\times n$ not to $i$ but to the set $n\setminus\{i\}$, so long as its elements are pairwise non-equivalent via $\sim$.\
For a set $X$, $\mathcal{B}(X)$ denotes the boolean algebra $\langle\wp(X), \cup, \setminus\rangle$. We write $a\cap b$ for $-(-a\cup-b)$.
\[our algebra\] Let $\mathfrak{B}(\Gamma)=\langle\mathcal{B}(\eta(\Gamma)), c_{i},
s^{i}_{j}, s_{ij}, d_{ij}\rangle_{i, j<n}$ be the algebra, with extra non-Boolean operations defined as follows:
$d_{ij}=D_{ij}$,\
$c_{i}X=\{c: \exists a\in X, a\equiv_{i}c\}$,\
$s_{ij}X=\{c: \exists a\in X, a\equiv_{ij}c\}$,\
$s^{i}_{j}X=\begin{cases}
c_{i}(X\cap D_{ij}), &\text{if $i\not=j$,}\\
X, &\text{if $i=j$.}
\end{cases}$ For all $X\subseteq \eta(\Gamma)$.
For any $\tau\in\{\pi\in n^{n}: \pi \text{ is a bijection}\}$, and any $(K, \sim)\in\eta(\Gamma)$. We define $\tau(K,
\sim)=(K\circ\tau, \sim\circ\tau)$.
The proof of the following two Lemmas is straightforward.
\[Lemma 1\] \
For any $\tau\in\{\pi\in n^{n}: \pi \text{ is a bijection}\}$, and any $(K, \sim)\in\eta(\Gamma)$. $\tau(K, \sim)\in\eta(\Gamma)$.
\[Lemma 2\] \
For any $(K, \sim)$, $(K', \sim')$, and $(K'', \sim'')\in\eta(\Gamma)$, and $i, j\in n$:
1. [$(K, \sim)\equiv_{ii}(K', \sim')\Longleftrightarrow (K, \sim)=(K', \sim')$.]{}
2. [$(K, \sim)\equiv_{ij}(K', \sim')\Longleftrightarrow (K, \sim)\equiv_{ji}(K', \sim')$.]{}
3. [If $(K, \sim)\equiv_{ij}(K', \sim')$, and $(K, \sim)\equiv_{ij}(K'', \sim'')$, then $(K', \sim')=(K'', \sim'')$.]{}
4. [If $(K, \sim)\in D_{ij}$, then\
$(K, \sim)\equiv_{i}(K', \sim')\Longleftrightarrow\exists(K_{1},
\sim_{1})\in\eta(\Gamma):(K, \sim) \equiv_{j}(K_{1},
\sim_{1})\wedge(K', \sim')\equiv_{ij}(K_{1}, \sim_{1})$.]{}
5. [$s_{ij}(\eta(\Gamma))=\eta(\Gamma)$.]{}
\[it is qea\] For any graph $\Gamma$, $\mathfrak{B}(\Gamma)$ is a simple $QEA_{n}$.
We follow the axiomatization in [@thompson] except renaming the items by $Q_i$. Let $X\subseteq\eta(\Gamma)$, and $i, j, \kappa\in n$:
- [$s^{i}_{i}=ID$ by definition \[our algebra\], $s_{ii}X=\{c:\exists a\in X, a\equiv_{ii}c\}=\{c:\exists a\in X, a=c\}=X$ (by Lemma \[Lemma 2\] (1));\
$s_{ij}X=\{c:\exists a\in X, a\equiv_{ij}c\}=\{c:\exists a\in X,
a\equiv_{ji}c\}=s_{ji}X$ (by Lemma \[Lemma 2\] (2)).]{}
- [Axioms $Q_{1}$, $Q_{2}$ follow directly from the fact that the reduct $\mathfrak{Rd}_{ca}\mathfrak{B}(\Gamma)=\langle\mathcal{B}(\eta(\Gamma)), c_{i}$, $d_{ij}\rangle_{i, j<n}$ is a cylindric algebra which is proved in [@hirsh].]{}
- [Axioms $Q_{3}$, $Q_{4}$, $Q_{5}$ follow from the fact that the reduct $\mathfrak{Rd}_{ca}\mathfrak{B}(\Gamma)$ is a cylindric algebra, and from [@tarski] (Theorem 1.5.8(i), Theorem 1.5.9(ii), Theorem 1.5.8(ii)).]{}
- [$s^{i}_{j}$ is a boolean endomorphism by [@tarski] (Theorem 1.5.3). $$\begin{aligned}
s_{ij}(X\cup Y)=&&\{c:\exists a\in(X\cup Y), a\equiv_{ij}c\}\\
=&&\{c:(\exists a\in X\vee\exists a\in Y), a\equiv_{ij}c\}\\
=&&\{c:\exists a\in X, a\equiv_{ij}c\}\cup\{c:\exists a\in Y,
a\equiv_{ij}c\}\\
=&&s_{ij}X\cup s_{ij}Y.\end{aligned}$$ $s_{ij}(-X)=\{c:\exists a\in(-X), a\equiv_{ij}c\}$, and $s_{ij}X=\{c:\exists a\in X, a\equiv_{ij}c\}$ are disjoint. For, let $c\in(s_{ij}(X)\cap s_{ij}(-X))$, then $\exists a\in X\wedge b\in
(-X)$, such that $a\equiv_{ij}c$, and $b\equiv_{ij}c$. Then $a=b$, (by Lemma \[Lemma 2\] (3)), which is a contradiction. Also, $$\begin{aligned}
s_{ij}X\cup s_{ij}(-X)=&&\{c:\exists a\in X,
a\equiv_{ij}c\}\cup\{c:\exists a\in(-X), a\equiv_{ij}c\}\\
=&&\{c:\exists a\in(X\cup-X), a\equiv_{ij}c\}\\
=&&s_{ij}\eta(\Gamma)\\
=&&\eta(\Gamma). \text{ (by Lemma \ref{Lemma 2} (5))}\end{aligned}$$ therefore, $s_{ij}$ is a boolean endomorphism.]{}
- [$$\begin{aligned}
s_{ij}s_{ij}X=&&s_{ij}\{c:\exists a\in X, a\equiv_{ij}c\}\\
=&&\{b:(\exists a\in X\wedge c\in\eta(\Gamma)), a\equiv_{ij}c, \text{ and }
c\equiv_{ij}b\}\\
=&&\{b:\exists a\in X, a=b\}\\
=&&X.\end{aligned}$$]{}
- [$$\begin{aligned}
s_{ij}s^{i}_{j}X=&&\{c:\exists a\in s^{i}_{j}X, a\equiv_{ij}c\}\\
=&&\{c:\exists b\in(X\cap d_{ij}),a\equiv_{i}b\wedge
a\equiv_{ij}c\}\\
=&&\{c:\exists b\in(X\cap d_{ij}), c\equiv_{j}b\} \text{ (by Lemma
\ref{Lemma 2} (4))}\\
=&&s^{j}_{i}X.\end{aligned}$$]{}
- [We need to prove that $s_{ij}s_{i\kappa}X=s_{j\kappa}s_{ij}X$ if $|\{i, j, \kappa\}|=3$. For, let $(K, \sim)\in s_{ij}s_{i\kappa}X$ then $\exists(K', \sim')\in\eta(\Gamma)$, and $\exists(K'', \sim'')\in X$ such that $(K'', \sim'')\equiv_{i\kappa}(K', \sim')$ and $(K',
\sim')\equiv_{ij}(K, \sim)$.\
Define $\tau:n\rightarrow n$ as follows: $$\begin{aligned}
\tau(i)=&&j\\
\tau(j)=&&\kappa\\
\tau(\kappa)=&&i, \text{ and}\\
\tau(l)=&&l \text{ for every } l\in(n\setminus\{i, j,
\kappa\}).\end{aligned}$$ Now, it is easy to verify that $\tau(K',
\sim')\equiv_{ij}(K'', \sim'')$, and $\tau(K',
\sim')\equiv_{j\kappa}(K, \sim)$. Therefore, $(K, \sim)\in
s_{j\kappa}s_{ij}X$, i.e., $s_{ij}s_{i\kappa}X\subseteq
s_{j\kappa}s_{ij}X$. Similarly, we can show that $s_{j\kappa}s_{ij}X\subseteq s_{ij}s_{i\kappa}X$.]{}
- [Axiom $Q_{10}$ follows from [@tarski] (Theorem 1.5.7)]{}
- [Axiom $Q_{11}$ follows from axiom 2, and the definition of $s^{i}_{j}$.]{}
Since $\mathfrak{Rd}_{ca}\mathfrak{B}$ is a simple $CA_{n}$, by [@hirsh], then $\mathfrak{B}$ is simple.
Let $\mathfrak{C}(\Gamma)$ be the subalgebra of $\mathfrak{B}(\Gamma)$ generated by the set of atoms.
Note that the cylindric algebra constructed in [@hirsh] is $\mathfrak{Rd}_{ca}\mathfrak{B}(\Gamma)$ not $\mathfrak{Rd}_{ca}\mathfrak{C}(\Gamma)$, but all results in [@hirsh] can be applied to $\mathfrak{Rd}_{ca}\mathfrak{C}(\Gamma)$. Therefore, since our results depends basically on [@hirsh], we will refer to [@hirsh] directly when we apply it to catch any result about $\mathfrak{Rd}_{ca}\mathfrak{C}(\Gamma)$.
$\mathfrak{C}(\Gamma)$ is a simple $QEA_{n}$ generated by the set of the $n-1$ dimensional elements.
$\mathfrak{C}(\Gamma)$ is a simple $QEA_{n}$ from Theorem \[it is qea\]. It remains to show that $\{(K, \sim)\}=\prod\{c_{i}\{(K,
\sim)\}: i<n\}$ for any $(K, \sim)\in H$. Let $(K, \sim)\in H$, clearly $\{(K, \sim)\}\leq\prod\{c_{i}\{(K, \sim)\}: i<n\}$. For the other direction assume that $(K', \sim')\in H$ and $(K,
\sim)\not=(K', \sim')$. We show that $(K',
\sim')\not\in\prod\{c_{i}\{(K, \sim)\}: i<n\}$. Assume toward a contradiction that $(K', \sim')\in\prod\{c_{i}\{(K, \sim)\}:i<n\}$, then $(K', \sim')\in c_{i}\{(K, \sim)\}$ for all $i<n$, i.e., $K'(i)=K(i)$ and $\sim'\upharpoonright(n\setminus\{i\})=\sim\upharpoonright(n\setminus\{i\})$ for all $i<n$. Therefore, $(K, \sim)=(K', \sim')$ which makes a contradiction, and hence we get the other direction.
\[chr. no.\] Let $\Gamma$ be a graph.
1. [Suppose that $\chi(\Gamma)=\infty$. Then $\mathfrak{C}(\Gamma)$ is representable.]{}
2. [If $\Gamma$ is infinite and $\chi(\Gamma)<\infty$ then $\mathfrak{Rd}_{df}\mathfrak{C}$ is not representable.]{}
<!-- -->
1. [We have $\mathfrak{Rd}_{ca}\mathfrak{C}$ is representable (c.f., [@hirsh]). Let $X=\{x\in \mathfrak{C}:\Delta x\not=n\}$. Call $J\subseteq \mathfrak{C}$ inductive if $X\subseteq J$ and $J$ is closed under infinite unions and complementation. Then $\mathfrak{C}$ is the smallest inductive subset of $\mathfrak{C}$. Let $f$ be an isomorphism of $\mathfrak{Rd}_{ca}\mathfrak{C}$ onto a cylindric set algebra with base $U$. Clearly, by definition, $f$ preserves $s^{i}_{j}$ for each $i, j<n$. It remains to show that $f$ preserves $s_{ij}$ for every $i, j<n$. Let $i, j<n$, since $s_{ij}$ is boolean endomorphism and completely additive, it suffices to show that $fs_{ij}x=s_{ij}fx$ for all $x\in At\mathfrak{C}$. Let $x\in At\mathfrak{C}$ and $\mu\in
n\setminus\Delta x$. If $\kappa=\mu$ or $l=\mu$, say $\kappa=\mu$, then$$fs_{\kappa l}x=fs_{\kappa
l}c_{\kappa}x=fs^{\kappa}_{l}x=s^{\kappa}_{l}fx=s_{\kappa l}fx.$$ If $\mu\not\in\{\kappa, l\}$ then$$fs_{\kappa
l}x=fs^{l}_{\mu}s^{\kappa}_{l}s^{\mu}_{\kappa}c_{\mu}x=s^{l}_{\mu}s^{\kappa}_{l}s^{\mu}_{\kappa}c_{\mu}fx=s_{\kappa
l}fx.$$]{}
2. [Assume toward a contradiction that $\mathfrak{Rd}_{df}\mathfrak{C}$ is representable. Since $\mathfrak{Rd}_{ca}\mathfrak{C}$ is generated by $n-1$ dimensional elements then $\mathfrak{Rd}_{ca}\mathfrak{C}$ is representable. But this contradicts Proposition 5.4 in [@hirsh].]{}
\
Let $2<n<\omega$ and $\mathcal{T}$ be any signature between $Df_{n}$ and $QEA_{n}$. Then the class of strongly representable atom structures of type $\mathcal{T}$ is not elementary.
By Erdös’s famous 1959 Theorem [@Erdos], for each finite $\kappa$ there is a finite graph $G_{\kappa}$ with $\chi(G_{\kappa})>\kappa$ and with no cycles of length $<\kappa$. Let $\Gamma_{\kappa}$ be the disjoint union of the $G_{l}$ for $l>\kappa$. Clearly, $\chi(\Gamma_{\kappa})=\infty$. So by Theorem \[chr. no.\] (1), $\mathfrak{C}(\Gamma_{\kappa})=\mathfrak{C}(\Gamma_{\kappa})^{+}$ is representable.\
Now let $\Gamma$ be a non-principal ultraproduct $\prod_{D}\Gamma_{\kappa}$ for the $\Gamma_{\kappa}$. It is certainly infinite. For $\kappa<\omega$, let $\sigma_{\kappa}$ be a first-order sentence of the signature of the graphs. stating that there are no cycles of length less than $\kappa$. Then $\Gamma_{l}\models\sigma_{\kappa}$ for all $l\geq\kappa$. By [Ł]{}oś’s Theorem, $\Gamma\models\sigma_{\kappa}$ for all $\kappa$. So $\Gamma$ has no cycles, and hence by, [@hirsh] Lemma 3.2, $\chi(\Gamma)\leq 2$. By Theorem \[chr. no.\] (2), $\mathfrak{Rd}_{df}\mathfrak{C}$ is not representable. It is easy to show (e.g., because $\mathfrak{C}(\Gamma)$ is first-order interpretable in $\Gamma$, for any $\Gamma$) that$$\prod_{D}\mathfrak{C}(\Gamma_{\kappa})\cong\mathfrak{C}(\prod_{D}\Gamma_{\kappa}).$$ Combining this with the fact that: for any $n$-dimensional atom structure $\mathcal{S}$
$\mathcal{S}$ is strongly representable $\Longleftrightarrow$ $\mathfrak{Cm}\mathcal{S}$ is representable,
the desired follows.
[49]{} Leon Henkin, J.Donald Monk, and Alfred Tarski, Cylindric algebras, part I, II, North-Holland, publishing company, Amesterdam London. I. Sain and R. Thompson. Strictly finite schema axiomatization of Quasi-Polyadic algebras. In “Algebraic Logic”. Editors: H. Andreka, J. Monk and I. Nemeti. North Holland 1989. R. Hirsch and I. Hodkinson. Strongly representable atom structures of cylindric algebras. J. Symbolic Logic, 74:811-828, 2009. P. Erdös, Graph theory and probability, Canadian Journal of Mathematics, vol. 11 (1959), pp. 34-38. R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1997.
|
---
abstract: 'Motivated by problems in search and detection we present a solution to a Combinatorial Multi-Armed Bandit (CMAB) problem with both heavy-tailed reward distributions and a new class of feedback, filtered semibandit feedback. In a CMAB problem an agent pulls a combination of arms from a set $\{1,...,k\}$ in each round, generating random outcomes from probability distributions associated with these arms and receiving an overall reward. Under semibandit feedback it is assumed that the random outcomes generated are all observed. Filtered semibandit feedback allows the outcomes that are observed to be sampled from a second distribution conditioned on the initial random outcomes. This feedback mechanism is valuable as it allows CMAB methods to be applied to sequential search and detection problems where combinatorial actions are made, but the true rewards (number of objects of interest appearing in the round) are not observed, rather a filtered reward (the number of objects the searcher successfully finds, which must by definition be less than the number that appear). We present an upper confidence bound type algorithm, Robust-F-CUCB, and associated regret bound of order $\mathcal{O}(\ln(n))$ to balance exploration and exploitation in the face of both filtering of reward and heavy tailed reward distributions.'
author:
- |
James A Grant\
*Lancaster University*
- |
David S Leslie\
*Lancaster University*
- |
Kevin Glazebrook\
*Lancaster University*
- |
Roberto Szechtman\
*Naval Postgraduate School*
bibliography:
- 'coltpaperrefs.bib'
title: 'Combinatorial Multi-Armed Bandits with Filtered Feedback'
---
Introduction
============
In this paper we present a solution to Combinatorial Multi-Armed Bandit (CMAB) problem with both heavy-tailed reward distributions and filtered semi-bandit feedback. This work is motivated by an application in search and detection (afforded a more detailed description in Section \[Motivation\]), where an agent sequentially selects combinations of cells to search, aiming to detect some objects in each cell. The number of objects in a given cell in a given round is randomly drawn from a Poisson distribution. The generalisation of previous work on CMAB problems to allow heavy tailed reward distributions is required to accommodate Poisson distributed counts of objects of interest. However due to the imperfect nature of search, this Poisson observation will not necessarily be observed as some events may go undetected. Moreover, the larger the area chosen for search, the less efficient the search will be. The filtered semibandit feedback allows for the observed outcomes to be a second random outcome, drawn from a distribution conditioned on the *true* or initial outcome. This new feedback class allows us to model the imperfect detection of objects of interest.
In a CMAB problem with semibandit feedback an agent is faced with $k$ bandit arms representing basic actions and may select some subset of these arms to play at each time step. Each arm $i \in \{1,...,k\}$ has an associated probability distribution $\nu_i$ with finite mean $\mu_i$, both unknown to the agent. Playing a combination of arms $S \subseteq \{1,..,k\}$ reveals random outcomes sampled independently from distributions $\nu_i : i \in S$ and grants the agent a *reward* $R(S)$ which is a function of the random outcomes observed. The agent’s goal is to maximise her cumulative reward (or equivalently minimise her cumulative regret) through time.
The work in this paper extends the CMAB framework of [@Chen2013] in two directions. Firstly, to one where the underlying probability distributions $\nu_i$ are only restricted to have a bounded moment of order $1+\epsilon$, for $\epsilon \in (0,1]$. Secondly, we introduce *filtered semibandit feedback* where observed outcomes associated with an arm $i$ need not be drawn from $\nu_i$ but can be *filtered observations* drawn from a related *filtering distribution* $\tilde{\nu}_i$ whose parameters depend on the true outcome $X_{i,t}$ drawn from $\nu_i$ and the combination of arms $S_t$ selected in a given round. The reward received is a function of the filtered observations. As in other versions of the CMAB problem the agent’s goal is to maximise her cumulative reward through time.
In addition to introducing this expanded view of the existing CMAB framework, we propose a general class of *upper confidence bound algorithms* for CMAB problems with filtered semibandit feedback, which achieve $\mathcal{O}(\ln n)$ regret subject to the identification of a suitable mean estimator for the specific problem. We include illustrative examples of specific algorithms within this general class for particular distributional families and filtering mechanisms.
So far as we are aware, the notion of observing filtered rewards in a CMAB problem is a new one and no previous work on algorithms for use under filtered semibandit feedback exists. For CMAB problems with semibandit feedback, the majority of work has focussed on a (relatively) simple CMAB framework where (at most) $m < k$ arms are selected in each round and the overall reward observed is simply the sum of the outcomes generated from these $m$ arms. This configuration is sometimes called Learning with Linear rewards or a Multiple play Bandit and has been considerd by authors including [@Gai2012], [@Combes2015], and [@Luedtke2016]. [@Chen2013] was the first paper to consider a more general class of reward functions, allowing all functions that satisfy certain smoothness assumptions. It is upon this work that our research is based. [@Chen2016a] permits an even less restrictive class of reward functions while [@Chen2016b] considers a variant of the usual semibandit feedback where arms not selected in a particular round may still be triggered with a certain probability. We have not attempted to incorporate the two latter innovations in to our work, principally because it was not relevant to our motivating application.
Most Multi-Armed Bandit (MAB) research deals with compact or sub-Gaussian reward distributions, however there are several notable exceptions. In particular [@Bubeck2013] present Robust-UCB algorithms suitable for heavy tailed (non sub-Gaussian) reward distributions with a bounded $1+\epsilon$ moment. Extending these algorithms to Robust-CUCB algorithms will be one of our contributions in this work. The Bayes-UCB method of [@Kaufmann2012] and KL-UCB method of [@Cappe2013] have recently been improved in [@Kaufmann2016] to versions with provable regret bounds of optimal order in MAB problems with exponential family rewards. However, as these algorithms are based upon quantile-type UCB indices, rather than UCB indices which take the form of a mean estimate plus an inflation term, the existing analysis from [@Chen2013] cannot be so easily exploited and we do not consider a combinatorial extension of these methods in this work.
The rest of the paper is organised as follows. In Section \[Motivation\] we outline the aforementioned motivating application and justify its link to the CMAB problem. Section \[Framework\] defines more rigorously our generalisation of the CMAB to include filtering of reward and heavy-tailed reward distributions. Section \[Algorithms Section\] introduces our main Robust-F-CUCB algorithm for this generalised CMAB problem along with a performance guarantee in the form of a bound on expected regret. We conclude by revisiting the motivating example in light of our theoretical work and providing a short discussion.
Motivating Example - Learning in Search {#Motivation}
=======================================
Our inspiration to study these CMAB problems with filtered semibandit feedback comes from a real world problem in search and detection. In this section we describe this motivating problem and its link to Combinatorial bandits.
Problem Specification
---------------------
This research is motivated by the problem of searching for objects over a large area [@Stone1976]. The main assumption is that the target objects appear in the search area according to a nonhomogeneous spatial Poisson process.
Repeated searches are conducted over this search area which is split into a finite number of cells. At each time $t=1,2,...,n$ objects appear according to the Poisson process and the agent selects a subset of the cells to search over (with objects disappearing whether detected or not at time $t+1$). However, the more cells the agent opts to search, the less effective her search can be in any one cell. The key operational question is: How should the cells be searched in order to maximize the expected number of detections over a finite time horizon? We assume that to aid in answering this question, the probability of detecting an object that has appeared in a particular cell given the set of cells the agent opts to search is known.
If the intensity function of the Poisson process were known, the analyst could formulate a mixed integer linear optimization problem to find the optimal subset of cells to patrol. The challenge for the agent is to come up with a patrolling scheme that judiciously balances exploration and exploitation. Specifically, in this example, a patrolling scheme should take the form of a choice of cells to search in rounds $t=1,2,...,n$, where choices may be made after observing the detections from previous rounds.
Link to Combinatorial Multi-Armed Bandits
-----------------------------------------
Clearly this problem in search with an unknown intensity function is a sequential decision problem, where the action space in each round is formed of different combinations of cells that the agent may patrol. A CMAB problem is therefore an appropriate model. In each round the agent will choose a set of cells in which to search, so that cells are viewed as bandit arms. Due to the spatio-temporal Poisson process model we specify, the number of objects appearing in a given cell $i$ over a fixed time window will be Poisson distributed with parameter $\mu_i$, independently of the number of objects in other cells.
However, an added complication comes from the fact that not all objects which appear are detected. Under the choice of combination of arms $S_t$ at time $t$, each object is detected with a certain probability $\gamma_{i,S_t}$ - assumed constant within a round - such that the number of objects observed given the number of objects appearing is Binomially distributed. i.e. if $X_{i,t} \sim Pois(\mu_i)$ is the number of objects appearing in cell $i$ during round $t$, then conditional on $X_{i,t}$ and $S_t$, the number of objects detected $Y_{i,t}$ will have a Binomial distribution such that $Y_{i,t}|X_{i,t},S_t \sim Bin(X_{i,t},\gamma_{i,S_t})$. A consequence of $X_{i,t}$ being Poisson is that the marginal distribution of $Y_{i,t}|S_t$ will follow a $Pois(\gamma_{i,S_t}\mu_i)$ distribution.
So while there is a clear link between the search problem and the CMAB problem in terms of sequential decision making with a combinatorially structured action space, the original CMAB model of [@Chen2013] does not apply directly to the search problem. In the search problem, draws from the underlying reward distribution are not observed. Further, due to the varying detection probabilities from round to round (as different combinations of arms are played) the distribution from which rewards are observed does not remain constant either. Additionally, Poisson rewards have heavier tails than can be accommodated within the framework considered by [@Chen2013]. This motivates us to develop an extended CMAB framework allowing for a broader range of underlying reward distributions and a feedback mechanism where the observed rewards are a filtered version of the true outcomes from the underlying distributions. With such a model design, algorithms to approach the search problem can be developed.
Framework {#Framework}
=========
In a CMAB problem, an agent is faced with $k$ arms each associated with some unknown, underlying probability distribution $\nu_i$ with expectation $\mu_i$. At each time step $t=1,2,...$ the agent selects a combination of arms $S_t$ from a set of possible combinations $\mathcal{S} \subseteq \mathcal{P}\big(\{1,...,k\}\big)$ where $\mathcal{P}\big(\{1,...,k\}\big)$ denotes the power set of the set of arms. When a combination of arms is selected in a round, we say that all the arms within that combination have been played in the round. Letting $T_{i,t}=\sum_{j=1}^t\mathds{I}\{i \in S_j\}$ denote the number of times an arm is played in the first $t$ rounds, we introduce the *filtered semi bandit feedback* framework as follows.
When a combination of arms $S_t$ is selected in round $t$, a random outcome $X_{i,T_{i,t}}$ is generated (independently) from underlying distribution $\nu_i$ for each $i \in S_t$. However, these outcomes remain unobserved. Instead, for each $i \in S_t$, a *filtered observation* $Y_{i,T_{i,t}}$ is drawn from a *filtering distribution* $\tilde{\nu}_{i,T_{i,t}}=\tilde{\nu}_i(X_{i,T_{i,t}},S_t)$ conditioned on the random outcome from the underlying distribution and the combination of arms played. These filtered observations *are* seen by the agent. Let $\mathbf{X}_{S_t}$ and $\mathbf{Y}_{S_t}$ respectively denote the vectors of true outcomes and filtered observations in round $t$ where combination of arms $S_t$ is selected.
In addition to observing $\mathbf{Y}_{S_t}$, playing the combination of arms $S_t$ grants the agent a reward $R(\mathbf{Y}_{S_t})$ which is a function of the filtered observations (and thus is a random variable). The expectation of the reward obtained by playing combination of arms $S_t$, with respect to a particular vector of underlying means ${\boldsymbol}\mu$, is denoted $r_{{\boldsymbol}\mu}(S_t)=\mathds{E}(R(\mathbf{Y}_{S_t})|S_t)$. The function $r_{{\boldsymbol}\mu}: \mathcal{S} \rightarrow \mathds{R}$ is referred to as a *reward function*.
One example of a filtering model is the binomial filtering of discrete non-negative integer data, as seen in the search example of Section \[Motivation\]. In such a model, $Y_{i,T_{i,t}}|X_{i,T_{i,t}},S_t$ follows a $Bin(X_{i,T_{i,t}},\gamma_{i,S_t})$ distribution where $\gamma_{i,S_t}$ is a success probability dependent on the combination of arms played.
Filtered semibandit feedback can be contrasted with bandit, semibandit and full information feedbacks where there is no filtering, or in our terms where the filtering distributions are such that $\mathbf{Y}_{S_t}=\mathbf{X}_{S_t}$ with probability 1 (so true outcomes and filtered observations can be treated as the same thing). In bandit feedback, only $R(\mathbf{X}_{S_t})$ is observed. In semibandit feedback $R(\mathbf{X}_{S_t})$ and $\mathbf{X}_{S_t}$ are observed. In full information feedback $R(\mathbf{X}_{S_t})$, $\mathbf{X}_{S_t}$, and a draw from $\nu_i$ for $i \notin S_t$ are observed. Our model applies filtering to the semibandit feedback case. We do not consider filtered variants of bandit or full information feedback. We note that the classical stochastic Multi-Armed Bandit (MAB) problem is a special case of the CMAB problem with (non-filtered) bandit or semibandit feedback, where $\mathcal{S}=\big\{\{1\},...,\{k\}\big\}$ and the reward observed is simply equal to the observation $X_{i,T_{i,t}}$ drawn from the arm $i$ selected.
A CMAB problem with filtered semibandit feedback is therefore defined by a set of underlying probability distributions ${\boldsymbol}\nu=(\nu_1,...,\nu_k)$ with means ${\boldsymbol}\mu=(\mu_1,...,\mu_k)$, a set of possible combinations $\mathcal{S}$, a reward function $r_{{\boldsymbol}\mu}(\cdot)$, and a set of filtering distributions $\tilde{{\boldsymbol}\nu}=(\tilde{\nu}_1,...,\tilde{\nu}_k)$ with variable parameters. To aid in the analysis in this paper, we make assumptions on the expected reward $r_{{\boldsymbol}\mu}(S)$ as in [@Chen2013].\
***Assumption 1** - Monotonicity*:
The expected reward of playing any combination of arms $S \in \mathcal{S}$ is monotonically nondecreasing with respect to the expectation vector, i.e. if for all $i \in \{1,...,k\},$ $\mu_i \leq \mu_i'$, we have $r_{{\boldsymbol}\mu}(S) \leq r_{{\boldsymbol}\mu'}(S)$ for all $S \in \mathcal{S}$.\
***Assumption 2** - Bounded Smoothness*:
There exists a strictly increasing function $f(\cdot)$ called a *bounded smoothness function*, such that for any two expectation vectors ${\boldsymbol}\mu$ and ${\boldsymbol}\mu'$ with $\max_{i \in S}|\mu_i - \mu_i'|\leq \Lambda$ we have $|r_{{\boldsymbol}\mu}(S)-r_{{\boldsymbol}\mu'}(S)|\leq f(\Lambda)$.\
With these assumptions in place we will be able to construct bounds on the performance of UCB-type algorithms for CMAB problems with filtered semibandit feedback. A CMAB algorithm will, in a round $t$, consider the rewards observed in previous rounds and select a combination of arms $S_t$ to be played. Its objective is to maximise cumulative expected reward over $n$ rounds, $\mathds{E}\big(\sum_{t=1}^n r_{{\boldsymbol}\mu}(S_t)\big)$ - where the expectation is taken with respect to the actions selected by the algorithm.
We investigate the performance of UCB type algorithms for the CMAB problems. Typically, UCB algorithms make decisions based on indices formed by adding an inflation term to a data-driven estimator of the underlying mean $\mu_i$. Successful algorithms are obtained by selecting the inflation term appropriately to match the convergence rate of the mean estimator thereby encouraging an appropriate balance of exploration and exploitation.
In simple CMAB or MAB problems with bounded or sub-Gaussian reward distributions, an empirical mean has convergence of a suitable rate to yield UCB algorithms with $\mathcal{O}(\ln n)$ bounded regret. However with non-sub-Gaussian (or *heavy tailed*) reward distributions the empirical mean lacks this same rate of convergence. As in [@Bubeck2013], we turn to more robust mean estimators to find the correct convergence rate. A further challenge is that our mean estimators must be based on observations from filtered distributions but converge to the mean of the underlying distributions. We seek estimators $\hat{\mu}(Y_{i,1},...,Y_{i,n})$ of $\mu_i$ based on filtered observations $Y_{i,1},...,Y_{i,n}$ which satisfy the following assumption for the relevant distributions in the particular CMAB problems we consider.\
***Assumption 3** - Concentration of Mean Estimator*:
The mean estimator $\hat{\mu}_{i,n}=\hat{\mu}(Y_{i,1},...,Y_{i,n})$ is such that for positive parameter $\epsilon \in (0,1]$, positive values $c,v$, and independent random variables $Y_{i,1},...,Y_{i,n}$ drawn from filtering distributions $\tilde{\nu}_{i,1},...,\tilde{\nu}_{i,n}$ we have for all $n\geq 1$ and $\delta \in (0,1)$ $$\begin{aligned}
\mathds{P}\bigg(\hat{\mu}_{i,n} \geq \mu_i + v^{\frac{1}{1+\epsilon}}\Big(\frac{c\ln(1/\delta)}{n}\Big)^{\frac{\epsilon}{1+\epsilon}}\bigg) & \leq \delta \label{Assumption 3 Pt. 1} \\
\mathds{P}\bigg({\mu}_i \geq \hat{\mu}_{i,n} + v^{\frac{1}{1+\epsilon}}\Big(\frac{c\ln(1/\delta)}{n}\Big)^{\frac{\epsilon}{1+\epsilon}}\bigg) & \leq \delta. \label{Assumption 3 Pt. 2}\end{aligned}$$
Robust-F-CUCB {#Algorithms Section}
=============
For the stochastic CMAB with filtered semibandit feedback as introduced in Section \[Framework\], we propose the Robust-F-CUCB algorithm, described in Algorithm 1. The Robust-F-CUCB algorithm is both a generalisation and combination of the Robust-UCB algorithm of [@Bubeck2013] and CUCB algorithm of [@Chen2013]. This section proceeds as follows: We begin by introducing the necessary language and notation to express performance guarantees in the form of regret bounds. We then present our Robust-F-CUCB algorithm for a general mean estimator satisfying Assumption 3, before considering versions of the algorithm tailored to more specific reward and filtering distributions. With each version of the algorithm we also present a regret bound.
Regret Notation
---------------
The regret of a CMAB algorithm in $n$ rounds with respect to an expectation vector ${\boldsymbol}\mu$ can be written $$Reg_{n,{\boldsymbol}\mu} = n\cdot\text{opt}_{{\boldsymbol}\mu} - \mathds{E}\bigg(\sum_{t=1}^n r_{{\boldsymbol}\mu}(S_t)\bigg). \label{Regret Definition}$$ where $\text{opt}_{{\boldsymbol}\mu}=\max_{S \in \mathcal{S}}r_{{\boldsymbol}\mu}(S)$ denotes the highest attainable expected reward in a single round of the CMAB problem with respect to a expectation vector ${\boldsymbol}\mu$, and the expectation in (\[Regret Definition\]) is taken with respect to the actions selected by the algorithm. The aim to maximise expected reward is equivalent to minimising regret. The quality of an algorithm is usually measured by determining an analytical bound on $Reg_{n,{\boldsymbol}\mu}$, the order of which is the principal consideration. Algorithms with bounds of $\mathcal{O}(\ln n)$ are said to be of optimal order in CMAB and MAB problems.
In line with Chen et al. we define $$\begin{aligned}
\Delta_{max}&=\text{opt}_{{\boldsymbol}\mu}-\min_{S \in \mathcal{S}}\{r_{{\boldsymbol}\mu}(S) \enspace | \enspace r_{{\boldsymbol}\mu}(S) \neq \text{opt}_{{\boldsymbol}\mu}\} \\
\Delta_{min}&=\text{opt}_{{\boldsymbol}\mu}-\max_{S \in \mathcal{S}}\{r_{{\boldsymbol}\mu}(S) \enspace | \enspace r_{{\boldsymbol}\mu}(S) \neq \text{opt}_{{\boldsymbol}\mu}\} \end{aligned}$$ The quantity $\Delta_{max}$ is then the difference in expected reward between an optimal combination of arms and the worst possible combination of arms, while $\Delta_{min}$ is the difference in expected reward between an optimal combination of arms and the closest to optimal suboptimal combination of arms. These quantities will be important in defining bounds on expected regret.
General Algorithm Statement and Regret Bound
--------------------------------------------
We first describe the Robust-F-CUCB algorithm for a general mean estimator satisfying Assumption 3, before considering more specific results later in this section. Like the CUCB algorithm, our Robust-F-CUCB algorithm consists firstly of an initialisation stage where a combination of arms containing each arm is played, to initialise mean estimates and $T_{i,t}$ counters. Thereafter, in each round, upper confidence bounds (UCBs) are calculated for each arm and these UCBs are passed to a combinatorial optimisation to identify the best combination of arms to play from our optimistic perspective. The approach is presented in full detail in Algorithm 1.
\[RobustFCUCBAlgorithm\]
------------------------------------------------------------------------
------------------------------------------------------------------------
**Inputs:** Parameter $\epsilon \in (0,1]$, positive constants $c,v$, and mean estimator $\hat{\mu}_{i,n}$ satisfying Assumption 3.\
**Initialisation Phase:**\
For each arm $i$, play an arbitrary combination of arms $S \in \mathcal{S}$ such that $i \in S$.\
Set $t$ $\leftarrow$ $k$.\
**Loop Phase:**\
For $t = k+1, k+2,....$\
$\quad$ For each arm $i$, calculate $\bar{\mu}_{i,T_{i,t-1},t} = \hat{\mu}_{i,T_{i,t-1}} + v^\frac{1}{1+\epsilon}\bigg(\frac{c\ln t^3}{T_{i,t-1}}\bigg)^{\frac{\epsilon}{1+\epsilon}}$.\
$\quad$ Play a combination of arms $S_t$ with $r_{\bar{{\boldsymbol}\mu}_t}(S_t)=\max_{S\in \mathcal{S}}r_{\bar{{\boldsymbol}\mu}_t}(S)=\text{opt}_{\bar{{\boldsymbol}\mu}_t}$.\
$\quad$ Update $T_i$ and $\hat\mu_i$ for all $i \in S_t$.
------------------------------------------------------------------------
The following theorem provides our performance bound for the Robust-F-CUCB algorithm.\
***Theorem 1:*** Let $\epsilon \in (0,1]$ and let $\hat{\mu}_{i,n}$ be a mean estimator. Suppose that the underlying distributions $\nu_1,...,\nu_k$ and filtering distributions $\tilde{\nu}_1,...,\tilde{\nu}_k$ are such that the mean estimator satisfies Assumption 3 for all $i=1,...,k$. Then the regret of the Robust-F-CUCB policy satisfies $$Reg_{n,{\boldsymbol}\mu} \leq \Bigg(3cv^{\frac{1}{\epsilon}}\bigg(\frac{2}{f^{-1}(\Delta_{min})}\bigg)^{\frac{1+\epsilon}{\epsilon}}\ln n + \frac{\pi^2}{3} +1 \Bigg)\cdot k \cdot \Delta_{max}. \label{Theorem 1 Bound}$$
*Proof:* Since the mean estimators are assumed to satisfy Assumption 3, the proof is an adaptation of those given by [@Chen2013] and [@Bubeck2013] and is given in Appendix \[Theorem1Proof\].\
A particular instance of the Robust-F-CUCB algorithm can be defined by a specific mean estimator and particular values of $\epsilon,$ $c,$ $v$. In the remainder of this section, we consider particular cases of the CMAB problem and particular Robust-F-CUCB algorithms which are appropriate to these problems.
Semibandit Feedback with Heavy Tails
------------------------------------
Firstly, we consider a CMAB problem without filtering - i.e. where the filtering distributions are such that $Y_{i,t}=X_{i,t}$ for all $i$ and $t$ and we simply have semibandit feedback. In this situation we are still considering a model not previously studied in the literature as we have permitted a more general class of reward distribution whose support is not solely contained within $[0,1]$. For this problem class we propose the Robust-F-CUCB algorithm with truncated empirical mean - a direct extension of the Robust-UCB policy with truncated empirical mean specified by [@Bubeck2013]. The truncated empirical mean, given some parameters $u>0$, and $\epsilon \in (0,1]$, and based on observations $X_1,...,X_n$ is defined as $$\hat{\mu}_{i,n}^{Trunc}=\frac{1}{n}\sum_{t=1}^n X_t \mathds{I}\bigg\{|X_t|\leq \bigg(\frac{ut}{\ln(t)}\bigg)^{\frac{1}{1+\epsilon}}\bigg\}. \label{TruncatedEmpiricalMean}$$
The following Proposition provides a bound on the regret of the Robust-F-CUCB algorithm where the underlying distributions ${\boldsymbol}\nu$ have suitably bounded $1+\epsilon$ moments. The Robust-F-CUCB algorithm with truncated empirical mean works for this problem class because in [@Bubeck2013] the truncated empirical mean has already been shown to satisfy Assumption 3.\
***Proposition 2***: Let $\epsilon \in (0,1]$ and $u > 0$. Let the reward distributions $\nu_1,...,\nu_k$ satisfy $$\mathds{E}_{X \sim \nu_i}|X_i|^{1+\epsilon} \leq u \quad \forall \enspace i \in \{1,...,k\}.$$ Then the regret of the Robust-F-CUCB algorithm used with the truncated empirical mean estimator defined in (\[TruncatedEmpiricalMean\]) satisfies $$Reg_{n,{\boldsymbol}\mu} \leq \Bigg(12(4u)^{\frac{1}{\epsilon}}\bigg(\frac{2}{f^{-1}(\Delta_{min})}\bigg)^{\frac{1+\epsilon}{\epsilon}}\ln n + \frac{\pi^2}{3} + 1\Bigg)\cdot k \cdot \Delta_{max}. \label{Lemma2RegretBound}$$
*Proof*: [@Bubeck2013] shows that Assumption 3 holds with $c=4$ and $v=4u$. The main result then follows from Theorem 1.
Binomially filtered Poisson rewards {#BinFilt}
-----------------------------------
We now wish to consider a CMAB problem with filtering. As we mentioned in Section 3, one possible filtering framework is the Binomial filtering of count data, i.e. where if in round $t$ $X_{i,T_{i,t}}$ is a draw from $\nu_i$ and $S_t$ is the combination of arms selected then the filtering distribution for arm $i$, $\tilde{\nu}_i(X_{i,T_{i,t}},S_t)$ is $Bin(X_{i,T_{i,t}},\gamma_{i,S_t})$. We also mentioned that if $\nu_i$ follows a Poisson distribution with parameter $\mu_i$ then the marginal distribution of the filtered observation $Y_{i,T_{i,t}}|S_t$ will be Poisson with parameter $\gamma_{i,S_t}\mu_i$. We consider this example, with the additional assumption that for some $\gamma_{min}>0$ we have $\gamma_{i,S}>\gamma_{min}$ for all $i$ and $S \in \mathcal{S}$.
To define a Robust-F-CUCB algorithm that satisfies the logarithmic order regret bound for this problem, we must have a mean estimator which satisfies Assumption 3 for the filtering distributions specified above. Consider the following *filtered* truncated empirical mean estimator, and the associated Lemma, demonstrating that a version of Assumption 3 holds for this estimator when applied to Poisson reward distributions.\
***Lemma 3***: Let $\delta\in (0,1)$ and $\mu_{max}>0$, and define $u_{max}=\mu_{max}^2 + \mu_{max}$. Consider a series of filtered Poisson observations $Y_{i,1},...,Y_{i,n}$ with means $\gamma_{i,1}\mu_i,...,\gamma_{i,n}\mu_i$ where $\gamma_{i,t} \in (\gamma_{min},1]$ for $t=1,...,n$ and $\gamma_{min} > 0$. Consider the filtered truncated empirical mean estimator, $$\hat{\mu}_{i,n}^{TruncF} = \frac{1}{n}\sum_{t=1}^n \frac{Y_{i,t}}{\gamma_{i,t}} \mathds{I}{\Bigg\{Y_{i,t} \leq \gamma_{i,t} \sqrt{\frac{u_{max}t}{\ln \delta^{-1}}}\Bigg\}}. \label{FiltTruncMean}$$ If $\mu_i \leq \mu_{max}$ then $$\begin{aligned}
\mathds{P}\Bigg(\hat{\mu}_{i,n}^{TruncF} \geq \mu_i + \Big(\frac{2}{\gamma_{min}} + \sqrt{\frac{2}{\gamma_{min}}} +\frac{1}{3}\Big)\sqrt{\frac{u_{max} \ln \delta^{-1}}{n}}\Bigg) &\leq \delta, \label{Lemma4Pt1} \\
\mathds{P}\Bigg(\mu_i \geq \hat{\mu}_{i,n}^{TruncF} + \Big(\frac{2}{\gamma_{min}} + \sqrt{\frac{2}{\gamma_{min}}} +\frac{1}{3}\Big)\sqrt{\frac{u_{max} \ln \delta^{-1}}{n}}\Bigg) &\leq \delta. \label{Lemma4Pt2}\end{aligned}$$\
We present a proof of Lemma 3 in Appendix \[Lemma3Proof\]. Proposition 4 below specifies the regret bound which holds if the filtered truncated empirical mean estimator is used to define a Robust-F-CUCB algorithm and this algorithm is applied to the CMAB problem with the filtering structure defined above.\
***Proposition 4***: Let $\epsilon=1$ and $\mu_{max}>0$. Let the reward distributions $\nu_1,...,\nu_k$ be Poisson satisfying $\mu_i \leq \mu_{max}$ for $i=1,...,k$. Let the filtering distributions $\tilde{\nu}_1,...,\tilde{\nu}_k$ be Binomial as described above. Then the regret of the Robust-F-CUCB algorithm used with the filtered truncated empirical mean estimator defined in (\[FiltTruncMean\]) satisfies $$Reg_{n,{\boldsymbol}\mu} \leq \Bigg(\frac{12(\mu_{max}^2+\mu_{max})\Big(\frac{2}{\gamma_{min}} + \sqrt{\frac{2}{\gamma_{min}}} +\frac{1}{3}\Big)^2}{\Big(f^{-1}(\Delta_{min})\Big)^2}\ln n + \frac{\pi^2}{3} + 1\Bigg)\cdot k \cdot \Delta_{max}. \label{Lemma4RegretBound}$$
*Proof*: Lemma 3 shows that Assumption 3 holds with $\epsilon=1$, $c=u_{max}$ and $v=\big(\frac{2}{\gamma_{min}} + \sqrt{\frac{2}{\gamma_{min}}} +\frac{1}{3}\big)^2$. The main result then follows from Theorem 1.
Discussion
==========
Within this paper we have presented a generalisation (in two senses) of the Combinatorial Multi-Armed Bandit framework, by considering unbounded reward distributions and filtered semibandit feedback. Our Robust-F-CUCB algorithm, presented in a general form, can be shown to have an associated logarithmic order bound on regret and we have specified this bound for particular CMAB problem instances. In particular we have shown that in a filtering free context, the truncated mean estimator can be used to provide an algorithm for a CMAB problem with heavy tails with a logarithmic order bound on regret. We developed a generalisation of the truncated mean estimator to deal with binomially filtered Poisson data and showed that for this class of data, it has the required concentration properties - a result which could of course be applied in the study of other problems, not just bandits.
We can apply the Robust-F-CUCB algorithm with filtered truncated empirical mean discussed in Section \[BinFilt\] in the sequential search problem as long as we have knowledge of some upper bound $\mu_{max}$ such that the average rate of the underlying Poisson process in each cell is below the upper bound in each round. As the reward function in this problem is the expected number of detected events, $r_{{\boldsymbol}\mu}(S_t)=\sum_{i \in S_t} \gamma_{i,S_t}\mu_i$, and $\gamma_{min} \leq \gamma_{i,S_t} \leq 1$ for all $i$ and $S_t \in \mathcal{S}$, Assumption 1 (monotonicity) holds and Assumption 2 (bounded smoothness) holds for a bounded smoothness function $f(\Lambda)=k\Lambda$. Thus the we can bound the regret of the Robust-F-CUCB as $\mathcal{O}(k^3\ln n)$.
We note also, that we could readily extend this work to a more complex application where there are multiple agents searching collaboratively. If multiple searchers were each to select a combination of cells to search (in such a way that the combinations do not overlap), one could still identify a best combination in the full information problem by formulating and solving a Integer Linear Program. In the sequential decision variant of the problem would still observe filtered rewards from the multiple combinations of cells in each round and could still apply an Upper Confidence Bound algorithm to balance exploration and exploitation.
The key difference with this more complex application is that the mapping between combinations of arms and an interpretable allocations of searchers would not be one-to-one. The added combinatorial aspect of the problem means that, while a combination of arms would be interpretable as the union of the sets of cells picked by the different searchers, different sets of sets could lead to the same overall combination of arms. Therefore, crucially, a combination of arms may have multiple different sets of filtering distributions associated with it, and the most appropriate *way to play* the combination of arms may vary as the arm indices do. So, to approach this more complex version of the problem, a definition of the set of possible combinations $\mathcal{S}$ that includes labellings of the partitions within combinations of cells $S \in \mathcal{S}$ is required.
We note that it would be possible to improve the leading order coefficients of all our regret bounds by applying the more sophisticated analysis used in the proof of Theorem 1 of [@Chen2013] to our Robust-F-CUCB framework. Said analysis would improve on our presented analysis by noting the discrepancy between defining sufficiency of sampling with respect to $\Delta_{min}$ but bounding regret with respect to $\Delta_{max}$. The more sophisticated analysis would remain usable in our more complex framework, as any intricacies due to filtering can be truly captured within the concentration inequality based step, which yields the $\frac{\pi^2}{3}k\Delta_{max}$ term, a step which the more sophisticated analysis does not alter. We have refrained from making these improvements in this work, as they do not affect the order of the bound and the omission permits an easier explanation of our key results. Furthermore although we did not directly consider the more general $(\alpha, \beta)$-*approximation regret* considered by Chen et al. (which allows for the CMAB algorithm to be a randomised algorithm with a small failure probability), the results presented in our paper can be trivially generalised to incorporate this by reintroducing the $\alpha$ and $\beta$ parameters which we have effectively fixed to equal 1.
Acknowledgements {#acknowledgements .unnumbered}
================
[We gratefully acknowledge the support of the EPSRC funded EP/L015692/1 STOR-i Centre for Doctoral Training]{}
Proof of Theorem 1 {#Theorem1Proof}
==================
*Proof of Theorem 1:*
For each arm maintain $T_{i,t}$ as the a count of the number of times arm $i$ has been played in the first $t$ rounds. We also maintain a second set of counters $\{N_i\}_{i=1}^k$, one associated with each arm. These counters, which collectively count the number of suboptimal plays, are updated as follows. Firstly, after the $k$ initialisation rounds set $N_{i,k}=1$ for all $i \in \{1,...,k\}$. Thereafter, in each round $t >k$, let $S_t$ be the combination of arms played in round $t$ and let $i=\arg\min_{j \in S_t}N_{j,t}$, if $i$ is non-unique then we choose randomly from the minimising set. If $r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}$ then we increment $N_{i}$, i.e. set $N_{i,t}=N_{i,t-1}+1$, . The key results of these updating rules are that $\sum_{i=1}^k N_{i,t}$ provides an upper bound on the number of suboptimal plays in $t$ rounds and $T_{i,t}\geq N_{i,t}$ for all $i$ and $t$.
Define $$l_t = 3cv^{\frac{1}{\epsilon}}\bigg(\frac{2}{f^{-1}(\Delta_{min})}\bigg)^{\frac{1+\epsilon}{\epsilon}}\ln(t).$$
We consider a round $t$ in which $S_t: r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}$ is selected and counter $N_i$ of some arm $i \in S_t$ is updated. We have $$\begin{aligned}
\sum_{i=1}^k N_{i,n} - k &= \sum_{t=k+1}^n \mathds{I}\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}\} \nonumber\\
\Rightarrow \enspace \sum_{i=1}^k N_{i,n} - k\cdot(l_n+1) &= \sum_{t=k+1}^n \mathds{I}\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}\}-kl_n \nonumber \\
&\leq \sum_{t=k+1}^n \sum_{i=1}^k \mathds{I}\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, N_{i,t} > N_{i,t-1}, N_{i,t-1} > l_n\} \nonumber \\
&\leq \sum_{t=k+1}^n \sum_{i=1}^k \mathds{I}\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, N_{i,t} > N_{i,t-1}, N_{i,t-1} > l_t\} \nonumber \\
&= \sum_{t=k+1}^n \mathds{I}\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, N_{i,t-1} > l_t \enspace \forall i \in S_t\} \label{Indicator1} \\
&\leq \sum_{t=k+1}^n \mathds{I}\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, T_{i,t-1} > l_t \enspace \forall i \in S_t\} \label{Indicator2}\end{aligned}$$ Here, the initial equations come from the updating rules for counters. The first inequality holds because there are at most $kl_n$ occasions where the specified conditions do not hold - i.e. once each counter has been updated $l_n$ times, none of the counters will be $< l_n$. The second inequality is true because $l_t \leq l_n$ for $t\leq n$, and equation (\[Indicator1\]) holds because of our rule that we always update only one of the smallest counters in the selected combination of arms. The final inequality follows from $N_{i,t} \leq T_{i,t}$.
We wish to show $\mathds{P}(r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, T_{i,t-1}>l_t \enspace \forall i \in S_t) \leq 2kt^{-2}$, so that the summation in (\[Indicator2\]) converges. As a consequence of Assumption 3, for any arm $i=1,..,k$ we have: $$\begin{aligned}
\mathds{P}\Bigg(|\hat{\mu}_{i,T_{i,t-1}}-\mu_i| \geq v^{\frac{1}{1+\epsilon}}\bigg(\frac{c\ln t^3}{T_{i,t-1}}\bigg)^{\frac{\epsilon}{1+\epsilon}}\Bigg)
&=\sum_{s=1}^{t-1} \mathds{P}\Bigg(\Bigg\{|\hat{\mu}_{i,s}-\mu_i| \geq v^{\frac{1}{1+\epsilon}}\bigg(\frac{c\ln t^3}{s}\bigg)^{\frac{\epsilon}{1+\epsilon}},T_{i,t-1}=s\Bigg\}\Bigg) \nonumber \\
&\leq \sum_{s=1}^{t-1} \mathds{P}\Bigg(|\hat{\mu}_{i,s}-\mu_i| \geq v^{\frac{1}{1+\epsilon}}\bigg(\frac{c\ln t^3}{s}\bigg)^{\frac{\epsilon}{1+\epsilon}}\Bigg) \nonumber \\
&\leq t\cdot 2t^{-3} \leq 2t^{-2}. \label{ProbBound}\end{aligned}$$
Define a random variable $\Lambda_{i,t} = v^{\frac{1}{1+\epsilon}}\bigg(\frac{c\ln t^3}{T_{i,t-1}}\bigg)^{\frac{\epsilon}{1+\epsilon}}$ and event $E_t =\{|\hat\mu_{i,T_{i,t-1}} - \mu_i| \leq \Lambda_{i,t}, \forall \enspace i=1,..,k\}$. It is clear, by a union bound on Eq. (\[ProbBound\]) that $\mathds{P}(\neg E_t) \leq 2kt^{-2}$. In the loop phase of the Robust-F-CUCB algorithm we have $\bar\mu_{i,t} - \hat\mu_{i,T_{i,t-1}} = \Lambda_{i,t}$. Thus, $E_t$ implies $\bar{\mu}_{i,t}\geq \mu_i$ for all $i$.
Let $\Lambda=v^{\frac{1}{1+\epsilon}}\bigg(\frac{c\ln t^3}{l_t}\bigg)^{\frac{\epsilon}{1+\epsilon}}$ (not a random variable) and define $\Lambda_t = \max\{\Lambda_{i,t}|i \in S_t\}$ (which is a random variable). The following results can then be written: $$\begin{aligned}
E_t \Rightarrow \enspace |\bar{\mu}_{i,t}-\mu_i| \leq 2&\Lambda_t \enspace \forall i \in S_t\label{EventImply}\\
\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu} , T_{i,t-1} > l_t \enspace \forall i \in S_t \} \Rightarrow \Lambda > &\Lambda_t \label{EventImply2}\end{aligned}$$ which follow from the definitions of the various $\Lambda$ terms.
We can then present the following derivation, true if $\{E_t,r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, T_{i,t-1} > l_t \enspace \forall i \in S_t\}$ holds: $$\begin{aligned}
r_{{\boldsymbol}\mu}(S_t) + f(2\Lambda) &> r_{{\boldsymbol}\mu}(S_t)+f(2\Lambda_t), \\
&\geq r_{\bar{{\boldsymbol}\mu}_t}(S_t) = \text{opt}_{\bar{{\boldsymbol}\mu}_t}, \\
&\geq r_{\bar{{\boldsymbol}\mu}_t}(S^*_{{\boldsymbol}\mu}), \\
&\geq r_{{\boldsymbol}\mu}(S^*_{{\boldsymbol}\mu}) = \text{opt}_{{\boldsymbol}\mu},\end{aligned}$$ where $S^*_{{\boldsymbol}\mu}$ is an combination of arms with optimal expected reward with respect to the true mean vector. The first inequality follows from the monotonicity of the bounded smoothness function specified in Assumption 2 and Eq. (\[EventImply2\]). The second is a result of the bounded smoothness property of Assumption 2 and Eq. (\[EventImply\]). The third inequality follows from the definition of $\text{opt}_{\bar{{\boldsymbol}\mu}_t}$ and the fourth from the monotonicity of $r_{{\boldsymbol}\mu}(S)$ assumed in Assumption 1 and the result that $E_t \Rightarrow \bar{{\boldsymbol}\mu}_t \geq {\boldsymbol}\mu$. In summary, this derivation says that if $\{E_t,r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, \forall i \in S_t, T_{i,t-1} > l_t\}$ holds then $$r_{{\boldsymbol}\mu}(S_t)+f(2\Lambda) > \text{opt}_{{\boldsymbol}\mu}. \label{DerivResult}$$ The definitions of $l_t$ and $\Lambda$ mean that $f(2\Lambda)= \Delta_{min}$, and we can rewrite Eq. (\[DerivResult\]) as $$r_{{\boldsymbol}\mu}(S_t) + \Delta_{min} > \text{opt}_{{\boldsymbol}\mu}.$$ This however, is a direct contradiction of the definition of $\Delta_{min}$ and the assumption that $r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}$. This means that $\mathds{P}(\{E_t,r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, T_{i,t-1} > l_t \enspace \forall i \in S_t\})=0$ and thus $$\mathds{P}(\{r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, T_{i,t-1} > l_t \enspace \forall i \in S_t\}) \leq \mathds{P}(\neg E_t) \leq 2kt^{-2}$$ as derived previously.
From (\[Indicator2\]) we can thus write: $$\begin{aligned}
\mathds{E}\bigg(\sum_{i=1}^k N_{i,n}\bigg) &\leq k(l_n+1) + \sum_{t=k+1}^n \mathds{P}\big(r_{{\boldsymbol}\mu}(S_t) \neq \text{opt}_{{\boldsymbol}\mu}, T_{i,t-1} > l_t \enspace \forall i \in S_t\big) \nonumber \\
&\leq k(l_n+1) + \sum_{t=1}^n \frac{2k}{t^{2}} \nonumber \\
&\leq \frac{2^{\frac{1+\epsilon}{\epsilon}} \cdot c \cdot k \cdot v^{\frac{1}{\epsilon}} \cdot \ln n^3}{(f^{-1}(\Delta_{min}))^{\frac{1+\epsilon}{\epsilon}}}+ \bigg(\frac{\pi^2}{3} + 1\bigg) \cdot k. \label{BadRoundBound}\end{aligned}$$ Since the expected reward from playing a suboptimal combination of arms is at most $\Delta_{max}$ from $\text{opt}_{{\boldsymbol}\mu}$ we can trivially reach the required result by assuming the suboptimal rounds are all as far from optimality as they could be. $\square$
Proof of Lemma 3 {#Lemma3Proof}
================
*Proof of Lemma 3*: The proof will show (\[Lemma4Pt2\]) to be true, and then proving (\[Lemma4Pt1\]) is just a simple modification of the same steps. Define $B_t = \sqrt{\frac{u_{max} t}{\ln \delta^{-1}}}$. We have $$\begin{aligned}
\mu_i - \hat{\mu}_{i,n}^{TruncF} &= \frac{1}{n}\sum_{t=1}^n \bigg(\mu_i - \frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\bigg) \nonumber \\
&= \frac{1}{n}\sum_{t=1}^n \Bigg(\mathds{E}\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\bigg) - \mathds{E}\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\bigg)\Bigg) \nonumber \\
&\quad \quad \quad \quad + \frac{1}{n}\sum_{t=1}^n \Bigg(\mathds{E}\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\bigg) - \frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\Bigg) \nonumber \\
&= \frac{1}{n}\sum_{t=1}^n \mathds{E}\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} > \gamma_{i,t}B_t\Big\}}\bigg) + \frac{1}{n}\sum_{t=1}^n Z_t \label{BoundableLemma4line}\end{aligned}$$ where $Z_t = \mathds{E}\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\bigg) - \frac{Y_{i,t}}{\gamma_{i,t}}\mathds{I}{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}$.
We bound the first sum in (\[BoundableLemma4line\]) by noting that $$\mathds{E}\bigg(Y_{i,t}\mathds{I}{\Big\{Y_{i,t} > \gamma_{i,t}B_t\Big\}}\bigg) \leq \mathds{E}\bigg(\frac{Y_{i,t}^2}{\gamma_{i,t}B_t}\bigg) \leq \frac{\gamma_{i,t}u_{max}}{\gamma_{i,t}B_t} = \frac{u_{max}}{B_t},$$ since $\mathds{I}{\Big\{Y_{i,t} > \gamma_{i,t}B_t\Big\}} \leq \frac{Y_{i,t}}{\gamma_{i,t}B_t}$ and $\mathds{E}(Y_{i,t}^2)\leq \gamma_{i,t}u_{max}$ because $Y_{i,t} \sim Pois(\gamma_{i,t}\mu_i)$. To bound the second sum in (\[BoundableLemma4line\]), we will use Bernstein’s inequality for bounded random variables:\
**Bernstein’s Inequality:** Let $X_1,X_2,...,X_n$ be independent bounded random variables such that $\mathds{E}(X_i)=0$ and $|X_i|\leq \varsigma$ with probability 1 and let $\sigma^2 = \frac{1}{n}\sum_{i=1}^n Var(X_i)$ then for any $a>0$ we have $$\mathds{P}\bigg(\frac{1}{n}\sum_{i=1}^n X_i \geq a\bigg) \leq \exp \Bigg\{ \frac{-na^2}{2\sigma^2 + \frac{2\varsigma a}{3}}\Bigg\}. \label{BernsteinInequality}$$\
The $Z_t$ have zero mean, bounded support ($|Z_t| \leq B_t \leq B_n$), and bounded variances
$$Var(Z_t) = Var\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\mathds{1}_{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\bigg) \leq \frac{1}{\gamma_{i,t}^2}\mathds{E}\bigg(Y_{i,t}^2 \mathds{1}_{\Big\{Y_{i,t} \leq \gamma_{i,t}B_t\Big\}}\bigg) \leq \frac{u_{max}}{\gamma_{i,t}}$$
for $t=1,...,n$. Thus we have a bounded $\sigma^2 = \frac{1}{n}\sum_{t=1}^n Var(Z_t) \leq \frac{u_{max}}{\gamma_{min}}$ also. Therefore applying Bernstein’s inequality for bounded random variables, with our upper bounds on $\sigma^2$ and $\varsigma$, have $$\mathds{P}\bigg(\frac{1}{n}\sum_{t=1}^n Z_t > a\bigg) \leq \exp \Bigg\{\frac{-na^2}{2\frac{u_{max}}{\gamma_{min}}+\frac{2B_na}{3}} \Bigg\}.$$ Plugging in $$a = \sqrt{\frac{2u_{max}\ln \delta^{-1}}{\gamma_{min} n}} + \frac{B_n}{3n}\ln \delta^{-1}$$ we see that $\mathds{P}\bigg(\frac{1}{n}\sum_{t=1}^n Z_t > a\bigg) < \delta$ and therefore $\mathds{P}\bigg(\frac{1}{n}\sum_{t=1}^n Z_t \leq a\bigg) \geq 1-\delta$. With these results we can place the following bound on (\[BoundableLemma4line\]) that holds with at least probability $1-\delta$: $$\begin{aligned}
&\quad \frac{1}{n}\sum_{t=1}^n \mathds{E}\bigg(\frac{Y_{i,t}}{\gamma_{i,t}}\mathds{1}_{\Big\{Y_{i,t} > \gamma_{i,t}B_t\Big\}}\bigg) + \frac{1}{n}\sum_{t=1}^n Z_t \\
&\leq \frac{1}{n}\sum_{t=1}^n \frac{u_{max}}{\gamma_{i,t} B_t} + \sqrt{\frac{2u_{max}\ln \delta^{-1}}{\gamma_{min} n}} + \frac{B_n}{3n}\ln \delta^{-1} \\
&= \frac{1}{n}\sum_{t=1}^n \frac{1}{\gamma_{i,t}}\sqrt{\frac{u_{max}\ln \delta^{-1}}{t}} + \sqrt{\frac{2}{\gamma_{min}}}\sqrt{\frac{u_{max}\ln \delta^{-1}}{n}} + \frac{1}{3}\sqrt{\frac{u_{max}\ln \delta^{-1}}{n}} \\
&\leq \bigg(\frac{1}{\gamma_{min}\sqrt{n}}\sum_{t=1}^n \frac{1}{\sqrt{t}} + \sqrt{\frac{2}{\gamma_{min}}} + \frac{1}{3}\bigg)\sqrt{\frac{u_{max}\ln \delta^{-1}}{n}} \\
&\leq \Big(\frac{2}{\gamma_{min}} + \sqrt{\frac{2}{\gamma_{min}}} +\frac{1}{3}\Big)\sqrt{\frac{u_{max} \ln \delta^{-1}}{n}}. \quad \quad \square\end{aligned}$$ This proves that $$\mathds{P}\Bigg(\mu_i \leq \hat{\mu}_{i,n}^{TruncF} + \Big(\frac{2}{\gamma_{min}} + \sqrt{\frac{2}{\gamma_{min}}} +\frac{1}{3}\Big)\sqrt{\frac{u_{max} \ln \delta^{-1}}{n}}\Bigg) \geq 1-\delta.$$
|
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address: |
^1^Department of Physics and Astronomy, VU University, 1081 HV Amsterdam, Netherlands\
^2^Max Planck Institute for Molecular Genetics, 14195 Berlin, Germany\
^3^Theoretical Biology and Bioinformatics, Utrecht University, Padualaan 8, 3584 CH Utrecht, the Netherlands\
^4^Drittes Physikalisches Institut, Georg-August-Universitat, 37073 G$\ddot{o}$ttingen, G$\ddot{o}$ttingen, Germany
author:
- 'M. Sheinman^1,2,3^, A. Sharma^1,4^, F. C. MacKintosh^1^'
title: 'Reply to Comment on “Anomalous Discontinuity at the Percolation Critical Point of Active Gels”'
---
The authors of the preceding Comment [@Comment] raise an interesting question about ambiguities in defining the Fisher exponent $\tau$. Ordinarily, such critical exponents are determined by the behavior in the thermodynamic limit. In the percolation theory context the number of connected clusters with mass $s$ scales as [@fisher1967theory; @stauffer1994introduction] $$n_s \propto s^{-\tau}
\label{god}$$ in the infinite system size limit, $M \rightarrow \infty$, up to possible logarithmic corrections. To estimate the value of $\tau$ numerically, however, one must consider systems with finite $M$, together with an appropriate finite-size scaling consistent with Eq. as $M \rightarrow \infty$. As in the Comment, one approach often used in the percolation literature [@stauffer1994introduction] is $$n_s=M s^{-\tau} f\left(\frac{s}{M^{d_f/d}} \right),
\label{def1}$$ where $d$ is the dimensionality ($d=2$ here) and $d_f$ is the fractal dimension of the clusters. The function $f(s/M^{d_f/d})$ is constrained to have no power-law dependence is the regime $1 \ll s \ll M$ and has to vanish for $s>M$. In random percolation (RP) $d_f<2$ and $\tau=d/d_f+1>2$ [@stauffer1994introduction]. Demanding conservation, $$\int_1^\infty s n_s ds = M,
\label{MassCond}$$ means that Eq. is consistent with only for $\tau \geq 2$. Thus, the approach in the Comment presupposes that $\tau \geq 2$ and is incapable of identifying possible values of $\tau<2$.
![Collapse attempts of the cluster masses distribution of the NEP model [@sheinman2015anomalous] at $p=p_c$ using $\tau=1.82<2$ (main figure) with definition and $\tau=2$ with equivalent (for this value of $\tau$) definitions and (inset) for different system sizes (see the values of $\sqrt{M}$ in the legend). The line in the inset corresponds to the power law with $0.18=2-1.82$ exponent.[]{data-label="collapse"}](3200.pdf){width="45.00000%"}
For this reason, in addition to the standard RP ansatz, we also used an ansatz consistent with Eq. , while allowing for possible $\tau< 2$: $$n_s=M^{\tau-1} s^{-\tau} f\left( \frac{s}{M} \right).
\label{def2}$$ This is consistent with Eq. , while satisfying Eq. for $\tau< 2$. In general, with no information about $\tau$ being larger or smaller than $2$, one should analyze the numerical data for both cases. We do this in Fig. \[collapse\], e.g., by plotting $s^\tau n_s/M^{\tau-1}$ vs. $s/M$ for the case $\tau< 2$. We find good collapse and near constancy of $s^\tau n_s/M^{\tau-1}$ for $\tau=1.82$ and over a wide range of $s/M$ up to $\sim 0.1$. By contrast, attempting the same collapse for $\tau=2$, where both our ansatz and that of the Comment are equivalent, we do not find the expected near constancy of $s^2 n_s/M$. Thus, while it may not be possible to entirely rule out $\tau=2$ with significant logarithmic corrections, our results appear to be more consistent with $\tau=1.82$. In the inset, however, we have plotted the distribution log-linear, in a way closely analogous to the Comment. Here, we do not find evidence of a logarithmic dependence. Our data are, in fact, consistent with a weak exponent $0.18$, as indicated by the thick line.
We thank the authors of the Comment for their interest and the useful discussion of subtleties in interpreting the numerical data. But, we fundamentally disagree with their approach that tacitly assumes $\tau \geq 2$.
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---
abstract: 'We classify the simple modules of the exceptional algebraic supergroups over an algebraically closed field of prime characteristic.'
address:
- 'Institute of Mathematics, Academia Sinica, Taipei, Taiwan 10617'
- 'Department of mathematics, East China Normal University, Shanghai, China 200241'
- 'Department of Mathematics, University of Virginia, Charlottesville, VA 22904'
author:
- 'Shun-Jen Cheng'
- Bin Shu
- Weiqiang Wang
title: Modular representations of exceptional supergroups
---
Introduction {#introduction .unnumbered}
============
Among the simple Lie superalgebras over the complex field ${\mathbb C}$, the basic Lie superalgebras distinguish themselves by admitting a non-degenerate super-symmetric even bilinear form (see, e.g., [@CW12]), and they include 3 exceptional Lie superalgebras: ${D(2|1;\zeta)}, G(3)$ and $F(3|1)$; cf. [@FK76]. The classification of finite-dimensional simple modules of complex simple Lie superalgebras was achieved by Kac [@Kac77 Theorem 8]. Note that the simple highest weight modules whose highest weights are dominant integral (with respect to the even subalgebra) are not all finite dimensional. This is one of several aspects that super representation theory differs from the classical representation theory dramatically. This classification theorem of Kac can be reformulated as a classification for simple modules over the corresponding supergroups over ${\mathbb C}$.
There are algebraic supergroups associated to the basic Lie superalgebras, valid over an algebraically closed field $k$ of prime characteristic $p\neq 2$. A general theory of Chevalley supergroups was systematically developed by Fioresi and Gavarini [@FG12] (also see [@G14]). In representation theory of algebraic supergroups $G$ over $k$, one of the basic questions is to classify the simple $G$-modules. For type $A$, the answer is immediate as it is the same as for the even subgroup $G_{{{\overline}0}}$. For type $Q$ such a classification was obtained in [@BK03], and it has applications to classification of simple modules of spin symmetric groups over $k$. For type $\frak{osp}$, the classification was obtained in [@SW08] in terms of the Mullineux involution by using odd reflections; also see Remark \[rem:osp\].
The goal of this paper is to classify the simple $G$-modules, when $G$ is a simply connected supergroup of exceptional type. We shall assume throughout the paper that $p>2$ for ${D(2|1;\zeta)}$ and $p>3$ for $G(3)$ or $F(3|1)$ (except in §\[sec:p=3\]). Under these assumptions, their corresponding supergroups admit non-degenerate super-symmetric even bilinear forms. We treat $G(3)$ for $p=3$ in §\[sec:p=3\].
Let us outline the approach of this paper. An equivalence of categories ([@SW08]; also cf. [@MS17]) reduces the classification of simple $G$-modules to the classification of the highest weights of finite-dimensional simple modules $L({\lambda}) =L^{{\mathfrak{b}}} ({\lambda})$ over the distribution superalgebra ${\text{Dist}(G)}$, where ${\mathfrak{b}}$ is the standard Borel subalgebra. We then reduce the verification of finite-dimensionality of $L({\lambda})$ to verifying that $L({\lambda})$ is locally finite over its [*even*]{} distribution subalgebra. The local finiteness criterion for $L({\lambda})$ is finally established by means of odd reflections (see [@LSS]), and is based on the following observation which seems to be well known to experts (see [@Se11]):
[*For every positive even root $\alpha$ in the standard positive system, either $\alpha/2$ (if it is a root) or $\alpha$ appears as a simple root in some simple system $\Pi'$ associated to some ${\mathfrak{b}}'$, where ${\mathfrak{b}}'$ is a Borel subalgebra obtained via a sequence of odd reflections from ${\mathfrak{b}}$.*]{}
For the exceptional Lie superalgebras, we make this observation explicit in this paper. We compute the highest weight $L^{{\mathfrak{b}}'} ({\lambda}')$ for all possible Borel subalgebras ${\mathfrak{b}}'$ as mentioned above. Requiring ${\lambda}'$ to be dominant integral for all possible ${\mathfrak{b}}'$ gives the local finiteness criterion for $L({\lambda})$.
Recently, an approach to obtain characters of projective and simple modules in the BGG category ${\mathcal O}$ for the exceptional Lie superalgebras over ${\mathbb C}$ has been systematically developed; see [@CW17] for ${D(2|1;\zeta)}$. Building on this and the current work, one may hope to better understand the characters of projective and simple modules of the exceptional supergroups over a field of prime characteristic in the future.
The organization of this paper is as follows. In Section \[sec:general\], we review the equivalence between the category of finite-dimensional modules over a supergroup $G$ and the category of finite-dimensional $({\text{Dist}(G)}, T)$-modules, where $T$ is a maximal torus of $G$. We develop a criterion for the finite-dimensionality of simple ${\text{Dist}(G)}$-modules $L({\lambda})$ via odd reflections. We also review the formula for the Euler characteristic, which implies that a ${\text{Dist}(G)}$-module $L({\lambda})$, with ${\lambda}$ dominant integral and ${\lambda}+\rho$ is regular, is always finite dimensional.
In Section \[sec:D\], we analyze the highest weight constraints given by odd reflections of a simple finite-dimensional ${\text{Dist}(G)}$-module when $G$ is of type ${D(2|1;\zeta)}$. Here ${D(2|1;\zeta)}$ is a family depending on a parameter $\zeta \in k\backslash \{0,-1\}$. We then classify the simple $G$-modules in Theorem \[thm:D\].
In Section \[sec:G\], we analyze the highest weight constraints given by odd reflections of a simple finite-dimensional ${\text{Dist}(G)}$-module when $G$ is of type $G(3)$. We then classify the simple $G$-modules in Theorem \[thm:G3\].
In Section \[sec:F\], we study the supergroup $G$ of type $F(3|1)$. When the highest weight ${\lambda}=a{\omega}_1 +b{\omega}_2 +c{\omega}_3+ d{\omega}_4$ with $a, b, c\in {\mathbb N}, d\ge 4$ is dominant integral, the weight ${\lambda}+\rho$ is regular and hence the Euler character formula implies that the ${\text{Dist}(G)}$-module $L({\lambda})$ is finite dimensional. For $d\leq 3$, it is rather involved to analyze the highest weight changes under sequences of odd reflections and formulate sufficient and necessary conditions for $L({\lambda})$ to be finite dimensional. We finally classify the simple $G$-modules in Theorem \[thm:F4\].
Finally we remark that, although in this article we deal with an algebraically closed field of positive characteristic, the results also make sense in characteristic zero and give the known classification in this case; cf. [@Kac77; @Ma14].
S.-J.C. is partially supported by a MoST and an Academia Sinica Investigator grant; B.S. is partially supported by the National Natural Science Foundation of China (Grant Nos. 11671138, 11771279) and Shanghai Key Laboratory of PMMP (No. 13dz2260400); W.W. is partially supported by an NSF grant DMS-1702254. We thank East China Normal University and Institute of Mathematics at Academia Sinica for hospitality and support.
Modular representations of algebraic supergroups {#sec:general}
================================================
Algebraic supergroups and $({\text{Dist}(G)},T)$-$\mathfrak{mod}$
-----------------------------------------------------------------
Throughout the paper, the ground field $k$ is assumed to be algebraically closed and of characteristic $p>2$ (sometimes we will specify a stronger assumption $p>3$).
We shall review briefly some generalities on algebraic supergroups; cf. [@BK03; @SW08; @FG12; @MS17]. An (affine) algebraic supergroup $G$ is an affine superscheme whose coordinate ring $k[G]$ is a Hopf superalgebra that is finitely generated as a $k$-algebra, and gives rise to a functor from the category of commutative $k$-superalgebras to the the category of groups. The underlying purely even group ${G_{{{\overline}0}}}$ is a closed subgroup of $G$ corresponding to the Hopf ideal generated by $k[G]_{\overline 1}$, and it is an algebraic group in the usual sense. For an algebraic supergroup $G$, the distribution superalgebra ${\text{Dist}}(G)$, which is by definition the restricted dual of the Hopf superalgebra $k[G]$, is a cocommutative Hopf superalgebra.
We denote by $G$-$\mathfrak{mod}$ the category of rational $G$-modules with (not necessarily homogeneous) $G$-homomorphisms. Note that a $G$-module is always [*locally finite*]{}, i.e., it is a sum of finite-dimensional $G$-modules. Given a closed subgroup $T$ of $G$, a ${\text{Dist}(G)}$-module $M$ is called a $({\text{Dist}(G)},T)$-module if $M$ has a structure of a $T$-module such that the ${\text{Dist}}(T)$-module structure on $M$ induced from the actions of ${\text{Dist}(G)}$ and of $T$ coincide. We denote by $({\text{Dist}(G)},T)$-$\mathfrak{mod}$ the category of locally finite $({\text{Dist}(G)},T)$-modules, and denote by ${\text{Dist}(G)}$-$\mathfrak{mod}$ the category of locally finite ${\text{Dist}(G)}$-modules. (We shall always take $T$ to be a maximal torus of $G$ when $G$ is of basic type.)
Modules of basic algebraic supergroups
--------------------------------------
Let ${\mathfrak{g}}$ be a basic Lie superalgebra over $k$ [@CW12; @FG12; @G14], including the three exceptional types: ${D(2|1;\zeta)}$, $G(3)$, and $F(3|1)$. The non-degenerate bilinear form $(\cdot, \cdot)$ of ${\mathfrak{g}}$ over $k$ exists when the characteristic $p$ of $k$ satisfies $p>2$ for type $\mathfrak{gl}, \mathfrak{osp}$ and ${D(2|1;\zeta)}$, and $p>3$ for $G(3)$ and $F(3|1)$.
Algebraic supergroups over $k$ associated with basic (including exceptional) Lie superalgebras are constructed in analogy to Chevalley’s construction of semisimple algebraic groups (see [@FG12] and [@G14]); we shall use the same terminologies (such as basic type, exceptional type) to refer to Lie superalgebras and corresponding supergroups. We shall call $G$ simply connected if ${G_{{{\overline}0}}}$ is a simply connected algebraic group, taking advantage of [@Mas12 Proposition 35]. Simple-connected supergroups of basic type exist, and we shall assume the exceptional supergroups in this paper to be simply connected.
The assumption on Chevalley bases in [@SW08 Theorem 2.8] is satisfied for all algebraic supergroups of basic type, by the constructions in [@FG12; @G14]. Hence we have the following.
[@SW08 Theorem 2.8] [@MS17] \[prop:equiv\] Let $G$ be an algebraic supergroup of basic type. Then there is a natural equivalence of categories between $G$-$\mathfrak{mod}$ and $({\text{Dist}(G)},T)$-$\mathfrak{mod}$.
If we further assume $G$ is simply connected, then $({\text{Dist}(G)},T)$-$\mathfrak{mod}$ in Proposition \[prop:equiv\] above can be replaced by ${\text{Dist}(G)}$-$\mathfrak{mod}$; cf. [@Jan03 II.1.20].
A supergroup $G$ of basic type can be constructed as a Chevalley supergroup through a Chevalley basis associated with a standard positive root system $\Phi^+$ as described in [@SW08 §3.4 and §3.5], [@FG12 3.3] and [@G14 §3]. Therefore, we have a standard Borel subgroup $B$ corresponding to $\Phi^+$, which contains a maximal torus $T$. The distribution superalgebra ${\text{Dist}(G)}$ contains ${\text{Dist}}(B)$ as a subalgebra. Set ${\mathsf{Lie}}(B) ={\mathfrak{b}}$. Let $X(T)$ be the character group of $T$. For $ {\lambda}\in X(T)$, we denote the Verma module of ${\text{Dist}(G)}$ by $$M({\lambda}) = {\text{Dist}(G)}\otimes_{{\text{Dist}}(B)} k_{\lambda},$$ where $k_{\lambda}$ is the one-dimensional ${\text{Dist}}(B)$-module of weight ${\lambda}$. The ${\text{Dist}(G)}$-module $M({\lambda})$ has a unique simple quotient $L({\lambda})$, and furthermore the ${\text{Dist}(G)}$-modules $L({\lambda})$ are non-isomorphic for distinct ${\lambda}\in X(T)$. By definition, $L({\lambda})$ is $X(T)$-graded and thus a $T$-module. Denote by $X^+(T)$ the set of $G_{{{\overline}0}}$-dominant integral weights (with respect to $\Phi^+$).
[@SW08 Lemma 4.1] \[lem:simple\] Every simple module in the category $({\text{Dist}(G)},T)$-$\mathfrak{mod}$ is isomorphic to a finite-dimensional highest weight module $L({\lambda})$ for some ${\lambda}\in X^+(T)$, and vice versa.
By Proposition \[prop:equiv\] and Lemma \[lem:simple\], the classification of simple $G$-modules can be reformulated as the determination of the following set: $$\label{X+T}
X^\dagger(T) =\big\{{\lambda}\in X^+(T) ~\big |~ L({\lambda}) \text{ is finite dimensional} \big\}.$$ For general supergroups of basic type, $X^\dagger(T)$ turns out to be a nontrivial proper subset of $X^+(T)$.
\[rem:osp\] For a supergroup $G$ of type $\mathfrak{spo}(2n|\ell)$, the subset $X^\dagger(T) \subset X^+(T)$ was determined explicitly in [@SW08]. Note the supergroup $G$ therein has even subgroup ${G_{{{\overline}0}}}= \text{Sp}_{2n} \times \text{SO}_\ell$ and hence is not simply connected. For a simply connected group of type $\mathfrak{spo}(2n|\ell)$, one would have additional simple modules $L({\lambda})$, where ${\lambda}\in X^+(T)$ is of the form ${\lambda}\in \sum_{i<0} {\mathbb Z}\delta_i + \sum_{j>0} ({{\Small \frac12}}+{\mathbb Z})\delta_j$ in the notation of [@SW08 §3.3-3.4]; This follows from Proposition \[prop:EulerFinite\] below.
We denote by $L'({\lambda})$ and $L''({\lambda})$ the highest weight ${\text{Dist}(G)}$-modules with respect to positive systems $\Phi'^{+}$ and $\Phi''^{+}$, respectively.
[@BKu03 Lemma 4.2] [@SW08 Lemma 5.7] \[lem:oddref\] Let ${\lambda}\in X(T)$, and let $\beta$ be an odd isotropic root for ${\mathfrak{g}}$. Suppose that $\Phi'^{+}$ and $\Phi''^{+}$ are two positive systems of ${\mathfrak{g}}$ such that $\Phi''^{+} =\Phi'^{+} \cup \{-\beta\} \backslash \{\beta \}$. Then, $$L'' ({\lambda}) \cong \left \{
\begin{array}{ll}
L'({\lambda}) & \text{ if } ({\lambda}, \beta) \equiv 0 \pmod p, \\
L'({\lambda}-\beta) & \text{ if } ({\lambda}, \beta) \not \equiv 0 \pmod p.
\end{array} \right.$$
We shall say $\Phi''^{+}$ is obtained from $\Phi'^{+}$ by an odd reflection in the setup of Lemma \[lem:oddref\]. Often we shall abbreviate $a\equiv b\pmod p$ as $a\equiv b$ later on. In the coming sections dealing with exceptional supergroups, we shall be very explicit about the (positive) root systems and odd reflections.
\[lem:rational\] Let $L=L({\lambda})$, for ${\lambda}\in X^+(T)$. Suppose that $L$ is isomorphic to $L^{{\mathfrak{b}}'}({\lambda}')$ with ${\lambda}' \in X^+(T)$, for every Borel subalgebra ${\mathfrak{b}}'$ that is obtained from ${\mathfrak{b}}$ by a sequence of odd reflections. Then $L$ is locally finite as a ${\text{Dist}(G_{\bar 0})}$-module, i.e., it is a rational $G_{{{\overline}0}}$-module.
We recall the following observation (cf., e.g., [@Se11; @Ma14]):
[*For every positive even root $\alpha$ in $\Phi_{{{\overline}0}}^+$, either $\alpha/2$ (if it is a root) or $\alpha$ appears as a simple root in some simple system $\Pi'$ associated to ${\mathfrak{b}}'$.*]{}
Denote by $SL_{2,{\alpha}}$ the root subgroup of $G$ associated to $\alpha$. Then by the assumption of the lemma, ${\text{Dist}}(SL_{2,{\alpha}})$ acts on $L$ locally finitely (i.e., $L$ is a rational $SL_{2,{\alpha}}$-module). It follows that $L$ is a rational $G_{{{\overline}0}}$-module, or equivalently, $L$ is locally finite as a ${\text{Dist}(G_{\bar 0})}$-module by Proposition \[prop:equiv\].
\[lem:lf=fd\] If a finitely generated ${\text{Dist}(G)}$-module $M$ is locally finite as a ${\text{Dist}(G_{\bar 0})}$-module, then $M$ is finite dimensional.
Since ${\text{Dist}(G)}$ is finitely generated over the algebra ${\text{Dist}(G_{\bar 0})}$, as a ${\text{Dist}(G_{\bar 0})}$-module $M$ is also finitely generated. Together with the locally finiteness assumption, this implies that $M$ is finite dimensional.
The combination of Proposition \[prop:equiv\], Lemmas \[lem:oddref\], \[lem:rational\] and \[lem:lf=fd\] provides us with an effective approach of classifying simple $G$-modules. Indeed, the problem of determining the finite-dimensional irreducible modules is thus reduced to determining the weights that remain to be $G_{\bar 0}$-dominant integral when transformed to highest weights with respect to any Borel (with fixed even part).
Euler characteristic
--------------------
Let $H$ be a closed subgroup of an algebraic supergroup $G$ such that the quotient superscheme $G/H$ is locally decomposable (cf. [@B06 the paragraph above Lemma 2.1]) and $G_{{{\overline}0}}/H_{{{\overline}0}}$ is projective; that is, the superscheme $X=G/H$ satisfies the assumptions (Q5)-(Q6) in [@B06 §2].
We refer to [@Jan03 II.2] and [@BK03 §6] for the precise definitions for induction and restriction functors below. Below, for a superspace $M$, we shall use $S(M)$ to denote the corresponding supersymmetric algebra.
([@B06 Corollary 2.8]) \[lem:Euler\] For any finite-dimensional $H$-module $M$, we have $$\sum_{i\geq 0} (-1)^i [{\mathsf{res}}^G_{{G_{{{\overline}0}}}} R^i {{\mathsf{ind}}}^G_H M] =\sum_{i\geq 0}(-1)^i [R^i{{\mathsf{ind}}}^{{G_{{{\overline}0}}}}_{{H_{{{\overline}0}}}}S\big(\left({\mathsf{Lie}}G \slash{\mathsf{Lie}}H\right)^*_{\bar 1}\big)\otimes M],$$ where the equality is understood in the Grothendieck group of ${G_{{{\overline}0}}}$-modules.
Now we take $G$ to be an algebraic supergroup of basic type, $H=B^-$ to be the opposite Borel subgroup. Since $G_{{{\overline}0}}/B^-_{{{\overline}0}}$ is projective and $G/B^-$ is locally decomposable (cf. [@MZ17] and [@Z18]), Lemma \[lem:Euler\] is applicable. For $M=k_\lambda$, we define $H^i(\lambda):=R^i {{\mathsf{ind}}}^G_{B^-}(k_\lambda)$ and then the Euler characteristic $$\chi({\lambda}):= \sum_{i\geq 0} (-1)^i\; {\mathsf{ch}}\, H^i(\lambda).$$ By Lemma \[lem:Euler\] we have the following formula for the Euler characteristic $$\begin{aligned}
\label{eq:Euler}
\chi ({\lambda})
= \sum_{i\geq 0}(-1)^i\; {\mathsf{ch}}\, R^i {{\mathsf{ind}}}^{{G_{{{\overline}0}}}}_{{B_{{{\overline}0}}}}S\big(\left({\mathfrak{g}}\slash {\mathfrak{b}}^-\right)^*_{\bar 1}\big)\otimes k_\lambda,
\end{aligned}$$ where ${\mathfrak{b}}^-$ is the opposite Borel subalgebra. Since the Euler characteristic is additive on short exact sequences, it suffices to determine the Euler characteristic on the composition factors of the ${B_{{{\overline}0}}}$-module $S\big(({\mathfrak{g}}\slash {\mathfrak{b}}^-)^*_{\bar 1}\big)\otimes k_\lambda$. Recall that the supersymmetric algebra of a purely odd space is the exterior algebra in the usual sense. Let $W$ be the Weyl group of ${\mathfrak{g}}$. Since $\Pi_{\beta\in \Phi^+_{\bar 1}}(e^{\beta\over 2}+e^{-\beta\over 2})$ is $W$-invariant, it follows by and Lemma \[lem:Euler\] that $$\begin{aligned}
\chi({\lambda})
=& {\sum_{w\in W}(-1)^{\ell(w)}w(e^{(\lambda+\rho_{\bar 0})}(\Pi_{\beta\in \Phi^+_{\bar 1}}(1+e^{-\beta}))) \over \Pi_{\alpha\in \Phi^+_{\bar 0}}(e^{\alpha\over 2}-e^{-\alpha\over 2})}
\cr
=& {\sum_{w\in W}(-1)^{\ell(w)}w(e^{(\lambda+\rho)}(\Pi_{\beta\in \Phi^+_{\bar 1}}(e^{\beta\over 2}+e^{-\beta\over 2}))) \over \Pi_{\alpha\in \Phi^+_{\bar 0}}(e^{\alpha\over 2}-e^{-\alpha\over 2})}
\cr
=&{\Pi_{\beta\in \Phi^+_{\bar 1}}(e^{\beta\over 2}+e^{-\beta\over 2})
\over \Pi_{\alpha\in \Phi^+_{\bar 0}}(e^{\alpha\over 2}-e^{-\alpha\over 2})}
\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)}.
\end{aligned}$$ Here as usual $\ell(w)$ denotes the length of $w\in W$, and $\rho$ is the Weyl vector given by $$\rho=\rho_{\bar 0}-\rho_{\bar 1},
\quad \text{ where }\rho_{\bar 0}={{\Small \frac12}}\sum_{\alpha\in\Phi^+_{\bar 0}} \alpha,\quad
\rho_{\bar 1} ={{\Small \frac12}}\sum_{\beta\in\Phi^+_{\bar 1}}\beta.$$
\[prop:EulerWeyl\] Let ${\lambda}\in X^+(T)$. The Euler characteristic is given by $$\begin{aligned}
\chi ({\lambda})
={\Pi_{\beta\in \Phi^+_{\bar 1}}(e^{\beta\over 2}+e^{-\beta\over 2})
\over \Pi_{\alpha\in \Phi^+_{\bar 0}}(e^{\alpha\over 2}-e^{-\alpha\over 2})}
\sum_{w\in W}(-1)^{\ell(w)}e^{w(\lambda+\rho)}.
\end{aligned}$$
\[prop:EulerFinite\] Let ${\lambda}\in X^+(T)$ be such that ${\lambda}+\rho$ is $G_{{{\overline}0}}$-dominant and regular. Then $L({\lambda})$ is finite dimensional.
By the same arguments as in [@B06 Corollary 2.8, Lemma 4.2], all $H^i(\lambda)$ are finite-dimensional $G$-modules. By assumption $\lambda+\rho$ is $G_{{{\overline}0}}$-dominant and regular, and hence, the highest weight of the Euler characteristic in Proposition \[prop:EulerWeyl\] equals $\lambda+\rho +(\rho_1-\rho_0)=\lambda$. The proposition now follows from Proposition \[prop:equiv\] and Lemma \[lem:simple\].
Modular representations of the supergroup of type ${D(2|1;\zeta)}$ {#sec:D}
==================================================================
Weights and roots for ${D(2|1;\zeta)}$
--------------------------------------
The Lie superalgebra ${\mathfrak{g}}={D(2|1;\zeta)}$ is a family of simple Lie superalgebras of basic type, which depends on a parameter ${\zeta}\in k\setminus\{0,-1\}$. There are isomorphisms of Lie superalgebras with different parameters $$\label{D:iso}
D(2|1;{\zeta}) \cong
D(2|1; -1-{\zeta}^{-1}) \cong
D(2|1;{\zeta}^{-1}).$$ Then ${\mathfrak{g}}={\mathfrak{g}}_{{{\overline}0}}\oplus {\mathfrak{g}}_{{{\overline}1}}$, where ${\mathfrak{g}}_{{{\overline}0}}\cong {\mathfrak{sl}_2}\oplus {\mathfrak{sl}_2}\oplus {\mathfrak{sl}_2}$ and, as a ${\mathfrak{g}}_{\bar 0}$-module, ${\mathfrak{g}}_{{{\overline}1}}\cong k^2\boxtimes k^2 \boxtimes k^2$. Here $k^2$ is the natural representation of ${\mathfrak{sl}_2}$.
Let ${\mathfrak{h}}^*$ be the dual of the Cartan subalgebra with basis $\{\delta, {\epsilon}_1,{\epsilon}_2\}$. We equip ${\mathfrak{h}}^*$ with a $k$-valued bilinear form $(\cdot,\cdot)$ such that $\{\delta, {\epsilon}_1,{\epsilon}_2\}$ are orthogonal and $$\begin{aligned}
\label{form:D}
(\delta, \delta) = -(1+{\zeta}),
\quad
({\epsilon}_1, {\epsilon}_1) = 1,
\quad
({\epsilon}_2, {\epsilon}_2) = {\zeta}.\end{aligned}$$ The root system for ${\mathfrak{g}}={\mathfrak{g}}_{{{\overline}0}}\oplus {\mathfrak{g}}_{{{\overline}1}}$ is denoted by $\Phi =\Phi_{\bar 0} \cup \Phi_{\bar 1}$. The set of simple roots of the standard simple system in ${\mathfrak{h}}^*$ of ${D(2|1;\zeta)}$ is chosen to be $$\begin{aligned}
\Pi =\{\alpha_0=\delta-{\epsilon}_1-{\epsilon}_2,\alpha_1=2{\epsilon}_1,\alpha_2=2{\epsilon}_2\}.\end{aligned}$$ The Dynkin diagram associated to $\Pi$ is depicted as follows:
-2cm
at (0,0) [$\bigotimes$]{}; (0,0)–(1,1); (0,0)–(1,-1); at (1.15,1.1) [$\bigcirc$]{}; at (1.15,-1.1) [$\bigcirc$]{}; at (-1.1,0) [$\delta-{\epsilon}_1-{\epsilon}_2$]{}; at (1.7,1.1) [$2{\epsilon}_1$]{}; at (1.7,-1.1) [$2{\epsilon}_2$]{}; at (-3,0) [ $\Pi$:]{};
The set of positive roots is $\Phi^+ =\Phi^+_{{{\overline}0}}\cup \Phi^+_{{{\overline}1}}$, where $$\begin{aligned}
\Phi^+_{\bar 0}=\{2\delta,2{\epsilon}_1,2{\epsilon}_2\},\quad
\Phi^+_{\bar 1}=\{\delta-{\epsilon}_1-{\epsilon}_2,\delta+{\epsilon}_1-{\epsilon}_2,\delta-{\epsilon}_1+{\epsilon}_2,\delta+{\epsilon}_1+{\epsilon}_2\}.\end{aligned}$$ One computes the Weyl vector $$\begin{aligned}
\rho =-\delta+{\epsilon}_1+{\epsilon}_2 \; (=-\alpha_0).\end{aligned}$$ Let $$X={\mathbb Z}\delta+{\mathbb Z}{\epsilon}_1+{\mathbb Z}{\epsilon}_2$$ denote the weight lattice of ${\mathfrak{g}}$. We denote the positive odd roots by $$\label{eq:betaD}
\beta_1=\delta-{\epsilon}_1-{\epsilon}_2, \quad
\beta_2=\delta+{\epsilon}_1-{\epsilon}_2, \quad
\beta_3=\delta-{\epsilon}_1+{\epsilon}_2, \quad
\beta_4=\delta+{\epsilon}_1+{\epsilon}_2.$$
There are 4 conjugate classes of positive systems under the Weyl group action. The 4 positive systems containing $\Phi^+_{{{\overline}0}}$ admit the following simple systems $\Pi^i$ $(0\le i\le 3)$, which are obtained from one another by applying odd reflections (see [@CW12 §1.4] for an introduction to odd reflections): $$\begin{aligned}
&\Pi^{0}:=\Pi=\{\delta-{\epsilon}_1-{\epsilon}_2, \; 2{\epsilon}_1, \; 2{\epsilon}_2\},
\cr
&\Pi^{1}:=r_{\beta_1}(\Pi)=\{-\delta+{\epsilon}_1+{\epsilon}_2,\; \delta+{\epsilon}_1-{\epsilon}_2, \; \delta-{\epsilon}_1+{\epsilon}_2\},
\cr
&\Pi^{2}:=r_{\beta_2}(\Pi^{1})=\{2{\epsilon}_1, \; - \delta-{\epsilon}_1+{\epsilon}_2, \; 2\delta \},
\cr
&\Pi^{3}:=r_{\beta_3}(\Pi^{1})=\{2{\epsilon}_2, \; 2\delta, \; -\delta+{\epsilon}_1-{\epsilon}_2\}.
\end{aligned}$$ The Dynkin diagrams of $\Pi^1$, $\Pi^2$, and $\Pi^3$ are respectively as follows:
-1cm
at (-6,0) [$\bigotimes$]{}; (-6,0)–(-5,1); (-6,0)–(-5,-1); (-4.85,-.9)–(-4.85,.9); at (-4.85,1.1) [$\bigotimes$]{}; at (-4.85,-1.1) [$\bigotimes$]{}; at (-7.1,0) [$-\delta+{\epsilon}_1+{\epsilon}_2$]{}; at (-6,1.1) [$\delta+{\epsilon}_1-{\epsilon}_2$]{}; at (-6,-1.1) [$\delta-{\epsilon}_1+{\epsilon}_2$]{}; at (-6.2,-2) [ $\Pi^1$]{}; at (-2,0) [$\bigotimes$]{}; (-2,0)–(-1,1); (-2,0)–(-1,-1); at (-.85,1.1) [$\bigcirc$]{}; at (-.85,-1.1) [$\bigcirc$]{}; at (-3.1,0) [$-\delta-{\epsilon}_1+{\epsilon}_2$]{}; at (-.3,1.1) [$2\delta$]{}; at (-.3,-1.1) [$2{\epsilon}_1$]{}; at (-2,-2) [ $\Pi^2$]{}; at (2,0) [$\bigotimes$]{}; (2,0)–(3,1); (2,0)–(3,-1); at (3.15,1.1) [$\bigcirc$]{}; at (3.15,-1.1) [$\bigcirc$]{}; at (.9,0) [$-\delta+{\epsilon}_1-{\epsilon}_2$]{}; at (3.7,1.1) [$2\delta$]{}; at (3.7,-1.1) [$2{\epsilon}_2$]{}; at (2,-2) [ $\Pi^3$]{};
The corresponding positive systems are denoted by $\Phi^{i+}$, for $0\le i \le 3$, with $\Phi^{0+}=\Phi^+$, and the corresponding Borel subalgebras of ${\mathfrak{g}}$ are denoted by ${\mathfrak{b}}^i$.
Highest weight computations {#subsec:comp}
---------------------------
The simply connected algebraic supergroup $G$ of type ${D(2|1;\zeta)}$ was constructed in [@G14]. With respect to the standard Borel subalgebra ${\mathfrak{b}}$ (associated to $\Phi^+$), we have $$X^+(T) =\{{\lambda}=d\delta + a {\epsilon}_1 + b {\epsilon}_2 \in X~|~a,b,d\in {\mathbb N}\}.$$ Denote the simple ${\text{Dist}(G)}$-module of highest weight ${\lambda}$ by $L({\lambda})$, where ${\lambda}=d\delta + a {\epsilon}_1 + b {\epsilon}_2 \in X^+(T)$. We denote by ${\lambda}^i$ the highest weight of $L({\lambda})$ with respect to $\Pi^i$, for $0\le i \le 3$. So ${\lambda}^0={\lambda}$. We shall apply and Lemma \[lem:oddref\] repeatedly to compute ${\lambda}^{i}$, for $1\le i\le 3$.
By using we have $$({\lambda}, \beta_1) = -d(1+{\zeta}) -a -b {\zeta}=-(a+d) -(b+d){\zeta}.$$ We now divide into 2 cases (1)-(2).
1. Assume $x_1:=(a+d) +(b+d){\zeta}\not\equiv 0 \pmod p$. Then $${\lambda}^{1} ={\lambda}-\beta_1 =(d-1){\delta}+(a+1){\epsilon}_1 +(b+1){\epsilon}_2.$$
First we compute $({\lambda}^1, \beta_2) = -(d-1)(1+{\zeta}) +(a+1) -(b+1){\zeta}=(a-d+2) -(b+d){\zeta}$, and then further divide into 2 subcases (a)-(b).
1. If $y_1:=(a-d+2) -(b+d){\zeta}\not\equiv 0\pmod p$, then $${\lambda}^{2} ={\lambda}^1 -\beta_2 =(d-2){\delta}+a {\epsilon}_1 + (b+2) {\epsilon}_2.$$
2. If $y_1 \equiv 0\pmod p$, then $ {\lambda}^{2} ={\lambda}^1 =(d-1){\delta}+(a+1){\epsilon}_1 +(b+1){\epsilon}_2.$
We also have $({\lambda}^1, \beta_3) = -(d-1)(1+{\zeta}) -(a+1) +(b+1){\zeta}=-(a+d) +(b-d+2){\zeta}$, and then divide into 2 subcases (a$'$)-(b$'$).
1. If $z_1:=-(a+d) +(b-d+2){\zeta}\not\equiv 0\pmod p$, then $${\lambda}^{3} ={\lambda}^1 -\beta_3 =(d-2){\delta}+(a+2) {\epsilon}_1 +b{\epsilon}_2.$$
2. If $z_1 \equiv 0\pmod p$, then ${\lambda}^{3} ={\lambda}^{1} =(d-1){\delta}+(a+1){\epsilon}_1 +(b+1){\epsilon}_2.$
2. Assume $x_1\equiv 0 \pmod p$. Then ${\lambda}^{1} ={\lambda}=d\delta + a {\epsilon}_1 + b {\epsilon}_2.$
First we compute $({\lambda}^1, \beta_2) = -d(1+{\zeta}) +a -b{\zeta}=(a-d) -(b+d){\zeta}$, and then divide into 2 subcases (a)-(b).
1. If $y_2:=(a-d) -(b+d){\zeta}\not\equiv 0\pmod p$, then $${\lambda}^{2} ={\lambda}^1 -\beta_2 =(d-1){\delta}+(a-1) {\epsilon}_1 + (b+1) {\epsilon}_2.$$
2. If $y_2 \equiv 0\pmod p$, then $ {\lambda}^{2} ={\lambda}^1 =d\delta + a {\epsilon}_1 + b {\epsilon}_2.$
We also have $({\lambda}^1, \beta_3) = -d(1+{\zeta}) -a +b{\zeta}=-(a+d) +(b-d){\zeta}$, and then divide into 2 subcases (a$'$)-(b$'$).
1. If $z_2:=-(a+d) +(b-d){\zeta}\not\equiv 0\pmod p$, then $${\lambda}^{3} ={\lambda}^1 -\beta_3 =(d-1){\delta}+(a+1) {\epsilon}_1 +(b-1){\epsilon}_2.$$
2. If $z_2 \equiv 0\pmod p$, then ${\lambda}^{3} ={\lambda}^{1} =d{\delta}+a{\epsilon}_1 +b{\epsilon}_2.$
Simple modules for the supergroup ${D(2|1;\zeta)}$
--------------------------------------------------
\[thm:D\] Let $p>2$. Let $G$ be the supergroup of type ${D(2|1;\zeta)}$. A complete list of inequivalent simple $G$-modules consists of $L({\lambda})$, where ${\lambda}=d\delta + a{\epsilon}_1 + b{\epsilon}_2$, with $d,a,b \in {\mathbb N}$, such that one of the following conditions is satisfied:
1. $d=0$, and $a\equiv b\equiv 0 \pmod p$;
2. $d =1$, and $(a+1) -(b+1){\zeta}\equiv 0 \pmod p$;
3. $d =1$, and $(a+1) +(b+1){\zeta}\equiv 0 \pmod p$;
4. $d\ge 2$, (and $a,b \in {\mathbb N}$ are arbitrary).
From the computations in §\[subsec:comp\] on the highest weights ${\lambda}^i$ $(1\le i \le 3)$ and their associated conditions, we obtain the following (mutually exclusive) sufficient and necessary conditions for $L({\lambda})$ to be finite dimensional:
1. $d=0$, $(a+d) +(b+d){\zeta}\equiv 0$, $(a-d) -(b+d){\zeta}\equiv 0$, $-(a+d) +(b-d){\zeta}\equiv 0$;
2. $d =1$, $(a+d) +(b+d){\zeta}\not\equiv 0$, $(a-d+2) -(b+d){\zeta}\equiv 0$, $-(a+d) +(b-d+2){\zeta}\equiv 0$;
3. $d =1$, $(a+d) +(b+d){\zeta}\equiv 0$, $(a-d) -(b+d){\zeta}\not\equiv 0$ with $a \ge 1$, $-(a+d) +(b-d){\zeta}\not\equiv 0$ with $b \ge 1$;
4. $d =1$, $(a+d) +(b+d){\zeta}\equiv 0$, $(a-d) -(b+d){\zeta}\equiv 0$, $-(a+d) +(b-d){\zeta}\not\equiv 0$ with $b \ge 1$;
5. $d =1$, $(a+d) +(b+d){\zeta}\equiv 0$, $(a-d) -(b+d){\zeta}\not\equiv 0$ with $a \ge 1$, $-(a+d) +(b-d){\zeta}\equiv 0$;
6. $d =1$, $(a+d) +(b+d){\zeta}\equiv 0$, $(a-d) -(b+d){\zeta}\equiv 0$, $-(a+d) +(b-d){\zeta}\equiv 0$;
7. $d\ge 2$, (and $a,b \in {\mathbb N}$ are arbitrary).
In Case (i), we obtain $d=0$ and $a\equiv b \equiv 0$, that is, Condition (1) in the theorem. Case (v) is the same as Condition (4).
Condition (ii) with the help of Condition (iii-a) simplifies to Condition (2).
We note that the seemingly additional constraints $a\ge 1$ and $b\ge 1$ in (iii-a)-(iii-b) as well as (iv-a)-(iv-b) follow automatically from the other conditions. Therefore, Conditions (iii-a)-(iv-b) simplify to Condition (3).
The theorem is proved.
Theorem \[thm:D\] makes sense over ${\mathbb C}$, providing an odd reflection approach to the classification of finite-dimensional simple modules over ${\mathbb C}$ (due to [@Kac77]). Indeed this classification can be read off from Theorem \[thm:D\] by regarding $p=\infty$.
Modular representations of the supergroup of type $G(3)$ {#sec:G}
========================================================
Weights and roots for the supergroup $G(3)$
-------------------------------------------
Let ${\mathfrak{g}}={\mathfrak{g}}_{{{\overline}0}}\oplus{\mathfrak{g}}_{{{\overline}1}}$ be the exceptional simple Lie superalgebra $G(3)$. We assume ${\epsilon}_1, {\epsilon}_2, {\epsilon}_3$ satisfy the linear relation $${\epsilon}_1 +{\epsilon}_2 +{\epsilon}_3 =0.$$ The root system is $\Phi =\Phi_{{{\overline}0}}\cup \Phi_{{{\overline}1}}$. We choose the standard simple system $\Pi =
\{\alpha_1, \alpha_2, \alpha_3\}$, where $$\alpha_1 = {\epsilon}_2 -{\epsilon}_1,
\quad
\alpha_2={\epsilon}_1,
\quad
\alpha_3 = \delta +{\epsilon}_3.$$ The Dynkin diagram associated to $\Pi$ is depicted as follows: .5cm
at (0,0) [$\bigcirc$]{}; (0.2,0)–(1.155,0); (0.15,0.1)–(1.2,0.1); (0.15,-0.1)–(1.2,-0.1); at (1.35,0) [$\bigcirc$]{}; at (0.75,0) [$>$]{}; (1.52,0)–(2.52,0); at (2.7,0) [$\bigotimes$]{}; at (-0.3,-.5) [${\alpha}_1={\epsilon}_2-{\epsilon}_1$]{}; at (1.3,-.5) [${\alpha}_2={\epsilon}_1$]{}; at (2.9,-.5) [${\alpha}_3=\delta+{\epsilon}_3$]{}; at (-2,0) [ $\Pi$:]{};
Then the standard positive roots are $\Phi^+ =\Phi_{{{\overline}0}}^+ \cup \Phi_{{{\overline}1}}^+$, where $$\Phi_{{{\overline}0}}^+ =\{2\delta, {\epsilon}_1, {\epsilon}_2, -{\epsilon}_3, {\epsilon}_2 -{\epsilon}_1, {\epsilon}_1 -{\epsilon}_3, {\epsilon}_2 -{\epsilon}_3 \},
\qquad
\Phi_{{{\overline}1}}^+ =\{\delta, \, \delta \pm {\epsilon}_i\mid 1\le i \le 3\}.$$ The Weyl vector for ${\mathfrak{g}}$ is $$\label{rho:ep}
\rho =- \frac52\, \delta +2{\epsilon}_1 +3{\epsilon}_2, \qquad \rho_{\bar 1} =\frac{7}{2}\delta.$$
We have ${\mathfrak{g}}_{{{\overline}0}}\cong \ G_2 \oplus {\mathfrak{sl}_2}$ and ${\mathfrak{g}}_{{{\overline}1}}\cong k^7 \boxtimes k^2$ as an adjoint ${\mathfrak{g}}_{{{\overline}0}}$-module, where $k^7$ denotes denotes the $7$-dimensional simple $G_2$-module and, as before, $k^2$ the natural ${\mathfrak{sl}_2}$-module. Note that $\{{\alpha}_1, {\alpha}_2\}$ forms a simple system of $G_2$, and we denote by ${\omega}_1, {\omega}_2$ the corresponding fundamental weights of $G_2$. We have $$\begin{aligned}
{\omega}_1 ={\epsilon}_1 + 2{\epsilon}_2, \qquad
{\omega}_2 &={\epsilon}_1 +{\epsilon}_2;
\\
{\epsilon}_1= 2{\omega}_2 -{\omega}_1, \qquad
{\epsilon}_2 &= {\omega}_1 -{\omega}_2.\end{aligned}$$ We can rewrite the formulae for $\rho$ in as $$\label{rho}
\rho =-\frac52 {\delta}+{\omega}_1 +{\omega}_2.$$
Denote the weight lattice of ${\mathfrak{g}}$ by $$X ={\mathbb Z}\delta \oplus X_2,$$ where $$X_2={\mathbb Z}{\omega}_1 \oplus {\mathbb Z}{\omega}_2 ={\mathbb Z}{\epsilon}_1 \oplus {\mathbb Z}{\epsilon}_2$$ is the weight lattice of $G_2$.
The bilinear form $(\cdot, \cdot)$ on $X$ is given by $$(\delta, \delta) =-2, \quad (\delta, {\epsilon}_i) =0,
\quad
({\epsilon}_i, {\epsilon}_i) =2, \quad ({\epsilon}_i, {\epsilon}_j) =-1, \quad \text{ for } 1\le i \neq j \le 3.$$ It follows that $$\begin{aligned}
\label{eq:omep}
\begin{split}
({\omega}_1, {\epsilon}_1) =0, \quad
({\omega}_1, {\epsilon}_2) =3, \quad ({\omega}_1, {\epsilon}_3) =-3,
\\
({\omega}_2, {\epsilon}_1) =1, \quad ({\omega}_2, {\epsilon}_2) =1,
\quad ({\omega}_2, {\epsilon}_3) =-2.
\end{split}\end{aligned}$$
Denote the following positive odd roots of $G(3)$ by $$\label{eq:beta}
\beta_1=\delta+{\epsilon}_3, \quad \beta_2=\delta-{\epsilon}_2, \quad \beta_3=\delta-{\epsilon}_1.$$
There are 4 conjugate classes of positive systems under the Weyl group action. The 4 positive systems containing $\Phi^+_{{{\overline}0}}$ admit the following simple systems $\Pi^i$ $(0\le i\le 3)$, which are obtained from one another by applying odd reflections (cf. [@CW12 §1.4]): $$\begin{aligned}
&\Pi^{0}:=\Pi=\{\varepsilon_2-\varepsilon_1, \varepsilon_1, \delta+\varepsilon_3\},
\cr
&\Pi^{1}:=r_{\beta_1}(\Pi)=\{{\epsilon}_2-{\epsilon}_1, \delta-{\epsilon}_2, -\delta-{\epsilon}_3 \},
\cr
&\Pi^{2}:=r_{\beta_2}(\Pi^{1})=\{\delta-{\epsilon}_1, -\delta+{\epsilon}_2, {\epsilon}_1 \},
\cr
&\Pi^{3}:=r_{\beta_3}(\Pi^{2})=\{-\delta+{\epsilon}_1, {\epsilon}_2-{\epsilon}_1, \delta\}.
\end{aligned}$$ The Dynkin diagrams of $\Pi^1$, $\Pi^2$, and $\Pi^3$ are respectively as follows:
at (0,0.5) [$\bigcirc$]{}; (0.2,0.5)–(1.155,0.5); (0.15,0.6)–(1.2,0.6); (0.15,0.4)–(1.2,0.4); at (1.35,0.5) [$\bigotimes$]{}; at (0.75,0.5) [$>$]{}; (1.52,0.5)–(2.52,0.5); at (2.7,0.5) [$\bigotimes$]{}; at (0,0) [${\epsilon}_2-{\epsilon}_1$]{}; at (1.3,0) [$\delta-{\epsilon}_2$]{}; at (2.6,0) [$-\delta-{\epsilon}_3$]{}; at (1.3,-.8) [ $\Pi^1$]{}; at (5,0) [$\bigotimes$]{}; (5.2,.1)–(6.155,1); at (6.35,1) [$\bigotimes$]{}; (6.52,1)–(7.52,.1); (5.2,0)–(7.52,0); at (7.7,0) [$\bigcirc$]{}; at (5,-.5) [$\delta-{\epsilon}_1$]{}; at (6.3,1.5) [$-\delta+{\epsilon}_2$]{}; at (7.6,-.5) [${\epsilon}_1$]{}; at (6.3,-.8) [ $\Pi^2$]{}; at (10,0) [$\bigcirc$]{}; (10.2,.1)–(11.155,1); at (11.35,1) [$\bigotimes$]{}; (11.52,1)–(12.52,.1); (10.2,0)–(12.52,0); (10.2,-.1)–(12.52,-.1); (10.2,.1)–(12.52,.1); at (11.5,0) [$>$]{}; (12.7,0) circle (1ex); at (10,-.5) [${\epsilon}_2-{\epsilon}_1$]{}; at (11.3,1.5) [$-\delta+{\epsilon}_1$]{}; at (12.7,-.5) [$\delta$]{}; at (11.3,-.8) [ $\Pi^3$]{};
The corresponding positive systems are denoted by $\Phi^{i+}$, for $0\le i \le 3$, with $\Phi^{0+}=\Phi^+$, and the corresponding Borel subalgebras of ${\mathfrak{g}}$ are denoted by ${\mathfrak{b}}^i$.
Highest weight computations {#subsec:comp}
---------------------------
The (simply connected) algebraic supergroup $G$ of type $G(3)$ was constructed in [@FG12]. With respect to the standard Bore subalgebra ${\mathfrak{b}}$ (associated to $\Phi^+$), we have $$X^+(T) =\{ {\lambda}=n\delta + r {\omega}_1 +s \omega_2 \in X \mid n,r,s\in {\mathbb N}\}.$$ Denote by $L({\lambda}) =L^{\mathfrak{b}}({\lambda})$ the irreducible ${\text{Dist}(G)}$-module of highest weight ${\lambda}$ with respect to the standard Borel subalgebra ${\mathfrak{b}}$, where $${\lambda}=d\delta + r {\omega}_1 +s \omega_2 \in X^+(T).$$ Assume the simple module $L({\lambda}) =L^{\mathfrak{b}}({\lambda})$ has ${\mathfrak{b}}^{i}$-highest weight ${\lambda}^{i}$, for $i=1,2,3$. We shall apply and Lemma \[lem:oddref\] repeatedly to compute ${\lambda}^{i}$, for $1\le i\le 3$. We have $({\lambda}, \beta_1) =-2d-3r-2s.$ We now divide into 2 cases (1)–(2).
1. Assume $\it{x_1:=2d+3r+2s}$ $\not\equiv 0 \pmod p$. Then $${\lambda}^{1} ={\lambda}-\beta_1 =(d-1){\delta}+r{\omega}_1 +(s+1){\omega}_2.$$ We obtain $({\lambda}^1, \beta_2) =-2d-3r-s+1.$ We then divide into 2 subcases (a)-(b).
1. Assume $\it y_1:=2d+3r+s-1$ $\not\equiv 0\pmod p$. Then $${\lambda}^{2} ={\lambda}^1 -\beta_2 =(d-2){\delta}+(r+1){\omega}_1 +s{\omega}_2.$$ We have $({\lambda}^2, \beta_3) =-2d-s+4.$
1. If $\it z_1:=2d+s-4$ $\not\equiv 0\pmod p$, then $${\lambda}^{3} ={\lambda}^2 -\beta_3 =(d-3){\delta}+r{\omega}_1 +(s+2){\omega}_2.$$
2. If $z_1 \equiv 0\pmod p$, then ${\lambda}^{3} ={\lambda}^2 =(d-2){\delta}+(r+1){\omega}_1 +s{\omega}_2.$
2. Assume $y_1=2d+3r+s-1 \equiv 0\pmod p$. Then $${\lambda}^{2} ={\lambda}^1 =(d-1){\delta}+r{\omega}_1 +(s+1){\omega}_2.$$ We have $({\lambda}^2, \beta_3) =-2d-s+1.$
1. If $\it z_2:=2d+s-1$ $\not\equiv 0\pmod p$, then $${\lambda}^{3} ={\lambda}^2 -\beta_3 =(d-2){\delta}+(r-1){\omega}_1 +(s+3){\omega}_2.$$
2. If $z_2 \equiv 0\pmod p$, then ${\lambda}^{3} ={\lambda}^2 ={\lambda}^1 =(d-1){\delta}+r{\omega}_1 +(s+1){\omega}_2.$
2. Assume $x_1=2d+3r+2s \equiv 0\pmod p$. Then ${\lambda}^{1} ={\lambda}=d{\delta}+r{\omega}_1 +s{\omega}_2.$ We have $({\lambda}^1, \beta_2) =-2d-3r-s.$ We then divide into 2 subcases (a)-(b).
1. Assume $\it y_2:=2d+3r+s$ $\not\equiv 0\pmod p$. Then $${\lambda}^{2} ={\lambda}^1 -\beta_2 ={\lambda}-\beta_2 =(d-1){\delta}+(r+1){\omega}_1 +(s-1){\omega}_2.$$ We have $({\lambda}^2, \beta_3) =-2d-s+3.$
1. If $\it z_3:=2d+s-3$ $\not\equiv 0\pmod p$, then $${\lambda}^{3} ={\lambda}^2 -\beta_3 =(d-2){\delta}+r{\omega}_1 +(s+1){\omega}_2.$$
2. If $z_3 \equiv 0\pmod p$, then ${\lambda}^{3} ={\lambda}^2 =(d-1){\delta}+(r+1){\omega}_1 +(s-1){\omega}_2.$
2. Assume $y_2=2d+3r+s \equiv 0\pmod p$. Then $${\lambda}^{2} ={\lambda}^1 ={\lambda}=d{\delta}+r{\omega}_1 +s{\omega}_2.$$ We have $({\lambda}^2, \beta_3) =-2d-s.$
1. If $\it z_4:=2d+s$ $\not\equiv 0\pmod p$, then $${\lambda}^{3} ={\lambda}^2 -\beta_3 =(d-1){\delta}+(r-1){\omega}_1 +(s+2){\omega}_2.$$
2. If $z_4 \equiv 0\pmod p$, then ${\lambda}^{3} ={\lambda}^2 ={\lambda}=d{\delta}+r{\omega}_1 +s{\omega}_2.$
\[prop:nec\] Assume ${\lambda}=d\delta + r {\omega}_1 +s \omega_2$, for $d,r,s\in {\mathbb N}$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions holds:
1. 1. 1. $d\ge 3$, $2d+3r+2s \not\equiv 0$, $2d+3r+s-1\not\equiv 0$, $2d+s-4\not\equiv 0$;
2. $d\ge 2$, $2d+s-4 \equiv 0$, ${3}(r+1) \not \equiv 0$, $2d+3r+2s\not\equiv 0$;
2. 1. $d\ge 2$, ${3}r\not\equiv 0$, $2d+3r+s-1 \equiv 0$, $2d+3r+2s \not\equiv 0$;
2. $d\ge 2$, ${3}r \equiv 0$, $2d+s-1\equiv 0$, $2d+3r+2s\not\equiv 0$;
2. 1. 1. $d\ge 2$, $s\not\equiv 0$, $2d+3r+2s\equiv 0$, $3r+s+3\not\equiv 0$;
2. $d\ge 1$, $s\not\equiv 0$, $2d+3r+2s\equiv 0$, $3r+s+3\equiv 0$;
2. 1. $s\equiv 0$, ${2}d\not\equiv 0$, $2d+3r \equiv 0$;
2. ${2}d\equiv {3}r\equiv s \equiv 0$.
The conditions in the proposition are summary of the dominant conditions for the new highest weights after odd reflections, which were computed in §\[subsec:comp\]. We remark that the natural condition on $d$ from the summary in §\[subsec:comp\] for the case (1b)(ii) is “$d\ge 1$", but “$d=1$" is quickly ruled out by the other conditions $3r\equiv 0, 2d+s-1\equiv 0, 2d+3r+2s\not\equiv 0$.
It follows by Lemmas \[lem:rational\] and \[lem:lf=fd\] that these conditions are also sufficient for $L({\lambda})$ to be finite dimensional.
Note the conditions in Proposition \[prop:nec\] are obtained without using any division on the conditions arising from odd reflections; some scalars $2, 3$ therein appear to be unnecessary for $p>3$, and they are kept for the case when $p=3$ below.
Simple modules for the supergroup $G(3)$ for $p>3$
--------------------------------------------------
We assume the characteristic of the ground field $k$ is $p>3$ in this subsection. We shall reformulate the conditions in Proposition \[prop:nec\] in a more useful form. We first analyze the case when $d\ge 3$.
\[prop:d3\] For ${\lambda}=d \delta + r {\omega}_1 +s \omega_2 \in X^+(T)$ with $d\ge 3$, the module $L({\lambda})$ is always finite dimensional.
Recall $\rho$ from . The proposition now follows by Proposition \[prop:EulerFinite\] since ${\lambda}+\rho =(d-\frac52) {\delta}+(r+1){\omega}_1 + (s+1){\omega}_2$ is in $X^+(T)$. (Alternatively, the proposition also follows from the analysis in §\[subsec:comp\].)
We then analyze the case when $d=2$.
\[prop:d=2\] Let $p>3$. The module $L({\lambda})$ is finite dimensional, for ${\lambda}=2 \delta + r {\omega}_1 +s \omega_2 \in X^+(T)$, if and only if one of the following 3 conditions are satisfied:
1. $s\equiv 0 \pmod p$;
2. $3r+ s +3 \equiv 0 \pmod p$;
3. $3r+2s+4\equiv 0 \pmod p$.
The three conditions (i)-(iii) in Proposition \[prop:d=2\] are not mutually exclusive. Three mutually exclusive conditions are given in - below.
Let us set $d=2$ in Proposition \[prop:nec\].
Condition (1a)(ii) becomes $s\equiv 0, r+1\not\equiv 0, 3r+4 \not\equiv 0$, while Condition (2b)(i) becomes $s\equiv 0, 3r+4\equiv 0$ (and it follows that $r+1\not\equiv 0$). Hence the combination of Conditions (1a)(ii) and (2b)(i) gives us the following conditions: $$\label{eq:1aii+2bi}
s\equiv 0, \qquad r+1\not\equiv 0.$$
Condition (1b)(i) becomes $r\not\equiv 0, 3r+s+3 \equiv 0, 3r+2s+4 \not\equiv 0$, while Condition (1b)(ii) becomes $r \equiv 0, 3r+s+3\equiv 0, 3r+2s+4\not\equiv 0$. Hence the combination of Conditions (1b)(i)-(ii) gives us the following conditions: $$\label{eq:1bi+1bii} 3r+s+3\equiv 0, \qquad 3r+2s+4\not\equiv 0.$$
Condition (2a)(i) becomes $s\not\equiv 0, 3r+2s+4 \equiv 0, 3r+s+3\not\equiv 0$, while Condition (2a)(ii) becomes $s\not\equiv 0, 3r+2s+4\equiv 0, 3r+s+3\equiv 0$. Hence the combination of Conditions (2a)(i)-(ii) gives us the following conditions: $$\label{eq:2ai+2aii} 3r+2s+4\equiv 0, \qquad s\not\equiv 0.$$ So by Proposition \[prop:nec\], $L({\lambda})$ is finite dimensional, for ${\lambda}=2 \delta + r {\omega}_1 +s \omega_2 \in X^+(T)$, if and only if one of the 3 (mutually exclusive) conditions , , holds.
Let us show that Conditions - are equivalent to Conditions (i)-(iii) in the proposition. Clearly if $r,s$ satisfy one of Conditions -, then they satisfy one of Conditions (i)-(iii). On the other hand, if $r,s$ satisfy Condition (i) but not , that is, $s\equiv r+1\equiv 0$, then is satisfied. If $r,s$ satisfy Condition (ii) but not , that is, $3r+s+3\equiv 3r+2s+4\equiv 0$, then is satisfied. Finally, if $r,s$ satisfy Condition (iii) but not , that is, $s\equiv 3r+2s+4\equiv 0$, then is satisfied.
The proof of Proposition \[prop:d=2\] is completed.
We finally analyze the case when $d=1$.
\[prop:d=1\] Let $p>3$. The module $L({\lambda})$ is finite dimensional, for ${\lambda}= \delta + r {\omega}_1 +s \omega_2 \in X^+(T)$, if and only if one of the following 2 conditions are satisfied:
1. $s-1\equiv 3r+4 \equiv 0 \pmod p$;
2. $s \equiv 3r+2 \equiv 0 \pmod p$.
Let us set $d=1$ in Proposition \[prop:nec\]. The case $d=1$ only occurs in Cases (2a)(ii) and (2b)(i). Condition (2a)(ii) reads $s\not\equiv 0, 3r+2s+2\equiv 0, 3r+s+3\equiv 0$, which is clearly equivalent to (i) in the proposition. Condition (2b)(i) is the same as (ii) above.
Summarizing Propositions \[prop:nec\], \[prop:d3\], \[prop:d=2\] and \[prop:d=1\] (and recalling Proposition \[prop:equiv\], Lemma \[lem:simple\]), we have established the following.
\[thm:G3\] Let $p>3$. Let $G$ be the supergroup of type $G(3)$. A complete list of inequivalent simple $G$-modules consists of $L({\lambda})$, where ${\lambda}= d \delta + r {\omega}_1 +s \omega_2$, with $d,r,s \in {\mathbb N}$, such that one of the following conditions is satisfied:
1. $d=0$, and ${3} r\equiv s\equiv 0 \pmod p$.
2. $d=1$, and $r,s$ satisfy either of (i)-(ii) below:
1. $s-1\equiv 3r+4 \equiv 0 \pmod p$;
2. $s \equiv 3r+2 \equiv 0 \pmod p$.
3. $d=2$, and $r,s$ satisfy either of (i)-(iii) below:
1. $s\equiv 0 \pmod p$;
2. $3r+ s +3 \equiv 0 \pmod p$;
3. $3r+2s+4\equiv 0 \pmod p$.
4. $d\ge 3$, (and $r,s\in {\mathbb N}$ are arbitrary).
Theorem \[thm:G3\] makes sense over ${\mathbb C}$, providing an odd reflection approach to the classification of finite-dimensional simple modules over ${\mathbb C}$ (due to [@Kac77]; also cf. [@Ma14]). Indeed this classification can be read off from Theorem \[thm:G3\] (by regarding $p=\infty$) as follows. [*The ${\mathfrak{g}}$-modules $L({\lambda})$ over ${\mathbb C}$ is finite dimensional if and only ${\lambda}= d \delta + r {\omega}_1 +s \omega_2$, for $d, r, s \in {\mathbb N}$, satisfies one of the 3 conditions: $
(1) \; d=r=s=0; \;
(2) \; d=2, s=0; \;
(3) \; d \ge 3. $* ]{}
Simple modules for the supergroup $G(3)$ for $p=3$ {#sec:p=3}
--------------------------------------------------
The assumption $p>3$ is not really necessary for the definition of $G$ and classification of simple $G$-modules. The (less polished) conditions in Proposition \[prop:nec\] remain valid for $p=3$. When one works it through, it turns out to be the same as setting $p=3$ in Theorem \[thm:G3\]; note the scalar $3$ in (1) therein. We summarize this in the following.
\[thm:p=3\] Let $p=3$. Let $G$ be the supergroup of type $G(3)$. A complete list of inequivalent simple $G$-modules consists of $L({\lambda})$, where ${\lambda}= d \delta + r {\omega}_1 +s \omega_2$, with $d,r,s \in {\mathbb N}$, such that one of the following conditions is satisfied:
1. $d=0$, and $s\equiv 0 \pmod 3$;
2. $d=2$, $s\equiv 0$ or $1 \pmod 3$;
3. $d\ge 3$, (and $r,s\in {\mathbb N}$ are arbitrary).
Modular representations of the supergroup of type $F(3|1)$ {#sec:F}
==========================================================
We assume the characteristic of the ground field $k$ is $p>3$ in this section.
Weights and roots for $F(3|1)$
------------------------------
Let ${\mathfrak{g}}={\mathfrak{g}}_{{{\overline}0}}\oplus{\mathfrak{g}}_{{{\overline}1}}$ be the exceptional simple Lie superalgebra $F(3|1)$ (which is sometimes denoted by $F(4)$ in the literature). We have ${\mathfrak{g}}_{{{\overline}0}}\cong {\mathfrak{sl}_2}\oplus\mathfrak{so}_7 $ and ${\mathfrak{g}}_{{{\overline}1}}\cong k^2 \boxtimes k^8$ as ${\mathfrak{g}}_{{{\overline}0}}$-module, where $k^8$ here is the $8$-dimensional spin representation of $\mathfrak{so}_7$. The root system of ${\mathfrak{g}}$ can be described via the basis $\{\epsilon_1,\epsilon_2,\epsilon_3,\delta\}$ in ${\mathfrak{h}}^*\cong {\mathbb{C}}^4$ with a non-degenerate bilinear form $(\cdot, \cdot)$ as follows: $$\begin{aligned}
\label{bilinear form}
(\delta,\delta)=-3, (\delta,\epsilon_i)=0, (\epsilon_i,\epsilon_i)=1, (\epsilon_i,\epsilon_j)=0,\quad i,j=1,2,3, i\ne j.\end{aligned}$$ The root system $\Phi=\Phi_{{{\overline}0}}\cup \Phi_{{{\overline}1}}$ is as below: $$\Phi_{{{\overline}0}}=\{\pm\delta;\pm\epsilon_i\pm\epsilon_j;\pm\epsilon_i\mid i,j=1,2,3,i\ne j \};\;\;
\Phi_{{{\overline}1}}=\{{1\over 2}(\pm\delta\pm \epsilon_1\pm\epsilon_2\pm\epsilon_3) \}$$ The standard Borel subalgebra ${\mathfrak{b}}$ corresponds to the simple root system $$\Pi=\big\{\alpha_1:=\epsilon_1-\epsilon_2, \; \alpha_2:=\epsilon_2-\epsilon_2,\; \alpha_3:=\epsilon_3, \;\alpha_4:={1\over 2}(\delta-\epsilon_1-\epsilon_2-\epsilon_3) \big\}.$$ The fundamental weights of ${\mathfrak{g}}_{{{\overline}0}}$ associated with the ${\mathfrak{g}}_{{{\overline}0}}$-simple roots $\alpha_1,\alpha_2,\alpha_3,\delta$ are: $$\omega_1 :=\epsilon_1,\quad
\omega_2 :=\epsilon_1+\epsilon_2,\quad
\omega_3 :={1\over 2}(\epsilon_1+\epsilon_2+\epsilon_3),\quad
\omega_4 :={1\over 2}\delta.$$ Denote the weight lattice by $$X =\{{\lambda}=a\omega_1 +b\omega_2 +c\omega_3 +d\omega_4 ~|~a,b,c,d\in {\mathbb Z}\}.$$ Sometimes we simply denote ${\lambda}=a\omega_1 +b\omega_2 +c\omega_3 +d\omega_4 \in X$ as $${\lambda}=(a,b,c,d)\in {\mathbb Z}^4.$$ With respect to ${\mathfrak{b}}$, the Weyl vector $\rho$ can be expressed in terms of the fundamental weights as $$\begin{aligned}
\rho=\omega_1+\omega_2+\omega_3-3\omega_4.\end{aligned}$$
The Dynkin diagram associated to $\Pi$ is depicted as follows:
at (0,0.5) [$\bigcirc$]{}; (0.16,0.5)–(1.15,0.5); at (1.35,0.5) [$\bigcirc$]{}; at (2,0.5) [$>$]{}; (1.5,0.55)–(2.5,0.55); (1.5,0.45)–(2.5,0.45); at (2.7,0.5) [$\bigcirc$]{}; (2.9,0.5)–(3.85,0.5); at (4.05,0.5) [$\bigotimes$]{}; at (0,0) [${\epsilon}_1-{\epsilon}_2$]{}; at (1.3,0) [${\epsilon}_2-{\epsilon}_3$]{}; at (2.7,0) [${\epsilon}_3$]{}; at (4.6,0) [${{\Small \frac12}}(\delta-{\epsilon}_1-{\epsilon}_2-{\epsilon}_3)$]{}; at (-1.8,0.5) [ $\Pi$:]{};
From (\[bilinear form\]), we have $$\begin{aligned}
\label{bracet for fundmental weight}
\begin{split}
&(\omega_1,\epsilon_1)=1, \; (\omega_1,\epsilon_2)=0,\;(\omega_1,\epsilon_3)=0,\; (\omega_1,\delta)=0,\cr
&(\omega_2,\epsilon_1)=1, \; (\omega_2,\epsilon_2)=1,\;(\omega_2,\epsilon_3)=0,\; (\omega_2,\delta)=0,\cr
&(\omega_3,\epsilon_1)={1\over 2}, \; (\omega_3,\epsilon_2)={1\over 2},\;(\omega_3,\epsilon_3)={1\over 2},\; (\omega_3,\delta)=0,\cr
&(\omega_4,\epsilon_1)=0, \; (\omega_4,\epsilon_2)=0,\;(\omega_4,\epsilon_3)=0,\; (\omega_4,\delta)=-{3\over 2}.
\end{split}\end{aligned}$$
Denote the positive odd roots for $F(3|1)$ by $$\begin{aligned}
{\gamma}_1 &={1\over 2}(\delta-\epsilon_1-\epsilon_2-\epsilon_3), \quad
{\gamma}_2={1\over 2}(\delta-\epsilon_1-\epsilon_2+\epsilon_3), \quad
{\gamma}_3={1\over 2}(\delta-\epsilon_1+\epsilon_2-\epsilon_3),
\\
{\gamma}_4 &={1\over 2}(\delta-\epsilon_1+\epsilon_2+\epsilon_3), \quad
{\gamma}_5={1\over 2}(\delta+\epsilon_1-\epsilon_2-\epsilon_3).\end{aligned}$$ In terms of the fundamental weights, we can reexpress the odd roots ${\gamma}_i$ as follows: $$\begin{aligned}
\gamma_1&={1\over 2}\delta - \omega_3,\quad \gamma_2={1\over 2}\delta - \omega_2+\omega_3,\quad \gamma_3={1\over 2}\delta - \omega_1+\omega_2-\omega_3,\cr
\gamma_4&={1\over 2}\delta - \omega_1+\omega_3,\quad \gamma_5={1\over 2}\delta + \omega_1-\omega_3.\end{aligned}$$ Besides the conjugate class of the standard simple system $\Pi^0:=\Pi=\{\epsilon_1-\epsilon_2, \epsilon_2-\epsilon_2,\epsilon_3,{\gamma}_1\}$ there are five other conjugate classes of simple systems under the Weyl group action as listed below. They all are obtained via sequences of odd reflections from $\Pi^0$ (cf. [@CW12 §1.4]): $$\begin{aligned}
\label{rep of simple roots}
\begin{split}
&\Pi^1=r_{{\gamma}_1}(\Pi^0)=\{\epsilon_1-\epsilon_2,\epsilon_2-\epsilon_3,{\gamma}_2,-{\gamma}_1\},\cr
&\Pi^2=r_{{\gamma}_2}(\Pi^1)=\{\epsilon_1-\epsilon_2,{\gamma}_3,-{\gamma}_2,\epsilon_3\}, \cr
&\Pi^3=r_{{\gamma}_3}(\Pi^2)=\{{\gamma}_5, -{\gamma}_3,\epsilon_2-\epsilon_3,{\gamma}_4\}, \cr
&\Pi^4=r_{{\gamma}_4}(\Pi^3)=\{\delta, \epsilon_3,\epsilon_2-\epsilon_3,-{\gamma}_4\}, \cr
&\Pi^5=r_{{\gamma}_5}(\Pi^3)=\{-{\gamma}_5, \epsilon_1-\epsilon_2,\epsilon_2-\epsilon_3,\delta\}.
\end{split}\end{aligned}$$ Their corresponding Dynkin diagrams are listed as follows:
at (0,0.5) [$\bigcirc$]{}; (0.2,0.5)–(1.15,0.5); at (1.35,0.5) [$\bigcirc$]{}; (1.52,0.5)–(2.52,0.5); at (2.7,0.5) [$\bigotimes$]{}; (2.9,0.5)–(3.85,0.5); at (4.05,0.5) [$\bigotimes$]{}; at (0,0) [${\epsilon}_1-{\epsilon}_2$]{}; at (1.3,0) [${\epsilon}_2-{\epsilon}_3$]{}; at (2.7,0) [${\gamma}_2$]{}; at (4,0) [$-{\gamma}_1$]{}; at (2.3,-.8) [ $\Pi^1$]{}; at (6,0.5) [$\bigcirc$]{}; (6.2,0.5)–(7.155,0.5); at (7.35,0.5) [$\bigotimes$]{}; (7.52,0.5)–(8.53,1.18); (7.52,0.5)–(8.52,-.17); (8.68,1.1)–(8.68,-.12); at (8.7,1.3) [$\bigcirc$]{}; at (8.7,-.3) [$\bigotimes$]{}; at (6,0) [${\epsilon}_1-{\epsilon}_2$]{}; at (7.3,0) [${\gamma}_3$]{}; at (8.6,-0.8) [$-{\gamma}_2$]{}; at (8.7,1.8) [${\epsilon}_3$]{}; at (7.3,-.8) [ $\Pi^2$]{}; at (11,0.5) [$\bigcirc$]{}; (11.2,0.5)–(12.155,0.5); at (12.35,0.5) [$\bigotimes$]{}; (12.52,0.5)–(13.53,1.18); (12.52,0.5)–(13.52,-.17); (13.68,1.1)–(13.68,-.12); at (13.7,1.3) [$\bigotimes$]{}; at (13.7,-.3) [$\bigotimes$]{}; at (11,0) [${\epsilon}_2-{\epsilon}_3$]{}; at (12.3,0) [$-{\gamma}_3$]{}; at (13.7,-0.8) [${\gamma}_5$]{}; at (13.7,1.8) [${\gamma}_4$]{}; at (12.3,-.8) [ $\Pi^3$]{};
at (0,0.5) [$\bigcirc$]{}; (0.2,0.5)–(1.15,0.5); at (1.35,0.5) [$\bigotimes$]{}; (1.52,0.5)–(2.52,0.5); at (2.7,0.5) [$\bigcirc$]{}; (2.9,0.55)–(3.85,0.55); (2.9,0.45)–(3.85,0.45); at (4.05,0.5) [$\bigcirc$]{}; at (0,0) [$\delta$]{}; at (1.3,0) [$-{\gamma}_4$]{}; at (2.7,0) [${\epsilon}_3$]{}; at (4,0) [${\epsilon}_2-{\epsilon}_3$]{}; at (3.4,0.5) [$<$]{}; at (2.3,-.5) [ $\Pi^4$]{}; at (6,0.5) [$\bigcirc$]{}; (6.2,0.5)–(7.15,0.5); at (7.35,0.5) [$\bigotimes$]{}; (7.52,0.5)–(8.52,0.5); at (8.7,0.5) [$\bigcirc$]{}; (8.9,0.5)–(9.85,0.5); at (10.05,0.5) [$\bigcirc$]{}; at (6,0) [$\delta$]{}; at (7.3,0) [$-{\gamma}_5$]{}; at (8.7,0) [${\epsilon}_1-{\epsilon}_2$]{}; at (10,0) [${\epsilon}_2-{\epsilon}_3$]{}; at (8.3,-.5) [ $\Pi^5$]{};
The corresponding positive systems are denoted by $\Phi^{i+}$, for $0\le i \le 5$, with $\Phi^{0+}=\Phi^+$, and the corresponding Borel subalgebras of ${\mathfrak{g}}$ are denoted by ${\mathfrak{b}}^i$.
Constraints on highest weights
------------------------------
Let $G$ be the simply connected algebraic supergroup of type $F(3|1)$ whose even subgroup is $SL_2(k) \times \text{Spin}_7(k)$. With respect to the standard Borel subalgebra ${\mathfrak{b}}$ (associated to $\Phi^+$), we have $$X^+(T) =\{ {\lambda}=a\omega_1 +b\omega_2 +c\omega_3 +d\omega_4 \in X~|~a,b,c,d\in {\mathbb N}\}.$$ Denote the simple ${\text{Dist}(G)}$-module of highest weight ${\lambda}$ by $L({\lambda})$, where ${\lambda}\in X^+(T)$. Assume that the simple module $L({\lambda}) =L^{\mathfrak{b}}({\lambda})$ has ${\mathfrak{b}}^{i}$-highest weight ${\lambda}^{i}$, for $0\le i\le 5$, where we have set ${\lambda}^0={\lambda}, {\mathfrak{b}}^0={\mathfrak{b}}$.
### The cases of $d\ge 4$ and $d=0$
\[lem:zero\] For any fixed $0\le i\le 3$, assume the module $L^{{\mathfrak{b}}^i}({\lambda}^{i})$ is finite dimensional and ${\lambda}^{i}$ is of the form $(x,y,z,0)$. Let $j=i+1$ if $i\le 2$, and let $j=4$ or $5$ if $i=3$. Then $$({\lambda}^i, {\gamma}_j) \equiv 0 \pmod p, \qquad \text{ and }\quad {\lambda}^j ={\lambda}^i.$$
The second equality is an immediate consequence of the first one by Lemma \[lem:oddref\].
Assume that $({\lambda}^i, {\gamma}_j) \not \equiv 0.$ Then, by applying the odd reflection $r_{{\gamma}_j}$ and Lemma \[lem:oddref\], we have $L^{{\mathfrak{b}}^i}({\lambda}^{i}) = L^{{\mathfrak{b}}^j}({\lambda}^{j})$, where ${\lambda}^{j} = {\lambda}^{i} -{\gamma}_j$ is of the form $(*, *, *, -1)$. But then $L^{{\mathfrak{b}}^j}({\lambda}^{j})$ cannot be finite dimensional due to the fact ${\lambda}^{j} \not \in X^+(T)$, which is a contradiction.
\[prop F4 First\] Let ${\lambda}=a\omega_1 +b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$.
- If $d\ge 4$, then $L({\lambda})$ is finite dimensional for arbitrary $a,b,c\in{\mathbb N}$.
- If $d=0$, then $L({\lambda})$ is finite dimensional if and only if $a\equiv b\equiv c\equiv 0 \pmod p$.
\(1) Let $d\ge 4$. Then ${\lambda}+\rho=(a+1,b+1,c+1,d-3)\in X^+(T)$ and it is regular. Hence $L({\lambda})$ is finite dimensional by Proposition \[prop:EulerFinite\].
\(2) Assume $L({\lambda})$ is finite dimensional, for ${\lambda}=(a,b,c,0)$. Lemma \[lem:zero\] is applicable and gives us $({\lambda}, {\gamma}_1) \equiv ({\lambda}, {\gamma}_2) \equiv ({\lambda},{\gamma}_3) \equiv 0\pmod p$. A direct computation shows $$({\lambda}, {\gamma}_1) = -{1\over 2}a-b-{3\over 4}c,
\qquad
({\lambda}, {\gamma}_2) = -{1\over 2}a-b-{1\over 4}c,
\qquad
({\lambda},{\gamma}_3) = -{1\over 2}a- \frac14 c.$$ From these we conclude that $a\equiv b\equiv c\equiv 0 \pmod p$. In this case we have ${\lambda}^5={\lambda}^4={\lambda}^3 ={\lambda}^2={\lambda}^1={\lambda}$.
By Lemma \[lem:rational\], we see the condition $a\equiv b\equiv c\equiv 0 \pmod p$ is also sufficient for $L({\lambda})$ to be finite dimensional (this also follows easily by Steinberg tensor product theorem).
### The case of $d=1$
\[prop:d=1simple\] Let ${\lambda}=a\omega_1 +b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$, with $d=1$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions holds.
- $a\equiv 2b +3\equiv c-1 \equiv 0 \pmod p$;
- $2a +1\equiv 2b+1 \equiv c\equiv 0 \pmod p$;
- $2a+3\equiv b\equiv c\equiv 0 \pmod p$.
Assume $L({\lambda})$ is finite dimensional, for ${\lambda}=(a,b,c,1) \in X^+(T)$. We compute $$({\lambda}, {\gamma}_1)=-{1\over 2}a-b-{3\over 4}(c+1).$$
For now let us assume $-{1\over 2}a-b-{3\over 4}(c+1)\not\equiv 0 \pmod p$. Then ${\lambda}^1={\lambda}-{\gamma}_1=(a,b,c+1,0).$ Hence Lemma \[lem:zero\] is applicable and gives us $({\lambda}^1, {\gamma}_2) \equiv ({\lambda}^1,{\gamma}_3) \equiv ({\lambda}^1, {\gamma}_4) \equiv 0$. A direct computation shows $$\begin{aligned}
({\lambda}^1, {\gamma}_2) =-{1\over 2}a-b-{1\over 4}(c+1),
\quad
({\lambda}^1, {\gamma}_3) =-{1\over 2}a-{1\over 4}(c+1),
\quad
({\lambda}^1,{\gamma}_4) =-{1\over 2}a+{1\over 4}(c+1).\end{aligned}$$ From these we conclude that $a\equiv b \equiv c+1 \equiv 0$. This contradicts the assumption $-{1\over 2}a-b-{3\over 4}(c+1)\not\equiv 0$.
So we always have $$\begin{aligned}
\label{eq:d=1:laga1=0}
-{1\over 2}a-b-{3\over 4}(c+1) \equiv 0 \pmod p,
\qquad
\text{and}
\qquad
{\lambda}^1 ={\lambda}=(a,b,c,1).\end{aligned}$$ Using the above equations, we compute $$({\lambda}^1,{\gamma}_2)=-{1\over 2}a-b-{1\over 4}c-{3\over 4}\equiv {1\over 2}c \pmod p.$$ We now divide into 2 cases (1)-(2).
\(1) Assume $c\not\equiv 0 \pmod p$. Then ${\lambda}^2={\lambda}^1-{\gamma}_2=(a,b+1,c-1,0)$; we necessarily have $c\ge 1$. Hence Lemma \[lem:zero\] is applicable and gives us that $({\lambda}^2,{\gamma}_3) \equiv ({\lambda}^2, {\gamma}_4) \equiv 0$. A direct computation shows $$({\lambda}^2, {\gamma}_3) =-{1\over 2}a-{1\over 4}(c-1),
\qquad
({\lambda}^2,{\gamma}_4) =-{1\over 2}a+{1\over 4}(c-1).$$ From these we conclude $a \equiv c-1\equiv 0$; a revisit of then gives us $b\equiv -{3\over 2}$. This gives us Condition (i) in the proposition. (Note the conditions $c\ge 1$ and are automatically satisfied.) In this case, we have ${\lambda}^5={\lambda}^4={\lambda}^3 ={\lambda}^2=(a,b+1,c-1,0)$ and ${\lambda}^1 ={\lambda}$.
\(2) Assume $c\equiv 0 \pmod p$. So ${\lambda}^2={\lambda}^1={\lambda}=(a,b,c,1)$. We compute $$({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}(c+3)\equiv -{1\over 2}a-{3\over 4} \pmod p.$$ Now we divide (2) into two subcases (2a)-(2b).
1. Assume $-{1\over 2}a-{3\over 4}\not\equiv 0 \pmod p$. Then ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1,b-1,c+1,0)$; we necessarily have $b\ge 1$. Hence Lemma \[lem:zero\] is applicable and gives us that $({\lambda}^3, {\gamma}_4) \equiv 0$. A direct computation shows $({\lambda}^3,{\gamma}_4) =-{1\over 2}a +{1\over 4}c -{1\over 4}$. Recalling $c\equiv 0$, we conclude that $a+{{\Small \frac12}}\equiv 0$. A revisit of then gives us $b\equiv -{{\Small \frac12}}$. This gives us Condition (ii) in the proposition. (Note the conditions $b\ge 1$ and are automatically satisfied.) In this case, we have ${\lambda}^5={\lambda}^4={\lambda}^3=(a+1,b-1,c+1,0)$ and ${\lambda}^2={\lambda}^1 ={\lambda}$.
2. Assume $-{1\over 2}a-{3\over 4}\equiv 0 \pmod p$. Then $a\equiv -{3\over 2}$, and it follows by $c\equiv 0$ and that $b\equiv 0$. This gives us Condition (iii). In this case, we have ${\lambda}^4={\lambda}^3={\lambda}^2={\lambda}^1={\lambda}$, and ${\lambda}^5 =(a-1,b,c+1,0)$.
By Lemma \[lem:rational\] and by inspection that all weights ${\lambda}^i$ lie in $X^+(T)$ for all $i$ in all cases above, we see the conditions (i)-(iii) are sufficient for $L({\lambda})$ to be finite dimensional. The proposition is proved.
### The case of $d=2$
\[prop:d=2messy\] Assume ${\lambda}=a\omega_1 + b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$ with $d=2$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions hold:
1. 1. $a\equiv c\equiv 0$, $b\not\equiv -1$ and $b\not\equiv -{3\over 2}$;
2. $a\not\equiv -1$, $b\equiv -a-1$, $c\equiv 2a$, and $b\ge 1$;
3. $a\not\equiv-{3\over 2}$, $b\equiv 0$ and $c\equiv-2a-4$.
2. 1. $c\equiv 2a+2$, $b\equiv -2a-3$, $c\geq 1$, and $a\not\equiv -1$;
2. 1. $c\equiv -2a-2$, $b\equiv a$, $a\not\equiv -{3\over 2}$, $a\not\equiv -1$, $c\geq 2$, and $a\geq 1$;
2. $a\equiv -{3\over 2}$, $b\equiv -{3\over 2}$, $c\equiv 1$;
3. $c\equiv 0$, $b\equiv -{1\over 2}a-{3\over 2}$, and $a\not\equiv -3$;
4. $c\equiv b\equiv 0$ and $a\equiv -3$.
Assume $L({\lambda})$ is finite dimensional, for ${\lambda}=(a,b,c,2) \in X^+(T)$. We compute $({\lambda},{\gamma}_1) =-{1\over 2}a-b-{3\over 4}c-{3\over 2}$, and then divide into two cases (1)-(2) below.
\(1) Assume $-{1\over 2}a-b-{3\over 4}c-{3\over 2}\not\equiv 0\pmod p$. Then ${\lambda}^1={\lambda}-{\gamma}_1=(a,b,c+1,1)$. We compute $({\lambda}^1,{\gamma}_2) =-{1\over 2}a-b-{1\over 4}c-1$, and then divide into 2 subcases (1a)-(1b).
(1a) Assume $-{1\over 2}a-b-{1\over 4}c-1\not\equiv 0\pmod p$. Then ${\lambda}^2={\lambda}^1-{\gamma}_2=(a,b+1,c,0)$. Hence Lemma \[lem:zero\] is applicable and gives us that $({\lambda}^2,{\gamma}_3) \equiv ({\lambda}^2, {\gamma}_4) \equiv ({\lambda}^2, {\gamma}_5) \equiv 0$. From these and a direct computation of $({\lambda}^2, {\gamma}_3)=-{1\over 2}a-{1\over 4}c$, $({\lambda}^2, {\gamma}_4)=-{1\over 2}a+{1\over 4}c$, and $({\lambda}^2, {\gamma}_5)={1\over 2}a-{1\over 4}c$, we conclude that $a\equiv c\equiv 0, b\not\equiv -1, b\not\equiv -{3\over 2}$, whence Condition 1.1. In this case, we have ${\lambda}^5={\lambda}^4 ={\lambda}^3={\lambda}^2=(a,b+1,c,0)$.
(1b) Assume $-{1\over 2}a-b-{1\over 4}c-1\equiv 0\pmod p$. Then ${\lambda}^2={\lambda}^1=(a,b,c+1,1)$. We compute $({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}c-1$, and then again divide into 2 subcases (1b-1)-(1b-2):
- Assume $-{1\over 2}a-{1\over 4}c-{1}\not\equiv 0$. Then ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1,b-1,c+2,0)$. Hence Lemma \[lem:zero\] is applicable and gives us that $({\lambda}^3,{\gamma}_4)\equiv ({\lambda}^3,{\gamma}_5) \equiv 0$. From these and a direct computation of [ $({\lambda}^3, {\gamma}_4)=-{1\over 2}(a+1)+{1\over 4}(c+2)$ and $({\lambda}^3, {\gamma}_5)={1\over 2}(a+1)-{1\over 4}(c+2)$]{}, we conclude that $c\equiv 2a$. Combining with the conditions on (1), (1b) and (1b-1), this gives us $b\equiv -a-1$ and $a\not\equiv -1$, whence Condition 1.2. In this case we have ${\lambda}^2={\lambda}^1=(a,b,c+1,1)$, and ${\lambda}^5={\lambda}^4={\lambda}^3=(a+1,b-1,c+2,0)$.
- Assume $-{1\over 2}a-{1\over 4}c-{1}\equiv 0$. Then ${\lambda}^3={\lambda}^2={\lambda}^1=(a,b,c+1,1)$. We deduce from the conditions on (1), (1b) and (1b-2) that $b\equiv 0,c\equiv-2a-4,a\not\equiv-{3\over 2}$, whence Condition 1.3. (We then compute $({\lambda}^3,{\gamma}_4)=-{1\over 2}a+{1\over 4}c-{1\over 2} \equiv -a-{3\over 2}\not\equiv0$. Thus, ${\lambda}^4={\lambda}^3-{\gamma}_4=(a+1,b,c,0)$. Note that $({\lambda}^3,{\gamma}_5)={1\over 2}a -{1\over 4} c-1$ ($\equiv a$). Hence ${\lambda}^5={\lambda}^3=(a,b,c+1,1)$ if $a\equiv 0$; ${\lambda}^5 =(a-1,b,c,0)$ if $a\not\equiv 0$.)
We again divide into 2 subcases, depending on the value of $({\lambda}^3,{\gamma}_4)$.
- Assume $-{1\over 2}a+{1\over 4}c-{1\over 2}\not\equiv 0$. Then [[$a\not\equiv -{3\over 2}$]{}]{}. In this case, ${\lambda}^4={\lambda}^3-{\gamma}_4=(a+1,b,c,0)$. If $a\not\equiv -{3\over 4}$, we have ${\lambda}^5=(a-1,b,c+2,0)$; otherwise, ${\lambda}^5={\lambda}^3=(a,b,c+1,1)$. As an upshot, the only new necessary condition in this subcase is $a\geq 1$. All these are summarized into Condition 1.3.
- Assume $-{1\over 2}a+{1\over 4}c-{1\over 2}\equiv 0$. Then $a\equiv -{9\over 4}$, $c\equiv-{5\over 2}$ and $b\equiv {3\over 4}$, whence Condition 1.4. In this case, we have ${\lambda}^4={\lambda}^3={\lambda}^2={\lambda}^1=(a,b,c+1,1)$ and ${\lambda}^5=(a-1,b,c+2,0)$.
Case (1b) and hence Case (1) are completed.
\(2) Assume $-{1\over 2}a-b-{3\over 4}c-{3\over 2}\equiv 0\pmod p$. We have ${\lambda}^1={\lambda}=(a,b,c,2)$. Then we compute $({\lambda}^1,{\gamma}_2)=-{1\over 2}a-b-{1\over 4}c-{3\over 2}$, and divide into 2 subcases (2a)-(2b).
(2a) Assume $-{1\over 2}a-b-{1\over 4}c-{3\over 2}\not\equiv 0$. Then we compute ${\lambda}^2={\lambda}^1-{{\gamma}_2}=(a,b+1,c-1,1)$; we necessarily have $c\geq 1$. (Note the combination of the condition $c\ge 1$ and the condition on (2) implies the condition on (2a).) We further compute $({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}c-{1\over 2}$, and then divide into 2 subcases (2a-1)-(2a-2).
- Assume $-{1\over 2}a-{1\over 4}c-{1\over 2}\not\equiv 0$. Then ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1, b,c,0)$. Hence Lemma \[lem:zero\] is applicable and gives us that $({\lambda}^3,{\gamma}_4)\equiv ({\lambda}^3,{\gamma}_5) \equiv 0$. Combining with the computations of [$({\lambda}^3,{\gamma}_4)=-{1\over 2}(a+1)+{1\over 4}c$ and $({\lambda}^3,{\gamma}_5)={1\over 2}(a+1)-{1\over 4}c$]{}, this implies $c\equiv 2a+2$ and $b\equiv -2a-3$; moreover the condition on (2a-1) becomes $a\not\equiv -1$. Thus, we have obtained Condition 2.1. In this case, we have ${\lambda}^1={\lambda}$, ${\lambda}^2=(a,b+1,c-1,1)$, and ${\lambda}^5={\lambda}^4={\lambda}^3=(a+1, b,c,0)$.
- Assume $-{1\over 2}a-{1\over 4}c-{1\over 2}\equiv 0$. The conditions on (2), (2a) and (2a-2) can be rephrased as $c\equiv -2a-2$, $b\equiv a$ and $a\not\equiv -1$. We have ${\lambda}^3={\lambda}^2=(a,b+1,c-1,1)$; we necessarily have $c\geq 1$. We further compute $({\lambda}^3,{\gamma}_4)=-{1\over 2}a+{1\over 4}c-1\equiv -a-{3\over 2}$, and again divide into 2 subcases:
- Assume $a\not\equiv -{3\over 2}$. Then we have ${\lambda}^4={\lambda}^3-{\gamma}_4=(a+1,b+1,c-2,0)$; we necessarily have $c\geq 2$. Moreover, if $({\lambda}^3,{\gamma}_5)=a\not\equiv 0$, then ${\lambda}^5={\lambda}^3-{\gamma}_5=(a-1,b+1,c,0)$, requiring $a\geq 1$; otherwise, ${\lambda}^5={\lambda}^3$. This gives us Condition 2.2(i).
- Assume $a\equiv -{3\over 2}$. Then we have $b\equiv -{3\over 2}$ and $c\equiv 1$, whence Condition 2.2(ii). In this case, we have ${\lambda}^1={\lambda}$, ${\lambda}^4={\lambda}^3={\lambda}^2=(a,b+1,c-1,1)$, and ${\lambda}^5={\lambda}^3-{\gamma}_5=(a-1,b+1,c,0)$.
This completes Case (2a).
(2b) Assume $-{1\over 2}a-b-{1\over 4}c-{3\over 2}\equiv 0\pmod p$. Then ${\lambda}^2={\lambda}^1=(a,b,c,2)$. We compute [$({\lambda}^2,{\gamma}_3) =-{1\over 2}a-{1\over 4}c-{3\over 2}$]{}, and divide into 2 subcases (2b-1)-(2b-2).
- Assume $-{1\over 2}a-{1\over 4}c-{3\over 2}\not\equiv 0\pmod p$. Then we have $c\equiv 0$, $b\equiv -{1\over 2}a-{3\over 2}$, and $a\not\equiv -3$, whence Condition 2.3. In this case, we have ${\lambda}^2={\lambda}^1={\lambda}=(a,b,c,2)$, ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1,b-1,c+1,1)$, and then $({\lambda}^3,{\gamma}_4)=-{1\over2}a+{1\over 4}c-1\equiv -{1\over 2}a-1$ and $({\lambda}^3,{\gamma}_5)\equiv{1\over2}a-{1\over 2}$. So ${\lambda}^4={\lambda}^3- {\gamma}_4=(a+2,b-1,c,0) $ if $a\not\equiv -2$, and ${\lambda}^4={\lambda}^3$ otherwise; moreover, if $a\not\equiv 1$ then ${\lambda}^5={\lambda}^3-{\gamma}_5=(a,b-1,c+2,0)$; otherwise ${\lambda}^5={\lambda}^3$.
- Assume $-{1\over 2}a-{1\over 4}c-{3\over 2}\equiv 0\pmod p$. Then we have $a\equiv -3$, $b\equiv 0$ and $c\equiv 0$, whence Condition 2.4. In this case, we have ${\lambda}^i={\lambda}$ for $1\le i \le 5$.
Case (2b) and then Case (2) are hence completed. Therefore, we have established the necessary conditions as listed in the proposition for $L({\lambda})$ to be finite dimensional.
By inspection, we have all weights ${\lambda}^i \in X^+(T)$ for all $i$ in every case above. Hence by Lemma \[lem:rational\] we conclude that the conditions as listed in the proposition are also sufficient for $L({\lambda})$ to be finite dimensional.
Now we simplify the above conditions by removing all inequalities. We caution that the resulting conditions are no longer mutually exclusive.
\[prop:d=2simple\] Set $d=2$. Assume ${\lambda}=a\omega_1 + b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions (i)–(vi) holds:
1. $a\equiv c\equiv 0\pmod p$;
2. $2a-c \equiv a+b+1\equiv 0\pmod p$;
3. $b\equiv 2a+c+4\equiv 0\pmod p$;
4. $2a-c+2 \equiv 2a+ b +3 \equiv 0\pmod p$, and $c\geq 2$;
5. $2a+c+2\equiv a-b\equiv 0\pmod p$, and $a\geq 1$;
6. $a+2b+3 \equiv c\equiv 0\pmod p$.
One first observes that all conditions listed in Proposition \[prop:d=2messy\] are part of conditions listed above in this proposition. Indeed the conditions above are basically obtained by removing the inequalities in the conditions in Proposition \[prop:d=2messy\]; the cases (1.4) and (2.4) with no inequalities in Proposition \[prop:d=2messy\] are part of (iii) and (vi) above, respectively.
It remains to show that all conditions above in this proposition are included in the list of conditions (1.1)–(1.3) and (2.1)–(2.4) in Proposition \[prop:d=2messy\].
If Condition (i) is satisfied but (1.1) in Proposition \[prop:d=2messy\] is not, then either (A) $b\equiv -1$, in which case $a\equiv c\equiv 0$, and so (1.2) is satisfied, or (B) $b \equiv -{3\over 2}$, in which case $a\equiv c\equiv 0$, and so (2.3) is satisfied.
If Condition (ii) is satisfied but (1.2) in Proposition \[prop:d=2messy\] is not, then either (A) $a\equiv -1$, in which case $b\equiv0$ and $c\equiv -2$, and hence (1.3) is satisfied, or (B) $b=0$, in which case, $a\equiv -1, c\equiv -2$, and so (1.3) is satisfied.
If Condition (iii) is satisfied but (1.3) in Proposition \[prop:d=2messy\] is not, then $a\equiv -{3\over 2}$, in which case $b\equiv 0, c\equiv -{1}$, and so (2.1) is satisfied.
If Condition (iv) is satisfied but (2.1) in Proposition \[prop:d=2messy\] is not, then $a\equiv -1$, in which case $b\equiv -1, c\equiv 0$, and so (2.3) is satisfied.
If Condition (v) is satisfied but (2.2)(i) in Proposition \[prop:d=2messy\] is not, then either (A) $a \equiv -{3\over 2}$, in which case $b \equiv -{3\over 2}, c\equiv 1$, and so (2.2)(ii) is satisfied; or (B) $a \equiv -1$, in which case $b \equiv -1, c\equiv 0$, and so (2.3) is satisfied; or (C) $c=0$, in which case $a\equiv b \equiv -1$, and so (2.3) is satisfied.
If Condition (vi) is satisfied but (2.3) in Proposition \[prop:d=2messy\] is not, then $a\equiv -3$, $b \equiv c\equiv 0$, and so (2.4) is satisfied.
The proposition is proved.
### The case of $d=3$
\[prop:d=3messy\] Assume ${\lambda}=a\omega_1 + b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$, with $d=3$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions holds:
1. 1. $c\equiv 2a+1$, and $b\not\equiv -2a-{3}$, $b\not\equiv -a-{2}$, $a\not\equiv -{1}$;
2. $c\equiv -2a-3$, and $b\not\equiv -1$, $b\not\equiv a$;
3. $b\equiv -{1\over 2}a-{1\over 4}c-{7\over 4}$, and $b\not\equiv 0$, $c \not\equiv -1$;
4. $b\equiv 0$, $c\equiv -2a-7$, and $a\not\equiv -3$.
2. 1. $b\equiv -{1\over 2}a-{3\over 4}c-{9\over 4}$, and $c\not\equiv 0$, $c\not\equiv -2a-5$;
2. $b\equiv a+{3\over 2}$, $c\equiv -2a-5$, and $c\not\equiv 0$;
3. $b\equiv-{1\over 2}a-{9\over 4}$, $c\equiv 0$, and $b\not\equiv 0$;
4. $a\equiv -{9\over 2}$, $b\equiv c\equiv 0$.
\[d=3messyBin\] Set $d=3$. Assume ${\lambda}=a\omega_1 +b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions holds:
[**[Case 1]{}**]{} $-{1\over 2}a-b-{3\over 4}c-{9\over 4}\not\equiv 0\pmod p$. Then one of the following conditions is satisfied
- $c\equiv 2a+1$ but $b\not\equiv -2a-{3}$ and $b\not\equiv -a-{2}$, with $a\not\equiv -{1}$.
- $c\equiv -2a-3$, $b\not\equiv -1$, $b\not\equiv a$, and $c\geq 1$ whenever $a\not\equiv -{3\over 2}$, and $a\geq 1$ whenever $a\not\equiv 0$.
- $b\equiv -{1\over 2}a-{1\over 4}c-{7\over 4}$ with $b\not\equiv 0$ and $a\not\equiv -2b-3$, satisfying $b\geq 1$ whenever either $c\not\equiv 2a-3$, or $c\not\equiv 2a+3$. [[Better to replace Condition $a\not\equiv -2b-3$ by $c\not\equiv -1$, thanks to $b\equiv -{1\over 2}a-{1\over 4}c-{7\over 4}$]{}]{}
- $b\equiv 0$, $c\equiv -2a-7$, with $a\not\equiv -3$ and $a\geq 1$ whenever $a\not\equiv -0$.
[**[Case 2]{}**]{} $-{1\over 2}a-b-{3\over 4}c-{9\over 4}\equiv 0\pmod p$. Then one of the following conditions is satisfied
- $b\equiv -{1\over 2}a-{3\over 4}c-{9\over 4}$, $c\not\equiv 0$ and $c\not\equiv -2a-5$ with $c\geq 1$.
- $b\equiv a+{3\over 2}$, $c=-2a-5$ with $a\not\equiv -{5\over 2}$ and $c\geq 1$, additionally $a\geq 1$ whenever $a\not\equiv 0$, $c\geq 2$ whenever $a\not\equiv -3$.
- $b\equiv-{1\over 2}a-{9\over 4}$, $c\equiv 0$ with $b\not\equiv 0$, and $b\geq 1$.
- $a\equiv -{9\over 2}$, $b\equiv c\equiv 0$.
Assume $L({\lambda})$ is finite dimensional, for ${\lambda}=(a,b,c,3) \in X^+(T)$. We compute $({\lambda},{\gamma}_1)= -{1\over 2}a-b-{3\over 4}c-{9\over 4}$, and divide into 2 cases (1)-(2).
\(1) Assume $-{1\over 2}a-b-{3\over 4}c-{9\over 4}\not\equiv 0\pmod p$. We have ${\lambda}^1={\lambda}-{\gamma}_1=(a,b,c+1,2)$. We compute $({\lambda}^1,{\gamma}_2) =-{1\over 2}a-b-{1\over 4}(c+1)-{3\over 2}$, and then divide into 2 cases (1a)-(1b).
(1a) Assume $-{1\over 2}a-b-{1\over 4}(c+1)-{3\over 2}\not\equiv 0\pmod p$. Then ${\lambda}^2={\lambda}^1-{\gamma}_2=(a,b+1,c,1)$. We compute $({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}c-{3\over 4}$, and again divide into two subcases (1a-i)-(1a-ii):
- Assume $-{1\over 2}a-{1\over 4}c-{3\over 4}\not\equiv 0$. Then ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1,b,c+1,0)$. Hence Lemma \[lem:zero\] is applicable and gives us that $({\lambda}^3,{\gamma}_4)\equiv ({\lambda}^3,{\gamma}_5) \equiv 0$. Combining with the computation of [$({\lambda}^3,{\gamma}_4)=-{1\over 2}(a+1)+{1\over 4}(c+1)$ and $({\lambda}^3,{\gamma}_5)={1\over 2}(a+1)-{1\over 4}(c+1)$]{}, this implies $c\equiv 2a+1$. The conditions on (1), (1a) and (1a-i) can be simplified to $a\not\equiv -{1}$, $b\not\equiv -a-{ 2}$ and $b\not\equiv {-2a}-{3}$, whence Condition 1.1. In this case, we have ${\lambda}^1=(a,b,c+1,2)$, ${\lambda}^2=(a,b+1,c,1)$, and ${\lambda}^5={\lambda}^4={\lambda}^3= (a+1,b,c+1,0)$.
- Assume $-{1\over 2}a-{1\over 4}c-{3\over 4}\equiv 0$. Then ${\lambda}^3={\lambda}^2=(a,b+1,c,1)$. The conditions on (1), (1a) and (1a-ii) can be simplified to $c\equiv -2a-{3}$, $b\not\equiv -1$ and $b\not\equiv a$, whence Condition 1.2. In this case, we have ${\lambda}^1=(a,b,c+1, 2)$, ${\lambda}^3={\lambda}^2=(a,b+1,c,1)$. If $a\not\equiv -{3\over 2}$, then ${\lambda}^4={\lambda}^3-{\gamma}_4=(a+1,b+1,c-1,0)$; otherwise, ${\lambda}^4={\lambda}^3$. If $a\not\equiv 0$, then ${\lambda}^4={\lambda}^3-{\gamma}_5=(a-1,b+1,c+1,0)$; otherwise, ${\lambda}^5={\lambda}^3$.
This completes Subcase (1a).
(1b) Assume $-{1\over 2}a-b-{1\over 4}(c+1)-{3\over 2}\equiv 0\pmod p$. Then ${\lambda}^2={\lambda}^1=(a,b,c+1,2)$. We compute $({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}c-{7\over 4}$, and again divide into two subcases (1b-i)-(1b-ii):
- Assume $-{1\over 2}a-{1\over 4}c-{7\over 4}\not\equiv 0$. We compute ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1,b-1,c+2,1)$. The conditions on (1), (1b) and (1b-i) become $b\equiv -{1\over 2}a-{1\over 4}c-{7\over 4}$, $b\not\equiv 0$, and $c\not\equiv -1$, whence Condition 1.3. In this case, we have ${\lambda}^2={\lambda}^1=(a,b,c+1,2)$, and ${\lambda}^3=(a+1,b-1,c+2,1)$. Moreover, if $c\not\equiv 2a+3$, then ${\lambda}^4={\lambda}^3-{\gamma}_4=(a+2,b-1,c+1,0)$; otherwise ${\lambda}^4={\lambda}^3$. If $c\not\equiv 2a-3$, then ${\lambda}^5={\lambda}^3-{\gamma}_5=(a,b-1,c+3,0)$; otherwise ${\lambda}^5={\lambda}^3$.
- Assume $-{1\over 2}a-{1\over 4}c-{7\over 4}\equiv 0$. Then ${\lambda}^3={\lambda}^2=(a,b,c+1,2)$. The conditions on (1), (1b) and (1b-ii) become $a\not\equiv -3$, $b\equiv 0$ and $c\equiv -2a-7$, whence Condition 1.4. In this case, we have ${\lambda}^3={\lambda}^2={\lambda}^1=(a,b,c+1,2)$. Moreover, If $a\not\equiv -3$, then ${\lambda}^4={\lambda}^3-{\gamma}_4=(a+1,b,c,1)$; otherwise ${\lambda}^4={\lambda}^3$. If $a\not\equiv 0$, then ${\lambda}^5={\lambda}^3-{\gamma}_5=(a-1,b,c+2,1)$; otherwise ${\lambda}^5={\lambda}^3$.
This completes Subcase (1b) and then Case (1).
\(2) Assume $-{1\over 2}a-b-{3\over 4}c-{9\over 4}\equiv 0$. Then ${\lambda}^1={\lambda}=(a,b,c,3)$. We compute $({\lambda}^1,{\gamma}_2)=-{1\over 2}a-b-{1\over 4}c-{9\over 4}$, and divide into 2 subcases (2a)-(2b).
(2a) Assume $-{1\over 2}a-b-{1\over 4}c-{9\over 4}\not\equiv 0$. Then ${\lambda}^2={\lambda}^1-{\gamma}_2=(a,b+1,c-1,2)$. We compute $({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}c-{5\over 4}$, and again divide into 2 subcases (2a-i)-(2a-ii):
- Assume $-{1\over 2}a-{1\over 4}c-{5\over 4}\not\equiv 0$. Then the conditions on (2), (2a) and (2a-i) become $b\equiv -{1\over 2}a-{3\over 4}c-{9\over 4}$, $c\not\equiv 0$ and $c\not\equiv -2a-5$, whence Condition 2.1. In this case, we have ${\lambda}^1={\lambda}$, ${\lambda}^2=(a,,b+1,c-1,2)$, ${\lambda}^3=(a+1,b,c,1)$. If $c\not\equiv 2a+5$, then ${\lambda}^4= (a+2,b,c-1,0)$; otherwise ${\lambda}^4={\lambda}^3$. If $c\not\equiv 2a-1$, then ${\lambda}^5= (a,b,c+1,0)$; otherwise ${\lambda}^5={\lambda}^3$.
- Assume $-{1\over 2}a-{1\over 4}c-{5\over 4}\equiv 0$. Then the conditions on (2), (2a) and (2a-ii) become $c\equiv -2a-5$, $b\equiv a+{3\over 2}$, and $c\not\equiv 0$, whence Condition 2.2. In this case, we have ${\lambda}^1={\lambda}$, ${\lambda}^3={\lambda}^2=(a,b+1,c-1,2)$. If $a\not\equiv -3$, then ${\lambda}^4=(a+1,b+1,c-2,1)$ (and $c\ge 2$ is guaranteed by Condition 2.2); otherwise, ${\lambda}^4={\lambda}^3$. If $a\not\equiv 0$, then ${\lambda}^5= (a-1,b+1,c,1)$; otherwise, ${\lambda}^5={\lambda}^3$.
This completes Subcase (2a).
(2b) Assume $-{1\over 2}a-b-{1\over 4}c-{9\over 4}\equiv 0$. Then ${\lambda}^2={\lambda}^1={\lambda}$. We compute $({\lambda}^2,{\gamma}_3)=-{1\over 2}a-{1\over 4}c-{9\over 4}$, and divide into 2 subcases (2b-i)-(2b-ii):
- Assume $-{1\over 2}a-{1\over 4}c-{9\over 4}\not\equiv 0$. Then ${\lambda}^3={\lambda}^2-{\gamma}_3=(a+1,b-1,c+1,2)$. The conditions on (2), (2b) and (2b-i) become $c\equiv 0$, $b\equiv-{1\over 2}a-{9\over 4}$, and $b\not\equiv 0$, whence Condition 2.3. In this case, we have ${\lambda}^2={\lambda}^1={\lambda}$, and ${\lambda}^3=(a+1,b-1,c+1,2)$. If $a\not\equiv -{7\over 2}$, then ${\lambda}^4= (a+2,b-1,c,1)$; otherwise, ${\lambda}^4={\lambda}^3$. If $a\not\equiv {5\over 2}$, then ${\lambda}^5= (a,b-1,c+2,1)$; otherwise, ${\lambda}^5={\lambda}^3$.
- Assume $-{1\over 2}a-{1\over 4}c-{9\over 4}\equiv 0$. The conditions on (2), (2b) and (2b-ii) become $b\equiv 0$, $c\equiv 0$ and $a\equiv -{9\over 2}$, whence Condition 2.4. In this case, we have ${\lambda}^i={\lambda}$ for $1\le i\le 4$ and ${\lambda}^5={\lambda}^3-{\gamma}_5=(a-1,b,c+1,2)$.
This completes Case (2). Therefore, we have established the necessary conditions as listed in the proposition for $L({\lambda})$ to be finite dimensional.
By inspection, we see that ${\lambda}^i\in X^+(T)$ for all $i$ in every case above. By Lemma \[lem:rational\], the conditions listed in the proposition are also sufficient for $L({\lambda})$ to be finite dimensional.
Now we simplify the conditions in Proposition \[prop:d=3messy\] by removing all inequalities.
\[prop:d=3simple\] Set $d=3$. Assume ${\lambda}=a\omega_1 + b\omega_2 +c\omega_3 +d\omega_4 \in X^+(T)$. Then $L({\lambda})$ is finite dimensional if only if one of the following conditions (i)–(v) holds:
1. $2a-c +1 \equiv 0\pmod p$;
2. $2a+c+3\equiv 0\pmod p$;
3. $2a+4b+c+7 \equiv 0\pmod p$;
4. $2a+c+7 \equiv b\equiv 0\pmod p$;
5. $2a +4b +3c +9\equiv 0\pmod p$.
One first observes that all conditions listed in Proposition \[prop:d=3messy\] are part of conditions listed above in this proposition. Indeed the conditions above are basically obtained by removing the inequalities in the conditions in Proposition \[prop:d=3messy\], and the case (2.4) with equalities only is included in (v).
It remains to show that all conditions above in this proposition are included in the list of conditions (1.1)–(1.4) and (2.1)–(2.4) in Proposition \[prop:d=3messy\]. We first check that the 4 subcases (2.1)–(2.4) of Proposition \[prop:d=3messy\] are equivalent to Condition (v). If Condition (v) is satisfied but (2.1) of Proposition \[prop:d=3messy\] is not, then we have 2 cases (A)-(B):
- $c\not \equiv 0$ and $c\equiv -2a-5$, in which case $b\equiv a+{3\over 2}$, $a\not\equiv -{5\over 2}$, and so (2.2) is satisfied;
- $c\equiv 0$. Then $b\equiv-{1\over 2}a-{9\over 4}$. We further divide into 2 subcases:
- $b\not\equiv 0$, in which case (2.3) is satisfied,
- $b\equiv 0$, in which case $a\equiv -{9\over 2}$, and so (2.4) is satisfied.
If Condition (i) is satisfied but (1.1) of Proposition \[prop:d=3messy\] is not, then we have the following 3 cases (A)-(B)-(C):
- $b \equiv -2a-3$, in which case $c \equiv 2a+1$, and so (v) is satisfied;
- $a\equiv -1$ and $b \not\equiv -2a-3$, in which case $c\equiv -1$ but $b \not \equiv -1$, and so (1.2) is satisfied;
- $b\equiv -a -2$ and $a\not \equiv -1$. Hence $c\not\equiv -1$ thanks to $c \equiv 2a+1$. We further divide into 2 subcases below:
- $b\not\equiv 0$, in which case $c\equiv 2a+1$, and so (1.3) is satisfied,
- $b\equiv 0$, in which case $a\equiv -2, c\equiv -3$, and so (1.4) is satisfied.
If Condition (ii) is satisfied but (1.2) of Proposition \[prop:d=3messy\] is not, then either (A) $b\equiv a$, in which case $c\equiv -2a-3$, and so (v) is satisfied, or (B) $b\equiv -1$ and $b\not \equiv a$, in which case $c\equiv -2a-3$ and then $c\not\equiv -1$, and so (1.3) is satisfied.
If Condition (iii) is satisfied but (1.3) of Proposition \[prop:d=3messy\] is not, then either (A) $c \equiv -1$, in which case the equality $b\equiv -{1\over 2}a-{1\over 4}c-{7\over 4}$ implies that (v) is satisfied; or (B) $c\not \equiv -1$ and $b\equiv 0$, in which case $a\not \equiv -3$, and so (1.4) is satisfied.
If Condition (iv) is satisfied but (1.4) of Proposition \[prop:d=3messy\] is not, then $a\equiv -3$, in which case $b\equiv 0, c\equiv -1$, and so (v) is satisfied.
The proposition is proved.
Simple modules of the supergroup $F(3|1)$
-----------------------------------------
Summarizing Propositions \[prop F4 First\], \[prop:d=1simple\], \[prop:d=2simple\], and \[prop:d=3simple\], we have established the following classification of simple modules for type $F(3|1)$.
\[thm:F4\] Let $p>3$. Let $G$ be the simply connected supergroup of type $F(3|1)$. A complete list of inequivalent simple $G$-modules consists of $L({\lambda})$, where ${\lambda}= a {\omega}_1 +b \omega_2 +c \omega_3 +d \frac{\delta}{2}$, with $a,b,c,d \in {\mathbb N}$, such that one of the following conditions is satisfied:
1. $d=0$, and $a\equiv b\equiv c\equiv 0 \pmod p$.
2. $d=1$, and $a,b,c$ satisfy either of (i)-(iii) below:
- $a\equiv 2b +3\equiv c-1 \equiv 0\pmod p$;
- $2a +1\equiv 2b+1 \equiv c\equiv 0\pmod p$;
- $2a+3\equiv b\equiv c\equiv 0\pmod p$.
3. $d=2$, and $a,b,c$ satisfy either of (i)-(vi) below:
1. $a\equiv c\equiv 0\pmod p$;
2. $2a-c \equiv a+b+1\equiv 0\pmod p$;
3. $b\equiv 2a+c+4\equiv 0\pmod p$;
4. $2a-c+2 \equiv 2a+ b +3 \equiv 0\pmod p$ and $c\geq 2$;
5. $2a+c+2\equiv a-b\equiv 0\pmod p$ and $a\geq 1$;
6. $a+2b+3 \equiv c\equiv 0\pmod p$.
4. $d=3$, and $a,b,c$ satisfy either of (i)-(v) below:
1. $2a-c +1 \equiv 0\pmod p$;
2. $2a+c+3\equiv 0\pmod p$;
3. $2a+4b+c+7 \equiv 0\pmod p$;
4. $2a+c+7 \equiv b\equiv 0\pmod p$;
5. $2a +4b +3c +9\equiv 0\pmod p$.
5. $d\ge 4$, (and $a,b,c\in {\mathbb N}$ are arbitrary).
We do no attempt the classification of simple $G$-modules for $p=3$ in this paper, and leave it to the reader.
Theorem \[thm:F4\] makes sense over ${\mathbb C}$, providing an odd reflection approach to the classification of finite-dimensional simple modules over ${\mathbb C}$ (due to [@Kac77]; also cf. [@Ma14]). Indeed this classification can be read off from Theorem \[thm:F4\] (by regarding $p=\infty$) as follows. [*The simple ${\mathfrak{g}}$-modules $L({\lambda})$ over ${\mathbb C}$ are finite dimensional if and only if ${\lambda}= a {\omega}_1 +b \omega_2 +c \omega_3 +d \frac{\delta}{2}$, for $a,b,c,d \in {\mathbb N}$, satisfies one of the 3 conditions: (1) $a=b=c=d=0$; (2) $d=2$ and $a=c=0$; (3) $d\ge 4$.* ]{}
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abstract: 'In the present work we demonstrate how to realize 1d-optical closed lattice experimentally, including a [*tunable*]{} boundary phase-twist. The latter may induce “persistent currents”, visible by studing the atoms’ momentum distribution. We show how important phenomena in 1d-physics can be studied by physical realization of systems of trapped atoms in ring-shaped optical lattices. A mixture of bosonic and/or fermionic atoms can be loaded into the lattice, realizing a generic quantum system of many interacting particles.'
author:
- 'Luigi Amico$^{(a)}$, Andreas Osterloh$^{(b)}$, and Francesco Cataliotti$^{(c)}$'
title: 'Quantum many particle systems in ring-shaped optical lattices'
---
Studies of one dimensional systems constitute an intense research activity both in experimental and theoretical physics. They are particularly interesting mainly because quantum effects are strongest at low dimensionality and peculiar phenomena emerge. Prominent examples are the spin-charge separation in Luttinger liquids[@LL], one dimensional persistent currents in mesoscopic rings[@PERSISTENT], and transmutation of quantum statistics[@FRACTIONAL]. Most of the approximate schemes working in higher dimensions break down in 1-d. Only for a restricted class of model Hamiltonians, physical properties can be obtained analytically resorting to powerful techniques as Bethe ansatz[@TAKAHASHI] or conformal field theory[@CONFORMAL]. For more generic 1d systems, numerical analysis is the standard route to extract physical information. Degenerate atoms in optical lattices could constitute a further tool for the investigations[@CIRAC], thus rediscovering Feynman’s ideas[@FEYNMAN] suggesting that an ideal system with a “quantum logic” can be used to study open problems in quantum physics. Precise knowledge of the model Hamiltonian, manipulation of its coupling constants, possibility of working with controllable disorder are some of the great advantages of atomic systems in optical lattices compared with solid state devices to experimentally realize Feynman’s ideas. The upsurge of interest of the scientific community has been remarkable, and some perspectives disclosed by trapped–atom “labs” have been already explored: the observation of the superfluid–Mott insulator quantum phase transition[@GREINER], the analysis of the Tonks-Girardeau regime in strongly interacting bosons[@PAREDES], and the physical realization of a $1d$-chain of Josephson junctions[@CATALIOTTI] were relevant achievements for condensed matter physics. The two most widely used methods to trap and manipulate atoms are based on the conservative interaction of atoms with either magnetic fields or with far off–resonant laser beams. For our purposes the magnetic trapping potential has a parabolic symmetry. Laser light interacts with the atomic induced dipoles creating attractive or repulsive potentials depending on the sign of the detuning $\Delta$ from resonance [@Grimm]. This can be used to create different potentials for different atoms, but with a single tunable laser beam. Notice that no light absorption occurs in creating the potential; therefore the medium can be considered transparent to the laser.
So far open optical lattices have been studied. This constitutes a limitation of optical apparata since a variety of studies for finite 1d lattices with Periodic Boundary Conditions (PBC) exists in the literature, that cannot be accesed with them. In the same way as Gaussian laser beams are useful to produce open optical lattices, we shall take advantage of the rotational symmetry of Laguerre–Gauss (LG) laser modes to produce closed optical lattices. LG beams, obtained experimentally making use of computer generated holograms [@CHAVEZ], have already been used in the field of ultra-cold atoms[@ERTMER]. A LG mode with frequency $\omega$, wave-vector $k$ and amplitude $E_0$ propagating along the $z$ axis can be written in cylindrical coordinates $(r,\varphi,z)$ as [@Santamato] $
E\left(r,\varphi\right)= E_0 f_{pl}(r) e^{il\varphi} e^{i(\omega
t
- k z)} $, $f_{pl}(r)={\displaystyle (-1)^p\sqrt{\frac{2p!}{\pi
\left(p+|l|\right)!}}\xi^{|l|} L_p^{|l|}\left(\xi^2\right)
e^{-\xi^2}}$, $\xi=\sqrt{2}r/r_0 $, where $r_0$ is the waist of the beam. and $L_p^{|l|}$ are associate Laguerre polynomials $L_m^n(x):= (-)^m d^m/dx^m
[L_{n+m}(x)]$, $L_{n+m}(x)$ being the Laguerre polynomials themselves. The numbers $p$ and $l$ label the radial and azimuthal quantum-coordinates, respectively. The lattice modulation is obtained by interference of a LG beam with a plane wave $E_0e^{i(\omega t -kz)}$: in the far field, the interferogram is periodic in $\varphi$ with $l$ wells. For even $l$ a perfect $1d$-ring with $L=l$ lattice sites is obtained. By reflecting the combined beam (LG beam plus plane wave) back on itself one achieves confinement also along $z$. Indeed a series of disk shaped traps are obtained. We point out that tunneling between the disks $t_z$ can be made much weaker than the corresponding tunneling within each ring $t_\phi$ adjusting $r_0/\lambda$ ([*i. e.*]{} focusing the LG beam). Such a parameter depends monotonously only on $L$; for $L\gtrsim 15$, $t_z/t_\phi \lesssim 10^{-2}$ can be achieved with $r_0/\lambda
\sim 100$. The resulting lattice potential (see Fig.(\[potenziale\])) is described by $$V_{latt}=4 E_0^2\left[1+ f^2_{pl} +2 f_{pl} \cos(l\varphi) \right ] \cos\left(k z\right)^2
\label{potondo}$$ Note that, contrary to what was done in [@ERTMER], here we need the laser frequency to be tuned below the atomic resonance since we want to trap atoms into the ring.
![The optical potential resulting from the interference of a plane wave with an LG mode with $L=14$, $p=0$. For $p\neq 0$ the potential is virtually unaltered[]{data-label="potenziale"}](prova3.eps "fig:"){width="6cm"}\
For example with a laser intensity of $I=5W/cm^2$ and $\Delta=-10^6MHz$ the potential wells would be separated by a barrier of $\sim 5 \mu K$ much larger than the chemical potential of a standard condensate (whose temperature can reach few $nK$); with these parameters the scattering rate is $\ll 1$ photon/$sec$. It is worth noting that, because of the relation: $L_m^{|n-m|}(r)\leftrightarrow
H_n[(x-y)/\sqrt{2}]H_m[(x+y)/\sqrt{2}]$, LG modes can be realized also from Hermite–Gauss modes (modulo a $\pi/2$ phase change). Such a “mode-converter”, realized experimentally in [@BEIJERSBERGEN], can switch from an open to a closed lattice potential with the same periodicity and $L$. As we shall discuss further, this device might be useful in the experiments.
We have just illustrated how to realize an optical lattice with PBC. Now we show how to twist them. The task can be achieved by applying an external, cone-shaped magnetic field ${\bf
B}=B_\varphi {\bf e}_\varphi+B_z {\bf e}_z$. In this way the atomic magnetic dipoles ${\bf \mu}_{m_F}$ experience a field varying along the ring, eventually equipping the periodic lattice by a twist factor: $\Psi \rightarrow \displaystyle{e^{i
\phi_{m_F}}}\Psi $ at each winding, $\Psi$ being a generic wave function. The phase factor $\phi_{m_F}=m_F \pi \cos{\theta}$, with $\tan{\theta}=B_\varphi/B_z$, is the analog of the Berry phase[@NOTE] of the two state system corresponding to the Zeeman splitting of the hyperfine atomic ground states; the role of time is played by the angle $\varphi$. We can adjust $\phi_{m_F}$ using an additional laser beam (with a suitable frequency), relying on the AC-Stark shift: $A_E (m_F)$, where the function $A_E$ depends on the intensity of the laser and on the Clebsch-Gordan coefficients corresponding to the matrix element of the electric dipole interaction energy[@AC-STARK]. The resulting phase twist is $\Phi_\sigma\doteq A_E (m_F)+ m_F\pi \cos \theta$ where $\sigma=m_F$. Whereas boundary twists induced by a magnetic field pierceing the ring are “symmetric”, $\Phi_+ \equiv \Phi_-$, our protocol realizes $1d$-models with a tunable $\Phi_\sigma$, thus opening the way to novel investigations discussed below.
For OBC, $\Phi_\sigma$ can be “gauged away” completely from the system. In contrast, the boundary phase cannot be eliminated for closed loops and alters the phase diagram of the system [@SHASTRY-SUTHERLAND]. Infact $\Phi_\sigma$ emerges from the sum of site dependent phases causing an increase of the velocity field ($\propto$ to the tight binding amplitude $t$) that, in absence of dissipation, may set a persistent current. Therefore different regions in the phase diagram are identified depending on the dynamical response of the system by perturbing $\Phi_\sigma$. The effect is reflected in the curvature of the ${\cal N}$-particle energy levels $E_n$ respect to the phase twist: $\rho_\sigma={L^2}\sum_n p_n \left
[E_n(\Phi_\sigma)-E_n(0)\right ]/({\cal N} t\Phi_\sigma^2) $, where $p_n=e^{-\beta E_n }/Z$ are the Boltzmann weights. For (spinless) bosons $\rho_+=\rho_-=\rho$ is proportional to the superfluid fraction. Persistent currents are studied analyzing the charge stiffness $D_c\propto \rho_+ +\rho_-$ (for electrons, it is the zero frequency conductivity or Drude weight); a non vanishing $D_c$ sets a persistent current, visible by releasing the condensate for a time much longer than the typical atomic oscillation period in the lattice wells.
![Interference pattern for condensates released by the lattice (\[potondo\]), obtained resorting to the analog of light diffraction from a circular grating[@PEDRI]. The figures show the square of the order parameter $|\psi(k_x,k_y)|^2=|
\psi_0(k_x,k_y)\sum_{j=0}^{L-1} \cos\left [i (k_x\cos(2\pi j/L)+
k_y\sin(2\pi j/L) +\phi_j)\right ]|^2$. On the left $\phi_j=0$ for all the condensates; on the right the atoms move along the ring with velocity $\propto \nabla \phi$; the interference pattern reflects a loss of matter at the trap center caused by centrifugal effects.[]{data-label="interf"}](prova.eps "fig:"){width="6cm"} ![Interference pattern for condensates released by the lattice (\[potondo\]), obtained resorting to the analog of light diffraction from a circular grating[@PEDRI]. The figures show the square of the order parameter $|\psi(k_x,k_y)|^2=|
\psi_0(k_x,k_y)\sum_{j=0}^{L-1} \cos\left [i (k_x\cos(2\pi j/L)+
k_y\sin(2\pi j/L) +\phi_j)\right ]|^2$. On the left $\phi_j=0$ for all the condensates; on the right the atoms move along the ring with velocity $\propto \nabla \phi$; the interference pattern reflects a loss of matter at the trap center caused by centrifugal effects.[]{data-label="interf"}](prova1.eps "fig:"){width="6cm"}
Then the spatial distribution of the condensates $ |\Psi({\bf r} ={\bf k}t,t)|$ is indicative of the initial atomic momentum distribution $|\Psi ({\bf k},0)|$[@PEDRI]; in particular the phase difference between atoms trapped in different sites, produces characteristic interference patterns in the released condensates. In Fig. (\[interf\]) we show such a pattern for condensates released from the potential of Fig. (\[potenziale\]) in mean-field approximation (see also Fig.(\[fourier\])). Supercurrent/superfluid fractions can be studied looking at the response of the system under imprinting of a dynamical phase $\alpha_d(j,\sigma)\delta\tau \,$ to the atomic wave functions, flashing the atoms with an additional Gaussian laser beam (can be much closer to resonance than those creating the potential) with a waist larger than the LG mode and with $\varphi$–dependent intensity. The time $\delta\tau $ must be too short to induce atomic motion by absorption during the pulse).
The case $\Phi_+=-\Phi_-$ is useful to study the spin stiffness $D_s\propto\rho_+-\rho_-$ indicating long range spin correlations in the system (for charged particles $D_s$ would be proportional to the inverse bulk spin susceptibility[@SHASTRY-SUTHERLAND]). Generic values of $\Phi_+\neq \pm \Phi_- $ can be seen also as a result of certain correlated–hopping processes (on the untwisted models)[@SCHULZ-SHASTRY] and correspond to more exotic cases that, as far as we know, have not been realized yet in physical systems. To be specific we consider ${\cal N}$ fermions described by the H ubbard model with particle-density modulated kinetic energy $$\begin{aligned}
\label{Hubbard}
H_{Hub} &=& -\sum_{j,\sigma} \mu_{j,\sigma} N_{{ j},\sigma} - \sum_{j,\sigma }(
\tilde{t}_j (\sigma) c_{j+1,\sigma}^\dagger c_{j,\sigma} +h.c.) +U\, \sum_{ j}N_{{j},+} N_{{ j},-} \\
&\tilde{t}_j& (\sigma) =t \exp\Bigl[{\rm i}\gamma_j (\sigma)+ {\rm i} \sum_{l}^{}\bigl(\alpha_{j,l}(\sigma)N_{l,-\sigma}
+ A_{j,l}(\sigma)N_{l,\sigma}\bigr)\Bigr] \;,
\label{correlated}\end{aligned}$$ where $c_{j,\sigma}$’s are fermionic operators, and $N_{l,\sigma}:= c_{l,\sigma}^\dagger c_{l,\sigma}$. $U=\pi b_s \int dx |w(x)|^4/m$ and $t=\int dx w(x)[-\frac{1}{2m} \nabla^2+ V_{latt}] w(x+a)$, ($b_s$, $a$, and $w(x)$ indicates the scattering length, the lattice spacing, and Wannier functions respectively) play the role of the Coulomb and hopping amplitudes respectively; $\mu_{j,\sigma}$ is of the order of the Bloch band separation[@SPIN]; the site dependence can be achieved by tuning the magnetic confinement out of the symmetry axis of the optical ring. For the model (\[Hubbard\]) in a closed lattice, (\[correlated\]) can be gauged away everywhere but at the boundary; therefore (\[Hubbard\]), (\[correlated\]) is equivalent to the ordinary Hubbard model, but with twisted BC[@SCHULZ-SHASTRY]. The phase twist is $
\Phi_\sigma:=\phi(\sigma)+ \phi^{(1)}_{+-}(\sigma) {\cal
N}_{-\sigma} +\phi^{(1)}_{++}(\sigma)({\cal N}_\sigma-1) $, where $
\phi^{(1)}_{+-}(\sigma)
= \sum_{j=1}^{L}\alpha_{j,m}(\sigma)\quad $, $\phi(\sigma)
=\sum_{j=1}^{L}\left (\gamma_j(\sigma)\;+\, A_{j,j}(\sigma)\right ) $, $ \phi^{(1)}_{++}(\sigma)
=\sum_{{j=1\atop j\neq m-1,m}}^{L} A_{j,m}(\sigma)+A_{m,m-1}(\sigma)+A_{m-1,m+1}(\sigma)$.
![The zero temp. momentum distribution for fermions with Hubbard dynamics is presented: $|\Psi (k_x,k_y)|^2\propto |w(k_x,k_y)|^2 \sum_{i,j}e^{i {\bf k} \cdot ({\bf x}_i-{\bf x}_j)} \sum_{k_\phi}
e^{i k_\phi(\phi_i-\phi_j)} \langle n_{k_\phi} \rangle
$; ${\cal N}/L=32/16$ (${\cal N}/L=1$ is the less advantageous case to discern the effects of $\Phi$ at finite size, since the metal-insulator transition strongly suppresses $D_c$[@SCALAPINO]); $\langle n_{k_\phi} \rangle$ is calculated perturbatively, at second order in $U/t$. For $\Phi=0$ (left), $k_\phi= \{-\pi({\cal N}-1)/L\dots \pi
({\cal N}-1)/L\}$. For $\Phi\neq 0$, $ k_\phi= \{-\pi ({\cal
N}-1)/L+\Phi/L \dots \pi ({\cal N}-1)/L+\Phi/L\}$; the asymmetry in $|\Psi (k_x,k_y)|^2$ is due to the offset of $\langle
n_{k_\phi} \rangle$ caused by $\Phi$.[]{data-label="fourier"}](Hubbard32.16.0.eps "fig:"){width="5.5cm"} ![The zero temp. momentum distribution for fermions with Hubbard dynamics is presented: $|\Psi (k_x,k_y)|^2\propto |w(k_x,k_y)|^2 \sum_{i,j}e^{i {\bf k} \cdot ({\bf x}_i-{\bf x}_j)} \sum_{k_\phi}
e^{i k_\phi(\phi_i-\phi_j)} \langle n_{k_\phi} \rangle
$; ${\cal N}/L=32/16$ (${\cal N}/L=1$ is the less advantageous case to discern the effects of $\Phi$ at finite size, since the metal-insulator transition strongly suppresses $D_c$[@SCALAPINO]); $\langle n_{k_\phi} \rangle$ is calculated perturbatively, at second order in $U/t$. For $\Phi=0$ (left), $k_\phi= \{-\pi({\cal N}-1)/L\dots \pi
({\cal N}-1)/L\}$. For $\Phi\neq 0$, $ k_\phi= \{-\pi ({\cal
N}-1)/L+\Phi/L \dots \pi ({\cal N}-1)/L+\Phi/L\}$; the asymmetry in $|\Psi (k_x,k_y)|^2$ is due to the offset of $\langle
n_{k_\phi} \rangle$ caused by $\Phi$.[]{data-label="fourier"}](Hubbard32.16.60.eps "fig:"){width="5.5cm"}\
Hence, loading the Hubbard model into the twisted ring effectively leads to the physical realization of the model (\[Hubbard\]), (\[correlated\]). To point out the effects of $U$ (smearing of the Fermi distribution with algebraic singularity at $k_F$) in the persistent current, $|\Psi ({\bf k})|^2$ is calculated for the Hubbard ground state at small $U/t$, and with $\Phi_\sigma=\phi(\sigma)$, $\phi(+)=\phi(-)=\phi$ (see Fig. (\[fourier\])).
The proposed setups could be used to study several issues in one dimensional systems.
I. The concept of conformal invariance plays a central role in $1+1$ dimensional critical phenomena: universality is characterized by a single parameter, the conformal anomaly $c$. The physical meaning of $c$ resides in the concept of Casimir energy, namely the variation of the vacuum energy density to a change in the BC. For PBC it was shown [@CONFORMAL] that the finite size correction to the bulk ground state energy is related to $c$: ${\cal
E}_{PBC}-{\cal E}_{bulk}=-\pi c v/6L$; resorting the modular invariance this correction should be visible in the specific heat of the system, at low temperature: $C(T)=\pi
c L T/3v $, for each collective mode of the system; the speed of sound $v$ can be extracted from the dispersion curve, at small $k$: $v=\Delta {\cal E}/\Delta k$, for sufficiently large $L$ (for the XXZ model, numerical analysis suggests that $L \gtrsim 15$[@BONNER]). Except for integrable models, it is hard to measure or even have numerical estimates of $c$ in solid state systems[@EXPERIMENTAL-c; @CONFORMAL]. With the presented setups for highly controllable loaded models these measurements can be done with unique accuracy. Both $C(T)$ and $\Delta {\cal E}/\Delta k$ can be measured following the techniques employed by Cornell [*et al.*]{}[@JIN]. To discern finite size effects in $C(T)$ the PBC to OBC converter, discussed above, could be a valid tool. Indeed, the finite size correction to ${\cal E}_{bulk}$ for OBC is also proportional to $c$, but with a [*different*]{} coefficient[@CONFORMAL]. Then: $
{\displaystyle c v=\frac{8 L}{\pi} ({\cal E}_{PBC}-{\cal E}_{OBC})+ {\cal F}_S}
$, where ${\cal F}_S$ is the bulk limit of the surface energy that, being non-universal, can be fixed by performing the measurements for different $L$. (mimicking a “finite size scaling analysis”). Remarkably, both the energies ${\cal E}_{PBC}$ and ${\cal E}_{OBC}$ might be accessible measuring the second moment of the velocity of the released condensate[@GROUND].
II\. A general model we can engineer in the ring shaped lattice is $$\label{mixture}
H=H_{BH}+H_{Hub}+H_I$$ where $H_{BH}$ is the Bose-Hubbard Hamiltonian [@GREINER] and $H_I$ describes a density-density, fermion-boson interaction[@MIXED]. By tuning $\Delta$ within the fine structure of the fermionic atoms, a spin dependence can be inserted in the hopping amplitude of the Hubbard model: $t\rightarrow t_\sigma$. At $ {\cal N}/L=1$ and $t_\sigma\ll U $ the Hubbard ring effectively accounts for the physical realization of the twisted $XXZ$ model with anisotropy $\gamma=(t_+^2+t_-^2)/(2
t_+ t_-) $ and external field $h=4 \sum_\sigma \sigma
t_\sigma^2/\mu_\sigma$[@SPIN]. Loading quantum systems described by Hamiltonians of the type (\[mixture\]) in lattices with twisted BC could serve to study charge and spin stiffness in physical systems with tunable interaction and/or disorder. For example a mixture of $^{87} Rb$ and $^{40} K$ atoms constitutes an ideal system to check the recent experimental evidence suggesting that the supersolid order[@SARO-REV] would be effectively favoured by the insertion of fermionic degrees of freedom into homogenous bosonic systems. The off-diagonal long range order manifests in superfluid currents. Jumps between non-vanishing supercurrents should reveal the existence of the supersolid phase[@SS-HELIUM]. This should be accompanied by a macroscopic occupation in the condensate at a non vanishing wave vector ($\sim \pi/(na)$, $n\ge 2$) signalling the charge-density-wave instability[@BLATTER]. The two condensates should be traced in the interference fringes.
It was proved that exactly solvable twisted Hubbard/$XXZ$ rings [@SHASTRY-SUTHERLAND; @OSTERLOH-DEF] are equivalent to untwisted models for particles with intermediate statistics; this results in modifications of the exponents of the (low energy) correlation functions[@SCHULZ-SHASTRY]. The spatial profile of the latter might be detected by photoassociation techniques, as suggested in [@GARCIA].
III Another interesting issue we can study is the conjecture[@CASTELLA-PRE] that Poisson $or$ Wigner-Dyson level-statistics manifest in that the thermal Drude weights have qualitatively different slopes for integrable (smooth algebraic temperature-decrease, universal behaviour of $D(T)/D(0)$) $or$ non-integrable (sharp, non-universal suppression of $D(T)/D(0)$) systems. Due to the precise knowledge of the model-Hamiltonian under analysis we can address the problem directly in a physical system. For example, we could consider $^{40}K$ pure-$XXZ$ rings with twisted BC, for different $L$’s; using the Feshbach resonance one could tune $b_s\sim 2a$; the resulting $XXZ$ model with next-nearest neighbor density-density interaction is non-integrable (another way is to destroy the integrability introducing disorder into the ring by site-dependent $h_j$). In short: integrability can be switched on and off by tuning the Feshbach resonance (or adjusting the energy offsets $h_j$). The presence of persistent currents can be detected along the lines described above (see Fig.(\[interf\]), (\[fourier\])). Numerical investigations for the $XXZ$ model suggest that the effect should be visible for $ T/L\gtrsim 0.1 \gamma $ [@CASTELLA-ZOTOS].
In summary we have suggested a number of protocols to realize closed rings of many quantum particles, by optical means. This is possible by employing slight variations and combinations of techniques already developed within the current experimental activity in atomic physics. We have discussed how several open problems in condensed matter physics can be enlightened by such a setup. We finally observe that the clockwise/anticlockwise currents in a few-wells-ring constitutes a controllable two state-system analogous to the flux-qubit realized by a SQUID. As the current is neutral, the corresponding decoherence-rate is much lower compared to solid state devices (charged currents). Information transfer could be mediated by an induced-dipole–dipole atomic interaction.
[**Acknowledgments**]{}. We thank A. Cappelli, M. Inguscio, G. Falci, R. Fazio, and H. Frahm for support and discussions.
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abstract: 'We give several new moduli interpretations of the fibers of certain Shimura varieties over several prime numbers. As a corollary we obtain that for every prescribed odd prime characteristic $p$ every bounded symmetric domain possesses quotients by arithmetic groups whose models have good reduction at a prime divisor of $p$.'
author:
- Oliver Bültel
title: 'PEL modulispaces without $\mathbb C$-valued points'
---
To my parents, Heinrich and Karola
Introduction
============
Let $(G,X)$ be a Shimura datum in the sense of all five axioms [@deligne3 (2.1.1.1)-(2.1.1.5)] of Deligne. One knows that there exists a canonical variation $\sy_{Hodge}(\rho)$ of pure Hodge structures over the quasi-projective algebraic variety $$\label{bekanntV}
_KM(G,X)=G(\q)\backslash(X\times G(\a^\infty)/K)$$ for every $\q$-rational linear representation $\rho:G\rightarrow\GL(V/\q)$, where we tacitly assume that $K$ is a neat compact open subgroup. In rare cases these yield so-called “moduli interpretations”, more specifically if there exists an injective map $\rho:G\rightarrow\GSp_{2g}$ with $\rho(X)\subset\gh_g$, then $_KM(G,X)$ is a moduli space of $g$-dimensional abelian varieties with additional structure, i.e. equipped with additional Hodge cycles (on suitable powers). This result is not only theoretically significant, but it also has an enormous practical meaning, because it is this particular class of Shimura varietes of Hodge type which are, at least in principle, amenable to the methods of arithmetic algebraic geometry. For instance, it is this way that Milne has reobtained Deligne’s canonical models over the reflex field $E\subset\c$ in [@milne2]. Outside this class of Shimura varieties these methods fail, but the work of many people (e.g. [@kazhdan], [@borovoi]) has culminated in more general results. Canonical models are finally shown to exist unconditionally in [@milne3]. However, the proof uses a very amazing reduction to the $\GL(2)$-cases, and again these are treated by means of abelian varieties whose endomorphism rings have a suitable structure.\
Fix a rational prime $p$ and write $\O_{E_\gp}$ for the complete valuation rings corresponding to prime divisors $\gp|p$. The Shimura variety is expected to possess a certain smooth integral $\O_{E_\gp}$-model $_K\M$, provided that $K$ can be written as $K^p\times K_p$, where $K^p$ is a compact open subgroup of $G(\a^{\infty,p})$, and $K_p$ is a hyperspecial subgroup of $G(\q_p)$. Apart from being canonical in the sense of Milne-Vasiu one wants this model to satisfy a number of other nice properties, for example every $\q_p$-rational representation $\rho:G\times\q_p\rightarrow\GL(V/\q_p)$ should determine a $F$-isocrystal $\sy_{Cris}(\rho)$ over the reduction $_K\M\times_{\O_{E_\gp}}\O_E/\gp=:_K\Mbar$, and a variant should exist for the category of Hodge-$F$-isocrystals over the $p$-adic completion of $_K\hat\M$ of $_K\M$. In either case $K_p$-invariant lattices in $V$ should determine interesting lattices in $\sy_{Cris}(\rho)$. For Shimura varieties of Hodge type see [@kisin] and the references therein. Even without any of the above additional requirements the existence of these models is highly conjectural, in fact the following weakening is already interesting: From now on we fix an embedding $W(\fc)\hookrightarrow\c$, we fix the hyperspecial subgroup $K_p$, and we write $\T$ for the set of compact open subgroups $K^p\subset G(\a^{\infty,p})$ such that $K^p\times K_p$ is neat. Write $X'$ for one of the connected components of $X$, and let us say that a discrete and cocompact subgroup $\tilde\Delta\subset G^{ad}(\r)^\circ\times G(\a^{\infty,p})$ is a lattice of good reduction if there exists a family of smooth and projective $W(\fc)$-schemes $\{_{K^p}\M\}_{K^p\in\T}$ together with biholomorphic bijections $$\label{uniformizationII}
\tilde\Delta\backslash(X'\times G(\a^{\infty,p})/K^p)\stackrel{\cong}{\rightarrow}{_{K^p}\M}(\c),$$ such that every inclusion of the form $K_1\subset\Int(\gamma^p)K_2$ with $K_1,K_2\in\T$, gives rises to a commutative diagram $$\begin{CD}
\tilde\Delta\backslash(X'\times G(\a^{\infty,p})/K_1^p)@>{\cong}>>{_{K_1^p}\M}(\c)\\
@V{\gamma^p}VV@VVV\\
\tilde\Delta\backslash(X'\times G(\a^{\infty,p})/K_2^p)@>{\cong}>>{_{K_2^p}\M}(\c)
\end{CD}$$ in which the right-hand side vertical arrow is the complexification of some étale covering map ${_{K_1^p}\M}\rightarrow{_{K_2^p}\M}$ while the left-hand vertical arrow is multiplication by the group element $\gamma^p\in G(\a^{\infty,p})$, from the right. If $X'$ does not contain factors of rank one, i.e. irreducible bounded symmetric domains of type $I_{n,1}$, then the left hand side of is a finite union of arithmetic varieties in the sense of [@kazhdan], simply by one of Margulis’ arithmeticity theorems, namely [@margulis Theorem(1.11), chapter IX]. The focus of the present paper is on domains of type $V$ and $VI$, and its outcome is the following result:
\[uniformizationIII\] Let $W(\fc)\hookrightarrow\c$ be as above, and assume in addition that $p$ is odd. Let $D$ be an irreducible bounded symmetric domain, which is not of type $II_n$ for any $n\geq5$. Let $z\geq1$ and $r\geq4z+1$ be integers. Then there exists at least one Shimura datum $(G,X)$ having a $\q$-simple adjoint group $G^{ad}$ and fulfilling:
- The connected components of $X$ decompose into a product of $z$ copies of $D$.
- The complexification of $G^{ad}$ decomposes into a product of $r$ simple factors.
- There exists a lattice of good reduction for $(G,X)$.
Among other things, those requirements on $(G,X)$ imply the anisotropicity of $G$, in which case one can view the smoothness and the projectivity as utmost minimal requirements in the pursuit of whatever notion of integral models to . In the projective case it turns out, that one can use deformation theory to deduce the existence of $_{K^p}\M$ from the existence of its special fiber $_{K^p}\Mbar$, which is a particular smooth projective algebraic variety over a particular finite extension, say $\f_{p^f}\supset\O_{E_\gp}/\gp$. The construction of $_{K^p}\Mbar$, which we give in this paper, is more involved and relies on discoveries which go a little bit beyond the world of good reduction: Under the assumptions of theorem \[uniformizationIII\] we introduce a (projective) moduli-space $_{K^p}\gM$ parametrizing abelian varieties with a certain kind of additional structure, which is a slightly more elegant variant of a notion that appeared already in [@habil Definition 5.3]. The very same kind of additional structure poses an analogous moduli problem in the category of $p$-divisible groups, which is used to introduce a certain fpqc-stack $\gB$. There exists an important (formally étale) $1$-morphism $_{K^p}\gM\rightarrow\gB$, which is easily defined by the ’passage to the underlying $p$-divisible group’. Now our search for $_{K^p}\Mbar$ is narrowed down by a certain hypothetical morphism: $$\label{magic}
_{K^p}\Mbar\rightarrow{_{K^p}\gM}.$$ This morphism is radicial and finite, but it fails to have an extension to $_{K^p}\M$, actually the generic fiber of $_{K^p}\gM$ has a strong tendency to be the emtpy scheme, and in fact it is only its fiber over (one single prime divisor of) $p$, that forms the starting point of several subsequent investigations.\
In order to complete the construction of we make a systematic use of displays [@zink2], in fact we make use of a certain, modest generalization, the basic motivating idea is this: At least over a local ring a display $(P,Q,F,V^{-1})$ possesses a normal decomposition $P=T\oplus L$ and bases $e_1,\dots,e_d$ of $T$ and $e_{d+1},\dots,e_h$ of $L$ relative to which there are structural equations $\sum_{i=1}^h\alpha_{i,j}e_i=\begin{cases}Fe_j&j\leq d\\V^{-1}e_j&j>d\end{cases}$, and hence an invertible $h\times h$-display matrix $(\alpha_{i,j})=U$, cf. [@zink2 Lemma 9]. Suppose that $U'$ is the matrix of another display $(P',Q',F,V^{-1})$ of possibly different height $h'$ and dimension $d'$, again taken with respect to a normal decomposition, and a choice of its bases. A linear map from $P'$ to $P$ can be visualized as $h\times h'$- matrix over the ring of Witt vectors with a block decomposition $\left(\begin{matrix}A&B\\C&D\end{matrix}\right)$ with the upper left block having $d$ rows and $d'$ columns. This map is a morphism $P'\stackrel{k}{\rightarrow}P$ in the category of displays if and only if:
- the entries in $B$ are Witt vectors with vanishing $0$th ghost component, and
- $U^{-1}\left(\begin{matrix}A&B\\C&D\end{matrix}\right)U'=
\left(\begin{matrix}^FA&^{V^{-1}}B\\p{^FC}&^FD\end{matrix}\right)$ holds.
It is enough to check the commutation of $k$ with the maps $V^{-1}$ on $Q$ and $Q'$ and these are given by $U$ and $U'$ precomposed with $\left(\begin{matrix}x\\y\end{matrix}\right)\mapsto\left(\begin{matrix}^{V^{-1}}x\\^Fy\end{matrix}\right)$. In particular the isomorphism classes of displays of height $h$ and dimension $d$ are simply the Witt vector-valued elements of $\GL(h)$ modulo the equivalence relation $$\label{transform}
k^{-1}U{^\Phi k}\sim U,$$ here we write ${^\Phi k}$ for the right hand side of (ii) which is well defined if the condition (i) on the upper right block of $k$ is valid. In the body of the text we will show that a similar map $\Phi$ exists on the inverse image of every minuscule parabolic subgroup, which gives rise to a definition of a stack $\B(\G,\mu)$ of $3n$-display with $\G$-structure. Slightly less obvious is, that a working notion of display with $\G$-structure can be defined by recourse to the structure of the crystalline realization in the adjoint representation. Once this has been done one can easily introduce an explicit map $$\label{trick}
\fx:\B(\G,\mu)\times_{\O_{E_\gp}}\f_{p^f}\rightarrow\gB$$ which is, rougly speaking a formal analog of the map . This adds a lot of content to the whole picture, and it gives a clue to interpret ${_{K^p}\Mbar}$ as the scheme which represents the upper left entry in the $2$-cartesian diagram $$\begin{CD}
{_{K^p}\Mot}@>>>{_{K^p}\gM}\\
@VVV@VVV\\
\B(\G,\mu)\times_{\O_{E_\gp}}\f_{p^f}@>>>\gB
\end{CD}$$ of fpqc-stacks. The bulk of this paper is devoted to the proof of partial representability properties of $_{K^p}\Mot$. We start by proving that it is radicial over $_{K^p}\gM$, i.e. its diagonal morphism is relatively representable by dominant closed immersions. It follows that $_{K^p}\Mot(R)$ possesses at most one element if $R$ is a reduced noetherian $\O_{_{K^p}\gM}$-algebra. The next step is devoted to the study of the image of $_{K^p}\Mot$: Outside a certain Zariski closed subset which we call $_{K^p}\gM_{can}$ we can show that $_{K^p}\Mot$ is always empty, and every point in $_{K^p}\gM_{can}$ gives rise to a non-empty $_{K^p}\Mot$ when regarded over a certain finite purely inseparable extensions of the residue field. In this manner, the generic point of $_{K^p}\gM_{can}$ should yield the function field of $_{K^p}\Mbar$, and the normalisation of $_{K^p}\gM_{can}$ in this particular field should yield the variety $_{K^p}\Mbar$, at least heuristically. However, additional complications arise because there could be several candidates for the function field of $_{K^p}\Mbar$. The details can be found at the end of section \[bootstrap\], and eventually we do obtain a variety ${_{K^p}\Mbar}$, which represents the restriction of $_{K^p}\Mot$ on the category of test schemes that are of finite type over an algebraic extension of $\f_{p^f}$, and the existence of is then clear.\
It is a pleasure to thank Prof. M. Kisin for encouragement and for pointing out the references [@sansuc] and [@prasad]. Moreover, I thank Prof. W. Kohnen, Prof. R. Coleman, Prof. R. Pink, Prof. R. Taylor, and Prof. T. Zink for interest and for invitations to Berkeley, Zürich, Harvard, and Bielefeld, where some predecessors of this work could be discussed. I thank Prof. J. Milne, Prof. G. Prasad, and Prof. Y. Varshavsky for many explanations, and finally I owe further thanks to the audience of my talk [@bultel3], in which I presented a special case of the map for the orthogonal group $\SO(h-2,2)\times\SO(h)$. The research on this paper was begun in Berkeley during the DFG funded scholarship BU-1382/1-1.
$3n$-Displays with additional structure
=======================================
In this section we introduce displays with additional structure. We prefer to work with arbitrary smooth, not necessarily reductive group schemes. This gives great conceptual clarity and could be of some use in other contexts.
Preliminaries
-------------
Write $\f_p$ for the prime field of characteristic $p$ and $\f_{p^f}$ for its extension of degree $f$. Recall the ringscheme $W$ of Witt vectors along with its truncated version $W_n$ for each finite length $n$. There are canonical maps: $$F_n:W_{n+1}\rightarrow W_n$$ and $$V_n:W_n\stackrel{\cong}{\rightarrow}I_{n+1}\subset W_{n+1},$$ where the ideal $I_n$ is defined to be the kernel of $w_0$, the $0$th coordinate of the ghost map, and, as usual, we shall suppress “$n$” in the notation if we mean ”$n=\infty$”. We need to make a number of remarks on set-valued covariant functors $\F$ on the category of $W(\f_{p^f})$-algebras: Let us write $\Res_{W_n}\F$ for the functor taking any $S/W(\f_{p^f})$ to $\F(W_n(S))$, where the $W(\f_{p^f})$-algebra structure on $W_n(S)$ is induced by the natural map $$\Delta_n:W(\f_{p^f})\rightarrow W_n(W(\f_{p^f}))$$ which one gets by truncating the usual functorial diagonal $\Delta$ from $W$ to $W\circ W$. Let us also write $^F\F$ for the pull-back via the Frobenius $F:W(\f_{p^f})\rightarrow W(\f_{p^f})$, i.e. $^F\F(S)=\F(S_{[F]})$, here the subscripted $[F]$ indicates the change of $W(\f_{p^f})$-algebra structure on the ring $S$ by means of $F$. Observe that for every ring $S$ there exist functorially truncated Frobenii: $F_n:W_{n+1}(S)\rightarrow W_n(S)$ which give rise to natural maps $$\label{FrobI}
F_n:\Res_{W_{n+1}}\F\rightarrow\Res_{W_n}{^F\F}.$$ This is due to the commutativity of the diagram: $$\begin{CD}
W_{n+1}(W(\f_{p^f}))@>F_n>>W_n(W(\f_{p^f}))\\
@A\Delta_{n+1}AA@A\Delta_nAA\\
W(\f_{p^f})@>F>>W(\f_{p^f})\\
\end{CD},$$ which is merely a special case of the remarkable compatibility between $\Delta$ and $F$, a fact that holds for just any commutative ring. In the special case of a functor $\F$ that “descends” (i.e. extends) to a functor over $W(\f_p)$ we may certainly drop the “$^F$” in the right hand side of , however do notice that we can always write $^F\Res_{W_n}\F$ instead of $\Res_{W_n}{^F\F}$, because of the commutativity of the diagram $$\begin{CD}
W_n(W(\f_{p^f}))@>W_n(F)>>W_n(W(\f_{p^f}))\\
@A\Delta_nAA@A\Delta_nAA\\
W(\f_{p^f})@>F>>W(\f_{p^f})\\
\end{CD},$$ which, this time, does use that $\f_{p^f}$ has an absolute Frobenius endomorphism which becomes $F$ if one applies $W$. Finally the ghost maps give rise to a canonical: $$\label{GhostI}
w:\Res_{W_n}\F\rightarrow\prod_{i=0}^{n-1}{^{F^i}\F}.$$ Let us set $w_0$ for its $0$th component. Finally we denote the functor $S\mapsto M\otimes_{W(\f_{p^f})}S$ by $\underline M$, provided that $M$ is a torsionfree. Notice that there is a natural short exact sequence: $$\label{VerI}
0\rightarrow\Res_{W_n}{^F\underline M}\stackrel{V_n}{\hookrightarrow}
\Res_{W_{n+1}}\underline M
\stackrel{w_0}{\twoheadrightarrow}\underline M\rightarrow0$$ We will write $\Res_{I_{n+1}}\underline M$ for the kernel of the right-hand side map and $$\label{VerIII}
V_n^{-1}:\Res_{I_{n+1}}\underline M\rightarrow\Res_{W_n}{^F\underline M}$$ for the inverse of the left-hand side one. Needless to say that the functor $^F\underline M$ is no other than $\underline{W(\f_{p^f})\otimes_{F,W(\f_{p^f})}M}$, finally observe $F_n\circ V_n=p$.
$\Phi$-data {#pivotalIII}
-----------
Let $S$ be a scheme, let $\G$ be a group-valued contravariant functor on the category of $S$-schemes, and let $P$ be a set-valued contravariant functor on the category of $X$-schemes, where $X$ is a scheme over $S$. If $P$ is endowed with a right $\G$-action, such that each non-empty $P(Y)$ becomes a principal homogeneous $\G(Y)$-space, then we will say that $P$ is a formal principal homogeneous space for $\G$ over $X$. We will say that $P$ is a locally trivial principal homogeneous space for $\G$ over $X$ if the following holds in addition:
- $P$ satisfies the sheaf axiom for fpqc (i.e. faithfully flat and quasicompact) morphisms.
- $P(Y)$ is non-empty for at least one fpqc map $Y\rightarrow X$.
Notice that locally trivial principal homogeneous spaces for affine group schemes are affine schemes, by descent theory (and [@egaiv Proposition (2.7.1.xiii)]). In this case it is interesting to consider the groupoid $\C_X(\G)$ of all locally trivial principal homogeneous spaces for $\G$ over $X$. An obvious notion of pull-back defines a fibration over the category of $S$-schemes, and we obtain the stack $\C(\G)$ if we think of the base category as the big fpqc site to $S$. In case $\G$ is flat and affine over $S=\Spec B$, where $B$ is a Dedekind ring, we will utilize the additive rigid $\otimes$-category $\bRep_0(\G)$ of ’representations’, i.e. $\G-B$-modules which are finitely generated and projective over $B$, this has been introduced and studied in [@rivano II.4.1.2.1], along with various functors defined thereon: Write $\ect_X$ for the additive rigid $\otimes$-category of vector bundles on $X$, i.e. of locally free, finitely generated $\O_X$-modules. Then there is a natural forgetful fiber functor $$\varpi_0^\G:\bRep_0(\G)\rightarrow\ect_S,$$ which takes a $\G-B$-module to its underlying $B$-module, furthermore for every $P\in\C_X(\G)$ there is also a twisted fiber functor $$\varpi_P:\bRep_0(\G)\rightarrow\ect_X,$$ which takes a representation $(\V,\rho)$ to the locally free $\O_X$-module $P\times^\G\V$, which is obtained by the usual extension of structure group (the map $\rho$ takes $\C_X(\G)$ naturally to $\C_X(\GL(\V/B))$, which in turn is naturally contained in $\ect_X$, cf. [@rivano II.3.2.3.4]).\
To fix ideas further consider a smooth group scheme over $W(\f_{p^f})$, together with a cocharacter $\mu:\g_m\times W(\f_{p^f})\rightarrow\G$ and observe that $\varpi_0^\G$ inherits a graduation of type $\z$ from it, [@rivano Corollaire IV.1.2.2.2]. Let $\G^-$ be the group of ascendingly filtered $\otimes$-automorphisms of $\varpi^\G$, and let $\G^+$ be the group of descendingly filtered $\otimes!$-automorphisms of $\varpi^\G$, in the sense of [@rivano IV.2.1.3]. These group functors are representable by (and will be identified with) closed subschemes of $\G$ and these are smooth over $W(\f_{p^f})$, by [@rivano Proposition IV.2.1.4.1]. Let $\Ad$ and $\Ad^-$ stand for the natural adjoint representations of $\G$ and $\G^-$ on their respective Lie algebras, which we denote by $\gg$ and $\gg^-$. We need notations for two more $\G^-$-representations of some significance: Observe that $\Ad^-$ is a subrepresentation of $\Ad|_{\G^-}$, and write $\Ad_-$ for the quotient of these two $\G^-$-representations. Its underlying $W(\f_{p^f})$-module may be identified with the Lie algebra of $\G^+$, which we denote by $\gg^+$, this is due to $\gg\cong\gg^-\oplus\gg^+$ (one should be cautioned to not confuse $\Ad_-$ with the adjoint representation of $\G^+$). By passage to the determinant one obtains a character $\modul:\G^-\stackrel{\det\circ\Ad_-}{\rightarrow}\g_m$, or equivalently a one-dimensional $\G^-$-representation on the invertible module $\gf:=(\det\gg)\otimes_{W(\f_{p^f})}(\det\check\gg^-)$.\
It also follows that the group multiplication identifies $\G^-\times_{W(\f_{p^f})}\G^+$ with an open subscheme $\G^\times\subset\G$ (use the implication “b)$\Rightarrow$a)” in [@egaiv Théorème (17.9.1)] followed by the “d)$\Rightarrow$b)”-one in the result [@egaiv Théorème (17.11.1)]). There is a canonical homomorphism $$\mu^{-\Int}:\a_{W(\f_{p^f})}^1\rightarrow\End(\G^-),$$ of multiplicative monoids which extends the interior $\g_m\times W(\f_{p^f})$-action $\Int_{\G^-}\circ\mu^{-1}$ on $\G^-$, and $\mu$ will be called a $\Phi$-datum for $\G$ if and only if the following holds:
- The only $\mu$-weight of $\gg^+$ is $1$.
- $\G$ has connected fibers.
If (M1) holds, then there exists an isomorphism $$e^+:\underline{\gg^+}\stackrel{\cong}{\rightarrow}\G^+,$$ such that the interior $\g_m\times W(\f_{p^f})$-operation of $\mu$ on $\G^+$ becomes the scalar multiplication of this vector group scheme.\
We write $\H_n^\pm$ and $\H_n$ for the affine group schemes $\Res_{W_n}\G^\pm$ and $\Res_{W_n}\G$, which are smooth of relative dimensions $n\dim\G^\pm/W(\f_{p^f})$ and $n\dim\G/W(\f_{p^f})$. Clearly its non-finite type affine proalgebraic pendants will be denoted by $\H^\pm$ and $\H$.\
The closed subgroup $W(\f_{p^f})$-scheme $\I_n\subset\H_n$ is defined to be the inverse image of $\G^-$ by means of the $0$ th component $w_0$ of the ghost map $w$ and the subgroup $\J_n\subset\H_n^+$ is defined to be the inverse image of the trivial group. We will need the following:
\[jacobson\] Suppose that $R$ is a commutative ring.
- If $p\in\rad(R)$ then $\rad(W_n(R))$ is equal to the inverse image of $\rad(R)$ via the $0$th ghost coordinate.
- If $n$ is finite and if $p\in\sqrt{0_R}$, then $I_n(R)$ is nilpotent and $\sqrt{0_{W_n(R)}}$ is equal to the inverse image of $\sqrt{0_R}$ via the $0$th ghost coordinate.
By a limit process it is enough to check the assertion (i) for $n\neq\infty$. By induction it suffices to check that $^{V^{n-1}}W_n(R)$ is contained in $\rad(W_n(R))$, which is implied by $$1-(1-^{V^{n-1}}x)^{-1}=^{V^{n-1}}(\frac{x_0}{p^{n-1}x_0-1}),$$ where $x_0$ is the $0$th ghost coordinate of $x$. The assertion (ii) is trivial.
\[factor\] Let $R$ be a $W(\f_{p^f})$-algebra whose radical contains $p$, then $\I_n(R)$ is the product of $\H_n^-(R)$ and $\J_n(R)$ (in any order for any $n$).
The diagram $$\begin{CD}
\Spec W_n(R)@>g>>\G\\
@A{\Spec w_0}AA@AAA\\
\Spec R@>{g_0}>>\G^-
\end{CD}$$ assures us of $g_0\in\G^\times(R)$ and $g\in\G^\times(W_n(R))$, by lemma \[jacobson\]. The equation $\G^\times=\G^-\times_{W(\f_{p^f})}\G^+$ finishes the proof.
For example it now follows that the natural group multiplication identifies the $W_m(\f_{p^f})$-scheme $\I_n\times_{W(\f_{p^f})}W_m(\f_{p^f})$ with the cartesian product of the $W_m(\f_{p^f})$-schemes $\J_n\times_{W(\f_{p^f})}W_m(\f_{p^f})$ and $\H_n^-\times_{W(\f_{p^f})}W_m(\f_{p^f})$.
Twisted Frobenius maps, part I {#pivotalI}
------------------------------
Let us denote the natural Frobenius groupscheme homomorphisms from $\H_{n+1}^\pm$ and $\H_{n+1}$ to ${^F}\H_n^\pm$, and ${^F}\H_n$ by $F_n^\pm$ and $F_n$. There are yet several more interesting homomorphisms $\Phi_n^\pm$ and $\Phi_n$ with targets one of the groups ${^F}\H_n^\pm$ and ${^F}\H_n$: Notice that $\mu^{-\Int}$ induces a functorial action of the multiplicative monoid $W(\f_{p^f})$-schemes $W_n$ on the group $W(\f_{p^f})$-schemes $\H_n^-$. When evaluated at the element $p=p[1_{W(\f_{p^f})}]_n\in W_n(W(\f_{p^f}))$ one obtains a map $$\mu^{-\Int}(p):\H_n^-\rightarrow\H_n^-; g\mapsto\mu(p)^{-1}g\mu(p),$$ which gives us further $$\Phi_n^-:\H_{n+1}^-\rightarrow{^F}\H_n^-$$ ones, when composed with $F_n^-$. Finally Zink’s “$V^{-1}$” makes sense as an $F$-morphism, of group $W(\f_{p^f})$-schemes i.e. $$\Phi_n^+:\J_{n+1}\rightarrow{^F}\H_n^+,$$ as in . Now we are in a position to introduce the twisted Frobenius map $$\Phi_n:\I_{n+1}\rightarrow{^F}\H_n$$ that will appear to have pivotal importance: Write $A$ for the $W(\f_{p^f})$-algebra of which the spectrum is $\I_{n+1}$, and consider the faithfully flat quasicompact covering $\Spec(1+pA)^{-1}A\cup\Spec A[\frac1p]$. On the first chart the corollary \[factor\] on factorisations allows us to define $\Phi_n$ to be the product of $\Phi_n^-$ and $\Phi_n^+$, and on the second chart we can most easily define it by the same formula as $\Phi_n^-$, namely by $F_n\circ\Int_{\I_{n+1}\times_{W(\f_{p^f})}K(\f_{p^f})}(\mu(p)^{-1})$. The ghost coordinates may be used to check that these definitions coincide on $\Spec(1+pA)^{-1}A[\frac1p]$, so that we may descent.\
By a $\Phi$-datum we mean a pair $(\G,\mu)$ where $\G$ is a $W(\f_p)$-group, and $\mu$ is a $\Phi$-datum for some scalar extension $\G\times_{W(\f_p)}W(\f_{p^f})$. If a $\Phi$-datum is given we continue to denote $\Res_{W_n}\G$ by $\H_n$, however do notice that the aforementioned twisted Frobenius map reads $\Phi_n:\I_{n+1}\rightarrow\H_n\times_{W(\f_p)}W(\f_{p^f})$, so that the following definition is meaningful:
\[concept\] Let $\mu:\g_m\times W(\f_{p^f})\rightarrow\G\times_{W(\f_p)}W(\f_{p^f})$ be a $\Phi$-datum, and let $X$ be a $W(\f_{p^f})$-scheme. A length $n$-truncated $3n$-display over $X$ with $\G$-structure is a quadruple $\P=(P_n,P_{n+1},Q_{n+1},\Phi_n)$ consisting of the following data:
- $X$-schemes $P_n$ and $P_{n+1}$ having the structure of a locally trivial principal homogeneous space over $X$ for the groups $\H_n$ and $\H_{n+1}$, and being equipped with a $\H_{n+1}$-equivariant forgetful $X$-map $P_{n+1}\rightarrow P_n$
- A closed subscheme $Q_{n+1}\hookrightarrow P_{n+1}$ that is $\I_{n+1}$-invariant and acquires the structure of a locally trivial principal homogeneous space over $X$ for this group
- A $X$-morphism $\Phi_n$ from $Q_{n+1}$ to the length $n$ truncation $P_n$ such that the diagram $$\begin{CD}
\label{Froblinear}
Q_{n+1}\times_{W(\f_{p^f})}\I_{n+1}@>>>Q_{n+1}\\
@V\Phi_n\times_{W(\f_{p^f})}\Phi_n VV@V\Phi_nVV\\
P_n\times_{W(\f_p)}\H_n@>>>P_n
\end{CD}$$ of $X$-schemes commutes.
A $3n$-display over $X$ with $\G$-structure is defined to be an equivariant closed immersion $Q\hookrightarrow P$ of locally trivial principal homogeneous spaces under $\H\times_{W(\f_p)}X$ and $\I\times_{W(\f_{p^f})}X$, together with an additional map $\Phi:Q\rightarrow P$ satisfying an appropriate version of (D3).
Adopting the convention of [@zink2] we denote the $\Phi$ (resp. $\Phi_n$)-operation on elements of the group $\I$ (resp. $\I_{n+1}$) by “$^\Phi g$” (resp. “$^{\Phi_n}g$”) in order to distinguish it from the map $\Phi$ which determines the display. Isomorphisms between the possibly truncated $3n$-displays over $X$ with $\G$-structure are defined in the obvious way and give rise to groupoids $\B_X^n(\G,\mu)$. There exists an obvious notion of pull back, giving rise to categories $\B^n(\G,\mu)$, which are fibered over the base category of $W(\f_{p^f})$-schemes. One more piece of notation will prove useful: In case that $\epsilon:\g_m\times W(\f_{p^f})\rightarrow\G\times_{W(\f_p)}W(\f_{p^f})$ is a cocharacter which lies in the center of $\G$, there is another $\Phi$-datum $(\G,\mu\epsilon)$ giving rise to the same twisted Frobenius map and hence to an absolutely identical notion of length $n$-truncated $3n$-display over $X$ with $\G$-structure. The (slightly tautological) $1$-morphism $\B^n(\G,\mu)\rightarrow\B^n(\G,\mu\epsilon)$, thus defined shall be denoted by $\P\mapsto\P(\epsilon)$, and is called twist.\
For all $n$ including $\infty$ one may interpret $\B^n(\G,\mu)$ as the unique fibered category rendering the diagram $$\label{posh}
\begin{CD}
\B^n(\G,\mu)@>>>\C(\I_{n+1})\\
@VVV@V{\Phi_n\times\Psi_n}VV\\
\C(\H_n)@>{\Delta_{\C(\H_n)}}>>\C(\H_n)\times\C(\H_n)
\end{CD}$$ $2$-cartesian, here $\Psi_n$ is the composition of the inclusion of $\I_{n+1}$ into $\H_{n+1}$ and the truncation $\H_{n+1}\rightarrow\H_n$. It follows immediately that $\B^n(\G,\mu)$, being the fibered product of stacks over a (pre)stack is indeed a stack for the fpqc topology.\
Furthermore, it is noteworthy that the fiber of the upper horizontal arrow over any $S$-valued point of $\C(\I_{n+1})$ is just the fiber of $\Delta_{\C(\H_n)}$ over that point, in particular we obtain a morphism $$b^n(\G,\mu):=c(\I_{n+1})\times_{\C(\I_{n+1})}\B^n(\G,\mu):
\Spec W(\f_{p^f})\times_{W(\f_p)}\H_n\rightarrow\B^n(\G,\mu).$$
\[elementary\] Later on we need some facts on the structure of the underlying locally trivial principal homogeneous spaces: If $(P_n,P_{n+1},Q_{n+1},\Phi_n)$ is as in definition \[concept\], then the natural truncations $Q_{n+1}\rightarrow Q_n$ and $P_{n+1}\rightarrow P_n$ are smooth morphisms of smooth $X$-schemes for all finite $n$, this follows from [@egaiv Proposition (2.7.1.iv)] in conjunction with [@egaiv Proposition (17.7.1)]. If $\P=(P,Q,\Phi)$ is a $3n$-display with $\G$-structure over $X$, then neither $Q$ nor $P$ are of finite type, however the natural truncations $Q\rightarrow Q_n$ and $P\rightarrow P_n$ are still formally smooth and flat morphisms of formally smooth and flat $X$-schemes: This follows from an elementary argument as $Q\cong\lim_\leftarrow Q_n$ and $P\cong\lim_\leftarrow P_n$ (use [@egaiv Proposition (2.7.1.viii)] in conjunction with [@egaiv Proposition (8.2.5)]).
The fibered groupoid of twisted conjugacy classes, part I
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An object in $\B^n(\G,\mu)$ will be called banal if the underlying principal homogeneous space $Q_{n+1}$ is the trivial one. The fibered full subcategory of banal displays will be denoted by $\bB^n(\G,\mu)$, it is a prestack, and it allows a very concrete description: The objects are maps $$\Phi_n:\I_{n+1}\times_{W(\f_{p^f})}X\rightarrow\H_n\times_{W(\f_p)}X,$$ of the form $g\mapsto U^{\Phi_n}g$, and thus indexed by the group elements $\Phi_n(1)=U\in\H_n(X)$. Consider a morphism $\psi_{n+1}$ from the object that has $\Phi_n(1)=U'$ to the object that has $\Phi_n(1)=U$. In the first place $\psi_{n+1}$ must be an automorphism of $\I_{n+1}\times_{W(\f_{p^f})}X$ (regarded as a trivial principal homogeneous space), hence equal to multiplication by some element $k\in\I_{n+1}(X)$. This morphism preserves the $\Phi_n$-structures if and only if the diagram $$\begin{CD}
Q_{n+1}'@>{\Phi_n}>>P_n'\\
@V{\psi_{n+1}}VV@V{\psi_n}VV\\
Q_{n+1}@>{\Phi_n}>>P_n
\end{CD}$$ commutes. Here $\psi_n$ is the truncation of $\psi_{n+1}$, hence it is equal to the multiplication by the same element $k$. However $Q_{n+1}'=Q_{n+1}=\I_{n+1}\times_{W(\f_{p^f})}X$, and therefore one must have $$U'=k^{-1}U{^{\Phi_n}k},$$ because the upper horizontal arrow is given by $g\mapsto U'{^{\Phi_n}g}$ and the lower one by $g\mapsto U{^{\Phi_n}g}$. Our explicit description entails an important equivalence of categories: $$\label{laber}
\bB_X^n(\G,\mu)\stackrel{\cong}{\rightarrow}\bB_{\Gamma(X,\O_X)}^n(\G,\mu).$$
For later use we note two more lemmas:
\[etale\] Let $R$ be a $W(\f_{p^f})$-algebra, and let $\P$ be a length $n$-truncated $3n$-display with $\G$-structure over $\Spec R$.
- If $I\subset R$ is a nilpotent ideal, then $\P$ is banal if and only if $\P\times_{\Spec R}\Spec R/I$ is banal.
- If $pR$ is nilpotent then there exists an étale and faithfully flat $R$-algebra over which $\P$ becomes banal.
The first of these two assertions follows from remark \[elementary\], and in the finite length case both of them are trivial, due to [@egaiv Corollaire (17.16.3.ii)]. In order to prove the infinite length case of the remaining assertion we may assume $p1_R=0$, due to [@egaiv Théorème 18.1.2]. Now we can clearly find an étale and faithfully flat $R$-algebra $R'$ over which the level $0$-truncation of $\P':=\P\times_RR'$ is banal. However, notice that there exists a short exact sequence $$0\rightarrow\underline{\gg\otimes\f_{p^f}}
\hookrightarrow\I_{n+1}\times\f_{p^f}\twoheadrightarrow\I_n\times\f_{p^f}\rightarrow1.$$ By induction on $n\in\n$ we deduce the banality of all of the level $n$-truncations of $\P'$, due to [@milne1 Chapter III, Proposition 3.7]. Finally it follows that $\P'$ is banal.
\[ordinary\] Let $k$ be a perfect field that contains $\f_{p^f}$, and let $\P$ and $\P'$ be length $n$-truncated $3n$-display with $\G$-structure over $\Spec k$, where $n\in\n\cup\{\infty\}$. Then there exists a finite Galois extension $l/k$, a polynomial $\hbar\in l[t]$ with $\hbar(0)\neq0\neq\hbar(1)$, and a length $n$-truncated $3n$-display $\tilde \P$ with $\G$-structure over $\Spec l[t]_{\hbar}$, such that $\P\times_kl\cong\tilde\P|_{t=0}$ and $\P'\times_kl\cong\tilde\P|_{t=1}$.
Pick a smooth map $\Spec W(\f_p)[x_1,\dots,x_m]_f\stackrel{r}{\rightarrow}\G$, where $f$ is a polynomial which is not contained in $pW(\f_p)[x_1,\dots,x_m]$, and let $\Spec W(\f_p)[x_1,\dots,x_{2m}]_g=:S\stackrel{q}{\rightarrow}\G$ be defined by $q(x_1,\dots,x_{2m})=r(x_1,\dots,x_m)r(x_{m+1},\dots,x_{2m})$, where $g(x_1,\dots,x_{2m})$ is the polynomial defined by $f(x_1,\dots,x_m)f(x_{m+1},\dots,x_{2m})$. The map $q$ is smooth and surjective. Now choose a finite Galois extension over which $\P$ and $\P'$ become banal, and write $U$ and $U'$ for representatives. Over a possibly further finite Galois extension $k'$, these can be lifted to $S$, in order to achieve $U=q(x_1,\dots,x_{2m})$, and $U'=q(x'_1,\dots,x'_{2m})$, where $x_1,\dots,x_{2m},x'_1,\dots,x'_{2m}\in W_n(l)$. Put $z_i:=x_i+[t](x'_i-x_i)\in W_n(l[t])$, and write $\hbar\in l[t]$ for the $\pmod p$ reduction of the polynomial $h(t):=g(x_1+t(x'_1-x_1),\dots,x_{2m}+t(x'_{2m}-x_{2m}))\in W_n(l)[t]$. Observe that $z:=(z_1,\dots,z_{2m})$ is a $W_n(l[t]_{\hbar})$-valued point of $S$ (the $0$th ghost coordinate of $g(z_1,\dots,z_{2m})$ is equal to $\hbar$). Note $\hbar(0)\neq0\neq\hbar(1)$, and let $\tilde\P$ be the length $n$-truncated $3n$-display with $\G$-structure over $\Spec l[t]_{\hbar}$ which is represented by the element $\tilde U:=q(z)\in\G(W_n(l[t]_{\hbar}))$.
Separatedness, part I {#representI}
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Let $\phi$ be an automorphism of a length $n$-truncated $3n$-display $\P$ over $S$ with $\G$-structure. In the sequel we need to know that the condition “$\phi=\id$” defines a closed subscheme of $S$. If $S=\Spec A$, for instance, then there should exist an ideal $I\subset A$ such that $\phi\times_AB=\id_{\P\times_AB}$ holds if and only if $IB$ vanishes. The following proposition achieves this:
\[sepI\] Suppose that $\P'$ and $\P$ are $n$-truncated $3n$-displays over $S$ with $\G$-structure. Then there exists a relatively affine $S$ scheme which represents the set-valued contravariant functor $X\mapsto\Hom(\P'\times_SX,\P\times_SX)$ on the category of $S$-schemes.
The functor in question satisfies the sheaf axiom for fpqc morphisms. Descent theory implies that one may assume $\P'$ and $\P$ to be banal, hence represented by display matrices $U',U\in\H_n(\Gamma(S,\O_S))$. In this case the morphisms from $\P'\times_SX$ to $\P\times_SX$ are the solutions of the equation $U'=h^{-1}U{^{\Phi_n}}h$ in $h\in\I_{n+1}(\Gamma(X,\O_X))$. This is obviously representable by a closed subscheme of $\I_{n+1}\times_{W(\f_{p^f})}S$.
The result may be also be interpreted by saying that the diagonal $1$-morphism $\Delta_{\B^n(\G,\mu)}:\B^n(\G,\mu)\rightarrow\B^n(\G,\mu)\times_{W(\f_{p^f})}\B^n(\G,\mu)$ is schematic and affine, hence separated and quasicompact. Moreover, for finite $n$ one may regard $\B^n(\G,\mu)$ as an algebraic stack over $W(\f_{p^f})$, basically because the $1$-morphism $b^n(\G,\mu):\Spec W(\f_{p^f})\times_{W(\f_p)}\H_n\rightarrow\B^n(\G,\mu)$ is a smooth presentation. We refrain from doing that, due to our (non-standard) choice of Grothendieck topology being the fpqc one. It should also be noticed, that the prime object of this work is the limit stack $\B(\G,\mu):=\B^\infty(\G,\mu)$, which is by no means algebraic.
Compatibility with limits
-------------------------
\[existence\] Let $R$ be a complete local noetherian $W(\f_{p^f})$-algebra with maximal ideal $\gm$. Then the natural forgetful functor $$\B_R^n(\G,\mu)\rightarrow2-\lim_{\leftarrow}\B_{R/\gm^\nu}^n(\G,\mu)$$ is an equivalence of categories, for all $n$ including $\infty$
The computation of the essential image is the only issue, so consider an object in the right hand side category, which is a sequence $\P^{(\nu)}\in\B_{R/\gm^\nu}^n(\G,\mu)$ endowed with transition maps $\psi^{(\nu)}:\P^{(\nu)}\rightarrow\P^{(\nu+1)}\times_{R/\gm^{\nu+1}}R/\gm^\nu$. Under the additional assumption that every $\P^{(\nu)}$ is banal one can easily construct a preimage: Let $U^{(\nu)}\in\H_n(R/\gm^\nu)$ represent $\P^{(\nu)}$, so that giving $\psi^{(\nu)}$ boils down to the equation $U^{(\nu)}=(h^{(\nu)})^{-1}\Ubar^{(\nu+1)}{^\Phi}h^{(\nu)}$ where $\Ubar^{(\nu+1)}$ stands for the image of $U^{(\nu+1)}$ in $\H_n(R/\gm^\nu)$, and $h^{(\nu)}\in\I_{n+1}(R/\gm^\nu)$. Next we choose lifts $\tilde h^{(\nu)}\in\I_{n+1}(R)$ of the $h^{(\nu)}$’s, and we define sequences of elements $m^{(\nu)}:=\tilde h^{(\nu-1)}.\cdots.\tilde h^{(1)}\in\I_{n+1}(R)$ and $O^{(\nu)}:=(\mbar^{(\nu)})^{-1}U^{(\nu)}{^\Phi}\mbar^{(\nu)}\in\H_n(R/\gm^\nu)$, where $\mbar^{(\nu)}$ stands for the image of $m^{(\nu)}$ in $\I_{n+1}(R/\gm^\nu)$. It is clear that the sequence of $O^{(\nu)}\in\H_n(R/\gm^\nu)$ together with neutral transitions lies in the image of the functor in question, and it is also clear that the sequence of $\mbar^{(\nu)}\in\I_{n+1}(R/\gm^\nu)$ defines an isomorphism $\lim_\leftarrow O^{(\nu)}\rightarrow\lim_\leftarrow U^{(\nu)}$.\
In the general case there still exists a finite Galois extension $l$ of $k:=R/\gm$ such that $\P^{(1)}\times_kl$ is banal, according to part (ii) of lemma \[etale\]. Write $B$ for the unique finite Galois extension over $R$ of which the special fiber is $l$, by [@egaiv Proposition (18.3.2)]. For every index $\nu$ we can deduce the banality of $\P^{(\nu)}\times_{R/\gm^\nu}B/B\gm^\nu$, according to part (i) of lemma \[etale\]. This completes the proof because one can use descent along $R\rightarrow B$.
Example {#beispiel}
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We give a brief sketch of an example: Fix a $p$-adically separated and complete ring $R$, and let us coin the term $n$-truncated $3n$-display over $R$ for the following class of quadruples $(M,N,F,V^{-1})$:
- $M$ is a finitely generated projective $W_{n+1}(R)$-module
- $N\subset M$ is a submodule which contains $I_{n+1}(R)P$
- $M/N$ is projective, when regarded as a module over $R$
- $F:M\rightarrow W_n(R)\otimes_{W_{n+1}(R)}M$ and $V^{-1}:N\rightarrow W_n(R)\otimes_{W_{n+1}(R)}M$ are $^{F_n}$-linear maps, satisfying $V^{-1}({^Va}x)=aV^{-1}(x)$ for all $x\in M$ and $a\in W_n(R)$ (of which the Verschiebung is an element in $W_{n+1}(R)$).
- the image of $V^{-1}$ generates $M$ as a $W_{n+1}(R)$-module.
A morphism between the $n$-truncated $3n$-displays $(M',N',F,V^{-1})$ and $(M,N,F,V^{-1})$ is a $W_{n+1}(R)$-linear mapping $M'\rightarrow M$ sending $N'$ to $N$ and such that the maps $F$ and $V^{-1}$ are preserved. We will write $\Dis_R^n$ for the additive category of $n$-truncated $3n$-displays. We will write $\Dis_R^n(d,h)$ for the full subcategory made up of quadruples $(M,N,F,V^{-1})$ with $\rk_{W_{n+1}(R)}M=h$ and $\rk_RM/N=d$. When discarding all morphisms other than the isomorphisms in $\Dis_R^n(d,h)$ we obtain a groupoid: $\Dis_R^n(d,h)^*$. Consider the cocharacter $$\mu_{d,h}:\g_m\rightarrow\GL(h);z\mapsto\diag(\underbrace{z,\dots,z,}_d\underbrace{1,\dots,1}_{h-d}),$$ then there is a natural equivalence of categories $$\Co_{d,h}:\B_R^n(\GL(h),\mu_{d,h})\stackrel{\cong}{\rightarrow}\Dis_R^n(d,h)^*.$$ By the theory of Witt-descent it suffices to construct the restriction of $\Co_{d,h}$ to the subcategory $\bB_R^n(\GL(h),\mu_{d,h})$ of banal displays. Whenever some object of $\bB_R^n(\GL(h),\mu_{d,h})$ is represented by a matrix $U\in\GL(h,W_n(R))$ the effect of $\Co_{d,h}$ on it is defined as follows: Write $U\stackrel{\Co_{d,h}}{\mapsto}(M,N,F,V^{-1})$ where: $$\begin{aligned}
&&M:=W_{n+1}(R)^{\oplus h}\\
&&F:M\rightarrow W_n(R)\otimes_{W_{n+1}(R)}M;
\left(\begin{matrix}x\\y\end{matrix}\right)\mapsto U\left(\begin{matrix}{^Fx}\\p{^Fy}\end{matrix}\right)\\
&&N:=I_{n+1}(R)^{\oplus d}\oplus W_{n+1}(R)^{\oplus h-d}\\
&&V^{-1}:N\rightarrow W_n(R)\otimes_{W_{n+1}(R)}M;
\left(\begin{matrix}x\\y\end{matrix}\right)\mapsto U\left(\begin{matrix}{^{V^{-1}}x}\\{^Fy}\end{matrix}\right)\end{aligned}$$ Whenever $k$ represents a morphism between two banal displays which are represented by the matrices $U'=k^{-1}U{^{\Phi_n}k}$ and $U$, then the effect of $\Co_{d,h}$ on it is defined as follows: Write $k$ as a $h\times h$-block matrix $\left(\begin{matrix}A&B\\C&D\end{matrix}\right)$ with $B\equiv0\mod{I_{n+1}(R)}$, and let $\Co_{d,h}(k)$ act on $\Co_{d,h}(U')=W_{n+1}(R)^{\oplus h}$ by plain matrix multiplication.\
The functor $\Co_{d,h}$ is fully faithful. In order to determine the set of $n$-truncated $3n$-displays $(M,N,F,V^{-1})$ which lie in the essential image one has to observe that the ring $W_{n+1}(R)$ is separated and complete with respect to the $I_{n+1}(R)$-adic topology, so that the modules $N$ can always be written as $I_{n+1}(R)T\oplus L$ for suitable ($n$-truncated) normal decompositions $M=T\oplus L$. This allows one to consider the map $$\label{U}
F\oplus V^{-1}:M\rightarrow W_n(R)\otimes_{W_{n+1}(R)}M;x+y\mapsto F(x)+V^{-1}(y)$$ where $x\in T$, and $y\in L$. Following the ideas in, and using the language of [@zink2 Lemma 9] one proves to be a $^{F_n}$-linear isomorphism. Over some fpqc extension there exist $W_{n+1}(R)$-bases of $T$ and of $L$, with respect to which can be written as $F_n$ composed with some $U\in\GL(h,W_n(R))$ and we are done.
Graded $(A,\tau)$-modules
=========================
In the sequel we need a number of rather harmless constructions taking place within the category of graded Frobenius modules. We collect them in this section, primarily in order to introduce some notation. Let $A$ be a $p$-adically separated and complete ring and let $\tau$ be an endomorphism on $A$, which agrees with the absolute Frobenius on $A/pA$. By a $\z/r\z$-graded $(A,\tau)$-module we mean:
- a finitely generated projective $\z/r\z$-graded $A$-module $M=\bigoplus_{\sigma\in\z/r\z}M_\sigma$,
- a pair of $A$-linear maps $F^\sharp:A\otimes_{\tau,A}M\rightarrow M$ and $V^\sharp:M\rightarrow A\otimes_{\tau,A}M$ of degrees $-1$ and $1$, and
- a $r$-tuple of non-negative integers $w_\sigma$, such that $$V^\sharp\circ F^\sharp|_{A\otimes_{\tau,A}M_\sigma}=p^{w_\sigma}\id_{A\otimes_{\tau,A}M_\sigma},$$ and $$F^\sharp\circ V^\sharp|_{M_\sigma}=p^{w_{\sigma+1}}\id_{M_\sigma}$$ too.
The invariant $\underline w$ is called the graded width and should be considered to be part of the structure. By a homomorphism between $\z/r\z$-graded $(A,\tau)$-modules $N$ and $M$ of the same width $\underline w$ we mean $A$-linear homomorphisms $f_\sigma:N_\sigma\rightarrow M_\sigma$ for each $\sigma\in\z/r\z$, such that the diagrams: $$\begin{CD}
A\otimes_{\tau,A}M_{\sigma+1}@>{F^\sharp}>>M_\sigma\\
@A{\tau(f_{\sigma+1})}AA@A{f_\sigma}AA\\
A\otimes_{\tau,A}N_{\sigma+1}@>{F^\sharp}>>N_\sigma\\
\end{CD}\qquad\begin{CD}
A\otimes_{\tau,A}M_{\sigma+1}@<{V^\sharp}<<M_\sigma\\
@A{\tau(f_{\sigma+1})}AA@A{f_\sigma}AA\\
A\otimes_{\tau,A}N_{\sigma+1}@<{V^\sharp}<<N_\sigma\\
\end{CD}$$ are commutative. These form abelian groups $\Hom_{\z/r\z}^{\underline w}(N,M)$, so that the class of $\z/r\z$-graded $(A,\tau)$-modules of some width $\underline w$ forms an additive category $(A,\tau)-\bMod_{\z/r\z}^{\underline w}$. Observe that there is a natural duality defined on it, moreover the graded tensor product defines bi-additive functors: $$(A,\tau)-\bMod_{\z/r\z}^{\underline w}\times(A,\tau)-\bMod_{\z/r\z}^{\underline w'}
\stackrel{\dot{\otimes}}{\rightarrow}(A,\tau)-\bMod_{\z/r\z}^{\underline w+\underline w'}.$$
\[duality\] If one precomposes the map $F^\sharp$ with $x\mapsto1\otimes_{\tau,A}x$, then one obtains a semilinear endomorphism, i.e. a map $F^\flat:M\rightarrow M$, that satisfies $$F^\flat(ax)=\tau(a)F^\flat(x)$$ (and sends $M_{\sigma+1}$ into $M_\sigma$). Dually there exists a semilinear endomorphism ${\check F}^\flat$ on the dual $A$-module $\check M=\Hom(M,A)$, which (sends $\check M_{\sigma+1}$ to $\check M_\sigma$ and) satisfies $${\check F}^\flat(ay)=\tau(a){\check F}^\flat(y)$$ and $$(F^\flat x,{\check F}^\flat y)=p^{w_{\sigma+1}}\tau(x,y),$$ for $x\in M_{\sigma+1}$ and $y\in\check M_{\sigma+1}$. If there exists an integer $n$ such that $(F^\flat)^{\circ n}(M)\subset pM$, then $M$ is called $F$-nilpotent, and it is called $V$-nilpotent, if $({\check F}^\flat)^{\circ n}(\check M)\subset p\check M$.
Let us also introduce the additive, rigid $\otimes$-category $(A,\tau)-\bMod_{\z/r\z}$ consisting of finitely generated projective $\z/r\z$-graded $A$-modules together with a pair of mutually inverse $\z/r\z$-graded maps $F^\sharp:\q\otimes A\otimes_{\tau,A}M\rightarrow\q\otimes M$ and $V^\sharp:\q\otimes M\rightarrow\q\otimes A\otimes_{\tau,A}M$ of degrees $-1$ and $1$. Finally notice, that whenever $A$ is a $W(\f_{p^r})$-algebra, then the grading can be interpreted as an action of $W(\f_{p^r})\rightarrow\End(M)$, by declaring $M_\sigma$ to be the $F^{-\sigma}$-eigenspace.
Flexibility, part I
-------------------
By a $\mod r$-multidegree we mean a map $\bd:\z\rightarrow\z$ with the following properties:
- $\bd(\omega)+r=\bd(\omega+r)$
- $\omega\leq\bd(\omega)$
- $\bd$ is monotone
Let us denote $\max\{\omega|\bd(\omega)\leq\sigma\}=:\bd^*(\sigma)$, and $\max\{\bd(\omega)-\omega|\omega\in\z\}=\max\{\sigma-\bd^*(\sigma)|\sigma\in\z\}=:|\bd|$. Let $\Sigma\subset\z$ be the image of $\bd$, and let $\sigma_1<\dots<\sigma_z(=:\sigma_0+r)$ be the elements of a ($z$-element) set of representatives for the $\pmod r$ congruence classes of $\Sigma$. Now write $[\omega_{j-1}+1,\dots,\omega_j]$ for the intervals $\bd^{-1}(\{\sigma_j\})$, so that $\bd(\omega_j)=\sigma_j$ and $\bd^*(\sigma_j)=\omega_j$. In this context we declare $\omega_{j+z}$ to be $\omega_j+r$, and $\sigma_{j+z}$ to be $\sigma_j+r$ for $j\in\{1,\dots,z\}$. The purpose of this subsection is to define a certain functor $$\fx^\bd:(A,\tau)-\bMod_{\z/r\z}^{\underline w}\rightarrow(A,\tau)-\bMod_{\z/r\z}^{\underline w^\bd},$$ where $w_\omega^\bd:=\sum_{\sigma=\bd(\omega-1)+1}^{\bd(\omega)}w_\sigma$, let us start out by putting: $$\fx^\bd(M)_\omega=A\otimes_{\tau^{\bd(\omega)-\omega},A}M_{\bd(\omega)}.$$ For any $\bd$-stationary index $\omega$, i.e. one with $\bd(\omega+1)=\bd(\omega)$ we have $A\otimes_{\tau,A}\fx^\bd(M)_{\omega+1}\cong\fx^\bd(M)_\omega$ and thus we may take this very isomorphism to constitute the $\omega$’th map $F^\sharp$ on $\fx^\bd(M)$. Otherwise we have to come up with a map from $A\otimes_{\tau,A}\fx^\bd(M)_{\omega_j+1}$ to $\fx^\bd(M)_{\omega_j}$ which boils down to: $$A\otimes_{\tau^{\sigma_{j+1}-\omega_j},A}M_{\sigma_{j+1}}\rightarrow A\otimes_{\tau^{\sigma_j-\omega_j},A}M_{\sigma_j}$$ Our definition here is the image under $\tau^{\sigma_j-\omega_j}$ of the map $$A\otimes_{\tau^{\sigma_{j+1}-\sigma_j},A}M_{\sigma_{j+1}}\rightarrow M_{\sigma_j}$$ which arises naturally from composition of $\sigma_{j+1}-\sigma_j$ many maps $F^\sharp$. One proceeds analogously for the $V^\sharp$’s, use duality for example. The upshot is a graded $(A,\tau)$-module $\fx^\bd(M)$ of width equal to $w_\omega^\bd$. There are natural isomorphisms $$\fx^\bd(M\dot{\otimes}M')\cong\fx^\bd(M)\dot{\otimes}\fx^\bd(M')$$ (in the category $(A,\tau)-\bMod_{\z/r\z}^{(\underline w+\underline w')^\bd}$), whenever $M$ and $M'$ are $\z/r\z$-graded $(A,\tau)$-modules of widths $\underline w$ and $\underline w'$ and analogously for duality.
\[isogI\] Write $u_\omega:=\sum_{\sigma=\omega+1}^{\bd(\omega)}w_\sigma$ (this only depends on the $\pmod r$ congruence class of $\omega$) and write $u:=\max\{u_\omega|\omega\in\z/r\z\}$. Then there exists a canonical map $$\Hom_{\z/r\z}^{\underline w^\bd}(\fx^\bd(N),\fx^\bd(M))\rightarrow\Hom_{\z/r\z}^{\underline w}(N,M)$$ of which the composition with $$\fx^\bd:\Hom_{\z/r\z}^{\underline w}(N,M)\rightarrow\Hom_{\z/r\z}^{\underline w^\bd}(\fx^\bd(N),\fx^\bd(M))$$ is equal to multiplication by $p^u$, for any $\z/r\z$-graded $(A,\tau)$-module $M_\sigma$ of graded width $\underline w$.
Assume that $\tilde f$ is some endomorphism of $\tilde M=\fx^\bd(M)$ with components $\tilde f_\omega$. We simply define a map $f_\omega$ by the commutative diagram: $$\begin{CD}
M_\omega@>{(V^\sharp)^{\bd(\omega)-\omega}}>>
{A\otimes_{\tau^{\bd(\omega)-\omega},A}M_{\bd(\omega)}=\tilde M_\omega}\\
@V{f_\omega}VV@V{p^{v_\omega}\tilde f_\omega}VV\\
M_\omega@<{(F^\sharp)^{\bd(\omega)-\omega}}<<
{A\otimes_{\tau^{\bd(\omega)-\omega},A}M_{\bd(\omega)}=\tilde M_\omega}
\end{CD}$$ where $v_\omega$ is $u-u_\omega$. Now the whole point is the equality $w_\omega^\bd-w_\omega=v_{\omega-1}-v_\omega$. The proof is completed by writing down the diagrams: $$\begin{CD}
N_\omega@>{p^{v_\omega}(V^\sharp)^{\bd(\omega)-\omega}}>>\tilde N_\omega
@>{\tilde f_\omega}>>\tilde M_\omega\\
@A{F^\sharp}AA@A{\tilde F^\sharp}AA@A{\tilde F^\sharp}AA\\
A\otimes_{\tau,A}N_{\omega+1}@>{p^{v_{\omega+1}}(V^\sharp)^{\bd(\omega+1)-\omega-1}}>>
A\otimes_{\tau,A}\tilde N_{\omega+1}@>{\tilde f_{\omega+1}}>>A\otimes_{\tau,A}\tilde M_{\omega+1}
\end{CD}$$ and $$\begin{CD}
\tilde N_\omega@>{\tilde f_\omega}>>\tilde M_\omega
@>{p^{v_\omega}(F^\sharp)^{\bd(\omega)-\omega}}>>M_\omega\\
@V{\tilde V^\sharp}VV@V{\tilde V^\sharp}VV@V{V^\sharp}VV\\
A\otimes_{\tau,A}\tilde N_{\omega+1}@>{\tilde f_{\omega+1}}>>A\otimes_{\tau,A}\tilde M_{\omega+1}
@>{p^{v_{\omega+1}}(F^\sharp)^{\bd(\omega+1)-\omega-1}}>>A\otimes_{\tau,A}M_{\omega+1}
\end{CD},$$ which are commutative because the outer rectangles are commutative. Notice that this proof is free of any assumptions on the $p$-torsion.
\[stupid\] One has $u\leq|\bd|\max\{w_\sigma|\sigma\in\z\}$.
Windows with additional structure, part I {#winone}
-----------------------------------------
If $A$ and $\tau$ are as above then some triple $(A,J,\tau)$ is called a frame over $W(\f_{p^f})$ if and only if $A$ is a torsionfree $W(\f_{p^f})$-algebra and $J$ is a pd-ideal that contains some power of $p$. Notice that these conditions imply $J\subset\rad(A)$ and $\tau(J)\subset pA$. Now let $(A,J,\tau)$ be as above and consider the $\Phi$-datum $\g_m\rightarrow\G\times_{W(\f_p)}W(\f_{p^f})$. It is easy to see that the product of $\G^-(A)$ and the kernel of $\G^+(A)\rightarrow\G^+(A/J)$ is a group which will be denoted by $\hat\G_J^-(A)$. It follows that we obtain a map $$\hat\Phi_J^-:\hat\G_J^-(A)\rightarrow\G(A),$$ gotten from the restriction of the map $\tau\circ\Int_{\G\times_{W(\f_p)}A[\frac1p]}(\mu(p)^{-1})$, which sends $\G(A[\frac1p])$ to itself. On the set $\G(A)$ we introduce the structure of a category $\hat\CAS_{A,J}(\G,\mu)$ by the constraint $U'\stackrel{h}{\rightarrow}U$ whenever $U'=h^{-1}U{^{\hat\Phi_J^-}h}$ with $h\in\hat\G_J^-(A)$. Clearly all morphisms are invertible, i.e. $\hat\CAS_{A,J}(\G,\mu)$ is a groupoid. If we suppress the ideal in the notation it shall be understood that $J=pA$, in particular we simply write $\hat\CAS_A(\G,\mu)$ if we mean $\hat\CAS_{A,pA}(\G,\mu)$, and we also set $\CAS_A(\G,\mu)$ for the faithful subcategory of $\hat\CAS_A(\G,\mu)$ in which the morphisms are elements of $\G^-(A)$. For this paper it is vital to have various notions of realizations in $\G$-spaces. To this end consider a finitely generated torsionfree $W(\f_{p^r})$-module $\V$ where $r$ divides $f$ and suppose that $$\rho:\G\times W(\f_{p^r})\rightarrow\GL(\V/W(\f_{p^r}))$$ is a group homomorphism. Pulling back with the $-\sigma$ fold iterates of Frobenius results in modules: $$W(\f_{p^f})\otimes_{F^{-\sigma},W(\f_{p^r})}\V=\V_\sigma,$$ and let us write $\rho_\sigma$ for the natural representation of $\G\times W(\f_{p^f})$ on that space. Finally we set $\upsilon_\sigma$ for the image of $\mu$ under $\rho_\sigma$. From now on we assume that the weights of $\upsilon_\sigma$ are contained in an interval of integers $[a_\sigma,b_\sigma]$. We define $$\alpha_\sigma:=\frac{\upsilon_\sigma}{z^{a_\sigma}}:\a^1\times W(\f_{p^f})\rightarrow
\underline{\End_{W(\f_{p^f})}(\V_\sigma)}$$ and $$\beta_\sigma:=\frac{z^{b_\sigma}}{\upsilon_\sigma}:\a^1\times W(\f_{p^f})\rightarrow
\underline{\End_{W(\f_{p^f})}(\V_\sigma)},$$ note that $(\alpha_\sigma\beta_\sigma)(z)=z^{b_\sigma-a_\sigma}$. We want to set up a functor $$\sy^{\underline a,\underline b}(\rho):\hat B_{A,J}(\G,\mu)\rightarrow(A,\tau)-\bMod_{\z/r\z}^{\underline b-\underline a}.$$ Let $U\in\Ob_{\hat B_{A,J}(\G,\mu)}=\G(A)$ be given. We start by defining $\sy_\sigma^{\underline a,\underline b}(\rho):=A\otimes_{F^{-\sigma},W(\f_{p^r})}\V=A\otimes_{W(\f_{p^f})}\V_\sigma$, and we set $F_\sigma^\sharp:A\otimes_{F,W(\f_{p^f})}\V_{\sigma+1}\rightarrow A\otimes_{W(\f_{p^f})}\V_\sigma$ equal to $\rho_\sigma(U){^F\beta_{\sigma+1}(p)}$, and we set $V_\sigma^\sharp:A\otimes_{W(\f_{p^f})}\V_\sigma\rightarrow A\otimes_{F,W(\f_{p^f})}\V_{\sigma+1}$ equal to ${^F\alpha_{\sigma+1}(p)}\rho_\sigma(U)^{-1}$. We decree the effect of $\sy^{\underline a,\underline b}$ on a morphism $U'\stackrel{h}{\rightarrow}U$ in $\Mor_{\hat\CAS_{A,J}(\G,\mu)}$ as follows: The diagrams: $$\begin{CD}
A\otimes_{W(\f_{p^f})}\V_\sigma
@<{\rho_\sigma(h)}<<A\otimes_{W(\f_{p^f})}\V_\sigma\\
@A{\rho_\sigma(U){^F\beta_{\sigma+1}(p)}}AA
@A{\rho_\sigma(U'){^F\beta_{\sigma+1}(p)}}AA\\
A\otimes_{F,W(\f_{p^f})}\V_{\sigma+1}
@<{^F{\rho_{\sigma+1}(h)}}<<A\otimes_{F,W(\f_{p^f})}\V_{\sigma+1}
\end{CD}$$ are commutative, as can be seen from $U'=h^{-1}U{^{\hat\Phi_J^-}h}$ in conjunction with $$\begin{aligned}
&&^F\rho_\sigma(h)\alpha_{\sigma+1}(p)=\alpha_{\sigma+1}(p)\rho_\sigma({^{\hat\Phi_J^-}h})\\
&&\beta_{\sigma+1}(p){^F\rho_\sigma(h)}=\rho_\sigma({^{\hat\Phi_J^-}h})\beta_{\sigma+1}(p),\end{aligned}$$ and this exhibits an isomorphism $\sy^{\underline a,\underline b}(\rho,U')
\stackrel{\cong}{\rightarrow}\sy^{\underline a,\underline b}(\rho,U)$, namely the multiplication by $\rho_\sigma(h)$-map on the $\sigma$-eigenspace $\sy_\sigma^{\underline a,\underline b}(\rho,U')$. There are natural isomorphisms $$\label{fiberfunk}
\sy_\sigma^{\underline a+\underline a',\underline b+\underline b'}
(\rho\otimes_{W(\f_{p^r})}\rho',U)\cong
\sy_\sigma^{\underline a,\underline b}(\rho,U)
\otimes_A\sy_\sigma^{\underline a',\underline b'}(\rho',U),$$ whenever the terms are well-defined and analogously for duality. Now we turn to the category of $n$-truncated $3n$-displays over $R$ with $\G$-structure, where we assume $pR=0$. In this case there exists a canonical absolute Frobenius $F:W_n(R)\rightarrow W_n(R)$ and again there is a canonical realization functor $$\sy^{\underline a,\underline b}(\rho):\B_R^n(\G,\mu)\rightarrow(W_n(R),F)-\bMod_{\z/r\z}^{\underline b-\underline a},$$ whose existence is justified in a similar manner: Use Witt descent ([@zink2 Proposition 33]) to reduce to the banal situation and then argue as before.\
The sole reason for the introduction of the regularized cocharacters $\alpha_\sigma$ and $\beta_\sigma$, is to make the above functor sensitive to torsion phenomena. If one is willing to invert the prime $p$ and restrict to $n=\infty$, then there is no need to keep track of $\underline a$ and $\underline b$ and one could much more easily just work with $\alpha_\sigma:=\upsilon_\sigma$ and $\beta_\sigma:=\upsilon_\sigma^{-1}$, to arrive at the completely self-explanatory functor $$\sy(\rho):\B_R(\G,\mu)\rightarrow(W(R),F)-\bMod_{\z/r\z},$$ which has the evident advantage of being a $W(\f_{p^r})$-linear rigid $\otimes$-functor in the variable $\rho$.\
Unitarity becomes incorporated as follows: Let $r$ be even and let $\chi:\G\times W(\f_{p^\frac r2})\rightarrow\g_m$ be a character such that $\chi_\sigma\circ\mu$ is of weight $c_\sigma$, where $$(a_{\sigma+\frac r2},b_{\sigma+\frac r2})=(c_\sigma-b_\sigma,c_\sigma-a_\sigma).$$ Suppose there is a pairing $\Psi:\V\times\V\rightarrow W(\f_{p^r})$, which satisfies $$-\Psi(x,y)=\tau^\frac r2(\Psi(y,x))$$ and is $W(\f_{p^r})$-linear in the first variable. The representation $\rho$ is called unitary if $\Psi(\gamma(x),\gamma(y))=\chi(\gamma)\Psi(x,y)$ holds (for all $\gamma\in\G(W(\f_{p^\frac r2}))$ and likewise functorially). We can think of the pairing as a morphism $\Psi_\sigma:\rho_{\sigma+\frac r2}\rightarrow\chi_\sigma\otimes\check\rho_\sigma$ of $\G\times W(\f_{p^f})$-representations. This endows the $(A,\tau)$-module $\sy^{\underline a,\underline b}(\rho)$ with maps $$\sy_{\sigma+\frac r2}^{\underline a,\underline b}(\rho)\rightarrow{\check\sy}_\sigma^{\underline a,\underline b}(\rho).$$
\[nilpotence\] If there exists at least one $\sigma_0\in\z$ such that the weights of $\upsilon_{\sigma_0}$ are actually contained in the smaller interval $[a_{\sigma_0},b_{\sigma_0}-1]$ (resp. $[a_{\sigma_0}+1,b_{\sigma_0}]$), then $\sy^{\underline a,\underline b}(\rho)$ is $F$-nilpotent (resp. $V$-nilpotent). This follows rather formally from if $\rho'$ is taken to be the trivial representation.
The fibered groupoid of twisted conjugacy classes, part II {#twisted}
==========================================================
Suppose that $\P=(P_n,P_{n+1},Q_{n+1},\Phi_n)$ is an object of $\B_X^n(\G,\mu)$. By slight abuse of notation we will write $\varpi_\P(\rho)$ for $\varpi_{Q_1}(\rho)$, where $\rho$ is a representation of $\G^-$ and $Q_1$ is the level-$0$ truncation of $\P$. One of the aims in this section is to show that $T_\P:=\varpi_\P(\Ad_-)$ behaves like a tangent bundle, provided that $n=\infty$. We have to start with a weak substitute for the result [@zink2 Theorem 44] in the language of our stack $\B(\G,\mu)$.
Let $X$ be a $W(\f_{p^f})$-scheme. An (infinite length) $3n$-display with $\G$-structure over $X$ is called a display with $\G$-structure over $X$, if and only if $X\times_{W(\f_{p^f})}\f_{p^f}$ has an affine open covering $\bigcup_lU_l$, such that $$\sy^{a,1}(\P\times_XU_l,\Ad),$$ is $F$-nilpotent for all $l$, where $\Ad$ is the adjoint representation on the Lie algebra of $\G$, and $a$ is a sufficiently small integer.
The full fibered subcategory of $\B(\G,\mu)$ whose objects are the displays will be denoted by $\B'(\G,\mu)$. The variants $\bB'(\G,\mu)$ and $\hat\CAS'_{A,J}(\G,\mu)$ are defined completely analogously.\
For every ideal $\ga$ in an $W(\f_{p^f})$-algebra $S$ we will write $\hat\I_{\ga,n}$ (resp. $\hat\J_{\ga,n}$) for the covariant set-valued functors that take any $S$-algebra $R$ to the inverse image of $\I_n(R/\ga R)$ in $\H_n(R)$ (resp. of $\J_n(R/\ga R)$ in $\H_n^+(R)$). The functors $\hat\I_{\ga,n}$ and $\hat\J_{\ga,n}$ are sheaves for the fpqc topology. Notice also that $\hat\I_{\ga,n}(R)$ is the product of $\H_n^-(R)$ and $\hat\J_{\ga,n}(R)$ in any order for any $n$, if $pR+\ga R\subset\rad R$. If we suppress the ideal in the notation, then it shall be understood that $\ga=pS$, e.g. $\hat\I_n(R)$ is the inverse image of $\I_n(R/pR)$ in $\H_n(R)$. There exist Lie-theoretic analogs for these functors: Let us write $\gi_n$ for the functor (on $W(\f_{p^f})$-algebras $R$) $\Res_{I_n}\underline{\gg^+}\oplus\Res_{W_n}\underline{\gg^-}$ and $\gh_n$ for the functor (on $W(\f_p)$-algebras) $\Res_{W_n}\underline\gg$. Furthermore let $\hat\gi_{\ga,n}(R)\subset\gh_n(R)$ stand for the subfunctor obtained as the inverse image of $\gg^-\otimes_{W(\f_{p^f})}R/\ga R$, equivalently it is the sum of $\gi_n(R)$ and $\gh_n(\ga R)$ (for variable $R$). For every ideal $\gb\subset R$ of vanishing square there exists an exact sequence: $$\label{exact}
0\rightarrow\hat\gi_{\ga,n}(\gb)\hookrightarrow\hat\I_{\ga,n}(R)
\twoheadrightarrow\hat\I_{\ga,n}(R/\gb)\rightarrow0,$$ which follows from analogous sequences for the groups $\H_n$ and $\I_n$, moreover the functor $\hat\I_{\ga,n}$ acts on $\hat\gi_{\ga,n}$ by conjugation, basically because of analogous properties of $\H_n$ and $\I_n$.
Twisted Frobenius maps, part II {#pivotalII}
-------------------------------
We start with the simple observation that $\Phi_n$ has a Lie-theoretic analog $$\phi_n:\gi_{n+1}\rightarrow\gh_n\times_{W(\f_p)}W(\f_{p^f}),$$ which is defined by $V_n^{-1}$ (as in ) on the positive summand, and by $F_n$ precomposed with the derived action of the endomorphism $\mu^{-\Int}(p)\in\End(\G^-)$ on the negative summand $\gg^-=\Lie(\G^-)$ (as in ). In two instances we need to extend the maps $\Phi_n$ to maps $\hat\Phi_{\ga,n}$, first assume that $\ga^2+p\ga=0$. This implies that $F$ vanishes on $W(\ga)$ and that there exists a canonical splitting $$\label{norm}
W_n(\ga)=\ga\oplus I_n(\ga),$$ which was introduced by Norman in the slightly different context of [@norman Section 1]. The group $\I_{\ga,n}(R)$ is consequently the product of $\I_n(R)$ and $\gg\otimes_{W(\f_p)}\ga R$ and the latter subgroup is a normal one. Now we come to the definition of our functorial maps $\hat\Phi_{\ga,n}:\hat\I_{\ga,n+1}(R)\rightarrow\H_n(R)$ by $$\begin{aligned}
&&\hat\Phi_{\ga,n}|_{\I_{n+1}(R)}=\Phi_n\\
&&\gg\otimes_{W(\f_p)}R\ga\subset\ker(\hat\Phi_{\ga,n}).\end{aligned}$$ This is consistent with $\gg^-\otimes_{W(\f_{p^f})}R\ga\subset\ker(\Phi_n)$ and the same procedure applies to get an extension of $\phi_n$ to a Lie-theoretic map $\hat\phi_{\ga,n}$ on $\hat\gi_{\ga,n}$. Second, assume $n=\infty$ and that $\ga$ is a pd-ideal in a $p$-adically separated and complete torsionfree ring $S$. However, this implies that $(W(S),W(\ga)+I(S),F)$ is a frame over $W(\f_{p^f})$, so that we merely have to appeal to our definition $$\hat\Phi_{W(\ga)+I(S)}^-=:\hat\Phi_\ga,$$ which was already introduced in the earlier subsection \[winone\]. We need the following version of the Grothendieck-Messing crystalline functoriality:
\[GMZCF\] Let $S$ be a $W(\f_{p^f})$-algebra, and let $\ga$ be an ideal that contains some power of $p$. Suppose that one of the following three assumptions is in force:
- $\ga^2+p\ga=0$.
- $S$ is $p$-adically separated and complete, torsionfree and $\ga$ is a pd-ideal.
- $\gg^+=0$ and there exists a number $N$ such that $x^N=0$ for all $x\in\ga$.
Recall the group homomorphism $\hat\Phi_\ga:\hat\I_\ga(S)\rightarrow\H(S)$, which is well-defined in each of these cases (in case that (ii) holds this is the restriction of $F\circ\Int_{\G\times_{W(\f_p)}W(S)[\frac1p]}(\mu(p)^{-1})$, sending $\G(W(S)[\frac1p])$ to itself, and in case (iii) holds one has $\hat\Phi_\ga=\Phi=\Phi^-$ as $\hat\I_\ga=\I=\H=\H^-$). Now suppose that $O,U\in\H(S)$ satisfy the nilpotence condition and fulfil $$O\equiv h^{-1}U{^{\hat\Phi_\ga}}h\mod\ga$$ for some $h\in\hat\I_\ga(S)$. Then there exists a unique $k\in\hat\I_\ga(S)$ such that $$\begin{aligned}
\label{approx}
&&O=k^{-1}U{^{\hat\Phi_\ga}}k\\
&&h\equiv k\mod\ga\end{aligned}$$
Without loss of generality we assume that $h=1$. In the first case we set $C$ for the map on the abelian group $\hat\gi_\ga(\ga)=\gg\otimes_{W(\f_p)}W(\ga)$ that sends some random lift $\hat\I_\ga(S)\ni k\equiv1\mod\ga$ to $C(k)=U{^{\hat\Phi%%%%%%%
_\ga}k}O^{-1}$. It is easy to see that every element $x\in\gg\otimes_{W(\f_p)}W(\ga)$ satisfies $C(k+x)=C(k)+c(x)$, where $c$ is the map given by $\Ad(U)\circ\hat\phi_\ga=\Ad(O)\circ\hat\phi_\ga$, and where $\hat\phi_\ga$ is the previously introduced derivative of $\hat\Phi_\ga$ (N.B.: On the whole of $\gg\otimes_{W(\f_p)}W(S)$ the maps $\Ad(U)$ and $\Ad(O)$ may not necessarily agree but their restrictions to $W(\ga)$ do agree). It suffices to prove that $c$ is nilpotent on $\gg\otimes_{W(\f_p)}W(\ga)$! The logarithmic ghost map $w'$ (along with the inclusion $\ga\subset S$) gives rise to the diagram $$\begin{CD}
\gg\otimes_{W(\f_p)}W(\ga)@>{w'}>>\gg\otimes_{W(\f_p)}(S^{\n_0})\\
@A{\hat\phi_\ga}AA@A{\phi^*}AA\\
\gg\otimes_{W(\f_p)}W(\ga)@>{w'}>>\gg\otimes_{W(\f_p)}(S^{\n_0})
\end{CD},$$ where $\phi^*$ is defined using $[x_0',\dots]\mapsto[x_1',\dots]$ in place of $V^{-1}$. Now, given that the former map is exactly the effect of the absolute Frobenius on the usual, non-logarithmic ghost coordinates, we see immediately that the topological nilpotence of $\Ad(U)\circ\phi^*$ is implied by the $F$-nilpotence of $\sy^{a,1}(\P,\Ad)$. In the torsionfree case the proof is completely analogous. For the third variant one may assume that $p\ga=0$ and $N=p$ hold to then argue as in the proof of [@zink2 Lemma 42].
As a consequence we get the following result on the rigidity of automorphisms, it can be regarded as an analog of [@zink2 Proposition 40]:
\[sepII\] Let $A$ be an $\ga$-adically separated $W(\f_{p^f})$-algebra, where $\ga$ is an ideal that contains some power of $p$, and let $\phi$ be an automorphism of a display with $\G$-structure $\P/A$. Then $\phi\times_AA/\ga$ is the identity if and only if $\phi$ is the identity.
By proposition \[sepI\] and by an induction argument we can assume that $\ga^2+p\ga=0$. After passage to some affine fpqc covering we can also assume that $\P$ is banal. In this case, however, the result follows immediately from lemma \[GMZCF\].
Suppose $\ga$ is an ideal in a $W(\f_{p^f})$-algebra $S$ such that $p$ and $\ga$ are nilpotent, and fix $\P_0\in\B'_{S/\ga}(\G,\mu)$. For a scheme $X$ over $\Spec S$ we write $X_0$ for $X\times_{\Spec S}\Spec S/\ga$. By a lift of $\P_0$ over $X$ we mean a pair $(\P,\delta)$ with $\P\in\B'_X(\G,\mu)$ and $\delta:\P_0\times_{\Spec S/\ga}X_0\rightarrow\P\times_XX_0$. Isomorphisms between lifts are expected to preserve the respective $\delta$’s, in particular no lift has any automorphisms other than the identity, by corollary \[sepII\]. Let $\hat T_{\P_0}(X)$ be the set of isomorphism classes of lifts over $X$. By pull-back of lifts this is a contravariant functor on the category of schemes over $\Spec S$.
\[obstII\] Fix a display $\P_0$ over the ring $S/\ga S$ with $\G$-structure, where $\ga^2+p\ga=0$:
- The functor $\hat T_{\P_0}$ possesses the structure of a locally trivial principal homogeneous space for the fpqc-sheaf $$X\mapsto T_{\P_0}\otimes_{S/\ga S}\ga\Gamma(X,\O_X)=\Hom_{S/\ga S}(\check T_{\P_0},\ga\Gamma(X,\O_X)).$$
- $\hat T_{\P_0}(\Spec S)\neq\emptyset$
Let us check that $\hat T_{\P_0}$ satisfies the sheaf axiom for a fpqc map $Y\rightarrow X$: If the pull-backs $\pr_1^*(\P)$ and $\pr_2^*(\P)$ of some $(\P,\delta)\in\hat T_{\P_0}(Y)$ agree for the two projections $\pr_1,\,\pr_2:Y\times_XY\rightarrow Y$, then this means that there exists $\alpha:\pr_1^*(\P)\stackrel{\cong}{\rightarrow}\pr_2^*(\P)$ which restricts to the identity on $Y_0\times_{X_0}Y_0$. We easily deduce the cocycle condition for $\alpha$, because any equality of isomorphisms of displays over $Y\times_XY\times_XY$ can be checked over $Y_0\times_{X_0}Y_0\times_{X_0}Y_0$, by corollary \[sepII\]. This shows that $\P$ (resp. $\delta$) descent to $X$ (resp. $X_0$).\
We are now allowed to assume that $\P_0$ is banal, so choose a representing element $U_0\in\H(S/\ga)$, indeed any lift of $\P_0$ over any $X/\Spec S$ is going to be banal too, so we may restrict to the affine case $X=\Spec R$ from now on (cf. ), more specifically: Every lift over $R$ is determined by a pair $(U,h)\in\H(R)\times\I(R/\ga R)$, with $U_0=h^{-1}\Ubar{^\Phi h}$, and another pair $(U',h')$ determines the same element in $\hat T_{\P_0}(\Spec R)$ if and only if $U'=k^{-1}U{^\Phi k}$ holds for some $k\in\I(R)$ with $\kbar h'=h$ (where $\Ubar\in\H(R/\ga R)$ and $\kbar\in\I(R/\ga R)$ stand for the reductions $\mod\ga R$). Recall that we defined a natural embedding of $\gg\otimes_{W(\f_p)}\ga R$ into $\hat\I_\ga(R)\subset\H(R)$. It is now evident that for $N\in\gg\otimes_{W(\f_p)}\ga R$ the assignment $(U,h)\mapsto((\ad(h)N)U,h)$ defines an action on $\hat T_{\P_0}(\Spec R)$, and that for $N\in\gg^-\otimes_{W(\f_{p^f})}\ga R$ that action is the trivial one.\
Now suppose that $U_0$ is the $\mod\ga$-reduction of some fixed $U\in\H(S)$. Then lemma \[GMZCF\] tells us that every lift of $\P_0$ over $R$ can be written as $U'=NU$, for a unique $N\in\gg^+\otimes_{W(\f_{p^f})}\ga R$. This finishes the proof of the part (i), and the final supplement is a straightforward consequence of [@milne1 Chapter III, Proposition 3.7] .
\[smooth\] Let $\P$ be a display with $\G$-structure over a $\f_{p^f}$-algebra $A$, and let $d:A\rightarrow\Omega_{A/\f_{p^f}}^1$ be the universal derivation into the $A$-module of Kaehler-differentials. The map $$\theta_A:A\rightarrow A\oplus\Omega_{A/\f_{p^f}}^1;x\mapsto x+d(x),$$ is a ring homomorphism, as the augmentation ideal $\Omega_{A/\f_{p^f}}^1$ is given the trivial multiplication. Consequently there exists a unique $K_\P\in\Hom_A(\check T_\P,\Omega_{A/\f_{p^f}}^1)$ which measures the difference between the elements $\P\times_{A,\theta_A}A\oplus\Omega_{A/\f_{p^f}}^1$ and $\P$ of $\hat T_\P(\Spec A\oplus\Omega_{A/\f_{p^f}}^1)$. We will call $K_\P$ the Kodaira-Spencer element of $\P$. We will call $\P$ formally smooth (resp. formally étale) if and only if $A$ is a formally smooth $\f_{p^f}$-algebra, and $K_\P$ is a split injection (resp. bijection).
\[formal\] The methods of this section have the following consequence: Let $k$ be a perfect field extension of $\f_{p^f}$, and let $\P_0$ be a display with $\G$-structure over $k$. Let $\P_1$ be the unique display with $\G$-structure over the augmented $k$-algebra $k\oplus\check T_{\P_0}$, such that the difference between $\P_1$ and $\P_0$ is measured by the element $N=\id_{\check T_{\P_0}}$. Let $\P_{uni}$ be an arbitrary lift of $\P_1$ to the power series $k$-algebra $k[[\check T_{\P_0}]]$, the existence of such a lift follows from the part (ii) of corollary \[obstII\] together with lemma \[existence\], but it might fail to be a display. Still, it is easy to see that $\P_{uni}$ is the universal formal deformation of $\P_0$. This is completely analogous to [@zink2 2.4]. See also [@vasiu2 3.2.7 Remarks 8b)] for yet another setting.\
One can enliven this a bit more, suppose that $\P$ is a display with $\G$-structure over an algebraic variety $X/k$. Then $U=\{x\in X|\P\times_X\O_{X,x}\text{ is formally \'etale}\}$ is Zariski open. Moreover, for all closed points $x\in X$ we have $x\in U$ if and only if the display $\P\times_X\hat\O_{X,x}$ is a universal formal deformation of its special fiber $\P\times_Xk(x)$ (as usual $k(x):=\O_{X,x}/\gm_x$ and $\hat\O_{X,x}:=\lim_{\leftarrow}\O_{X,x}/\gm_x^\nu$).
\[obstI\] Suppose that $A$ is a sub-$W(\f_{p^f})$-algebra of $B$ and that $\ga$ is an ideal of $A$ containing some power of $p$. Assume that one of following holds:
- $\ga$ is nilpotent.
- $B$ is finite over $A$, and $A$ is a $\ga$-adically separated and complete noetherian ring.
Then for each pair $\P$ and $\T$ of displays over $\Spec A$ with $\G$-structure every two compatible isomorphisms $\phbar:\P\times_AA/\ga\rightarrow\T\times_AA/\ga$ and $\psi:\P\times_AB\rightarrow\T\times_AB$ are scalar extensions of some unique $\phi:\P\rightarrow\T$.
For the first assertion we may also assume $\ga^2+p\ga=0$, so that the map $\phbar$ allows to view $\T$ as a lift of $\P_0:=\P\times_AA/\ga$. Let $N\in T_{\P_0}\otimes_{A/\ga}\ga$ measure the difference between the elements $\T$ and $\P$ of $\hat T_{\P_0}(\Spec A)$. Its image in $T_{\P_0}\otimes_{A/\ga}\ga B$ has to vanish, according to the existence of $\psi$. Now the injectivity of $T_{\P_0}\otimes_{A/\ga}\ga\rightarrow T_{\P_0}\otimes_{A/\ga}\ga B$ yields $N=0$.\
Due to the Artin-Rees lemma and proposition \[sepI\] one gets the second assertion as a consequence of the first.
Newton points and their specializations {#supersingular}
---------------------------------------
Let us assume that the generic fiber $G:=\G\times_{W(\f_p)}K(\f_p)$ is a reductive algebraic group, and let $k$ be a pefect field of characteristic $p$. We have to recall and apply some of the key-concepts in [@rapoport]: On the set of $K(k)$-valued points of $G$ one defines the important equivalence relation of $F$-conjugacy by requiring that $b'\sim b$ if and only if $b'=g^{-1}b{^Fg}$, for some $K(k)$-valued point $g$, write $B_k(G):=G(K(k))/\sim$ for the set of $F$-conjugacy classes, and $B(G):=B_\fc(G)$. Recall that any $\P\in\Ob_{\bB_{\Spec k}(\G,\mu)}$ is represented by the twisted conjugacy class of an element $U\in\G(W(k))$, and observe that the $F$-conjugacy class of $U{^F\mu(p)}^{-1}$ defines therefore a canonical element $\barb_\P\in B_k(G)$. From now onwards we assume the algebraic closedness of $k$, so that $\bB_{\Spec k}(\G,\mu)=\B_{\Spec k}(\G,\mu)$ holds. In this case matters are simplified even more as $B_k(G)=B(G)$ by [@rapoport Lemma 1.3], and the same independence result is valid for the set of Newton points that we introduce next: Write $\q\otimes\g_m$ for the pro-algebraic torus whose character group is $\q$ and put $$\N(G):=(G(K(k))\backslash\Hom(\q\otimes\g_m\times K(k),G\times_{K(\f_p)}K(k)))^{<F>},$$ where the left $G(K(k))$-action is defined by composition with interior automorphisms, and where the action of the infinite cyclic group $<F>$ is defined by $\nu\mapsto{{^F}\nu}$. Every element $b\in G(K(k))$ gives rise to a so-called slope homomorphism $\nu_b:\q\otimes\g_m\times K(k)\rightarrow G\times_{K(\f_p)}K(k)$, for which we refer the reader to loc.cit. Its formation is canonical in the sense that $g^{-1}\nu_bg=\nu_{g^{-1}b{^Fg}}$. Consequently, one is in a position to introduce the Newton-map: $$\nbar:B(G)\rightarrow\N(G);\barb\mapsto\nbar(\barb):=\overline{\nu_b},$$ here notice that the fractionary cocharacters ${^F}\nu_b$ and $\nu_b$ are lying in the same conjugacy class.\
Now fix a maximal torus $T\subset G\times K(\f_p)^{ac}$. Consider its associated Weyl group $\Omega$, and its associated lattice of cocharacters $X_*(T)$, which is equipped with a left $\Omega\rtimes\Gal(K(\f_p)^{ac}/K(\f_p))$-action. There are natural inclusions $$\N(G)\subset(\Omega\backslash X_*(T)_\q)^{\Gal(K(\f_p)^{ac}/K(\f_p))}\subset\Omega\backslash X_*(T)_\q,$$ moreover, the set $\Omega\backslash X_*(T)_\r$ is partially ordered in a very natural way, as one puts $\Omega x\prec\Omega x'$ if and only if $x$ lies in the convex hull of the $\Omega$ orbit of $x'$, cf. [@rapoport Lemma 2.2(i)]. The same notation is used for the partial orders on $\N(G)$ and $B(G)$ which are immediately inherited via the Newton map.\
Now let $\P$ be an infinite length $3n$-display with $\G$-structure over an $\f_{p^f}$-algebra $R$. Consider the function $\Spec R\rightarrow B(G)$ defined by $\barb_\P(\gp):=\barb_{\P\times_Rk(\gp)^{ac}}$. The following variant of Grothendieck’s specialization theorem (cf. [@rapoport Theorem 3.6(ii)]) will be used in the sequel:
\[Grothendieck\] If $\barb_0\in B(G)$, and if $\P$ is as above, then there exists a finitely generated ideal whose zero set is $\{\gp\in\Spec R| \barb_\P(\gp)\prec\barb_0\}$.
We would also like to adjust the inequality of Mazur to the needs of our category $\B(\G,\mu)$: So assume that $\G$ is reductive, and write $B$ for a $K(\f_p)$-rational Borel group of $G$, and write $T$ for a maximal torus of $B$. Pick an element $\gamma\in G(K(\f_p^{ac}))$ such that $\gamma\mu\gamma^{-1}$ lies in $T$ and is dominant with respect to $B$. Pick a factorization $\gamma=\gamma_1\gamma_2$, where $\gamma_1\in B(K(\f_p^{ac}))$, and $\gamma_2\in\G(W(\f_p^{ac}))$, and define an infinite length banal $3n$-display with $\G$-structure $\P_{ord}/\f_p^{ac}$ by the representative $\gamma_2^{-1}{^F\gamma_2}\in\G(W(\f_p^{ac}))$, then we have (cf. [@rapoport Theorem 4.2(ii)]):
\[Mazur\] Under the above hypotheses on $\G$ we have $\barb_\P\prec\barb_{\P_{ord}}$, for all $\P\in\Ob_{\B_{\Spec k}(\G,\mu)}$.
Finally we arrive at the main result of this subsection:
\[Hodgepoint\] Let $k$ be a perfect field containing $\f_{p^f}$, and let $\P_0$ be a display with $\G$-structure over $k$. Let $\P_{uni}/k[[t_1,\dots,t_d]]$, be its universal formal deformation ($d$ being the $W(\f_{p^f})$-rank of $\gg^+$, see remark \[formal\]). Then the function $\barb_{\P_{uni}}$ attains the value $\barb(\P_{ord})$ at the generic point of $\Spec k[[t_1,\dots,t_d]]$.
Lemma \[ordinary\] provides us with a family $\tilde\P\in\Ob_{\B_U(\G,\mu)}$ specializing to $\P_0$ and $\P_{ord}$, when evaluated at the closed points $0$ and $1$ lying in an open subscheme $U\subset\Spec l[t]$, for some finite extension $l$. The theorems \[Grothendieck\] and \[Mazur\] imply that $\barb_{\tilde\P}$ attains the value $\barb(\P_{ord})$ at the generic point, clearly the same is true for $\tilde\P\times_U\Spec l[[t]]$. Finally notice that the latter object descents to $\Spec(k+tl[[t]])$, and thus provides a morphism to $\Spec k[[t_1,\dots,t_d]]$ showing that the function $\barb_{\P_{uni}}$ attains $\barb(\P_{ord})$ at some, or equivalently at the generic point of $\Spec k[[t_1,\dots,t_d]]$.
\[supercompact\] We may deduce a simple criterion for whether or not $\P_{uni}$ is a display, namely according to whether or not $\P_{ord}$ is a display. This happens if and only if every $K(\f_p)$-rational simple factor of $G$ contains a $K(\f_p^{ac})$-rational simple factor in which $\mu$ is trivial.
Flexibility, part II
====================
Gauges {#global}
------
Let $\V$ be a finitely generated torsionfree $W(\f_{p^{2r}})$-module. Suppose that the pairing $\Psi:\V\times\V\rightarrow W(\f_{p^{2r}})$ is $W(\f_{p^{2r}})$-linear in the first variable and satisfies: $$\begin{aligned}
&&\Psi(x,y)^*=-\Psi(y,x)\\
&&\V=\{y\in\q\otimes\V|\Psi(\V,y)\subset W(\f_{p^{2r}})\}.\end{aligned}$$ Let $\GU(\V/W(\f_{p^{2r}}),\Psi)\subset\Res_{W(\f_{p^{2r}})/W(\f_{p^r})}\GL(\V/W(\f_{p^{2r}}))$ be the $W(\f_{p^r})$-subgroup whose sets of $R$-valued points are defined by the condition $\Psi(\gamma(x),\gamma(y))=m\Psi(x,y)$ for all $x,y\in R\otimes_{W(\f_{p^r})}\V$, and let us write $\chi:\GU(\V/W(\f_{p^{2r}}),\Psi)\rightarrow\g_m;\gamma\mapsto m$ for the multiplier character. Let $f$ be a fixed multiple of $2r$. In this subsection we introduce a certain averaging processes for cocharacters: $$\upsilon:\g_m\times W(\f_{p^f})\rightarrow\Res_{W(\f_{p^r})/W(\f_p)}\GU(\V/W(\f_{p^{2r}}),\Psi).$$ First of all notice that giving $\upsilon$ is equivalent to giving a family of cocharacters $\upsilon_\sigma:\g_m\times W(\f_{p^f})\rightarrow\GL(\V_\sigma/W(\f_{p^f}))$ each of whose duals $\check\upsilon_\sigma$ (cocharacters of $\GL(\check\V_\sigma/W(\f_{p^f}))$) coincides with $\Psi_\sigma\circ\upsilon_{\sigma+r}$ up to a homothety. Here it is taken for granted that $\V_\sigma$ and $\Psi_\sigma:\V_{\sigma+r}\stackrel{\cong}{\rightarrow}\check\V_\sigma$ are defined as in subsection \[winone\]. For any $l\in\z$ we denote $$H_0(l):=\begin{cases}0&l\leq0\\1&l\geq1\end{cases},$$ and for any $\nu:\g_m\times W(\f_{p^f})\rightarrow\GL(\V_\sigma/W(\f_{p^f}))$ we will write $H_0(\nu)$ for the cocharacter acting with weight $H_0(l)$ on every direct summand of $\V_\sigma$ on which $\nu$ would act with weight $l$.\
Now fix a $\pmod r$ multidegree $\bd$, and let us retain the notation $\sigma_j$ and $\omega_j$ for the indices in the images of $\bd$ and $\bd^*$. Let us write $a_\sigma$ for the minimal, and $b_\sigma$ for the maximal weight of $\upsilon_\sigma$ on $\V_\sigma$. From now on we wish to assume that $(\Res_{W(\f_{p^r})/W(\f_p)}\chi)\circ\upsilon$ is equal to the diagonal cocharacter $(z,\dots,z)$ of $\Res_{W(\f_{p^r})/W(\f_p)}\g_m$, which is equivalent to $a_{\sigma+r}+b_\sigma=1$ for all $\sigma$. Finally we denote $b_\sigma-a_\sigma$ by $w_\sigma$.
\[qflex\] By a gauge $\bj$ of multidegree $\bd$ for $\upsilon$ we mean a function $\bj:\z\rightarrow\z$ such that:
- For each $\sigma\in\z$ and $l\in[a_\sigma,b_\sigma-1]$ there exists a unique $\omega\in\z$ such that $\bd(\omega)=\sigma$ and $\bj(\omega)=l$.
- $\bj(\omega+r)=-\bj(\omega)$
Notice that the first of the two conditions implies $\bd(\z)\supset\{\sigma|w_\sigma\neq0\}$. This allows us to define a cocharacter $$\label{soul}
\tilde\upsilon_\omega(z)=H_0(\frac{^{F^{\bd(\omega)-\omega}}\upsilon_{\bd(\omega)}}{z^{\bj(\omega)}})$$ for all $\omega$. Clearly, the weights of $\tilde\upsilon_\omega$ are contained in $[\tilde a_\omega,\tilde b_\omega]$, where $$\begin{aligned}
&&\tilde a_\omega=H_0(a_{\bd(\omega)}-\bj(\omega))\\
&&\tilde b_\omega=H_0(b_{\bd(\omega)}-\bj(\omega))\\
&&\tilde w_\omega=\tilde b_\omega-\tilde a_\omega=\begin{cases}
1&\bj(\omega)\in[a_{\bd(\omega)},b_{\bd(\omega)}-1]\\
0&\text{otherwise}\end{cases},\end{aligned}$$ Notice that $\tilde a_{\omega+r}=1-\tilde b_\omega$, due to $H_0(1-l)=1-H_0(l)$. Furthermore, one sees that $$\sum_{\omega=\omega_{j-1}+1}^{\omega_j}\tilde w_\omega=\sum_{\sigma=\sigma_{j-1}+1}^{\sigma_j}w_\sigma=w_{\sigma_j}$$ holds. Let us also define two “error homotheties” for the $W(\f_p)$-group $\Res_{W(\f_{p^r})/W(\f_p)}\GU(\V/W(\f_{p^{2r}}),\Psi)$ by $\epsilon_\sigma(z)=z^{a_\sigma}$ and $\tilde\epsilon_\omega(z)=z^{\tilde a_\omega}$, one has $$\label{normI}
\prod_{\omega=\omega_{j-1}+1}^{\omega_j}
{^{F^{\omega-\sigma_j}}(\tilde\upsilon_\omega\tilde\epsilon_\omega^{-1})}=
\prod_{\sigma=\sigma_{j-1}+1}^{\sigma_j}
{^{F^{\sigma-\sigma_j}}(\upsilon_\sigma\epsilon_\sigma^{-1})}=
\upsilon_{\sigma_j}\epsilon_{\sigma_j}^{-1}$$ moreover, the set of cocharacters occuring in the above product do mutually commute.
Faithfulness of $\fx$, part I
-----------------------------
Consider a homomorphism $\rho:\G\rightarrow\GU(\V/W(\f_{p^{2r}}),\Psi)$ of group schemes over $W(\f_{p^r})$. Notice that: $$\Res_{W(\f_{p^r})/W(\f_p)}\G\times W(\f_{p^f})=\prod_{\sigma=0}^{r-1}\G_\sigma$$ where $\G_\sigma=^{F^{-\sigma}}\G=\G\times_{W(\f_{p^r}),F^{-\sigma}}W(\f_{p^f})$ (here $F=$ Frobenius on $W(\f_{p^f})$). In order to give a $\Phi$-datum $\mu$ of $\Res_{W(\f_{p^r})/W(\f_p)}\G$ over $W(\f_{p^f})$ it is necessary and sufficient to give $\Phi$-data $\mu_\sigma$ for each of the groups $\G_\sigma$. It will not cause confusion to write $\B^n(\G,\mu)$ instead of $\B^n(\Res_{W(\f_{p^r})/W(\f_p)}\G,\mu)$. Let $\bj$ be a gauge of multidegree $\bd$ for $\upsilon:=(\Res_{W(\f_{p^r})/W(\f_p)}\rho)\circ\mu$, and let $\tilde\upsilon$ be as in , recall that we regarded it as a family of cocharacters $\tilde\upsilon_\sigma$ for the groups $\tilde\G_\sigma:=\GL(\V_\sigma/W(\f_{p^f}))$, where $\sigma\in\z/2r\z$. Recall the group schemes $\Res_{W_n}\tilde\G_\sigma=\tilde\H_{\sigma,n}\supset\tilde\I_{\sigma,n}$ that are associated to $\tilde\upsilon_\sigma$ according to subsection \[pivotalIII\]. In this subsection our aim is to introduce a family of fibered functors: $$\label{heart}
\fx_X^{\bd,\bj}(\rho):\B_X^{n+w}(\G,\mu)\rightarrow
\B_X^n(\GU(\V/W(\f_{p^{2r}}),\Psi),\tilde\upsilon),$$ where $X$ is a $\f_{p^f}$-scheme and where $w:=\max\{w_\sigma\}<p$. We begin with a version of $\fx$ for the fibered groupoid of $\Phi$-conjugacy classes over an affine scheme $X=\Spec R$. So consider some $U_\sigma\in\G_\sigma(W_{n+w}(R))=\H_{\sigma,n+w}(R)$ and define $$\label{obflex}
\tilde U_\omega:=
\rho_\omega(\prod_{\bd^*(\sigma)=\omega}{^{F^{\sigma-\omega}}U_\sigma})=
\begin{cases}\rho_\omega(^{F^{\sigma_j-\omega_j}}U_{\sigma_j}
\dots\,^{F^{\sigma_{j+1}-\omega_j-1}}U_{\sigma_{j+1}-1})&\omega=\omega_j\\
1&\omega\notin\Omega\end{cases}.$$ The order of multiplication is the one indicated and does matter. Next we consider a family of elements $k_\sigma\in\I_{\sigma,n+w+1}(R)$, say, representing an isomorphism between $U_\sigma$ and $O_\sigma=k_\sigma^{-1}U_\sigma{^{(\Phi_{n+w}^{\mu_{\sigma+1}})}k_{\sigma+1}}$. Before we proceed to an analogous definition of elements $\tilde k_\omega$ we need a lemma:
\[nullnullsieben\] Suppose $k\in\I_{\sigma_j}(R)$, then $$^{F^{\omega_j-\omega_{j-1}-1}}\rho_{\sigma_j-1}(^{\Phi^{\mu_{\sigma_j}}}k)=
^{\Phi^{\tilde\upsilon_{\omega_{j-1}+1}}\circ\dots\circ\Phi^{\tilde\upsilon_{\omega_j}}}\rho_{\sigma_j}(k)$$ holds. Here it shall be understood that the right hand side is well defined in the following iterative sense: For every index $\omega\in\{\omega_{j-1}+1,\dots,\omega_j\}$ the element $^{\Phi^{\tilde\upsilon_{\omega+1}}\circ\dots\circ\Phi^{\tilde\upsilon_{\omega_j}}}\rho_{\sigma_j}(k)$ lies in the subgroup $\tilde\I_\omega(R)$, so that one may apply $\Phi^{\tilde\upsilon_\omega}$ to it.
We begin with elements $k\in\gg_{\sigma_j}^+\otimes_{W(\f_{p^f})}I(R)=\J_{\sigma_j}(R)$. Let us exploit that they have a Lie theoretic meaning too, and thus could be exposed to the derived operation $$\gg_{\sigma_j}^+\otimes_{W(\f_{p^f})}\V_{j,l}\rightarrow\V_{j,l+1},$$ where $\bigoplus_{l=a_{\sigma_j}}^{b_{\sigma_j}}\V_{j,l}=\V_{\sigma_j}$ is the decomposition according to $\mu_{\sigma_j}$-weights, $\gg_{\sigma_j}^+$ being of weight one. This results in elements $$t_l\in\Hom_{W(\f_{p^f})}(\V_{j,l},\V_{j,l+1})\otimes_{W(\f_{p^f})}I(R),$$ which completely determine the image: $$\left(\begin{matrix}1&t_{b_{\sigma_j}-1}&\dots&\frac{t_{b_{\sigma_j}-1}\dots t_{a_{\sigma_j}}}{w_{\sigma_j}!}\\0&1&\ddots&\vdots\\
\vdots&\ddots&\ddots&t_{a_{\sigma_j}}\\0&\dots&0&1\end{matrix}\right)$$ of $k$ under $\rho_{\sigma_j}$. On the one hand the effect of $\Phi^{\tilde\upsilon_\omega}$ on block matrices can be visualised as $$\left(\begin{matrix}A&B\\C&D\end{matrix}\right)\mapsto\left(\begin{matrix}^FA&^{V^{-1}}B\\p{^FC}&^FD\end{matrix}\right)$$ on the other hand all interesting cuts $l\in[a_{\sigma_j},b_{\sigma_j}-1]$ occur exactly once (along with $\omega_j-\omega_{j-1}-w_{\sigma_j}$ many less interesting ones). This patches to give a matrix of the form: $$\left(\begin{matrix}1&s_{b_{\sigma_j}-1}&\dots
&\frac{s_{b_{\sigma_j}-1}\dots s_{a_{\sigma_j}}}{w_{\sigma_j}!}\\
0&1&\ddots&\vdots\\
\vdots&\ddots&\ddots&s_{a_{\sigma_j}}\\
0&\dots&0&1
\end{matrix}\right),$$ where we have used the identity $^{F^{n-1}}(\prod_{\nu=1}^n{^{V^{-1}}x_\nu})={^{V^{-n}}(\prod_{\nu=1}^nx_\nu)}$ which follows for arbitrary elements $x_\nu\in I(R)$ from $^V(p^{n-1}\prod_{\nu=1}^nx_\nu)=\prod_{\nu=1}^n{^Vx_\nu}$ together with $FV=VF=p$, which is valid in characteristic $p$ rings, $s_l$ means $^{F^{\omega_j-\omega_{j-1}-1}\circ V^{-1}}t_l$. In the case of elements $k$ in $\H_{\sigma_j}^-(R)$ one can utilize $$\rho_{\sigma_j}\circ\mu_{\sigma_j}^{-\Int}(p)=\upsilon_{\sigma_j}^{-\Int}(p)\circ\rho_{\sigma_j},$$ because $\rho_{\sigma_j}(k)$ is a $W(R)$-valued point of the parabolic subgroup of $\tilde\G_{\sigma_j}$ that is defined by the ascending filtration $\V_{j,a_{\sigma_j}}\subset\dots\subset\bigoplus_{l\neq
b_{\sigma_j}}\V_{j,l}$. This parabolic is easily seen to be preserved by the pairwise commuting endomorphisms $$H_0(\frac{\upsilon_{\sigma_j}}{z^{a_{\sigma_j}}})^{-\Int}(p),\dots,H_0(\frac{\upsilon_{\sigma_j}}{z^{b_{\sigma_j}-1}})^{-\Int}(p),$$ of which the composition is equal to $\upsilon_{\sigma_j}^{-\Int}(p)$. For a general $k\in\I_{\sigma_j}(R)$ one applies the corollary \[factor\] on factorizations.
The lemma allows us to define
$$\label{morflex}
\tilde k_\omega:=\begin{cases}
^{F^{\sigma_j-\omega_j}}\rho(k_{\sigma_j})&\omega=\omega_j\\
{^{(\Phi_?^{\tilde\upsilon_{\omega+1}})}\tilde k_{\omega+1}}
&\omega\notin\Omega\end{cases},$$
and it proves the required compatibility $$\tilde O_\omega=\tilde k_\omega^{-1}\tilde U_\omega{^{(\Phi_n^{\tilde\upsilon_{\omega+1}})}\tilde k_{\omega+1}}$$ too. In the non-banal case one completes the definition of with a simple sheafification argument, which we leave to the reader. With a little bit of extra work one sees that the assignments in equation come from actual algebraic group homomorphisms: $$\label{ProjI}
\tilde\rho_\omega:\I_{\bd(\omega)}\times\f_{p^f}
\rightarrow\tilde\I_\omega\times\f_{p^f}.$$ Notice that the diagrams $$\begin{CD}
\I_{\bd(\omega+1)}\times\f_{p^f}
@>{\tilde\rho_{\omega+1}}>>\tilde\I_{\omega+1}\times\f_{p^f}\\
@V{\Phi^{\mu_{\bd(\omega+1)}}}VV
@V{\Phi^{\tilde\upsilon_{\omega+1}}}VV\\
\H_{\bd(\omega+1)-1}\times\f_{p^f}
@>{F^{\bd(\omega+1)-\omega-1}\circ\rho_{\bd(\omega+1)-1}}>>
\tilde\H_\omega\times\f_{p^f}
\end{CD}$$and$$\begin{CD}
\I_{\bd(\omega)}\times\f_{p^f}
@>{\tilde\rho_\omega}>>\tilde\I_\omega\times\f_{p^f}\\
@VVV@VVV\\
\H_{\bd(\omega)}\times\f_{p^f}
@>{F^{\bd(\omega)-\omega}\circ\rho_{\bd(\omega)}}>>
\tilde\H_\omega\times\f_{p^f}
\end{CD}$$ are commutative, provided that $\bd(\omega)\neq\bd(\omega+1)$, and otherwise $$\begin{CD}
\I_{\bd(\omega)}\times\f_{p^f}
@>{\tilde\rho_\omega}>>\tilde\I_\omega\times\f_{p^f}\\
@V{\tilde\rho_{\omega+1}}VV@VVV\\
\tilde\I_{\omega+1}\times\f_{p^f}@>{\Phi^{\tilde\upsilon_{\omega+1}}}>>\tilde\H_\omega\times\f_{p^f}
\end{CD}$$ is a commutative diagram (the arrows without label stand for the inclusions $\I_{\bd(\omega)}\times\f_{p^f}\subset\H_{\bd(\omega)}\times\f_{p^f}$ and $\tilde\I_\omega\times\f_{p^f}\subset\tilde\H_\omega\times\f_{p^f}$).
### Compatibility with realizations {#rectify}
We need to point out a compatibility with the realization functor of subsection \[winone\]. Let $\P$ be a $n+w$-truncated display with $\G$-structure over a $\f_{p^f}$-algebra $R$, and let us write $\underline a=(\dots,a_\sigma,\dots)$ and $\underline b=(\dots,b_\sigma,\dots)$ for the invariants introduced in subsection \[global\], in the unitary case these are $2r$-tuples, and $r$-tuples otherwise. For any $\P$ we have a canonical isomorphism $$W_n(R)\otimes_{W_{n+w}(R)}(\fx^\bd(\sy^{\underline a,\underline b}(\rho,\P)))
\cong\fx^{\bd^*\circ\bd}(\sy^{\tilde{\underline a},\tilde{\underline b}}(\sd,\fx_R^{\bd,\bj}(\rho,\P))),$$ which follows from $\bd=\bd\circ\bd^*\circ\bd$. Applying the functor $\fx^{\bd^*\circ\bd}$ will be called the "rectification procedure”. In case $n=\infty$, we have a canonical isogeny: $$\q\otimes\sy(\rho,\P)\cong\q\otimes\sy(\sd,\fx_R^{\bd,\bj}(\rho,\P)(\frac{\epsilon}{\tilde\epsilon})),$$ which follows along the same lines.
### Compatibility with the modulus character
We also need to point out a useful compatibility with the determinant of the tangent bundle, which was introduced in section \[twisted\], if $$\tilde\rho_\omega^-:=(\prod_{\sigma=\omega+1}^{\bd^*(\bd(\omega))}
H_0(\frac{\upsilon_{\bd(\omega)}}{z^{\bj(\sigma)}}))^{-\Int}(p)\circ({\rho_{\bd(\omega)}}|_{\G_{\bd(\omega)}^-}):
\G_{\bd(\omega)}^-\rightarrow{^{F^{\omega-\bd(\omega)}}\tilde\G}_\omega^-\subset\tilde\G_{\bd(\omega)},$$ then the diagram $$\begin{CD}
\I_{\bd(\omega)}\times\f_{p^f}
@>{\tilde\rho_\omega}>>\tilde\I_\omega\times\f_{p^f}\\
@VVV@VVV\\
{\G_{\bd(\omega)}^-}\times\f_{p^f}
@>{F^{\bd(\omega)-\omega}\circ\tilde\rho_\omega^-}>>{\tilde\G_\omega^-}\times\f_{p^f}
\end{CD}$$ commutes (notice that the higher levels of truncation are not at all preserved by $\tilde\rho_\omega$, but they are rather shifted down by $\sum_{\sigma=\omega+1}^{\bd^*(\bd(\omega))}\tilde w_\sigma$). Now observe that there is a canonical $\G_{\bd(\omega)}^-$-action on the $W(\f_{p^f})$-module ${^{F^{\omega-\bd(\omega)}}\tilde\gg}_\omega^+$, simply by means of the composition $$\G_{\bd(\omega)}^-\stackrel{\tilde\rho_\omega^-}{\rightarrow}
{^{F^{\omega-\bd(\omega)}}\tilde\G}_\omega^-\rightarrow
\GL({^{F^{\omega-\bd(\omega)}}\tilde\gg}_\omega^+/W(\f_{p^f})),$$ in which the last arrow is the scalar extension of the adjoint action of $\tilde\G_\omega^-$ on the $W(\f_{p^f})$-module $\tilde\gg_\omega/\tilde\gg_\omega^-=\tilde\gg_\omega^+$. Consider the peculiar invertible $W(\f_{p^f})$-module $$\label{ample}
\gf_\omega^{\bd,\bj}:=({^{F^{\omega-\bd(\omega)}}}\det_{W(\f_{p^f})}\tilde\gg_\omega^+)^{\otimes_{W(\f_{p^f})}p^{\bd(\omega)-\omega}},$$ in order to put $\modul_\omega^{\bd,\bj}$ for the $\prod_{\sigma=0}^{r-1}\G_\sigma^-$-character which is defined by it. The upshot are isomorphisms $$\label{invertible}
\varpi_{\fx^{\bd,\bj}(\rho,\P)}(\modul)\cong\bigotimes_{\omega=0}^{r-1}\varpi_\P(\modul_\omega^{\bd,\bj}),$$ for any $w$-truncated $3n$-display $\P$ with $\G$-structure over any $\f_{p^f}$-scheme $X$.
### Compatibility with multipliers {#Weil}
There exists a canonical $2$-commutative diagram
$$\begin{CD}
\B_X^{n+w}(\G,\mu)@>{\chi\circ\rho}>>\B_X^{n+w}(\g_m\times W(\f_{p^r}),\mu_1)\\
@V{\fx_X^{\bd,\bj}(\rho)}VV@VVV\\
\B_X^n(\GU(\V/W(\f_{p^{2r}}),\Psi),\tilde\upsilon)@>{\chi}>>\B_X^n(\g_m\times W(\f_{p^r}),\mu_1),
\end{CD}$$
where the right vertical arrow is the truncation to the appropriate level.
### Faithfulness
From now onwards we will always assume that $\bd$ is not a translation, i.e. $\bd\neq|\bd|$. In view of remark \[nilpotence\] this implies $\B(\G,\mu)=\B'(\G,\mu)$. In the rest of this subsection we study a first faithfulness property of the product functor $\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i,\dots)$ where $\rho_i:\G\rightarrow\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)$ is a family of (possibly) unitary representations each of which is equipped with a gauge $\bj_i$. Let us begin with the following triviality:
\[faithfulII\] Let $A$ be a $\f_{p^f}$-algebra which is separated with respect to the $M$-adic topology where $M=\{x\in A|x^p=0\}$. Suppose that the product representation $\prod_{i\in\Lambda}\rho_i:\G\rightarrow\prod_{i\in\Lambda}\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)$ is a monomorphism. Then the product functor $\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i,\dots)$ is faithful (i.e. injective on morphisms).
In view of corollary \[sepII\] this is clear, because a close inspection of the formula shows that the kernel of $\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i,\dots)$ is killed by a power of Frobenius.
From now on we do assume that $\prod_{i\in\Lambda}\rho_i$ is a monomorphism, we have a very important corollary:
\[tensorsquare\] If $k$ is any field extension of $\f_{p^f}$, then the product functor $\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i,\dots)$ is faithful over the ring $R=k^{perf}\otimes_kk^{perf}$.
We must prove that $R$ satisfies the criterion of the previous lemma. In order to control the situation we use a $p$-basis $\{x_i|i\in I\}$, so that every element of $k$ has a unique representation as a sum $x=\sum_{\underline n}a_{\underline n}^p{\underline x}^{\underline n}$ where $\underline n=(\dots,n_i,\dots)$ runs through the set of multiindices with $p-1\geq n_i\geq0$ an $n_i=0$ for almost all $i$. It is easy to see that $k^{perf}$ is the quotient of the polynomial algebra $k[\{t_{i,e}|i\in I\,,e=1,\dots\}]$ by the ideal which is generated by $t_{i,e+1}^p-t_{i,e}$ and $t_{i,1}^p-x_i$. It follows that $k^{perf}\otimes_kk^{perf}$ is the quotient of the polynomial algebra $k^{perf}[\{s_{i,e}|i\in I\,,e=1,\dots\}]$ by the ideal which is generated by $s_{i,e+1}^p-s_{i,e}$ and $s_{i,1}^p$, so that $M$ is generated by the set $\{s_{i,1}\}$. We deduce that there exist ring endomorphisms $\theta_{I_0}:R\rightarrow R$, defined by $$s_{i,e}\mapsto\begin{cases}s_{i,e}&i\in I_0\\0&i\notin I_0\end{cases}$$ for every finite subset $I_0\subset I$. Now $M^\nu$ is generated by the set $\{{\underline s_1}^{\underline\nu}|\sum_i\nu_i=\nu\}$. The only multiindices which give rise to non-zero products are bounded by $\nu_i<p$, and these are seen to involve factors indexed by at least $\frac\nu{p-1}$ many elements of $I$. Consequently $\theta_{I_0}(M^\nu)=0$ if $\Card(I_0)<\frac\nu{p-1}$, and this shows $\bigcap_\nu M^\nu=0$.
The results of this subsection apply mutatis mutandis to the case of linear groups: $$\label{split}
\fx_R^{\bd,\bj}(\rho):\B_R^{n+w}(\G,\mu)\rightarrow
\B_R^n(\g_m\times\GL(\V/W(\f_{p^r})),\tilde\upsilon),$$ where $\V$ is a finitely generated torsion free $W(\f_{p^r})$-module. Do notice however that even in this case we are actually working with a highly non-split $W(\f_p)$-group namely: $\Res_{W(\f_{p^r})/W(\f_p)}(\g_m\times\GL(\V/W(\f_{p^r})))$. All further results in this paper will apply simultaneously in the unitary and in the linear situation (although some of our notation indicates emphasis to the unitary case).\
There is only one notable aspect, in which the unitary situation is very slightly different: Notice that $$\sum_{\sigma\in\z/2r\z}a_\sigma
-\sum_{\omega\in\z/2r\z}\tilde a_\omega=
\sum_{\sigma\in\z/2r\z}b_\sigma
-\sum_{\omega\in\z/2r\z}\tilde b_\omega=
\sum_{\omega\in\z/2r\z}\tilde a_{\omega+r}
-\sum_{\sigma\in\z/2r\z}a_{\sigma+r},$$ and therefore $\prod_{\sigma\in\z/2r\z}{^{F^\sigma}\epsilon}_\sigma=\prod_{\omega\in\z/2r\z}{^{F^\omega}\tilde\epsilon}_\omega$. Contrarily, the cocharacters $\prod_{\sigma\in\z/r\z}{^{F^\sigma}\epsilon}_\sigma$ and $\prod_{\omega\in\z/r\z}{^{F^\omega}\tilde\epsilon}_\omega$ may well be different in the linear case.
Separatedness, part II {#practical}
----------------------
We are now able to state our first serious result:
\[sepIII\] Let $\f_{p^f}\subset A\subset B$ be a ring extension. Let $\P$ and $\T$ be displays with $\G$-structure over $\Spec A$ and let $\psi_i$ be isomorphisms between $\fx^{\bd,\bj_i}(\rho_i,\P)$ and $\fx^{\bd,\bj_i}(\rho_i,\T)$. Assume that one of the following two assumptions is in force:
- The nilradical of $B$ is trivial and $$\label{primeI}
A=B\cap\sqrt[p]A$$ holds, the intersection taking place in $\sqrt[p]B$.
- The nilradical of $B$ is nilpotent and $A$ is noetherian.
Then the family of isomorphisms $(\dots,\psi_i,\dots)$ has a preimage under the product functor $\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i,\dots)$ if and only if the same statement holds for the scalar extensions $(\dots,\psi_i\times_AB,\dots)$.
The functor $\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i,\dots)$ is faithful over $B$ and over $A$, in both cases (i) and (ii), in particular a preimage over $A$ is always unique and it exists if and only if the given preimage $\phi_B\in\Hom(\P\times_AB,\T\times_AB)$ descents to an isomorphism between $\P$ and $\T$. It seems to be elementary to check the case (i) of the assertion directly from the formula . More general, let us say that the $\psi_i$’s become flexed over some $A$-algebra $B'$ if one has found a preimage of $(\dots,\psi_i\times_AB',\dots)$. We do the proof of (ii) in several steps:
\[pof\] $A$ is a product of finitely many fields.\
It is clear that we can reduce the situation to the case of a single field, in which case $B$ can be assumed to be a field as well, in fact $B$ can be assumed to be equal to the perfection of $A$, according to part (i) of our proposition. Now we merely have to prove that the scalar extensions by means of the two maps $B\rightrightarrows B\otimes_AB$ give the same answer when applied to the isomorphism $\phi_B$. As these are still preimages of the $\psi_i\times_A(B\otimes_AB)$’s we conclude by using corollary \[tensorsquare\].
$A$ is a reduced complete local noetherian ring.\
By the previous step and part (i) of proposition \[sepIII\] we know that the $\psi_i$’s become flexed over the normalization $A'$. Now part (ii) of corollary \[obstI\] and [@egaiv Théorème (23.1.5)] tell us that we only need to know that the $\psi_i$’s become flexed over $A/\rad(A)$ which is dealt with by another application of the previous step.
$A$ is a complete local noetherian ring.\
This case follows from the previous step and part (i) of corollary \[obstI\].
$A$ is a reduced local noetherian ring.\
By step \[pof\] we do know that the $\psi_i$’s become flexed over $K(A)$. By the previous step we have the same result over the completion $\hat A$. Due to $A=\hat A\cap K(A)$ this is sufficient.
$A$ is a general noetherian ring.\
As before we can assume reducedness. By (flatness of $A_\gp$ and) the previous step we have that the $\psi_i$’s become flexed over the local rings $A_\gp$. Now use $A=\bigcap_{\gp\in\Spec A}A_\gp$, the intersection taking place in $K(A)$.
Here is a separatedness property of , as promised in the title of this subsection:
\[sepV\] Let $A$ be a noetherian $\f_{p^f}$-algebra and let $\P$ and $\T$ be displays with $\G$-structure over $\Spec A$ and let $\psi_i$ be isomorphisms between $\fx^{\bd,\bj_i}(\rho_i,\P)$ and $\fx^{\bd,\bj_i}(\rho_i,\T)$. Then there exists an ideal $I\subset A$, such that the following two assertions are equivalent for all $A$-algebras $B$ with nilpotent nilradical:
- There exists an isomorphism $\phi_B:\P\times_AB\rightarrow\T\times_AB$ such that $\fx^{\bd,\bj_i}(\rho_i,\phi_B)=\psi_i\times_AB$
- $BI=0$
It is clear that the set $\I=\{I\subset A|\mbox{the }\psi_i\mbox{'s are flexed }\pmod I\}$ is stable under intersections of finitely many ideals, and it is also clear that $\bigcap\I=:I_0\in\I$ is all we need to know. The ring $\prod_{I\in\I}A/\sqrt I=:B_1$ is reduced, and so we know $\sqrt{I_0}\in\I$. The nilradical of $\prod_{\I\ni I\subset\sqrt{I_0}}A/I=:B_2$ is nilpotent and so we know that $I_0\in\I$.
Faithfulness of $\fx$, part II
------------------------------
Let $\{\rho_i\}_{i\in\Lambda}$ be a family of (possibly unitary) representations. If one has fixed gauges $\bj_i$ for $\upsilon_i=(\Res_{W(\f_{p^r})/W(\f_p)}\rho_i)\circ\mu$ of some common multidegree $\bd$ then we will say that the family is gauged. Recall that we have constructed a specific cocharacter $$\tilde\upsilon_i:\g_m\times W(\f_{p^f})\rightarrow\Res_{W(\f_{p^r})/W(\f_p)}\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)$$ from $\upsilon_i$ and $\bj_i$. For each subset $\pi\subset\Lambda$ of odd cardinality we need to introduce a further unitary group cocharacter $$\g_m\times W(\f_{p^f})\rightarrow\Res_{W(\f_{p^r})/W(\f_p)}\GU(\V_\pi/W(\f_{p^{2r}}),\Psi_\pi)$$ where $(\V_\pi,\Psi_\pi)$ is defined to be the skew-Hermitian $W(\f_{p^{2r}})$-module $\bigotimes_{i\in\pi}\V_i$, and where $\tilde\upsilon_\pi$ is defined to be the product of the central cocharacter $z\mapsto z^\frac{1-\Card(\pi)}2$ with the image of the diagonal cocharacter $(\dots,\tilde\upsilon_i,\dots)$ under the natural homomorphism $$g_\pi:\prod_{i\in\Lambda}\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)\rightarrow\GU(\V_\pi/W(\f_{p^{2r}}),\Psi_\pi),$$ that is given by the tensor product of the factors. Furthermore $\rho_\pi$ will stand for the natural representation $g_\pi\circ\prod_{i\in\Lambda}\rho_i$, finally write $\underline a=\sum_i{\underline a}_i$ and ditto for $\underline b$, $\underline w$, $\tilde{\underline a}$, $\tilde{\underline b}$, and $\tilde{\underline w}$, and let $w$ be $\max\{w_{i,\sigma}|i\in\Lambda,\,\sigma\in\z\}$.
Fix a gauged family $\{\rho_i\}_{i\in\Lambda}$ as above. A subset $\pi\subset\Lambda$ of odd cardinality is called multiplyable if the weights of $\tilde\upsilon_\pi$ are contained in $\{0,1\}$. A set $\Pi$ of subsets is called strict if the following statements are valid:
- Each element of $\Pi$ is multiplyable
- $\prod_{i\in\Lambda}\rho_i:\G\rightarrow\prod_{i\in\Lambda}\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)$ is a closed immersion
- The generic fiber $G$ is the stabilizer of $\bigcup_{\pi\in\Pi}\End_G(\V_\pi)$
\[hyperspecial\] If one assumes that $p$ is odd and that $\G$ is reductive, then $\prod_{i\in\Lambda}\rho_i$ is a closed immersion if and only if its generic fiber has that property ([@prasad Corollary 1.3]). Hence in this case the condition (S2) is a consequence of (S3).\
Furthermore (S2) implies that $\G$ agrees with the Zariski-closure in $\prod_{i\in\Lambda}\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)$ of its generic fiber, and in turn this implies that one can pin down a constant $c$ such that $\G$ is the stabilizer of $\O_{H_c}=\End_G(\bigotimes_{i\in\Lambda}\V_i^{\otimes c})$, cf. [@kisin Proposition(1.3.2)].
For every $n+w$-truncated $3n$-display $\P$ with $\G$-structure over $R/\f_{p^f}$, and for every multiplyable $\pi$ we obtain: $$\begin{aligned}
&&W_n(R)\otimes_{W_{n+w}(R)}(\fx^\bd(\sy^{\underline a,\underline b}(\rho_\pi,\P)))\cong\\
&&\fx^{\bd^*\circ\bd}\circ\sy^{\tilde{\underline a},\tilde{\underline b}}
(\sd,\dot\bigotimes_{i\in\pi}\fx^{\bd,\bj_i}(\rho_i,\P)),\end{aligned}$$ from subsubsection \[rectify\]. According to lemma \[isogI\] and remark \[stupid\] it follows that we obtain a canonical functorial homomorphism $$p^{|\bd^*\circ\bd|}\End_\G(\V_\pi)\rightarrow
\End_{\z/2r\z}^{\tilde{\underline w}}(\sy^{\tilde{\underline a},\tilde{\underline b}}(\sd,\dot\bigotimes_{i\in\pi}\fx^{\bd,\bj_i}(\rho_i,\P))),$$ whence it follows the existence of a canonical ring homomorphism: $$\label{akwardIII}
W(\f_{p^{2r}})+p^{|\bd^*\circ\bd|+1}\End_\G(\V_\pi)=:\O_{L_\pi}
\stackrel{s_\pi}{\rightarrow}
\End(\dot\bigotimes_{i\in\pi}\fx^{\bd,\bj_i}(\rho_i,\P)).$$ The right-hand side graded tensor product is constructed along the lines of [@habil Proposition 4.3], equivalently observe that there is a natural functoriality $$\B(g_\pi):\prod_{i\in\Lambda}\B(\GU(\V_i/W(\f_{p^{2r}}),\Psi_i),\tilde\upsilon_i)
\rightarrow\B(\GU(\V_\pi/W(\f_{p^{2r}}),\Psi_\pi),\tilde\upsilon_\pi),$$ because the interior actions of $g_\pi(\dots,\tilde\upsilon_i,\dots)$ and $\tilde\upsilon_\pi$ agree. Now we return to the situation in theorem \[sepV\] and study some necessary and sufficient conditions on the $\psi_i$’s to arise from an isomorphism $\phi$:
\[faithfulIII\] Let the assumptions be as in theorem \[sepV\]. Let $\Pi$ be the set of multiplyable subsets $\pi\subset\Lambda$ such that the isomorphisms $$\dot\bigotimes_{i\in\pi}\psi_i:\dot\bigotimes_{i\in\pi}\fx^{\bd,\bj_i}(\rho_i,\P)\stackrel{\cong}{\rightarrow}\dot\bigotimes_{i\in\pi}\fx^{\bd,\bj_i}(\rho_i,\T)$$ are $\O_{L_\pi}$-linear. If $\Pi$ is strict, then the ideal $I$ which was proven to exist in theorem \[sepV\] is nilpotent.
By what we have shown already we can assume that $A$ is an algebraically closed field, in particular $\P$ and $\T$ are banal, and hence represented by graded display matrices $U_\sigma,O_\sigma\in\G_\sigma(W(A))$. We are allowed to assume that $U_\sigma=O_\sigma=1$ for all $\sigma$ outside the image of $\bd$, so that formula reads: $$\tilde U_{i,\omega}=\begin{cases}\rho_i(^{F^{\sigma_j-\omega_j}}U_{\sigma_j})&\omega=\omega_j\\1&\omega\notin\Omega\end{cases}$$ and similarily for $\tilde O_{i,\omega}$. Now assume that we have $$\tilde O_{i,\omega}=\tilde k_{i,\omega}^{-1}\tilde U_{i,\omega}{^F(\tilde\upsilon_{i,\omega+1}(p)^{-1}\tilde k_{i,\omega+1}\tilde\upsilon_{i,\omega+1}(p))}.$$ It is straightforward to check that the families $(\dots,\tilde k_{i,\omega_j}\dots)=\tilde k_{\omega_j}$ constitute elements in $\G_{\omega_j}$ and thus can be written in the form $^{F^{\sigma_j-\omega_j}}\rho_i(k_{\sigma_j})$. The difficulity lies in checking that indeed $k_{\sigma_j}\in\I_{\sigma_j}(A)$. Notice however, that $\tilde U_{i,\omega}=\tilde O_{i,\omega}=1$ for all $\omega$ outside the image of $\bd^*$, and consequently $$\begin{aligned}
&&\tilde U_{i,\omega_{j-1}}^{-1}\tilde k_{i,\omega_{j-1}}
\tilde O_{i,\omega_{j-1}}=\\
&&(\prod_{\omega=\omega_{j-1}+1}^{\omega_j}
{^{F^{\omega-\omega_{j-1}}}\tilde\upsilon_{i,\omega}(p)^{-1})}
({^{F^{\omega_j-\omega_{j-1}}}\tilde k_{i,\omega_j}})
(\prod_{\omega=\omega_{j-1}+1}^{\omega_j}
{^{F^{\omega-\omega_{j-1}}}\tilde\upsilon_{i,\omega}(p)})\\
&&={^{F^{\sigma_j-\omega_{j-1}-1}}\rho_i}(^{\Phi^{\mu_{\sigma_j}}}k_{\sigma_j})\end{aligned}$$ where we have used lemma \[nullnullsieben\]. Given that $\tilde U_{i,\omega_{j-1}}^{-1}$, $\tilde k_{i,\omega_{j-1}}$, and $\tilde O_{i,\omega_{j-1}}$ are elements in $\H_{\omega_{j-1}}(A)$ we get the integrality of $^{\Phi^{\mu_{\sigma_j}}}k_{\sigma_j}$ which was sought for.\
For elements $\omega$ outside the image of $\bd^*$ it is important to understand that the corresponding families $\tilde k_\omega=(\dots,\tilde k_{i,\omega},\dots)$ might not constitute elements in $\G_\omega$, in fact they are of no significance.
We have to work with two more fibered categories, assuming that a gauged family and a set $\Pi$ of multiplyable subsets of $\Lambda$ are fixed: For every $W(\f_{p^{2r}})$-scheme $X$ we define $\gB_X^n$ to be the groupoid of data $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi}\})$ where
- $\S_i$ is a $n$-truncated $3n$-display with $\GU(\V_i/W(\f_{p^{2r}}),\Psi_i)$-structure over $X$, for every $i\in\Lambda$.
- $s_\pi:\O_{L_\pi}\rightarrow\End(\dot\bigotimes_{i\in\pi}\S_i)$ is a $*$-preserving ring homomorphism for every $\pi\in\Pi$, where $\O_{L_\pi}$ is as in ,
This is a again a stack over $W(\f_{p^{2r}})$ for the fpqc topology (the isomorphisms in $\gB_X^n$ are tuples $\zeta_i:\S'_i\stackrel{\cong}{\rightarrow}\S_i$ such that the diagrams:
$$\begin{CD}
\label{desingI}
\dot\bigotimes_{i\in\pi}\S_i@<{\dot\bigotimes_{i\in\pi}\zeta_i}<<\dot\bigotimes_{i\in\pi}\S'_i\\
@A{s_\pi(a)}AA@A{s_\pi(a)}AA\\
\dot\bigotimes_{i\in\pi}\S_i@<{\dot\bigotimes_{i\in\pi}\zeta_i}<<\dot\bigotimes_{i\in\pi}\S'_i
\end{CD}$$
commute for all $\pi\in\Pi$ and $a\in\O_{L_\pi}$). Analogously we define $\gC_X$ to be the groupoid of data $(\{M_i\}_{i\in\Lambda},\{m_\pi\}_{\pi\in\Pi}\})$ where
- $M_i\in\ect_X$ and $\V_i\in\ect_{\Spec W(\f_{p^{2r}})}$ have the same constant rank for every $i\in\Lambda$.
- $m_\pi:\O_{L_\pi}\rightarrow\End_{\O_X}(\bigotimes_{i\in\pi}M_i)$ is a $W(\f_{p^{2r}})$-linear ring homomorphism for every $\pi\in\Pi$, where $\O_{L_\pi}$ is as in .
The significance of $\gC$ stems from the fact, that for every $p$-adically separated and complete ring $R$, there are natural forgetful functors $$\gsy_{\sigma,R}^n:\gB_R^n\rightarrow\gC_{W_{n+1}(R)},$$ which are induced by passage to the $\sigma$-eigenspaces of the underlying modules (in the sense of subsection \[beispiel\]) for all $n$ including $\infty$. The significance of $\gB^n$ stems from the product functor $$\label{akwardV}
\prod_{i\in\Lambda}\fx^{\bd,\bj_i}(\rho_i):\B^{n+w}(\G,\mu)\times_{W(\f_{p^f})}\f_{p^f}
\rightarrow\gB^n\times_{W(\f_{p^{2r}})}\f_{p^f},$$ which is naturally induced according to . Let us now fix a $\f_{p^f}$-scheme $\gX$ and an object $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi}\})$ of $\gB_\gX^n$. By a flexibilisator for $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi}\})$ over an $\gX$-scheme $X$ we mean a $2$-commutative diagram $$\begin{CD}
\label{akwardVI}
X@>>>\gX\\
@VVV@VVV\\
\B^{n+w}(\G,\mu)\times_{W(\f_{p^f})}\f_{p^f}@>>>\gB^n\times_{W(\f_{p^{2r}})}\f_{p^f}
\end{CD},$$ and we will simply say that $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi})$ becomes flexed over $X$ if there exists at least one such diagram. If $X=\Spec R$ and $n=\infty$, then the following holds:
- If $R$ is separated with respect to the $M$-adic topology, where $M=\{x\in R|x^p=0\}$, then no flexibilisator has non-trivial automorphisms,
- whenever $R$ is perfect or noetherian and reduced then flexibilisators are unique up to unique isomorphism.
It is practical to have the following principle as a reference:
\[construct\] As above, we fix a gauged family of representations $\{\rho_i\}_{i\in\Lambda}$ and a strict set of subsets $\Pi$. Let $\f_{p^f}\subset A\subset B$ be an extension of reduced noetherian rings and let $C$ be a faithfully flat noetherian reduced $A$-algebra. Assume that $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi}\})$ is an object of $\gB_{\Spec A}$ which becomes flexed over $B$ and over $C$. Suppose that one of the following two assumptions is in force:
- The condition is satisfied, $B$ is essentially finitely generated over $A$, and $$\label{primeII}
C\otimes_A\sqrt[p]A=\sqrt[p]C$$ holds (as $C$-algebras).
- Every maximal point of $\Spec B$ is mapped to a maximal point of $\Spec A$ and belongs to a separable residue field extension, and $C$ is essentially finitely generated over $A$.
Then $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi}\})$ is already flexed over $A$.
Choose preimages $\P/B$ and $\T/C$ of $\S_i\times_AB$ and $\S_i\times_AC$ under the product functor $\prod_i\fx^{\bd,\bj_i}(\rho_i,\dots)$. Notice that in the (i)-case the maximal points of $\Spec C$ are mapped to the maximal points of $\Spec A$ and belong to separable residue field extensions, by a theorem of MacLane, hence the $A$-algebra $B\otimes_AC$ is noetherian and reduced in both cases, so that $\P\times_B(B\otimes_AC)\cong\T\times_C(B\otimes_AC)$ canonically. Now use $\P/B$ to construct a descent datum for $\T\times_C(B\otimes_AC)$ relative to the fpqc morphism $B\rightarrow(B\otimes_AC)$ and use proposition \[sepIII\] to restrict it to a descent datum for $\T$ relative to the fpqc morphism $A\rightarrow C$.
\[constructI\] In the same vein one would obtain a similarily satisfactory variant of the part (i) of corollary \[construct\] if $A\rightarrow C$ was required to be a normal morphism with $A$ integrally closed in $B$. See [@egaiv Proposition (6.14.4)].\
Suppose a ring $B$ was reduced and faithfully flat, essentially finitely generated and generically separable (in the sense of part (ii) of the previous corollary) over some noetherian subring $A$. It follows that every flexibilisator over $B$ comes from $A$.
We have used several times that property is preserved under flat base change to $A$-algebras $C$ with property . The same is true for the morphism $A\rightarrow\prod_\gm A_\gm$ (which may or may not satisfy ). This leads to the following analog of part (i) of \[construct\]:
\[chaotic\] Suppose that $A$ is a reduced noetherian $\f_{p^f}$-algebra which happens to be equal to $K(A)\cap\sqrt[p]A$. Then the previous assumptions on $\{\rho_i\}_{i\in\Lambda}$ and $\Pi$ imply that an object $(\{\S_i\}_{i\in\Lambda},\{s_\pi\}_{\pi\in\Pi}\})$ of $\gB_{\Spec A}$ is flexed over $A$ if and only if the same holds over all localisations $A_\gm$.
The whole point is that $C=\prod_\gm A_\gm$ is faithfully flat over $A$ and that $$(\prod_\gm A_\gm)\otimes_A(\prod_\gm A_\gm)\subset(\prod_\gm K(A))\otimes_{K(A)}(\prod_\gm K(A))$$ satisfies condition : Given that the inclusion $\prod_\gm A_\gm\subset\prod_\gm K(A)$ satisfies this already by assumption, this fact is easily implied by the fact that the base change by means of the morphism $A\rightarrow\prod_\gm A_\gm$ preserves condition , as does $A\rightarrow\prod_\gm K(A)$.
Windows with additional structure, part II
==========================================
Fix a $\Phi$-datum $(\G,\mu)$. The following is an analog of the main result in [@zink1], for our category $\bB'(\G,\mu)$:
\[wintwo\] Let $(A,J,\tau)$ be a frame over $W(\f_{p^f})$. The canonical functor $$\hat\delta:\hat\CAS'_{A,J}(\G,\mu)\rightarrow\hat\CAS'_{W(A),W(J)+I(A)}(\G,\mu)\cong\bB'_{A/J}(\G,\mu)$$ that is induced from Cartier’s diagonal map $A\rightarrow W(A)$ is an equivalence of categories.
In the torsionfree situation the ghost map is injective, and it is well known that one can use the Frobenius lift to give a precise description of the image: $$w:W(A)\stackrel{\cong}{\rightarrow}\{(w_0,\dots)\in A^{\n_0}|w_{i+1}\equiv\tau(w_i)\pmod{p^{i+1}}\},$$ This implies the congruence $^Fx\equiv\tau(x)\mod W(pA)$, recall that Witt vectors $x,x'\in W(A)$ have the same image in $W(A/p^sA)$ if and only if the components of their respective ghost images $w,w'\in A^{\n_0}$ satisfy $w_i\equiv w_i'\pmod{p^{i+s}}$, cf. [@zink2 Lemma 4].\
The functor is fully faithful. Now consider some $U\in\Ob_{\hat\CAS'_{W(A),W(J)+I(A)}(\G,\mu)}$, i.e. some $U\in\H(A)$ which satisfies the nilpotence condition. Observe that $^FU\equiv\tau(U)\mod W(pA)$ holds. Part (ii) of lemma \[GMZCF\] implies the existence of some $\hat\I_{pA}(A)\ni h\equiv1\mod W(pA)$ with $$^FU=h^{-1}\tau(U){^{\hat\Phi_{pA}}}h.$$ By a recursion argument one easily finds $h=\tau(\hat m){^F\hat m^{-1}}$, for some $\hat m\in\hat\I_{pA}(A)$ and thus $$^F(\hat m^{-1}U{^{\hat\Phi_{pA}}}\hat m)=\tau(\hat m^{-1}U{^{\hat\Phi_{pA}}}\hat m).$$ According to the introductory remarks of section \[twisted\] and the generalization of the canonical splitting due to [@zink2 1.4] one can find a factorization $\hat m=mm^+$ where $m\in\I(A)$, and $m^+\in\gg^+\otimes_{W(\f_{p^f})}pA$. Write $U':=m^{-1}U{^{\Phi}m}\in\Ob_{\hat\CAS'_{A,J}(\G,\mu)}$, this is the required preimage of $U$, as $m^+$ lies in the image of $\delta$.
\[Witt\] The same argument shows that there exists an equivalence $\CAS'_A(\G,\mu)\rightarrow\CAS'_{W(A),I(A)}(\G,\mu)\cong\bB'_A(\G,\mu)$.
Algebraic Connections
---------------------
Let $X$ be a smooth $S$-scheme with sheaf of Kähler differentials $\Omega_{X/S}^1$. As usual we extend the differentiation $d:\O_X\rightarrow\Omega_{X/S}^1$ to an endomorphism (still denoted by $d$) on the exterior algebra $\bigoplus_m\Omega_{X/S}^m$ in such a way that the Leibniz rule $d(\eta_1\wedge\eta_2)=d(\eta_1)\wedge\eta_2+(-1)^{\deg(\eta_1)}\eta_1\wedge d(\eta_2)$ holds, so that in particular $d\circ d=0$. By a connection (relative to $S$) on some $\F\in\ect_X$ one means a map $\nabla:\F\rightarrow\Omega_{X/S}^1\otimes_{\O_X}\F$ satisfying an analogous Leibniz rule, and again this extends canonically to an endomorphism on the whole of $\bigoplus_m\Omega_{X/S}^m\otimes_{\O_X}\F$, the sheaf of so-called $\F$-valued differential forms on $X$, see [@deligne2]. Sections killed by $\nabla$ are called horizontal, and the map $\nabla\circ\nabla$ can be written as $\eta\mapsto R\wedge\eta$, where $R$ is called the curvature of $\nabla$, it is a global section in the $\F\otimes_{\O_X}\check\F$-valued $1$-forms on $X$ (in fact a horizontal one). If $R$ vanishes, then one calls $\nabla$ integrable, write $\yst_{X/S}$ for the class of vector bundles on $X$, equipped with an integrable connection, relative to $S$. The sets of horizontal $\O_X$-module homomorphisms define an additive rigid $\otimes$-category structure on $\yst_{X/S}$ (and so they would without the integrability condition).\
Suppose, for instance, that $\F\in\ect_X$ is isomorphic to $\O_X^h$. The connections on $\F$ can be written in the form $d+D$, for some $D\in\Gamma(X,\Omega_{X/S}^1\otimes_{\O_X}\F\otimes_{\O_X}\check\F)$, furthermore the horizontal isomorphisms from some $(\O_X^h,d+E)$ to $(\O_X^h,d+D)$ are those $u\in\GL(h,\Gamma(X,\O_X))$, such that $$\label{synthI}
uEu^{-1}-d(u)u^{-1}=D$$ Finally, notice that $R=d(D)+D\wedge D$, furthermore the above implies $$\label{synthII}
uQu^{-1}=R$$ where $Q$ is the curvature of $d+E$. From now on we assume that $S$ is a scheme over a Dedekind ring $B$, and that $\G$ is a smooth affine group scheme over $B$. By an integrable connection $\nabla$ on some locally trivial principal homogeneous space $P$ over $X$, we simply mean a $\yst_{X/S}$-valued rigid $\otimes$-functor factorizing the twisted fiber functor $\varpi_P$ (see e.g. [@rivano VI.1.2.3.1]). Suppose, for instance that $P$ is trivial, so that $\varpi_\P\cong\O_X\otimes_B\varpi_0^\G$. It is immediate that every $D\in\Omega_{X/S}^1\otimes_B\gg$ gives rise to a connection on each $\varpi_\P(\rho)$, namely $$\label{synthV}
\nabla_D(\rho):=d+\rho^{der}(D),$$ clearly one has $\nabla_D(\rho_1\otimes_B\rho_2)=\nabla_D(\rho_1)\otimes_{\O_X}\nabla_D(\rho_2)$ for any $\rho_1,\rho_2\in\bRep_0(\G)$, and similarly for duality, furthermore the curvature of $\nabla_D(\rho)$ is equal to $\rho^{der}(d(D)+\frac{[D,D]}2)$, and in the sequel we will refer to the expression $d(D)+\frac{[D,D]}2$ as the curvature of the $\gg$-valued $1$-form $D$. Now assume that $E$ and $D$ are both $\gg$-valued $1$-forms with vanishing curvature, and consider some element $u\in\G(X)$. The resulting automorphism of $\varpi_\P(\rho)$ is an isomorphism $(\varpi_\P(\rho),\nabla_E(\rho))\rightarrow(\varpi_\P(\rho),\nabla_D(\rho))$ in $\yst_{X/S}$ if and only if $$\rho^{der}(\Ad(u)(E)-D)=d((\rho(u))\rho(u)^{-1}.$$ The right-hand side of this equation can be rewritten in an aesthetically pleasing way: Consider $\eta_\rho=d(f_1)\wedge f_2$ where $f_1$ is the inclusion of $\GL(\varpi_0^\G(\rho)/B)$ into $\End_B(\varpi_0^\G(\rho))$, and $f_2(g)=f_1(g^{-1})$. It is worthy of remark that the $1$-form $\eta_\rho$ is right-invariant, so let $\eta_\G$ be the canonical, so called Cartan-Maurer, right-invariant $\gg$-valued $1$-form on $\G$. One has $\rho^{der}(\eta_\G)=\eta_\rho\circ\rho$. Hence $u$ is a horizontal, provided that $$\Ad(u)E-D=\eta_\G\circ u$$ (where $\circ$ denotes the pull-back of a differential form by means of a morphism). Connections are related to displays:
\[synthIV\] Let $\P$ be a display with $\G$-structure over a smooth $W_\nu(k)$-scheme $X$, where $\nu$ is a positive integer and $k$ is a perfect field containing $\f_{p^f}$. Then there exists a canonical connection $\nabla_\P$ on the composite of the $\otimes$-functors: $$\bRep_0(\G\times W(\f_{p^f}))\rightarrow\bRep_0(\G^-)\stackrel{\varpi_\P}{\rightarrow}\ect_X.$$ The formation of $\nabla_\P$ is functorial in $\P$ and it commutes with base change in the following sense: Whenever there is a diagram $$\begin{CD}
X'@>>>X\\
@VVV@VVV\\
\Spec W_{\nu'}(k')@>>>\Spec W_\nu(k)
\end{CD},$$ where $\nu'\leq\nu$ and $k'$ is a perfect field extension of $k$, the canonical isomorphism $\O_{X'}\otimes_{\O_X}\varpi_\P\cong\varpi_{\P'}$ is a horizontal one, if the left-hand side is endowed with $\nabla_\P$ and the right-hand side is endowed with $\nabla_{\P'}$, where $\P':=\P\times_XX'$.
It clearly suffices to consider an affine basis $X=\Spec R$, and in fact the part (ii) if lemma \[etale\] tells us that it is sufficient to consider a banal display, so that the $\otimes$-functor in question is just $R\otimes_{W(\f_p)}\omega_0^\G$. So choose a $p$-adically formally smooth, separated and complete ring $A$ with $A/p^\nu A\cong R$, such a ring is unique up to non-unique isomorphism. Let $\hat\Omega_A^1$ be the formal Kähler differentials of $A$. Now one can reduce the construction of $\nabla_\P$ from the following, slightly stronger problem: Given any $U\in\G(W(A))$ (regarded as a banal display with an auxiliary lift to $A$), find an associated element $D_U\in\hat\Omega_A^1\otimes\gg$, such that the following holds:
- For all $h\in\hat\I(A)$ we have $\Ad(w_0(h))(D_{U'})-D_U=\eta_\G\circ w_0(h)$, where $w_0$ denotes the $0$th component of the ghost map and $U'$ stands for $h^{-1}U{^{\hat\Phi}h}$
- $d(D_U)+\frac{[D_U,D_U]}2=0$
Let $\theta_A:A\rightarrow A\oplus\hat\Omega_A^1$ be the map $x\mapsto x+d(x)$, and let $U'$ be the image of $U$ under the map $W(\theta_A)$. Notice that $\hat\Omega_A^1$ is a pd-ideal in a $p$-adically separated and complete ring, so that the part (ii) of lemma \[GMZCF\] grants the existence of a unique $k\in\H(A\oplus\hat\Omega_A^1)$ with $k^{-1}U{^{\hat\Phi_\ga}k}=U'$ and $1\equiv k\mod{\hat\Omega_A^1}$ (this particular pd-ideal contains no powers of $p$, so that one might rather look at the pd-ideals $\ga:=p^nA\oplus\hat\Omega_A^1$ in the first place, and afterwards use the uniqueness of $k$ in order to pass to their intersection). That equation can be rewritten as: $$U{^{\hat\Phi_\ga}k}U^{-1}=k(U'U^{-1}),$$ and given that all of the elements $U{^{\hat\Phi_\ga}k}U^{-1}$, $k$, and $U'U^{-1}$ are $\equiv1\mod{\hat\Omega_A^1}$ the above problem is solved by $D_U:=w_0(k)$, which can be regarded as an element of $\hat\Omega_A^1\otimes\gg$. The vanishing of the curvature is postponed until lemma \[synthVII\].
Suppose that $A$ is a $p$-adically separated, complete, and formally smooth $W(\f_{p^f})$-algebra with module of formal Kähler differentials $\hat\Omega_A^1$. If $\tau$ is a Frobenius lift, then so is the map $$A\oplus\hat\Omega_A^1\rightarrow A\oplus\hat\Omega_A^1;x+\sum_ix_idy_i\mapsto\tau(x)+\sum_i\tau(x_i)d\tau(y_i),$$ which we continue to denote by $\tau$ (here notice that $py_i^{p-1}dy_i=dy_i^p\equiv d\tau(y_i)\pmod p$). We would like to translate some of the previous techniques into the language of windows, so suppose that $U\in\G(A)$ satisfies the nilpotence condition, define $U'\in\G(A\oplus\hat\Omega_A^1)$ as above, and suppose for a while that the equation $$U{^{\hat\Phi_\ga^-}k}U^{-1}=k(U'U^{-1})$$ held for some element $\hat\G_\ga^-(A\oplus\hat\Omega_A^1)\ni k\equiv1\mod{\hat\Omega_A^1}$, where $\ga=pA\oplus\hat\Omega_A^1$. Plugging $U{^{\hat\Phi_\ga^-}k}U^{-1}$, $k$, and $U'U^{-1}$ into Cartier’s diagonal map $\hat\delta$, would reveal that the construction of the above $\gg$-valued $1$-form reads: $$D_{\hat\delta(U)}=w_0(\hat\delta(k))=k$$ (regarded as an element in the kernel $\ker(\hat\G_\ga^-(A\oplus\hat\Omega_A^1)\rightarrow\hat\G_\ga^-(A))=\hat\Omega_A^1\otimes\gg$.) The existence and uniqueness of that $\gg$-valued formal $1$-form, can be justified independently of lemma \[GMZCF\]:
\[synthVII\] Fix $U\in\G(A)$, where $(A,pA,\tau)$ is a formally smooth frame over $W(\f_{p^f})$. Let us write $\phi^*$ for the $\tau$-linear map on $\bigoplus_{m\neq0}\hat\Omega_A^m\otimes\gg$ which is given by the compositon of $\tau\otimes\id$ with $\id\otimes\Ad(\mu(p)^{-1})$ (notice that “$m=0$” is omitted). If $U$ satisfies the nilpotence condition, then there exists a unique element $D_U\in\hat\Omega_A^1\otimes\gg$ with $\Ad(U)({^{\phi^*}D_U})-D_U=\eta_\G\circ U$, furthermore, the curvature $R_U=d(D_U)+\frac{[D_U,D_U]}2\in\hat\Omega_A^2\otimes\gg$ vanishes.
This is analogous to [@habil Lemma 2.8]. The nilpotence condition implies that $D\mapsto\Ad(U)({^{\phi^*}D})-\eta_\G\circ U$ is a contractive map for the $p$-adic topology, so there exists a unique fixedpoint $D_U$. We have $R_U=\Ad(U)({^{\phi^*}R_U})$, again this has a unique solution, namely $R_U=0$.
Formal Connections
------------------
We want to look at $A:=W(k_0)[[t_1,\dots,t_d]]$, where the field $k_0$ is an algebraically closed or an algebraic extension of $\f_{p^f}$ and $d:=\rk_{W(\f_{p^f})}\gg^+$. Let us fix the Frobenius lift determined by $\tau(t_i):=t_i^p$. Any display $\P_0$ with $\G$-structure over $k_0$ is automatically banal, and hence represented by some $U_0\in\G(W(k_0))$. Choose a $W(\f_{p^f})$-basis $\{N_1,\dots,N_d\}$ of $\gg^+$. Assume that $\G$ is reductive and that $\mu$ fulfills the criterion in remark \[supercompact\]. Then we have $U_1:=e^+(t_1N_1+\dots+t_dN_d)U_0\in\Ob_{\CAS'_A(\G,\mu)}$, so that the element $U_{uni}:=\hat\delta(U_1)\in\Ob_{\B'_A(\G,\mu)}$ represents the universal formal mixed characteristic deformation $\P_{uni}$ of $\P_0$. This object has a rich amount of symmetry: For every $\gamma\in\Aut(\P_0)$, there exists a unique pair $(h_\gamma,s_\gamma)\in\I(A)\rtimes\Aut(A/W(k_0))$ with:
- $s_\gamma(U_{uni})=h_\gamma^{-1}U_{uni}{{^\Phi}h_\gamma}$
- $h_\gamma$ is a lift of $\gamma\in\I(k_0)$
We will be particularly interested in the truncated version $(u_\gamma,s_\gamma)\in\G^-(A)\rtimes\Aut(A/W(k_0))$, where $u_\gamma:=w_0(h_\gamma)$. Let $D_1$ be the $1$-form that corresponds to $U_1$, as has been pointed out before. Dwork’s trick provides us with an element $\Theta\in\G(K(k_0)\{\{t_1,\dots,t_d\}\})$ satisfying: $$\begin{aligned}
&&\Theta(0,\dots,0)=1\\
&&\Theta^{-1}U_1{^F\mu(p)^{-1}}\tau(\Theta)=U_0{^F\mu(p)^{-1}}\\
&&\eta_\G\circ\Theta=-D_1\end{aligned}$$ (the last equation has to be interpreted in $K(k_0)\{\{t_1,\dots,t_d\}\}\otimes_A\hat\Omega_A^1=:\hat\Omega_{K(k_0)\{\{t_1,\dots,t_d\}\}}^1$). Note the following consequence: An arbitrary element $(u,s)\in\G(K(k_0)\{\{t_1,\dots,t_d\}\})\rtimes\Aut(A/W(k_0))$ lies in $\Int(\Theta)(\G(K(k_0))\times\Aut(A/W(k_0)))$ if and only if $$\label{synthIII}
\Ad(u)s(D_1)-D_1=\eta_\G\circ u,$$ here the right-hand side is again the image of $\eta_\G$ under the map $\Omega_\G^1\rightarrow\hat\Omega_{K(k_0)\{\{t_1,\dots,t_d\}\}}^1$, and the left-hand side utilizes the natural action of $\Aut(A/W(k_0))$ on $\hat\Omega_{K(k_0)\{\{t_1,\dots,t_d\}\}}^1$, from the left. Let us call the element $(u,s)$ horizontal with respect to $D_1$ if holds. The canonicity of $D_1$ shows that the elements $(u_\gamma,s_\gamma)$ are indeed horizontal with respect to $D_1$, for all $\gamma\in\Aut(\P_0)$.\
After these preparatory remarks we are able to state and prove the important technical fact that $D_1$ tends to be complicated as can be. More specifically, write $G$ and $G^+$ for the generic fibers of $\G$ and $\G^+$ and let $Z\subset G$ be the center, and let $G^{spc}\subset G$ be the smallest algebraic subgroup containing the commutator of the $K(\f_{p^f})$-groups $G^+$ and $G\times_{K(\f_p)}K(\f_{p^f})$, then we have $G^+\subset G^{spc}\times_{K(\f_p)}K(\f_{p^f})$, and $G^{spc}\triangleleft G$. In the following result we call any $b\in B_{\f_q}(G)$ basic if $b{^F}b\cdots{^{F^{s-1}}}b\in Z(K(\f_p))$ for a suitable $s\in\n$.
\[maximalholonomy\] Under the assumptions and with the notations at the beginning of this subsection we have $D_1\in\hat\Omega_A^1\otimes_{K(\f_p)}\Lie(G^{spc})$.\
Let $\rho:\G\rightarrow\GL(\V/W(\f_p))$ be an object in $\bRep_0(\G)$ and recall that there is an associated $(A,\tau)$-module $\varpi_{\P_{uni}}(\rho|_{\G^-})=\sy(\rho,U_1)$. Let $\nabla:\sy(\rho,U_1)\rightarrow\hat\Omega_A^1\otimes_A\sy(\rho,U_1)$ be derived from the formal $1$-form $D_1$, by means of the formal analogue of , and assume that the following holds:
- $k_0$ is a finite field.
- $\barb_{\P_0}\in B_{k_0}(G)$ is a basic element.
- $\V$ has no non-zero $G^{spc}$-invariants.
Then $M:=\q\otimes\sy(\rho,U_1)$ has no $\nabla$-stable $\q\otimes A$-submodules of rank one.
By the right-invariance of the Cartan-Maurer form one has: $\eta_\G\circ U_1=\eta_\G\circ e^+(t_1N_1+\dots+t_dN_d)\in\hat\Omega_A^1\otimes_{K(\f_p)}\Lie(G^{spc})$. Hence the $p$-adically contractive map $D\mapsto\Ad(U_1)({^{\phi^*}D})-\eta_\G\circ U_1$ preserves $\hat\Omega_A^1\otimes_{K(\f_p)}\Lie(G^{spc})$, because $\Lie(G^{spc})$ is a Lie ideal.\
It does no harm to replace $k_0$ by a finite extension, and hence we can assume $\Aut(\P_0\times_{k_0}k_0^{ac})=\Aut(\P_0)$, and that $Z$ splits over $K(k_0)$, i.e. there exists a finitely generated $\Gal(K(k_0)/K(\f_p))$-module $X$, whose Cartier-dual is $Z$. Now let $\F$ be a $\nabla$-stable submodule contained in $M=\q\otimes A\otimes_{W(\f_p)}\V$. The $Z$-action on $\q\otimes\V$ induces a decomposition $M=\bigoplus_{\lambda\in X}M_\lambda$, we will proceed with the study of the $\nabla$-stable projections of $\F$ onto each $M_\lambda$, which we denote by $\F_\lambda$. The assumptions on $\P_0$ and $k_0$ imply that there exists some $s\in\n$ such that the $s$-fold iteration of $F^\flat$ preserves $M_\lambda$ and acts on $W(k_0)\otimes_AM_\lambda$ by multiplication with a scalar. It follows that $(F^\flat)^{\circ s}$, being horizontal, must preserve $\F_\lambda$, simply because it does that at the specific point $t_1=\dots=t_d=0$. We can deduce that each non-zero $\F_\lambda$ contains a non-zero horizontal section.\
The proof of the lemma is complete if we show the vanishing of the $K(k_0)$-vector space $H=\ker(\nabla)$. Let $k_1$ be any algebraically closed field containing $A/pA$. Cartier’s diagonal map identifies $A$ with a subring of $W(k_1)$, so let us write $b_1\in G(K(k_1))$ for the image under that map of the element $U_1{^F\mu(p)^{-1}}$ and let us write $\nu_1:\q\otimes\g_m\times K(k_1)\rightarrow G\times_{K(\f_p)}K(k_1)$ for the slope homomorphism of $b_1$. Every $x\in K(k_1)\otimes_{K(k_0)}H$ is surely fixed by $\nu$, as the inclusion $A\hookrightarrow W(k_1)$ preserves the Frobenius lifts.\
The automorphism group $\Aut(A/W(k_0))$ acts from the left on both $\G(A)$ and $M$, so the semidirect product $\G(A)\rtimes\Aut(A/W(k_0))$ acts thereon too. The horizontal elements preserve $H$, moreover $\Aut(\P_0)$ is a (rather small) compact open subgroup in an inner form of $\G$, by [@rapoport Proposition 1.12], and consequently Zariski-dense. We deduce that $K(k_1)\otimes_{K(k_0)}H$ must be invariant under the whole conjugacy class of $\nu_1$, and therefore it is invariant under all elements of $G^{spc}(K(k_1))$, by corollary \[Hodgepoint\].
We finish this subsection with a quantitative version of corollary \[obstII\]:
\[obstIII\] Let $N$ be a discretely valued complete field containing $K(\f_{p^f})$, and let the residue field $k$ of its ring of integers $\O_N$ be perfect of characteristic $p$. Let $\P$ be a banal display with $\G$ structure over $\O_N$, and let $\P_0:=\P\times_{\O_N}k$ be its special fiber. Then there exists an element $m\in\G(N)$ such that every $h\in\Aut(\P)\hookrightarrow\I(\O_N)$ satisfies the equation $$w_0(h)=\Int(m)(h_0),$$ of elements in $\G(N)$, where $h_0\in\Aut(\P_0)\hookrightarrow\I(k)\subset\G(W(k))$ stands for the special fiber of $h$ (N.B.: if $N=W(k)$ the element $m=1$ satisfies this, however in the general case $m$ might not even be in $\G(\O_N)$).
We may assume that $k$ is algebraically closed. Notice that $m$ is independent of the choice of the banality, even though the two inclusions $\Aut(\P)\hookrightarrow\I(\O_N)$ and $\Aut(\P_0)\hookrightarrow\I(k)$ do depend on it. Now recall the aforementioned universal formal mixed characteristic deformation $\P_{uni}\in\B_{W(k)[[t_1,\dots,t_d]]}(\G,\mu)$ of $\P_0$, and let $\kappa:W(k)[[t_1,\dots,t_d]]\twoheadrightarrow\O_N$ be the classifying morphism to $\P$. Recall the canonical embedding of $\Aut(\P_0)$ into the intersection: $$\begin{aligned}
&&\G^-(W(k)[[t_1,\dots,t_d]])\rtimes\Aut(W(k)[[t_1,\dots,t_d]]/W(k))\\
&&\cap\Int(\Theta)(\G(K(k))\times\Aut(W(k)[[t_1,\dots,t_d]]/W(k)),\end{aligned}$$ of subgroups in $\G(K(k)\{\{t_1,\dots,t_d\}\})\rtimes\Aut(W(k)[[t_1,\dots,t_d]]/W(k))$. Notice also that the image of any element in $\Aut(\P_0)$ in the ambient group $\I(k)\subset\G(K(k))$ can be recovered as the $\G(K(k))$-component in the second of the two groups above, for example by applying the lemma \[wintwo\] to the specific lift $t_1=\dots=t_d=0$. Moreover, the subgroup $\Aut(\P)\subset\Aut(\P_0)$ is mapped into: $$\G^-(W(k)[[t_1,\dots,t_d]])\rtimes\Aut_{\O_N}(W(k)[[t_1,\dots,t_d]]/W(k)),$$ because it preserves the augmentation $\kappa$. It remains to look at the homomorphic image of $\Aut(\P)$ under the canonical homomorphism $$\G(K(k)\{\{t_1,\dots,t_d\}\})\rtimes\Aut_{\O_N}(W(k)[[t_1,\dots,t_d]]/W(k))\rightarrow\G(N),$$ to find that the image of $\Theta$ is the group element $m$ that we have sought for.
Moduli spaces of abelian varieties with mock $G^0$-structure
============================================================
Let $(G,X)$ be a Shimura datum in the sense of the axioms [@deligne3 (2.1.1.1)-(2.1.1.5)], let us assume that the rational weight homomorphism $w_X:\g_m\rightarrow G$ is injective, write $G^1\subset G$ for the complement to its image, and assume that this group can be written as $\Res_{L^+/\q}G^0$, and fix a totally imaginary quadratic extension $L/L^+$. It is convenient to fix a specific element $h\in X$ and we also want to choose $\sqrt{-1}\in\s(\r)$ once and for all. Notice that $h(\sqrt{-1})\in G^1(\r)$ and $w_X(-1)\in G^0(L^+)$. By a polarization on some $(V,\rho)\in\bRep_0(G^0\times_{L^+}L)$, we mean a $G^0$-equivariant $L$-linear map $\Psi:L\otimes_{*,L}V\rightarrow\check V$ such that:
- $(\tr_{L/\q}\Psi)(\rho(h(\sqrt{-1}))x,y)$ is positive definite on $\r\otimes V$
- $\Psi(\rho(w_X(-1))x,y)^*=\Psi(y,x)$.
By slight abuse of notation we will also write $\rho:G^0\rightarrow\UL(V/L,\Psi)$ to denote the corresponding group homomorphism, here it is understood that $\UL(V/L,\Psi)$ represents the group functor $$R\mapsto\{\gamma\in\GL_{L\otimes_{L^+}R}(V\otimes_{L^+}R)|
\Psi(\gamma x,\gamma y)=\Psi(x,y)\,\forall x,y\in V\otimes_{L^+}R\}.$$ Notice that the tensor product of polarized representations carries a natural polarization. Let us say that $\rho$ is even (resp. odd) if it maps the group element $w_X(-1)$ to $1$ (resp. to $-1$). We call $(G,X)$ ’poly-unitary’ if there exist:
- a family of odd representation $(V_i,\rho_i)$, indexed by some set $\Lambda$, and each endowed with a polarization $\Psi_i$, and
- a family of $G^0$-invariant $L$-$*$-subalgebras $L_\pi\subset\End_L(\bigotimes_{i\in\pi}V_i)$, indexed by some set $\Pi$ of index subsets $\pi\subset\Lambda$ of odd cardinality, such that
- $G^0$ is the stabilizer of $\bigcup_{\pi\in \Pi}L_\pi$.
See part \[hilfssaetze\] of the appendix for a couple of examples. In this scenario we write $G_i$ for the subgroup generated by $G_i^1:=\Res_{L^+/\q}\UL(V_i/L,\Psi_i)$ and $\g_m$ (i.e. the centre of $\GL(V_i/\q)$), and we write $\varrho_i:G\rightarrow G_i$ for the extension of $\Res_{L^+/\q}\rho_i$ with $\varrho_i(w_X(z))=z^{-1}$. We call our poly-unitary structure ’unramified at $p$’ if $L$ is unramified at $p$, and if there exist hyperspecial subgroups $K_{i,p}$ of $G_i(\q_p)$ such that $$\label{twistingII}
K_p=\bigcap_{i\in\Lambda}\varrho_i^{-1}(K_{i,p})$$ is a hyperspecial subgroup of $G(\q_p)$. Let us denote the composition $\s\stackrel{h}{\rightarrow}G\times\r\stackrel{\varrho_i}{\rightarrow}G_i\times\r$ by $h_i$, it gives rise to Hodge decompositions $V_{i,\iota}=\bigoplus_{p+q=-1}V_{i,\iota}^{p,q}$, where $V_{i,\iota}=\c\otimes_{\iota,L}V$ stands for the eigenspace, and where $\iota$ runs through the set $L^{an}=\Spec L(\c)=\Hom(L,\c)$ of embeddings of $L$ into $\c$. Let us set $[a_{i,\iota},b_{i,\iota}]$ for the smallest intervals of integers such that these decompositions are of type $\{(-b_{i,\iota},b_{i,\iota}-1),\dots,(-a_{i,\iota},a_{i,\iota}-1)\}$. The lengths $w_{i,\iota}:=b_{i,\iota}-a_{i,\iota}$ of theses intervals are particularly important invariants, and their maximum $w=\max\{w_{i,\iota}|i\in\Lambda,\,\iota\in L^{an}\}$ will be called the width of the poly-unitary structure. Finally notice that $\overline V_{i,\iota}^{p,q}=V_{i,\iota\circ*}^{q,p}$, so that $b_{i,\iota}=1-a_{i,\iota\circ*}$. We also need to introduce certain combinatorial data: Observe that $L^{an}$ carries a natural left $\Gal(R/\q)$-action commuting with the complex conjugation which could be viewed as acting from the right, the subfield $R$ stands for the normal closure. Fix an element $\vartheta\in\Gal(R/\q)$. In this global context we proceed with the following variant of definition \[qflex\]:
\[vqflex\] By a $\vartheta$-multidegree we mean a function $\bd^+:L^{an}\rightarrow\n_0$ with $\bd^+(\vartheta\circ\iota)\leq\bd^+(\iota)+1$ and $\bd^+(\iota\circ*)=\bd^+(\iota)$. By a $\vartheta$-poly-gauge for $\{h_i\}_{i\in\Lambda}$ we mean a family of functions $\bj_i:L^{an}\rightarrow\z$, together with a $\vartheta$-multidegree $\bd^+$ such that the function $\bd(\iota):=\vartheta^{-\bd^+(\iota)}\circ\iota$ satisfies the following properties:
- For every $\iota\in L^{an}$ and $l\in[a_{i,\iota},b_{i,\iota}-1]$ there exists a unique $\kappa\in L^{an}$ such that $\bd(\kappa)=\iota$ and $\bj_i(\kappa)=l$.
- $\bj_i(\iota\circ*)=-\bj_i(\iota)$
- For every given $\pi$ and $\iota$, the cardinality of both sets $\{i\in\pi|\bj_i(\iota)<a_{i,\bd(\iota)}\}$, and $\{i\in\pi|\bj_i(\iota)\geq b_{i,\bd(\iota)}\}$ is at least $\frac{\Card(\pi)-1}2$.
- For every given $\pi$ each $\vartheta$-orbit of $L^{an}$ contains at least one element $\iota$ which satisfies $\bj_i(\iota)\notin[a_{i,\bd(\iota)},b_{i,\bd(\iota)}-1]$ for all $i\in\pi$.
These properties are rather restrictive and they can only be satisfied if the inequalities $$\begin{aligned}
&&\lim_{N\to\infty}\frac{\sum_{k=1}^N\max\{\sum_{i\in\pi}w_{i,\vartheta^k\circ\iota}|\pi\in\Pi\}}N\leq1\\
&&\max\{\sum_{i\in\pi}\lim_{N\to\infty}\frac{\sum_{k=1}^Nw_{i,\vartheta^k\circ\iota}}N|\pi\in\Pi\}<1\end{aligned}$$ hold for every $\iota$. Conversely, in the key case $\Pi=\{\Lambda\}\cup\{\{i\}|i\in\Lambda\}$ it is not difficult to prove that $\vartheta$-poly-gauges exist if and only if one has $\sum_{i\in\Lambda}\lim_{N\to\infty}\frac{\sum_{k=1}^Nw_{i,\vartheta^k\circ\iota}}N<1$. We call a closed point $\gr\in\Spec\O_R$ to be of type $\vartheta$, if $\vartheta$ fixes $\gr$ and induces the absolute Frobenius automorphism on $\O_R/\gr$. If a $\vartheta$-poly-gauge is fixed, then there exist Hodge structures $\tilde V_i$, which are endowed with $L$-operations and $L$-linear polarizations $\tilde\Psi_i$ such that:
- $\dim\tilde V_{i,\iota}^{\tilde p,\tilde q}=\begin{cases}
\sum_{p<-\bj_i(\iota)}\dim V_{i,\bd(\iota)}^{p,q}&(\tilde p,\tilde q)=(-1,0)\\
\sum_{p\geq-\bj_i(\iota)}\dim V_{i,\bd(\iota)}^{p,q}&(\tilde p,\tilde q)=(0,-1)\\
0&\text{otherwise}\end{cases}$
- There exists a similarity $$\label{twistingI}
\a^\infty\otimes\tilde V_i\stackrel{\varepsilon_i}{\rightarrow}\a^\infty\otimes V_i$$ of skew-Hermitian $L\otimes\a^\infty$-modules,
- When calculating in the $\otimes_L$-category of Hodge structures with coefficients in $L$ (see [@habil section 3.1]), one finds that $\tilde V_\pi:=(\bigotimes_{i\in\pi}\tilde V_i)(\frac{1-\Card(\pi)}2)$ is an object of type $\{(-1,0),(0,-1)\}$ for every $\pi\in\Pi$, and by means of $\tilde\Psi_\pi:=\bigotimes_{i\in\pi}\tilde\Psi_i$ it is canonically $L$-linearly polarized as well.
See [@habil section 6.2] for a related consideration, in this scenario we write $\tilde G_\pi$ for the subgroup generated by $G_\pi^1:=\Res_{L^+/\q}\UL(\tilde V_\pi/L,\tilde\Psi_\pi)$ and $\g_m$ (i.e. the centre of $\GL(\tilde V_\pi/\q)$). This sets up a family of canonical PEL-type Shimura data $(\tilde G_\pi,\tilde X_\pi)$, where $\tilde X_\pi$ is the $\tilde G_\pi(\r)$-conjugacy class of Hodge-structures on $\tilde V_\pi$ specified by (iii) above. Notice that their reflex fields are contained in $R$. Moreover, regarding $\tilde V:=\bigoplus_{i\in\Lambda}\tilde V_i$ as a skew-Hermitian module over the $*$-algebra $L^\Lambda=\underbrace{L\oplus\dots\oplus L}_\Lambda$ yields yet another canonical PEL-type Shimura datum $(\tilde G,\tilde X)$, where $\tilde G$ is the $\q$-group of $L^\Lambda$-linear similitudes of $\tilde V$, and $\tilde X$ is the $\tilde G(\r)$-conjugacy class of Hodge-structures on $\tilde V$ specified by (i) above. For later reference we also put $\ge_i\subset\O_L^\Lambda$ for the ideal generated by the idempotent $(1,\dots,1,0,1,\dots,1)$ with the “$0$” in the $i$ th position. Finally observe that there exist canonical morphisms of Shimura data $g_\pi:(\tilde G,\tilde X)\rightarrow(\tilde G_\pi,\tilde X_\pi)$, and hence canonical morphisms of Shimura varieties: $$\label{bekanntI}
{_{\tilde K}M}(\tilde G,\tilde X)\stackrel{g_\pi}{\rightarrow}{_{\tilde K_\pi}M}(\tilde G_\pi,\tilde X_\pi),$$ under the assumption $g_\pi(\tilde K)\subset\tilde K_\pi$ of course. We need to collect further facts on integrality, first notice that a decomposition $\tilde K=\tilde K^p\times\tilde K_p$ for some hyperspecial $\tilde K_p\subset\tilde G(\q_p)$ implies an analogous decomposition for $\tilde K_\pi$ basically by (iii) above, and second we have a canonical extension (cf. [@habil Theorem 4.8]) $$\label{bekanntII}
{_{\tilde K}\U}\stackrel{g_\pi}{\rightarrow}{_{\tilde K_\pi}\U}_\pi,$$ of , where $_{\tilde K}\U$ and $_{\tilde K_\pi}\U_\pi$ are the usual $\O_R\otimes\z_{(p)}$-models of these unitary group Shimura varieties, which are smooth (cf. [@kottwitz1]) and proper, by the above property (G4). We write $Y_\pi$ for the pull-back of the universal abelian scheme on ${_{\tilde K_\pi}\U}_\pi$ to ${_{\tilde K}\U}$ and tacitly omit the mentioning of level structures. Now and again we need to invoke the projective limit $_{\tilde K_p}\U=\lim_{\tilde K^p\to1}{_{\tilde K}\U}$, which is a scheme with a right $\tilde G(\a^{\infty,p})$-action. We will now consider a prime $\gp\subset\O_{ER}$ which restricts to a prime of type $\vartheta$. Once and for all we also fix a set $\R$ of extensions to $L$ of the primes of $L^+$ over $p$, and for each $\gq\in\R$ we let $r_\gq$ be the degree of $\gq^+:=\gq\cap\O_{L^+}$, and we let $r$ be the degree of $\gp\cap\O_R$, and $f$ the degree of $\gp$. For each $\gq\in\R$ we fix an embedding $\iota_\gq:L\hookrightarrow R$ with $\iota_\gq(\gq)\subset\gp$, so that $$\tau^{r_\gq}\circ\iota_\gq=\begin{cases}\iota_\gq\circ*&\gq^*=\gq\\\iota_\gq&\text{otherwise}\end{cases}$$ ($*$ is the conjugation, and $\tau$ is the Frobenius acting on $K(\f_{p^f})$, which contains $R$). In the local settings the function $\bd^+$ translates into $\pmod{r_\gq}$ multidegrees $$\bd_\gq(\omega):=\omega+\bd^+(\vartheta^{-\omega}\circ\iota_\gq).$$ One can find selfdual $\z_{(p)}$-lattices $\V_i$ whose stabilizers in the $\q_p$-points of $G_i$, are the hyperspecial groups $K_{i,p}$. Set $\V_{i,\gq}$ for the modules: $$\V_i\otimes_{\O_L,\iota_\gq}\begin{cases}W(\f_{p^{2r_\gq}})&\gq^*=\gq\\W(\f_{p^{r_\gq}})&\text{otherwise}\end{cases},$$ on which we have perfect pairings with $\Psi_{i,\gq}(x,y)=-\Psi_{i,\gq}(y,x)^*$ in case $\gq^*=\gq$, and ditto for $\V_{\pi,\gq}$. Let $\G_\gq/W(\f_{p^{r_\gq}})$ (resp. $\G_\gq^1/W(\f_{p^{r_\gq}})$) be the Zariski closure of $(\g_m\times_{L^+}G^0)/\{\pm1\}$ (resp. $G^0$) in $$\prod_{i\in\Lambda}\begin{cases}\GU(\V_{i,\gq}/W(\f_{p^{2r_\gq}}),\Psi_{i,\gq})&\gq^*=\gq\\
\g_m\times\GL(\V_{i,\gq}/W(\f_{p^{r_\gq}}))&\text {otherwise}\end{cases},$$ and denote $(\g_m\times\prod_{\gq\in\R}\Res_{W(\f_{p^{r_\gq}})/W(\f_p)}\G_\gq^1)/\{\pm1\}$ by $\G/W(\f_p)$, these groups are reductive, as $K_p$ was assumed to be hyperspecial. Choose any $*$-invariant $\O_L$-order of $L_\pi$ with: $$\O_{L_\pi}\subset\O_L+(L_\pi\cap\bigoplus_{\gq\in\R}p^{|\bd_\gq^*\circ\bd_\gq|+1}\End_{L_\gq}(\V_{\pi,\gq})).$$ Now consider the family of compact open subgroups $K\subset G(\a^\infty)$ which satisfy $$\begin{aligned}
\label{twistingIII}
&&K=K^p\times K_p\\
\label{twistingIV}
&&K^p=\bigcap_{i\in\Lambda}\varrho_i^{-1}(K_i^p),\end{aligned}$$ for suitable $K_i^p\subset G_i(\a^{\infty,p})$ and for fixed choices of hyperspecial subgroups $K_{i,p}$ and $K_p$ satisfying . Finally choose $\tilde K=\tilde K^p\times\tilde K_p$ compatible with the groups $K_i=K_i^p\times K_{i,p}$ with respect to . Let the scheme $_K\gM/W(\f_{p^r})$ represent the functor that sends a connected pointed base scheme $(S,s)$ to the set of $4+\Card(\Pi)$-tuples $(Y,\lambda,\iota,\ebar,y_\pi)$ with the following properties:
- $(Y,\lambda,\iota,\ebar)$ is a $\z_{(p)}$-isogeny class of: homogeneously $p$-principally polarized abelian schemes $(Y,\lambda)$ together with a $*$-invariant action $\iota:\O_L^\Lambda\rightarrow\z_{(p)}\otimes\End(Y)$ satisfying the determinant condition with respect to the skew-Hermitian $L^\Lambda$-module $\tilde V$, and a $\pi_1(S,s)$-invariant $\tilde K^p$-orbit $\ebar$ of $\O_L^\Lambda$-linear similitudes $$\tilde\eta^p:\a^{\infty,p}\otimes\tilde V\stackrel{\cong}{\rightarrow}H_1^{\mathaccent19 et}(Y\times_Ss,\a^{\infty,p}).$$
- $y_\pi:\O_{L_\pi}\rightarrow\z_{(p)}\otimes\End_L(Y_\pi)$ is a $\O_L$-linear $*$-preserving homomorphism such that $\ebar$ contains at least one element $\tilde\eta^p$ rendering the diagrams $$\begin{CD}
\a^{\infty,p}\otimes\bigotimes_{i\in\pi}\End_L(V_i)@<<<\O_{L_\pi}\\
@V{\bigotimes_{i\in\pi}\eta_i^p}VV@V{y_\pi}VV\\
\End(\bigotimes_{i\in\pi}H_1^{\mathaccent19 et}(Y[\ge_i]\times_Ss,\a^{\infty,p})(\frac{1-\Card(\pi)}2))
@<{H_1^{\mathaccent19 et}}<<\End_L(Y_\pi\times_Ss)
\end{CD}$$ commutative, simultaneously for all $\pi\in\Pi$ where $\eta_i^p$ stands for the unique $\pi_1(S,s)$-invariant $\O_L$-linear similitude $\a^{\infty,p}\otimes V_i\stackrel{\cong}{\rightarrow}H_1^{\mathaccent19 et}(Y[\ge_i]\times_Ss,\a^{\infty,p})$ with $\tilde\eta^p=\sum_{i\in\Lambda}\eta_i^p\circ\varepsilon_i$.
Notice that the projective limit $_{K_p}\gM=\lim_{K^p\to1}{_K\gM}$ is equipped with a right $G(\a^{\infty,p})$-action and with a $G(\a^{\infty,p})$-equivariant morphism to ${_{\tilde K_p}\U}\times_{\O_R}W(\f_{p^r})$, which at the finite levels recovers the tautological forgetful morphism $_K\gM\rightarrow{_{\tilde K}\U}\times_{\O_R}W(\f_{p^r});(Y,\lambda,\iota,\ebar,y_\pi)\mapsto(Y,\lambda,\iota,\ebar)$. We write $\O_{H_c}=\End_G((\bigotimes_i\V_i)^{\otimes_{\O_L}c})$, where $c$ is a fixed constant which is sufficiently large for all of the $\G_\gq$’s in the sense of remark \[hyperspecial\]. Let us also fix cocharacters $\mu:\g_m\times W(\f_{p^f})\rightarrow\G$, and $\mu_\gq:\g_m\times W(\f_{p^f})\rightarrow\Res_{W(\f_{p^{r_\gq}})/W(\f_p)}\G_\gq$ in the conjugacy class of $\mu_h$, and let us write ${\underline a}_{i,\gq}$,and ${\underline b}_{i,\gq}$ for the weight invariants that go with it, according to subsection \[global\]. We have shown that the gauges $\bj_{i,\gq}(\omega):=\bj_i(\vartheta^{-\omega}\circ\iota_\gq)$ defined cocharacters $\tilde\upsilon_{i,\gq}$ and functors $$\begin{aligned}
&&\fx^{\bd_\gq,\bj_{i,\gq}}(\rho_i):\B(\G_\gq,\mu_\gq)\times\f_{p^f}\rightarrow\B_{i,\gq}\\
&&\B_{i,\gq}:=
\begin{cases}\B(\GU(\V_{i,\gq}/W(\f_{p^{2r_\gq}}),\Psi_{i,\gq}),\tilde\upsilon_{i,\gq})/W(\f_{p^{2r_\gq}})&\gq^*=\gq\\
\B(\g_m\times\GL(\V_{i,\gq}/W(\f_{p^{r_\gq}})),\tilde\upsilon_{i,\gq})/W(\f_{p^{r_\gq}})&\text{otherwise}\end{cases}.\end{aligned}$$ And recall also, that there are associated invariants $\tilde{\underline a}_{i,\gq}$ and $\tilde{\underline b}_{i,\gq}$, again according to subsection \[global\]. For future reference we will write $\S_{i,\gq}:{_K\gM}\rightarrow\B_{i,\gq}$ for the classifying $1$-morphisms of the graded displays of the universal families $Y[\gq^\infty][\ge_i]$ over the moduli space $_K\gM$ (unitary ones if $\gq^*=\gq$). The tuple $\{\S_{i,\gq}\}_{i\in\Lambda}$ will be denoted by $\S_\gq$ and we will think of it as a $1$-morphism $$\label{SerreTate}
_K\gM\rightarrow\gB_\gq,$$ of stacks, where the $\gB_\gq/W(\f_{p^r})$’s are defined as in subsection \[practical\].
The canonical locus
-------------------
A morphism of $\f_{p^f}$-schemes $X\rightarrow{_K\gM}\times_{W(\f_{p^r})}\f_{p^f}$ will be called canonical if $X$ is noetherian and reduced, and becomes flexed over $X$ for every $\gq\in\R$. Our first result in this section deals with fields:
\[properI\] Let $k$ be a finitely generated field extension of $\f_{p^f}$, and let $l$ be an algebraically closed extension of $k$. If the image of $\xi\in{_K\gM}(k)$ in ${_K\gM}(l)$ is canonical, then the following is true too:
- There exists a finitely generated subfield $k_0\subset l$, containing $k$ and such that the image of $\xi$ in ${_K\gM}(k_0)$ is canonical.
- The image of $\xi$ in ${_K\gM}(k_0^{ac})$ is canonical.
Fix a flexibilisator $$\zeta_{i,\gq}^\circ:\fx^{\bd_\gq,\bj_{i,\gq}}(\V_{i,\gq}, \P_\gq^\circ)\cong\S_{i,\gq}\times_{_K\gM,\xi}l.$$ The displays with $\G_\gq$-structure $\P_\gq^\circ$ are banal, and thus we are allowed to choose representing group elements $U_{\gq,\sigma}\in\G_{\gq,\sigma}(W(l))$. Analogously we write $\tilde M_{i,\gq}$ for the graded windows of $Y_{i,\xi}[\gq^\infty]$ relative to a choice of Frobenius lift $\tau:C\rightarrow C$ on the Cohen $W(\f_{p^f})$-frame for $k$. Let $\tilde N_{i,\gq}$ be $\sy^{{\underline a}_{i,\gq},{\underline b}_{i,\gq}}(\V_{i,\gq},\P_\gq^\circ)$, and let $M_{i,\gq}$ be $\fx^{\bd_\gq^*\circ\bd_\gq}(\tilde M_{i,\gq})$. For varying $i$ the family $$N_{i,\gq,\sigma}:=\fx^{\bd_\gq}(\tilde N_{i,\gq})_\sigma
\stackrel{\zeta_{i,\gq,\sigma}^\circ}{\rightarrow}W(l)\otimes_CM_{i,\gq,\sigma},$$ of isomorphisms is easily seen to be a $W(l)$-valued point in the moduli scheme $\gS_{\gq,\sigma}$ of the $\O_{H_c}$-linear families of isomorphisms between $\V_{i,\gq,\sigma}=C\otimes_{F^{-\sigma}\circ\iota_\gq,\O_L}\V_i$ and $M_{i,\gq,\sigma}$. As $W(l)$ is faithfully flat over $C$ we get the local triviality of this principal homogeneous space for $\G_{\gq,\sigma}^1$. This gives us the existence of a point $\zeta_{i,\gq,\sigma}'$ over some finite unramified subextension $C'\subset W(l)$. Consider the unique $W(l)$-valued group element $h'_{\gq,\sigma}$ turning the family $(\dots,\zeta_{i,\gq,\sigma}^\circ,\dots)$ into $(\dots,\zeta_{i,\gq,\sigma}',\dots)$, and choose further elements $\G_{\gq,\sigma}^1(W(l))\ni h_{\gq,\sigma}\equiv1\pmod p$ such that $h_{\gq,\sigma}'':=h_{\gq,\sigma}^{-1}\circ h_{\gq,\sigma}'$ are contained in $\G_{\gq,\sigma}^1(W(k''))$, where $k''$ is a finitely generated field extension of $k'=C'/pC'$ in $l$: $$\begin{CD}
W(l)\otimes_CN_{i,\gq,\sigma}
@>{W(l)\times_{C'}\zeta_{i,\gq,\sigma}'}>>W(l)\otimes_CM_{i,\gq,\sigma}\\
@V{W(l)\times_{W(k'')}\rho_i(h_{\gq,\sigma}'')}VV
@A{\zeta_{i,\gq,\sigma}^\circ}AA\\
W(l)\otimes_CN_{i,\gq,\sigma}
@>{\rho_i(h_{\gq,\sigma})}>>W(l)\otimes_CN_{i,\gq,\sigma}
\end{CD},$$ and in case $\gq^*=\gq$ it does no harm to assume $h_{\gq,\sigma+r_\gq}$ proportional to $h_{\gq,\sigma}$. A suitable twisted conjugation endows the family $\zeta_{i,\gq,\sigma}^\circ\circ\rho_i(h_{\gq,\sigma})$ with the structure of a flexibilisator, and we merely need to show its rationality over a finitely generated field extension! To this end one checks that some $O_{\gq,\sigma}\in\G_{\gq,\sigma}(W(l))$ which possesses $W(k'')$-rational images under the functors $\fx^{\bd_\gq,\bj_{i,\gq}}(\V_{i,\gq})$ is surely $W(\sqrt[p^d]{k''})$-rational for some $d$, at least if one arranges $O_{\gq,\sigma}=1$ for $\sigma$ outside the image of $\bd$.\
The second assertion of the proposition follows easily from the first one in conjunction with remark \[constructI\], please notice that one cannot apply remark \[constructI\] directly to $l$.
A two-fold application of this proposition yields the significant strengthening that $k_0$ may be taken to be finite over $k$. We are now in a position to give a set theoretic definition for what will later turn out to be our Shimura variety: We set $_K\gM_{can}\subset{_K\gM}\times_{W(\f_{p^r})}\f_{p^f}$ for the locus of points that are canonical over some finite (or equivalently over any algebraically closed) extension of the residue field. The following proposition proves that the set $_K\gM_{can}$ is stable under specialization:
\[properII\] Consider an algebraically closed field extension $k$ of $\f_{p^f}$. Then for every $\eta\in{_K\gM}(k[[t]])$, the following are equivalent.
- The image of $\eta$ in ${_K\gM}(R)$ is canonical for some finite extension $k[[t]]\subset R$.
- There exists a finite extension of $k((t))$ with that property.
Throughout the whole proof we are working with the Frobenius lift $\tau:t\mapsto t^p$ on the $W(\f_{p^f})$-frames $W(k)[[t]]$ and $\widehat{W(k)((t))}$. For the non-trivial direction we may assume that there exist flexibilisators: $$\zeta_{i,\gq}^\circ:\fx^{\bd_\gq,\bj_{i,\gq}}(\V_{i,\gq}, \P_\gq^\circ)\stackrel{\cong}{\rightarrow}\S_{i,\gq}\times_{_K\gM,\eta}k((t)),$$ and we may assume that the $\P_\gq^\circ/k((t))$’s are banal, and hence represented by the graded window matrices $U_{\gq,\sigma}\in\G_{\gq,\sigma}(\widehat{W(k)((t))})$ by means of lemma \[wintwo\]. We start by defining the graded $(\widehat{W(k)((t))},\tau)$-module $$\tilde N_{i,\gq}=\sy^{{\underline a}_{i,\gq},{\underline b}_{i,\gq}}(\V_{i,\gq}, \P_\gq^\circ),$$ and analogously we write $\tilde\M_{i,\gq}$ for the graded $W(k)[[t]]$-windows of $Y_{i,\eta}[\gq^\infty]$ and $\tilde M_{i,\gq}$ for its generic fiber. We have canonically induced isomorphisms: $$N_{i,\gq,\sigma}:=\fx^{\bd_\gq}(\tilde N_{i,\gq})_\sigma\stackrel{\zeta_{i,\gq,\sigma}^\circ}{\rightarrow}
\fx^{\bd_\gq^*\circ\bd_\gq}(\tilde M_{i,\gq})_\sigma=:M_{i,\gq,\sigma},$$ and clearly we want to denote $\fx^{\bd_\gq^*\circ\bd_\gq}(\tilde\M_{i,\gq})_\sigma$ by $\M_{i,\gq,\sigma}$. Now we should look at the moduli scheme $\gS_{\gq,\sigma}/W(k)[[t]]$ of $\O_{H_c}$-preserving isomorphisms $$\dots\otimes\V_{i,\gq,\sigma}\stackrel{\cong}{\rightarrow}\dots\otimes\M_{i,\gq,\sigma},$$ where $\V_{i,\gq,\sigma}$ means $W(k)[[t]]\otimes_{F^{-\sigma}\circ\iota_\gq,\O_L}\V_i$, and where the $\O_{H_c}$-structure on the right arises from the $\O_{H_c}$-structure on $M_{i,\gq,\sigma}$. It is important to understand that the lattice families $\tilde\M_{i,\gq,\sigma}$ and $\tilde M_{i,\gq,\sigma}$ may not have a $\O_{H_c}$-structure and they have the wrong Hodge numbers too. By Dwork’s trick $\gS_{\gq,\sigma}$ has a point over $K(k)\{\{t\}\}$, so that the pull-back of $\gS_{\gq,\sigma}$ to $W(k)[[t]][\frac1p]$ is an honest locally trivial principal homogeneous space. Now we can follow the ideas in [@kisin], given the existence of $\zeta_{i,\gq,\sigma}^\circ$ we see immediately that the same is true over $\widehat{W(k)((t))}$, i.e. the pull back of $\gS_{\gq,\sigma}$ to $W(k)((t))_{(p)}$ is again a locally trivial principal homogeneous space under $\G_{\gq,\sigma}^1$. This glues to a principal homogeneous space over the scheme $\Spec W(k)[[t]]-\Spec k$, and by [@sansuc] we get a principal homogeneous space over $W(k)[[t]]$, in fact a trivial one, because the basis is strictly henselian. In particular we are allowed to choose isomorphisms: $$\V_{i,\gq,\sigma}\stackrel{\zeta_{i,\gq,\sigma}'}{\rightarrow}\M_{i,\gq,\sigma}.$$ Consider the unique $\widehat{W(k)((t))}$-valued group element $h_{\gq,\sigma}'$ turning the family $(\dots,\zeta_{i,\gq,\sigma}^\circ,\dots)$ into $(\dots,\zeta_{i,\gq,\sigma}',\dots)$. By adjusting with an element in $\G_{\gq,\sigma}^1(W(k)[[t]])$ we may assume the $\pmod p$-reductions of $h_{\gq,\sigma}'$ are contained in ${^{F^{\bd_\gq(\sigma)-\sigma}}\G}_{\gq,\bd_\gq(\sigma)}^{1-}(k((t)))$, given that these subgroups are parabolic ones. The rest of the proof runs similar to the arguments in proposition \[properI\], and again we arrange $h_{\gq,\sigma+r_\gq}'$ to be proportional to $h_{\gq,\sigma}'$, in case $\gq^*=\gq$. When working over a large finite covering of the form $k[[\sqrt[p^d]t]]$ there exists one and only one graded window matrix $O_{\gq,\sigma}\in\G_{\gq,\sigma}(\widehat{W(k)((\sqrt[p^d]t))})$ and a family of elements $k_{\gq,\sigma}\in\hat\G_{\gq,\sigma}^{1-}(\widehat{W(k)((\sqrt[p^d]t))})$ describing a morphism from $\{O_{\gq,\sigma}\}_\sigma$ to $\{U_{\gq,\sigma}\}_\sigma$, such that:
- If $\sigma$ lies in the image of $\bd_\gq$, then $k_{\gq,\sigma}=\tau^{\bd_\gq^*(\sigma)-\sigma}(h_{\gq,\bd_\gq^*(\sigma)}')$.
- If $\sigma$ lies outside the image of $\bd_\gq$, then $O_{\gq,\sigma}=1$.
The associated change of coordinates gives rise to $W(k)[[\sqrt[p^d]t]]$-rational maps $\zeta_{i,\gq,\sigma}^\circ$, and finally guarantees the existence of the requested integral models of $\P_\gq^\circ$ and $\zeta_{i,\gq}^\circ$.
Now consider the scheme $${_K\gM}\times_{\prod_{\gq\in\R}\gB_\gq^n}
\prod_{\gq\in\R}\prod_{\sigma\in\z/r_\gq\z}\H_{\gq,\sigma,n+w},$$ where the structural morphism of the second factor is given by: $$\prod_{\gq\in\R}(\prod_{i\in\Lambda}\fx^{\bd_\gq,\bj_{i,\gq}}(\rho_i)\circ b^{n+w}(\G_\gq,\mu_\gq)),$$ and write $_K\gM_{can}^n$ for its image under the projection to $${_K\gM}\times_{W(\f_{p^r})}\f_{p^f},$$ i.e. the first factor. For each finite $n$ this is a constructible subset, according to Chevalley’s theorem ([@egaiv Théorème(1.8.4)]). The constructibility of ${_K\gM_{can}}$ is harder to get by and requires some digressions: For every perfect field $k$ we denote the algebraic closure of $K(k)$ by $K(k)^{ac}$ and we denote the ring of integers in $K(k)^{ac}$ by $\O_{K(k)^{ac}}$ and we denote the $p$-adic completion of $\O_{K(k)^{ac}}$ by $\hat\O_{K(k)^{ac}}$, and we denote the fraction field of $\hat\O_{K(k)^{ac}}$ by $\hat K(k)^{ac}$. An object $(\{M_i\}_{i\in\Lambda},\{m_\pi\}_{\pi\in\Pi})$ of $\gC_{W(k)}$ is called nearly trivial for $\gq$ if there exists a sequence of families of $\hat K(k)^{ac}$-linear isomorphisms $\alpha_{i,\nu}:\hat K(k)^{ac}\otimes_{\iota_\gq,\O_L}\V_i\rightarrow\hat K(k)^{ac}\otimes_{W(k)}M_i$ such that: $$\lim_{\nu\to\infty}(g_\pi(\dots,\alpha_{i,\nu},\dots)\circ a\circ g_\pi(\dots,\alpha_{i,\nu},\dots)^{-1})=m_\pi(a)$$ holds for all $\pi\in\Pi$ and $a\in\O_{L_\pi}$. An object of $\gB_{\gq,k}$ is called nearly trivial its image under $\gsy_{\gq,\sigma,k}$ is nearly trivial for $\gq$, for every $\sigma$. Let us write $_K\gM_{triv}$ for the locus of points $x\in{_K\gM}$ such that the $\gB_{\gq,k(x)^{ac}}$-objects $\S_{i,\gq}\times_{_K\gM}k(x)^{ac}$ are nearly trivial for all $\gq$. It is easy to see that $_K\gM_{triv}$ is Zariski-closed, hence constructible.
\[aciso\] Let $k$ be an algebraically closed extension of $\f_{p^r}$. For every integer $m\geq0$ there exists an integer $\mbar\geq m$ such that the following is true: Suppose that $(\{M_i\}_{i\in\Lambda},\{m_\pi\}_{\pi\in\Pi})\in\Ob_{\gC_{W(k)}}$ is nearly trivial for $\gq$. Then to any isomorphism $$\alfbar_i:W_{\mbar}(k)\otimes_{\iota_\gq,\O_L}\V_i\rightarrow W_{\mbar}(k)\otimes_{W(k)}M_i$$ (whose source and target are regarded as objects of $\gC_{W_{\mbar}(k)}$) there exists another isomorphism $$\alpha_i:W(k)\otimes_{\iota_\gq,\O_L}\V_i\rightarrow M_i$$ (between objects of $\gC_{W(k)}$) such that $\{\alfbar_i\}_{i\in\Lambda}$ and $\{\alpha_i\}_{i\in\Lambda}$ have the same pull-backs to $W_m(k)$.
Fix $\gq$ and write $V_i^{ac}:=\hat K(k)^{ac}\otimes_{\iota_\gq,\O_L}\V_i$. By a $p$-adic analytic consideration there exists a nondecreasing function $\n_0\rightarrow\n_0;m\mapsto\mbar$ with the following property:\
For all families $\gamma_i\in\GL_{\hat K(k)^{ac}}(V_i^{ac})$ fulfilling: $$g_\pi(\dots,\gamma_i,\dots)\circ a\circ g_\pi(\dots,\gamma_i,\dots)^{-1}
\equiv m_\pi(a)\mod{p^{\mbar}}$$ ($\forall\pi\in\Pi,a\in\O_{L_\pi}$) there exist families $\tilde\gamma_i\in\GL_{\hat K(k)^{ac}}(V_i^{ac})$ fulfilling: $$\begin{aligned}
&&g_\pi(\dots,\gamma_i,\dots)\circ a\circ g_\pi(\dots,\gamma_i,\dots)^{-1}=\\
&&g_\pi(\dots,\tilde\gamma_i,\dots)\circ a\circ g_\pi(\dots,\tilde\gamma_i,\dots)^{-1}\\
&&\tilde\gamma_i\equiv1\mod{p^m}\end{aligned}$$ ($\forall\pi\in\Pi,a\in\O_{L_\pi}$). The isomorphisms $\alfbar_i$ allow lifts $\alpha'_i$, which however might not respect the tensors in $\O_{L_\pi}$. By induction on $\nu$ we will now define a sequence of families of $\hat K(k)^{ac}$-linear isomorphisms such that:
- $g_\pi(\dots,\tilde\alpha_{i,\nu},\dots)\circ a\circ g_\pi(\dots,\tilde\alpha_{i,\nu},\dots)^{-1}
\equiv m_\pi(a)\mod{p^{\nbar}}$
- $\tilde\alpha_{i,\nu+1}\equiv\tilde\alpha_{i,\nu}\mod{p^{\nu}}$
- $\tilde\alpha_{i,m}=\alpha'_i$,
in fact all of these families are automatically integral, i.e. already defined over $\hat\O_{K(k)^{ac}}$, simply because $\tilde\alpha_{i,m}$ is. Suppose that the family $\tilde\alpha_{i,\nu}$ has been found. The existence of a family, say, $\alpha_{i,\nu+1}$ that satisfies property (i) alone is clear, applying the above to $\gamma_i:=\tilde\alpha_{i,\nu}^{-1}\circ\alpha_{i,\nu+1}$ gives a family of isomorphisms $\tilde\gamma_i=\tilde\alpha_{i,\nu}^{-1}\circ\tilde\alpha_{i,\nu+1}$, and we have $\tilde\alpha_{i,\nu+1}\equiv\tilde\alpha_{i,\nu}\mod{p^{\nu}}$. Consider the family $\tilde\alpha_i=\lim_{\nu\to\infty}\tilde\alpha_{i,\nu}$. One can establish a canonical $\O_{H_c}$-action on $\bigotimes_{i\in\Lambda}M_i^{\otimes c}$, just use the isomorphism $\{\tilde\alpha_i\}_{i\in\Lambda}$ for transport of structure and observe that the left $G_0$-coset of this element is $K(k)$-rational. The existence of $\{\tilde\alpha_i\}_{i\in\Lambda}$ shows, that the $\O_{H_c}$-linear isomorphisms from $W(k)\otimes_{\iota_\gq,\O_L}\V_i$ to $M_i$ form a locally trivial principal homogeneous space for $\G_\gq^1$. As the latter one is smooth so are its locally trivial principal homogeneous spaces, and consequently there exists a lift of the $\mod{p^m}$-reduction of $\{\tilde\alpha_i\}_{i\in\Lambda}$.
There exists a constant $b$ with: $$\label{boundedII}
{_K\gM}_{can}=({_K\gM}_{triv}\times_{\f_{p^r}}\f_{p^f})\cap{_K\gM}_{can}^b$$
Write $\V_{i,\gq,\sigma}:=W(k)\otimes_{F^{-\sigma}\circ\iota_\gq,\O_L}\V_i$. By the crystalline boundedness principle ([@vasiu1]) there exists a positive integer $m$ with the following property: Given any congruence $$F_{i,\gq,\sigma}^{(1)}\equiv F_{i,\gq,\sigma}^{(2)}\mod{p^m}$$ between families of $\tau$-linear maps $$F_{i,\gq,\sigma}^{(1)},\,F_{i,\gq,\sigma}^{(2)}:\V_{i,\gq,\sigma+1}\rightarrow\V_{i,\gq,\sigma},$$ such that:
- there exist $\check F_{i,\gq,\sigma}^{(\nu)}:\check\V_{i,\gq,\sigma+1}
\rightarrow\check\V_{i,\gq,\sigma}$ with $(F_{i,\gq,\sigma}^{(\nu)}x,\check F_{i,\gq,\sigma}^{(\nu)}y)=p\tau(x,y)$ for $x\in\V_{i,\gq,\sigma+1}$ and $y\in\check\V_{i,\gq,\sigma+1}$,
- the diagrams: $$\begin{CD}
\V_{\pi,\gq,\sigma+1}@>{\bigotimes_{i\in\pi}F_{i,\gq,\sigma}^{(\nu)}}>>\V_{\pi,\gq,\sigma}\\
@A{v_\pi(a)}AA@A{v_\pi(a)}AA\\
\V_{\pi,\gq,\sigma+1}@>{\bigotimes_{i\in\pi}F_{i,\gq,\sigma}^{(\nu)}}>>\V_{\pi,\gq,\sigma}
\end{CD}$$ commute ($\forall\pi\in\Pi,\,a\in\O_{L_\pi},\,\nu\in\{1,2\}$), and so do: $$\begin{CD}
\check\V_{i,\gq,\sigma+1}@>{\check F_{i,\gq,\sigma}^{(\nu)}}>>\check\V_{i,\gq,\sigma}\\
@A{\Psi_{i,\gq,\sigma+1}}AA@A{\Psi_{i,\gq,\sigma}}AA\\
\V_{i,\gq,\sigma+r_\gq+1}@>{F_{i,\gq,\sigma}^{(\nu)}}>>\V_{i,\gq,\sigma+r_\gq}
\end{CD},$$ in case $\gq^*=\gq$ ($\forall i\in\Lambda,\,\nu\in\{1,2\}$),
one always obtains further commutative diagrams: $$\begin{CD}
\V_{i,\gq,\sigma+1}@>{F_{i,\gq,\sigma}^{(2)}}>>\V_{i,\gq,\sigma}\\
@A{\gamma_{i,\gq,\sigma+1}}AA@A{\gamma_{i,\gq,\sigma}}AA\\
\V_{i,\gq,\sigma+1}@>{F_{i,\gq,\sigma}^{(1)}}>>\V_{i,\gq,\sigma}
\end{CD},$$ for a suitable family of elements $\{\gamma_{i,\gq,\sigma}\}_{i\in\Lambda}=\gamma_{\gq,\sigma}\in\G_{\gq,\sigma}(W(k))$. The whole point of lemma \[aciso\] is to yield another positive integer $b:=\mbar$ for which we are going to verify our corollary, so pick any $\xi\in{_K\gM}(k)$ lying in the right-hand side of . Choose a $\mbar$-truncated flexibilisator $$\zeba_{i,\gq}:\fx^{\bd_\gq,\bj_{i,\gq}}(\V_{i,\gq},\peebar_\gq)
\stackrel{\cong}{\rightarrow}\essbar_{i,\gq}\times_{_K\gM,\xi}k,$$ where $\essbar_{i,\gq}$ is the truncation of $\S_{i,\gq}$ at level $\mbar$. Fix a banalisation of $\peebar$ with a corresponding $\mbar+w$-truncated graded display matrix $\{U_{\gq,\sigma,\mbar+w}\}_{\sigma\in\z/r_\gq\z}$. Lift it by $U_{\gq,\sigma}\in\G_{\gq,\sigma}(W(k))$, let $\P_\gq$ be the banal display with matrix $\{U_{\gq,\sigma}\}_{\sigma\in\z/r_\gq\z}$, and consider $\S'_{i,\gq}:=\fx^{\bd_\gq,\bj_{i,\gq}}(\V_{i,\gq},\P_\gq)$, and notice that the underlying eigenspaces of this object are canonically identified with $S'_{i,\gq,\sigma}=\V_{i,\gq,\sigma}$. Write $S_{i,\gq,\sigma}$ for the underlying eigenspaces of the displays $\S_{i,\gq}\times_{_K\gM,\xi}k$. The previous lemma finds us a $v_\pi$-preserving family of isometries $\zeta_{i,\gq,\sigma}:\V_{i,\gq,\sigma}\rightarrow S_{i,\gq,\sigma}$ such that $\zeta_{i,\gq,\sigma}\equiv\zeba_{i,\gq,\sigma}\mod{p^m}$. The crystalline boundedness principle tells us that $\{\S'_{i,\gq}\}_{i\in\Lambda}\cong\{\S_{i,\gq}\times_{_K\gM,\xi}k\}_{i\in\Lambda}$ and we are done.
It follows immediately that $_K\gM_{can}$ is constructible, hence closed, because it is also stable under specialization. This enables us to endow it with the reduced subscheme structure that is induced by the ambient scheme, namely $_K\gM\times_{W(\f_{p^r})}\f_{p^f}$. One should appreciate that $_K\gM_{can}$ does not depend on the choice of the orders $\O_{L_\pi}$.
Construction of ${_K\M}_\gp$ {#bootstrap}
----------------------------
Consider the stack rendering the diagram: $$\begin{CD}
{_K\Mot_\gp}@>>>{_K\gM\times_{W(\f_{p^r})}\f_{p^f}}\\
@VVV@VVV\\
\prod_{\gq\in\R}\B(\G_\gq,\mu_\gq)\times_{W(\f_{p^f})}\f_{p^f}
@>>>\prod_{\gq\in\R}\gB_\gq\times_{W(\f_{p^r})}\f_{p^f}
\end{CD}$$ $2$-cartesian, where the lower horizontal morphism is given by $$\prod_{\gq\in\R}(\prod_{i\in\Lambda}\fx^{\bd_\gq,\bj_{i,\gq}}(\rho_i)),$$ one could expect that $_K\Mot_\gp$ is representable (in particular discrete), here is a result in that direction:
There exists a $2$-commutative diagram $$\begin{CD}
{_K\Mbar}_\gp@>>>{_K\gM}_{can}\\
@V{\prod_{\gq\in\R}\P_\gq}VV@VVV\\
\prod_{\gq\in\R}\B(\G_\gq,\mu_\gq)\times_{W(\f_{p^f})}\f_{p^f}
@>>>\prod_{\gq\in\R}\gB_\gq\times_{W(\f_{p^r})}\f_{p^f}
\end{CD}$$ with a finite, surjective, and radicial upper horizontal morphism, and $\prod_{\gq\in\R}\P_\gq$ formally étale (in the sense of definition \[smooth\]).
For every closed point $x\in{_K\gM}_{can}$ there exists a flexibilisator which is defined over the residue field $k(x)$, because it is perfect. Let us write $\tilde A_x$ for the complete local ring which prorepresents the product of the universal formal deformation spaces of the underlying displays with $\G_\gq$-structure, and write $\tilde\P_{\gq,x}$ for the universal objects over either $\Spf\tilde A_x$ or $\Spec\tilde A_x$, which according to lemma \[existence\] makes no difference. Clearly $\tilde A_x$ is a power series algebra over $k(x)$. The Serre-Tate theorem yields naturally a finite map $$\Spec\tilde A_x\rightarrow\Spec\hat\O_{_K\gM_{can},x},$$ which is also radicial, the diagonal being dominant (corollary \[faithfulIII\]). We will need the dominance of this very map, which can be established as follows: Every minimal prime ideal is the kernel of some local homomorphism $\hat\O_{_K\gM_{can},x}\rightarrow k[[t]]$, and possibly after going to a finite extension one obtains a canonical one, again by proposition \[properII\] and by the definition of $_K\gM_{can}$. The morphism $\tilde A_x\rightarrow k[[t]]$, which we can deduce from that, tells us that the given minimal prime comes from a minimal prime of $\tilde A_x$.\
Consequently it is also clear that there exists a regular scheme $\tilde U_x$ together with a finite and surjective map onto an affine open neighbourhood $U_x\subset{_K\gM}_{can}$ of $x$, such that the completion at $x$ gives us back $\tilde A_x$: Just look at the fraction fields $K_x$, $L_x$, and $\tilde L_x$ of the rings $\O_{_K\gM_{can},x}$, $\hat\O_{_K\gM_{can},x}=A_x$, and $\tilde A_x$, observe that there exists a unique finite purely inseparable field extension $\tilde K_x/K_x$ with $\tilde K_x\otimes_{K_x}L_x=\tilde L_x$, and consider the normalization $\tilde U_x$ in $\tilde K_x$ of some affine open neighbourhood $U_x$. This reasoning is justified by applying the result [@egaiv Proposition (6.14.4)] to the normal morphism $\O_{_K\gM_{can},x}\rightarrow A_x$. Passage to the completion in $x$ shows that this scheme is at least regular in $x$, so that a further shrinkage of $U_x$ enforces the regularity of the whole of $\tilde U_x$. Moreover, the existence of the rings $\tilde A_x$ tells us that the stalk $\O_{\tilde U_x,x}$ is radicial over $U_x$ by [@egaiv Proposition(6.15.3.1)(ii)].\
Now we come to the canonicity of the maps $$\Spec\tilde K_x\rightarrow{_K\gM_{can}}.$$ This is easy, for by very definition of the canonical locus and by the lemma \[properI\] one can find a flexibilisator over some finitely generated field extension $C$ of $\tilde K_x$, and then one applies part (ii) of corollary \[construct\] with $\tilde A_x$ playing the role of the ring $B$.\
We next study the canonicity of $\tilde U_x$: It is easy to see that the stalks $$\Spec\O_{\tilde U_x,x}\rightarrow{_K\gM_{can}}$$ are canonical, as this follows immediately from part (i) of corollary \[construct\], applied to $B=\tilde K_x$ and $C=\tilde A_x$. Let us write $\P_{\gq,x}$ and $\zeta_{i,\gq,x}$ for the display with $\G_\gq$-structure and the flexibilisator over $\O_{\tilde U_x,x}$. Remember that our aim was to prove that $\tilde U_x\rightarrow{_K\gM_{can}}$ is canonical! To this end we continue to argue stalkwise: For every closed point $y\in U_x$ the stalk $\O_{\tilde U_x,y}$ is by definition equal to the normalization of the local ring $\O_{_K\gM_{can},y}$ in the field extension $\tilde K_x$, and the stalk $\O_{\tilde U_y,y}$ is by definition equal to the normalization of the local ring $\O_{_K\gM_{can},y}$ in the field extension $\tilde K_y$. Unfortunately we must not use that the fields $\tilde K_x$ and $\tilde K_y$ are equal, but given they are both finite purely inseparable field extensions of $K_x=K_y$ we can find an integer $n$ such that $\tilde K_y\subset\sqrt[p^n]{\tilde K_x}$, and this gives rise to inclusions: $$\O_{\tilde U_x,y}\subset\sqrt[p^n]{\O_{\tilde U_x,y}}\supset\O_{\tilde U_y,y},$$ which show us the canonicity of the map $$\Spec\sqrt[p^n]{\O_{\tilde U_x,y}}\rightarrow{_K\gM_{can}}.$$ Now apply part (ii) of corollary \[construct\], to the rings $B=\tilde A_x$ and $C=\sqrt[p^n]{\O_{\tilde U_x,y}}$, which is faithfully flat because $\O_{\tilde U_x,y}$ is regular. The last step towards the canonicity over $\tilde U_x$ is accomplished by the corollary \[chaotic\] (whose proof is morally yet another application of part (i) of corollary \[construct\]). By slight abuse of notation we will denote the extensions of $\P_{\gq,x}$ and $\zeta_{i,\gq,x}$ over the whole of $\tilde U_x$ by the same symbols. Remark \[formal\] provides a further shrinkage of $\tilde U_x$ that gains the formal étaleness of $\prod_{\gq\in\R}\P_{\gq,x}$, so that the pull-backs to all of the equicharacteristic completed local rings (at the closed points other than $x$) are universal formal deformations. It follows that these agree canonically with the $\prod_{\gq\in\R}\tilde\P_{\gq,y}$’s, and as a by-product we do obtain that indeed $\tilde K_x=\tilde K_y$. Surely all those $\tilde U_x$’s glue coherently and we finished the construction of a smooth and projective $\f_{p^f}$-scheme $_K\Mbar$ that is finite and radicial over $_K\gM_{can}$ and over which there is a formally étale flexibilisator, alternatively $_K\Mbar$ is (componentwise) the normalization of $_K\gM_{can}$ in the function fields $\tilde K_x$, which only depend on the irreducible component that passes through $x$.
If $X$ is a scheme of finite type over an algebraic extension of $\f_{p^f}$, then the natural map from the set of $X$-valued points of $_K\Mbar_\gp$ to the fiber of $_K\Mot_\gp$ over $X$ is an equivalence of discrete groupoids (i.e. a bijection of sets).
Let $\nu$ be a positive integer, let $\bigcup_lU_l$ be an open affine covering of $_K\Mbar_\gp$, and let $\P_{l,\gq}$ be the restriction of $\P_\gq$ to $U_l$. Choose smooth affine $W_\nu(\f_{p^f})$-schemes $U_l^{(\nu)}$ lifting the smooth affine $\f_{p^f}$-schemes $U_l$, and write $U_{l,m}^{(\nu)}\subset U_l^{(\nu)}$ for the open subscheme whose underlying point set is $U_l\cap U_m$. Pick lifts $\P_{l,\gq}^{(\nu)}\in\hat T_{\P_{l,\gq}}(U_l^{(\nu)})$, and write $\P_{l,m,\gq}^{(\nu)}$ for the restriction of $\P_{l,\gq}^{(\nu)}$ to $U_{l,m}^{(\nu)}$. Let finally be $\psi_{l,m}^{(\nu)}:U_{l,m}^{(\nu)}\rightarrow U_{m,l}^{(\nu)}$ the unique lift of the identity which pulls-back $\P_{m,l,\gq}^{(\nu)}$ to $\P_{l,m,\gq}^{(\nu)}$. The transition maps $\psi_{l,m}^{(\nu)}$ satisfy the cocycle condition and define a formally smooth lift ${_K\M}_\gp^{(\nu)}/W_\nu(\f_{p^f})$ along with a family of displays $\P_\gq^{(\nu)}$ with $\G_\gq$-structure over it. The locally trivial principal homogeneous spaces for $\G_\gq^-$ over ${_K\M}_\gp^{(\nu)}$ which arise by truncating $\P_\gq^{(\nu)}$ at level $0$ shall be denoted by ${_K\F}_\gq^{(\nu)}=\prod_{\sigma=0}^{r_\gq-1}{_K\F}_{\gq,\sigma}^{(\nu)}$. Naturally these give rise to invertible sheafs $$\L_{i,\gq,\omega}^{(\nu)}:=\varpi_{_K\F_{\gq,\bd_\gq(\omega)}^{(\nu)}}(\modul_\omega^{\bd_\gq,\bj_{i,\gq}})^{\otimes-1}.$$ Let $\L^{(\nu)}$ be their product. The line bundle $\L^{(1)}$ is just the pull-back of the canonical sheaf on the ambient Shimura variety $_{\tilde K}\Ubar_\gp$, according to , so it is ample ([@mori]). By [@egaiii Théorème (5.4.5)], this completes the construction of a smooth projective $W(\f_{p^f})$-scheme $_K\M_\gp$ together with an isomorphism of formal schemes $$_K\hat\M_\gp\cong\lim_{\nu\to\infty}{_K\M}_\gp^{(\nu)},$$ where the left-hand side is the $p$-adic completion. Let $\L_{i,\gq,\omega}$ and $\L$ be the projective limits of the $\L_{i,\gq,\omega}^{(\nu)}$’s and $\L^{(\nu)}$’s. We need a limit of the ${_K\F}_{\gq,\sigma}^{(\nu)}$’s too, which can be justified as follows: Write $$\begin{aligned}
&&{_K\F}_{i,\gq,\sigma}^{(\nu)}:=\varpi_{_K\F_{\gq,\sigma}^{(\nu)}}(\rho_i)\\
&&{_K\F}_{i,\gq,\sigma}:=\lim_\leftarrow{_K\F}_{i,\gq,\sigma}^{(\nu)}\end{aligned}$$ and let ${_K\F}_{\gq,\sigma}$ be the formal principal homogeneous space of ascended filtered $\O_{H_c}$-preserving maps from $\V_{i,\gq,\sigma}=\O_{_K\M_\gp}\otimes_{\tau^{-\sigma}\circ\iota_\gq,\O_L}\V_i$ to ${_K\F}_{i,\gq,\sigma}$. In order to verify the local triviality of ${_K\F}_{\gq,\sigma}$ it suffices to construct points over the completed local rings at each minimal point of ${_K\M}_\gp$, which one can do using the smoothness of the ${_K\F}_{\gq,\sigma}^{(\nu)}$’s. Another limit process yields connections $\nabla_{\gq,\sigma}$ on the composite of the $\otimes$-functors $$\bRep_0(\G_{\gq,\sigma})\rightarrow\bRep_0(\G_{\gq,\sigma}^-)
\stackrel{\varpi_{_K\F_{\gq,\sigma}}}{\rightarrow}\ect_{_K\M_\gp},$$ i.e. connections on ${_K\F}_{\gq,\sigma}\times^{\G_{\gq,\sigma}^-}\G_{\gq,\sigma}$.
Stabilizers
-----------
Notice that the display of the multiplicative $p$-divisible group $\mu_{p^\infty}$ is canonically identified with the mulipliers of all of the $\S_{i,\gq}$’s, being the displays of the $p$-divisible groups $Y[\gq^\infty][\ge_i]$. One sees immediately that the diagram in \[Weil\] gives rise to a canonical display $\P$ with $\G$-structure over ${_K\Mbar}_\gp$ that maps to the tuple $\{\P_\gq\}_{\gq\in\R}$, under the upper horizontal map in the $2$-cartesian diagram below: $$\begin{CD}
\B(\G,\mu)@>>>\prod_{\gq\in\R}\B(\G_\gq,\mu_\gq)\\
@VVV@VVV\\
\B(\g_m,\mu_1)@>>>\prod_{\gq\in\R}\B(\g_m\times W(\f_{p^{r_\gq}}),\mu_1)
\end{CD}.$$ Analogously, one sees that the there is a canonical locally trivial principal homogeneous space $_K\F$ over ${_K\M}_\gp$ yielding $_K\F_{\gq,\sigma}$ upon an extension of structure groups from $\G^-$ to $\G_{\gq,\sigma}^-$. From now onwards it is more comfortable to work with $\P$ and $_K\F$ rather then $\P_\gq$ and $_K\F_{\gq,\sigma}$.\
Let $k_0$ be a perfect field. Let $\d(Y)$ denote the covariant Grothendieck-Messing crystalline Dieudonné theory of an abelian variety $Y/k_0$, and let $\d^0(Y)$ denote the (non-effective) $F$-isocrystal whose underlying $K(k_0)$-space is $\q\otimes\d(Y)$, and whose Frobenius operator $K(k_0)\otimes_{F,K(k_0)}\d^0(Y)\rightarrow\d^0(Y)$ is the inverse of $\d(F_{Y/k_0})$, where $F_{Y/k_0}:Y\rightarrow Y\times_{k_0,F}k_0;y\mapsto y^p$ is the relative Frobenius.\
Let $(Y,\lambda,\iota)$ be a homogeneously $p$-principally polarized $\z_{(p)}$-isogeny class of abelian $\fc$-varieties with a $*$-invariant action $\iota:\O_L^\Lambda\rightarrow\End(Y)$ satisfying the determinant condition with respect to the skew-Hermitian $L^\Lambda$-module $\tilde V$. Consider the eigenspaces $Y[\ge_i]$. Any choice of a full $\tilde K^p$-level structure completes this to a PEL-quadruple, say $(Y,\lambda,\iota,\tilde\eta^p)\in{_{\tilde K}\U}(\fc)$. From now onwards we will write $\dot\bigotimes_{i\in\pi}Y[\ge_i]$ to denote the homogeneously $p$-principally polarized $\z_{(p)}$-isogeny class of abelian $\fc$-varieties derived by discarding the level structure from its image quadruple $g_\pi(Y,\lambda,\iota,\tilde\eta^p)\in{_{\tilde K_\pi}\U}_\pi(\fc)$, under the map . The formation of $\dot\bigotimes_{i\in\pi}Y[\ge_i]$ is independent of the choice of $\tilde\eta^p$, and the canonical $*$-invariant action $\iota_\pi:\O_L\rightarrow\z_{(p)}\otimes\End(\dot\bigotimes_{i\in\pi}Y[\ge_i])$ satisfies the determinant condition with respect to the skew-Hermitian $L$-module $\tilde V_\pi$, moreover we have canonical isomorphisms $$\bigotimes_{L\otimes\a^{\infty,p}}H_1^{\mathaccent19 et}(Y[\ge_i],\a^{\infty,p})\cong
H_1^{\mathaccent19 et}(\dot\bigotimes_{i\in\pi}Y[\ge_i],\a^{\infty,p})(\frac{\Card(\pi)-1}2),$$ and similarily for $\d^0$ (Sketch: once one chooses a $\c$-lift $Y'$ of $Y$ one obtains an isomorphism of the following Hodge structures $$\bigotimes_{\O_L}H_1(Y'[\ge_i](\c),\z)\cong
H_1(\dot\bigotimes_{i\in\pi}Y'[\ge_i](\c),\z)(\frac{\Card(\pi)-1}2)$$ with coefficients in $\O_L$, and $\dot\bigotimes_{i\in\pi}Y'[\ge_i]$ turns out to be a $\c$-lift of $\dot\bigotimes_{i\in\pi}Y[\ge_i]$). We need two more functoriality properties which are mediate consequences of Deligne’s theory of absolute Hodge cycles:
\[bekanntIV\] To every quasi-isogeny $\gamma:Y_1\dashrightarrow Y_2$ there is a canonical quasi-isogeny $g_\pi(\gamma):\dot\bigotimes_{i\in\pi}Y_1[\ge_i]\dashrightarrow\dot\bigotimes_{i\in\pi}Y_2[\ge_i]$, such that application of $\d^0$ and $H_1^{\mathaccent 19 et}(\dots,\a^{\infty,p})$ recovers the usual tensor-products (with coefficients in $L$).
\[bekanntIII\] There is a canonical $*$-preserving and $L$-linear homomorphism of algebras: $$\bigotimes_L\End_L^0(Y[\ge_i])\rightarrow\End_L^0(\dot\bigotimes_{i\in\pi}Y[\ge_i]);
(\dots,f_i,\dots)\mapsto\dot\bigotimes_{i\in\pi}f_i,$$ such that application of $\d^0$ and $H_1^{\mathaccent 19 et}(\dots,\a^{\infty,p})$ recovers the usual tensor-products (with coefficients in $L$).
For a situation, which is slightly different from the one at hand the proofs have been carried out in full detail (cf. [@habil Theorem 4.10] and [@habil Proposition 5.1]). Let $(Y,\lambda,\iota)$ be a homogeneously polarized $\q$-isogeny class of abelian $\fc$-varieties with a $*$-invariant action $\iota:L^\Lambda\rightarrow\End^0(Y)$. By fact \[bekanntIV\] the homogeneously polarized $\q$-isogeny class of $\dot\bigotimes_{i\in\pi}Y[\ge_i]$ has a well-defined meaning, provided only that the homogeneously polarized $\q$-isogeny class of $Y$ contains a member for which
- the polarization $\lambda:Y\rightarrow\check Y$ is $p$-principal
- the action $L^\Lambda\rightarrow\End^0(Y)$ is integral and satisfies the determinant condition with respect to the skew-Hermitian $L^\Lambda$-module $\tilde V$.
From now on we assum that the $\vartheta$-poly-gauge $\{\bj_i\}_{i\in\Lambda}$ is ’normalized’ in the sense that ${\underline a}_{i,\gq}=\tilde{\underline a}_{i,\gq}$ whenever $\gq^*\neq\gq\in\R$. There is a $G(\a^{\infty,p})$-equivariant bijection between the $G(\a^{\infty,p})$-set $_{K_p}\Mbar_\gp(\fc)$ and tuples $(Y,\lambda,\iota,y_\pi,\eta^p,\eta_p,\P_0)$ where:
- $(Y,\lambda,\iota)$ is a homogeneously polarized $\q$-isogeny class of abelian $\fc$-varities with $*$-invariant action $\iota:L^\Lambda\rightarrow\End^0(Y)$ containing at least one member for which $\lambda$ is $p$-principal and for which $\iota$ is integral and satisfies the determinant condition with respect to the skew-Hermitian $L^\Lambda$-module $\tilde V$.
- $y_\pi:L_\pi\rightarrow\End_L^0(\dot\bigotimes_{i\in\pi}Y[\ge_i])$ is a homomorphism which commutes with $L$ and preserves $*$, for every $\pi\in\Pi$
- $\P_0$ is a display with $\G$-structure over $\fc$ and $\eta_p$ is a $L^\Lambda$-linear isomorphism $\bigoplus_{i\in\Lambda}\eta_{i,p}:\q\otimes\sy(\bigoplus_{i\in\Lambda}\varrho_i,\P_0)
\stackrel{\cong}{\rightarrow}\d^0(Y)$ of $F$-isocrystals, preserving the pairings on either side up to a factor in $K(\fc)$ and such that the diagrams $$\begin{CD}
\End^0(\sy(\otimes_{i\in\pi}\varrho_i,\P))@<<<L_\pi\\
@V{\bigotimes_{i\in\pi}\eta_{i,p}}VV@V{y_\pi}VV\\
\End_L(\bigotimes_{i\in\pi}\d^0(Y[\ge_i])(\frac{1-\Card(\pi)}2))
@<{\d^0}<<\End_L^0(\dot\bigotimes_{i\in\pi}Y[\ge_i])
\end{CD}$$ are commutative for all $\pi\in\Pi$.
- $\eta^p=\bigoplus_{i\in\Lambda}\eta_i^p:\a^{\infty,p}\otimes V
\stackrel{\cong}{\rightarrow} H_1^{\mathaccent19 et}(Y,\a^{\infty,p})$ is a $L^\Lambda$-linear similitude such that the diagrams $$\begin{CD}
\a^{\infty,p}\otimes\bigotimes_{i\in\pi}\End_L(V_i)@<<<L_\pi\\
@V{\bigotimes_{i\in\pi}\eta_i}VV@V{y_\pi}VV\\
\End(\bigotimes_{i\in\pi}H_1^{\mathaccent19 et}(Y_i,\a^{\infty,p})(\frac{1-\Card(\pi)}2))
@<{H_1^{\mathaccent19 et}}<<\End_L^0(\dot\bigotimes_{i\in\pi}Y_i)
\end{CD}$$ are commutative for all $\pi\in\Pi$.
Let $y=(Y,\iota,\lambda,\iota_\pi)$ be data consisting of (i) and (ii). Write $I_y/\q$ (resp. $J_y/\q_p$) for the algebraic group representing the functor:
$$\begin{aligned}
R\mapsto&&\{(f,\dots,f_i,\dots)\in R^\times\times\prod_{i\in\Lambda}R\otimes\End_L^0(Y[\ge_i])^\times|\\
&&\forall\pi\in\Pi:\,\dot\bigotimes_{i\in\pi}f_i\in R\otimes\End_{L_\pi}^0(\dot\bigotimes_{i\in\pi}Y[\ge_i])^\times\\
&&\forall i\in\Lambda:\,f_i^*f_i=f\}\mbox{ (resp.}\\
R\mapsto&&\{(f,\dots,f_i,\dots)\in R^\times\times\prod_{i\in\Lambda}R\otimes_{\q_p}\End_L(\d^0(Y[\ge_i]))^\times|\\
&&\forall\pi\in\Pi:\,\bigotimes_{i\in\pi}f_i\in R\otimes_{\q_p}\End_{L_\pi}(\bigotimes_{i\in\pi}\d^0(Y[\ge_i]))^\times\\
&&\forall i\in\Lambda:\,f_i^*f_i=f\})\end{aligned}$$
Notice that every choice of full $K^p$-level structure $\eta^p$ as in (iv) furnishes $I_y\times\a^{\infty,p}$ with a group homomorphism, say $i_{y,\eta^p}$, to $G\times\a^{\infty,p}$ and that one has $i_{y,\eta^p\circ\gamma^p}=\Int(\gamma^p)^{-1}\circ i_{y,\eta^p}$ for all $\gamma^p\in G(\a^{\infty,p})$. Notice also that there is a natural embedding $I_y\times\q_p\subset J_y$, and that choices of $\eta_p$ and $\P\in\B_\fc(\G,\mu)$ as in (iii) do in fact identify the group $\Aut(\P_0)$ with a subgroup of $J_y(\q_p)$, so let us eventually write $\Gamma_{y,\eta_p,\P}\subset I_y(\q)$ for the inverse image of $\Aut(\P_0)$ under this embedding. Whenever $\P_0$ is lifted by a display $\P_1$ with $\G$-structure over a complete local noetherian $W(\f_{p^f})$-algebra $A$, then we will also write $\Gamma_{y,\eta_p,\P_1}$ to denote the inverse image of $\Aut(\P_1)$ under that embedding (N.B.: the lifts of $\P_0$ are in one-one correspondence with lifts of the $\fc$-valued point of $_{K_p}\M_\gp$ that corresponds to the tuple $(Y,\lambda,\iota,y_\pi,\eta^p,\eta_p,\P_0)$).\
Once $\P_0$ is represented by a concrete element $U\in\G(W(\fc))$ there exists a map $j_{y,\eta_p,U}:J\times K(\fc)\rightarrow G\times K(\fc)$, which is analogous to $i_{y,\eta^p}$. However a change of coordinates of the form $h^{-1}U{^\Phi h}=U'$ will alter $j_{y,\eta_p,U}$ and turn it into: $\Int(h)^{-1}\circ j_{y,\eta_p,U}=j_{y,\eta_p,U'}$.\
We need one more concept: Let us call $\P_0\in\B_\fc(\G,\mu)$ elliptic if it possesses a representative $U\in\G(W(\fc))$ which can be written in the form $U=g^{-1}{^Fg}$, where $g\in G(K(\fc))$ is such that the image of $\Int(g)\circ\mu$ lies in a maximal $\q_p$-elliptic torus $T\subset G\times\q_p$. If this holds then elements of $\G(W(\fc))\cap\Int(g^{-1})T(K(\f_p))$ are automorphisms of $\P_0$. There is a very nice lift $\P_1\in\B_{W(\fc)}(\G,\mu)$ for example by regarding the very same representative $U$ as an object in the category $\CAS_{W(\fc)}(\G,\mu)$ (use remark \[Witt\]), $\G(W(\fc))\cap\Int(g^{-1})T(K(\f_p))$ acts on $\P_1$ too.
\[automorphisms\] Let $\xi\in{_{K_p}\Mbar}_\gp(\fc)$ correspond to $(y,\eta^p,\eta_p,\P_0)$. The stabilizer of $\xi$ in $G(\a^{\infty,p})$ agrees with $i_{y,\eta^p}(\Gamma_{y,\eta_p,\P_0})$.\
Suppose that $\P_0$ is elliptic. Then there exists a display $\P_1$ with $\G$-structure over $W(\fc)$ which lifts $\P_0$ such that $\Gamma_{y,\eta_p,\P_1}$ contains the $\z_{(p)}$-points of a $\z_{(p)}$-model of a $\q$-rational, $\q_p$-elliptic, maximal torus in $I_y$.
The first assertion is trivial. To prove the second recall that giving a lift is effectively equivalent to giving a pair $(\P_1,\delta)$ with $\P_1\in\Ob_{\B_{W(\fc)}(\G,\mu)}$ and $\delta\in\Hom(\P_0,\P_1\times_{W(\fc)}\fc)$. Let $g^{-1}{^F g}=U\in\G(W(\fc))$ stand for a representative that qualifies $\P_0$ as an elliptic display, so that $\Aut(\P_0)$ reads $\{h\in\I(\fc)|h^{-1}U{^\Phi}h\}$ and contains $\G(W(\fc))\cap\Int(g^{-1})T(K(\f_p))$. It causes no harm to denote the endomorphism $F\circ\Int_G(\mu(p)^{-1}):\G(K(\fc))\rightarrow\G(K(\fc))$ by $\Phi$ (so-named as it extends the twisted Frobenius map $\I(\fc)\rightarrow\H(\fc)$ as introduced at the beginning of subsection \[pivotalI\]). In this language one sees that $$\Aut(\P_0)\subset J_y(\q_p)\cong\{h\in\G(K(\fc))|h^{-1}U{^\Phi}h\}\subset\G(K(\fc)).$$ If $T^\times\subset T(\q_p)$ stands for the subset of regular elements, then the subset $\bigcup_{j\in\Aut(\P_0)}\Int(j^{-1}g^{-1})T^\times$ is open in the $p$-adic topology of $I_y(\q_p)$, so that it contains a rational element $\Int(j^{-1}g^{-1})t\in I_y(\q)$. It follows that the image of $T(\q_p)$ under $\Int(j^{-1}g^{-1})$ is equal to $S(\q_p)$ for some $\q$-rational torus $S\subset I_y$ (namely the centralizer of $\Int(j^{-1}g^{-1})t$). The lift $(\P_1,j)$ solves our problem, as $\P_1$ has already been exhibited before.
From now on we study exclusively the union of those connected components of $_{K_p}\M_\gp$ which contain at least one elliptic point. We write $_{K_p}\M_\gp^*$ for this open and closed $G(\a^{\infty,p})$-invariant subscheme, and $_{K_p}\F^*$ is defined similarily, as are their analogs $_K\M_\gp^*$ and $_K\F^*$ for finite level structures $K\subset G(\a^{\infty,p})$. Whether one actually has ${_{K_p}\M_\gp}={_{K_p}\M_\gp^*}$ is a difficult question whose answer has no significance for the present paper (the answer is probably: ’yes’).\
Let $G^-/K(\f_{p^f})$ be the generic fiber of $\G^-$, and let $G'/K(\f_{p^f})$ be the quotient of $G\times K(\f_{p^f})$ by the largest normal subgroup that is entirely contained in $G^-$, and let $G'^-/K(\f_{p^f})$ be the image of $G^-$ under the canonical projection $G\times K(\f_{p^f})\rightarrow G'$. Choose an embedding $\iota_\infty:W(\f_{p^f})\hookrightarrow\c$. It is known that $_{K_p}\M_\gp^*(\c)$ can be written as a disjoint union $$\coprod_l\tilde\Delta_l\backslash(\tilde M_l\times G(\a^{\infty,p}))$$ for certain connected and simply connected complex manifolds $\tilde M_l$ with certain discrete cocompact subgroups $\tilde\Delta_l\subset\Aut(\tilde M_l)\times G(\a^{\infty,p})$. The locally trivial principal homogeneous space $_K\F^*$ together with the connection on $_K\F^*\times^{\G^-}\G$ induce homomorphisms $\phi_l:\tilde\Delta_l\rightarrow G(\c)$ and locally biholomorphic $\tilde\Delta_l$-equivariant maps $\rho_l:\tilde M_l\rightarrow G/G^-(\c)$. The composition of $\phi_l$ with the canonical projection $G\times K(\f_{p^f})\rightarrow G'$ shall be denoted by $\phi'_l$. For every $l$ one obtains a bounded period morphism $(\tilde\Delta_l,\tilde M_l,G',G'^-,\phi'_l,\rho_l)$ in the sense that all properties (P1), (P2), (P3), and (P4) of the part \[characterization\] of the appendix are valid. The reasons are as follows:
- property (P1) holds because the holonomy group is already maximal in the formal category by lemma \[maximalholonomy\]. To utilize this we only need a single basic point, for example an elliptic one.
- property (P2) holds because the corollary \[obstIII\] allows to describe the stabilizers in characteristic $0$ in terms of stabilizers in characteristic $p$, which are given by the $\q$-groups self-isogenies, by the first part of proposition \[automorphisms\]
- property (P3) holds because of the second part of the proposition \[automorphisms\].
- property (P4) holds because $\L$ is ample.
The fact \[bekanntIII\] implies that $_K\M_\gp^*\neq\emptyset$. We have shown the following result:
\[uniformizeIV\] Fix a Shimura-datum $(G,X)$ with a poly-unitary structure of width $w$ relative to a CM field $L$ whose normal closure is $R\subset\c$. Fix $\vartheta\in\Gal(R/\q)$ and a $\vartheta$-poly-gauge as in the beginning of this section. Let $\gp\subset\O_{ER}$ be a prime of norm $p^f$. Assume that $\gp\cap\O_R$ is of type $\vartheta$, that the poly-unitary structure is unramified at $p$, and that $p>\max\{2,w\}$. There exist data ${_K\M}_\gp^*,\,{_K\F}^*,\,\nabla$ that are canonically associated to each of the compact open subgroups in the family outlined in and such that:
- ${_K\M}_\gp^*$ is a non-empty projective and smooth $W(\f_{p^f})$-scheme, of which the special fiber is a finite ${_{\tilde K}\U}\times_{\O_R}\f_{p^f}$-scheme in a canonical way, where $_{\tilde K}\U$ is the integral model of the unitary group Shimura variety described at the beginning of this section.
- $_K\F^*$ is a locally trivial principal homogeneous space for $\G^-$ over ${_K\M}_\gp^*$
- $\nabla$ is a connection on the composite of the $\otimes$-functors: $$\bRep_0(\G\times W(\f_{p^f}))\rightarrow\bRep_0(\G^-)
\stackrel{\varpi_{_K\F^*}}{\rightarrow}\ect_{{_K\M}_\gp^*}$$
- The association of ${_K\M}_\gp^*,\,{_K\F}^*,\,\nabla$ is functorial in the sense that each pair of compact open subgroups $K_1$ and $K_2$ satisfying $K_1\subset\Int(\gamma^p)K_2$ for some $\gamma^p\in G(\a^{\infty,p})$, gives rise to a commutative diagram: $$\begin{CD}
{_{K_1}\F}^*@>>>{_{K_1}\M}_\gp^*@<<<{_{K_1}\Mbar}_\gp^*@>>>{_{\tilde K_1}\U}\times_{\O_R}\f_{p^f}\\
@VVV @VVV @VVV @VVV\\
{_{K_2}\F}^*@>>>{_{K_2}\M}_\gp^*@<<<{_{K_2}\Mbar}_\gp^*@>>>{_{\tilde K_2}\U}\times_{\O_R}\f_{p^f}
\end{CD}$$ where the vertical arrow on the left is a $\G^-$-equivariant étale covering which is compatible with the connections on ${_{K_1}\F}^*\times^{\G^-}\G$ and ${_{K_2}\F}^*\times^{\G^-}\G$ while the vertical arrow on the right is the $\mod{\gp\cap\O_R}$-reduction of the canonical action of the group element $(\dots,\varepsilon_i^{-1}\circ\varrho_i(\gamma^p)\circ\varepsilon_i,\dots)\in\tilde G(\a^{\infty,p})$.
- Let $X'$ be a connected component of $X$, and let $\Aut(X')$ be the real Lie group of its biholomorphic transformations. Then there exists a finite family of discrete cocompact lattices $\tilde\Delta_l\subset\Aut(X')^\circ\times G(\a^{\infty,p})$ and a biholomorphic map between the disjoint union: $$\coprod_l\coprod_{\gamma^p\in\tilde\Delta_l\backslash G(\a^{\infty,p})/K^p}(\tilde\Delta_l\cap\Int(\gamma^p)K^p)\backslash X'$$ and the complexification of $_K\M_\gp^*$.
Existence of poly-unitary Shimura data {#hilfssaetze}
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Let $k$ be an algebraically closed field of characteristic $0$, and let $\gg$ be a semi-simple Lie-algebra over $k$. We call a finite-dimensional representation $\rho:\gg\rightarrow\End_k(U)$ asymmetrical if for every non-inner automorphism $\alpha:\gg\rightarrow\gg$ one can find some sub-representation $\sigma$ of $\rho$ which is nonequivalent to $\sigma\circ\alpha$. The raw material for sample poly-unitary Shimura data is supplied by:
\[triple\] Let $\rho_0:G_0\rightarrow\GL(U_0/k)$ be a linear representation of the semi-simple $k$-group $G_0$ and write $\rho'_0$ for the direct sum of $\rho_0$ and $\Ad$ on the representation space $U'_0=U_0\oplus\gg_0$. Assume that $\rho'_0$ is faithful and has an asymmetrical derived representation ${\rho'_0}^{der}:\gg_0\rightarrow\End_k(U'_0)$. Write $U':=U\oplus\gg_0$, where $U$ is the trivial one-dimensional representation of $G_0$, and fix an element $e\in U$. Consider the sub-$k$-algebras $k[a]\subset\End_{G_0}(U\otimes_kU'\otimes_kU'_0)$, and $k[a']\subset\End_{G_0}(U')$ that are generated by the single endomorphisms $$x_1\otimes(x+x')\otimes(x_0+x'_0)\stackrel{a}{\mapsto}
x_1\otimes(x'_0\otimes x'+e\otimes{\rho'_0}^{der}(x')(x_0+x'_0)),$$ and $$x+x'\stackrel{a'}{\mapsto}x,$$ where $x_0\in U_0$, $x,\,x_1\in U$, and $x'_0,\,x'\in\gg_0$. Write $G^0$ for the stabilizer in $\GL(U/k)\times_k\GL(U'/k)\times_k\GL(U'_0/k)$ of $k[a]$, $k[a']$ and $\End_{G_0}(U'_0)$, and let $T^0$ be the center of $G^0$. Then one has: $$T^0G_0=G^0$$ More specifically, write $T'_0$ for the center of $\End_{G_0}(U'_0)^\times$, and write $T'\subset T'_0$ for the sub-torus consisting of elements whose action on $\gg_0$ is a scalar. Then $T'$ is naturally contained in $T^0$, and indeed one has $\g_m^2\times T'=T^0$, where the two copies of $\g_m$ act as scalars on $U$ and $U'$. The rank of $T^0$ is equal to $$3+\Card\{\text{isotypic components of }U'_0\}-\Card\{\text{simple factors of }\gg_0\},$$ and it is connected.
The proposition and its proof are both similar to [@habil Lemma 7.3]. Fix an element of $G^0$, according to the presence of $\End_{G_0}(U'_0)$ and $a'$ we can write $\left(\begin{matrix}g_0&0\\0&g'_0\end{matrix}\right)$, $\left(\begin{matrix}g&0\\0&g'\end{matrix}\right)$, and $g_1$, for the induced maps on $U'_0$, $U'$, and $U$. Notice that $g'_0$ and $g'$ are proportional, and that: $$\begin{aligned}
&&\frac{g_0}g\rho^{der}(x')x_0=\rho^{der}(\frac{g'}gx')(\frac{g_0}gx_0)\\
&&\frac{g'_0}g[x',x'_0]=[\frac{g'}gx',\frac{g'_0}gx'_0]\end{aligned}$$ according to the presence of $a$. This means that $\alpha:=\frac{g'}g$ is an automorphism of the Lie algebra $\gg_0$, and that $\beta:=\frac1g\left(\begin{matrix}g_0&0\\0&g'_0\end{matrix}\right)$ intertwines ${\rho'_0}^{der}$ and ${\rho'_0}^{der}\circ\alpha$. Using the asymmetry of ${\rho'_0}^{der}$, and again the presence of $\End_{G_0}(U'_0)$, we see that $\alpha$ must be an inner automorphism, so that it is induced from an element in $G_0(k)$. Upon an adjustment we are allowed to assume $\alpha=1$, so that $g'$ is equal to the multiplication by the scalar $g$. Consequently $\beta$ lies in the center of $\End_{G_0}(U'_0)^\times$. Upon a further adjustment we are allowed to assume that each of $g_0$, $g'_0$ and $g'$ is equal to the multiplication by the scalar $g$. The remaining degrees of freedom are $g,\,g_1\in k^\times$.
In the special case of a simply connected algebraic group $G_0$, the condition on the asymmetry of the derived representation ${\rho'_0}^{der}$ may be removed from the assumptions of the previous proposition, because this is an automatic consequence of the faithfulness of $\rho'_0$. In fact, if $G_0$ is simply connected and semisimple without simple factors of type $E_8$, $F_4$ or $G_2$, then the faithfulness of $\rho_0$ implies already the asymmetry of $\rho_0^{der}$ (Sketch: every automorphism of $\gg_0$ comes from an automorphism of $G_0$, which is inner if and only if it restricts to the identity on the center).\
In the special case of an algebraic group of adjoint type over an algebraically closed field, one can easily give a specific example of a representation $\rho_0$ such that ${\rho'_0}^{der}$ is faithful and asymmetrical: Let $G_0=\prod_{i=1}^dG_i$ be the decomposition into simple factors. Let us write $\rho_i:G_i\rightarrow\GL(U_i/k)$ for the following asymmetrical representations:
- If $G_i\cong\PGL(n)$, then $U_i:=\sm^n(\sd)\otimes\bigwedge^n(\check\sd)$,
- If $G_i\cong\SO(2n)/\{\pm1\}$, then $U_i:=\begin{cases}\bigwedge^n(\sd)&n\equiv0\pmod2\\
\sm^2(\bigwedge^n(\sd))&n\equiv1\pmod 2\end{cases}$,
- If $G_i$ is of type $E_6$, then $U_i:=\sm^3(\j)$,
- $U_i:=0$ in all remaining cases,
and let us write $\rho_0:G_0\rightarrow\GL(U_0/k)$ for the natural representation on the (exterior) direct sum $U_0:=\bigoplus_{i=1}^dU_i$ (here $\sd$ means standard representation and $\j$ is the $27$-dimensional exceptional Jordan algebra). In general $U_0$ need not be faithful nor asymmetrical, but it is easy to see that $U'_0=U_0\oplus\gg_0$ does indeed possess both of these properties. In the sequel this is our prime example, we also need the following $\n_0$-valued function on the set of isomorphism classes of connected groups of adjoint type, which we call the “radius”:
- $r(G_1\times\dots\times G_d):=\max\{r(G_1),\dots,r(G_d)\}$, if $G_1,\dots,G_d$ are simple.
- $r(\PGL(n)):=n-1$.
- $r(\SO(2n)/\{\pm1\}):=\begin{cases}\frac n2&n\equiv0\pmod2\\n&n\equiv1\pmod2\end{cases}$.
- If $G_0$ is of type $E_6$, then $r(G_0):=4$.
- If $G_0$ is of type $E_7$, $C_l$ or $B_l$, then $r(G_0):=1$.
- If $G_0$ is of type $E_8$, $F_4$ or $G_2$, then $r(G_0):=0$.
The number $r(G_0)$ has the following significance: If $U'_0$ is the representation above, then no minuscule cocharacter of $G_0$ possesses a weight on $U'_0$ which is strictly greater than $r(G_0)$ (or strictly smaller than $-r(G_0)$). For a totally imaginary quadratic extension $L$ of a totally real field $L^+$ we have to introduce the following algebraic tori: $$\begin{aligned}
&&C^0:=\ker(\Res_{L/L^+}\g_m\stackrel{\n_{L/L^+}}{\rightarrow}\g_m)\\
&&C^1:=\Res_{L^+/\q}C^0\\
&&C:=(\g_m\times C^1)/\{\pm1\},\end{aligned}$$ Every CM-type for $L$ endows $C$ with the structure of a Shimura datum. Now we turn to our poly-unitary examples:
\[polyI\] Fix an $\r$-algebra homomorphism $c:\c\rightarrow\r\otimes L$, i.e. a CM type. Let $G_0/L^+$ be an algebraic group of adjoint type, where $L^+$ is as above. Let $(\Res_{L^+/\q}G_0,X_0)$ be a Shimura datum. Then there exists an embedding of connected $L^+$-tori $C^0\hookrightarrow T^0$ together with a poly-unitary Shimura datum of the form $$(G,X)=((\g_m\times\Res_{L^+/\q}T^0)/\{\pm1\},\{c\})\times(\Res_{L^+/\q}G_0,X_0),$$ with the following additional features:
- The torus $T^0$ splits over the composite of $L$ with the smallest field over which $G_0$ becomes an inner form of a split form, and $\Res_{L^+/\q}T^0$ is compact over $\r$.
- The poly-unitary structure is unramified outside the set of primes for which at least one of $L$ or $\Res_{L^+/\q}G_0$ is ramified, and its width is $2r$, where $r$ is the radius of $G_0$.
Finally, this particular poly-unitary structure allows a $\vartheta$-poly-gauge if and only if the automorphism $\vartheta$ in the Galois group of the normal closure $R^+\subset\r$ of $L^+$ satisfies: $$\frac{2r+1}{2r+2}<
\lim_{N\to\infty}\frac{\Card\{k\in\{1,\dots,N\}|G_0\times_{L^+,\vartheta^k\circ\iota}\r\text{ is compact }\}}N$$ for all embeddings $\iota:L^+\rightarrow\r$.
We begin with any representation $\rho_0:G_0\rightarrow\GL(W_0/L)$ whose scalar extension $L^{ac}\otimes_LW_0$ is a direct sum of any number of copies of $\Gal(L^{ac}/L)$-conjugates of the previously described $U_0$. Let $\rho_1:G_0\rightarrow\GL(W_1/L)$ be the trivial one-dimensional representation. Pick polarizations $\Psi_i:L\otimes_{*,L}W_i\rightarrow\check W_i$ for $i\in\{0,1\}$, notice that $\Psi_i(x,y)^*=\Psi_i(y,x)$, as the $\rho_i$’s are even. Consider the prolongation of the Killing form on $\gg_0$ to a sesquilinear form $\Psi'$ on $W':=L\otimes_{L^+}\gg_0$. The triple of polarized $G_0\times_{L^+}L$-representations, that we wish to work with is: $W_1$, $W_2:=W_1\oplus W'$, and $W_3:=W_0\oplus W'$. The $G_0$-invariant sub-algebras $L_\pi\subset\End_L(V_\pi)$ are constructed along the lines of proposition \[triple\] and are easily seen to be $*$-invariant, here $\Lambda:=\{1,2,3\}$ and $\Pi:=\{\{1\},\{2\},\{3\},\{1,2,3\}\}$. The same proposition exhibits a connected $L^+$-torus $T^0$ such that $T^0\times_{L^+}G_0$ is the stabilizer in $\UL(W_1/L,\Psi_1)\times_{L^+}\UL(W_2/L,\Psi_2)\times_{L^+}\UL(W_3/L,\Psi_3)$ of $\bigcup_{\pi\in\Pi}L_\pi$, moreover the proof thereof shows that $T^0\times_{L^+}L$ splits over the splitting field of $G_0\times_{L^+}L$. Write $T^1:=\Res_{L^+/\q}T^0$ and $T:=(\g_m\times T^1)/\{\pm1\}$, and let us note in passing that $C$ is embedded diagonally into $T$, as $C^1$, the complement in $C$ to $\g_m$, is embedded diagonally into $T^1$. Notice that $w_X=w_{\{c\}}$ maps $-1$ to $-1$, and thus we obtain a poly-unitary structure if we only divide out each of the pairings $\Psi_i$ by $\sqrt{-1}$, for $i\in\{1,2,3\}$.\
In order to obtain an unramified poly-unitary structure we have to work a little bit harder: To fix ideas pick a compact and open subgroup of the form $K=K_S\times\prod_{p\notin S}K_p\subset\Res_{L^+/\q}G_0(\a^\infty)$ such that every $K_p$ is a hyperspecial subgroup of $\Res_{L^+/\q}G_0(\q_p)$. Sufficiently small $K$-invariant $\O_L[\prod_{p\in S}p^{-1}]$-lattices $\W_i\subset W_i$ for $i\in\Lambda$, and a sufficiently large integer $d$ would satisfy: $\W_i\subset\W_i^\perp\subset\frac{\W_i}d$. Now recall Zarhin’s trick: Choose a lattice $Z$ of rank $8$ containing a direct factor $D$ of rank $4$, such that:
- $Z^\perp=Z$
- $D^\perp\supset\frac Dd$.
Replace $W_i$ by $V_i:=Z\otimes W_i$. The $L$-algebras $\End(Z)^{\otimes\Card(\pi)}\otimes L_\pi$ are evidently $*$-invariant subalgebras of $\End_L(\bigotimes_{i\in\pi}V_i)$ for $\pi\in\Pi$, and the lattices $\V_i:=Z\otimes\W_i+D\otimes\W_i^\perp\subset V_i$ are $K$-invariant and unimodular for $i\in\Lambda$. The groups of $\z_p$-points in the unitary similitude groups of these lattices are hyperspecial for $p\notin S$, and we are done.
Let us round off the discussion with another family of examples:
\[polyII\] Let $G_0/L^+$ be an absolutely simple connected, algebraic group, where $L^+$ is as above. Let $(\Res_{L^+/\q}G_0,X_0)$ be a Shimura datum without factors of type $D_l^\h$ for any $l\geq5$. Then there exists a certain poly-unitary Shimura datum $(G,X)$ of width $2$, relative to a certain totally imaginary quadratic extension $L$, such that both of them are unramified outside the set of odd primes for which at least one of $L^+$ or $\Res_{L^+/\q}G_0$ is ramified, and they enjoy the following properties:
- There exist a certain embedding of connected $L^+$-tori $C^0\hookrightarrow T^0$ together with a certain central extension of algebraic $L^+$-groups $$1\rightarrow T^0\hookrightarrow G^0\twoheadrightarrow G_0\rightarrow 1,$$ where $\Res_{L^+/\q}T^0$ is unramified outside the set of odd primes for which at least one of $L^+$ or $\Res_{L^+/\q}G_0$ is ramified, and this $\q$-torus is compact over $\r$.
- The quotient of $\g_m\times\Res_{L^+/\q}G^0$ by the subgroup $\{\pm1\}\subset\g_m\times C^1$ agrees with $G$, the canonical epimorphism $G\twoheadrightarrow\Res_{L^+/\q}G_0$ which is derived from that sends $X$ to $X_0$, and $w_X$ agrees with the canonical inclusion $\g_m\hookrightarrow G$.
Finally, this particular poly-unitary structure allows a $\vartheta$-poly-gauge if and only if the automorphism $\vartheta$ in the Galois group of the normal closure $R^+\subset\r$ of $L^+$ satisfies: $$\frac34<
\lim_{N\to\infty}\frac{\Card\{k\in\{1,\dots,N\}|G_0\times_{L^+,\vartheta^k\circ\iota}\r\text{ is compact }\}}N$$ for all embeddings $\iota:L^+\rightarrow\r$.
In the cases of type $A_l$ there is nothing to prove. In the cases of type $E_7$, $C_l$, $B_l$ we can just apply the proposition \[polyI\] to the quadratic extension $L:=L^+[\sqrt{-1}]$. In the $E_6$ or $D_l$-situations our proof starts with a description of a provisional central extension: Let $\Gbar_0$ be the simply connected covering group of $G_0$ and let $\Cbar_0$ be its center. This group is of order $3$ or $4$ and it is cyclic, except for the $D_l$-case with $l\equiv0\pmod2$. Excluding this for while one may observe that there is one and only one choice of a totally imaginary quadratic extension $L$, such that there exists a rational embedding $\Cbar_0\hookrightarrow C^0$, because in those cases the Weyl-opposition is a non-inner automorphism, and there are no other ones. Given that $\Gbar^0:=(C^0\times\Gbar_0)/\Cbar_0$ has connected center there exists a Shimura datum $(\Res_{L^+/\q}\Gbar^0,\Xbar)$ which lifts $X_0$ and has a trivial weight homomorphism. Let $\rho_0:\Gbar^0\times_{L^+}L\rightarrow\GL(W_0/L)$ be a $L$-rational representation which is a direct sum of minuscule representations each of whose restrictions to the center $C^0\times_{L^+}L=\g_m\times L$ agree with scalar multiplication. The latter one is unique so that $L\otimes_{*,L}W_0$ is equivalent to $\check W_0$. The sets of $\mu$-weights of each eigenspace (to the embeddings $L\hookrightarrow\c$) can be read off from [@deligne3 Table 1.3.9], so these are translates of $\{-\frac23,\frac13,\frac43\}$ or $\{\pm\frac12\}$, depending on whether or not we are in the $E_6$-situation. Notice that $\Xbar$ can be multiplied with an arbitrary homomorphism $\s\rightarrow C^1\times_\q\r$ without changing $X_0$, so upon a careful adjustment the $\mu$-weights of all eigenspaces are in $\{0,\pm1\}$. The proof is completed by following the arguments in the proof of proposition \[polyI\], notice that the $L^+$-structure of the group $G^0$ (or equivalently of $T^0$) may depend on the choice of the polarizations which in turn assumes a choice of $\Xbar$. Finally it remains to do the $D_l$-case with $l\equiv0\pmod2$, in this case $\Cbar_0$ is a $2$-dimensional $\f_2$-vector space scheme and $\Res_{L^+/\q}\Cbar_0$ splits over $\r$. Let $C^+$ be a connected $L^+$-torus satisfying:
- $C^+[2]\cong\Cbar_0$
- The $\q$-torus $\Res_{L^+/\q}C^+$ splits over $\r$.
Such a torus exists because $\GL(2,\z)\twoheadrightarrow\GL(2,\f_2)$ is a split epimorphism. Write $\Cbar$ for the connected $L^+$-torus $C^0\otimes X^*(C^+)$, where we use $L:=L^+[\sqrt{-1}]$ again for the above definition of $C^0$. Notice that the group scheme of its $2$-torsion points satisfies $\Cbar[2]\cong\Cbar_0$, so let us put $\Gbar^0:=(\Cbar\times\Gbar_0)/\Cbar_0$ and let us choose a Shimura datum $(\Res_{L^+/\q}\Gbar^0,\Xbar)$ which lifts $X_0$ and has a trivial weight homomorphism (notice that $\Res_{L^+/\q}\Cbar$ is compact over $\r$). Now pick a selfdual, faithful, $L$-rational representation $\rho_0:\Gbar^0\times_{L^+}L\rightarrow\GL(W_0/L)$ decomposing into any direct sum of minuscule ones. In case $l\neq4$, assume in addition that $\rho_0$ does not contain the unique representation of dimension $2l$, i.e. the standard representation. The proof is again completed by the arguments in the proof of proposition \[polyI\].
Varshavsky’s characterization method {#characterization}
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The paper [@varshavskyII] deals with characterizations of Shimura varieties. In this work we want to be slighty more basic and confine to characterizations of symmetric Hermitian spaces, here are the objects under consideration:
- Let $\tilde\Delta$ be a group and let $\tilde M$ be a separated, non-empty, connected, complex manifold with a holomorphic left $\tilde\Delta$-action.
- Let $G_1,\dots,G_r$ be connected, reductive, simple algebraic groups over $\c$, and let $P_i$ be a proper minuscule parabolic subgroup of $G_i$. Denote the associated irreducible symmetric Hermitian domains of compact type by $X_i:=G_i(\c)/P_i(\c)$, and write $G:=\prod_{i=1}^rG_i$ and $P:=\prod_{i=1}^rP_i$ and $X:=\prod_{i=1}^rX_i$.
- Let $\phi:\tilde\Delta\rightarrow G(\c)$ be a group homomorphism, and let $\rho_0:\tilde M\rightarrow X$ be a locally biholomorphic $\tilde\Delta$-equivariant map (use $\phi$ to define a left action of $\tilde\Delta$ on $X$)
A sextuple $(\tilde\Delta,\tilde M,G_i,P_i,\phi,\rho_0)$ as above is called a period map if:
- The image of $\tilde\Delta$ in $G(\c)$ is Zariski-dense.
- The image in $G(\c)$ of the stabilizer in $\tilde\Delta$ of any element in $\tilde M$ possesses a compact closure.
- There exists some $y_0\in\tilde M$ whose stabilizer in $\tilde\Delta$ contains some subgroup $\Delta_0$ the closure of whose image in $G(\c)$ is equal to $\prod_{i=1}^r\bT_i(\r)$, where each $\bT_i$ is a maximal compact torus in $\Res_{\c/\r}G_i$ (i.e. the $\r$-rank of $\bT_i$ is equal to the $\c$-rank of $G_i$).
Without any attempt of originality we wish to give a slight reformulation of [@varshavsky p.89,p.92-94] in the above axiomatic setup:
\[uniformizationI\] Suppose that $(\tilde\Delta,\tilde M,G_i,P_i,\phi,\rho_0)$ is a period map. Then there exist real forms $\bJ_i$ of $G_i$ for every $i\in\{1,\dots,r\}$, such that $\overline{\phi(\tilde\Delta)}=\bJ(\r)^\circ$, where $\bJ:=\prod_{i=1}^r\bJ_i$. The locally biholomorphic map $\rho_0$ is actually an injection and the $\tilde\Delta$-action on $\tilde M$ can be extended to a continuous and transitive $\bJ(\r)^\circ$-action thereon.\
Pick a base point $\tilde y\in\tilde M$, and assume for notational convenience that $\rho_0(\tilde y)$ is the canonical base point of $\prod_{i=1}^rX_i$, i.e. equal to $(1,\dots,1)$. Write $\bU_i:=P_i\cap\Pbar_i$, where $\Pbar_i$ denotes the complex conjugate of $P_i$ with respect to the real form $\bJ_i$. Consider the homogeneous spaces $\tilde M_i:=\bJ_i(\r)^\circ/\bU_i(\r)$, so that and $\tilde M=\prod_{i=1}^r\tilde M_i$. Then for each $i\in\{1,\dots,r\}$ one and only one of the following alternatives hold:
- $\bJ_i$ is compact, and $\bU_i$ is a maximal proper connected subgroup,
- or $\bJ_i$ is not compact, and $\bU_i$ is a maximal compact subgroup,
and in any case $\bU_i$ has indiscrete center so that $\tilde M_i$ is a symmetric Hermitian domain of compact or of non-compact type.
We give a synopsis of the proof: The solution to the 5th problem of Hilbert implies that $J:=\overline{\phi(\tilde\Delta)}$ is a Lie-group. Note that $\c\otimes_\r\Lie J$ is semisimple, because $\Lie G$ is semisimple and both of $\Lie J\cap\sqrt{-1}\Lie J$ and $\Lie J+\sqrt{-1}\Lie J$, being $G$-invariant $\c$-subspaces of $\Lie G$ in view of (P1), are semisimple too. Moreover, there exists a semi-simple real algebraic group $\bJ\subset\Res_{\c/\r}G$ such that $\Lie\bJ=\Lie J$, as semisimple Lie-algebras are algebraic. Finally the existence of $\bT$ tells us that $\bJ=\prod_{i=1}^r\bJ_i$ where each single $\bJ_i$ contains $\bT_i$ and it is either a real form of $G_i$ or it is equal to $\Res_{\c/\r}G_i$. Let us write $\Aut(\tilde M)$ for the the homeomorphism group of $\tilde M$, and let us endow it with the compact-open topology. Let $\tilde J$ (resp. $\tilde T$) denote the closure of the image of $\tilde\Delta$ (resp. $\Delta_0$) in $\Aut(\tilde M)$. The group $\tilde T$ is compact because it fixes a point and preserves a suitable Riemannian metric, [@kobayashi II, Theorem 1.2]. It is straightforward to see that $\phi$ extends to a continuous group homomorphism, say $\tilde\phi:\tilde J\rightarrow J$, the kernel of which is a discrete subgroup of $\tilde J$, cf. [@varshavsky Lemma 3.1]. The following argument shows that $\tilde\phi$ is surjective: We clearly have $\overline{\tilde\phi(\tilde J)}=J$ and note also that $\tilde\phi(\tilde T)=\bT(\r)$ because $\tilde T$ is compact. Now there exist elements $\gamma_1,\dots,\gamma_n\in\tilde\Delta$ such that $\Ad(\gamma_1)\Lie\bT+\dots+\Ad(\gamma_n)\Lie\bT=\Lie J$. It follows that the product $\tilde\phi(\gamma_1\tilde T\gamma_1^{-1})\cdot\dots\cdot\tilde\phi(\gamma_n\tilde T\gamma_n^{-1})$ contains an open neighborhood of the identity in $J$ and whence it follows that $\tilde\phi(\tilde J)=J$. One shows the openness of $\tilde\phi$ along the same lines.\
If a Cartan subgroup $C\subset J$ fixes some point in the image of $\tilde M$ in $X$, then it must be compact. This can be shown as in [@varshavsky Lemma 3.6] using the property (P2) only, together with the fact that $\tilde J$ is a Lie-group, which is implied by the openness of $\tilde\phi$. Finally two more facts follow easily from that observation: First, no $\bJ_i$ is equal to $\Res_{\c/\r}G_i$ so that $\bJ$ is actually a real form of $G$ and second, the stabilizer in $J$ of any point on $X$ has to contain some Cartan subgroup $C$ of $J$, this is because it is equal to the intersection of two mutually complex conjugate parabolics, namely the stabilizer in $G$ of that very point and its complex conjugate.\
We note in passing that $(\tilde J,\tilde M,G_i,P_i,\tilde\phi,\rho_0)$ is a period map too, by [@varshavsky Proposition 3.5]. One next establishes the transitivity of the $\tilde J$-action, which is accomplished by a simple dimension count (look at the stabilizers and notice that all of the maximal compact subgroups of $G$ act transitively on $G/P$).
\[boundedI\] Let $\GR_{d,h}$ be the scheme whose $S$-valued points are $h+1$-tuples $(\F,e_1,\dots,e_h)$ where $\F\in\ect_S$ is of rank $d$ and $e_1,\dots,e_h\in\Gamma(S,\F)$ generate $\F$ as an $\O_S$-module. This $\z$-scheme is projective and smooth of relative dimension $d(h-d)$. Write $\omega_{d,h}:=\Omega_{\GR_{d,h}/\z}^{d(h-d)}$ for its canonical line bundle. Let us say that a period map $(\tilde\Delta,\tilde M,G_i,P_i,\phi,\rho_0)$ is bounded if
- there exists a finite family of projective representations $\rho_j:G\rightarrow\PGL(h_j)\times\c$, together with points $y_j\in\GR_{d_j,h_j}(\c)$ such that the intersection of the stabilizers in $G$ of the points $y_j$ is equal to $P$, and there exist positive integers $p_j$ such that the pull-back of some line bundle of the form $\bigotimes_j\omega_j^{\otimes p_j}$ by means of $\rho_0$ is generated by its holomorphic global sections, where $\omega_j$ denotes the pull-back of $\omega_{d_j,h_j}$ by means of the map $G/P\rightarrow\GR_{d_j,h_j}\times\c;gP\mapsto g(y_j)$.
It is clear that bounded period maps give rise to bounded symmetric Hermitian domains $\tilde M$.
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---
abstract: 'We present an alternative approach to the result of Guentner, Higson, and Weinberger concerning the Baum-Connes conjecture for finitely generated subgroups of $SL(2,\C)$. Using finite-dimensional methods, we show that the Baum-Connes assembly map for such groups is an isomorphism.'
address: 'Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal'
author:
- Dmitry Matsnev
bibliography:
- 'bibliography-data.bib'
title: 'The Baum-Connes conjecture for countable subgroups of SL(2)'
---
Introduction {#chapterIntroduction}
============
The Baum-Connes conjecture, introduced in the early 80’s by Paul Baum and Alain Connes, connects the $K$-theory of the reduced crossed product of a $C^*$-algebra by a group acting on such algebra and the $K$-homology of the corresponding classifying space of proper actions of that group (for a formal account see [@BaumConnesHigson]).
Let $\Gamma$ be a discrete group acting on a $C^*$-algebra $A$ by automorphisms. The Baum-Connes conjecture proposes that the “assembly” morphism $$\mu : KK^{\Gamma}(\underline{E}\Gamma,A)\to K(A\rtimes_r\Gamma)$$ from the $K$-homology of the classifying space $\underline{E}\Gamma$ of proper actions of $\Gamma$ to the $K$-theory of the reduced crossed product of $A$ by $\Gamma$ is an isomorphism.
While the conjecture is formulated in terms of pairs $(\Gamma,A)$, it is possible to state it purely in terms of group $\Gamma$: one can ask whether the original conjecture holds for the group $\Gamma$ and any $C^*$-algebra $A$ on which $\Gamma$ acts. This version of the conjecture is called *the Baum-Connes conjecture with coefficients*[^1], and this will be our main concern in this work.
To indicate some connections of this conjecture with other areas, we mention that the injectivity of the assembly map implies the Novikov’s higher signature conjecture, while the surjectivity of the assembly map has to do with the Idempotents conjecture of Kadison and Kaplansky. A more comprehensive account on various versions of the Baum-Connes conjecture and the ambient areas consult [@MislinValette].
The starting point of this work is the following
\[theoremGL2Std\] For any field $K$, the Baum-Connes conjecture with coefficients holds for any countable subgroup of $GL(2,K)$.
The original proof of this theorem was based on the existence of a metrically proper and isometric action of such group on a Hilbert space and then appealing to the result of Higson and Kasparov in [@HigsonKasparov]. In this construction the group action was not constructed explicitly, and the Hilbert space on which the groups act can easily be infinite-dimensional. We are interested in giving a more elementary proof of Theorem \[theoremGL2Std\], without appealing to Hilbert space techniques and with a more direct description of the group action.
As a result, we prove the following
\[theoremMain2\] Let $K$ be any field of characteristic 0, and $\Gamma$ be a finitely generated subgroup of $SL(2,K)$. Then $\Gamma$ satisfies the Baum-Connes conjecture with coefficients.
Of course, one can also extend this result to any countable subgroup $\Gamma$ of $SL(2,K)$, since $\Gamma$ is a directed union of its finitely generated subgroups, and the Baum-Connes conjecture holds for directed unions of groups satisfying the conjecture (see [@MislinValette]).
Technical tools {#sectionTechnicalTools}
===============
In this section we collect some technical tools to be used later in the discussion.
Discrete valuations {#sectionPreliminariesDiscreteValuations}
-------------------
\[definitionDiscreteValuation\] Let $R$ be an integral domain (a commutative ring without zero divisors in which $0\ne1$). A map $\nu:R\to\Z\cup\{+\infty\}$ is called a *discrete valuation* if it satisfies the following properties for any $a, b\in R$:
- $\nu(a)=+\infty$ if and only if $a=0$
- $\nu(ab)=\nu(a)+\nu(b)$
- $\nu(a+b)\geq\min(\nu(a),\nu(b))$
Given a discrete valuation $\nu$ of an integral domain $R$, it can be extended to the field of fractions $\fracc(R)$ by $$\nu\left(\frac ab\right)=\nu(a)-\nu(b),\qquad\qquad a, b\in R.$$
Starting from a field $K$, one can regard $K$ as a ring and define the corresponding notion of a discrete valuation, following Definition \[definitionDiscreteValuation\]. Notice that the extension to the field of fractions is consistent with such treatment.
To any discrete valuation $\nu$ one associates its *ring of integers* $\mathcal O_{\nu}$. It consists of all elements of the field $K$ with non-negative valuation. Any element $\pi$ of $\mathcal O_{\nu}$ with $\nu(\pi)=1$ is called a *uniformizer* of $\nu$.
For any prime number $p$ the $p$-adic valuation $\nu_p$ on $\Q$ is defined by $$\nu_p\left(p^n\frac ab\right)=n, a, b, n\in\mathbb Z \mbox{ and } (a,p)=(b,p)=1.$$ The ring of integers of $\nu_p$ consists of all rational numbers without any occurrence of $p$ in the denominator, $p$ itself serves as a uniformizer, and the residue field $\mathcal O_{\nu}/\pi\mathcal O_{\nu}$ is isomorphic to $\Z/p\Z$.
Another construction which we shall employ is an extension of the $p$-adic valuation $\nu_p$ to an algebraic field extension of $\Q$. Suppose that such an extension is $\Q(\gamma)$, where $\gamma$ has degree $m$. Then the extension of $\nu_p$ is defined as $$\widetilde{\nu_p}(q_{m-1}\gamma^{m-1}+q_{m-2}\gamma^{m-2}+\dots+q_0)=\min\{\nu_p(q_{m-1}),\nu_k(q_{m-2}),\dots,\nu_k(q_0)\}.$$ (here $q_{m-1},q_{m-2},\dots,q_0\in\Q$.)
Simplicial tree for $SL(2,K)$ {#subsectionSimplicialTree}
-----------------------------
Suppose $K$ is a field with a discrete valuation $\nu$, its ring of integers $\mathcal O$, and a uniformizer $\pi$.
A simplicial tree $T_{\nu}$ of equivalence classes of lattices in $K^2$ is constructed as follows. The vertices of the tree are the homothety classes of $\mathcal O_{\nu}$-lattices in $K^2$. Two such vertices are connected by an edge if there exist some lattice representatives $L$ and $M$ of these vertices such that $$\pi L\subset M\subset L.$$
Notice that the valence of each vertex of $T_{\nu}$ is equal to the cardinality of the residue field.
The action of $SL(2,K)$ on $T_{\nu}$ is defined via a natural action on the corresponding lattices. If we equip the tree with a standard simplicial metric then the action is an isometry. For more in-depth discussion of this construction, consult [@Serre].
Groups of integral characteristic and Alperin-Shalen reduction
--------------------------------------------------------------
In their study of the cohomological dimension of linear groups in [@AlperinShalen], Alperin and Shalen introduced a reduction technique from a given linear group to a family of its subgroups of a special type, which we shall summarize here.
A subgroup $\Gamma$ of $SL(2,\C)$ is said to have an *integral characteristic* if the coefficients of the characteristic polynomial of every element of $\Gamma$ are algebraic integers.
We shall rely on the following technical fact proven in [@AlperinShalen].
\[theoremAlperinShalen\] Let $\Gamma$ be a finitely generated subgroup of $SL(2, \C)$ and let $A$ be a (finitely generated) ring of the matrix entries of the elements of $\Gamma$ with its field of fractions being $K$. Then there exist finitely many discrete valuations $\nu_1, \nu_2, \dots, \nu_m$ on $K$ and a finite sequence (“hierarchy”) $$\mathcal H_0, \mathcal H_1, \dots, \mathcal H_m$$ of families of subgroups of $\Gamma$ such that $\mathcal H_m$ consists of $\Gamma$ only, $\mathcal H_0$ consists of some subgroups of $\Gamma$ of integral characteristic, and for each $i$ with $1\leq i\leq m$ any group $G$ from $\mathcal H_i$ acts on a simplicial tree of $\mathcal O_{\nu_i}$-equivalence classes with the isometry of the action belonging to $\mathcal H_{i-1}$.
Metrically proper actions
-------------------------
Let $\Gamma$ be a discrete group acting on a metric space $X$ by isometries. The action is called *metrically proper* if for every bounded subset $B$ of $X$ the set $$\{g\in\Gamma | g.B\cap B\ne\emptyset \}$$ is finite.
We remark here that the condition $$\forall x\in X \; \forall C \; \{g\in\Gamma | \dist(x,g.x)<C \} \mbox{ is finite}$$ implies that the action is metrically proper (to see this, one can take $C$ to be three times the diameter of $B$ and use the fact that the action is an isometry.)
Proof of the theorem
====================
In what follows, we shall extensively use the fact that the Baum-Connes conjecture with coefficients passes to subgroups (see [@ChabertEchterhoff]), without explicitly mentioning it.
The argument will be structured as follows. In Section \[sectionSubgroupsAlgebraic\] we prepare the ground by constructing a metrically proper action for finitely generated subgroups over an algebraic extension of $\Q$.
In Section \[sectionReductionToIntegral\] we prove Theorem \[theoremReductionToIntChar\], which shows that in order to prove the Baum-Connes conjecture for a given finitely generated subgroup of $SL(2,\C)$, it is sufficient to prove it for the subgroups of integral characteristic of the original group.
In Section \[sectionZariskiDense\] we treat the case of subgroups of integral characteristic which are Zariski-dense in $SL(2,\C)$. Namely, we prove in Theorem \[theoremZariskiDense\] that for all such subgroups coming from a finitely generated group in $SL(2,\C)$ we started with, the Baum-Connes conjecture holds by means of using the results from Section \[sectionSubgroupsAlgebraic\].
Finally, in Section \[sectionZariskiNondense\] we prove in Theorem \[theoremZariskiNondense\] that all countable subgroups of $SL(2,\C)$ of integral characteristic which are not Zariski-dense in it satisfy the Baum-Conjecture.
Subgroups over algebraic fields {#sectionSubgroupsAlgebraic}
-------------------------------
We start with a technical construction of a certain proper action which will be used later.
\[lemmaProperActionAlgebraic\] Let $\Gamma$ be a finitely generated subgroup of $SL(2,K)$, where $K$ is a finite extension of $\Q$. Then $\Gamma$ acts metrically properly on a finite-dimensional space (a finite product of simplicial trees and a hyperbolic plane).
We can treat $\Gamma$ as defined over a finitely generated ring which we can take, enlarging it if necessary, to be $A=\Z[\frac1s,\gamma]$, where $s$ is a natural number (the l.u.b. of all the denominators of the rationals participating in the entries of the generating set) and $\gamma$ is an algebraic integer. We assume that prime factorization of $s$ is $p_1\cdot p_2\cdots p_n$ with nonrepeating terms.
For each $p_k$, $k=1,2,\dots,n$ consider the $p_k$-adic valuation $\nu_k$ on $\Q\cap A$ and extend it to the whole $A$ and its fraction field by the usual rule. Denote by $T_k$ the simplicial tree corresponding to $\widetilde{\nu_k}$ and denote by $\alpha_k$ the induced action of $\Gamma$ on this tree.
Define an action $\alpha_H$ of $\Gamma$ on the 2-dimensional real hyperbolic space $\HH_2$ via the natural isometric action of $SL(2,\R)$ on $\HH_2$.
We claim that the diagonal action $$\alpha=\alpha_1\times\alpha_2\times\dots\times\alpha_n\times\alpha_H$$ on the product of trees and a hyperbolic space $\HH_2$ $$T=T_1\times T_2\times\dots\times T_n\times\HH_2$$ is metrically proper. To clarify this, we shall show that for any bounded set $B$ $$\#\{g\in\Gamma|g.B\cap B\neq\emptyset\}<\infty.$$
Since it is enough to prove the statement for sufficiently large sets $B$ only, we shall enlarge $B$ in the following way: let $B_k$ be the projection of $B$ to the $k$-th tree, and $B_{\mathbb H_2}$ be the projection of $B$ to the hyperbolic space. Clearly $B\subseteq B_1\times\dots\times B_n\times B_{\mathbb H_2}$, and we shall be working with the latter set instead of the original $B$.
Now the set $\{g\in\Gamma|g.B\cap B)\neq\emptyset\}$ is actually $$\bigcap_{k=1}^n\{g\in\Gamma|g.B_k\cap B_k\neq\emptyset\}\cap\{g\in\Gamma|g.B_{\mathbb H_2}\cap B_{\mathbb H_2}\neq\emptyset\},$$ therefore it is enough to show that for any number $C$ and any points $v_k\in T_k$ ($k=1,\dots, n$) and $x\in\mathbb{H}_2$, $$\begin{gathered}
\label{propernessWithC}
\#\{g\in\Gamma|\dist_{T_k}(g.v_k,v_k)<C \;\mbox{for all }k=1,\dots,n,\\\;\mbox{and}\;\dist_{\mathbb{H}_2}(g.x,x)<C\}<\infty.\end{gathered}$$
Further, by triangle inequality it is sufficient to check condition (\[propernessWithC\]) for the “root” vertices $v_0$ of the trees and the “center” point $x_0$ of the hyperbolic space.
Now let $g=(g_{ij})\in\Gamma$ be an element from the set in question. Here each matrix entry $g_{ij}$ belongs to $\Z[\frac1{p_1},\dots, \frac1{p_n},\gamma]$, say $$g_{ij}=\sum_{0\leq l\leq m-1}\frac{a_{ijl}}{\prod_{k=1}^np_k^{n_{ijkl}}}\gamma^l,\qquad a_{ijl}, n_{ijkl}\in\Z,\; (a_{ij},p_k)=1, \; k=1,\dots, n.$$
The distance from $g.v_0$ to $v_0$ in the tree $T_{p_k}$ is bounded only if each matrix entry has limited from above powers $n_{ijkl}$ in the denominator, which means the denominator itself should be bounded.
Since $\dist_{\mathbb{H}_2}(g.x_0,x_0)<C$ means $\cosh(\dist_{\mathbb{H}_2}(g.x_0,x_0))<\cosh C$, and $$\cosh(\dist_{\mathbb{H}_2}(g.x_0,x_0))=\sum g_{ij}^2,$$ each matrix entry should be bounded. Together with the previous observation, this leads to a finite number of choices for $a_{ijl}$ and $n_{ijkl}$, and thus there are finitely many such elements $g$ in the intersection.
Reduction to groups of integral characteristic {#sectionReductionToIntegral}
----------------------------------------------
In this section we applythe Alperin-Shalen hierarchy construction which will allow us to reduce the proof of the Baum-Connes conjecture for any finitely generated subgroup of $SL(2,\C)$ to subgroups of integral characteristic.
The main motivation for the study of the isotropy of group actions on trees is the following result:
\[theoremOyonoOyono\] Let $\Gamma$ be a discrete countable group acting on a tree. Then the Baum-Connes conjecture holds for $\Gamma$ if and only of it holds for all the isotropy subgroups of the action on vertices of the tree.
\[theoremReductionToIntChar\] Let $\Gamma$ be a finitely generated subgroup of $SL(2,\C)$. Then the Baum-Connes conjecture with coefficients holds for $\Gamma$ if and only if it holds for all subgroups of $\Gamma$ of integral characteristic.
Given $\Gamma$, apply Theorem \[theoremAlperinShalen\] and consider a hierarchy of families of subgroups of $\Gamma$ together with the actions on trees which that theorem furnishes. Repeated applications of Theorem \[theoremOyonoOyono\] allow one to reduce the Baum-Connes conjecture for the top level of the hierarchy (for $\Gamma$ itself, that is) to the one for the bottom of the hierarchy, which contains subgroups of $\Gamma$ of integral characteristic only.
Zariski-dense subgroups {#sectionZariskiDense}
-----------------------
Now we need to prove the Baum-Connes conjecture for subgroups of integral characteristic. In this section we concentrate on integral characteristic subgroups $\Gamma$ of $SL(2,\C)$ whose Zariski closure is the entire $SL(2,\C)$.
The following result essentially goes back to Zimmer (cf. [@ZimmerErgodicTheory Lemma 6.1.7]).
\[lemmaZimmerEmbedding\] Let $\Gamma$ be a Zariski-dense subgroup of $SL(2,\C)$ of integral characteristic. Then there exists a faithful representation $$\alpha: SL(2,\C)\to GL(4,\C)$$ such that the matrix entries of every element of $\alpha(\Gamma)$ is an algebraic integer.
We shall write $G$ for the Zariski-closure of $\Gamma$ in $SL(2,\C)$, that is, $G=SL(2,\C)$. Let $f_g$ be a map $G\to\C$ defined by $$f_g:h\mapsto\tr(gh), \qquad h\in G.$$ Notice that $f_{g_1}(h)+f_{g_2}(h)=\tr((g_1+g_2)h)$ and $\lambda f_g(h)=\tr((\lambda g)h)$ for any $g_1, g_2, h\in G$ and $\lambda\in\C$. This allows us to consider $f_g(h)$ as a short-hand notation for $\tr(gh)$ for any $g\in\C G$ and $h\in G$. Let $$V=\Span_{\C}\langle f_g \rangle_{g\in\C G}.$$ Since the conditions defining $f_g$ are linear with respect to the entries of $g$, the linear space $V$ has finite dimension (more precisely, its dimension is $4$.)
Consider the following action of $G$ on $V$: $$\label{equationSL2ActionZDense}
g.f_h=f_{gh}, \qquad g\in G, h\in\C G.$$ Since this action is linear, we have a representation of $G$.
Let $$W=\Span_{\C}\langle f_g \rangle_{g\in\C\Gamma}.$$ This subspace is $\Gamma$-invariant and, since $\Gamma$ is Zariski-dense in $G$, is also $G$-invariant. Thus $W=V$.
Let $g_1, g_2, g_3, g_4\in\Gamma$ be such that $\left\lbrace f_{g_1}, f_{g_2}, f_{g_3}, f_{g_4}\right\rbrace $ is a basis of $V$ (we can arrange this because $V$ is generated by $f_g$ for $g\in\Gamma$). With respect to this basis the action is given by a matrix $(\alpha_{ij}^g)$, such that $$\label{equationSL2ActionMatrix}
g.f_{g_i}(h)=\sum_{j=1}^4\alpha_{ij}^gf_{g_j}(h), \qquad g, h\in G, i=1,\dots,4.$$ Thus we obtain a representation $\alpha: G\to GL(4,\C)$.
We confirm that $\alpha$ is faithful by showing that from the identity $g.f_1(h)=f_1(h)$ for all $h\in G$ follows that $g=1$. To do this, take elementary matrices for $h$ and write this condition entries-wise.
If $g\in\Gamma$, means that in particular $$\tr(gg_ig_k)=g.f_{g_i}(g_k)=\sum_{j=1}^4\alpha_{ij}^gf_{g_j}(g_k)=\sum_{j=1}^4\alpha_{ij}^g\tr(g_jg_k), \qquad i, k=1,\dots,4.$$ Then $\{\alpha_{ij}^g\}$ are the solutions of a system of linear equations with algebraic coefficients, and therefore the matrix entries $(\alpha_{ij}^g)$ of the representation $\alpha$ are algebraic.
\[theoremZariskiDense\] Let $G$ be a finitely generated subgroup of $SL(2,\C)$. Then the Baum-Connes conjecture with coefficients holds for any subgroup $\Gamma$ of $G$, provided that $\Gamma$ has integral characteristic and is Zariski-dense in $SL(2,\C)$.
We may assume that given $G$ contains at least one subgroup $\Gamma$ which is both of integral characteristic and is Zariski-dense. Select one such $\Gamma$ and apply Lemma \[lemmaZimmerEmbedding\], yielding an embedding $\alpha: SL(2,\C)\to GL(4,\C)$. Notice that if $\Gamma'$ is any Zariski-dense subgroup of $G$ of integral characteristic then its image $\alpha(\Gamma')$ is conjugate to a subgroup whose matrix entries are algebraic. Moreover, since $\alpha(G)$ is finitely generated, all matrix entries of its elements belong to a finitely generated subring of $\C$, and this is true for its subgroup $\alpha(\Gamma')$ as well, which means that the entries of the elements of $\alpha(\Gamma')$ belong to a certain finitely generated subring of algebraic numbers which depends on the original $G$ only, rather than on $\Gamma'$.
By the Primitive Element Theorem $\alpha(\Gamma')\subseteq GL(4,K)$ for some field $K$ with $[K:\Q]<\infty$. Take $H=\alpha(SL(2,\C))$. We see that both $\alpha(\Gamma')$ and $H\cap GL(4,K)$ are Zariski-dense in $H$, thus they are both defined over $K$ by [@ZimmerErgodicTheory Proposition 3.1.8]. This means $\alpha(\Gamma')$ is locally isomorphic to a subgroup of $SL(2,K)$ (see [@Zimmer Theorem 7]). We can represent $\alpha(\Gamma')$ as a directed union of its finitely generated subgroups and for each such subgroup apply Lemma \[lemmaProperActionAlgebraic\], thus obtaining a metrically proper action of on a finite-dimensional space. Via a finite-dimensional version of the result of Higson and Kasparov in [@HigsonKasparov], the Baum-Connes conjecture for such subgroup follows, hence it follows for $\Gamma'$ as well.
Zariski-non-dense subgroups {#sectionZariskiNondense}
---------------------------
Finally, we discuss the case of Zariski-non-dense subgroups of integral characteristic. The rest of this section is devoted to the proof of the following
\[theoremZariskiNondense\] Let $\Gamma$ be a countable non-Zariski-dense subgroup of $SL(2,\C)$ of integral characteristic. Then the Baum-Connes conjecture with coefficients holds for $\Gamma$.
Let us start with some preliminary remarks on algebraic Lie groups. Suppose $G$ is a Zariski–closed proper subgroup of $SL(2,\C)$. We write $G_0$ for the Zariski-connected component of the unit of $G$. It is known that $G_0$ is a normal subgroup of $G$ of finite index [@Borel I.1.2].
Since for algebraic groups the notions of connected and irreducible components coincide [@Borel AG.17.2], $G_0$ is abelian if and only if its Lie algebra is commutative [@Hochschild IV.4.3]. Since $G$ is a proper subgroup of $SL(2,\C)$, its dimension is strictly less than $3$. In the subsections below we shall address each dimension case separately.
Dimension $0$ {#dimension-0 .unnumbered}
-------------
In this case $\dim G_0=0$ as well, and, since $G_0$ is connected, we conclude that it is trivial. The group $G$ itself, being a finite extension of $G_0$, is finite, whence the Baum-Connes conjecture for $G$ holds trivially.
Dimension $1$ {#dimension-1 .unnumbered}
-------------
The Lie algebra of $G$ (and $G_0$ as well) has to be $1$-dimensional. In particular, it has to be commutative, thus $G_0$ is abelian. There are only two (up to conjugacy) connected abelian $1$-dimensional groups, namely $\left\{\left. \m{a}{0}{0}{a^{-1}}\right| a\in\C^{\times}\right\}$ and $\left\{\left. \m{1}{b}{0}{1}\right| b\in\C\right\}$. We shall treat them separately.
Suppose $G_0=\left\{\m{a}{0}{0}{a^{-1}}\right\}$ (up to conjugacy). Then, since $G_0$ is normal in $G$, any conjugate of $\m{a}{0}{0}{a^{-1}}$ by any element in $G$, say $\m{g_{11}}{g_{12}}{g_{21}}{g_{22}}$, has to have the same diagonal form: $$\begin{gathered}
\m{g_{11}}{g_{12}}{g_{21}}{g_{22}}\m{a}{0}{0}{a^{-1}}\m{g_{11}}{g_{12}}{g_{21}}{g_{22}}^{-1}=\\
\m{*%ag_{11}g_{22}-a^{-1}g_{12}g_{21}
}{(a^{-1}-a)g_{11}g_{12}}{(a-a^{-1})g_{21}g_{22}}{*%a^{-1}g_{11}g_{22}-ag_{12}g_{21}
}=
\m{b}{0}{0}{b^{-1}}.\end{gathered}$$ This means that $g_{11}g_{12}=0$ and $g_{21}g_{22}=0$. To satisfy the first condition, we need to take either $g_{11}$ to be zero or $g_{12}$ to be zero. Thus $G$ may contain only matrices with zeros on the diagonal, or off the diagonal: $$G\subseteq\left\{\m{a}{0}{0}{a^{-1}}, \m{0}{a}{-a^{-1}}{0}\right\}=H.$$ This group $H$ is amenable, and, modifying Theorem \[theoremOyonoOyono\], it is possible to show (see [@MislinValette Theorems 5.18 and 5.23]) that any countable subgroup of $H$ satisfies the Baum-Connes conjecture with coefficients[^2]. Now suppose $G_0=\left\{\m{1}{b}{0}{1}\right\}$ (again, up to conjugacy). Since $G_0$ is normal in $G$, any conjugate of $\m{1}{b}{0}{1}$ by an arbitrary element $\m{g_{11}}{g_{12}}{g_{21}}{g_{22}}$ element of $G$ should have the same form: $$\m{g_{11}}{g_{12}}{g_{21}}{g_{22}}\m{1}{b}{0}{1}\m{g_{11}}{g_{12}}{g_{21}}{g_{22}}^{-1}=\m{*}{*}{-bg_{21}^2}{*}=\m{*}{*}{0}{*}.$$ Thus we have $g_{21}=0$, which means that $G$ contains only matrices of the form $\m{a}{b}{0}{a^{-1}}$, and we shall discuss this group in the following subsection.
Dimension $2$ {#dimension-2 .unnumbered}
-------------
Let $H$ denote the subgroup of $SL(2,\mathbb C)$ consisting of all the matrices of the form $\m{a}{b}{0}{a^{-1}}$, where $a\in\C^{\times}, b\in\C$. Note that its Lie algebra consists of the matrices of the form $\m{a}{b}{0}{-a}$, $a, b\in\C$.
\[MaximalityOfH\] Any subgroup $K$ of $SL(2,\mathbb C)$, which includes $H$ and some element not in $H$, coincides with the whole group $SL(2,\mathbb C)$.
Suppose $K$ contains some element with $g_{21}\ne0$. Then we can multiply this element by an element of $H$ on the right to get .
Now for any complex numbers $a$ and $b$ with $a\ne0$ we can multiply on the left by to get . Since $K$ is a group, it ought to contain all inverses as well, in particular . This means that $K$ contains all matrices of determinant $1$ with $0$ in the upper left corner.
Finally, let us take an arbitrary element of $SL(2,\mathbb C)$, say . Since we already know that all matrices with $s_{21}=0$ belong to $H$, and therefore to $K$, the essential part of the argument is to show that any such matrix with $s_{21}\ne0$ belongs to $K$. The identity $$\m{s_{11}}{s_{12}}{s_{21}}{s_{22}}=\m{1}{s_{11}s_{21}^{-1}}{0}{1}\m{0}{-s_{21}^{-1}}{s_{21}}{s_{22}}$$ completes the proof, since all matrices on the right-hand-side belong to $K$.
Now we provide some technical results about Lie subalgebras of $\mathfrak{sl}(2,\C)$.
For any 2-dimensional noncommutative Lie algebra there exists a basis $\{x_1, x_2\}$ with multiplication rule $[x_1x_2]=x_1$.
\[OnlyOne\] The Lie algebra $\mathfrak{sl}(2,\C)$ contains only one (up to conjugation) $2$-dimensional Lie subalgebra, namely $\left\{\left.\m{a}{b}{0}{-a}\right| a, b\in\C\right\}$.
Suppose we have a $2$-dimensional noncommutative Lie subalgebra $\mathfrak h$ of $\mathfrak{sl}(2,\C)$. Let $\{x_1, x_2\}$ denote the basis of $\mathfrak h$ constructed in the lemma above. A priori there could be two possibilities: both eigenvalues of the matrix $x_2$ coincide (and therefore are zeros) or they are distinct. In the first case $x_2$ is conjugate to its Jordan form, namely $\m{0}{1}{0}{0}$, and if $x_1$ after same conjugation has the form $\m{a}{b}{c}{d}$, then the multiplication condition $[x_1x_2]=x_1$ is $$\left[\m{a}{b}{c}{d},\m{0}{1}{0}{0}\right]=\m{-c}{a-b}{0}{c}=\m{a}{b}{c}{d},$$ from where we conclude that $a=b=c=d=0$, which means $x_1=0$ and therefore can not serve as basis element, so that the case where both eigenvalues of the matrix $x_2$ coincide can not happen. Now suppose that the eigenvalues of $x_2$ are distinct, say $\lambda$ and $-\lambda$. Conjugating $x_1$ and $x_2$, we write the multiplication condition as $$\left[\m{a}{b}{c}{d},\m{\lambda}{0}{0}{-\lambda}\right]=\m{0}{-2b\lambda}{2c\lambda}{0}=\m{a}{b}{c}{d},$$ so that we have $a=d=0$ and $2c\lambda=c$, $-2b\lambda=b$. We are looking for solutions with at least one of the coefficients $b$ and $c$ being non-zero, therefore we end up with two possibilities:
1. $b=0\ne c$, $\lambda=\frac12$
2. $c=0\ne b$, $\lambda=-\frac12$
Thus any non-commutative $2$-dimensional Lie subalgebra of $\mathfrak{sl}(2,\mathbb C)$ is conjugate-equivalent to $\mathfrak h_1=\mathbb C\m{0}{0}{1}{0}\oplus\mathbb C\m{\frac12}{0}{0}{-\frac12}$ or $\mathfrak h_2=\mathbb C\m{0}{1}{0}{0}\oplus\mathbb C\m{-\frac12}{0}{0}{\frac12}$. Finally, $\mathfrak h_1$ and $\mathfrak h_2$ are conjugate to each other via the matrix . By scaling the second matrix in $\mathfrak h_2$ , we obtain the representation $\left\{\left. \m{a}{b}{0}{-a} \right| a, b\in\mathbb C\right\}$.
Now we show that $\mathfrak{sl}(2,\C)$ does not contain any commutative $2$-dimensional subalgebras. Suppose one such exists and has a basis $\{x, y\}$. Conjugating by some matrix, we can put $y$ into Jordan form, and let $x$ be represented by under the same conjugation. We have two possibilities: the eigenvalues of the matrix, representing $y$, coincide (and therefore are zeros) or they are distinct, and by scaling the matrix we assume that they are $1$ and $-1$. In the first case the commutativity condition can be written as $$\m{x_{11}}{x_{12}}{x_{21}}{-x_{11}}\m{0}{1}{0}{0}=\m{0}{1}{0}{0}\m{x_{11}}{x_{12}}{x_{21}}{-x_{11}},$$ which leads to $x_{11}=x_{21}=0$, so that $x$ is a scalar multiple of $y$, and this can not happen. In the second case we have $$\m{x_{11}}{x_{12}}{x_{21}}{-x_{11}}\m{1}{0}{0}{-1}=\m{1}{0}{0}{-1}\m{x_{11}}{x_{12}}{x_{21}}{-x_{11}},$$ this means $x_{12}=x_{21}=0$, and again we have a contradiction with linear independence of $x$ and $y$.
Getting back to the group $G_0$, we see that Lemma \[OnlyOne\] describes the Lie algebra of $G_0$, up to conjugacy. Therefore $G_0$ and $H$ are conjugate to each other. Lemma \[MaximalityOfH\] confirms that there are no proper subgroups of $SL(2,\C)$ larger than $H$, therefore we conclude that $G=G_0$.
Finally, $G$ is a semidirect product $$\left\{\left. \begin{pmatrix}1&b\\0&1\end{pmatrix} \right| b\in\C\right\}\rtimes\left\{\left. \begin{pmatrix}a&0\\0&a^{-1}\end{pmatrix} \right| a\in\C^{\times}\right\}$$ of two abelian groups, hence it is amenable, and we conclude that any finitely generated subgroup of $G$ (and hence any countable one) satisfies the Baum-Connes conjecture with coefficients by applying [@MislinValette Theorem 5.23].
Now Theorem \[theoremMain2\] has been proven completely.
Acknowledgements {#acknowledgements .unnumbered}
================
This paper is based on one part of the author’s thesis completed under the supervision of Nigel Higson at Penn State University. The author is very grateful to Professor Higson for his invaluable guidance, comments, and suggestions. Also the author is thankful to an anonymous referee for her or his comments and suggestions.
[^1]: Some authors refer to it as to the Baum-Connes *property* with coefficients, in the view of the counterexamples (modulo a statement due to Gromov) by Higson, Lafforgue, and Skandalis in [@HigsonLafforgueSkandalis].
[^2]: Theorem 5.18 in [@MislinValette] shows that $H$ satisfies the Baum-Connes conjecture with trivial coefficients, while Theorem 5.23 proves that any countable subgroup of such group satisfies the conjecture with arbitrary coefficients.
|
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abstract: abstract
author:
- |
Jacob Steinhardt\
Computer Science Department\
Stanford University\
[[email protected]]{} Percy Liang\
Computer Science Department\
Stanford University\
[[email protected]]{}
bibliography:
- 'refdb/all.bib'
title: '[Unsupervised Risk Estimation Using Only Conditional Independence Structure]{}'
---
introduction framework extensions learning experiments discussion
matching-details tensor-proof robustness-proof gradient-proof learning-extensions
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---
abstract: 'Quantum entanglement, the essential resource for quantum information processing, has rich dynamics under different environments. Probing different entanglement dynamics typically requires exquisite control of complicated system-environment coupling in real experimental systems. Here, by a simple control of the effective solid-state spin bath in a diamond sample, we observe rich entanglement dynamics, including the conventional asymptotic decay as well as the entanglement sudden death, a term coined for the phenomenon of complete disappearance of entanglement after a short finite time interval. Furthermore, we observe counter-intuitive entanglement rebirth after its sudden death in the same diamond sample by tuning an experimental parameter, demonstrating that we can conveniently control the non-Markovianity of the system-environment coupling through a natural experimental knob. Further tuning of this experimental knob can make the entanglement dynamics completely coherent under the same environmental coupling. Probing of entanglement dynamics, apart from its fundamental interest, may find applications in quantum information processing through control of the environmental coupling.'
author:
- 'F. Wang$^{1}$, P.-Y. Hou$^{1}$, Y.-Y. Huang$^{1}$, W.-G. Zhang$^{1}$, X.-L. Ouyang$^{1}$, X. Wang$^{1}$, X.-Z. Huang$^{1}$, H.-L. Zhang$^{1}$, L. He$^{1}$, X.-Y. Chang$^{1}$, L.-M. Duan[^1]'
title: |
Observation of entanglement sudden death and rebirth\
by controlling solid-state spin bath
---
Besides its significance as a fundamental concept in quantum mechanics, entanglement has been recognized as an essential resource for quantum computation and communication [@1; @2; @3; @4]. In any real experimental system, due to its inevitable coupling to the surrounding environment, entanglement degrades with time, leading to various kinds of entanglement dynamics under different environmental couplings [@5; @6; @7; @8; @9]. The most common one is the asymptotic decay of the entanglement, where the entanglement approaches zero (typically exponentially) as the time goes to infinity. This behavior is similar to the quantum coherence decay and arises when the environmental coupling is dominated by pure dephasing [@5; @6]. Under more complicated dissipation, entanglement may completely vanish in a finite (typically short) time interval, a phenomenon called the entanglement sudden death (ESD) [@7; @8; @9; @10; @11]. The ESD is identified to be more disruptive to quantum information processing due to the fast disappearance of entanglement [@7; @8; @9]. Under more unusual situations which require non-Markovianity of the system-environment coupling, the entanglement can reappear after its sudden death for a while, which is termed as the entanglement rebirth [@9]. The entanglement sudden death and rebirth have been extensively studied theoretically [12,14,15,16,17,18,19]{}. On the experimental side, all-optical quantum experiments can simulate environmental couplings of photonic qubits by linear optics elements to demonstrate the ESD [@20; @21] as well as the non-Markovian coupling [@22; @23; @24]. ESD is also observed between atomic ensembles but with no entanglement rebirth or non-Markovian behavior [25]{}. It is desirable to find an experimental system where the natural system-environment coupling and its Markovianity can be controlled to probe different kinds of entanglement dynamics in a single system, including the ESD and the entanglement rebirth. Probe of the non-Markovian dynamics plays an important role in control of open quantum systems [@26; @27; @28].
In this paper, we demonstrate that rich entanglement dynamics can be observed in a single diamond sample by controlling the effective coupling of spin qubits to the solid-state spin bath. The system-environment coupling is controlled by the dynamical decoupling pulses, which provide a tunable filter function to modify the contributing noise spectrum of the bath. Depending on the control parameter in this filter function, we observe the Markovian asymptotic decay as well as the sudden death of the entanglement. Tuning of the same parameter allows us to go to non-Markovian region of the dissipation, where we observe the entanglement rebirth after the ESD. Further tuning of this parameter can make the entanglement dynamics completely coherent under this spin bath, showing entanglement Rabi oscillations.
Our experimental demonstration makes use of the hybrid spin system composed of the electron spin and the host $^{14}N$ nuclear spin associated with the nitrogen-vacancy center in diamond (Fig. 1(a)). The surrounding $^{13}C$ nuclear spins in the crystal lattice act as a natural reservoir and are coupled to the electron spin via varying anisotropic hyperfine interactions (Fig. 1(a)). The impact of the $^{13}C$ bath on the nitrogen nuclear spin can be ignored due to the lower interaction strength compared to that of the electron spin.
{width="160mm"}
The experiments are performed at room temperature on a bulk diamond sample with a natural $^{13}C$ abundance. Under an external magnetic field of $479$ Gauss along the NV symmetry axis, both the electron and the nitrogen nuclear spin are polarized via optical pumping [@29]. We prepare entangled state between the two spins using a rf-induced $\pi$ rotation of the nuclear spin conditional on the state of the electron spin (Fig. 1(b)). Coherence of the electron spin is protected by two fast $\pi$ rotations symmetric on both sides of the rf pulse with the form $\tau_{1}-\pi-2\tau_{1}-\pi-\tau_{1}$, where $2\tau_{1}$ is the rf pulse duration. Final state density matrix is reconstructed from measurements by two-bit state tomography [@30].
Entanglement for a two-qubit state $\rho$ can be quantified by concurrence C [@31], which is given by $C=max\{0,\Gamma\}$ where $\Gamma=\sqrt{%
\lambda_1}-\sqrt{\lambda_2}-\sqrt{\lambda_3}-\sqrt{\lambda_4}$ with $%
\lambda_i$ denoting the eigenvalues of the matrix $\rho(\sigma_y\otimes%
\sigma_y)\rho^*(\sigma_y\otimes\sigma_y)$ in decreasing order, and $\rho^*$ denoting the complex conjugate of $\rho$ in the computational basis $%
\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$. This quantity can be interpreted as a measure for the quantum correlation present in the system and takes the value from $0$ to $1$.
We start by studying the dynamics of entanglement under free evolution of the two spins. Under this condition, the electron spin is subject to fast decoherence noise induced by the bath. Therefore, asymptotic decay of entanglement is observed by fitting the concurrence to $exp(-(t/T_c)^2)$ (Fig. 1(c)), where $T_c$ is the electron spin coherence time. A Hahn echo can cancel part of the interaction with the spin bath and thus significantly increase the coherence time. As a result, the observed decay of entanglement under the Hahn echo is also asymptotic but with a much slower decay rate as shown in Fig. 1(d).
{width="160mm"}
A more complicated behavior of entanglement dynamics can be observed if one extends the Hahn echo to periodic repetitions of the Carr-Purcell-Meiboom-Gill (CPMG) sequence (Fig. 2(a)). The CPMG sequence acts as a filter of the bath in the frequency domain, whose center frequency can be tuned by changing the inter-pulse duration $2\tau$. In experiments we selectively apply four specific filters to the system by choosing corresponding $\tau$ and monitor the evolution of entanglement by gradually increasing the number of pulses $N
$ (and therefore the total evolution time). Three distinct behaviors of entanglement dynamics are observed: (i) asymptotic decay with $\tau =2$ $\mu s$ (Fig. 2(b)), (ii) entanglement sudden death with $\tau =0.47$ $\mu s$ (Fig. 2(c)), (iii) non-Markovian behavior which shows entanglement sudden death and rebirth with $\tau=0.44$ $\mu s$ and $\tau =0.51$ $\mu s$ (Fig. 2(d,e)).
![Analysis of the entanglement evolution under CPMG sequences. (a) Upper panel: decay of electron coherence as a function of half inter-pulse duration $\protect\tau$ with $N=16$. Black dots and solid line are experimental data taken every $10$ ns. Red dashed line denotes the simulation result under six identified carbon nuclear spins in the bath with their hyperfine parameters calibrated by the method described in the supplementary material [@41]. Arrows indicate corresponding $%
\protect\tau$ in Fig. 2(b,c,d,e) respectively. Lower panel: calculated entanglement concurrence from electron coherence in the upper figure. Inset: experimental results with the same time range in the blue dashed square. Blue circles are experimental data taken every $50$ ns. (b) Upper panel: noise spectrum constructed from electron spin coherence in Fig. 3(a). Central components correspond to the first order resonance of the nuclear spin bath. Higher order resonance terms are shown as peaks at lower frequencies. Lower panel: filter function of the CPMG sequences with $N=16$. Peaks appear at $\protect\omega_0/(2\protect\pi)=1/(4\protect\tau)$. (c,d,e) Simulation of entanglement decay under the same condition described in Fig. 2(c,d,e) respectively. Dashed lines are results corresponds to single identified carbon nuclear spin. Solid line is the result with effects of six carbon nuclear spins combined together.](Fig3.pdf){width="160mm"}
The phenomena can be explained by the spectral filtering of the spin bath through the CPMG sequence, which either suppresses [@32; @33; @34] or resonantly amplifies [@35] the interactions of the electron spin with the structured bath. The power spectrum $S(\omega)$ of the bath includes two parts [@36]: thermal noise due to the random orientations and flip-flops of the bath spins at room temperature, and dynamical quantum noise caused by the evolution of nuclear spins. Under the CPMG sequence, both components of the surrounding bath can be detected by measuring the coherence decay $W(t)$ of the electron spin [@37; @38] $$W(t)=exp[-\chi(t)]=exp\left[-\int_{0}^{\infty}\frac{d\omega}{\pi}\frac{S(\omega)}{%
\omega^2}F_N(\omega t)\right]$$ where $$F_N(\omega t)=8sin^2(\frac{\omega t}{2})\frac{sin^4(\frac{\omega t}{4N})}{%
cos^2(\frac{\omega t}{2 N})}$$ is the filter function [@39] associated with the pulse number $N$ and the total evolution time $t=2N\tau$ of the CPMG sequence. The detected probability corresponds to $P_0=(1/2)(W(t)+1)$.
We focus on the dynamical quantum noise of the bath, as the effects of static thermal noise are removed by the CPMG sequence. According to equation (1), the precession of the $^{13}C$ spin bath produces a coherence dip of electron spin centered at the nuclear Larmor frequency $\omega_L=\gamma_C B_z
$, where $\gamma_C$ is the nuclear spin gyromagnetic ratio and $B_z$ is the external magnetic field. Due to the frequency shift caused by the hyperfine interactions of multiple nuclear spins in the bath, the observed signal is shown as a broad collapse instead of a dip (upper panel of Fig. 3(a)). Higher order resonances of the bath are observed as broadening of the collapses and splitting of isolated dips. The influence of the bath on the electron spin coherence takes effect on the entangled state shared between the electron and the nuclear spin and leads to disruption of entanglement as shown in lower panel of Fig. 3(a), where entanglement concurrence is calculated from the detected $P_0$ in the upper panel. We experimentally measure the entanglement concurrence around the bath and observe a consistent behavior with the calculation prediction (Inset of Fig. 3(a)).
Spectrum of the dynamical quantum noise can be constructed from the coherence decay of the electron spin under the CPMG sequence with a spectrum decomposition method [@37]. As shown in the upper panel of Fig. 3(b), central components in the spectrum correspond to first order resonance of the nuclear spin bath and peaks in the lower frequencies are higher order resonance terms. With fixed inter-pulse duration $2\tau$, filter function of the CPMG sequence covers a narrow frequency region centered at $%
\omega_0=\pi/(2\tau)$ (See lower panel of Fig. 3(b) for filter function). If we control the filter peak to be off resonant from the bath (for example $%
\tau=2\mu s$), bath influence on the electron spin is sufficiently suppressed, thus concurrence of entanglement is shown as Markovian asymptotic decay and can be kept to the limit of electron spin $T_2$ (Fig. 2(b)). On the contrary, if we control the filter to be on resonant with the bath, interactions of the electron spin with multiple carbons are amplified, leading to information flow between the electron spin and the bath. To understand the mechanism of the information flow, we experimentally identify six carbon nuclear spins in the bath which give strong influence on the electron spin coherence [@40] (Fig. 3(a) and Fig. 1 in [@41]) and perform simulations with the $8$-qubit system [@41]. From Fig. 3(c), when the filter amplifies the center of the bath ($\tau=0.47\mu s$), information exchanges between the electron spin and several carbons with separate exchange frequency are mixed together and on average are shown as dissipation from the electron spin to the bath. Consequently, entanglement vanishes completely in a finite time and ESD is observed. Comparatively, when the filter amplifies the edge of the bath ($\tau=0.44\mu s$ and $%
\tau=0.51\mu s$), information flow is dominant by the interaction of the electron spin with the resonant carbon (C1 with $\tau=0.44\mu s$, C2 with $%
\tau=0.51\mu s$), thus revival of entanglement is observed after the concurrence drops to the minimal value (Fig. 3(d,e)). Full revival is prohibited by the interactions with other carbons in the bath.
{width="160mm"}
For the initially entangled states evolving under the CPMG sequence, concurrence coincides with the measure for quantum non-Markovianity of the electron-bath system [@22], which in the standard approach is defined by the trace difference between two quantum states $D(\rho_1(t),\rho_2(t)=(1/2)||%
\rho_1(t)-\rho_2(t)||$, where $\rho_{1,2}(t)=|\phi_{1,2}(t)\rangle\langle
\phi_{1,2}(t)|$ and $|\phi_{1,2}(0)\rangle=(1/\sqrt{2})(|0\rangle_e\pm|1%
\rangle_e)$. Therefore, the observed revival of entanglement also quantifies the memory effect of the system.
To further probe the non-Markovianity of the system, we control the filter to be on resonant with isolated single carbon where influence of the unwanted spin bath is suppressed (Fig. 4(a)). Under this condition, the isolated carbon acts as a register in a way that coherence information flows back and forth between the electron-nuclear pair and the electron-carbon pair, thus leads to entanglement Rabi oscillations. As can be observed from Fig. 4(b,c), after dropping to $0$, the concurrence immediately begins to grow and recovers to the initial value after a half cycle. The period of this entanglement Rabi oscillation is dependent on the transverse component of the hyperfine parameters ($A_{xz 1}=114.5(1)kHz$, $A_{xz 2}=58.7(3)kHz$).
In summary, beyond demonstrating a perspective insight into the entanglement dynamics, our observations provide a possible method to control entanglement in a hybrid system coupled to a natural reservoir. Moreover, measurement for quantum non-Markovianity allows for the study of memory effects emerging from correlations within the environment.
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Supplementary Material.
[^1]: To whom correspondence should be addressed. E-mail: [email protected].
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title: 'What season suits you best? Seasonal light changes and cyanobacterial competition.'
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Nearly all living organisms, including some bacterial species, exhibit biological processes with a period of about 24 h called circadian (from the Latin circa, about, dies, day) rhythms. These rhythms allow living organisms to anticipate the daily alternation of light and darkness. Experiments carried out in cyanobacteria have shown the adaptive value of circadian clocks. In these experiments a wild type cyanobacterial strain (with a 24 h circadian rhythm) and a mutant strain (with a longer or shorter period) grow in competition. In different experiments the external light dark cycle was changed in order to match the circadian period of the different strains, revealing that the strain whose circadian period matches the light-dark has a larger fitness. As a consequence the initial population of one strain grows while the other decays.
These experiments were made under fixed light and dark intervals. However, in Nature this relationship changes according to the season. Therefore, seasonal changes in light could affect the results of the competition. Using a theoretical model we analyze how modulation of light can change the survival of the different cyanobacterial strains. Our results show that there is a clear shift in the competition due to the modulation of light, which could be verified experimentally.
Introduction
============
Circadian rhythms, oscillations with approximately 24 h period in many biological processes, are found in nearly all living organisms. Until the mid-1980s, it was thought that only eukaryotic organisms had a circadian clock, since it was assumed that an endogenous clock with a period of $\tau=24$ h would not be useful to organisms that often divide more rapidly [@Johnson]. However, in 1985, several research groups discovered that in cyanobacteria there was a daily rhythm of nitrogen fixation [@Stal; @Grobbelaar; @Mitsui]. Huang and co-workers were the first to recognize that a strain of Synechococcus, a unicellular cyanobacterium, had circadian rhythms [@Huang]. This transformed Synechococcus in one of the simplest models for studying the molecular basis of the circadian clock.
The ubiquity of circadian rhythms suggests that they confer an evolutionary advantage. The adaptive functions of biological clocks are divided into two hypotheses. The external advantage hypothesis supposes that circadian clocks allow living organisms to anticipate predictable daily changes, such as light/dark, so they can schedule their biological functions like feeding and reproduction at appropriate times. In contrast to this hypothesis, it has been suggested that circadian clocks confer adaptive benefit to organisms through temporal coordination of their internal physiology (intrinsic advantage) [@Sharma]. In this case, the circadian clock should be of adaptive value in constant conditions as well as in cyclic environments.
In order to study if circadian clocks provide evolutionary advantages Woelfle and co-workers tested the relative fitness under competition between various strains of cyanobacteria [@Woelfle]. They carried out experiments where a wild-type strain ($\tau=25$ h) of cyanobacteria and mutant strains, with shorter ($\tau=22$ h) and longer ($\tau=30$ h) periods, were subjected to grow in competition with each other under light-dark (LD) cycles of different periodicity. They found that the strain which won the competition was the one whose free-running period matched closely the period of the LD cycle. This difference in fitness was observed despite the fact that the growth rates were not significantly different when each strain was grown with no competition. Also, mutant strains could outcompete wild-type strains under continuous light (LL) conditions, suggesting that endogenous rhythms are advantageous only in rhythmic environments [@Woelfle]. This study provided one of the most convincing evidence so far in support of fitness advantages of synchronization between the endogenous period and the period of environmental cycles.
Ouyang et al. suggested an explanation for fitness differences: this could be due to competition for limiting resources or secretion of diffusible factors that inhibit the growth of other cyanobacterial strains [@Ouyang]. Roussel et al. proposed mathematical models in order to test which of these hypotheses was more plausible [@Roussel]. They found that the model based on mutual inhibition was consistent with the experimental observations of [@Ouyang]. In this model the mechanism of competition between cells involves the production of a growth inhibitor, which is produced only during the subjective day (sL) phase, and that growth occurs only in light phase.
Each of the experiments and computational simulations mentioned before had equal amounts of light and dark exposure. However, in Nature the relationship is not constant, and the duration of sunlight in a day changes according to the season and the latitude. The circadian system has to adapt to day length variation in order to have a functional role in optimizing seasonal timing and generating the capacity to survive at different latitudes [@Hut].
In this work we will test how day length variation plays a role in the competition between different strains of cyanobacteria.
The model
=========
For modelling the growth of each cyanobacterial strain we use the model introduced by Gonze et al. [@Gonze], that is based on a diffusible inhibitor with a light sensitive oscillator to represent the cellular circadian oscillator. The evolution equations of cell population $N_i$ and the level of inhibitor $I$ are:
$$\begin{array}{rl}
\displaystyle \frac{dN_i}{dt}=k_iN_i(1-\sum_{j=1}^n N_j), & \displaystyle \frac{dI}{dt}=\sum_{i=1}^n N_i (p_i- \frac{V_{max}I}{K_M+I})
\end{array}$$
$$\begin{array}{rl}
k_i=\begin{cases}
k & \mbox{in L and $i$ in sL or $I<I_c$}\\ 0 & \mbox{otherwise,}
\end{cases} & p_i=\begin{cases}
p & \mbox{if $i$ in sL}\\ 0 & \mbox{otherwise.}
\end{cases}
\end{array}$$
In these equations, $N_i$ is the number of cells of strain $i$, $k_i$ is the growth rate of each strain, $p$ measures the rate of inhibitor production, $V_{max}$ is the maximum rate and $K_M$ is the Michaelis constant characterizing the enzymatic degradation of the inhibitor.
Following the work of Gonze et al. we use a modified version of the Van der Pol oscillator to produce sustained circadian oscillations [@Gonze]. Also, we use the same parameters which were found to be in agreement with experimental data obtained by Woelfle et al. [@Woelfle].
In Fig. \[fig:model\] we present a schematic plot of the model, that shows how the growth of each population is coupled with the circadian oscillator. As can be seen in this figure, when we modify the length of the light (L) phase the overlaps 1 and 2 change, so the growth of each strain is altered affecting the competition.
![Following the schematic explanation of Gonze et al. [@Gonze], we show how the model works. The inhibitor $I$ is produced in 3, during the sL phase, and it is degraded during the entire day. Each strain grows in 1, if its sD phase overlaps with L and $I<I_c$, and in 2, when its SL phase overlaps with L.[]{data-label="fig:model"}](model_v3.eps){width="8"}
Results
=======
We want to test how day length variation can modify the competition between different cyanobacteria strains. In many organisms, a photoperiodic response is reflected in a physiological change. Photoperiodic responses are common amongst organisms from the equator to high latitudes and have been observed in different types of organisms, from arthropods to plants. Diapause (a suspension of development done by insects), migration and gonadal maturation are examples of these annual changes controlled by photoperiod. These biological processes are triggered as soon as the day length reaches certain duration, known as the critical photoperiod. Even near the equator, where day length changes are very small through the whole year, they are used to synchronize reproductive activities with annual events.
In Fig. \[fig:daylength\] we show how day length varies depending on the latitude. The figure compares the day length on the twenty-first day of each month in three cities in South America. We show Quito (Ecuador), which sits near the equator in latitude $0^{\circ}15'$, Jujuy (Argentina), which is located near the tropic of Capricorn in latitude $24^{\circ} 01'$ and Ushuaia, the southernmost city in Argentina which lies in latitude $54^{\circ} 48'$. We can see that during the equinoxes, all places receive 12 hours of daylight.
![Day length over the course of 2012 at different latitudes in South America.[]{data-label="fig:daylength"}](daylength.ps){width="6"}
We simulate the seasonal fluctuations in day length by adding or subtracting minutes of light-time every day to the external LD cycle. For example, if we add 12 minutes of light per day in a LD12:12 cycle, after five days the external cycle of LD has 13 hours of light and 11 of darkness. We initiate competition between equal fractions of wild-type strain and long-period mutant and equal amounts of light and dark. We dilute the culture after 8 light-dark cycles by dividing by a factor of 100 the variables $N_1$, $N_2$ and $I$. In this way we mimic experiments in cultures, that were diluted and sampled every 8 days [@Woelfle].
First, we analysed the case in which there was a phase of coexistence. For $T=28$, the period of the LD cycle has an intermediate value between the free-running period (FRP) of the two strains and both strains can coexist for a long time. However, when we allowed the days to become longer and the nights shorter, after some days the coexistence was broken, as we show in Fig. \[fig:fig2\]. This is due to the external period $T$ that starts to get closer to the FRP of the long-period mutant.
![(A) The outcome of competition between wild-type (continuous line) and long-period mutant (dashed line) shows coexistence between the two strains for $T=28$ h. (B) Coexistence ends after $\approx 8$ days of adding $3$ minutes of light-time per day. The long-period mutant can win competition as its free-running period becomes closer to the LD cycle.[]{data-label="fig:fig2"}](fig2_T28convivencia.ps){width="7"}
In Fig. \[fig:n1\] we show in the left panel the fraction of cells belonging to the wild-type strain as a function of time with fixed LD cycles (red continuous curve) and in the case in which we added 30 minutes of light time each day (blue dashed curve). The first days both curves are similar, but from the second day, the long-period strain has a FRP closest to the period of the LD cycle. The fraction of wild-type strain starts to decrease and is out-competed. In the right panel we show the corresponding fraction of long-period mutant cells in the same cases.
The results of our numerical simulations could be tested experimentally. This would be very simple, since the cultures do not need to be diluted. It is only needed to sample the culture at regular intervals to determine the composition of the population and verify a difference of about ten percent in the two cultures after eight days.
Starting from this simple test, we looked for non trivial effects in a longer experiment. We found an interesting effect that can be observed in Fig. \[fig:fig3\]. In this simulation we added 12 minutes of light time each day. In the first days, the growth was the expected. The wild-type strain could out-compete the long period mutant strain, since the external cycle was LD12:12. But after eight days, when the day was longer than 13 hours, a crossover was observed. The mutant strain started to win the competition because its endogenous period was closer now to the external cycle. This effect could also be tested experimentally, however, in this case the dilution of the culture every 8 days would be needed.
![Effect of modulation in light time (30 minutes per day) on the outcome of competition between strain 1 (wild-type, $\tau=25 h$) and strain 2 (long-period mutant, $\tau=30 h$) in (A) LD12:12 and (B) LD15:15 cycles. Fraction of strain 1 (left panel) and strain 2 (right panel) are shown as a function of time; red with modulation and blue with fixed cycles.[]{data-label="fig:n1"}](fig1.ps){width="9"}
![Competition between long-period mutant and wild-type strains in a LD12:12 cycle for the same parameters as in Fig. \[fig:n1\], but adding 12 minutes per day the light time. A crossover is observed after 8 days.[]{data-label="fig:fig3"}](fig3.ps){width="8"}
Conclusions
===========
The mechanisms underlying the enhancement of reproductive fitness remain still unknown. Despite numerous models have been tested, each has some evidence that supports it and none can be excluded at this time [@JohnsonReview]. In this work we used a diffusible inhibitor model, so our predictions in the growth rates changes could be useful to test the validity of this mechanism.
Our study is motivated by fluctuations in the day length throughout the year which are reflected in organisms behaviour. We studied how these fluctuations affect the competition between different strains of cyanobacteria. We found non-trivial effects which could be tested experimentally. In the first case we determinate the composition of two strains under competition after eight days when the light is modulated. The prediction of these numerical simulations can be tested in a simple experiment where no dilution is needed. We also propose a second experiment where dilution in the cultures is necessary, which allows for a non trivial effect such as a crossover to be observed.
[99]{}
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Ma, P., M. A. Woelfle, and C. H. Johnson. An evolutionary fitness enhancement conferred by the circadian system in cyanobacteria. *Chaos, Solitons & Fractals* 50:65-74 (2013).
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abstract: 'We investigate up to the fourth order normalized factorial moments of free-propagating and pulsed single photons displaced in phase space in a phase-averaged manner. Due to their loss independence, these moments offer expedient methods for quantum optical state characterization. We examine quantum features of the prepared displaced states, retrieve information on their photon-number content and study the reliability of the state reconstruction method used.'
address:
- '$^{1}$Applied Physics, University of Paderborn, Warburgerstr. 100, 33098 Paderborn, Germany.'
- '$^{2}$Max Planck Institute for the Science of Light, Guenther-Scharowsky-Str. 1/ Bldg. 24, 91058 Erlangen, Germany.'
author:
- 'Kaisa Laiho$^{1,2}$, Malte Avenhaus$^{1,2}$, and Christine Silberhorn$^{1,2}$'
bibliography:
- 'references.bib'
title: Characteristics of displaced single photons attained via higher order factorial moments
---
Introduction
============
The observation of quantum light in phase space is typically associated with homodyne detection [@Raymer2009]. However, photon counting also provides attractive approaches for studying the distinct quantum character of the quasi-probability distributions [@Cahill1969; @Cahill1969a]. Using the photon-parity operator, for example, even individual points of the Wigner function can be directly probed despite the Heisenberg’s uncertainty relation of the field quadratures [@Englert1993; @K.Banaszek1996; @Wallentowitz1996]. In the optical regime, this technique has already been utilized for the characterization of cavity confined optical states [@Lutterbach1997; @Bertet2002; @Lougovski2003; @Hofheinz2009] as well as free propagating light fields [@K.Banaszek1999; @Allevi2009; @Bondani2009; @Laiho2010]. Even though the photon-number parity can be straightforwardly deduced from the measured photon statistics, the detection losses of practical photon counters tend to destroy the real quantum characteristics featured in the photon-number distribution. Therefore, sophisticated methods for testing the nonclassical character of the loss-degraded states via their photon-number content have been developed [@Waks2006a; @Jezek2011], and techniques allowing a loss-tolerant reconstruction of photon statistics are of great importance [@Hong1986; @Waks2004; @Zambra2005; @Achilles2006; @Avenhaus2008; @Wasilewski2008]. Still, an accurate determination of the state’s properties from the loss-degraded data is challenging, especially when the studied state incorporates higher-photon number contributions [@K.Banaszek1997; @Achilles2004; @Laiho2009].
Genuine quantum features can fortunately be recognized even without access to the state’s complete phase-space representation. One of the pioneering techniques was introduced by Hanbury Brown and Twiss [@Hanbury1956], and in an extended form [@Shchukin2005] their experiment allows us to access the higher order factorial moments of photon number [@Assmann2009; @Avenhaus2010; @Stevens2010]. In general the $m$-th order normalized factorial moment is determined as $$g^{(m)} = \braket{n^{(m)}} / \braket{n}^{m},
\label{eq_g}$$ where $\braket{n^{(m)}} = \braket{: \hat{n}^{m}:} = \sum_{n} n (n-1)\dots(n-m+1) \ \varrho(n)$ and $\braket{n} = \braket{n^{(1)}}$ can be evaluated either as normally ordered ($: :$) moments of the photon number operator $\hat{n}$ or via the photon statistics $\varrho(n)$ with $n$ being the photon number [@L.Mandel1995]. Usually, the normalized form of the factorial moments can be extracted loss independently [@S.M.Barnett1997; @Avenhaus2010]. However, care has to be taken when detecting multimode states since the individual modes may suffer from different amounts of losses [@Laiho2011]. Nonetheless, these moments provide versatile alternatives for investigating the properties of quantum optical states.
Regarding single photons [@Waks2006], already their second order normalized factorial moment, which ideally takes the value $g^{(2)} = 0$, can be employed as a valuable characterization tool. The real measured values are widely used to classify the practical single-photon sources [@Eisaman2011]. More generally, however, observing $g^{(2)}<1$ can be regarded as a signature of the nonclassicality [@Korbicz2005] and as an indication of the sub-Poissonian photon-number characteristics [@Short1983; @Rarity1987; @U'Ren2005]. Nonetheless, when single photons are displaced their $g^{(2)}$ values gradually increase and finally exceed unity, which signalizes a super-Poissonian photon-number distribution [@Moya-Cessa1995]. In this region also a more sophisticated test for the nonclassicality should be found [@Agarwal1992; @W.Vogel2008]. Further, once having accessed the photon-number content of displaced single photons [@A.I.Lvovsky2002; @Laiho2010] many other intriguing phase-space features can be directly scrutinized such as the nonclassical oscillations in the photon statistics [@W.Schleich1987; @Oliveira1990]. Moreover, the factorial moments of *displaced* states indeed provide routes for accessing more elaborate moments of the photon creation and annihilation operators [@Shchukin2005].
Here, we measure up to the fourth order normalized factorial moments of phase-averaged and displaced single photons directly in a coincidence counting experiment by employing the time-multiplexed detector (TMD) [@Achilles2003; @Fitch2003] that has proven to be a powerful tool for measuring the higher order moments of pulsed quantum states of light [@Avenhaus2010]. Even without access to the complete photon statistics we can loss-independently observe quantum features in the prepared states. Further, by calibrating the mean photon number of the loss-degraded states we gain information about their photon-number content. At low detection efficiencies, it is generally a highly nontrivial task to invert the action of losses [@Kiss1995]. Further, in order to estimate the reliability of the state reconstruction often numerical methods are applied such that the effects of statistical fluctuations, losses, and highest resolved photon number can also be taken into account [@Laiho2009; @Laiho2010]. In contrast to the ordinary loss inversion, our technique allows us to study the effect of different experimental limitations separately from each other. As a consequence, we can directly define the boundaries that an experimental realization sets for the state reconstruction in phase space. Further, the artifacts introduced by the nonideal detection can be recognized in a straightforward manner.
This paper is organized as follows. In Sec. \[sec\_gen\_function\] we review the properties of factorial moments, which are connected to the photon statistics via the moment generating function. In Sec. \[sec\_displaced\_single\_photon\] we survey the properties of displaced single photons and review the effects caused by experimental imperfections. In Sec. \[sec\_experiment\] we utilize the factorial moments to investigate the characteristics of the prepared displaced single photons.
\[sec\_gen\_function\]Retrieving state characteristics via factorial moments
============================================================================
The normalized factorial moments can be directly utilized for different state characterization tasks such as discrimination between sub- ($g^{(2)} < 1$) and super-Poissonian ($g^{(2) }> 1$) photon-number distributions [@Grosse2007; @Bartley2012] or classification of different quantum states [@Avenhaus2010]. Additionally, they provide several alternatives for the direct examination of quantum features [@Avenhaus2010; @Allevi2011]. One option for investigating phase-insensitive nonclassical behavior in a single mode is to study whether it is possible to violate the criterion $$g^{(m+1)} \ge g^{(m)} \ge 1
\label{eq_classic}$$ that classical states obey [@Sudarshan2006], and we treat the multimode states equivalently. For this purpose, also the measurement of the normalized factorial moments with orders higher than two becomes relevant when regarding states with super-Poissonian characteristics [@Agarwal1992]. A more detailed investigation of the state’s photon-number content via the factorial moments is possible by employing the moment generating function [@S.M.Barnett1997; @Barnett1998] that is described in terms of a real valued variable $0\le \mu \le2$ as $$M(\mu) = \sum_{n}\varrho(n)(1-\mu)^{n}.
\label{eq_MGF}$$ At the upper bound, that is when $\mu = 2$, the moment generating function provides information on the photon-number parity. Nevertheless, equally interesting is the lower bound since the derivatives of Eq. (\[eq\_MGF\]) at $\mu =0$ are directly connected with the factorial moments by $$\braket{n^{(m)}} = \left( -\frac{\textrm{d} }{\textrm{d}\mu}\right)^{m} \left. M(\mu) \right |_{\mu = 0}.
\label{eq_n_m}$$
Once having accessed the factorial moments the expression in Eq. (\[eq\_MGF\]) can be re-written as an expansion $$M(\mu) = \sum_{m} \frac{(-1)^{m} \braket{n^{(m)}}}{m!} \mu^{m},
\label{eq_M_taylor}$$ and the photon statistics is again reconstructed from the moment generating function in Eq. (\[eq\_M\_taylor\]) by $$\varrho(n) = \frac{1}{n!}\left( -\frac{\textrm{d} }{\textrm{d}\mu}\right)^{n} \left. M(\mu) \right |_{\mu = 1}
= \sum_{m\ge n} \frac{(-1)^{m+n}}{n!(m-n)!} \braket{n^{(m)}},
\label{eq_p_n_rec}$$ which obeys the conventional normalization $\sum_{n}\varrho({n})= 1$. Thus, the photon statistics is gained by summing up the factorial moments with proper weight factors. However, we note that this method for reconstructing the photon statistics can only be successful when the expansion in Eq. (\[eq\_M\_taylor\]) converges near $\mu = 1$.
The loss tolerance in Eq. (\[eq\_p\_n\_rec\]) is achieved after re-writing $ \braket{n^{(m)}}= g^{(m)}\braket{n}^{m} $ and deducing the mean photon number $\braket{n} $ from the loss-degraded measurement via $\braket{n}_{\textrm{lossy}} = \eta \braket{n}$, in which $\eta$ is the detection efficiency [@S.M.Barnett1997]. Thus, apart from recording the different orders of $g^{(m)}$, this method further entails a calibration of $\eta$ and measurement of $\braket{n}_{\textrm{lossy}}$. If the mean photon number can be calibrated accurately, the reconstruction of photon statistics via normalized factorial moments becomes especially expedient at low detection efficiencies. The only constraint lies in measuring enough orders of $g^{(m)}$ with good precision during a finite integration time. As a consequence, the highest statistically accessible moment, which is known from the experimental data, limits the possibilities to completely reconstruct the state’s photon-number content. Natural limits are determined by the physical bounds $0 \le \varrho(n) \le1$. More stringent conditions may be found by investigating the distinctive features in the photon statistics of the studied states.
\[sec\_displaced\_single\_photon\]Modeling free-propagating displaced single photons
====================================================================================
Even though displaced single photons have been studied in several experiments [@Bertet2002; @A.I.Lvovsky2002; @Hofheinz2009; @Laiho2010], the generated states are seldom ideal and imperfections in the preparation process have to be taken into account. We first investigate the properties of ideal displaced single photons and then regard the effect of experimental imperfections. We consider a displacement with a mode mismatch and take into account higher photon-number contributions of the prepared state.
In the single-mode picture, an ideal displaced single-photon state is described as $\hat{D}(\alpha)\ket{1}$, where $\hat{D}(\alpha)$ \[with the hermitian conjugate $\hat{D}^{\dagger}(\alpha)$\] defines the displacement by an amount of $\alpha$ and $\ket{1} $ is the single-photon Fock state. We deduce its factorial moments by evaluating the normally ordered mean values in Eq. (\[eq\_g\]) with the help of the transformations $\hat{D}^{\dagger}(\alpha)\hat{a}\hat{D}(\alpha) = \hat{a}+\alpha$ and $\hat{D}^{\dagger}(\alpha)\hat{a}^{\dagger}\hat{D}(\alpha) = \hat{a}^{\dagger}+\alpha^{*}$ of the photon annihilation ($\hat{a}$) and creation ($\hat{a}^{\dagger}$) operators. After a straightforward calculation, the normalized factorial moments of the ideal displaced single-photon state can be expressed as $$g^{(m)}_{\textrm{ideal}} = \frac{|\alpha|^{2(m-1)} (m^{2}+ |\alpha|^{2})}{(1+|\alpha|^{2})^{m}}.
\label{eq_g_ideal_fock}$$ These moments as depicted in Fig. \[fig\_1\](a) rapidly grow from zero when increasing the mean photon number given by $\braket{n}_{\textrm{ideal}} = 1+|\alpha|^2$. By following the behavior of the second normalized moment, one directly concludes the gradual transition in the photon-number characteristics of the displaced states [@Oliveira1990; @Moya-Cessa1995], and this moment reaches the maximal value of $g^{(2)}_{\textrm{ideal}} \approx 1.333$ at the displacement $|\alpha|_{\textrm{max}} = \sqrt{2}$. Further, the ideal displaced single-photon states always violate the criterion in Eq. (\[eq\_classic\]). However, as seen in Fig. \[fig\_1\](a), the verification becomes more and more challenging when increasing the displacement since moments with higher and higher orders have to be resolved.
In practical applications, the displacement can be implemented with an asymmetric beam splitter, at which the studied state is overlapped with a coherent reference state (see e.g. [@Laiho2009] and the references therein). The mismatch of the two overlapping modes in temporal, spectral or spatial degrees of freedom can be modeled with a simple overlap factor $\mathcal{M}$. In order to evaluate the required mean values in Eq. (\[eq\_g\]), the photon-number operator can be replaced with $$\begin{aligned}
\label{eq_N_eff}
\hat{n} &\rightarrow& \eta \ \hat{n}_{\textrm{eff}} \\
& = &\eta \left[ \hat{D}^{\dagger}(\sqrt{\mathcal{M}}\alpha)\hat{a}^{\dagger} \hat{a} \hat{D}(\sqrt{\mathcal{M}}\alpha) + (1-\mathcal{M})|\alpha|^{2}\right], \nonumber\end{aligned}$$ where $\eta$ is the total detection efficiency, $\mathcal{M}$ is the mode overlap, and $\alpha$ is the amount of the applied displacement [@Laiho2009]. The effective photon-number operator $\hat{n}_{\textrm{eff}}$ in Eq. (\[eq\_N\_eff\]) is a sum of a displaced photon-number operator and a background term.
The total detection efficiency cancels out when evaluating the normalized factorial moments, and we can write them loss independently in the form $$g^{(m)}_{\textrm{eff}} = \frac{\braket{: \hat{n}_{\textrm{eff}}^{m}:}}{\braket{\hat{n}_{\textrm{eff}}}^{m}} = \frac{ \sum_{k}\bigg ( \hspace{-1ex} \begin{array}{c} m \\ k \end{array} \hspace{-1ex} \bigg ) \braket{n^{(k)}}_{\textrm{D}} \braket{n^{(m-k)}}_{\textrm{bg} }}{ (\braket{n}_{\textrm{D}} + \braket{n}_{\textrm{bg}})^{m} },
\label{eq_g_two_modes}$$ in which $ \braket{n^{(m)}}_{\textrm{bg}} = [(1-\mathcal{M})|\alpha|^{2}]^{m}$ describes the properties of the background and $$\begin{aligned}
\braket{n^{(m)}}_{\textrm{D}} &=& \braket{ : [\hat{D}^{\dagger}(\sqrt{\mathcal{M}}\alpha)\hat{a}^{\dagger} \hat{a} \hat{D}(\sqrt{\mathcal{M}}\alpha)]^{m}:} \nonumber \\
&=& \braket{\hat{D}^{\dagger}(\sqrt{\mathcal{M}}\alpha)\hat{a}^{\dagger m} \hat{a}^{m} \hat{D}(\sqrt{\mathcal{M}}\alpha)}
\label{eq_O}
\end{aligned}$$ predicts the behavior of the displaced part. If the mode overlap is imperfectly aligned, the results of the measurement change drastically. As shown in Fig. \[fig\_1\](b), in the case of the single-photon Fock state the effective normalized factorial moments cannot take values larger than unity when $\mathcal{M} = 0$. Therefore, the super-Poissonian region becomes a loss-independent indicator that the displacement takes place, in other words that $\mathcal{M} \neq 0$. Nevertheless, our model in Eq. (\[eq\_g\_two\_modes\]) provides inaccurate results if only single-photon Fock states are regarded. This can be as corrected by considering the higher photon-number contributions of the real single-photon source. We assume that the single photon is prepared into a photon-number mixed state [@Christ2012], and we take the higher photon-number contributions of it into account in Eq. (\[eq\_O\]) when fitting Eq. (\[eq\_g\_two\_modes\]) against the values measured for the displaced single-photon states. The overlap factor $\mathcal{M}$ is held as fitting parameter. Moreover, the amount of displacement applied to the prepared single photon can be straightforwardly extracted in our model from the effective mean photon number. Even in the case of imperfect mode overlap this is given by $ \braket{ \hat{n}_{\textrm{eff}} }= \braket{n}_{\textrm{sp}}+|\alpha|^{2} $, in which $\braket{n}_{\textrm{sp}}$ is the mean photon number of the prepared single photon.
\[sec\_experiment\]Experimental investigation of displaced single photons
=========================================================================
In our experiment, shown in Fig. \[fig\_2\], we heralded single photons from a pulsed waveguided twin-beam source based on parametric downcoversion. For preparing displaced states, the heralded single photons were overlapped at an asymmetric beam splitter with coherent reference states in a phase-averaged manner. The amount of displacement was controlled by altering the mean photon number in the reference beam. The heralded displaced single photons were then coupled to a TMD for detection. Our experimental arrangement is similar to the one used [@Laiho2010] except that the TMD is employed for measuring manyfold coincidence counts and not the so-called click statistics. We note, however, that the values of the normalized factorial moments could also be deduced via the loss-degraded photon statistics that can be retrieved from the click statistics recorded with TMD by taking into account the intrinsic detector characteristics [@Bartley2012].
As described in [@Avenhaus2010], the higher order correlations in a light beam can be accessed with the TMD detection scheme, and the time-integrated measurement delivers the desired expectation values [@Christ2010]. Thence, the value of $g^{(m)}$ is extracted from the raw data by dividing the probability to measure a coincidence click between $m$ selected temporal TMD bins by the product of the single click probabilities in these bins—all of them conditioned on the detection of the herald. In order to apply the measured normalized moments for the loss-tolerant reconstruction of photon statistics, we further require a calibration of the detection efficiency such that the mean photon number can be determined. The perfect photon-number correlation between the twin beams empowers us to estimate the detection efficiency according to Klyshko [@Klyshko1977]. For this purpose we block the reference beam, subtract the amount of accidental counts from the coincidences between the twin beam detection, and compare this number to the amount of single counts in herald. The mean photon number of the displaced state merely follows from dividing the first *un*normalized factorial moment—the probability of measuring a single click conditioned on the detection of the herald—by the estimated detection efficiency.
We first study the values of the moments $g^{(2)}$ to $g^{(4)}$ in order to categorize the photon-number properties of the prepared states. For the prepared heralded single-photon state we extracted the values $g^{(2)} = 0.184(4)$ and $g^{(3)} = 0.04(2)$, the accuracy of which is limited by the statistical fluctuations only. In contrast to a genuine single-photon state, one clearly recognizes an additional two-photon contribution in the heralded state. This is the trade-off from a rather high pump power—on average , which nevertheless yielded a heralding rate of . When the prepared single photon is now displaced, the values of $g^{(2)}$ to $g^{(4)}$ gradually increase. This behavior is shown in Figs \[fig\_3\](a-b) with respect to the calibrated mean photon number. Maximally we observed the values $g^{(2)}_{\textrm{max}} = 1.148(6)$, $g^{(3)}_{\textrm{max}} = 1.18(3)$ and $g^{(4)}_{\textrm{max}} = 1.00(15)$. A comparison of these results with the ones gained for the case of vanishing mode overlap \[Figs. \[fig\_3\](c-d)\] reveals, as expected, the rapid appearance of the higher order normalized factorial moments, when the prepared single photon is displaced. In contrast to our earlier studies [@Laiho2010], we now fit our model in Eq. (\[eq\_g\_two\_modes\]) against the properties of the higher photon-number contributions. Our model assumes that all the photon-number components of the prepared single photon possess the same overlap factor and a reasonable match for the data in Figs \[fig\_3\](a-b) is found. If the overlap factor was drastically lower for the higher photon-number contributions—in our case the two-photon contribution—we would expect to encounter difficulties when fitting our model against the values measured for the third or higher order normalized factorial moments.
Even without information on the complete photon statistics, we conclude from our results in Fig. \[fig\_3\](a) that the expected transition between the sub- and super-Poissonian photon statistics takes place. Further, our displaced states show quantum features at small values of displacement. As seen in Fig. \[fig\_3\](a), the inequality $g^{(2)} \ngeq1$ guarantees the nonclassicality of the prepared states below the mean photon number of approximately 1.6. Our results in Fig. \[fig\_3\](a) clearly also violate the classicality via $g^{(3)} \ngeq g^{(2)} $. However, this criterion becomes inadequate close to the mean photon number of 1.9, beyond which we are unable to detect quantum features in the prepared displaced states. In order to do so, a more accurate measurement of the fourth normalized moment $g^{(4)}$ than the one shown in Fig. \[fig\_3\](b) is required.
Next, we examine, how much information on the displaced state’s photon-number content can be deduced via the measured moments. We estimate the photon-number contributions of the heralded and displaced states by plugging the measured values in Figs \[fig\_3\](a-b) into Eq. (\[eq\_p\_n\_rec\]) together with the calibrated mean photon numbers. For the heralded single photon, the photon statistics of which is shown Fig. \[fig\_4\](a), we extracted the loss-calibrated mean photon number of $\braket{n}_{\textrm{sp}} = 1.07(3)$ that was estimated in a measurement with little less than one percent detection efficiency per TMD bin. Further, we note that the accuracy at which the mean photon number is deduced depends not only on the statistical fluctuations but also on the approximative estimation of the detection efficiency. As shown in Fig. \[fig\_4\](b), we can recover the behavior expected for the displaced single photon—increasing vacuum contribution and decreasing one-photon contribution—at the mean photon number of $1.30(4)$.
However, the limited resolution restricts the acceptable phase-space displacement and sets a boundary to a region, in which the photon statistics of the prepared displaced state can reliably be reconstructed. Being able to statistically resolve moments only up to the fourth order, in other words the experiment delivers $g^{(n > 4)} = 0$, the two highest accessible photon-number contributions are deduced according to Eq. (\[eq\_p\_n\_rec\]) as $\varrho(4) = \braket{n}^{4}/4! \ g^{(4)}$ and $\varrho(3) = \braket{n}^{3} / (3)! \ [g^{(3)}- g^{(4)}\braket{n}]$. The non-negativity of the photon-number components provides us with a condition $ \braket{n} \le g^{(3)}/g^{(4)} $ for the accessible reconstruction range. By employing the fits in Figs \[fig\_3\](a-b) we get the bound $\braket{n} \lesssim1.4$. Looking closer at the individual photon-number contributions in Fig. \[fig\_5\], a second, stricter condition is obtained by studying the boundary, close to which the reconstructed photon-number contributions start to significantly deviate from the expected behavior. As the highest resolvable photon-number component, in our case $\varrho(4)$ \[Fig. \[fig\_5\](e)–dashed line\], gradually increases with respect to mean photon number, it eventually surpasses the contribution $\varrho(3)$ \[Fig. \[fig\_5\](d)–dashed line\]. This is an artifact not expected in the photon statistics of the displaced single photon at the studied region. Therefore, we use the limitation $ \varrho{(3)}\ge\varrho{(4)} $ as the boundary of the reliable reconstruction range for the displaced prepared states and gain the ultimatum $\braket{n} \lesssim1.3$. Furthermore, we note that the vacuum component \[Fig. \[fig\_5\](a)–symbols\] of the prepared displaced states could be fairly reliably extracted from our measurement even beyond the determined range, whereas the higher photon-number contributions \[Figs \[fig\_5\](b-e)–symbols\] are not as resilient.
In order to accurately reconstruct the complete photon statistics outside the determined region, one is required to resolve moments with orders higher than four. Thus, we predict the values of the moments $g^{(5)}$ and $g^{(6)}$ in order to interpret their influence on the photon statistics. As depicted in Fig. \[fig\_5\] the accessible reconstruction range gradually increases when moments with higher orders are resolved. Clearly, the method used provides us with insight into the reliability of the state characterization at low detection efficiencies. In our example it allows us to estimate the bounds of the reconstruction range in phase space and to recognize the artifacts caused by the nonideal detection. Systematical deviations in the reconstructed photon statistics become apparent if enough orders of the normalized factorial moments cannot be measured due to the limited detection time. Further, the statistical fluctuations in the measured moments and the precision at which the mean photon number is extracted affect the accuracy of the reconstructed photon statistics. Fortunately, their effect can be investigated separately from the systematical deviations and in our case the errorbars are dominated by the fluctuations in the resolved moments rather than by the inaccuracy in the mean photon number. In summary, our results can give direct specifications for the experimental parameters when studying the fine structure in the photon statistics of displaced single photons at low detection efficiencies.
Conclusions
===========
We measured up to the fourth order normalized factorial moments of displaced single photons in a loss-independent manner. By studying the second normalized moment, we confirmed as expected that the sub-Poissonian photon-number distribution of a single photon gradually moves toward the super-Poissonian photon statistics when the state is displaced. The prepared displaced states further violated the classicality even in the super-Poissonian regime. Moreover, the measured moments provide means for the loss-tolerant reconstruction of photon statistics after determining the mean photon number of the studied state. However, it is essential to accurately measure enough orders of the factorial moments in order to reliably reconstruct the investigated properties. Our results show the versatility of the factorial moments for the state characterization, and we believe they prove to be useful in examining genuine quantum features at low detection efficiencies.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank G. Harder for fruitful discussions. This work was supported by the EC under the grant agreements Corner (FP7-ICT-213681) and Q-Essence (248095).
References {#references .unnumbered}
==========
|
---
abstract: 'Classical sum rules arise in a wide variety of physical contexts. Asymptotic expressions have been derived for many of these sum rules in the limit of long orbital period (or large action). Although sum rule convergence may well be exponentially rapid for chaotic systems in a global sense with time, individual contributions to the sums may fluctuate with a width which diverges in time. Our interest is in the global convergence of sum rules as well as their local fluctuations. It turns out that a simple version of a lazy baker map gives an ideal system in which classical sum rules, their corrections, and their fluctuations can be worked out analytically. This is worked out in detail for the Hannay-Ozorio sum rule. In this particular case the rate of convergence of the sum rule is found to be governed by the Pollicott-Ruelle resonances, and both local and global boundaries for which the sum rule may converge are given. In addition, the width of the fluctuations is considered and worked out analytically, and it is shown to have an interesting dependence on the location of the region over which the sum rule is applied. It is also found that as the region of application is decreased in size the fluctuations grow. This suggests a way of controlling the length scale of the fluctuations by considering a time dependent phase space volume, which for the lazy baker map decreases exponentially rapidly with time.'
author:
- 'John R. Elton$^{1,2}$, Arul Lakshminarayan$^{1,3}$, and Steven Tomsovic$^{1,3,4}$'
bibliography:
- 'rmtmodify.bib'
- 'quantumchaos.bib'
- 'classicalchaos.bib'
- 'nano.bib'
title: Fluctuations in classical sum rules
---
Introduction {#sec:introduction}
============
Classical sum rules play an important role in a number of physical contexts. It is interesting to have a case of a non-uniformly spreading chaotic system where their limits, fluctuations, and convergences can be studied analytically, which is given in this paper. We focus on the sum rule of Hannay and Ozorio de Almeida, who were motivated by a desire to understand the two-point quantum density of states correlator [@Hannay84]. Their derivation relied on the Principle of Uniformity, which states that the periodic orbits, weighted naturally are uniformly dense in phase space. There are many others; see, for example, the sum rules in Refs. [@Argaman96; @Sieber99; @Richter02] which involve return probability, the connection of two points in coordinate space, and fixed orientations of initial and final velocities, respectively. They arise in mesoscopic conductance, diffraction contributions to spectral fluctuations, and chaotic quantum transport, also respectively.
In one possible form of the Hannay-Ozorio sum rule [@OzorioBook] for chaotic systems, it is the inverses of the stability matrix determinants for the periodic orbits of period $\tau$, $\left|{\rm Det}\left( {\mathbf M}_\tau - {\mathbf 1} \right) \right|^{-1}$, which are summed. For completely chaotic systems the finite-time stability exponents, which roughly determine the value of each determinant, converge as $\tau\rightarrow \infty$ to a single set. Nevertheless, the determinant’s fluctuations from one periodic orbit to the next grow without bound in the same limit. This feature is made even more curious by the expectation that the Hannay-Ozorio sum rule converges exponentially rapidly. See a preprint by Pollicott [@Pollicott01] for a theorem regarding the sum rule’s convergence.
It is known that smooth classical functions have exponential decays in fully chaotic systems toward their ergodic averages, which are governed by the Pollicott-Ruelle (PR) resonances [@Pollicott85; @Pollicott86; @Ruelle86; @Ruelle87] associated with the Perron-Frobenius operator. It is natural to ask whether the kinds of classical sum rules encountered always converge to their limiting values with corrections that decay exponentially according to the leading PR resonances. The work of Andreev et al. [@Andreev96] would suggest that as long as there is a gap in the spectra of the Perron-Frobenius operators, there is no other possibility.
A first step is taken here in approaching this general line of questioning by considering a form of the Hannay-Ozorio sum rule for fixed period in maps. Its convergence rate and local fluctuations are considered in detail in a simple version of a lazy baker map introduced by Balazs and one of us (AL) [@Lakshminarayan93; @Lakshminarayan93a]. Although the usual baker map is especially simple, it is also non-generic in that its lacks the essential fluctuations of interest, whereas the lazy baker map possesses fluctuations and still turns out to be analytically tractable. The fluctuations of the inverse determinant were studied briefly in Ref. [@Tomsovic07], where they were demonstrated, not surprisingly, to be extremely sensitive to even tiny islands of stability. In the form of the lazy baker map studied here, all orbits, with one exception of a marginal orbit of period two, are unstable, and there is no ambiguity concerning whether the system is completely chaotic.
The organization of this paper is as follows: in Sect. \[sec:notations\] notations are specified and definitions of various quantities of interest are given. Section \[sec:srmap\] introduces a simple version of a lazy baker map (named the SR map), which is studied in detail, and gives basic counting results for fixed points and a method of subdividing the phase space into local regions. This is followed in Sect. \[sec:statistics\] by a derivation of the main results for the local fluctuation and convergence properties of the Hannay-Ozorio sum rule.
Measurements of chaotic systems {#sec:notations}
===============================
Dynamical systems theory has generated a number of ways to specify the complexity of a chaotic mapping. Three of the more familiar concepts to physicists are the topological entropy, $h_T$, the metric or Kolmogorov-Sinai (KS) entropy, and the Lyapunov exponent, $\lambda_L$ [@Jacobs98; @Ott02]. The topological entropy is designed to measure the information content of the optimal partition of the dynamics. It turns out for that a class of systems known as Axiom A, the limiting value of the exponential rate of increase in the number of fixed points with iteration number gives this entropy. The KS entropy can be thought of in a similar way, except that it is weighted. And finally, the Lyapunov exponent measures the exponential separation of neighboring initial conditions.
For the purpose of studying the fluctuations of classical sum rules, the main quantities of interest tend to be the number of fixed points, their finite-time stability exponents, and their probability densities and moments, all of which can be considered in both a global and local phase space context. We do not worry as to what exact relations exist between each of these measures for a given system, but they are closely related where they do not have an identical counterpart and it is useful to relate our results to some of these quantities when they are known.
Basic quantities {#sec:notationsa}
----------------
The notation $N_{\tau}$ denotes the number of fixed points at integer time $\tau$ taken over the full phase space. This is distinguished from a local count of fixed points by writing $N_{\tau}(s,k)$ where the parameters $s$ and $k$ conveniently specify the location and size respectively of the local phase space volume in question for the SR map. Additionally, of great importance in this work are the sums over fixed points appearing in one form of the Hannay-Ozorio sum rule, which are given the notation $$\begin{aligned}
F_{\tau} &=& \sum_{f.p.} \frac{1}{\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|}\qquad\ \ \ ({\rm global})\nonumber \\
F_{\tau}(s,k) &=& \sum_{f.p.\in (s,k)} \frac{1}{\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|}\ \ \ ({\rm local})
\label{HannayOzorio}\end{aligned}$$ where “f.p.” denotes fixed points in the specified region of phase space and $$\label{stabilitymatrix}
{\bf M}_\tau(q_0,p_0) = \prod_{i=0}^{\tau-1}{{\bf M}(q_i,p_i)}$$ is the Jacobian stability matrix along the trajectory fixed by the set of iterates $\{ q_i,p_i \}$ of the initial conditions $(q_0,p_0)$; the notation for initial conditions is mostly left suppressed. A related quantity which is sometimes of interest is the finite-time stability exponent, given here for a map with single position and single momentum coordinates $$\begin{aligned}
\label{deflambda}
\lambda(q_0,p_0;\tau) &=& {1\over \tau} \ln \left ({\left|\mbox{\tr}[{\bf M}_\tau]\right| + \sqrt{\mbox{\tr}[{\bf M}_\tau]^2-4}\over 2}\right) \nonumber \\
& \approx& {1\over \tau} \ln \left| \tr[{\bf M}_\tau] \right| \approx \frac{1}{\tau}\ln {\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|} \nonumber \\\end{aligned}$$ where $\tr(...)$ denotes the trace operation. For long times, the approximate relations for $\lambda$ tend exponentially quickly to the first relation.
Convergence and measuring sum rule fluctuations {#sec:fluctuations}
-----------------------------------------------
The Hannay-Ozorio sum rule in this context and notation (also referred to as the uniformity principle) reads $F_{\tau}(s,k) \rightarrow {\cal V}_k$ where ${\cal V}_k$ is the phase space volume over which the fixed points are summed. The absence of an $s$ dependence is the absence of a dependence on the location in phase space. This result, which holds in both the local and global cases, emerges in the limit of long times. The simplest determination of its convergence amounts to calculating the leading corrections to the sum as a function of $\{\tau,s,k\}$. The expectation is that it should decrease exponentially with $\tau$ if $\{s,k\}$ are held fixed, although it is not obvious to us, a priori, at precisely which rate and with what length oscillations. Ahead, it turns out to be governed by the leading Pollicott-Ruelle resonance, which interestingly enough, has a real part equal to the Lyapunov exponent in the SR map.
### Local convergence boundary {#sect:averages}
It is also possible to consider the convergence with increasingly smaller local regions in the phase space. By controlling the size of a region, the number of terms contributing to the local sum can be tuned for a given $\tau$. Given that the individual stability determinants fluctuate with ever increasing width as $\tau$ increases, the question becomes, “at which time on average do the corrections become smaller than the sum rule expectation, i.e. the local phase space volume?” As the regions shrink in size this time extends later, thus making it possible to find a shrinking volume as a function of $\tau$ that offsets the exponentially decaying corrections. One can think of this “${\cal V}(\tau)$” as the boundary for the local application of the sum rule to be converged. Both the global and the local boundary of convergence is given ahead.
### Moments
For the subregions of phase space, subtracting the local volume from the sum itself, call it $\tilde F_{\tau}(s,k)$, gives the leading corrections and fluctuating components of the sum rule. A probability density for the values it takes on at fixed time over all regions can be defined, $P_{\tilde F_{\tau}}(x)$, which carries all information about convergence, local or global, and fluctuations. In particular, our focus in this paper is on central moments of the density. A very important case is the mean square deviation $$\sigma^2(\tilde F_{\tau},k) = \langle \tilde F_{\tau}(s,k)^2 \rangle_s = \int {\rm d} x\ \hat x^2 P_{\tilde F_{\tau}}(x)$$ where $\hat x = x- \bar x$ and for which an asymptotic expression is derived in the case of the SR map. More generally, the $n^{th}$ central moment is $${\cal M}_n(\tilde F_{\tau},k) = \langle \tilde F_{\tau}(s,k)^n \rangle_s = \int {\rm d}x\ \hat x^n P_{\tilde F_{\tau}}(x).$$ The moments $${\cal M}_n( e^{\lambda\tau},k) = \langle e^{n\lambda(s,k;\tau)\tau} \rangle_s = \int {\rm d}x\ x^n P_{\exp(\lambda\tau)}(x)
\label{moment}$$ which by Eq. is associated with the probability density of the finite-time stability determinants, are distinctly different from the moments of the sum rule fluctuations. Both cases are treated in this paper. However, as also shown, the probability density $P_{\tilde F_{\tau}}(x)$ asymptotically tends to a Gaussian density and only the first two moments (cumulants) are considered in detail.
The Lazy Baker SR map {#sec:srmap}
=====================
The Hannay-Ozorio sum rule fluctuations may be worked out exactly for the case of a simple dynamical system which is a modification of the usual baker’s map. Lazy baker maps have previously been introduced [@Lakshminarayan93] as a class of 2D area-preserving maps. We study here a particular case called the SR map (stretch-rotate) which is chaotic over the whole measure and is defined on the unit square in the usual position, momentum coordinates as follows: $$\begin{aligned}
&\left.
\begin{array}{l}
q' \hspace{2 mm}= \hspace{2 mm} 2q \\
p' \hspace{2 mm} = \hspace{2 mm} p/2 \\
\end{array} \right \} \hspace{5 mm} \textrm{if}\hspace{5 mm} 0 \leq q \leq 1/2, \nonumber \\
&\left. \begin{array}{l}
q' \hspace{2 mm} = \hspace{2 mm} 1-p \\
p' \hspace{2 mm} = \hspace{2 mm} q \\
\end{array} \right\} \hspace{5 mm} \textrm{if}\hspace{5 mm} 1/2 < q \leq
1. \end{aligned}$$ The action of the map can be pictured most easily by splitting the unit square into four equal subsquares, $\mathcal R_1$ through $\mathcal R_4$, shown in .
![Partition of the unit square into four subregions, $\mathcal R_1$ through $\mathcal R_4$ (upper panel) and where each subregion maps after one iteration (lower panel). As shown, $\mathcal R_1$ and $\mathcal R_2$ are squeezed and stretched whereas $\mathcal R_3$ and $\mathcal R_4$ are rotated.[]{data-label="fig:regions"}](Fig1.eps){width=".4\textwidth"}
The region $\mathcal R_4$ is rotated uniquely into $ \mathcal R_3$ on the next iteration and region $\mathcal R_3$ is rotated into $\mathcal R_2$. On the left half of the square, points in $\mathcal R_2$ and $\mathcal R_1$ are compressed by a factor two along the $p$ axis and stretched by the same factor along the $q$ axis.
The combination of rotation and stretching in the same dynamical system gives rise to the possibility of nonuniform hyperbolic motion or even non-hyperbolic motion. For the SR map defined as above with the vertical cut in the middle of the square at $q = 0.5$, the motion is of the former kind. As the cut is moved to the right it ceases to be completely hyperbolic at the golden mean. In this paper we will only study the case when there are equal regions stretching and rotating, which is arguably the simplest “exactly solvable" model of nonuniform hyperbolicity in an area preserving map. It admits a Markov partition of phase space and the dynamics is one of subshift of finite type on [*three*]{} symbols as shown in [@Lakshminarayan93a]. The atoms of the partition are $A=\mathcal{ R}_1 \bigcup \mathcal{ R}_2, \, B=\mathcal{ R}_3,$ and $C= \mathcal{ R}_4$.
The transition matrix is $$T_0= \left( \begin{array}{ccc}1&0&1\\1&0&0\\0&1&0 \end{array} \right)
\label{transition.matrix}$$ whose $ij$ element, $t_{ij}$, is $1$ if there is a transition from $i$ to $j$ and $0$ otherwise. Here $i,j\, \in \ \{A,B,C\}$. This has only topological information. Let $p_{ij}$ be the fraction of atom $i$ in atom $j$, on one evolution of the map. This gives the transition probabilities for the 3-state Markov chain that the SR map is equivalent to. The transition probabilities are $p_{AA}=1/2, \, p_{AC}=1/2, \, p_{BA}=1, \, p_{CB}=1$, the rest being zero. The Markov matrix is then $T_1$ whose matrix elements are $t_{ij} p_{ij}$ while for fluctuations to be studied below the matrix $T_2$ whose elements are $t_{ij} p_{ij}^2$ is also useful. These matrices can be used to study the uniformity principle at the global as well as the local scales as is shown below. To study what happens on restriction to smaller areas whose size can be controlled, as well as to find the actual locations of the orbits, it is useful to use a binary representation given ahead. This is not a symbolic dynamics in the sense that the dynamics is no more a left shift. However the dynamics is an [*eventual*]{} left shift even in the binary representation and is used extensively below.
Any point in phase space can be represented as a bi-infinite binary string $p.q$ representing its position $q$ and momentum $p$. The binary string representing the first $m$ bits of the position coordinate is labeled $\gamma_m$. The quantity $m$ also corresponds to the number of times an orbit visits the stretching region (left half) of the square after some number of iterations. The rules for the mapping equations on the binary string are given in [@Lakshminarayan93] and summarized here: if the most significant bit of position is a 0, the dynamics is that of a left shift. If the most significant bit is a 1, position and momentum coordinates are interchanged and in the *new* momentum coordinate all 0’s and 1’s are switched. For example, consider the period 3 orbit starting at $(q_0,p_0) = (2/3,1/3)$: (2/3,1/3) (2/3,2/3) (1/3,2/3) (2/3,1/3). The binary representations for the starting coordinates are $q = .\underline{10}$, $p = .\underline{01}$, where the underline indicates infinite repetition, and under the dynamics this point maps as . . . . while under the symbolic dynamics this orbit is $\underline{CBA} \rightarrow \underline{BAC} \rightarrow \underline{ACB}.$
Making use of the symbolic dynamics, it is possible to prove that the map contains a dense set of periodic orbits and is hence an ergodic transformation. It is also a straightforward argument to see that the positive Lyapunov exponent $\lambda_L$ for the SR map is \_L = 2. \[lyapunov\] This may be seen as follows: The unique smooth invariant density is uniform on the phase space. Therefore ergodicity implies that at a large number of iterations a typical orbit spends equal amounts of time on the left and right halves of the unit square. Since points along an orbit which are on the left half are stretched by a factor of two and points on the right half are not stretched at all, the Lyapunov exponent will be the average of $\ln 2$ and 0. The known symbolic dynamics, or binary representation, also allow the enumeration of all periodic orbits of any period as seen in Sect. \[enumerate\].
Periodic Orbits and Stability {#sect:postability}
-----------------------------
To begin the discussion of the periodic orbits, first note that there exist two exceptional periodic orbits on the boundary of the square: a period 1 fixed point at the origin and a period 2 orbit between the points $(1,1/2)$ and $(1/2,1)$. All other orbits must pass through the interior of the rotating region $\mathcal R_4$ and it is thus sufficient to count orbits originating in $\mathcal R_4$. The orbits come in two types depending on the parameter henceforth called $j$, which is the number of changes from 0 to 1 or 1 to 0 in the binary string $\gamma_{m}$ representing the first $m$ bits of the position coordinate. If $j$ is odd, the orbit may be represented as $p.q = \underline{\gamma_{m}}. \underline{\gamma_{m}}$. If $j$ is even the orbit may be written as $p.q = \underline{\gamma_{m}\overline{\gamma}_{m}}.\underline{\gamma_{m}\overline{\gamma}_{m}}$ where $\overline{\gamma}_{m}$ denotes the complement of $\gamma_{m}$. From the rules given for the symbolic dynamics one finds the period $\tau$ of an orbit in terms of its $\gamma_{m}$ string to be \[eqn:period\] = 2j + m + 2 where $j$ may take any integer value from 0 to $m-1$. Reference [@Lakshminarayan93] may be consulted for more details.
It is also useful to realize that it is possible to translate from the binary to the symbolic dynamics and vice-versa. Sticking to orbits originating from $\mathcal R_4$ these are of the form $\cdots 0.1x_2x_3\cdots$. This translates by replacing every transition (which is a 0-1 or 1-0 “bond” in the binary representation) including the initial 0.1 by $CBA$ and every other type ([*i.e.*]{} 0-0 and 1-1 bonds) by $AA$. Thus for example the orbit with the binary representation $\cdots 0.10100 \cdots$ translates to $\cdots CBACBACBACBAAA \cdots$.
The Jacobian stability matrix for a single time step is dependent on whether the point in question is in the left or right half of the unit square. Denoting ${\bf M}_{L}$ as the stability matrix for points in the left half and ${\bf M}_{R}$ as the stability matrix for points in the right half we have, following from the definition of the map, that \_[L]{} =
[2]{} & [0]{} & [1/2]{}
, \_[R]{} =
[0]{} & [-1]{} & [0]{}
. \[leftrightjacobian\] The stability matrix along a trajectory Eq. may then be calculated explicitly as a product of the matrices Eq. . Since points in $\mathcal R_4$ always map to $\mathcal R_3$ on the next time step, the product ${\bf M}_{R}{\bf M}_{R}$ will always come in pairs in the full product for ${\bf M}_{\tau}$. Because this product ${\bf M}_{R}{\bf M}_{R}$ produces a diagonal matrix, $-\bf{I}$, the full product Eq. contains only diagonal matrices, so it is commutative. This implies that up to a minus sign $\bf M_{\tau}$ is determined by the number of times an orbit visits the left half of the unit square, and the parity is determined by whether $j$ is even or odd. Putting all of this together gives the period $\tau$ Jacobian stability matrix of a particular periodic orbit as \[eqn:jacobian\] [**M**]{}\_ =
[2\^[m]{}(-1)\^[j+1]{}]{} & [0]{} & [2\^[-m]{}(-1)\^[j+1]{}]{}
with eigenvalues $$\begin{aligned}
|\Lambda_{c}| &= 2^{-m}, \hspace{4 mm} \text{contracting} \nonumber \\
|\Lambda_{e}| &= 2^{m}, \hspace{4 mm} \text{expanding}.
\end{aligned}$$
The explicit form of the inverse determinant that arises in semiclassical calculations is \[invdet\] = {
[\*[20]{}c]{} [(2\^[m]{} + 2\^[ - m]{} - 2)\^[ - 1]{} ]{}\
[(2\^[m]{} + 2\^[ - m]{} + 2)\^[ - 1]{} ,]{}\
. which may be written as with a $\pm$ for notational convenience, knowing that the sign in front of the 2 is determined by the parity of $j$. Since the main interest is in asymptotic calculations (large $\tau$) it is sufficient to keep the first two terms in the binomial expansion (2\^[m]{} + 2\^[ - m]{} 2)\^[ - 1]{} 2\^[-m]{} - ( - 1)\^j 2\^[ -2m + 1]{} + \[binapprox\] It is shown in Sect. \[sect:localdist\] that actually only the first term in the expansion is necessary to investigate certain asymptotic fluctuation properties, leaving the approximation 2\^[-m]{} \[detapprox\] Thus, for large period, to leading order $\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^{-1}$ coincides with the contracting stability eigenvalue $|\Lambda_{c}|$ of the stability matrix. A final note, although the determinant is the quantity which arises in semiclassical theory, some of the classical dynamical systems literature defines the uniformity principle with respect to the inverse of the stretching exponential [@Ott02], in which case the correction term of Eq. (\[binapprox\]) is not relevant.
Enumerating the periodic points {#enumerate}
-------------------------------
It is quite valuable to be able to count the periodic points of fixed period in a given subregion of the unit square. To do so we proceed as follows: divide the unit square into a grid of boxes of area $2^{-k} \times 2^{-k}$ , whose lower left corners are specified by $q =.x_{1}x_{2}...x_{k}$ and $p = .y_{1}y_{2}...y_{k}$. This is a binary expansion, so each $x_{i}$ and $y_{i}$ is either 0 or 1. This is a total of $4^{k}$ boxes. Membership of an orbit in a box with a specified lower left corner is simply that the orbit has the same first $k$ bits for $p$ and $q$ in its binary representation as the lower left corner, and arbitrary bits beyond the $k$th. To start, consider boxes in the lower right subsquare $\mathcal R_4$. The same counting results will hold for regions $\mathcal R_3$ and $\mathcal R_2$ since these are merely rotations of $\mathcal R_4$. The slightly more detailed counting arguments for region $\mathcal R_1$ can be found in Appendix \[appa\].
As in Sect. \[sect:postability\] the periodic points are written either in the form $\underline{\gamma_{m}}.\underline{\gamma_{m}}$ or $\underline{\gamma_{m}\overline{\gamma}_{m}}.\underline{\gamma_{m}\overline{\gamma}_{m}}$, the former if $j$, the number of 0-1 or 1-0 transitions in $\gamma_{m}$, is odd, and the latter if $j$ is even. The requirement on the string $\gamma_{m}$ for a point to be in $\mathcal R_4$ is that the first bit is 1. If the last bit is 0, then the first form represents a point in $\mathcal R_4$, and if the last bit is 1, then it is the second form that is a point in $\mathcal R_4$. In either case, $m$ is related to the period $\tau$ by Eq. . Note that $j \leq m-1$, so $j \leq \tau - 2j - 3$ and $3j \leq \tau - 3$, so that $j$ is at most $\lfloor(\tau/3)\rfloor - 1$ where $\lfloor \cdots \rfloor$ denotes the floor function. The first step is to count how many $\tau$-periodic points with a fixed value of $j$ there are in the box whose corner is specified by $q = .1x_{2}...x_{k}$ and $p =
.0y_{2}...y_{k}$ (note the 1 and the 0 are forced because the point must lie in $\mathcal R_4$). This point is also represented by combining the binary expansions into one expression $y_{k}y_{k-1}...y_{2}0.1x_{2}...x_{k}$, and similarly for other points.
For a $\tau$-periodic point of the first form, $\gamma_{m}$ must look like $1x_{2}...x_{k}w_{1}...w_{i}y_{k}...y_{2}0$ for some $i,k$ with the restriction $i
= \tau - 2j - 2k-2$ and the total number of transitions in this string is $j$. If the periodic point is of the second form, then $\gamma_{m} = 1x_2 ...x_k w_1 ...w_i \bar y_k ...\bar y_2 1$.
Let $s$ equal the number of 0-1 or 1-0 transitions in $1x_2...x_k$ plus the number of 0-1 or 1-0 transitions in $0y_2...y_k$. That is, $s$ is the total number of changes for the lower left corner point of the box. If $s$ and $j$ are both odd, a periodic point would be of the first form, and $\gamma_{m} =
1x_{2}...x_{k}w_{1}...w_{i}y_{k}...y_{2}0$. There are $s$ transitions from 1 to $x_k$ and from $y_k$ to 0 combined, so there must be $j - s$ transitions in the $i+1$ possible places in $x_{k}w_{1}...w_{i}y_{k}$. So there are $\left( {\begin{array}{*{20}c}
{i + 1} \\
{j - s} \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s} \\
\end{array} } \right)$ ways to do this. In the other cases in which $j$ and $s$ may be either even or odd, the same result holds and thus we have that for each possible value of $j$, the number of $\tau$-periodic points in a $2^{-k}$ $\times$ $2^{-k}$ box in $\mathcal R_4$ with corner value specified by $s$ is $ \left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s} \\
\end{array} } \right) $. The smallest possible value of $j$ is $s$, which occurs when all the $w$’s are the same as $x_{k}$, and the maximum attainable value of $j$ is $\lfloor(\tau - 2k + s -1)/3\rfloor$. In addition, for a given value of $s$, there are $\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array} } \right)$ possible $2^{-k}$ $\times$ $2^{-k}$ boxes with $s$ the number of transitions in the first $k$ $p$-bits plus transitions in the first $k$ $q$-bits of the corner point.
To summarize, the counting just given is for $\tau$-periodic points in a binary grid of boxes within $\mathcal R_4$, $\mathcal R_3$, and $\mathcal R_2$ where $k$ is the number of bits specifying a box side, and $s$ is the number of transitions in the $k$ bits of the $q$-coordinate of the corner of the box plus the number of transitions in the $k$ bits of the $p$-coordinate of the corner point. The index $j$ ranges from $s$ to $\lfloor(\tau - 2k + s -1)/3\rfloor$ and for a given $j$ the number of periodic points with that value of $j$ is given by $\left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s} \\
\end{array} } \right) $, so the total number of period-$\tau$ points in this box is N\_(s,k) = \_[j=s]{}\^[(-2k+s-1)/3]{}. \[eqn:numofpts1\] Note that for fixed period and box size, the statistics of the periodic points within a box are determined entirely by the value $s$ of its corner point. Any two boxes with the same value of $s$ will have exactly the same distribution, and for each $s$ there are $\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array} } \right)$ such boxes. Thus, the Hannay-Ozorio sum, Eq. , over all periodic points within a binary box in $\mathcal R_4$, $\mathcal R_3$, or $\mathcal R_2$ is $$\begin{aligned}
F_{\tau}(s,k) = \sum_{j=s}^{\lfloor (\tau-2k+s-1)/3\rfloor}&\VectorII{\tau-2j-2k-1}{j-s} \nonumber \\
& \times \frac{1}{\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|} \label{hannaysum}
\end{aligned}$$ and the global form of Eq. (excluding $\mathcal R_1$) by summing over all boxes in $\mathcal R_4$, $\mathcal R_3$, and $\mathcal R_2$ gives $$\begin{aligned}
&F_{\tau} = 3\sum_{s=0}^{2k-2}\VectorII{2k-2}{s} F_{\tau}(s,k).
\label{eqn:R4sum} \end{aligned}$$ The relation of the inverse determinant to the period and the symbolic representation of an orbit is, from Eq. and Eq. , given by 2\^[-+2j+2]{} \[inversedet\] which provides an explicit summable expression for looking at fluctuations in the uniformity principle.
The counting arguments for the subsquare $\mathcal R_1$ are slightly different, but similar in character, to those presented here and the details are given in Appendix \[appa\]. In fact, the resulting equations are quite close to the ones given in this section.
Figure \[fig:T28\] shows a plot of all periodic points in the unit square at $\tau=24$ and $28$. This visualization of the structure of the periodic points is interesting in its own right, as the points appear to have a fractal-like structure to them. In fact the checkered pattern created mimics the stable and unstable manifolds of the map.
![Fixed points of the time $\tau$ iterated map. The upper square is the plot for [$\tau$ = 24]{} and the lower square is the $R1$ region for [$\tau$ = 28]{}. The density of fixed points at $\tau=28$ is roughly 4.6 times that of $\tau=24$. Below, expanding $R1$ renders the lower plot’s fixed point density similar to the upper plot. The similarity of the expanded $R1$ fixed point’s structure to the full phase space at an earlier time illustrates the fractal-like structure mentioned in the text.[]{data-label="fig:T28"}](Fig2.eps){width="45.00000%"}
The symbolic dynamics and the Markov matrices can also be used to find the number of periodic orbits as well as to study the uniformity principle. Since the SR map is simple enough to permit both a combinatorial approach as well as a symbolic dynamics one, it is useful to present both. Given a periodic point of the first type in $\mathcal R _4$ whose $\gamma_m$ string has a binary representation $1x_2x_3\cdots w_1w_2\cdots w_i y_k \cdots y_2 0$, this translates into an orbit which is a repetition of symbol strings of length $\tau= 2j+2k+i+2$. Of these $3(s+1)+[2(k-1)-s]=2s+2k+1$ are utilized to specify the fixed corner point. Thus there are $n=\tau-(2s+2k+1)$ number of possible “free" symbols, say $S_1,\ldots S_n$. A little thought shows that the free symbol string [*has*]{} to end with $A$, that is $S_n=A$, and [*has*]{} to be prefixed by an $A$. Thus it can be either of the form $CBA \cdots A$ or $A \cdots A$. In both these cases the number of periodic points is then given by N\_(s,k)=\_[S\_i]{} (T\_0)\_[AS\_1]{} (T\_0)\_[S\_1S\_2]{} (T\_0)\_[S\_[n-1]{}A]{}, or N\_(s,k)=(T\_0\^[-2k-2s-1]{})\_[AA]{}. Recall that $T_0$ is the transition matrix in Eq. . Thus the combinatorial problem can be reduced to that of finding powers of a matrix. From this point the mathematical complexity is comparable as both lead to the analysis of cubic equations; see Appendix \[sect:sumformula\] for the combinatorial case.
The uniformity principle sum for the local area can also be written compactly in terms of a matrix power, this time the Markov matrix $T_1$. A similar reasoning as above leads to F\_(s,k)=(T\_1\^[-2k-2s-1]{})\_[AA]{}. \[symbdynFsk\] Here however, the approximation in Eq. is already used, as otherwise such a compact formula is not possible. Note that with this approximation the global (over the whole phase space) uniformity principle sum is simply the trace of the power of the Markov matrix. That is F\_ = (T\_1\^). This follows from the fact that the entries in the stochastic matrix $T_1$ are precisely the multipliers which are either $1/2$ or $1$. That the stochastic matrix has necessarily an eigenvalue 1, and therefore $F_{\tau} \rightarrow 1$ as $\tau \rightarrow \infty$ is an alternative formulation of the uniformity principle. The other eigenvalues of $T_1$ whose eigenvalues are less than 1 in modulus determine both the rate of decay of correlations as well as approach to uniformity of the periodic orbits. We will expand on this below shortly. Almost all of the analysis below follows the consequences of the binary representation and the combinatorial approach as detailed statistics is more transparently done this way.
Statistical Results {#sec:statistics}
===================
The results of the previous section and Appendix \[sect:sumformula\] can be used to evaluate sum rule fluctuations. Consider the local density of the inverse determinant $\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^{-1}$, which occurs as the natural weighting for periodic orbits in many semiclassical expressions. The first goal is to derive an asymptotic formula for its variance. This analysis leads naturally to discussing the density and convergence of the remaining component $\tilde F_{\tau}(s,k)$ of the Hannay-Ozorio sum rule introduced in Sect. \[moments\]. We give an analytic expression for $\tilde F_{\tau}(s,k)$ and compute its variance, as well as local and global boundaries of convergence for the sum rule.
Local distribution of the inverse determinant {#sect:localdist}
---------------------------------------------
Consider the regions $\mathcal R_4$, $\mathcal R_3$, and $\mathcal R_2$ (see ) of the unit square whose density of periodic orbits is described in Sect. \[enumerate\]. As before, the (similar) discussion for the region $\mathcal R_1$ is left to Appendix \[appa\]. There is a range of values taken by $\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^{-1}$ within a local patch of phase space, as described in Sect. \[enumerate\]. It was shown that the number of period $\tau$ fixed points with a fixed value of the determinant specified by $j$ in a box with parameters $s$ and $k$ is given by $\VectorII{\tau-2j-2k-1}{j-s}$ where $j \geq s$. For large period the combinatorial as a function of allowed values of $j$ (which can be thought of as a probability density for various stability determinant values of fixed points in the box) is approximately normally distributed (see Appendix \[sect:sumformula\]) with an exact mean given by Eq. , but written approximately here as + s - 0.162. Recall from Eq. that at period $\tau$ the inverse determinant may take on the values $2^{-\tau + 2j +2}$ as $j$ ranges from $s$ to $\lfloor(\tau - 2k + s - 1)/3\rfloor$. So in fact, for the discussion of $\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^{-1}$, the sum rule $F_\tau(s,k)$ is over terms of the form $\VectorII{\tau-2j-2k-1}{j-s}4^j$. It turns out that the density for these terms is normal as before for the $t=0$ case; note, oddly enough, that does not imply that the density for finding a particular value of the inverse determinant is lognormal as the convergence with $\tau-2j-2k-1\rightarrow\infty$ to normal is too slow.
Consider the density $g(i,t) = \VectorII{n-2i}{i}e^{ti}$ as a function of $i$ for a given $t$, where as in Appendix \[sect:sumformula\], $e^t = \alpha$. Using Stirling’s formula to approximate $g(i,t)$, and calculus, one finds that the maximum value of $g(i,t)$ occurs at the value $i_0$ of $i$ given by i\_0 n where $\beta_1(t)$ is the real root of the cubic equation $ \beta^3 - \beta^2 - e^t = 0$. Interestingly this same cubic equation arises here for a different problem from the one considered in Appendix \[sect:sumformula\]. This method does not give the exact transient terms as the recurrence method of Appendix \[sect:sumformula\] does, but the same structure exists and near the maximum at $i_0$, $g(i,t)$ is approximated continuously as a Gaussian with width on the order of $\sqrt{n}$ and so the values of $i$ which contribute to the sum are sharply peaked around the maximum. Specifically, $$\begin{aligned}
g(i,t) &= \left( {\begin{array}{*{20}c}
{n - 2i} \\
i \\
\end{array}} \right)e^{ti} \nonumber \\
&\simeq
\frac{\beta_1^{n+1} }{{3\beta_1 - 2}} \frac{1}{{\sqrt {2\pi \sigma^2_i} }}e^{ - \,(i - i_0 )^2 /(2\sigma^2_i)}
\end{aligned}$$ where $\sigma^2_i = n \frac{{\beta_1 (\beta_1 - 1)}}{{(3\beta_1 - 2)^3 }}$.
For the quantity $\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^{-1}$, for which $e^t = 4$, $\beta_1 = 2$ and so the mean of $g(i,t)$ occurs at $i_0 = n/4$, even though the mean $\mu(n)$ (for the unweighted combinatorial) occurs at about $n/5.148$. As $n \rightarrow \infty$, the two densities tend toward a vanishing overlap since the difference in the means grows faster than the widths. In considering values taken by the inverse determinant, only those periodic points with transition number $j$ which occur near j\_0 = + s \[j0det\] contribute to the sum $F_{\tau}(s,k)$, in spite of the fact that there is a vanishing relative fraction of fixed points associated with this value of $j$, as most points have a value of $j$ near $(\tau-2s-2k-1)/5.148 + s - 0.162$.
Important Moments
-----------------
We give here explicit expressions for some of the quantities of interest related to the inverse determinant and the SR map, using the results from the methods of Appendix \[sect:sumformula\]. First consider the number of period $\tau$ fixed points within a binary box specified by $s$ and $k$. The number $N_{\tau}(s,k)$ is given by Eq. , which is a special case of the sum formula $S(n,\alpha)$ from Appendix \[sect:sumformula\] with $\alpha = 1$ and $n = \tau - 2k - 2s - 1 $. The form of the solution is therefore specified by Eq. , with appropriate values for the constants. The topological entropy is given by $h_T=\ln \beta_1(0)$, and thus N\_(s,k) = c\_1 e\^[h\_T n]{} + 2e\^[-h\_T n/2]{}This equation contains the finite-time correction terms to the count of fixed points of a binary box, which cannot be given by specifying the entropy alone. If $\tau$ is large, the leading term $c_{1}e^{h_T n}$ dominates and may be used for asymptotic calculations.
The moments for the inverse determinant (which are different from the moments involved in the sum rule fluctuations, ahead) may be computed by averaging over powers of $\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^{-1}$. The most important case is the mean, which corresponds to $F_{\tau}(s,k)$, and gives an explicit finite-time correction term to the (infinite time) prediction of the uniformity principle. It is also an ingredient of other sum rule moments.
From Eq. and Eq. , the local form of the sum of the inverse determinant over all fixed points within a box reads F\_(s,k) = \_[j = s]{}\^[ ]{}
(
[\*[20]{}c]{} [- 2j - 2k - 1]{}\
[j - s]{}\
)
2\^[ - + 2j + 2]{} \[eqn:Fs\] which is predicted by the Hannay-Ozorio sum rule to asymptotically approach the area of the box, $4^{-k}$. A slight change of variables puts the sum in the generic form, Eq. , with $\alpha = 4$ and $n = \tau - 2k - 2s -1$. The solution is thus again of the form of Eq. , where the real root $\beta_1$ of the cubic is exactly 2. After some algebraic manipulation, it is seen that $F_{\tau}(s,k)$ may be written in a form which displays both its exponential dependence on the period as well as its relation to the phase space area as F\_(s,k) = 4\^[-k]{} + e\^[ - \_L ]{} 2\^[s - k + 5/2]{} \[a (n) - b (n)\] \[eqn:Fs2\] where $\lambda_L $ is the real part of the leading Pollicot-Ruelle resonance (here also equal to the positive Lyapunov exponent) whose value is given in Eq. . The numerical values of the constants $a$, $b$, and $\theta$ may be calulated using the formulas of Appendix. \[sect:sumformula\]. Subtracting the Hannay-Ozorio term leaves the oscillating part with time of the sum rule as F\_(s,k) = e\^[ - \_L ]{} 2\^[s - k + 5/2]{} \[a (n) - b (n)\]. \[eqn:fluctuating\] By factoring out the time dependence, it turns out that the rate of convergence towards uniformity with increasing period is exponential, as expected. It is clear from the discussion based on symbolic dynamics and Eq. that the rate is governed by the eigenvalues of a Markov matrix if it exists or generally by the Pollicott-Ruelle resonances. In the case of the SR map, the modulus of the eigenvalues of the Markov matrix, which are the resonances, have modulus equal to $1/\sqrt{2}$, which is $e^{-\lambda_L}$. It is well-known that the Pollicott-Ruelle resonances are generically not related to the Lyapunov exponents and therefore the equality for the SR map must be considered a coincidence. The close connections between mixing and uniformity principle makes the emergence of the Pollicott-Ruelle resonances as governing the rate of convergence to uniformity much more natural.
It is also interesting to consider the convergence boundary as mentioned in Sect. \[sect:averages\]. This amounts to determining the box size (phase space volume) for a given period and location in phase space at which the size of the correction term $\tilde F_{\tau}(s,k)$ is just the same order of magnitude as the local area itself. In particular, from Eq. , if $2^{-\tau/2 + s - k + 5/2} = 4^{-k}$ then $k = \tau/2 -s -5/2$ and the volume at the convergence boundary is $$\mathcal{V} (s;\tau) = 2^{-\tau + 2s + 5}.$$ In this way, the local sum rule fluctuations are equally as important as the mean, and hence to any results which invoke a sum rule on that local scale at that time. Given that $s$ varies in the domain $0\le s \le 2k-2$ or in terms of $\tau$, $0 \le 3s \le 2\tau -7$, the convergence boundary varies greatly from one location to another in the phase space, i.e. $2^{-\tau} \le \mathcal{V} (s;\tau) \le 2^{-\tau/3}$. Although the local convergence boundary vanishes everywhere as $\tau\rightarrow \infty$, its relative variation tends to infinity. The relatively larger boundaries are precisely linked to locally greater inverse determinant variation just ahead.
In Eq. , the leading correction term to the inverse determinant was given, but up to this point not included in the calculations. It is important to know if this error is subdominant relative to the fluctuating component just calculated. If the second expansion term is kept, this leads to a sum denoted $\tilde B_{\tau}(s,k)$ of the form $$\begin{aligned}
\tilde B_{\tau}(s,k) = \sum\limits_{j =s}^{\left\lfloor {(\tau - 2k + s - 1)/3} \right\rfloor } &{\left(
{\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s} \\
\end{array} } \right)} \nonumber \\
& \times 2^{ - 2\tau + 5}(-16)^{j}
\label{sumcorrection}
\end{aligned}$$ which once again is in the form of the sum discussed in Appendix \[sect:sumformula\] with $\alpha =
- 16$ and $n = \tau -2s - 2k -1$.
In fact, more properly, the two most dominant corrections to the local sum rule are F\_(s,k) = 4\^[-k]{} + F\_(s,k) - B\_(s,k) We know that the first correction term is governed by the Pollicott-Ruelle resonances, but the second term is something else. A priori, it is not obvious which of these two correction terms dominates for large period. Extracting only the exponential dependence on $\tau$ gives $\tilde F_{\tau}(s,k) \propto e^{-\lambda_L \tau} \approx (0.71)^\tau$. For $\tilde B_{\tau}(s,k)$, because $\alpha<0$, the dominant fluctuation term comes from the oscillatory $(\alpha/\beta)^{n/2}$, which is approximately $\tilde B_{\tau}(s,k) \approx (0.67)^\tau$. In this case, the first correction term eventually dominates over the second, and using the approximation of Eq. is justified. Had the situation turned out the opposite way, then the correction term would not have been a Pollicott-Ruelle resonance; we are not aware of an argument suggesting that this could not have happened and thus both sources of corrections must be considered in other cases.
Before continuing with the spatial fluctuations in the sum rule itself, consider the variation of the individual inverse determinants contributing to each sum. They vary wildly from one fixed point to the next and there is greater variation in some regions as opposed to others. This gives an $s$-dependence to their variation within any single box. This can be seen by computing the variance. The sum of squares of the inverse determinant, $Q_{\tau}(s,k)$, is given by $$\begin{aligned}
Q_{\tau}(s,k) &= \sum_{f.p.} \frac{1}{\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|^2} \nonumber \\
&= \sum\limits_{j =
s}^{\left\lfloor {(\tau - 2k + s - 1)/3} \right\rfloor } {\left(
{\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s} \\
\end{array} }
\right)} 2^{ - 2\tau + 4j + 4}
\end{aligned}$$ This is a sum of the form of Eq. with $\alpha = 16$ and $n = \tau - 2k - 2s -1$. Keeping only the leading term in Eq. , the solution is Q\_(s,k) = 4\^[-+2s+2]{}c\_[1Q]{}\_[1Q]{}\^[-2k-2s-1]{} where $\beta_{1Q}$, the real root of the cubic polynomial for $\alpha = 16$, is approximately 2.901 and $c_{1Q} \approx 0.433$. The subscripts on the numerical constant $c$ and $\beta$ are used here to make it clear that these refer to the case $Q$, for which $\alpha = 16$. The same result can also be derived from the symbolic dynamics as Q\_(s,k)= (T\_2\^[-2k-2s-1]{})\_[AA]{}.
In the notation of Eq. , the variance of the inverse determinants within a given box is $\sigma^2(e^{\lambda\tau}) = {\cal M}_{-2}( e^{\lambda\tau},s,k) - {\cal M}_{-1}( e^{\lambda\tau},s,k)^2$ and \^2(e\^,s,k) = - ( )\^[2]{} where the leading order of the mean is sufficient. The product $Q_{\tau}N_{\tau}$ depends on $\tau$ by a factor $(e^{h_T}\beta_{1Q}/4)^{\tau}$, which is greater than unity. Thus, this term diverges as $\tau\rightarrow\infty$. Asymptotically the local variance is just $Q_{\tau}/N_{\tau}$, or $$\sigma^2(e^{\lambda\tau},s,k) \longrightarrow
\frac{16c_{1Q}}{c_{1N}}\left( {\frac{{\beta_{1Q} }}
{{4e^{h_T}}}} \right)^{\tau-2s} \left(
{\frac{{\beta_{1Q} }} {{e^{h_T}}}} \right)^{ - 2k - 1}$$ which shows asymptotically how the variance varies with $s$, a local characteristic of a particular region of phase space (box). Here $\beta_{1Q} / e^{h_T}$ is about 1.98 and $\beta_{1Q} / 4 e^{h_T}$ is about 0.495. When the analogous details are worked out for boxes in the region $\mathcal R_1$, the variance differs only by a constant factor of $16 \left(\frac{\beta_{1Q}}{e^{h_T}}\right)^{2}$.
Note that boxes whose lower-left corner has a small number of transitions (small $s$) have smaller variances, as well as more $\tau$-periodic points, than boxes with large $s$. Precisely as found for the local convergence boundaries, the variation of $s$ for moderately large $k$ leads to an enormous difference in the variations within different boxes of the same size. Although, the variances vanish in the limit of $\tau\rightarrow\infty$, the ratios of the variances from one box to another another increase indefinitely as the box size shrinks.
Sum rule fluctuations {#sumfluct}
---------------------
Next the global variance of the local sum rule $F_{\tau}(s,k)$ is considered due to spatial variation. First we comment on the form of the density of $F_{\tau}(s,k)$. Recall that with the method of subdividing the phase space into a grid of binary boxes, the value of $F_{\tau}(s,k)$ locally within a box is specified by a parameter $s$ which counts the number of 0 to 1 changes in the binary representation of the lower left corner of the box (see Sect. \[enumerate\]). Thus, $\tilde F_{\tau}(s,k)$ depends exponentially on $s$, Eq. . Furthermore, the number of boxes throughout a quarter region of the unit phase space square with a given value of $s$ is $\VectorII{2k - 2}{s}$, as $s$ ranges from 0 to $2k-2$. Thus, the density of the logarithm of $\tilde F_{\tau}(s,k)$ follows a binomial centered at $k-1$. As with the inverse determinant, the distribution of $\tilde F_{\tau}(s,k)$ is described by the product of an exponential function (of $s$, here) and a combinatorial coefficient that is approximately normal. A qualitatively similar behavior to the discussion of Sect. \[sect:localdist\] arises in describing the density of values taken by the sum formula $\tilde F_{\tau}(s,k)$.
The variance of $\tilde F_{\tau}(s,k)$ is the average square deviation from the mean summing over all boxes and dividing by their total number, $4^{k}$. This is essentially the second moment defined in Sect. \[moments\] \_2(F\_,k) = \_[s=0]{}\^[2k-2]{} (
[c]{} 2k - 2\
s\
) F\_(s,k)\^2 \[variance2\] For calculational convenience, Eq. is rewritten in the form F\_(s,k) = Ae\^[s]{} + (Ae\^[s]{})\^\* \[Fgamma\] where $\gamma = \ln{2} - 2\theta i$ and $A = c_{2}2^{-\tau /2 - k +3/2}e^{i\theta (\tau - 2k - 1)}$. Recall that the constants $c_2$ and $\theta$ arise from the solutions of the sum formula in Appendix \[sect:sumformula\], in this case for $\alpha = 4$. This result holds for the regions $\mathcal R_4$, $\mathcal R_3$, and $\mathcal R_2$ of the unit square. For $\mathcal R_1$, the expression used for $F_{\tau}(s,k)$ differ only by a constant factor, as shown in Appendix \[appa\], and this factor is accounted for below in giving the variance over the entire unit square.
For large $k$, it is possible to find a simplified asymptotic expression for the variance. Let $c_2=|c_2|e^{i\zeta}$ and $\eta = \theta(\tau - 2k - 1) + \zeta$ giving \[Ae\^[s]{} + (Ae\^[s]{})\^\*\]\^2 = |c\_2|\^2 2\^[-- 2k + 3]{}\[2\^[2s+1]{} + 2 (e\^[2i]{}e\^[2s]{})\]. The expression for the variance becomes $$\begin{aligned}
{\cal M}_2(\tilde F_{\tau},k) = & \hspace{1 mm} 4^{-k}|c_2|^2 2^{-\tau - 2k +4} \left[\sum_{s=0}^{2k-2}{\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array} } \right)4^s}\right. \nonumber \\
& \hspace{5 mm} \left. + \text{Re}(e^{2i\eta}\sum_{s=0}^{2k-2}{\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array} } \right)e^{2\gamma s}})\right].
\end{aligned}$$ Recalling the binomial theorem, each term above may be summed explicitly to give $$\begin{aligned}
{\cal M}_2(\tilde F_{\tau},k) = &|c_2|^2 2^{-\tau}16^{-k+1} \left[ 5^{2k-2} \right.
\nonumber \\
& \hspace{5 mm} \left. + \text{Re}\left(e^{2i\eta}\left[1+e^{2\gamma}\right]^{2k-2}\right)\right].
\end{aligned}$$ Letting $1+e^{2\gamma} = 1 + 4e^{-i4\theta} = \rho e^{i\omega}$ where $\rho^2 = 17 + 8\cos(4\theta)$ gives $$\begin{aligned}
{\cal M}_2(\tilde F_{\tau},k) &= |c_2|^2 2^{-\tau-4k+4}\left[5^{2k-2} + \rho^2 \cos(2\eta + 2\omega[k-1])\right] \nonumber \\
&= |c_2|^2 2^{-\tau}\left(\frac{25}{16}\right)^{k-1} \nonumber \\
& \hspace{4 mm} \times\left[1 + \left(\frac{\rho^2}{25}\right)^{k-1} \cos(2\eta + 2\omega[k-1])\right].
\end{aligned}$$ Since $4\theta$ is not a multiple of $2\pi$, $\cos(4\theta)$ is less than unity and so is $\rho^2 / 25$. Thus, the oscillatory terms are subdominant as $k$ increases. For large $k$, or small local volume, the expression for the variance in each of regions $\mathcal R_4$, $\mathcal R_3$, and $\mathcal R_2$ becomes: \_2(F\_,k) (3/4)|c\_2|\^2 e\^[-2\_L ]{}(5/4)\^[2k-2]{}. For the region $\mathcal R_1$ the same equation for $\tilde F_{\tau}(s,k)$ as Eq. applies except that the coefficient $A$ is replaced by $A' = c_{2}2^{-\tau /2 - k +1/2}e^{i\theta (\tau - 2k + 1)}$. From this it follows that the contribution to the variance from the region $\mathcal R_1$ is simply one fourth the value for $\mathcal R_4$, $\mathcal R_3$, or $\mathcal R_2$. The asymptotic formula for the variance of $F_{\tau}$ taken over the entire unit square is \_2(F\_,k) |c\_2|\^2 e\^[-2\_L ]{}()\^[2k-2]{}. \[eqn:variance\] The variance thus decreases exponentially with time, again governed by the Pollicott-Ruelle resonance. It also increases with the decreasing local volumes. This gives a global convergence boundary for the sum rule on which the variance over the entire phase space remains a constant (rather than vanishing). From Eq. this would be given approximately by $k = \frac{\lambda_L\tau}{\ln{5/4}}$, and () = 2\^[-/]{} 2\^[-3.1 ]{}.
Concluding Remarks
==================
The convergences and fluctuations of classical sum rules are interesting in a multitude of ways. Although, their corrections may be exponentially suppressed with increasing time, the individual contributions can have a diverging variance themselves. Another interesting feature of local sum rules, as shown herein, is that certain fluctuations can be surprisingly large as the location of phase space is varied. Correction terms may be related to known properties of the system in more general dynamical systems, such as the topological entropy, the Pollicott-Ruelle resonances, or the Lyapunov exponent depending on the precise sum rule of interest. It would appear that a fluctuation quantity, which depends sensitively on some higher power of the stability determinant, if such a quantity exists, may be more likely to reflect the kinds of fluctuations that have been described here on a theoretical basis for the SR map. The results, however, may be suggestive of the type of behavior one might expect in a regime where sum rule fluctuations could arise. The asymptotic form of several different fluctuation measures derived for the Hannay-Ozorio sum, as well as their time and length scales, came from the solution of the same simple cubic polynomial. The origin of this lies in the symbolic dynamics which is a subshift of finite type on three symbols and thus there is an equivalent three state Markov chain. It may be the case that similar methods could be applied to other relatively simple systems, and that sum rule corrections could also be derived for these systems from the basis of their dynamics.
The main results for the SR map begin with the calculation of the two sources of fluctuations in the Hannay-Ozorio sum rule. The first source is governed by the dominant Pollicott-Ruelle resonances. It arises from the non-uniformity of the locations of fixed points and their non-uniform weighting by the leading behavior of their inverse stability determinants. The second source arises from the effects of next-to-leading order corrections to the inverse stability determinants. These corrections are not governed by the Pollicott-Ruelle resonances, but are also exponentially decreasing in time. The dominant correction here comes from the first source and hence the Pollicott-Ruelle resonances, however we do not currently know whether this must be the case for general chaotic dynamical systems.
It is a matter of how closely the stretching multipliers approximate the determinant in Eq. (\[invdet\]), and it could be that for some other chaotic system they are different enough to produce corrections that dominate the one due to the resonances, although these would still be present. In the specific case of the SR map the second term in Eq. (\[binapprox\]), the principal correction, is an oscillating sum because the periodic orbits are reflecting hyperbolic if the number of $0-1$,$1-0$ bonds are odd. This term can be written as a trace of the power of the matrix $$\left( \begin{array}{ccc}1/4&0&1/4\\-1&0&0\\0&1&0 \end{array} \right)$$ which is different from the $T_2$ matrix in that the element $(2,1)$ is $-1$ rather than $1$. This ensures that each time the orbit gets rotated, it acquires a negative sign. Alternately, for each CBA part of the symbolic string a negative sign is acquired.
The leading eigenvalue of this matrix has a modulus of $\approx 0.67$ which is smaller than the subleading eigenvalue, $\approx 0.707$, of $T_1$ that gives the Pollicot-Ruelle resonance. If the orbit were [*all*]{} direct hyperbolic then in the above matrix the $-1$ will change to $1$ and this will be same as $T_2$ whose leading eigenvalue is $0.725$ which is larger than $0.707$, and would have dominated the corrections. The fact that some of the orbits are reflecting hyperbolic seems to have been crucial to lower the contribution from corrections that come from the fact that a $\det(J-I)$ is present instead of just the multipliers.
However if the sum rule is weighted by (the inverse) of the largest eigenvalue of the stability matrix, the Pollicott-Ruelle resonances will govern the corrections to the sum rule, especially if there is a finite symbolic dynamics description of the system. The relevance of Markov and related matrices ($T_0,T_1,T_2, \ldots$) for the calculation of the fluctuations indicate possible connections with the Thermodynamic Formalism especially as applied to finite Markov processes [@Gaspard98]. A second result shows how the relative local variations of the inverse determinants varies infinitely broadly at long times. Finally, the relative variation of the sum rule applied locally also has an infinite width while maintaining an exponential convergence rate for fixed phase space volume. We gave convergence boundaries that show how small a local volume may be considered for a given time of propagation if one expects convergence to the asymptotic sum rule result. Again, the relative size of a converged local volume depended on location and varied infinitely broadly while maintaining exponential convergence with time at fixed volume
It would be extremely interesting to investigate other sum rules, especially those that connect to quantum fluctuation properties of eigenfunctions and transport. The various localizing effects giving rise to eigenfunction scarring [@Heller84], localization manifestations of time scales introduced by transport barriers [@Bohigas93], and interaction effects linked to Friedel oscillations [@Tomsovic08; @Ullmo09] give a few interesting directions for further studies. As mentioned earlier, the SR map is easily studied quantum mechanically and would be one possible way to study sum rules arising from quantum fluctuations properties involving eigenfunctions.
We would like to acknowledge very helpful discussions with J. H. Elton on several of the mathematical points involved, and the generous support of the U.S. National Science Foundation grants PHY-0855337 and PHY-0649023.
Region $\mathcal R_1$ {#appa}
=====================
Here the counting arguments and several results for the region $\mathcal R_1$ of the unit square, which has mostly been ignored in the body of the text, are presented. The reason for leaving this discussion here is that many of the derived results closely resemble those for the other regions, although the arguments are somewhat longer.
We begin with an extension of Sect. \[enumerate\] by counting the number of period $\tau$ points in a binary $2^{-k}$ by $2^{-k}$ box in the region $\mathcal R_1$ (), but not on the bottom row of boxes. The lower left corner of each box is defined by $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} y_k...y_2 0.0x_2 ...x_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} $, with the condition that not all of the $y$’s are zero. The upper right corner is given by $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} y_k...y_2 0.0x_2 ...x_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} $. It is easy to see that after applying the inverse transformation of the map some number of times, each square of area $4^{-k}$ will be mapped into a rectangle in $\mathcal R_4$ of the same area, although not square, and also that the upper right corner of the square in $\mathcal R_1$ gets mapped into the lower left corner of the rectangle in $\mathcal R_4$. Thus the box in $\mathcal R_1$ with lower-left corner $y_{k}...10...0 . 0x_2...x_k$ (where not all the $y$’s are zero) has upper right corner $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} y_k
...10...0\,.\,0x_2 ...x_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} $ which transforms under the inverse transformation as follows: $\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} y_k
...10...0\,.\,0x_2 ...x_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1}
\leftarrow
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} y_k
...1.0...\,0x_2 ...x_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1}
\leftarrow
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} \bar
x_k ...\bar x_2 1...1\,.\,1...y_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1}
\leftarrow
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} \bar
y_k ...0\,.\,1...1\bar x_2 ...\bar x_k
\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} $, so the lower left corner of the rectangle in $\mathcal R_4$ is $\bar y_k
...0\,.\,1...1\bar x_2 ...\bar x_k $, with transition number one less than the lower left corner of the box in $\mathcal R_1$.
Let $s$ be the number of transitions of the lower left corner of a box in $\mathcal R_1$. Then by the same counting argument that was used before (the fact that it is a rectangle instead of a square does not change things), the number of $\tau$-periodic points in this box in $\mathcal R_1$ with a given transition number $j$ is $\left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s + 1} \\
\end{array} } \right)$ as $j$ ranges from $s-1$ to $\lfloor(\tau - 2k + s -2)/3\rfloor$. So this looks just like it did for the $\mathcal R_4$ case except $s$ is replaced by $s$ - 1.
Now, the lower left corner of a box that we are considering in this region has, say, $t$ transitions in the position coordinate and $v$ transitions in the momentum coordinate, where $0 \leq t \leq k-1$, and $1 \leq v \leq k-1$, and $s = t + v$. The combinatorial expression above shows that the number of $\tau$-periodic points in a box depends only on the total $s$ and not how it is distributed between $t$ and $v$, however it is necessary to consider $t$ and $v$ separately because it is $v$ that is restricted to be greater than zero and not just the sum of $t$ and $v$. We separately choose $t$ transitions from $k-1$ places for position, and $v$ transitions from $k-1$ places for momentum, with $v$ restricted to be greater than zero. So, for example, the local form of the sum of the inverse determinant for boxes in $\mathcal R_1$ excluding the bottom row (denoted $\mathcal R_{1a}$), analogous to expression Eq. , is $$\begin{aligned}
F_{\tau}(t+v,k;\mathcal R_{1a}) =
\sum\limits_{j = t + v - 1}^{\left\lfloor {(\tau - 2k + t + v - 2)/3} \right\rfloor } &{\left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - (t + v - 1)} \\
\end{array} } \right)} \nonumber \\
& \times \frac{1}{\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|}
\label{R1local}
\end{aligned}$$ and the sum over all boxes in $\mathcal R_{1a}$, analogous to expression Eq. , is \_1F\_[a]{} = \_[v = 1]{}\^[k - 1]{} \_[t = 0]{}\^[k - 1]{} (
[\*[20]{}c]{} [k - 1]{}\
v\
)(
[\*[20]{}c]{} [k - 1]{}\
t\
) F\_(t+v,k;R\_[1a]{}). Considering the bottom row of boxes in $\mathcal R_1$ (denoted $\mathcal R_{1b}$) a similar, but more tedious argument, gives the number of period $\tau$ points again as $\left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s + 1} \\
\end{array} } \right)$ as this time $j$ ranges from $s$ to $\lfloor(\tau-2k+s-2)/3\rfloor$. So the local sum of the inverse determinant for boxes on the bottom row of $\mathcal R_1$ is $$\begin{aligned}
F_{\tau}(s,k;\mathcal R_{1b}) = \sum\limits_{j = s }^{\left\lfloor {(\tau - 2k + s - 2)/3} \right\rfloor } &{\left( {\begin{array}{*{20}c}
{\tau - 2j - 2k - 1} \\
{j - s + 1} \\
\end{array} } \right)} \nonumber \\ &\times \frac{1}{\left|{\rm Det}({\bf M}_\tau-{\bf 1})\right|} \label{bottomlocal}\end{aligned}$$ and the sum over all boxes on the bottom row of $\mathcal R_1$ is \_1F\_[b]{} = \_[s = 0]{}\^[k - 1]{}
(
[\*[20]{}c]{} [k - 1]{}\
s\
)
F\_(s,k;R\_[1b]{}).
The more complicated sums that arise for region $\mathcal R_1$ may be simplified for certain calculations of interest. In particular, consider the calculation of the variance for the Hannay-Ozorio sum in Sect. \[sumfluct\]. Equations and , like their $\mathcal R_4$ counterpart Eq. , are sums of the form of Eq. from Appendix \[sect:sumformula\] with $\alpha = 4$, and they may both be expressed as F\_(s; R\_[1]{}) = 4\^[-k]{} + A’e\^[s]{} + where $A' = c_{2}2^{-\tau /2 - k +1/2}e^{i\theta (\tau - 2k + 1)}$ and $\gamma = \ln2 - 2\theta i$.
For the sum of squared deviations over boxes in $\mathcal R_1$ the expression to evaluate is $$\begin{aligned}
\sigma^2(\tilde F_{\tau}) = & \hspace{1 mm} 4^{ - k} \sum\limits_{v = 1}^{k - 1} {\sum\limits_{t = 0}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)} \left( {\begin{array}{*{20}c}
{k - 1} \\
t \\
\end{array}} \right)\tilde{F}_{\tau}(t+v)^2 \quad } \nonumber \\
&+ 4^{-k}\sum\limits_{s = 0}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } \quad\end{aligned}$$ where $\tilde{F}_{\tau}(s) = F_{\tau}(s) - 4^{-k}$.
There is an identity due to Vandermonde [@Graham94] which simplifies the double sum above and gives a result that is almost exactly like the sum for region $\mathcal R_4$. Denoting the double sum by $W$ gives $$\begin{aligned}
W &= \sum\limits_{v = 1}^{k - 1} {\sum\limits_{t = 0}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)} \left( {\begin{array}{*{20}c}
{k - 1} \\
t \\
\end{array}} \right)\tilde{F}_{\tau}(t+v)^2 }\quad \nonumber \\
&= \sum\limits_{v = 1}^{k - 1} {\sum\limits_{s = v}^{v + k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{k - 1} \\
{s - v} \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } }\end{aligned}$$ where $s = t + v$. Interchanging the order of summation and breaking this into two sums generates $$\begin{aligned}
W = &\sum\limits_{s = 1}^{k - 1} {\tilde{F}_{\tau}(s)^2 \sum\limits_{v = 1}^s {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{k - 1} \\
{s - v} \\
\end{array}} \right)} } \nonumber \\
&+ \sum\limits_{s = k}^{2k - 2} {\tilde{F}_{\tau}(s)^2 \sum\limits_{v = s - k + 1}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{k - 1} \\
{s - v} \\
\end{array}} \right)} }. \,
\nonumber \end{aligned}$$ Vandermonde’s convolution identity is \_b
(
[\*[20]{}c]{} a\
b\
)
(
[\*[20]{}c]{} c\
[d - b]{}\
) = (
[\*[20]{}c]{} [a + c]{}\
d\
)where the sum is over all values of $b$ for which the summand is not zero. This gives $$\sum\limits_{v = 1}^s {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{k - 1} \\
{s - v} \\
\end{array}} \right)} \,\,\,\, = \,\,\,\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array}} \right) - \left( {\begin{array}{*{20}c}
{k - 1} \\
s \\
\end{array}} \right)$$ when $1 \leq s \leq k-1$, where the subtracted term corresponds to $v = 0$. Also note that $$\sum\limits_{v = s - k + 1}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
v \\
\end{array}} \right)\left( {\begin{array}{*{20}c}
{k - 1} \\
{s - v} \\
\end{array}} \right)} \,\,\,\, = \,\,\,\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array}} \right)$$ when $k \leq s \leq 2k - 2$. Combining the two gives $$W =
\sum\limits_{s = 1}^{2k - 2} {\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } \,\, - \sum\limits_{s = 1}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 },$$ and therefore, $$\begin{aligned}
\sigma^2(\tilde F_{\tau}) = 4^{ - k} [ &\sum\limits_{s = 1}^{2k - 2} {\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } \,\, \nonumber \\ &- \sum\limits_{s = 1}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } \,\, \nonumber \\ &+ \quad \sum\limits_{s = 0}^{k - 1} {\left( {\begin{array}{*{20}c}
{k - 1} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 }]\end{aligned}$$ or $$\begin{aligned}
\sigma^2(\tilde F_{\tau}) &= 4^{ - k} [ {\sum\limits_{s = 1}^{2k - 2} {\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } \,\,\,\, + \,\,\,\tilde{F}_{\tau}(0)^2 } ] \nonumber \\
&= 4^{ - k} \sum\limits_{s = 0}^{2k - 2} {\left( {\begin{array}{*{20}c}
{2k - 2} \\
s \\
\end{array}} \right)\tilde{F}_{\tau}(s)^2 } \,\quad\end{aligned}$$ which is simply \^2(F\_) = 4\^[ - k]{} \_[s = 0]{}\^[2k - 2]{}
(
[\*[20]{}c]{} [2k - 2]{}\
s\
)(A’e\^[s]{} + )\^2
. This sum is of exactly the same form as Eq. for computing the variance for the other regions of the unit square.
A sum formula for the SR map {#sect:sumformula}
============================
Upon examination of the form of Eq. and given the result of Eq. , it happens that in order to arrive at closed form expressions for fluctuations in the Hannay-Ozorio sum, Eq. , it turns out that several sums of the form S(n,) = \_[i=0]{}\^[n/3 ]{}
(
[\*[20]{}c]{} [n - 2i]{}\
i\
)
\^i \[appendixform\] are needed for various real values of $\alpha$, with $n$ a positive integer. This section gives a general discussion of such sums and presents a method of finding closed form expressions for them before moving on to the main results of interest. In the analysis of the SR map, the cases $\alpha$ = 1, 4, 16 and -16 show up naturally when considering the lower order moments of Sect. \[moments\].
Obtaining a closed form for the sum may begin by finding a recursion formula for it, and then using a standard technique for solving such recursions. Recall the recursion for building Pascal’s triangle: $\left( {\begin{array}{*{20}c}
k \\
i \\
\end{array} } \right) = \left( {\begin{array}{*{20}c}
{k - 1} \\
{i - 1} \\
\end{array} } \right) + \left( {\begin{array}{*{20}c}
{k - 1} \\
i \\
\end{array} } \right)$ where $k$ and $i$ are greater than one. Applying this gives $$\begin{aligned}
S(n,\alpha) &= \sum_{i=0}^{\lfloor n/3 \rfloor} {\left( {\begin{array}{*{20}c}
{n - 2i} \\
i \\
\end{array} } \right)} \alpha ^i \nonumber\\ &= \sum_{i=1}^{\lfloor n/3 \rfloor} {\left( {\begin{array}{*{20}c}
{n - 2i - 1} \\
{i - 1} \\
\end{array} } \right)\alpha ^i } \nonumber\\ & \hspace{1 cm} + \sum_{i=0}^{\lfloor (n-1)/3 \rfloor} {\left( {\begin{array}{*{20}c}
{n - 2i - 1} \\
i \\
\end{array} } \right)\alpha ^i } \nonumber\\
&= \alpha \sum_{j=0}^{\lfloor (n-3)/3 \rfloor} {\left( {\begin{array}{*{20}c}
{n - 3 - 2j} \\
j \\
\end{array} } \right)\alpha ^j} \nonumber \\
& \hspace{1 cm} + \sum_{i=0}^{\lfloor (n-1)/3 \rfloor} {\left( {\begin{array}{*{20}c}
{n - 1 - 2i} \\
i \\
\end{array} } \right)\alpha ^i }
\end{aligned}$$ where $j=i-1$ in the first summation. This produces the recursion relation S(n,) = S(n-1,) + S(n-3,) \[eqn:recursion\] for $n \geq 3$ with initial conditions $S(0,\alpha) = S(1,\alpha) = S(2,\alpha) = 1$. Of historical note, the 14$^{th}$ century Indian mathematician Narayana studied a problem of the proliferation of cows (each offspring gives birth after its third year) that leads to this very same recursion relation with $\alpha=1$ [@Datta93]. A standard technique for solving such recurrence relations is to look for solutions of the form $S(n,\alpha) = \beta^n$, and thus $S(n-1,\alpha) = \beta^{-1}\beta^{n}, S(n-3,\alpha) =
\beta^{-3}\beta^n$. Plugging these expressions into Eq. the factor $\beta^n$ cancels and leaves the cubic equation \^3 - \^2 - = 0. \[eqn:cubic\] The three roots of this cubic $\beta_1$, $\beta_2$, $\beta_3$ give three solutions of the recurrence $\beta_1^n$, $\beta_2^n$, and $\beta_3^n$ . The difference equation Eq. is third order, linear, and homogeneous, and the standard theory of such difference equations (analogous to that for differential equations) says that if there are three linearly independent solutions, then the general solution may be formed as a linear combination of these independent solutions. Naturally this cubic equation also appears when using the matrices from symbolic dynamics. Indeed the characteristic equations for $T_0$, $T_1$ and $T_2$ are, up to a scaling, the same as the cubic equations with $\alpha=1,4$ and 16, respectively.
The cubic polynomial $f(\beta) = \beta^3 - \beta^2 - \alpha$ has a local maximum of $-\alpha$ when $\beta = 0$ and a local minimum at $\beta = 2/3$. It has one real root, say $\beta_1$, when $\alpha > 0$ and also when $\alpha < -4/27$, which covers all of the cases of interest. The other two roots are complex conjugates, $\beta_2 = re^{i\theta} = \delta + i\gamma$ and $\beta_3 = re^{-i\theta} = \delta - i\gamma$. Since $\beta_{1}^2(\beta_1 - 1) = \alpha$, it implies that $\beta_1 > 1$ when $\alpha >0$, and $\beta_1 < -1/3$ when $\alpha < -4/27$, so that $\beta_1$ has the same sign as $\alpha$.
Because the three roots are distinct, the three solutions are independent and the general solution to Eq. may be written as S(n,) = c\_1 \_1\^n + c\_2 (re\^[i]{} )\^n + c\_3 (re\^[ - i]{} )\^n \[sum1\] where the coefficients $c_1$, $c_2$, $c_3$ may be found from the initial conditions on the recurrence (in all cases here the initial conditions on $S(n,\alpha)$ are real).
It is possible to express the two complex roots, as well as the coefficients, in terms of the real root $\beta_1$ and in terms of $\alpha$. The constant $c_1$ is real and $c_3$ is the conjugate of $c_2$, which can be denoted as $c_2$ = $a$ + $i b$, $c_3$ = $a$ - $i b$ where $a$ and $b$ are real. The polynomial of Eq. may be factored as $(\beta - \beta_1)(\beta - \delta - i\gamma)(\beta - \delta + i\gamma)$.
Comparing the constant terms of the polynomial written both ways gives $\beta_1(\delta^2 + \gamma^2) = \alpha$. Thus, the magnitude of the complex roots is $r = (\delta^2 + \gamma^2)^{1/2} = (\alpha / \beta_1)^{1/2}$. Because $\beta_{1}^2 - \beta_1 = \alpha/\beta_1$, it is also true that $r < \beta_1$ when $\alpha > 0$, but $r > \beta_1$ when $\alpha < -4/27$. The significance is that for large $n$ the oscillatory terms in Eq. are dominated by the first term when $\alpha > 0$, but the oscillatory terms are dominant when $\alpha < -4/27$. This observation is used ahead in deriving several asymptotic results.
Comparing the coefficients of the square terms gives $\delta = (1 - \beta_1)/2$, which is a negative number when $\alpha > 0$ and a positive number when $\alpha < -4/27$. The two complex roots are in the second and third quadrants when $\alpha > 0$, and in the first and fourth quadrants in the other case. For $\alpha > 0$, we can take $\theta = \pi + \arcsin((1 - \beta_1)/2r)$ which lies in the second quadrant, and for $\alpha < -4/27$ we can take $\theta = \arcsin((1 - \beta_1)/2r)$ which is in the first quadrant.
With the complex roots in terms of the one real root, it suffices to find the real root, which can be expressed straightforwardly for the regime of interest, i.e. either $\alpha>0$ or $\alpha<-4/27$. In that case, with $x=1+27\alpha/2$, $$\label{realroot}
\beta_1 = \frac{1}{3}\left[ 1+ \left(x+\sqrt{x^2-1} \right)^{1/3} + \left(x - \sqrt{x^2-1} \right)^{1/3} \right].$$ Rewriting Eq. in terms of the real constants $c_1$, $a$, $b$ gives S(n,) = c\_1 \_1\^n + 2 ()\^[n/2]{} \[genericform\] Putting in the initial conditions gives three real equations for the coefficients which may be solved in terms of $\alpha$ and $\beta_1$. Skipping the algebraic steps, one finds $$\begin{aligned}
c_1 &= \frac{{\alpha + \beta _1^2 }} {{3\alpha + \beta _1^2
}},\,\,\,a = \frac{\alpha } {{3\alpha + \beta _1^2 }},\,\,\, \nonumber \\
b &= - \frac{{\beta _1 }} {{(3\alpha + \beta _1^2 )}}\left(\frac{\alpha}{3\beta
_1 + 1}\right)^{1/2}, \end{aligned}$$ which gives a complete and explicit solution to Eq. .
The counting results for periodic points in the previous section beg the question, ‘what is the asymptotic density of the combinatorial expression $\left( {\begin{array}{*{20}c}
{n - 2i} \\
i \\
\end{array} } \right)$ as a function of $i$?’. It turns out that it is possible to find the asymptotic mean, variance, and density of $i$ (as $n$ approaches infinity) by essentially the same algebraic methods used in the recurrence relation. Using the moment-generating function, or by using Stirling’s approximation (essentially a saddle point expression), it can be shown that the density of $\left( {\begin{array}{*{20}c}
{n - 2i} \\
i \\
\end{array} } \right)$, when properly normalized, converges to a normal density. The moment-generating function technique also gives simple formulas for the asymptotic mean and variance. More specifically, S(n,e\^t) = \_[i=0]{}\^[n/3 ]{}
(
[\*[20]{}c]{} [n - 2i]{}\
i\
)
e\^[ti]{} where one substitutes $\alpha = e^t$, and the moment-generating function for this density is $\phi(n,t) = S(n,e^t)/S(n,0)$. Large $n$ gives $S(n,e^t) \rightarrow c(t)\beta_1(t)^n$ and $\phi(n,t) = c(t)\beta_1(t)^n\left[c(0)\beta_1(0)^n\right]^{-1}$ where $\beta_1(t)$ is the real root of $\beta(t)^3 - \beta(t)^2 - e^t = 0$. The details are omitted, but by differentiating the cubic equation all of the derivatives of $\phi(n,t)$ can be found, which can be used to find the moments of the density. When $\alpha = e^t = 1$ the real root of Eq. is $\beta_1(0) = 1.46557123\ldots$, which in the next section is seen to have special significance to this map. The asymptotic mean of this density can be shown to be $$\mu (n) = \frac{1}{3+\beta^2_1(0)} \left[ n-2 + \frac{6}{3+\beta^2_1(0)}\right]
\label{mean}$$ and the variance $$\sigma^2 (n) = \frac{\beta^5_1(0)}{\left[3+\beta^2_1(0)\right]^3}\left[n - 2 + \frac{12}{3+\beta^2_1(0)} \right]$$ The scaling of the mean is $n$ and the width is $n^{1/2}$. All the higher reduced cumulants (rescaled by the appropriate power of the width) vanish in the limit of $n\rightarrow \infty$.
|
---
abstract: 'I briefly review the three nonperturbative methods for the treatment of disordered systems — supersymmetry, replicas and dynamics — with a parallel presentation that highlights their connections and differences.'
author:
- Jorge Kurchan
title: 'Supersymmetry, replica and dynamic treatments of disordered systems: a parallel presentation.'
---
Running title: Supersymmetry, replicas and dynamics
Keywords: replica trick, supersymmetry, Langevin dynamics.
82C31 37A50 37N40
Disordered systems need to be treated with a method that allows to perform averages over the sample realisation. There is no universal way to do this that can be applied efficiently to all problems.
For Gaussian systems, the method of supersymmetry is as good as one can expect: it involves a minimum of variables, it is elegant and rigorous. Although one can still apply it for some non-Gaussian problems, in many of the interesting cases – as for example spin-glasses – it only gives limited information.
The replica trick was introduced to tackle such ‘complex’ problems. It has been extensively used and has yielded some of the most innovative solutions in disordered systems. It has however the problem that it is very far from being controlled, let alone rigorous. This is because the space itself – a vector space with noninteger dimension – does not have a general definition other than the ansatz itself - or perturbations around it.
The dynamic method consists of solving exactly the evolution of the system in contact with a heat bath. If the system reaches equilibrium one recovers all the thermodynamic information. Surprisingly enough, one can treat this way all the problems one can solve with replicas. A problem arises, however, when equilibrium cannot be achieved: then the long-time out of equilibrium regime may be of interest in itself (as in the case of glasses), or it may be viewed as an obstacle for exploring the deepest levels in phase-space (as for example in optimisation problems). Although the dynamic method was initially proposed as a way to obtain equilibrium results, this tendency has reverted in the last few years, at least in the field of glasses, where replicas are now used mostly to mimic the out of equilibrium dynamics.
The aim is of this paper is not to make a complete presentation of either of the three methods — there are very complete reviews of this [@Efetov; @Mepavi; @Bocukume] (including some very recent ones [@Ka]) but rather to put the three methods ‘side by side’ so that the connections can be better appreciated. To the best of my knowledge this has not been done for supersymmetry, replicas and dynamics simultaneously, as the practitioners of each method tend to belong to different communities.
[**The Problem**]{}
Consider an energy $$E_{\boldsymbol{J}}= \frac{i}{2} \sum_{ij} (\lambda \delta_{ij}-J_{ij} ) s_i s_j
\;\;\;
; \;\;\;
E_{\boldsymbol{J}}(h)=E_{\boldsymbol{J}} - \sum_i h_i s_i
\label{linear}$$ where $s_i$ ($i=1,...,N$) are real variables, and $J_{ij}$ is a random matrix. We take $\lambda$ with negative imaginary part. This energy can be used to calculate the averaged Green function: $${\overline{G(\lambda)}} \equiv {\overline{{
\mbox{Tr}} [\lambda {\boldsymbol{I}} -
{\boldsymbol{J}}]^{-1}
}}$$ from which one obtains the eigenvalue distribution. (Here and in what follows the overline denotes averages over the disorder ${\boldsymbol{J}}$). This is done by defining the partition function $$Z_{\boldsymbol{J}}(h) = \int
\boldsymbol{ds} \; e^{-\beta E_{\boldsymbol{J}}(h)}
\label{partition}$$ and computing: $$\begin{aligned}
{\overline{
G(\lambda)
}} &=& -iT \left.
\sum_k \frac{\partial^2}{\partial h_k^2}
{\overline{ \ln Z_{\boldsymbol{J}}}}(h)\right|_{h_i=0} =
\left. i \beta \sum_k
{\overline{
\left[\int \boldsymbol{ds}
\; s^2_k \; e^{-\beta E_{\boldsymbol{J}} (h)} \right]
\cdot \frac{1}{Z_{\boldsymbol{J}}(h)} }}\right|_{h_i=0} \nonumber\\
&{\boldsymbol{\neq}}&
\left. i \beta \sum_k
{\overline{
\left[\int \boldsymbol{ds}
\; s^2_k \; e^{-\beta E_{\boldsymbol{J}} (h)} \right]}}
\cdot
\frac{1}{{\overline{Z_{\boldsymbol{J}}(h)} }}\right|_{h_i=0}
\label{quenched}\end{aligned}$$ (We include the constant normalisations in the differential: ${\boldsymbol{d}}\equiv d/\sqrt(2\pi N)$).
The third expression in (\[quenched\]) is the correct (quenched) average, in general different from the last one, the annealed average. [*The problem is that in order to compute the average over the $J_{ij}$, we need to express $1/Z_{\boldsymbol{J}}$ in (\[quenched\]) in a tractable (i.e. exponential) form*]{}. Three methods to do so are:
- [*Supersymmetry*]{}: we can take advantage of the Gaussian nature of the partition function to write, in terms of two sets of Grassmann $\eta_i$ and $\eta_i^*$ and a set of ordinary variables $\sigma_i$: $$\frac{1}{Z_{\boldsymbol{J}}} = \int {d \boldsymbol{\eta}} \;
{d \boldsymbol{\eta}}^*
{\boldsymbol{d\sigma}}
\;
e^{
-\frac{\beta}{2} i \sum_{ij}
(\lambda \delta_{ij}- J_{ij}) (\eta_i^* \eta_j+
\sigma_i \sigma_j) }
\label{partition-1}$$ so that we get: $${\overline{G(\lambda)}} =
\left. \sum_k
{\overline{
\int
{\boldsymbol{d s}}
{\boldsymbol{d \sigma}}
\int \boldsymbol{d \eta} \; {\boldsymbol{d \eta^*}}
\; s^2_k \; e^{
-\frac{\beta}{2} i \sum_{ij}
(\lambda \delta_{ij}- J_{ij})
(\eta_i^*
\eta_j + \sigma_i \sigma_j + s_i s_j )}
}}
\right|_{h_i=0}
\label{susy}$$
- [*Replicas*]{}: we replicate $n$ times each variable $s_i \rightarrow
s_i^\alpha$ and compute: $$Z_{\boldsymbol{J}}^{n-1} = \int \Pi_{\alpha=1}^{n-1}
{\boldsymbol{d s^\alpha }} \; e^{
-\frac{\beta}{2} i \sum_{\alpha=1}^{n-1}
\sum_{ij} (J_{ij}-\lambda \delta_{ij}) s_i^{\alpha}
s_j^{\alpha} }
\label{rep1}$$ The calculation proceeds for every integer $n$, and finally we somehow take the limit: $$Z_{\boldsymbol{J}}^{-1} = \lim_{n \rightarrow 0} Z_{\boldsymbol{J}}^{n-1}
\label{rep2}$$ which should in principle be shown to be the correct analytic continuation over $n$. We hence have: $${\overline{G(\lambda)}}
\sim_{n \rightarrow 0} {\overline{
\sum_k \int \Pi_{\alpha=1}^{n}
d{\boldsymbol{s}}^\alpha
\; (s_k^{(1)})^2
\; e^{
-\frac{\beta}{2} \sum_{\alpha=1}^{n}
\sum_{ij} (J_{ij}-\lambda \delta_{ij}) s_i^{\alpha}
s_j^{\alpha} }}}
\label{rep3}$$ where we have chosen to take the expectation value of the first replica, although clearly any other replica will do.
- [*Dynamics*]{}: The dynamic method [@cirano; @Sozi] consists of calculating the average in (\[quenched\]) by considering the solution of the Langevin equation: $$\gamma {\dot{s}}_i= -\beta \frac{\partial E_{\boldsymbol{J}}}{\partial s_i} +
\beta h_i + \rho_i
\label{Langevin}$$ where the $\rho_i$ are independent Gaussian white noises with variance $=2\gamma$.
The energy might be complex: this poses no problem (at least for linear systems [@CLang]). Starting from $t=0$, we are guaranteed that at long times $t_o$: $$\langle A({\boldsymbol{s}}) \rangle =
\lim_{t \rightarrow \infty} \langle A({\boldsymbol{s}}(t_o))
\rangle_\rho
\label{timeavg}$$ where $\langle \bullet \rangle$ denotes thermodynamic average and $\langle \bullet \rangle_\rho$ average over the process, i.e. over the noise realisation. We obtain an expression for the average Green function (\[quenched\]) as: $${\overline{
G(\lambda)
}}=
\left. i \lim_{t_o \rightarrow \infty}
\sum_k {\overline{
\frac{\partial \langle s_k(t_o) \rangle_\rho}{\partial h_k}}} \right|_{h=0}
\label{response}$$ In practice, one calculates the dynamics averaged over both thermal noise and disorder and in the large $N$ limit, as we shall see below.
The problem of treating the denominator is not exclusive of Gaussian systems, it appears whenever we wish to obtain the correct quenched averages over disorder. For example, the energy (\[linear\]) can be modified to obtain the standard spin-glass model: $$E_{\boldsymbol{J}}^{nl}= i E_{\boldsymbol{J}}(h) +
m \sum_i s_i^2 + g \sum_i s_i^4
\label{nonlin}$$ and we may wish to calculate averages of any observable $A({\boldsymbol{s}})$. As soon as $g > 0$ the system becomes as complicated as can be, with all the subtelties of spin-glasses. Once we abandon the Gaussian world, the three methods encounter difficulties:
- [*Supersymmetry*]{}: there is no obvious way to write $1/Z_{\boldsymbol{J}}$ in general as an integral over an exponential. This [*does not*]{} mean that the supersymmetry method is entirely inapplicable for non-Gaussian systems: even though when the energy is not quadratic this method [*does not*]{} give the Boltzmann-Gibbs measure, it can still be useful in some cases, as we shall see below.
- [*Replicas*]{}: In contrast to (\[partition-1\]), expressions (\[rep1\]) and (\[rep2\]) are formally valid for non-quadratic energies. Thus, the replica trick has been applied successfully to the study of many complex systems, spin-glasses being the main example. The expectation values of an observable $A$ can in general be written as: $${\overline{\langle A \rangle }}
\sim_{n \rightarrow 0} {\overline{
\int \Pi_{\alpha=1}^{n}
{\boldsymbol{ds^\alpha }}
\; A({\boldsymbol{s^{(1)}}})
\; e^{-\beta \sum_\alpha E_{\boldsymbol{J}}(\boldsymbol{s^\alpha})}
}}
\label{rep4}$$
From the point of view of making the results rigorous (or even reliable), there is the following difficulty: a closed analytic expression in terms of $n$ can be obtained in some limit, typically large $N$. This poses the problem that the limits $N \rightarrow \infty$ and $n
\rightarrow 0$ may not commute – and indeed in most interesting cases they do not. In those cases we have to consider the assumed infinite-$N$ continuation valid around $n=0$ as a guess (see however Ref. [@KK]).
- The [*dynamic*]{} expression (\[timeavg\]) shares with the replica treatment the advantage of being equally valid for linear or nonlinear problems. There is however a problem also here: (\[timeavg\]) holds to the extent that we make $t_o\rightarrow
\infty$ before any other limit, in particular $N \rightarrow \infty$. Again, in many interesting (nonlinear) problems these limits do not commute: in physical terms this means that an infinite system is not able to equilibrate at finite times [@Bocukume]. This is indeed the physical situation one wishes to reproduce in glassy systems. However, one may still be interested in knowing what happens in times that diverge with the system size, and in particular to reproduce the equilibrium situation - even if it might be unreachable in a realistic situation [@Sozi1]. To do this, the $N \rightarrow \infty$ solutions must be supplemented with activated, ‘instanton’ solutions [@Loio]: this problem has not yet been solved in general.
[**Dynamics is a generalisation of supersymmetry.** ]{}
Let us see that the supersymmetry method is a [*‘time-less’*]{} version of dynamics (\[Langevin\]). We compute the solutions of the stochastic equation: $$0= -\beta \frac{\partial E_{\boldsymbol{J}}}{\partial s_i} + \beta h_i + \rho_i
\label{LL}$$ There is no time-dependence, and the $\rho_i$ are Gaussian variables of variance $2\gamma$. If $E_{\boldsymbol{J}}(h)$ is quadratic the system (\[LL\]) has a single solution $$s_i= -i T \sum_{j} [\lambda {\boldsymbol{I}} -
{\boldsymbol{J}}]^{-1}_{ij} (\beta h_j+\rho_j)$$ Denoting $\langle A({\boldsymbol{s}}) \rangle$ the average of $A$ evaluated over the ($\rho$-dependent) solutions, we have : $${\overline{
G(\lambda)
}}=
\left. i \sum_k
\frac{\partial {\overline{
\langle
s_k
\rangle_\rho}}}{\partial h_k} \right|_{h=0}
\label{response1}$$ to be compared with (\[response\]). To see that this gives back the supersymmetry method, let us write, for the Gaussian case: $$\langle
s_k \rangle_\rho =
\left< \int
{\boldsymbol{ds}}
\; s_k \; \Pi_i \delta\left(-i\beta \sum_j
(\lambda \delta_{ij} - J_{ij}) s_j +\beta h_i +\rho_i\right)
\det [i\beta (\lambda {\boldsymbol{I}} - {\boldsymbol{J}})] \right>
\label{cosa}$$ where the determinant guarantees that the solution for every realisation of $\rho$ is counted with the same weight. Exponentiating the delta function as usual [@delta]: $$\begin{aligned}
\langle
s_k \rangle_\rho &=&
\left< \int
{\boldsymbol{ds}}{\boldsymbol{d\hat{s}}}
{\boldsymbol{d\eta}}
{\boldsymbol{d\eta^*}}
\; s_k \; \right. \times \nonumber \\
& &\left. \exp \left\{
\sum_{ij}
-i \beta [\lambda \delta_{ij} - J_{ij}] (i {\hat{s}}_i s_j + \eta^*_i \eta_j)
+i \sum_i {\hat{s}}_i (\beta h_i +\rho_i)
\right\} \right> \nonumber \\
&=&\int
{\boldsymbol{ds}}{\boldsymbol{\hat{s}}}
{\boldsymbol{d \eta}}
{\boldsymbol{d \eta^*}}
\; s_k \; \times \nonumber \\
& &\exp \left\{
\sum_{ij}
-i \beta [\lambda \delta_{ij} - J_{ij}] (i {\hat{s}}_i s_j + \eta^*_i \eta_j)
+ i \sum_i \beta {\hat{s}}_i h_i - \gamma \sum_i {\hat{s}}_i^2
\right\}
\nonumber \\
\label{cosa1}\end{aligned}$$ which, using (\[response1\]) yields: $$\begin{aligned}
G(\lambda)
&=& i
\beta \sum_k \int
{\boldsymbol{ds}}{\boldsymbol{\hat{ds}}}
{\boldsymbol{d \eta}}
{\boldsymbol{d \eta^*}}
\; s_k {\hat{s}}_k \; \times \nonumber \\
& &\exp \left\{
\sum_{ij}
-i \beta [\lambda \delta_{ij} - J_{ij}] (i {\hat{s}}_i s_j + \eta^*_i \eta_j)
- \gamma \sum_i {\hat{s}}_i^2
\right\}
\label{ttt}\end{aligned}$$ This is an implementation of supersymmetry like (\[susy\]), with two ordinary $({\boldsymbol{s}},{\boldsymbol{\hat{s}}})$ and two Grassmann $({\boldsymbol{\eta}},{\boldsymbol{{\eta^*}}})$ sets of variables. For $\gamma \rightarrow 0$ it can be taken to the form (\[susy\]) by a rotation in the $({\boldsymbol{s}},{\boldsymbol{\hat{s}}})$.
The conclusion we draw from this exercise is that: [*i)*]{} Supersymmetry is just ‘dynamics without time’, which strongly suggests that any problem solvable with the former is solvable with the latter method. [*ii)*]{} Supersymmetry can be extended to treat certain nonlinear problems, as we shall now show.
[**Supersymmetry for nonlinear problems.**]{}
Equation (\[LL\]) is not restricted to linear energy functions. If (\[LL\]) is nonlinear, but still has one solution, it can be used to calculate the expectation value of any function $A({\boldsymbol{s}})$ in its root. The generalisation of eqs. (\[cosa\]) and (\[cosa1\]) is: $$\begin{aligned}
\langle
A({\boldsymbol{s}}) \rangle_\rho &=&
\left< \int d{\boldsymbol{s}} \; A \; \Pi_i \delta\left( -
\beta \frac{\partial E_{\boldsymbol{J}}}
{\partial s_i} +\rho_i\right)
\det \left[
\frac{\partial^2 E_{\boldsymbol{J}}}{\partial s_k \partial s_l}\right] \right>
\nonumber \\
&=&\int d{\boldsymbol{s}}d{\boldsymbol{\hat{s}}} d{\boldsymbol{\eta}}
d{\boldsymbol{\eta^*}}
\; A \; \times \nonumber \\
& &\exp \left\{- i\beta \sum_i {\hat{s}}_i \frac{\partial E_{\boldsymbol{J}}}
{\partial s_i}
+ \beta
\sum_{ij} \eta^*_i \frac{\partial^2 E_{\boldsymbol{J}}}{\partial s_i \partial s_j} \eta_j
- \gamma \sum_i {\hat{s}}_i^2
\right\}
\label{cosaa}\end{aligned}$$ This way of imposing a solution has its origin in the path-integral treatment of gauge theories [@ZJ], where the fermions are called ‘ghosts’ [@Paso0].
In many cases of interest the equation (\[LL\]) has many solutions for some realisations of $\rho$. If we wish to add the values of the observable in every solution we should take the absolute value of the determinant in (\[cosaa\]). In particular, we need to do this if we wish to calculate the average [*number*]{} of solutions. Writing this absolute value as an exponential is possible [@K1], although it involves introducing new fields.
An interesting situation we shall consider here and in what follows is when we [*do not*]{} take the absolute value. Each solution is then added with the sign of the determinant of the matrix of second derivatives [@Paso1]. In particular: $$\langle 1 \rangle = \sum_{solutions} (-1)^{sign}
\label{Mo}$$ which is an invariant only dependent on the topology of the space of the ${\boldsymbol{s}}$, and independent of the energy function $E_{\boldsymbol{J}}$ [@K1]. For the usual case of the ${\boldsymbol{s}}$ forming a flat space and $E_{\boldsymbol{J}}({\boldsymbol{s}}) \rightarrow \infty$ as $|{\boldsymbol{s}}|
\rightarrow \infty$ the invariant is [*one*]{}. In cases in which there are many solutions, the method does not select the lowest ones, but averages flatly (apart from the sign of the Hessian) over [*all solutions*]{} [@foot1]: it is in this sense that supersymmetry fails.
In any case, as mentioned above, one is not calculating the Gibbs measure, but just values over local minima and saddles. There are however interesting nonlinear problems having a finite number of solutions for which there is no reason to abandon supersymmetry.
One of the most interesting applications involving non-gaussian problems are the quantum systems. A note on terminology is necessary for what follows: In quantum systems, we can distinguish two ways in which nonlinearity may appear: in the wavefunction and/or in the Hamiltonian. In the former case, one has a nonlinear Schroedinger equation, containing for example terms cubic [*in the wavefunction*]{} (see Eq. (\[nlins\]) below). In the latter case, one generally considers a usual, [*linear*]{} Schroedinger problem, but the Hamiltonian contains terms of degree higher than two in the creation and destruction operators. It is then the path integral that is non-Gaussian, since the [*action*]{} is no longer quadratic. We shall discuss below both cases.
[**Functional expression for dynamics.**]{}
We can see more clearly the relation between supersymmetry and dynamics by constructing a functional expression for the equation (\[Langevin\]). We use exactly the same procedure as in (\[cosaa\]), with now delta-functions and Jacobians promoted to functionals of the trajectories. $$\begin{aligned}
\langle A({\boldsymbol{s}}(t_o)) \rangle_\rho
&=&
\int D{\boldsymbol{s}}
D{\boldsymbol{\hat{s}}}
D{\boldsymbol{\eta}}
D{\boldsymbol{\eta^*}}
\; A({\boldsymbol{s}}(t_o)) \; \times \nonumber \\
& & \exp \left\{
- i \beta\sum_i \int dt \; {\hat{s}}_i
\frac{\partial E_{\boldsymbol{J}}}{\partial s_i}
+\beta \sum_{ij}\int dt \;
\eta^*_i \frac{\partial^2 E_{\boldsymbol{J}}}{\partial s_i
\partial s_j}
\eta_j \right. \nonumber \\
& & \left. + \gamma \sum_i
\int dt (\eta_i^* {\dot{\eta}}_i- i {\hat{s}}_i {\dot{s}}_i \;- {\hat{s}}_i^2)
\right\}
\label{cosab}\end{aligned}$$
This functional equation can be viewed either as the de Dominicis-Janssen, Martin-Siggia-Rose [@DJ; @MSR] functional expression for the Langevin dynamics – with the determinant exponentiated through ghosts – or as the path-integral expression for supersymmetric quantum mechanics [@Witten].
[*Here we see clearly that by expressing expectation values dynamically the problem now becomes, just like in the case of supersymmetry and replicas, the computation of an integral of an exponential*]{}, albeit a functional one. This is the usual starting point for the developments in dynamics - at least within the physics literature.
This is a good place to see how one can calculate with the same method the localisation of wavefunctions in a nonlinear Schroedinger problem [@Tr]: $$i{\dot{\varphi}}_n= -\frac{1}{2}(\varphi_{n-1}+\varphi_{n+1})
+(\epsilon_n + \Lambda |\varphi_n|^2) \varphi_n
\label{nlins}$$ where $\epsilon_n$ is the site disorder. The system of equations (\[nlins\]) together with its conjugate can have a single solution (this will surely be the case if we fix the initial conditions). In fact, (\[nlins\]) can be viewed as a (noisless) dynamical equation for the variables ${\varphi_n}$. We can obtain a functional expression as in (\[cosab\]) for this case introducing complex Lagrange multiplier time-dependent fields ${\hat{\phi}}_n (t)$, and Grassmann fields $\eta_n(t)$, $\eta_n^*(t)$. The normalisation is guaranteed, even if the action is no longer quadratic in the $\phi_n(t)$.
[**Normalisation and symmetries.**]{}
We have three expressions for the expectation of an observable: using supersymmetry (\[cosaa\]), replicas (\[rep4\]) and dynamics (\[cosab\]). All three lend themselves to averaging over the disorder, and have no uncomfortable normalisations. Indeed, the three expressions yield $$\langle 1 \rangle =1
\label{norm}$$ but for apparently different reasons:
- Within the supersymmetric formalism (\[norm\]) arises because around each solution the Grassmann and the ordinary variables conspire, just as in the Gaussian case, to give $\pm 1$ (the sign of the determinant of the Hessian). Even when there are many solutions, these signs add up to [*one*]{} because of topological constraints [@foot].
Now, even if we did not know where the function (\[cosaa\]) came from, we could still see that the expectation value $ \langle 1 \rangle$ does not depend on $E_{\boldsymbol{J}}$ using the fact that the exponent has the [*two*]{} supersymmetries (which indeed give the name to the approach): $$\begin{aligned}
\delta{s_i}=\eta_i \;\; ; \;\; \delta {\eta^*}_i= i {\hat{s}}_i
\nonumber \\
\delta{s_i}=\eta^*_i \;\; ; \;\; \delta \eta_i=i {\hat{s}}_i
\label{susy1}\end{aligned}$$
- Within the replica formalism (\[norm\]) just expresses the fact that we have an integral to the $n^{th}$ power, and we let $n \rightarrow 0$. Again, if we did not know where (\[rep4\]) came from, we could show that $ \langle 1 \rangle=1$ using the fact that the exponent is symmetric with respect to replica permutations.
- In the [*causal*]{} dynamic treatment starting from an initial condition and letting the endpoint free, (\[norm\]) is just a statement of probability conservation [@ffnn]. Also in this case we can see directly from the action that $ \langle 1 \rangle=1$, for reasons of symmetry [@ZJ]. One has the following two supersymmetries, which are the generalisation of (\[susy1\]) to the case [*‘with time’*]{}: $$\begin{aligned}
\delta{s_i}&=&\eta_i \;\; \, \;\; \delta {\eta^*}_i= i {\hat{s}}_i
\nonumber \\
\delta{s_i}&=&\eta^*_i \;\; \, \;\; \delta \eta_i=i {\hat{s}}_i -
{\dot{s}}_i
\;\; \, \;\; \delta {\hat{s}}_i = -i {\dot{\eta}}^*_i
\label{susy2}\end{aligned}$$
which, together with time-translation invariance, constitute the full group of symmetry.
[**A unifying notation.**]{}
We have seen that the methods of supersymmetry and dynamics (itself also possessing a supersymmetry) are closely connected. In fact, we can uncover more algebraic correspondences between the three approaches by using a suitable notation [@K1; @Saclay]. This can be done by introducing two anticommuting Grassmann variables $\theta\,, \,\bar\theta$: $$[\theta\, , \bar\theta]_+ = \theta^2 = \bar \theta^2 = 0
\; \label{2.2}$$ The integrals over these variables are defined as: $$\int 1 d\theta = \int 1 d \bar \theta = 0
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\int \theta d\theta = \int \bar \theta d \bar \theta = 1
\;
\label{2.3}$$ We can encode the $s_i$, $\eta_i$, $\eta^*_i$ and ${\hat{s}}_i$ in a single [*superfield*]{}: $$\Phi_i
= s_i + \bar \theta \; \eta_i + \eta_i^* \; \theta + \hat s_i \;
\bar \theta
\;
\theta\label{2.4}$$ Using Eqs. (\[2.2\])-(\[2.3\]) and (\[2.4\]) one obtains, in terms of the superfields $\Phi_i$ $$\begin{aligned}
\langle A \rangle
&=&
\int \prod_i \; D[\Phi_i] \; A \;\; \exp \int \; d\alpha \; \left[
\frac{1}{2} \sum_i \Phi_i(\alpha) \Delta(\alpha,\alpha') \Phi_i(\alpha')
- \beta E_{\boldsymbol{J}}(\Phi(\alpha)) \right] \nonumber\\
&~&
\label{functio} \end{aligned}$$ where we have denoted $\alpha \equiv (\theta, \bar\theta)$, $d\alpha= d \theta \; d\bar\theta $ and $$\Delta =
\Delta^{SUSY}(\alpha,\alpha') \equiv 2 \gamma
\label{fg}$$ independent of $\alpha,\alpha'$.
The dynamics can be encoded in an expression formally identical to (\[functio\]), but now the field dependencies and integration variables include time: $\alpha \equiv (\theta, \bar\theta,t)$, $\alpha' \equiv
(\theta', \bar\theta',t')$, $d\alpha= d \theta \; d\bar\theta \; dt$ and $$\Delta =
\Delta^{Dyn} =
2 \gamma \delta(t-t')+
\gamma \delta'(t-t')(\bar \theta-\bar \theta')(\theta+\theta')
\label{2.9}$$ Finally, the replica expression is again formally (\[functio\]), but with the identification: $$\Delta^{Replica} =0 \;\;\; ; \;\;\;
\Phi_i(\alpha) \;\; \leftrightarrow s_i^\alpha \;\;\;\;
; \;\;\;\; \int \; d\alpha \;\; \leftrightarrow \sum_{\alpha=1}^{n}$$ (The correspondence between supersymmetry and replicas can be made to hold even for $\gamma \neq 0$ by using a term $\Delta^{Replica}= \gamma$ which does not affect the final result.)
In particular, the expectation values $\langle A \rangle$ associated with the calculation of the Green function (\[rep3\]),(\[response\]) and (\[ttt\]) can be written in terms of $ A $ which, in a notation that highlights the analogies, reads: $$A = i \beta
\sum_a \int d\alpha d \alpha' \; \Phi_a(\alpha) O(\alpha') \Phi_a(\alpha')
\label{AA}$$ with the identifications: $$O(\alpha)\equiv \delta_{\alpha,1} \;\;\; ; \;\;\;
O(\alpha)\equiv \delta (\bar \theta)\delta (\theta) \;\;\; ; \;\;\;
O(\alpha)\equiv \delta (t-t_o) \delta (\bar \theta)\delta (\theta)$$ for the replica, the supersymmetry and the dynamic cases, respectively. We see that the expressions are analogous to one another.
The important point about expression (\[functio\]) is that, apart from the first term in the exponent, it has the same form as the partition function. This unified notation is useful as a book-keeping device when we have a diagrammatic expansion [@K2; @Saclay], because diagrams on the three approaches have the same form. Internal lines involve integrations over the superspace/replica variable, and the effect of each method is the same due to relations like: $$\int \; 1 \; d\alpha
=0$$ valid in all three approaches (in the replica approach as $n \rightarrow 0$).
[**The correspondence at work.**]{}
[*One-point functions of random matrices.*]{}
We now work out the example of the one-point function for the Gaussian orthogonal ensemble in parallel with replicas and supersymmetry. (The problem has also been attacked with dynamics [@p2], but we will not review this here). The object is not to discuss how both methods can be used in this case (this has been done in detail long time ago [@Edw; @rep]), but rather to show how the equality of results follows from the formal correspondence.
We use the functional expression (\[functio\]) with the energy given by (\[linear\]), where the $J_{ij}$ are random Gaussian variables of variance $N^{-1/2}$. Averaging over the ${\boldsymbol{J}}$, and expressing everything in terms of the order parameter: $$Q(\alpha,\alpha')\equiv \frac{1}{N} \sum_i \Phi_i(\alpha) \Phi_i(\alpha')
\label{order}$$ we get, after a few standard steps (which can be borrowed either from the supersymmetry or from the replica literature): $$\begin{aligned}
{\overline{G(\lambda)}} &=& \int d\alpha d \alpha' O(\alpha')
\langle Q(\alpha,\alpha') \rangle \nonumber \\
\langle Q(\alpha,\alpha') \rangle &=&
\int\; D[Q]\; Q(\alpha,\alpha') \times \nonumber \\
& &\exp \left\{ -\frac{N}{2}\; Tr\; \ln
[\Delta {\boldsymbol{\delta}} + i \beta \lambda + \beta^2 Q]
+\frac{N}{4}\;\beta^2\; Tr\; Q^2 \right\} \nonumber \\
\label{expression}\end{aligned}$$ Here we have used (\[AA\]). The square and log functions are to be understood as functions of $Q$ considered as an operator (i.e. $Q^2(\alpha,\gamma) = \int d \alpha' Q(\alpha,\alpha')
Q(\alpha',\gamma)$, etc) and $Tr\;Q\equiv \int d \alpha Q(\alpha,\alpha)$. The delta function is either the Kroenecker function (in replica space) or the superspace delta ${\boldsymbol{\delta}}=(\bar \theta-\bar
\theta')(\theta-\theta')$.
Expression (\[expression\]) can be evaluated by saddle point integration. $$Q^{-1}=(\Delta+i\beta \lambda){\boldsymbol{\delta}}+\beta^2 Q
\label{sp}$$ We can now propose for the saddle point value the most general (replica and super) symmetric form for $Q$: $$Q(\alpha,\alpha') = \tilde q {\boldsymbol{\delta}} + q
\label{order1}$$ First note that under operator powers and traces (\[order1\]) behaves exactly in the same way whether we interpret it as being a replica matrix ($n \rightarrow 0$) or as a function of two superspace variables. The saddle point equation then becomes: $$\begin{aligned}
1&=&i \beta \lambda \tilde q + \beta^2 {\tilde{q}}^2 \nonumber \\
0&=& \gamma \tilde q +i \beta \lambda q +2\beta^2 q \end{aligned}$$ Using (\[AA\]) we have: $${\overline{G(\lambda)}} = i \beta
\int d\alpha d \alpha' \; Q(\alpha,\alpha') O(\alpha')=i\beta \tilde q
\label{AA1}$$ and this yields the semicircle law in the usual way.
The point worth noting here is that there is a close algebraic relation between the replica and the supersymmetric approaches. Indeed, as we shall stress below, all [*three*]{} approaches are essentially isomorphic when restricted to a symmetric ansatz.
[*Quantum systems with interactions.*]{}
As a second example, let us briefly see how dynamics can be used as an alternative to replicas in an interacting quantum system. Consider the system of interacting bosons in a random potential [@Chna] with imaginary-time action: $$\begin{gathered}
S = \int {d^2}x\,d\tau\:{\psi^*}\left({\partial_\tau}-
\frac{1}{2m}{\nabla^2}-\mu + V(x)\right)\psi\\
+ \int {d^2}x\,{d^d}x'\,d\tau\:
{\psi^*}(x)\psi(x) u(x-x'){\psi^*}(x')\psi(x')
\label{uunn}\end{gathered}$$ where $u(x-x')$ is the boson interaction and $V(x)$ is the random potential. In order to do the correct averaging over disorder, one can use the replica trick, thereby obtaining the averaged action: $$\begin{gathered}
S = \int {d^2}x\,d\tau\:{\psi_\alpha^*}(x,\tau)\left({\partial_\tau}-
\frac{1}{2m}{\nabla^2}-\mu\right){\psi_\alpha}(x,\tau)\\
- \int {d^2}x\,d\tau\,d\tau'\: \frac{1}{2} v_0\:
{\psi_\alpha^*}(x,\tau){\psi_\alpha}(x,\tau)
{\psi_\beta^*}(x,\tau){\psi_\beta}(x,\tau')\\
+ \int {d^2}x\,{d^2}x'\,d\tau\:
{\psi_\alpha^*}(x){\psi_\alpha(x)}
u(x-x'){\psi_\alpha^*}(x'){\psi_\alpha}(x')
\label{ddoo}\end{gathered}$$ $\alpha=1,2,\ldots,n$ is a replica index.
We can just as well apply a dynamic treatment here. Going back to (\[uunn\]), we can consider $x$ and $\tau$ as the site indices, $\psi(x,\tau)$ and $\psi^*(x,\tau)$ as the dynamic variables, and consider their Langevin evolution in an extra (unphysical) time $t$: $$\frac{d\psi(x,\tau;t)}{dt} = - \frac{\delta S}{\delta{\psi(x,\tau)}}+
\rho(x,\tau;t)$$ This ‘stochastic quantisation’ strategy can be implemented for fermions as well [@Pawu]. We can obtain an expression that is formally identical to (\[ddoo\]) (up to a term $\Delta$ as in (\[fg\])), but now interpreting the fields $\psi$ as superfields, functions of both $x,\tau$ and the superspace variable $\alpha \equiv \bar \theta,\theta,t$. Diagrams for superfields have the same form as the replica ones, and one can also study nonperturbative approximations.
Let us conclude this section by remarking that for this last case there is another (more physical) approach: the treatment of quantum dynamics with a thermal bath à la Schwinger-Keldysh (see the first of Refs. [@Ka]). This has the advantage of not having to introduce an extra time.
[**Order parameters, symmetry breaking.**]{}
Order parameters can be of vector nature $\Psi(\alpha)$, of matrix nature $Q(\alpha,\alpha')$ and of higher tensorial character. They may, of course, depend on space. A special case arises when one wishes to calculate the two-point correlation function of random matrices. One needs to introduce two sets of superfields, or of replicas $\Phi_i^{(1)}(\alpha),\Phi_i^{(2)}(\alpha)$, and ends up with an order parameter: $${\boldsymbol{Q}}= \left(
\begin{array}{cc}
Q^{(11)} & Q^{(12)} \\
Q^{(21)} & Q^{(22)}
\end{array} \right)$$ where $N Q^{(ab)}(\alpha,\alpha') \equiv \sum_i \langle \Phi_i^{(1)}(\alpha)
\Phi_i^{(2)}(\alpha') \rangle$ for $a,b=1,2$.
The different solutions can be classified according to the manner in which the symmetry is broken.
- [*Symmetric*]{} order parameters appear in the solution of Gaussian one-point problems. This corresponds, as we have seen in the previous section, to replica-symmetric/supersymmetric solutions. In the dynamic treatment, the fact that correlation functions satisfy supersymmetry (\[susy2\]) is equivalent to stating that [*the system is in equilibrium, and satifies stationarity as well as the fluctuation-dissipation theorem*]{}. The dynamics of glassy systems in the high temperature phase is of this kind, and can be solved easily [@Sozi] in all the cases in which the replica trick calculation can also be implemented. (For an explicit presentation of the algebraic connection between the two methods, see [@K1]).
- [*Vector breakings*]{}
Within the replica trick such form of symmetry breaking appears when the order parameter is a vector in replica space, and all components are not equal [@dopa]. For matrix order parameters, vector breakings are those such that the vector $\Psi$ defined as: $$\Psi(\alpha) \equiv \int d\alpha' Q(\alpha,\alpha')
\label{crt}$$ is itself non-symmetric, i.e. dependent on $\alpha$. [*The same definition can be applied to supersymmetric and dynamic solutions*]{}, with the substitution of ‘replica-symmetry’ by ‘super-symmetry’. There are several examples of such symmetry-breaking fields in the literature: [*i)*]{} vectors in replica space [@dopa1] were considered in the study of instantons in the random field Ising model, their supersymmetric and dynamic counterparts [@Cukuunp] have closely related properties. [*ii)*]{} Replica matrices with vector type were considered [@Cagapa] in the computation of saddles in free-energy landscapes, and also in [@Kame] for the two-point functions for random matrices. A related scheme with matrices is the ‘two block model’ [@Brmo], (the first attempt at replica symmetry breaking) used to count solutions of a spin-glass equations. For this last example there is a supersymmetry-breaking ansatz shown to have the same properties [@Brmo; @K1], and more recently a causality-breaking dynamics [@Biku].
- [*Matrix breakings*]{}: This appear only for two (or more) indexed correlations. They can be characterised by the fact that although $Q(\alpha,\alpha')$ breaks the symmetry, the integral $\Psi$ (Eq. (\[crt\])) is itself symmetric (independent of $\alpha$). The best known example of matrix breaking is the Parisi ansatz [@Mepavi] in replica space. In the context of dynamics the solution of the long-time out of equilibrium evolution of the same systems [@Cuku; @Bocukume] is of this kind. Both the Parisi ansatz and the dynamic solution have been generalised to order parameters of higher tensorial character [@tensor; @Cuse].
Whenever the replica trick is feasable, the dynamic treatment is also possible. They do not yield the same answers if the system is not ergodic, as one corresponds to the equilibrium situation and the other to the nonequilibrium dynamics. Only with the inclusion of all activated (instanton) processes will the dynamic solution reproduce all time regimes, and this is not yet available in general [@Loio; @Biku].
In several of the cases above, the equality between the solutions within the different methods stems from an algebraic correspondence, a generalisation of the kind of that we described in the previous section.
[**Conclusions**]{}
Having a dictionary that allows to translate developments from one method to the other, whenever this is possible, can be useful for several reasons. For example, in the field of structural glasses and supercooled liquids, arguably the most important theoretical challenge is the inclusion of solutions representing the activated processes responsible for the smearing of the purely [*dynamic*]{} transition. Once these solutions are found, one can envisage constructing formally analogous solutions in replica space, which one might conjecture would be responsible for the disapearence of the [*thermodynamic*]{} (Kauzman) glass transition, or for a change in its nature.
From the point of view of mathematical physics, the dynamic method seems a promising strategy, since everything that is involved is standard probability theory and analysis [@alice; @mathos]. Indeed, there seems to be no obstacle of principle for the rigorous derivation of the solution of out of equilibrium glass dynamics [@Cuku; @Cuku2; @Bocukume], at least at the mean-field level.
[**Aknowledgements**]{}
I wish to thank C. Chamon, L. Cugliandolo and G. Lozano for clarifying discussions and suggestions.
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|
---
author:
- Benjamin Dodson
title: 'Global well-posedness and scattering for the radial, defocusing, cubic nonlinear wave equation'
---
**Abstract:** In this paper we prove global well-posedness and scattering for the defocusing, cubic, nonlinear wave equation on $\mathbf{R}^{1 + 3}$ with radial initial data lying in the critical Sobolev space $\dot{H}^{1/2}(\mathbf{R}^{3}) \times \dot{H}^{-1/2}(\mathbf{R}^{3})$.
Introduction
============
In this paper we study the defocusing, cubic nonlinear wave equation $$\label{1.1}
u_{tt} - \Delta u + u^{3} = 0, \qquad u(0,x) = u_{0}, \qquad u_{t}(0,x) = u_{1}.$$ This problem is $\dot{H}^{1/2}$ critical, since the equation $(\ref{1.1})$ is invariant under the scaling symmetry $$\label{1.2}
u(t,x) \mapsto \lambda u(\lambda t, \lambda x).$$ This scaling symmetry completely determines local well-posedness theory for $(\ref{1.1})$. Positively, [@LS] proved
\[t1.1\] The equation $(\ref{1.1})$ is locally well-posed for initial data in $u_{0} \in \dot{H}^{1/2}(\mathbf{R}^{3})$ and $u_{1} \in \dot{H}^{-1/2}(\mathbf{R}^{3})$ on some interval $[-T(u_{0}, u_{1}), T(u_{0}, u_{1})]$. The time of well-posedness $T(u_{0}, u_{1})$ depends on the profile of the initial data $(u_{0}, u_{1})$, not just its size.
Additional regularity is enough to give a lower bound on the time of well-posedness. Therefore, there exists some $T(\| u_{0} \|_{\dot{H}^{s}}, \| u_{1} \|_{\dot{H}^{s - 1}}) > 0$ for any $\frac{1}{2} < s < \frac{3}{2}$.
Negatively, [@LS] proved
\[t1.1.1\] Equation $(\ref{1.1})$ is ill-posed for $u_{0} \in \dot{H}^{s}(\mathbf{R}^{3})$ and $u_{1} \in \dot{H}^{s - 1}(\mathbf{R}^{3})$ when $s < \frac{1}{2}$.
Local well-posedness is defined in the usual way.
\[d1.2\] The initial value problem $(\ref{1.1})$ is said to be locally well-posed if there exists an open interval $I \subset \mathbf{R}$ containing $0$ such that:
1. A unique solution $u \in L_{t}^{\infty} \dot{H}^{1/2}(I \times \mathbf{R}^{3}) \cap L_{t,loc}^{4} L_{x}^{4}(I \times \mathbf{R}^{3})$, $u_{t} \in L_{t}^{\infty} \dot{H}^{-1/2}(I \times \mathbf{R}^{3})$ exists.
2. The solution $u$ is continuous in time, $u \in C(I ; \dot{H}^{1/2}(\mathbf{R}^{3}))$, $u_{t} \in C(I ; \dot{H}^{-1/2}(\mathbf{R}^{3}))$.
3. The solution $u$ depends continuously on the initial data in the topology of item one.
Given this fact, it is natural to inquire as to the long-time behavior of solutions to $(\ref{1.1})$ with initial data at the $\dot{H}^{1/2}$-critical regularity. Do they continue for all time, and if they do, what is their behavior at large times?
Global well-posedness for initial data in $\dot{H}^{1/2} \cap \dot{H}^{1}(\mathbf{R}^{3}) \times \dot{H}^{-1/2} \cap L^{2}(\mathbf{R}^{3})$ follows from conservation of the energy $$\label{1.4}
E(u(t)) = \frac{1}{2} \int u_{t}(t,x)^{2} dx + \frac{1}{2} \int |\nabla u(t,x)|^{2} dx + \frac{1}{4} \int u(t,x)^{4} dx.$$ By the Sobolev embedding theorem and Hölder’s inequality, $$\label{1.4.1}
\| u(0) \|_{L_{x}^{4}(\mathbf{R}^{3})}^{4} \lesssim \| u(0) \|_{L_{x}^{3}(\mathbf{R}^{3})}^{2} \| u(0) \|_{L_{x}^{6}(\mathbf{R}^{3})}^{2} \lesssim \| u(0) \|_{\dot{H}^{1/2}(\mathbf{R}^{3})}^{2} \| u(0) \|_{\dot{H}^{1}(\mathbf{R}^{3})}^{2},$$ and therefore, $$\label{1.4.2}
E(u(0)) \lesssim_{\| u_{0} \|_{\dot{H}^{1/2}}} \| u_{0} \|_{\dot{H}^{1}(\mathbf{R}^{3})}^{2} + \| u_{1} \|_{L^{2}(\mathbf{R}^{3})}^{2}.$$ By $(\ref{1.4})$, $E(u(t)) = E(u(0))$ controls the size of $\| u(t) \|_{\dot{H}^{1}} + \| u_{t}(t) \|_{L^{2}}$, which by Theorem $\ref{t1.1}$ gives global well-posedness.
Comparing $(\ref{1.1})$ to the quintic wave equation in three dimensions, $$\label{1.4.3}
u_{tt} - \Delta u + u^{5} = 0, \qquad u(0,x) = u_{0}, \qquad u_{t}(0,x) = u_{1},$$ a solution to $(\ref{1.4.3})$ is invariant under the scaling symmetry $u(t,x) \mapsto \lambda^{1/2} u(\lambda t, \lambda x)$, a symmetry that preserves the $\dot{H}^{1} \times L^{2}$ norm of $(u_{0}, u_{1})$. Observe that the conserved energy for $(\ref{1.4.3})$, $$\label{1.4.4}
E(u(t)) = \frac{1}{2} \int u_{t}(t,x)^{2} dx + \frac{1}{2} \int |\nabla u(t,x)|^{2} dx + \frac{1}{6} \int u(t,x)^{6} dx$$ is also invariant under the scaling symmetry. For this reason, $(\ref{1.4.3})$ is called energy–critical, and it is possible to prove a result in the same vein as Theorems $\ref{t1.1}$ and $\ref{t1.1.1}$ at the critical regularity $\dot{H}^{1} \times L^{2}$.
This fact combined with conservation of the energy $(\ref{1.4.4})$ is insufficient to prove global well-posedness for $(\ref{1.4.3})$. The reason is because the time of local well-posedness depends on the profile of the initial data $(u_{0}, u_{1}) \in \dot{H}^{1} \times L^{2}$, and not just its size. Instead, the proof of global well-posedness for the quintic problem uses a non-concentration of energy argument. This result has been completely worked out, proving both global well-posedness and scattering, for both the radial ([@GSV], [@Struwe]) and the nonradial case ([@BS], [@Gril], [@Shatah; @-; @Struwe]).
\[d1.1\] A solution to $(\ref{1.4.3})$ is said to be scattering in some $\dot{H}^{s}(\mathbf{R}^{3}) \times \dot{H}^{s - 1}(\mathbf{R}^{3})$ if there exist $(u_{0}^{+}, u_{1}^{+}), (u_{0}^{-}, u_{1}^{-}) \in \dot{H}^{s} \times \dot{H}^{s - 1}$ such that $$\label{1.4.5}
\lim_{t \rightarrow +\infty} \| (u(t), u_{t}(t)) - S(t)(u_{0}^{+}, u_{1}^{+}) \|_{\dot{H}^{s} \times \dot{H}^{s - 1}} = 0,$$ and $$\label{1.4.6}
\lim_{t \rightarrow -\infty} \| (u(t), u_{t}(t)) - S(t)(u_{0}^{+}, u_{1}^{+}) \|_{\dot{H}^{s} \times \dot{H}^{s - 1}} = 0,$$ where $S(t)(f, g)$ is the solution operator to the linear wave equation. That is, if $(u(t), u_{t}(t)) = S(t)(f, g)$, then $$\label{1.4.7}
u_{tt} - \Delta u = 0, \qquad u(0,x) = f, \qquad u_{t}(0,x) = g.$$
Similar results for $(\ref{1.1})$ may also be obtained if one assumes a uniform bound over $\| u \|_{\dot{H}^{1/2}(\mathbf{R}^{3})} + \| u_{t} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})}$ for the entire time of existence of the solution.
\[t1.3\] Suppose $u_{0} \in \dot{H}^{1/2}(\mathbf{R}^{3})$ and $u_{1} \in \dot{H}^{-1/2}(\mathbf{R}^{3})$ are radial functions, and $u$ solves $(\ref{1.1})$ on a maximal interval $0 \in I \subset \mathbf{R}$, with $$\label{1.6}
\sup_{t \in I} \| u(t) \|_{\dot{H}^{1/2}(\mathbf{R}^{3})} + \| u_{t}(t) \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})} < \infty.$$ Then $I = \mathbf{R}$ and the solution $u$ scatters both forward and backward in time.
*Proof:* See [@DL]. $\Box$
In this paper we remove the a priori assumption on uniform boundedness of the critical norm in $(\ref{1.6})$, proving,
\[t1.2\] The initial value problem $(\ref{1.1})$ is globally well-posed and scattering for radial initial data $u_{0} \in \dot{H}^{1/2}(\mathbf{R}^{3})$ and $u_{1} \in \dot{H}^{-1/2}(\mathbf{R}^{3})$. Moreover, there exists a function $f : [0, \infty) \rightarrow [0, \infty)$ such that if $u$ solves $(\ref{1.1})$ with initial data $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$, then $$\label{1.5}
\| u \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} \leq f(\| u_{0} \|_{\dot{H}^{1/2}(\mathbf{R}^{3})} + \| u_{1} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})}).$$
The proof of Theorem $\ref{t1.2}$ combines the Fourier truncation method and hyperbolic coordinates. Previously, [@KPV] applied the Fourier truncation method to the cubic wave equation, $(\ref{1.1})$, proving global well-posedness of $(\ref{1.1})$ with initial data lying in the inhomogeneous Sobolev spaces $H_{x}^{s}(\mathbf{R}^{3}) \times H_{x}^{s - 1}(\mathbf{R}^{3})$ for $s > \frac{3}{4}$. This argument was improved and modified in many subsequent papers, for both radial and nonradial data. In particular, see [@D] for a proof of global well-posedness for $(\ref{1.1})$ with radial initial data lying in $$\label{1.7}
(\dot{H}^{s}(\mathbf{R}^{3}) \cap \dot{H}^{1/2}(\mathbf{R}^{3})) \times (\dot{H}^{s - 1}(\mathbf{R}^{3}) \cap \dot{H}^{-1/2}(\mathbf{R}^{3})),$$ for any $s > \frac{1}{2}$, as well as for a description of other results along this line.
**Remark:** The method used in [@D] was the I-method, a modification of the Fourier truncation method.
In this paper, using the Fourier truncation method, global well-posedness is proved for $(\ref{1.1})$ with radial initial data lying in $\dot{H}^{1/2}(\mathbf{R}^{3}) \times \dot{H}^{-1/2}(\mathbf{R}^{3})$. The idea behind the proof is that at low frequencies, the initial data has finite energy, and a solution to $(\ref{1.1})$ with finite energy is global. Meanwhile, at high frequencies, the $\dot{H}^{1/2} \times \dot{H}^{-1/2}$ norm is small, and for such initial data, $(\ref{1.1})$ may be treated using perturbative arguments. The mixed terms in the nonlinearity are then shown to have finite energy, proving global well-posedness.
Proof of scattering utilizes hyperbolic coordinates. Hyperbolic coordinates were used in [@Tataru] to prove weighted Strichartz estimates that were proved in [@GLS]. More recently, [@Shen], working in hyperbolic coordinates, was able to prove a scattering result for data lying in a weighted energy space. Later, [@D1] combined the result of [@Shen] with the I-method argument in [@D] to prove scattering data lying in the subspace of $\dot{H}^{1/2} \times \dot{H}^{-1/2}$, $$\label{1.8}
\| u_{0} \|_{\dot{H}^{1/2 + \epsilon}(\mathbf{R}^{3})} + \| |x|^{2 \epsilon} u_{0} \|_{\dot{H}^{1/2 + \epsilon}(\mathbf{R}^{3})} + \| u_{1} \|_{\dot{H}^{-1/2 + \epsilon}(\mathbf{R}^{3})} + \| |x|^{2 \epsilon} u_{1} \|_{\dot{H}^{-1/2 + \epsilon}(\mathbf{R}^{3})}.$$ Here, the Fourier truncation global well-posedness argument in hyperbolic coordinates shows that $(\ref{1.1})$ is globally well-posed and scattering for any $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$.
This fact still falls short of $(\ref{1.5})$, since the proof does not give any uniform control over the $\| u \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})}$ norm. To remedy this deficiency, and complete the proof of Theorem $\ref{t1.2}$, a profile decomposition is used. The profile decomposition shows that for any bounded sequence of initial data $$\label{1.9}
\| u_{0}^{n} \|_{\dot{H}^{1/2}(\mathbf{R}^{3})} + \| u_{1}^{n} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})} \leq A,$$ and if $u^{n}(t)$ is the global solution to $(\ref{1.1})$ with initial data $(u_{0}^{n}, u_{1}^{n})$, then $$\label{1.10}
\| u^{n} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} < \infty,$$ is uniformly bounded. Then by Zorn’s lemma, the proof of Theorem $\ref{t1.2}$ is complete.
The author believes this to be the first unconditional global well-posedness and scattering result for a nonlinear wave equation with initial data lying in the critical Sobolev space, with no conserved quantity that controls the critical norm. Previously, [@D2] proved global well-posedness and scattering for $(\ref{1.1})$ with radial initial data lying in the Besov space $B_{1,1}^{2} \times B_{1,1}^{1}$. These spaces are also invariant under the scaling $(\ref{1.2})$. Later, [@MYZ] proved a similar result in five dimensions.
There are two main improvements for this result over the results of [@D2] and [@MYZ]. The first is that, while scale invariant, the Besov spaces are only subsets of the critical Sobolev spaces. The second improvement is that the $\dot{H}^{1/2} \times \dot{H}^{-1/2}$ norm is invariant under the free evolution of the linear wave equation. Whereas, for initial data lying in a Besov space, the proof of scattering simply meant that the solution scattered in the $\dot{H}^{1/2} \times \dot{H}^{-1/2}$ norm.
**Acknowledgements:** The author was partially supported on NSF grant number $1764358$ during the writing of this paper. The author was also a guest of the Institute for Advanced Study during the writing of this paper.
Local well-posedness
====================
The local well-posedness result of [@LS] may be proved via the Strichartz estimates of [@Stri].
\[t6.1\] Let $I \subset \mathbf{R}$, $t_{0} \in I$, be an interval and let $u$ solve the linear wave equation $$\label{6.1}
u_{tt} - \Delta u = F, \hspace{5mm} u(t_{0}) = u_{0}, \hspace{5mm} u_{t}(t_{0}) = u_{1}.$$ Then we have the estimates $$\label{6.2}
\aligned
\| u \|_{L_{t}^{p} L_{x}^{q}(I \times \mathbf{R}^{3})} + \| u \|_{L_{t}^{\infty} \dot{H}^{s}(I \times \mathbf{R}^{3})} + \| u_{t} \|_{L_{t}^{\infty} \dot{H}^{s - 1}(I \times \mathbf{R}^{3})} \\
\lesssim_{p, q, s, \tilde{p}, \tilde{q}} \| u_{0} \|_{\dot{H}^{s}(\mathbf{R}^{3})} + \| u_{1} \|_{\dot{H}^{s - 1}(\mathbf{R}^{3})} + \| F \|_{L_{t}^{\tilde{p}'} L_{x}^{\tilde{q}'}(I \times \mathbf{R}^{3})},
\endaligned$$ whenever $s \geq 0$, $2 \leq p, \tilde{p} \leq \infty$, $2 \leq q, \tilde{q} < \infty$, and $$\label{6.3}
\frac{1}{p} + \frac{1}{q} \leq \frac{1}{2}, \hspace{5mm} \frac{1}{\tilde{p}} + \frac{1}{\tilde{q}} \leq \frac{1}{2}.$$
*Proof:* Theorem $\ref{t6.1}$ was proved for $p = q = 4$ in [@Stri] and then in [@GV] for a general choice of $(p, q)$. $\Box$
To prove local well-posedness of $(\ref{1.1})$, $(\ref{6.2})$ when $p = q = 4$ will suffice. Indeed, $(\ref{6.2})$ implies that for any $I$, $$\label{6.4}
\| u \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} \lesssim \| S(t)(u_{0}, u_{1}) \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} + \| u \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})}^{3}.$$ If $\| S(t)(u_{0}, u_{1}) \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} \leq \epsilon$, then $(\ref{6.4})$ implies that $(\ref{1.1})$ is locally well-posed on the interval $I$.
For $\| u_{0} \|_{\dot{H}^{1/2}} + \| u_{1} \|_{\dot{H}^{-1/2}}$ sufficiently small, $(\ref{6.2})$ and $(\ref{6.4})$ imply that $(\ref{1.1})$ is well-posed on $I = \mathbf{R}$. For generic $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$, the dominated convergence theorem and $(\ref{6.2})$ imply that for any fixed $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$, $$\label{6.5}
\lim_{T \searrow 0} \| S(t)(u_{0}, u_{1}) \|_{L_{t,x}^{4}([-T, T] \times \mathbf{R}^{3})} = 0,$$ which implies local well-posedness on some open interval $I$, where $0 \in I$.
Equation $(\ref{6.4})$ also implies that $(\ref{1.1})$ is locally well-posed on an interval $I$ on which an a priori bound $\| u \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} < \infty$ is obtained. This may be seen by partitioning $I$ into finitely many pieces $I_{j}$ on which $\| u \|_{L_{t,x}^{4}(I_{j} \times \mathbf{R}^{3})}$ is small, and then iterating local well-posedness arguments on each interval. This argument also shows that scattering is equivalent to $\| u \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} < \infty$.
Strichartz estimates also yield perturbative results.
\[l6.2\] Let $I \subset \mathbf{R}$ be a time interval. Let $t_{0} \in I$, $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$ and some constants $M$, $A$, $A' > 0$. Let $\tilde{u}$ solve the equation $$\label{6.6}
(\partial_{tt} - \Delta) \tilde{u} = F(\tilde{u}) = e,$$ on $I \times \mathbf{R}^{3}$, and also suppose $\sup_{t \in I} \| (\tilde{u}(t), \partial_{t} \tilde{u}(t)) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}} \leq A$, $\| \tilde{u} \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} \leq M$, $$\label{6.7}
\| (u_{0} - \tilde{u}(t_{0}), u_{1} - \partial_{t} \tilde{u}(t_{0})) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}} \leq A',$$ and $$\label{6.8}
\| e \|_{L_{t,x}^{4/3}(I \times \mathbf{R}^{3})} + \| S(t - t_{0})(u_{0} - \tilde{u}(t_{0}), u_{1} - \partial_{t} \tilde{u}(t_{0})) \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} \leq \epsilon.$$ Then there exists $\epsilon_{0}(M, A, A')$ such that if $0 < \epsilon < \epsilon_{0}$ then there exists a solution to $(\ref{1.1})$ on $I$ with $(u(t_{0}), \partial_{t} u(t_{0})) = (u_{0}, u_{1})$, $\| u \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} \leq C(M, A, A')$, and for all $t \in I$, $$\label{6.9}
\| (u(t), \partial_{t} u(t)) - (\tilde{u}(t), \partial_{t} \tilde{u}(t)) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}} \leq C(A, A', M)(A' + \epsilon).$$
*Proof:* The method of proof is by now fairly well-known. See for example lemma $2.20$ of [@KM]. $\Box$
The proof of Theorem $\ref{t1.2}$ also utilizes some additional Strichartz estimates that only appear for radially symmetric data. First, [@KlMa] proved that the endpoint case of Theorem $\ref{t6.1}$ also holds.
\[t6.3\] For $(u_{0}, u_{1})$ radially symmetric, and $u$ solves $(\ref{6.1})$ with $F = 0$, $$\label{6.10}
\| u \|_{L_{t}^{2} L_{x}^{\infty}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \| u_{0} \|_{\dot{H}^{1}(\mathbf{R}^{3})} + \| u_{1} \|_{L^{2}(\mathbf{R}^{3})}.$$
Additionally, the proof will rely very heavily on the estimates of [@Sterb] for radially symmetric initial data, extending the range of $(p, q)$ in $(\ref{6.3})$ for radial initial data.
\[t6.4\] Let $(u_{0}, u_{1})$ be spherically symmetric, and suppose $u$ solves $(\ref{6.1})$ with $F = 0$. Then if $q > 4$ and $$\label{6.11}
\frac{1}{2} + \frac{3}{q} = \frac{3}{2} - s,$$ then $$\label{6.12}
\| u \|_{L_{t}^{2} L_{x}^{q}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \| u_{0} \|_{\dot{H}^{s}(\mathbf{R}^{3})} + \| u_{1} \|_{\dot{H}^{s - 1}(\mathbf{R}^{3})}.$$
Virial identities for the wave equation
=======================================
The proof of Theorem $\ref{t1.2}$ will also use some weighted Strichartz-type estimates. These estimates could actually be proved using Proposition $3.5$ of [@Sterb] after making a Bessel function-type reduction from three dimensions to two dimensions using radial symmetry.
However, these estimates will instead be proved using virial identities. There are at least two reasons for doing this. The first is that, in the author’s opinion, the exposition is cleaner and more readable using virial identities. The second reason is that many of the computations may be applied equally well to defocusing problems as to linear problems.
Suppose $u$ solves the equation $$\label{2.0}
u_{tt} - \Delta + \mu u^{3} = 0, \qquad u(0,x) = u_{0}, \qquad u_{t}(0,x) = u_{1},$$ where $\mu = 0, 1$. The case when $\mu = 0$ is a solution to the linear wave equation and $\mu = 1$ is the defocusing nonlinear wave equation $(\ref{1.1})$.
\[t2.3\] If $u$ solves $(\ref{1.1})$ on an interval $[0, T]$, then $$\label{2.19}
\int_{0}^{T} \int \frac{\mu}{|x|} u^{4} dx dt \lesssim \| u \|_{L_{t}^{\infty} \dot{H}^{1}([0, T] \times \mathbf{R}^{3})} \| u_{t} \|_{L_{t}^{\infty} L_{x}^{2}([0, T] \times \mathbf{R}^{3})},$$ $$\label{2.20}
\sup_{R > 0} \frac{1}{R^{3}} \int_{0}^{T} \int_{|x| \leq R} u^{2} dx dt \lesssim \| u \|_{L_{t}^{\infty} \dot{H}^{1}([0, T] \times \mathbf{R}^{3})} \| u_{t} \|_{L_{t}^{\infty} L_{x}^{2}([0, T] \times \mathbf{R}^{3})},$$ and $$\label{2.21}
\sup_{R > 0} \frac{1}{R} \int_{0}^{T} \int_{|x| \leq R} [|\nabla u|^{2} + u_{t}^{2}] dx dt \lesssim \| u \|_{L_{t}^{\infty} \dot{H}^{1}([0, T] \times \mathbf{R}^{3})} \| u_{t} \|_{L_{t}^{\infty} L_{x}^{2}([0, T] \times \mathbf{R}^{3})}.$$
*Proof:* Define the generic Morawetz potential $$\label{2.22}
M(t) = \int u_{t} a(|x|) x \cdot \nabla u + \int u_{t} a(|x|) u.$$ Computing the time derivative, $$\label{2.23}
\aligned
\frac{d}{dt} M(t) = \int u_{t} a(|x|) x \cdot \nabla u_{t} + \int u_{t}^{2} a(|x|) \\ \int \Delta u a(|x|) x \cdot \nabla u + \int \Delta u a(|x|) u \\ - \mu \int u^{3} a(|x|) x \cdot \nabla u - \mu \int u^{3} a(|x|) u.
\endaligned$$ Integrating by parts, $$\label{2.24}
\aligned
\frac{d}{dt} M(t) = -\frac{1}{2} \int [a(|x|) + a'(|x|) |x|] u_{t}^{2} - \frac{1}{2} \int [a(|x|) + a'(|x|) |x|] |\nabla u|^{2} \\
+ \int a'(|x|) |x| [|\nabla u|^{2} - |\partial_{r} u|^{2}] + \frac{1}{2} \int u^{2} \Delta a(|x|) \\ - \frac{\mu}{4} \int a(|x|) u^{4} + \frac{\mu}{4} \int a'(|x|) |x| u^{4}.
\endaligned$$ If we choose $a(|x|) = \frac{1}{|x|}$, then $$\label{2.25}
a(|x|) + a'(|x|) |x| = 0.$$ When $u$ is radial, $|\nabla u|^{2} - |\partial_{r} u|^{2} = 0$. For a general $u$, $$\label{2.26}
|\nabla u|^{2} - |\partial_{r} u|^{2} \geq 0,$$ so since $a'(|x|) \leq 0$, $$\label{2.27}
a'(|x|) |x| [|\nabla u|^{2} - |\partial_{r} u|^{2}] \leq 0.$$ Also, by direct calculation, $\Delta \frac{1}{|x|} = - 2 \pi \delta(x)$, so when $a(|x|) = \frac{1}{|x|}$, $$\label{2.28}
\frac{d}{dt} M(t) \leq -\pi u(t,0)^{2} - \frac{\mu}{2} \int \frac{1}{|x|} u^{4} dx.$$ Now by Hardy’s inequality, when $a(x) = \frac{1}{|x|}$, $$\label{2.29}
|M(t)| \lesssim \| u_{t} \|_{L^{2}} \| \nabla u \|_{L^{2}}.$$ Therefore, $$\label{2.30}
\int_{0}^{T} u(t,0)^{2} dt + \int_{0}^{T} \int \frac{\mu}{|x|} u^{4} dx dt \lesssim \| u_{t} \|_{L_{t}^{\infty} L_{x}^{2}} \| \nabla u \|_{L_{t}^{\infty} L_{x}^{2}}.$$ This takes care of $(\ref{2.19})$.
Replacing $a(|x|)$ by $a(|x - y|)$ and $x$ with $x - y$, $(\ref{2.30})$ implies $$\label{2.31}
\aligned
\frac{1}{R^{3}} \int_{0}^{T} \int_{|y| \leq R} u(t,y)^{2} dy dt + \frac{1}{R^{3}} \int_{|y| \leq R} \int \frac{\mu}{|x - y|} u(t,x)^{4} dx dy \\ \lesssim \| u_{t} \|_{L_{t}^{\infty} L_{x}^{2}} \| \nabla u \|_{L_{t}^{\infty} L_{x}^{2}},
\endaligned$$ which takes care of $(\ref{2.20})$.
To prove $(\ref{2.21})$, choose a smooth function $\chi : [0, \infty) \rightarrow [0, \infty)$ satisfying $\chi(|x|) = 1$ for $|x| \leq 1$, $\chi(|x|) = \frac{3}{2 |x|}$ for $|x| \geq 2$, and such that $$\chi(|x|) + \chi'(|x|) |x| = \phi(|x|)$$ is a smooth function, $\phi(|x|) \geq 0$, $\phi(|x|) = 1$ for $|x| \leq 1$, and $\phi(|x|)$ is supported on $|x| \leq 2$. Take $a(|x|) = \frac{1}{R} \chi(\frac{|x|}{R})$. $$\label{2.32}
a(|x|) + a'(|x|) |x| = \frac{1}{R} \chi(\frac{|x|}{R}) + \frac{1}{R} \chi'(\frac{|x|}{R}) \frac{|x|}{R} = \frac{1}{R} \phi(\frac{|x|}{R}).$$ Therefore, $$\label{2.33}
\frac{d}{dt} M(t) = -\frac{1}{2R} \int \phi(\frac{|x|}{R}) [u_{t}^{2} + |\nabla u|^{2}] - \frac{\mu}{4R} \int a(\frac{|x|}{R}) u^{4} + \frac{1}{2R} \int u^{2} \Delta a(\frac{|x|}{R}).$$ Now, since $a(|x|) = \frac{3}{2} \frac{1}{|x|}$ when $|x| \geq 2$, $\Delta a(|x|)$ is supported on $|x| \leq 2$. Therefore, $$\label{2.34}
\frac{1}{2R} \int u^{2} \Delta a(\frac{|x|}{R}) \lesssim \sup_{R > 0} \frac{1}{R^{3}} \int_{|x| \leq R} u^{2}.$$ Also, $a(|x|) \lesssim \frac{1}{|x|}$, so again by Hardy’s inequality, $$\label{2.33.1}
|M(t)| \lesssim \| v_{t} \|_{L^{2}} \| \nabla v \|_{L^{2}}.$$ Plugging $(\ref{2.31})$ and $(\ref{2.34})$ into $(\ref{2.33})$ proves $(\ref{2.21})$. $\Box$
\[c2.4\] If $u$ is an approximate solution to the cubic wave equation, $$\label{2.35}
u_{tt} - \Delta u + u^{3} = F,$$ then $$\label{2.36}
\aligned
\frac{d}{dt} [\int u_{t} \frac{x}{|x|} \cdot \nabla u + \int u_{t} \frac{1}{|x|} u] \leq -2 \pi u(t,0)^{2} - \frac{1}{2} \int \frac{1}{|x|} u^{4} \\
+ \int F \frac{x}{|x|} \cdot \nabla u + \int F \frac{1}{|x|} u,
\endaligned$$ $$\label{2.37}
\aligned
\frac{d}{dt} \frac{1}{R^{3}} [\int_{|y| \leq R} \int u_{t} \frac{x}{|x|} \cdot \nabla u + \int_{|y| \leq R} \int u_{t} \frac{1}{|x|} u] \\ \leq -\pi \frac{1}{R^{3}} \int_{|y| \leq R} u(t,y)^{2} - \frac{1}{2} \frac{1}{R^{3}} \int_{|y| \leq R} \int \frac{1}{|x - y|} u^{4} \\
+ \frac{1}{R^{3}} \int_{|y| \leq R} \int F \frac{x - y}{|x - y|} \cdot \nabla u + \frac{1}{R^{3}} \int_{|y| \leq R} \int F \frac{1}{|x - y|} u,
\endaligned$$ and $$\label{2.38}
\aligned
\frac{d}{dt} [\frac{1}{R} \int u_{t} \chi(\frac{|x|}{R}) x \cdot \nabla u + \frac{1}{R} \int u_{t} \chi(\frac{|x|}{R}) u] \leq -\frac{1}{2R} \int \phi(\frac{|x|}{R}) [u_{t}^{2} + |\nabla u|^{2}] \\
- \frac{1}{4R} \int \chi(\frac{|x|}{R}) u^{4} + \frac{1}{4R} \int \chi'(\frac{|x|}{R}) \frac{|x|}{R} u^{4} \\ + \frac{1}{2R} \int u^{2} \Delta a(\frac{|x|}{R}) + \frac{1}{R} \int F \chi(\frac{|x|}{R}) x \cdot \nabla u + \frac{1}{R} \int F \chi(\frac{|x|}{R}) u.
\endaligned$$
Theorem $\ref{t2.3}$ also gives some nice estimates for the linear wave equation $(\mu = 0)$.
\[c2.1\] For any $j \in \mathbf{Z}$, let $w$ be the solution to the linear wave equation $$\partial_{tt} w - \Delta w = 0, \qquad w(0,x) = P_{j} u_{0}, \qquad w_{t}(0,x) = P_{j} u_{1}.$$ Then for any $2 < p < \infty$, $$\label{2.1}
\| |x|^{1/2} w \|_{L_{t}^{p} L_{x}^{\infty}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \| P_{j} u_{0} \|_{\dot{H}^{1/p'}(\mathbf{R}^{3})} + \| P_{j} u_{1} \|_{\dot{H}^{1/p' - 1}(\mathbf{R}^{3})},$$ where $\frac{1}{p'} = 1 - \frac{1}{p}$ is the Lebesgue dual of $p$. Also, for $p = 2$, for any $0 < R < 1$, $1 < R_{1} < \infty$, $$\label{2.2}
\| |x|^{1/2} w \|_{L_{t,x}^{2}(\mathbf{R} \times \{ x : R \leq |x| \leq R_{1} \})}^{2} \lesssim (\ln(R_{1}) - \ln(R) + 1) [\| P_{j} u_{0} \|_{\dot{H}^{1/2}(\mathbf{R}^{3})}^{2} + \| P_{j} u_{1} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})}^{2}].$$
*Proof:* Let $\psi$ be a smooth, radial function supported on an annulus, $\psi(r) = 1$ for $1 \leq r \leq 2$, and $\psi(r)$ is supported on $\frac{1}{2} \leq r \leq 4$. By Bernstein’s inequality, $$\label{2.11}
\| P_{k}(\psi(\frac{r}{R}) w) \|_{L^{2}} \lesssim 2^{-k} \| \partial_{r}(\psi(\frac{r}{R}) w) \|_{L^{2}} + 2^{-k} R^{-1} \| \psi'(\frac{r}{R}) w \|_{L^{2}}.$$ Therefore, by $(\ref{2.20})$, $(\ref{2.21})$, and the radial Sobolev embedding theorem, $$\label{2.12}
\sum_{k \geq j - 3} \| P_{k} (\psi(\frac{r}{R}) w) \|_{L_{t}^{2} L_{x}^{\infty}} \lesssim 2^{-j/2} R^{-1/2} (\| P_{j} u_{0} \|_{\dot{H}^{1}} + \| P_{j} u_{1} \|_{L^{2}}).$$ Next, by the Fourier support properties of $w$, $$\label{2.13}
\| P_{\leq j - 3} (\psi(\frac{r}{R}) w) \|_{L^{\infty}} \lesssim 2^{-j} R^{-1} \| w \|_{L^{\infty}}.$$ Combining $(\ref{2.13})$ with $(\ref{6.10})$, $$\label{2.14}
\| P_{\leq j - 3} (\psi(\frac{r}{R}) w) \|_{L_{t}^{2} L_{x}^{\infty}} \lesssim 2^{-j} R^{-1} (\| P_{j} u_{0} \|_{\dot{H}^{1}} + \| P_{j} u_{1} \|_{L^{2}}).$$ Then when $R \geq 2^{-j}$, $$\| P_{\leq j - 3} (\psi(\frac{r}{R}) w) \|_{L_{t}^{2} L_{x}^{\infty}} \lesssim 2^{-j/2} R^{-1/2} (\| P_{j} u_{0} \|_{\dot{H}^{1}} + \| P_{j} u_{1} \|_{L^{2}}).$$ Finally, when $R \leq 2^{-j}$, a straightforward application of the endpoint Strichartz estimate yields $$\label{2.15}
\| \psi(\frac{r}{R}) w \|_{L_{t}^{2} L_{x}^{\infty}} \lesssim (\| P_{j} u_{0} \|_{\dot{H}^{1}} + \| P_{j} u_{1} \|_{L^{2}}) \lesssim R^{-1/2} 2^{-j/2} (\| P_{j} u_{0} \|_{\dot{H}^{1}} + \| P_{j} u_{1} \|_{L^{2}}).$$ Since there are $\lesssim \ln(R_{1}) - \ln(R) + 1$ dyadic annuli overlapping $R \leq |x| \leq R_{1}$, $(\ref{2.12})$–$(\ref{2.15})$ directly yields $(\ref{2.2})$.
To prove $(\ref{2.1})$, interpolating $(\ref{2.15})$ with the radial Sobolev embedding theorem, for any $2 < p < \infty$, $$\label{2.16}
\aligned
\| \psi(\frac{r}{R}) w \|_{L_{t}^{p} L_{x}^{\infty}} \lesssim \| \psi(\frac{r}{R}) w \|_{L_{t}^{2} L_{x}^{\infty}}^{2/p} \| \psi(\frac{r}{R}) w \|_{L_{t,x}^{\infty}}^{1 - 2/p} \\ \lesssim R^{-1/2} R^{-\frac{1}{2}(1 - \frac{2}{p})} (\| P_{j} u_{0} \|_{\dot{H}^{1/2}} + \| P_{j} u_{1} \|_{\dot{H}^{-1/2}}),
\endaligned$$ which directly implies $$\label{2.17}
\| |x|^{1/2} w \|_{L_{t}^{p} L_{x}^{\infty}(\mathbf{R} \times \{ x : |x| \geq 2^{-j} \})} \lesssim (\| P_{j} u_{0} \|_{\dot{H}^{1/p'}} + \| P_{j} u_{1} \|_{\dot{H}^{1/p' - 1}}).$$ Meanwhile, by $(\ref{6.10})$ and the Sobolev embedding theorem, $$\label{2.18}
\| |x|^{1/2} w \|_{L_{t}^{p} L_{x}^{\infty}(\mathbf{R} \times \{ |x| \leq 2^{-j} \})} \lesssim 2^{-j/2} \| w \|_{L_{t}^{2} L_{x}^{\infty}}^{2/p} \| w \|_{L_{t,x}^{\infty}}^{1 - 2/p} \lesssim (\| P_{j} u_{0} \|_{\dot{H}^{1/p}} + \| P_{j} u_{1} \|_{\dot{H}^{1/p - 1}}).$$ This finally proves the theorem. $\Box$
**Remark:** Also observe that by the radial Sobolev embedding theorem, Corollary $\ref{c2.1}$ implies $$\label{2.18.1}
\| w \|_{L_{t}^{2} L_{x}^{\infty}([0, T] \times \{ |x| \geq R \})}^{2} \lesssim (1 + \ln(T) - \ln(R)) [\| P_{j} u_{0} \|_{\dot{H}^{1/2}} + \| P_{j} u_{1} \|_{\dot{H}^{-1/2}}].$$
The virial identities in Theorem $\ref{t2.3}$ commute very well with Littlewood–Paley projections.
\[l3.1\] For any $j$, $$\label{3.18}
\int \frac{1}{|x|} |P_{\leq j} v|^{4} dx + \int \frac{1}{|x|} |P_{\geq j} v|^{4} dx \lesssim \int \frac{1}{|x|} |v|^{4} dx.$$
*Proof:* Let $\psi$ be the Littlewood–Paley kernel. $$\label{2.19.1}
\frac{1}{|x|^{1/4}} P_{\leq j} v(x) = \frac{1}{|x|^{1/4}} \int 2^{3j} \psi(2^{j}(x - y)) v(y) dy.$$ When $|y| \lesssim |x|$, $$\label{2.20.1}
\frac{1}{|x|^{1/4}} 2^{3j} \psi(2^{j}(x - y)) \lesssim 2^{3j} \psi(2^{j}(x - y)) \frac{1}{|y|^{1/4}}.$$ When $|y| \gg |x|$ and $|x| \geq 2^{-j}$, since $\psi$ is rapidly decreasing, for any $N$, $$\label{2.21.1}
\aligned
\frac{1}{|x|^{1/4}} 2^{3j} \psi(2^{j}(x - y)) \lesssim_{N} \frac{1}{|x|^{1/4}} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N}} \\ \lesssim \frac{1}{|x|^{1/4} 2^{j} |y|} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N - 1}} \lesssim \frac{1}{|y|^{1/4}} \frac{2^{3j}}{(1 + 2^{j}|x - y|)^{N - 1}}.
\endaligned$$ Combining $(\ref{2.20.1})$ and $(\ref{2.21.1})$, $$\label{2.22}
\| \frac{1}{|x|^{1/4}} |P_{\leq j} v| \|_{L^{4}(|x| \geq 2^{-j})} \lesssim \| \frac{1}{|x|^{1/4}} v \|_{L^{4}(\mathbf{R}^{3})}.$$ When $|y| \gg |x|$ and $|x| \leq 2^{-j}$, since $\psi$ is rapidly decreasing, for any $N$, $$\label{2.23}
\aligned
\frac{1}{|x|^{1/4}} 2^{3j} \psi(2^{j}(x - y)) \lesssim_{N} \frac{1}{|x|^{1/4}} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N}} \\ \lesssim \frac{1}{|x|^{1/4}} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N - 1/4}} \frac{1}{2^{j/4} |y|^{1/4}}.
\endaligned$$ $$\label{2.24}
\| \frac{2^{11j/4}}{(1 + 2^{j}|x - y|)^{N}} \|_{L^{4/3}(\mathbf{R}^{3})} \lesssim 2^{j/2},$$ so by $(\ref{2.20.1})$, $(\ref{2.24})$, Young’s inequality, and H[ö]{}lder’s inequality, $$\label{2.25}
\| \frac{1}{|x|^{1/4}} |P_{\leq j} v| \|_{L^{4}(|x| \leq 2^{-j})} \lesssim \| \frac{1}{|x|^{1/4}} v \|_{L^{4}(\mathbf{R}^{3})}.$$ This proves $(\ref{3.18})$. $\Box$
\[l2.2\] $$\label{2.26}
\aligned
\| P_{\geq j} v \|_{L^{4}(|x| \leq \frac{R}{2})}^{2} \lesssim \| P_{\geq j} v \|_{L^{3}} [\| \nabla v \|_{L^{2}(|x| \leq R)} + \frac{1}{R} \| v \|_{L^{2}(|x| \leq R)}] \\ + 2^{-j/2} (\int \frac{1}{|x|} v^{4})^{1/2}.
\endaligned$$
*Proof:* Let $\phi \in C_{0}^{\infty}(\mathbf{R}^{3})$ be supported on $|x| \leq 1$ and $\phi(x) = 1$ for $|x| \leq \frac{1}{2}$. By H[ö]{}lder’s inequality, $$\label{2.27}
\| P_{\geq j} v \|_{L^{4}(|x| \leq \frac{R}{2})}^{2} \leq \| \phi(\frac{x}{R}) (P_{\geq j} v) \|_{L^{4}(\mathbf{R}^{3})}^{2}.$$ Then, by the triangle inequality, H[ö]{}lder’s inequality, and the Cauchy–Schwartz inequality, $$\label{2.28}
\aligned
\| \phi(\frac{x}{R}) (P_{\geq j} v) \|_{L^{4}(\mathbf{R}^{3})}^{2} \leq \| \phi(\frac{x}{R}) (P_{\geq j} v) \cdot P_{\geq j} (\phi(\frac{x}{R}) v) \|_{L^{2}(\mathbf{R}^{3})} \\ + \| \phi(\frac{x}{R}) (P_{\geq j} v) \cdot [\phi(\frac{x}{R}), P_{\geq j}] v \|_{L^{2}(\mathbf{R}^{3})} \leq \| P_{\geq j} \phi(\frac{x}{R}) v \|_{L^{6}(\mathbf{R}^{3})} \| P_{\geq j} v \|_{L^{3}(\mathbf{R}^{3})} \\ + \frac{1}{2} \| \phi(\frac{x}{R}) (P_{\geq j} v) \|_{L^{4}(\mathbf{R}^{3})}^{2} + \frac{1}{2} \| [\phi(\frac{x}{R}), P_{\geq j}] v \|_{L^{4}(\mathbf{R}^{3})}^{2},
\endaligned$$ where $$\label{2.29.1}
[\phi(\frac{x}{R}), P_{\geq j}] v = \phi(\frac{x}{R}) (P_{\geq j} v) - P_{\geq j} (\phi(\frac{x}{R}) v).$$ By the Littlewood–Paley theorem, $$\label{2.30.1}
\| \phi(\frac{x}{R}) (P_{\geq j} v) \|_{L^{4}(\mathbf{R}^{3})}^{2} \lesssim \| \phi(\frac{x}{R}) v \|_{L^{6}(\mathbf{R}^{3})} \| P_{\geq j} v \|_{L^{3}(\mathbf{R}^{3})} + \| [\phi(\frac{x}{R}), P_{\geq j}] v \|_{L^{4}(\mathbf{R}^{3})}^{2},$$ and by the Sobolev embedding theorem, $$\label{2.31.1}
\| \phi(\frac{x}{R}) v \|_{L^{6}(\mathbf{R}^{3})} \lesssim \| \nabla (\phi(\frac{x}{R}) v) \|_{L^{2}(\mathbf{R}^{3})} \lesssim \frac{1}{R} \| v \|_{L^{2}(|x| \leq R)} + \| \nabla v \|_{L^{2}(|x| \leq R)}.$$ This takes care of the first term on the right hand side of $(\ref{2.30.1})$.
To handle the commutator, observe that $$\label{2.32}
[\phi(\frac{x}{R}), P_{\geq j}] = -[P_{\leq j}, \phi(\frac{x}{R})].$$ Then compute $$\label{2.33.1.1}
[P_{\leq j}, \phi(\frac{x}{R})] v = 2^{3j} \int \psi(2^{j}(x - y)) [\phi(\frac{y}{R}) - \phi(\frac{x}{R})] v(y) dy.$$ When $|y| \gg |x|$, the kernel $$\label{2.34.1}
\aligned
2^{3j} \psi(2^{j}(x - y)) [\phi(\frac{y}{R}) - \phi(\frac{x}{R})] \lesssim_{N} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N}} \\ \lesssim 2^{-j/4} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N - 1/4}} \frac{1}{|y|^{1/4}}.
\endaligned$$ When $|y| \lesssim |x|$ and $|x| \leq R$, by the fundamental theorem of calculus, $$\label{2.35}
\aligned
2^{3j} \psi(2^{j}(x - y)) [\phi(\frac{y}{R}) - \phi(\frac{x}{R})] \lesssim_{N} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N}} \frac{|x - y|^{1/4}}{R^{1/4}} \\ \lesssim 2^{-j/4} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N - 1/4}} \cdot \frac{1}{|y|^{1/4}}.
\endaligned$$ When $|y| \lesssim |x|$ and $|x| > R$, interpolating $$\label{2.36.1}
\aligned
2^{3j} \psi(2^{j}(x - y)) [\phi(\frac{y}{R}) - \phi(\frac{x}{R})] = 2^{3j} \psi(2^{j}(x - y)) \phi(\frac{y}{R}) \lesssim_{N} \frac{2^{3j}}{(1 + 2^{j}|x - y|)^{N}} \frac{R^{1/2}}{|y|^{1/2}}
\endaligned$$ with the fact that $$\label{2.37.1}
\aligned
2^{3j} \psi(2^{j}(x - y)) [\phi(\frac{y}{R}) - \phi(\frac{x}{R})] \lesssim_{N} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N}} \frac{|x - y|^{1/2}}{R^{1/2}} \\ \lesssim 2^{-j/4} \frac{2^{3j}}{(1 + 2^{j} |x - y|)^{N - 1/2}} \cdot \frac{1}{2^{j/2} R^{1/2}},
\endaligned$$ implies $$\label{2.38.1}
2^{3j} \psi(2^{j}(x - y)) [\phi(\frac{y}{R}) - \phi(\frac{x}{R})] \lesssim_{N} 2^{-j/4} \frac{2^{3j}}{(1 + 2^{j}|x - y|)^{N}} \frac{1}{|y|^{1/4}}.$$ The kernel estimates $(\ref{2.34})$, $(\ref{2.35})$, and $(\ref{2.38})$ imply that $$\label{2.39}
\| [\phi(\frac{x}{R}), P_{\geq j}] v \|_{L^{4}(\mathbf{R}^{3})} \lesssim 2^{-j/4} \| \frac{1}{|x|^{1/4}} v \|_{L^{4}(\mathbf{R}^{3})},$$ proving Lemma $\ref{l2.2}$. $\Box$
Global well-posedness
=====================
To prove global well-posedness of $(\ref{1.1})$ using the Fourier truncation method, decompose the initial data into a finite energy piece and a small data piece, $u_{0} = v_{0} + w_{0}$ and $u_{1} = v_{1} + w_{1}$, where $$E(v_{0}, v_{1}) = \frac{1}{2} \int |\nabla v_{0}|^{2} dx + \frac{1}{2} \int |v_{1}|^{2} dx + \frac{1}{4} \int |v_{0}|^{4} dx < \infty,$$ and $$\| w_{0} \|_{\dot{H}^{1/2}} + \| w_{1} \|_{\dot{H}^{-1/2}} \ll 1.$$ A solution $u$ to $(\ref{1.1})$ may then be decomposed into $u = w + v$, where $w$ solves $$\label{3.2}
w_{tt} - \Delta w + w^{3} = 0, \qquad w(0,x) = w_{0}, \qquad w_{t}(0,x) = w_{1},$$ and $v$ solves $$\label{3.3}
v_{tt} - \Delta v + v^{3} + 3 v^{2} w + 3 v w^{2} = 0, \qquad v(0,x) = v_{0}, \qquad v_{t}(0,x) = v_{1}.$$ If $\| w_{0} \|_{\dot{H}^{1/2}} + \| w_{1} \|_{\dot{H}^{-1/2}} < \epsilon$ for some $\epsilon > 0$ sufficiently small, then the small data arguments in $(\ref{6.4})$ implies that $(\ref{3.2})$ is globally well-posed, and moreover, by Theorem $\ref{t6.4}$, $$\label{3.4.1}
\aligned
\| w \|_{L_{t}^{2} L_{x}^{6}(\mathbf{R} \times \mathbf{R}^{3})} + \| |\nabla|^{1/10} w \|_{L_{t}^{2} L_{x}^{5}(\mathbf{R} \times \mathbf{R}^{3})} \\ + \| |\nabla|^{1/6} w \|_{L_{t}^{6} L_{x}^{3}(\mathbf{R} \times \mathbf{R}^{3})} + \| w \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \epsilon, \\ \| w^{3} \|_{L_{t}^{1} L_{x}^{3/2}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \epsilon^{3}.
\endaligned$$ Following $(\ref{1.4})$, let $E(t)$ denote the energy of $v$, where $$\label{3.4}
E(t) = \frac{1}{2} \int v_{t}(t,x)^{2} dx + \frac{1}{2} \int |\nabla v(t,x)|^{2} dx + \frac{1}{4} \int v(t,x)^{4} dx.$$
To prove global well-posedness it suffices to prove that $E(t) < \infty$ for all $t \in \mathbf{R}$.
\[t3.1\] $(\ref{1.1})$ is locally well-posed on the time interval $[-\frac{c}{E(0)}, \frac{c}{E(0)}]$ for some fixed $c > 0$ sufficiently small.
*Proof:* To simplify notation let $I = [-\frac{c}{E(0)}, \frac{c}{E(0)}]$. By Theorem $\ref{t1.1}$, $(\ref{1.1})$ has a solution for initial data $(v_{0}, v_{1})$, and moreover, by conservation of energy, $$\label{3.6}
\| v \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})}^{4} \lesssim |I| E(0) \leq c.$$ Therefore, for $c > 0$ sufficiently small, independent of $E(0)$, $$\label{3.7}
\tilde{u}_{tt} - \Delta \tilde{u} + v^{3} + w^{3} = 0, \qquad \tilde{u}(0,x) = u_{0}, \qquad \tilde{u}_{t}(0,x) = u_{1},$$ has a solution satisfying $\| \tilde{u} \|_{L_{t,x}^{4}(I \times \mathbf{R}^{3})} \ll 1$. Applying the perturbation lemma (Lemma $\ref{l6.2})$ completes the proof of Theorem $\ref{t3.1}$. $\Box$
\[t3.2\] Equation $(\ref{1.1})$ is globally well-posed for radial $(u_{0}, u_{1}) \in \dot{H}^{1/2}(\mathbf{R}^{3}) \times \dot{H}^{-1/2}(\mathbf{R}^{3})$.
*Proof:* To compute the time derivative of $E(t)$, by H[ö]{}lder’s inequality, $$\label{3.8}
\frac{d}{dt} E(t) = -3 \int v_{t} v^{2} w - 3 \int v_{t} v w^{2} \lesssim \| v_{t} \|_{L^{2}} \| v \|_{L^{6}}^{2} \| w \|_{L^{6}} + \| v_{t} \|_{L^{2}} \| v \|_{L^{6}} \| w \|_{L^{6}}^{2}.$$ Therefore, by the Cauchy–Schwartz inequality, $$\label{3.10}
|\frac{d}{dt} E(t)| \lesssim E(t)^{2} + \| w \|_{L^{6}}^{2} E(t).$$ If only the second term on the right hand side of $(\ref{3.10})$ were present, global boundedness of $E(t)$ would be an easy consequence of $(\ref{3.4.1})$ and Gronwall’s inequality. However, the bound $|\frac{d}{dt} E(t)| \lesssim E(t)^{2}$ is not enough to exclude blow up in finite time. Instead, we will use a modification of $E(t)$, $\mathcal E(t)$, which has much better global derivative bounds, and satisfies $\mathcal E(t) \sim E(t)$.
To simplify notation, rescale by $(\ref{1.2})$ so that $$\label{3.1}
\| P_{\geq 1} u_{0} \|_{\dot{H}^{1/2}(\mathbf{R}^{3})} + \| P_{\geq 1} u_{1} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})} < \epsilon,$$ and then let $v_{0} = P_{\leq 1} u_{0}$ and $v_{1} = P_{\leq 1} u_{1}$.
Following $(\ref{2.36})$, $(\ref{2.37})$, and $(\ref{2.38})$, let $$\label{3.1.1}
\aligned
M_{1}(t) = c_{1} \int v_{t} \frac{x}{|x|} \cdot \nabla v + c_{1} \int v_{t} \frac{1}{|x|} v, \\
M_{2}(t) = \frac{c_{2}}{R^{3}} \int_{|y| \leq 2R} \int v_{t} \frac{(x - y)}{|x - y|} \cdot \nabla v dx dy \\ + c_{2} \int_{|y| \leq 2R} \int v_{t} \frac{1}{|x - y|} v dx dy, \\
M_{3}(t) = \frac{c_{3}}{R} \int v_{t} \chi(\frac{|x|}{R}) x \cdot \nabla v + \frac{c_{3}}{R} \int v_{t} \chi(\frac{|x|}{R}) v,
\endaligned$$ where $c_{1}, c_{2}, c_{3} > 0$ are small constants and let $$\mathcal E(t) = E(t) + M_{1}(t) + M_{2}(t) + M_{3}(t) + \int v^{3} w dx.$$ Then by $(\ref{2.29})$, $(\ref{2.31})$, and $(\ref{2.33.1})$, and the Sobolev embedding theorem, which implies $$\int v^{3} w dx \lesssim \| v \|_{L^{6}} \| w \|_{L^{3}} \| v \|_{L^{4}}^{2} \lesssim \epsilon E(t),$$ we have $$\label{3.1.2}
\mathcal E(t) \sim E(t).$$ Then by $(\ref{2.36})$, $(\ref{2.37})$, $(\ref{2.38})$, and $(\ref{3.8})$, $$\label{3.1.3}
\aligned
\frac{d}{dt} \mathcal E(t) \leq -c_{1} \pi v(t,0)^{2} - \frac{c_{2} \pi}{8 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ - \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} - \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ -\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{2R} \int v^{2} \Delta \chi(\frac{|x|}{R}) \\ - \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} + \frac{c_{3}}{4R} \int \chi'(\frac{|x|}{R}) \frac{|x|}{R} v^{4} \\
+ \frac{d}{dt} \int v^{3} w dx + \int F v_{t} + c_{1} \int F \frac{x}{|x|} \cdot \nabla v + c_{1} \int F \frac{1}{|x|} v \\ + \frac{c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{1}{|x - y|} v \\ + \frac{c_{3}}{R} \int F \chi(\frac{|x|}{R}) x \cdot \nabla v + \frac{c_{3}}{R} \int F \chi(\frac{|x|}{R}) v,
\endaligned$$ where $F = -3 v^{2} w - 3 v w^{2}$.
By the support properties of $\Delta \chi(\frac{|x|}{R})$, it is possible to choose $c_{2}, c_{3} > 0$ such that $$\label{3.1.4}
- \frac{c_{2} \pi}{8 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} + \frac{c_{3}}{2R} \int v^{2} \Delta \chi(\frac{|x|}{R}) \leq -\frac{c_{2}}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2}.$$ Also, since $\chi'(\frac{|x|}{R}) \leq 0$, $$\label{3.1.5}
\frac{c_{3}}{4R} \int \chi'(\frac{|x|}{R}) \frac{|x|}{R} v^{4} \leq 0.$$ Therefore, $$\label{3.1.6}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
\leq \frac{d}{dt} \int v^{3} w + \int F v_{t} + c_{1} \int F \frac{x}{|x|} \cdot \nabla v + c_{1} \int F \frac{1}{|x|} v \\ + \frac{c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{1}{|x - y|} v \\ + \frac{c_{3}}{R} \int F \chi(\frac{|x|}{R}) x \cdot \nabla v + \frac{c_{3}}{R} \int F \chi(\frac{|x|}{R}) v.
\endaligned$$
By Hardy’s inequality and the Sobolev embedding theorem, $$\label{3.26}
\int v^{2} w \frac{1}{|x|} v dx \lesssim (\int \frac{1}{|x|} v^{4} dx)^{1/2} \| \frac{1}{|x|^{1/2}} v \|_{L^{3}} \| w(t) \|_{L^{6}} \lesssim \delta (\int \frac{1}{|x|} v^{4} dx) + \frac{1}{\delta} E(t) \| w(t) \|_{L^{6}}^{2}.$$ Also by H[ö]{}lder’s inequality and Hardy’s inequality, $$\label{3.26.1}
\int v w^{2} \frac{1}{|x|} v \lesssim \| w \|_{L^{6}}^{2} \| \nabla v \|_{L^{2}} \| v \|_{L^{6}} \lesssim E(t) \| w \|_{L^{6}}^{2}.$$ Therefore, $$\label{3.26.2}
\int F \frac{1}{|x|} v dx \lesssim \delta (\int \frac{1}{|x|} v^{4} dx) + \frac{1}{\delta} E(t) \| w(t) \|_{L^{6}}^{2}.$$ Because $\chi(|x|) \lesssim \frac{1}{|x|}$, the same argument also implies $$\label{3.26.3}
\frac{1}{8 R^{3}} \int_{|y| \leq 2R} \int F \frac{1}{|x - y|} v + \frac{1}{R} \int F \chi(\frac{|x|}{R}) v \lesssim \delta (\int \frac{1}{|x|} v^{4} dx) + \frac{1}{\delta} E(t) \| w(t) \|_{L^{6}}^{2}.$$ Therefore, $$\label{3.26.4}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
- \frac{d}{dt} \int v^{3} w - \int F v_{t} - c_{1} \int F \frac{x}{|x|} \cdot \nabla v \\ - \frac{c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{(x - y)}{|x - y|} \cdot \nabla v - \frac{c_{3}}{R} \int F \chi(\frac{|x|}{R}) x \cdot \nabla v \\ \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2}.
\endaligned$$ Next, splitting $F = -3v^{2} w - 3v w^{2}$, the Sobolev embedding theorem implies that $$\label{3.11}
-3 \int v_{t} v w^{2} dx \lesssim \| w \|_{L_{x}^{6}(\mathbf{R}^{3})}^{2} \| v \|_{L_{x}^{6}(\mathbf{R}^{3})} \| v_{t} \|_{L_{x}^{2}(\mathbf{R}^{3})} \lesssim E(t) \| w(t) \|_{L_{x}^{6}(\mathbf{R}^{3})}^{2}.$$ Therefore, $$\label{3.26.5}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
- \frac{d}{dt} \int v^{3} w + 3 \int v^{2} w v_{t} - c_{1} \int F \frac{x}{|x|} \cdot \nabla v \\ - \frac{c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{(x - y)}{|x - y|} \cdot \nabla v - \frac{c_{3}}{R} \int F \chi(\frac{|x|}{R}) x \cdot \nabla v \\ \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2}.
\endaligned$$ Analysis of the other terms involving $-3 v^{2} w$ is similar. $$\label{3.25}
\int v w^{2} \frac{x}{|x|} \cdot \nabla v \lesssim \| w \|_{L^{6}}^{2} \| \nabla v \|_{L^{2}} \| v \|_{L^{6}} \lesssim E(t) \| w \|_{L^{6}}^{2},$$ and $$\label{3.25.1}
\frac{1}{8R^{3}} \int_{|y| \leq 2R} \int F \frac{(x - y)}{|x - y|} \cdot \nabla v \lesssim E(t) \| w \|_{L^{6}}^{2}.$$ Since $\chi(\frac{|x|}{R}) \frac{x}{R}$ is also uniformly bounded, $$\label{3.25.2}
\frac{1}{R} \int v w^{2} \chi(\frac{|x|}{R}) x \cdot \nabla v \lesssim E(t) \| w \|_{L^{6}}^{2}.$$ Therefore, $$\label{3.25.3}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
- \frac{d}{dt} \int v^{3} w + 3 \int v^{2} w v_{t} + 3c_{1} \int v^{2} w \frac{x}{|x|} \cdot \nabla v \\ + \frac{3 c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int v^{2} w \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{3 c_{3}}{R} \int v^{2} w \chi(\frac{|x|}{R}) x \cdot \nabla v \\ \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2}.
\endaligned$$
Next, by the product rule, $$\label{3.12}
3 \int v_{t} v^{2} w dx - \frac{d}{dt} \int v^{3} w dx = -\int v^{3} \partial_{t} w dx.$$ Making a Littlewood–Paley decomposition, $$\label{3.13}
\aligned
\int v^{3} w_{t} dx = \sum_{j} \int v^{3} \partial_{t} w_{j} dx.
%= \sum_{j} \int (P_{\geq j - 3} v)^{3} \partial_{t} w_{j} dx \\ + 3 \sum_{j} \int (P_{\geq j - 3} v)^{2} (P_{\leq j - 3} v) \partial_{t} w_{j} dx \\ + 3 \sum_{j} \int (P_{\geq j - 3} v)(P_{\leq j - 3} v)^{2} \partial_{t} w_{j} dx.
\endaligned$$ By Fourier support properties, $$\label{3.13.1}
\aligned
\int v^{3} \partial_{t} w_{j} dx = \int (v^{3} - (P_{\leq j - 3} v)^{3}) (\partial_{t} w_{j}) dx \\ = \int (P_{\geq j - 3} v)^{3} (\partial_{t} w_{j}) dx + 3 \int (P_{\geq j - 3} v)(P_{\leq j - 3} v) v \cdot \partial_{t} w_{j} dx.
\endaligned$$ Using Lemma $\ref{l3.1}$, $$\label{3.17}
\aligned
\sum_{j} \int_{|x| \geq \frac{R}{2}} [v^{3} - (P_{\leq j - 3} v)^{3}] (\partial_{t} w_{j}) dx \\ \lesssim \sum_{j} (\| \frac{1}{|x|^{1/4}} |P_{\leq j} v| \|_{L^{4}} + \| \frac{1}{|x|^{1/4}} |P_{\geq j} v| \|_{L^{4}}) \| P_{\geq j - 3} v \|_{L_{x}^{2}} \| |x|^{1/2} \partial_{t} w_{j} \|_{L_{x}^{\infty}(|x| \geq \frac{R}{2})} \\ \lesssim (\int \frac{1}{|x|} v^{4})^{1/2} \sum_{j} \| P_{\geq j - 3} v \|_{L_{x}^{2}} \| |x|^{1/2} \partial_{t} w_{j} \|_{L_{x}^{\infty}(|x| \geq \frac{R}{2})}.
\endaligned$$ By the Cauchy–Schwartz inequality, $$\label{3.15}
\aligned
(\ref{3.17}) \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} (\sum_{j} \| P_{\geq j - 3} v \|_{L^{2}} \| \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})})^{2}.
\endaligned$$ By Bernstein’s inequality and Young’s inequality, $$\label{3.21}
\aligned
(\sum_{j} \| P_{\geq j - 3} v \|_{L^{2}} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})})^{2} \\ \leq (\sum_{j} (\sum_{k \geq j - 3} 2^{k} 2^{j - k} \| P_{k} v \|_{L^{2}}) \cdot 2^{-j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})})^{2} \\
\lesssim (\sum_{k} 2^{2k} \| P_{k} v \|_{L^{2}}^{2}) (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}) \\
\lesssim E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}).
\endaligned$$ Therefore, $$\label{3.22}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
- \sum_{j} \int_{|x| \leq \frac{R}{2}} (v^{3} - (P_{\leq j - 3} v)^{3}) \cdot \partial_{t} w_{j} + 3c_{1} \int v^{2} w \frac{x}{|x|} \cdot \nabla v \\ + \frac{3 c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int v^{2} w \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{3 c_{3}}{R} \int v^{2} w \chi(\frac{|x|}{R}) x \cdot \nabla v \\ \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2} + \frac{1}{\delta} E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}).
\endaligned$$ By $(\ref{2.31})$, H[ö]{}lder’s inequality, and the Cauchy–Schwartz inequality, $$\label{3.24}
\aligned
\sum_{j} \int_{|x| \leq \frac{R}{2}} v (P_{\leq j - 3} v)(P_{\geq j - 3} v) \cdot \partial_{t} w_{j} dx \\ \leq \sum_{j} \| \partial_{t} w_{j} \|_{L^{6}} \| v \|_{L^{6}(|x| \leq \frac{R}{2})} \| P_{\geq j - 3} v \|_{L^{2}} \| P_{\leq j - 3} v \|_{L^{6}} \\ \lesssim \delta R E(t) (\frac{1}{R} \int_{|x| \leq R} |\nabla v|^{2} + \frac{1}{R^{3}} \int_{|x| \leq R} v^{2}) + \frac{1}{\delta} (\sum_{j} \| P_{\geq j - 3} v \|_{L^{2}} \| \partial_{t} w_{j} \|_{L^{6}})^{2}.
\endaligned$$ Following $(\ref{3.21})$, $$\label{3.24.1}
(\sum_{j} \| P_{\geq j - 3} v \|_{L^{2}} \| \partial_{t} w_{j} \|_{L^{6}})^{2} \lesssim E(t) (\sum_{j} 2^{-2j} \| \partial_{t} w_{j} \|_{L^{6}}^{2}).$$
Next, following $(\ref{3.21})$, by the Cauchy–Schwartz inequality, and Lemma $\ref{l2.2}$, $$\label{3.28}
\aligned
\sum_{j} \int_{|x| \leq \frac{R}{2}} (P_{\geq j - 3} v)^{3} \cdot \partial_{t} w_{j} dx \\ \lesssim \sum_{j} \| P_{\geq j - 3} v \|_{L^{4}(|x| \leq \frac{R}{2})}^{2} \| P_{\geq j - 3} v \|_{L^{3}(\mathbf{R}^{3})} \| \partial_{t} w_{j} \|_{L^{6}(\mathbf{R}^{3})} \\
\lesssim \delta R \| \nabla v(t) \|_{L^{2}}^{2} [\frac{1}{R} \| \nabla v \|_{L^{2}(|x| \leq R)}^{2} + \frac{1}{R^{3}} \| v \|_{L^{2}(|x| \leq R)}^{2}] + \delta (\int \frac{1}{|x|} v^{4}) \\ + \frac{1}{\delta} (\sum_{j} 2^{-j/2} \| P_{\geq j - 3} v \|_{L^{3}} \| \partial_{t} w_{j} \|_{L^{6}})^{2} \\
\lesssim \delta R E(t) [\frac{1}{R} \| \nabla v \|_{L^{2}(|x| \leq R)}^{2} + \frac{1}{R^{3}} \| v \|_{L^{2}(|x| \leq R)}^{2}] \\ + \delta (\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} E(t) [\sum_{j} 2^{-2j} \| \partial_{t} w_{j} \|_{L^{6}}^{2}].
\endaligned$$ Therefore, $$\label{3.26}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
+ 3c_{1} \int v^{2} w \frac{x}{|x|} \cdot \nabla v \\ + \frac{3 c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int v^{2} w \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{3 c_{3}}{R} \int v^{2} w \chi(\frac{|x|}{R}) x \cdot \nabla v \\ \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \delta R E(t) [\frac{1}{R} \| \nabla v \|_{L^{2}(|x| \leq R)}^{2} + \frac{1}{R^{3}} \| v \|_{L^{2}(|x| \leq R)}^{2}] \\ + \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2} + \frac{1}{\delta} E(t)(\sum_{j} 2^{-2j} \| \partial_{t} w_{j} \|_{L^{6}}^{2}) \\ + \frac{1}{\delta} E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}).
\endaligned$$ Integrating by parts, $$\label{3.30}
3c_{1} \int v^{2} w \frac{x}{|x|} \cdot \nabla v dx = -2c_{1} \int \frac{1}{|x|} v^{3} w - c_{1} \int v^{3} (\nabla w) \cdot \frac{x}{|x|}.$$ Following $(\ref{3.26.2})$, $$\label{3.31}
-2c_{1} \int \frac{1}{|x|} v^{3} w dx \lesssim \frac{1}{\delta} E(t) \| w(t) \|_{L_{x}^{6}}^{2} + \delta (\int \frac{1}{|x|} v^{4} dx).$$ The term $$-c_{1} \int (v^{3} - (P_{\leq j - 3} v)^{3}) (\nabla w_{j}) \cdot \frac{x}{|x|} dx$$ may be estimated using exactly the same arguments as in the estimates for $(\ref{3.17})$.
Now, the Fourier support of $(\nabla w_{j})(P_{\leq j - 3} v)^{3}$ is $|\xi| \sim 2^{j}$, so integrating by parts, $$\label{3.32}
\aligned
c \int (P_{\leq j - 3} v)^{3} (\nabla w_{j}) \cdot \frac{x}{|x|} dx = \int \frac{x_{l} x_{k}}{|x|^{3}} \frac{\partial_{k}}{\Delta} (P_{\leq j - 3} v)^{3} (\partial_{l} w_{j}) \\
\lesssim 2^{-j} \| \frac{1}{|x|^{1/4}} P_{\leq j - 3} v \|_{L^{4}}^{2} \| \frac{1}{|x|^{1/2}} P_{\leq j - 3} v \|_{L^{10/3}} \| \partial_{k} w_{j} \|_{L^{5}}.
\endaligned$$ Then by the Cauchy–Schwartz inequality, $$\label{3.33}
\aligned
\sum_{j} c \int (P_{\leq j - 3} v)^{3} (\nabla w_{j}) \cdot \frac{x}{|x|} dx \\ \lesssim \delta (\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} (\sum_{j} 2^{-j} \| \frac{1}{|x|^{1/2}} P_{\leq j - 3} v \|_{L^{10/3}} \| \nabla w_{j} \|_{L^{5}})^{2},
\endaligned$$ and then by Bernstein’s inequality, $$\label{3.34}
\lesssim \delta (\int \frac{1}{|x|} v^{4}) + \frac{1}{\delta} E(t) (\sum_{j} 2^{j/5} \| w_{j} \|_{L^{5}}^{2}).$$ Therefore, $$\label{3.35}
\aligned
\frac{d}{dt} \mathcal E(t) + c_{1} \pi v(t,0)^{2} + \frac{c_{2} \pi}{16 R^{3}} \int_{|y| \leq 2R} v(t,y)^{2} \\ + \frac{c_{1}}{2} \int \frac{1}{|x|} v^{4} + \frac{c_{2}}{8 R^{3}} \int_{|y| \leq 2R} \int \frac{1}{|x - y|} v^{4} \\ +\frac{c_{3}}{2R} \int \phi(\frac{|x|}{R}) [v_{t}^{2} + |\nabla v|^{2}] + \frac{c_{3}}{4R} \int \chi(\frac{|x|}{R}) v^{4} \\
+ \frac{3 c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int v^{2} w \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{3 c_{3}}{R} \int v^{2} w \chi(\frac{|x|}{R}) x \cdot \nabla v \\ \lesssim \delta(\int \frac{1}{|x|} v^{4}) + \delta R E(t) [\frac{1}{R} \| \nabla v \|_{L^{2}(|x| \leq R)}^{2} + \frac{1}{R^{3}} \| v \|_{L^{2}(|x| \leq R)}^{2}] \\ + \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2} + \frac{1}{\delta} E(t) (\sum_{j} 2^{j/5} \| w_{j} \|_{L^{5}}^{2})+ \frac{1}{\delta} E(t)(\sum_{j} 2^{-2j} \| \partial_{t} w_{j} \|_{L^{6}}^{2}) \\ + \frac{1}{\delta} E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}) \\
\frac{1}{\delta} E(t)(\sum_{j} 2^{-2j} \| \nabla w_{j} \|_{L^{6}}^{2}) + \frac{1}{\delta} E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \nabla w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}) .
\endaligned$$ Like $\frac{x}{|x|}$ the potentials $$\label{3.36}
a(x) = \chi(\frac{2 |x|}{R}) \frac{x}{R}, \qquad \text{and} \qquad a(x) = \int_{|y| \leq 2R} \frac{(x - y)}{|x - y|} dy$$ are also bounded, radial functions satisfying $$\label{3.37}
\nabla \cdot a(x) \lesssim \frac{1}{|x|},$$ and therefore, the analysis of $$\label{3.38}
+ \frac{3 c_{2}}{8R^{3}} \int_{|y| \leq 2R} \int v^{2} w \frac{(x - y)}{|x - y|} \cdot \nabla v + \frac{3 c_{3}}{R} \int v^{2} w \chi(\frac{|x|}{R}) x \cdot \nabla v$$ may be carried out in much the same manner as $$\label{3.39}
\int v^{2} w \frac{x}{|x|} \cdot \nabla v.$$ Therefore, $$\label{3.40}
\aligned
\frac{d}{dt} \mathcal E(t)
\lesssim \frac{1}{\delta} E(t) \| w \|_{L^{6}}^{2} + \frac{1}{\delta} E(t) (\sum_{j} 2^{j/5} \| w_{j} \|_{L^{5}}^{2})+ \frac{1}{\delta} E(t)(\sum_{j} 2^{-2j} \| \partial_{t} w_{j} \|_{L^{6}}^{2}) \\ + \frac{1}{\delta} E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}) \\
\frac{1}{\delta} E(t)(\sum_{j} 2^{-2j} \| \nabla w_{j} \|_{L^{6}}^{2}) + \frac{1}{\delta} E(t) (\sum_{j} 2^{-2j} \| |x|^{1/2} \nabla w_{j} \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}) .
\endaligned$$ Since $E(t) \sim \mathcal E(t)$, $$\label{3.41}
\aligned
\frac{d}{dt} \ln(\mathcal E(t))
\lesssim \frac{1}{\delta} \| w \|_{L^{6}}^{2} + \frac{1}{\delta} (\sum_{j} 2^{j/5} \| w_{j} \|_{L^{5}}^{2})+ \frac{1}{\delta} (\sum_{j} 2^{-2j} \| \partial_{t} w_{j} \|_{L^{6}}^{2}) \\ + \frac{1}{\delta} (\sum_{j} 2^{-2j} \| |x|^{1/2} \partial_{t} w_{j} \|_{L^{\infty}(|x| \geq \frac{1}{2 \mathcal E(T)})}^{2}) \\
\frac{1}{\delta} (\sum_{j} 2^{-2j} \| \nabla w_{j} \|_{L^{6}}^{2}) + \frac{1}{\delta} (\sum_{j} 2^{-2j} \| |x|^{1/2} \nabla w_{j} \|_{L^{\infty}(|x| \geq \frac{1}{2 \mathcal E(T)})}^{2}) .
\endaligned$$ Integrating in time and combining $(\ref{2.18.1})$ with $(\ref{3.4.1})$, $$\label{3.45}
\aligned
\ln(\mathcal E(T)) - \ln(\mathcal E(0)) \lesssim \frac{\epsilon^{2}}{\delta} \ln(T) + \frac{\epsilon^{2}}{\delta} \ln(\mathcal E(T)) + \epsilon.
\endaligned$$ Doing some algebra, $$\label{3.46}
\aligned
\ln(\mathcal E(T)) \leq (\frac{1}{1 - \frac{C \epsilon^{2}}{\delta}}) \ln(\mathcal E(0)) + \frac{C \epsilon^{2}}{\delta(1 - \frac{C \epsilon^{2}}{\delta})} \ln(T) + \frac{C \epsilon}{(1 - \frac{C \epsilon^{2}}{\delta})}.
\endaligned$$ This proves that for any $t$, $E(t) \sim \mathcal E(t) \lesssim (1 + t)^{C \epsilon}$. $\Box$
Proof of scattering
===================
By time reversal symmetry, it suffices to prove
\[t4.0\] For any radial initial data $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$, the solution to $(\ref{1.1})$ scatters forward in time.
This theorem is proved using hyperbolic coordinates. By the dominated convergence theorem, there exists $R(\epsilon) < \infty$ such that $$\label{4.1}
\| S(t)(u_{0}, u_{1}) \|_{L_{t,x}^{4}(|x| \geq R + |t|)} < \epsilon.$$ Then by finite propagation speed and small data arguments, if $u$ is a global solution to $(\ref{1.1})$, then $$\label{4.2}
\| u \|_{L_{t,x}^{4}(|x| \geq R + |t|)} \lesssim \epsilon.$$ Rescaling, $(u_{0}(x), u_{1}(x)) \mapsto (2 R u_{0}(2 R x), (2 R)^{2} u_{1}(2R x))$, $$\label{4.3}
\| u \|_{L_{t,x}^{4}(|x| \geq \frac{1}{2} + |t|)} \lesssim \epsilon.$$
The quantity $$\label{4.4}
\| u \|_{L_{t,x}^{4}([0, \infty) \times \{|x| \leq \frac{1}{2} + t \})}$$ is estimated using hyperbolic coordinates, which combined with $(\ref{4.3})$ proves $$\label{4.5}
\| u \|_{L_{t,x}^{4}([0, \infty) \times \mathbf{R}^{3})} < \infty.$$
Make a time translation so that $$\label{4.6}
u(1, x) = 2 R u_{0}(2 R x), \qquad u_{t}(1,x) = (2 R)^{2} u_{1}(2 Rx).$$ After time translation, $(\ref{4.3})$ implies $$\label{4.7}
\| u \|_{L_{t,x}^{4}([1, \infty) \times \{ |x| \geq t - \frac{1}{2} \})} \lesssim \epsilon.$$ Switching to hyperbolic coordinates for the region inside the cone, let $$\label{4.8}
\tilde{u}(\tau, s) = \frac{e^{\tau} \sinh s}{s} u(e^{\tau} \cosh s, e^{\tau} \sinh s).$$ Then making a change of variables, $$\label{4.9}
\aligned
\int_{0}^{\infty} \int_{0}^{\infty} \tilde{u}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} s^{2} ds d\tau \\ = \int_{0}^{\infty} \int_{0}^{\infty} u(e^{\tau} \cosh s, e^{\tau} \sinh s)^{4} e^{2 \tau} \sinh^{2} e^{2 \tau} ds d\tau \\ = \int_{1}^{\infty} \int_{t^{2} - r^{2} \geq 1} u(t,r)^{4} r^{2} dr dt \geq \int_{1}^{\infty} \int_{t \geq r} u(t,r)^{4} r^{2} dr dt.
\endaligned$$ Therefore, $$\label{4.10}
\int_{0}^{\infty} \int_{0}^{\infty} \tilde{u}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} s^{2} ds d\tau < \infty,$$ combined with $(\ref{4.7})$ implies $$\label{4.11}
\| u \|_{L_{t,x}^{4}([1, \infty) \times \mathbf{R}^{3})} < \infty,$$ which after undoing time translation, implies $(\ref{4.5})$. Also, by direct computation, $$\label{4.12}
(\partial_{\tau \tau} - \partial_{ss} - \frac{2}{s} \partial_{s}) \tilde{u}(\tau, s) + (\frac{s}{\sinh s})^{2} \tilde{u}^{3} = 0,$$ with $$\label{4.13}
\tilde{u}|_{\tau = 0} = \frac{e^{\tau} \sinh s}{s} u(e^{\tau} \cosh s, e^{\tau} \sinh s)|_{\tau = 0},$$ and $$\label{4.14}
\tilde{u}_{\tau}|_{\tau = 0} = \partial_{\tau} (\frac{e^{\tau} \sinh s}{s} u(e^{\tau} \cosh s, e^{\tau} \sinh s))|_{\tau = 0}.$$ A solution to $(\ref{4.12})$ has the conserved energy, $$\label{4.15}
E(\tau) = \frac{1}{2} \| \tilde{u}_{\tau} \|_{L^{2}}^{2} + \frac{1}{2} \| \tilde{u}_{s} \|_{L^{2}}^{2} + \frac{1}{4} \int \tilde{u}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} s^{2} ds.$$ For now, assume the following lemma.
\[l4.1\] There exists a decomposition $$\label{4.16}
\tilde{u}|_{\tau = 0} = \frac{e^{\tau} \sinh s}{s} u(e^{\tau} \cosh s, e^{\tau} \sinh s)|_{\tau = 0} = \tilde{v}_{0} + \tilde{w}_{0}$$ and $$\label{4.17}
\tilde{u}_{\tau}|_{\tau = 0} = \partial_{\tau} (\frac{e^{\tau} \sinh s}{s} u(e^{\tau} \cosh s, e^{\tau} \sinh s))|_{\tau = 0} = \tilde{v}_{1} + \tilde{w}_{1},$$ with $$\label{4.18}
\frac{1}{2} \int |\partial_{s} \tilde{v}_{0}|^{2} s^{2} + \frac{1}{2} \int |\tilde{v}_{1}|^{2} s^{2} + \frac{1}{4} \int \tilde{v}_{0}^{4} (\frac{s}{\sinh s})^{2} s^{2} < \infty,$$ and $$\label{4.19}
\| \tilde{w}_{0} \|_{\dot{H}^{1/2}} + \| w_{1} \|_{\dot{H}^{-1/2}} \leq \epsilon.$$
**Remark:** Following $(\ref{3.1})$, it is enough to prove $\tilde{u}_{0} \in \dot{H}^{1} + \dot{H}^{1/2}$ and $\tilde{u}_{1} \in L^{2} + \dot{H}^{-1/2}$ and then truncate in frequency.
*Proof of Theorem $\ref{t4.0}$:* Let $\tilde{v}$ and $\tilde{w}$ solve $$\label{4.19.2}
(\partial_{\tau \tau} - \Delta) \tilde{w} + (\frac{s}{\sinh s})^{2} \tilde{w}^{3} = 0, \qquad \tilde{w}(0,y) = \tilde{w}_{0}, \qquad \tilde{w}_{\tau}(0,y) = \tilde{w}_{1},$$ and $$\label{4.19.1}
(\partial_{\tau \tau} - \Delta) \tilde{v} + (\frac{s}{\sinh s})^{2} [\tilde{v}^{3} + 3 \tilde{v}^{2} \tilde{w} + 3 \tilde{v} \tilde{w}^{2}] = 0, \qquad \tilde{v}(0,y) = \tilde{v}_{0}, \qquad \tilde{v}_{\tau}(0,y) = \tilde{v}_{1}.$$ Define the energy, $$\label{4.18}
E(\tau) = \frac{1}{2} \int |\partial_{s} \tilde{v}|^{2} s^{2} + \frac{1}{2} \int |\partial_{\tau} \tilde{v}|^{2} s^{2} + \frac{1}{4} \int \tilde{v}^{4} (\frac{s}{\sinh s})^{2} s^{2}.$$ As in the proof of global well-posedness, define the quantity $$\label{4.19}
\mathcal E(\tau) = E(\tau) + M(\tau) + \int \tilde{v}^{3} \tilde{w} (\frac{s}{\sinh s})^{2} s^{2} ds,$$ where $$\label{4.20}
M(\tau) = c \int \tilde{v}_{\tau} \tilde{v}_{s} s^{2} ds + c \int \tilde{v}_{\tau} \tilde{v} ds.$$ Then by direct computation, making a slight modification of $(\ref{2.24})$ and $(\ref{2.36})$, $$\label{4.21}
\aligned
\frac{d}{d\tau} M(\tau) = -\frac{1}{2} \tilde{v}(\tau, 0)^{2} - \frac{1}{2} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds \\
-3 \int \tilde{v}^{2} \tilde{v}_{s} \tilde{w} (\frac{s}{\sinh s})^{2} s^{2} ds - 3 \int \tilde{v} \tilde{v}_{s} \tilde{w}^{2} (\frac{s}{\sinh s})^{2} s^{2} ds \\
- 3 \int \tilde{v}^{3} \tilde{w} (\frac{s}{\sinh s})^{2} s ds - 3 \int \tilde{v}^{2} \tilde{w}^{2} (\frac{s}{\sinh s})^{2} s ds.
\endaligned$$ Therefore, $$\label{4.22}
\aligned
\frac{d}{d\tau} \mathcal E(\tau) = -\frac{c}{2} \tilde{v}(\tau, 0)^{2} - \frac{c}{2} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds \\
-3c \int \tilde{v}^{2} \tilde{v}_{s} \tilde{w} (\frac{s}{\sinh s})^{2} s^{2} ds - 3c \int \tilde{v} \tilde{v}_{s} \tilde{w}^{2} (\frac{s}{\sinh s})^{2} s^{2} ds \\
- 3c \int \tilde{v}^{3} \tilde{w} (\frac{s}{\sinh s})^{2} s ds - 3c \int \tilde{v}^{2} \tilde{w}^{2} (\frac{s}{\sinh s})^{2} s ds \\
-3 \int (\frac{s}{\sinh s})^{2} \tilde{v}^{3} \tilde{w}_{\tau} s ds - 3 \int (\frac{s}{\sinh s})^{2} \tilde{v} \tilde{v}_{\tau} \tilde{w}^{2} s ds.
\endaligned$$ By Hardy’s inequality, $$\label{4.23}
\aligned
-3c \int \tilde{v} \tilde{v}_{s} \tilde{w}^{2} (\frac{s}{\sinh s})^{2} s^{2} ds - 3c \int \tilde{v}^{2} \tilde{w}^{2} (\frac{s}{\sinh s})^{2} s ds\\ - 3 \int (\frac{s}{\sinh s})^{2} \tilde{v} \tilde{v}_{\tau} \tilde{w}^{2} s^{2} ds \lesssim E(\tau) \| \tilde{w} \|_{L^{6}}^{2}.
\endaligned$$ Also, by Hardy’s inequality and the Cauchy–Schwartz inequality, $$\label{4.24.2}
\aligned
\int \tilde{v}^{3} \tilde{w} (\frac{s}{\sinh s})^{2} s ds \lesssim \delta(\int \tilde{v}^{4} (\frac{\cosh s}{\sinh s}) (\frac{s}{\sinh s})^{2} s^{2} ds) + \frac{1}{\delta} \| \tilde{w} \|_{L^{6}}^{2} \| \frac{1}{|x|^{1/2}} \tilde{v} \|_{L^{3}}^{2} \\ \lesssim \delta(\int (\frac{s}{\sinh s})^{2} \frac{\cosh s}{\sinh s} \tilde{v}^{4} s^{2} ds) + \frac{1}{\delta} \| \tilde{w} \|_{L^{6}}^{2} E(\tau).
\endaligned$$ Therefore, $$\label{4.24}
\aligned
\frac{d}{d\tau} \mathcal E(\tau) + \frac{c}{2} \tilde{v}(\tau, 0)^{2} + \frac{c}{2} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds \\
+ 3c \int \tilde{v}^{2} \tilde{w} (\frac{s}{\sinh s})^{2} \tilde{v}_{s} s^{2} ds + 3 \int (\frac{s}{\sinh s})^{2} \tilde{v}^{3} \tilde{w}_{\tau} s^{2} ds \\ \lesssim \frac{1}{\delta} E(\tau) \| w \|_{L^{6}}^{2} + \delta(\int \tilde{v}^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds).
\endaligned$$ Integrating by parts, $$\label{4.25}
3c \int \tilde{v}^{2} \tilde{v}_{s} \tilde{w} (\frac{s}{\sinh s})^{2} s^{2} ds = -c \int \tilde{v}^{3} \tilde{w}_{s} (\frac{s}{\sinh s})^{2} s^{2} ds - c \int \tilde{v}^{3} \tilde{w} \cdot \partial_{s}(\frac{s^{4}}{(\sinh s)^{2}}) ds.$$ Since $$\label{4.26}
\partial_{s}(\frac{s^{4}}{(\sinh s)^{2}}) \lesssim s,$$ by $(\ref{4.24.2})$, $$\label{4.27}
c \int \tilde{v}^{3} \tilde{w} \cdot \partial_{s} (\frac{s^{4}}{(\sinh s)^{2}}) \lesssim \delta(\int \tilde{v}^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds) + \frac{1}{\delta} \| \tilde{w} \|_{L^{6}}^{2} E(\tau).$$ Next, following $(\ref{3.13.1})$–$(\ref{3.15})$ and using Lemma $\ref{l3.1}$, $$\label{4.28}
\aligned
-c \sum_{j} \int_{s \geq \frac{R}{2}} [\tilde{v}^{3} - (P_{\leq j - 3} \tilde{v})^{3}] (\partial_{s} \tilde{w}_{j}) \cdot (\frac{s}{\sinh s})^{2} s^{2} ds \\ + 3 \sum_{j} \int_{s \geq \frac{R}{2}} [\tilde{v}^{3} - (P_{\leq j - 3} \tilde{v})^{3}] (\partial_{\tau} \tilde{w}_{j}) (\frac{s}{\sinh s})^{2} s^{2} ds \\
\lesssim \delta (\int (\frac{\cosh s}{\sinh s}) (\frac{s}{\sinh s})^{2} \tilde{v}^{4} s^{2} ds) \\ + \frac{1}{\delta} E(\tau) (\sum_{j} 2^{-2j} \| (\nabla_{\tau, x} \tilde{w}_{j}) (\frac{\sinh s}{\cosh s})^{1/2} (\frac{s}{\sinh s}) \|_{L^{\infty}(|x| \geq \frac{R}{2})}^{2}).
\endaligned$$ Next, by H[ö]{}lder’s inequality, $$\label{4.29}
\aligned
\sum_{j} \| (\tilde{v}^{3} - (P_{\leq j - 3} \tilde{v})^{3}) (\nabla_{\tau, x} \tilde{w}_{j}) \|_{L^{1}(|x| \leq \frac{R}{2})} \\ \lesssim \sum_{j} \| \tilde{v} \|_{L^{\infty}} \| P_{\geq j - 3} \tilde{v} \|_{L^{2}} \| \nabla_{\tau, x} \tilde{w}_{j} \|_{L^{6}} \| \tilde{v} \|_{L^{3}(|x| \leq \frac{R}{2})} \\
\lesssim E(\tau) (\sum_{j} 2^{-2j} \| \nabla_{\tau, x} \tilde{w}_{j} \|_{L^{6}}^{2}) + R E(\tau) \| v \|_{L^{\infty}}^{2}.
\endaligned$$ Following $(\ref{3.32})$ and $(\ref{3.33})$, $$\label{4.30}
\aligned
\int (P_{\leq j - 3} \tilde{v})^{3} (\partial_{s} \tilde{w}_{j}) \cdot (\frac{s}{\sinh s})^{2} s^{2} ds + \int (P_{\leq j - 3} \tilde{v})^{3} (\partial_{\tau} \tilde{w}_{j}) \cdot (\frac{s}{\sinh s})^{2} s^{2} ds \\
\lesssim \delta (\int \frac{1}{|x|} \tilde{v}^{4}) + \frac{1}{\delta} E(\tau) (\sum_{j} 2^{-8j/5} \| \nabla_{\tau, x} \tilde{w}_{j} \|_{L^{5}}^{2}).
\endaligned$$ Therefore, $$\label{4.24.1}
\aligned
\frac{d}{d\tau} \mathcal E(\tau) + \frac{c}{2} \tilde{v}(\tau, 0)^{2} + \frac{c}{2} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds \\ \lesssim E(\tau) (\sum_{j} 2^{-2j} \| \nabla_{\tau, x} w_{j} \|_{L^{6}}^{2}) + \frac{1}{\delta} E(\tau) (\sum_{j} 2^{-8j/5} \| \nabla_{\tau, x} \tilde{w}_{j} \|_{L^{5}}^{2}) + R E(\tau) \| v \|_{L^{\infty}}^{2} \\
+ \frac{1}{\delta} E(\tau) \| w \|_{L^{6}}^{2} + \delta(\int \tilde{v}^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds).
\endaligned$$ Absorbing $$\delta(\int \tilde{v}^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds)$$ into the left hand side, $$\aligned
\frac{d}{d\tau} \mathcal E(\tau) + \frac{c}{4} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds \\ \lesssim E(\tau) (\sum_{j} 2^{-2j} \| \nabla_{\tau, x} w_{j} \|_{L^{6}}^{2}) + \frac{1}{\delta} E(\tau) (\sum_{j} 2^{-8j/5} \| \nabla_{\tau, x} \tilde{w}_{j} \|_{L^{5}}^{2}) \\ + R E(\tau) \| v \|_{L^{\infty}}^{2} + \frac{1}{\delta} E(\tau) \| w \|_{L^{6}}^{2}.
\endaligned$$ Since $E(\tau) \sim \mathcal E(\tau)$, $$\aligned
\frac{d}{d\tau} \ln(\mathcal E(\tau)) + \frac{c}{4 \mathcal E(\tau)} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds \\ \lesssim (\sum_{j} 2^{-2j} \| \nabla_{\tau, x} w_{j} \|_{L^{6}}^{2}) + \frac{1}{\delta} (\sum_{j} 2^{-8j/5} \| \nabla_{\tau, x} \tilde{w}_{j} \|_{L^{5}}^{2}) \\ + R \| v \|_{L^{\infty}}^{2} + \frac{1}{\delta} \| w \|_{L^{6}}^{2} + \frac{1}{\delta} (\sum_{j} 2^{-2j} \| (\nabla_{\tau, x} \tilde{w}_{j}) (\frac{\sinh s}{\cosh s})^{1/2} (\frac{s}{\sinh s}) \|_{L^{\infty}(s \geq \frac{R}{2})}^{2}).
\endaligned$$ Suppose $T$ is such that $\mathcal E(T) = \sup_{0 < \tau < T} \mathcal E(\tau)$. Integrating in $\tau$, $$\label{4.28.1}
\aligned
\ln(\mathcal E(T)) - \ln(\mathcal E(0)) + \frac{c}{4} \int_{0}^{T} \frac{1}{\mathcal E(\tau)} \int \tilde{v}(\tau, s)^{4} (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) s^{2} ds d\tau \\ \lesssim \frac{\epsilon^{2}}{\delta} (1 - \ln(R)) + \epsilon^{2} + \int_{0}^{T} R \| \tilde{v} \|_{L^{\infty}}^{2} d\tau.
\endaligned$$ Now by direct computation, $$\label{4.29}
\| (\frac{s}{\sinh s})^{1/2} \tilde{u} \|_{L^{4}} \lesssim \| (\frac{s}{\sinh s})^{1/2} (\frac{\cosh s}{\sinh s})^{1/4} \tilde{v} \|_{L^{4}} + \| \tilde{w} \|_{L^{4}}.$$ If $I$ is an interval on which $\| (\frac{s}{\sinh s})^{1/2} \tilde{u} \|_{L_{\tau, x}^{4}(I)} \lesssim \epsilon$, then by $(\ref{6.10})$ and $(\ref{4.19.1})$, $$\label{4.30}
\| \tilde{v} \|_{L_{\tau}^{2} L_{x}^{\infty}(I \times \mathbf{R}^{3})} \lesssim \| \nabla \tilde{v} \|_{L_{\tau}^{\infty} L_{x}^{2}} + \| \tilde{v}_{\tau} \|_{L_{\tau}^{\infty} L_{x}^{2}} + \| \tilde{v} \|_{L_{\tau}^{2} L_{x}^{\infty}} (\int_{I} \int \tilde{u}^{4} (\frac{s}{\sinh s})^{2} s^{2} ds d\tau)^{1/2},$$ which implies $$\label{4.30.1}
\| \tilde{v} \|_{L_{\tau}^{2} L_{x}^{\infty}(I \times \mathbf{R}^{3})} \lesssim \| \nabla \tilde{v} \|_{L_{\tau}^{\infty} L_{x}^{2}} + \| \tilde{v}_{\tau} \|_{L_{\tau}^{\infty} L_{x}^{2}},$$ and therefore, $$\label{4.31}
\int_{0}^{T} R \| \tilde{v} \|_{L^{\infty}}^{2} d\tau \lesssim R \mathcal E(T) (\int_{0}^{T} \int \tilde{v}^{4} (\frac{s}{\sinh s})^{2} s^{2} ds d\tau).$$ Choosing $R = \delta \frac{1}{\mathcal E(T)^{2}}$, $(\ref{4.31})$ can be absorbed into the left hand side of $(\ref{4.28.1})$, proving $$\label{4.28.2}
\aligned
\ln(\mathcal E(T)) - \ln(\mathcal E(0)) \lesssim \frac{\epsilon^{2}}{\delta} (\ln(\frac{1}{\delta}) + \ln(\mathcal E(T))) + \epsilon^{2}.
\endaligned$$ This implies a uniform bound on $\mathcal E(T)$. Plugging the uniform bound on $\mathcal E(\tau)$ for all $\tau$ further implies a uniform bound on $$\label{4.28.3}
\int_{0}^{T} \int (\frac{s}{\sinh s})^{2} (\frac{\cosh s}{\sinh s}) \tilde{v}(\tau, s)^{4} s^{2} ds d\tau < \infty.$$ This proves scattering, assuming Lemma $\ref{l4.1}$ is true. $\Box$
*Proof of Lemma $\ref{l4.1}$:* For $t > 1$, $$\label{4.40}
u(t) = S(t)(u_{0}, u_{1}) + \int_{0}^{t} S(t - t')(0, u^{3}) dt' = u_{l} + u_{nl}.$$ First take the Duhamel term $u_{nl}$. Because the curve $t^{2} - r^{2} = 1$ has slope $\frac{dr}{dt} > 1$ everywhere, $$\label{4.42}
s \tilde{u}_{nl}(\tau, s)|_{\tau = 0} = \int_{1}^{e^{\tau} \cosh s} \int_{e^{\tau} \sinh s - e^{\tau} \cosh s + t}^{e^{\tau} \sinh s + e^{\tau} \cosh s - t} r u^{3}(t,r) dr dt.$$ By direct computation, $$\label{4.43}
\aligned
\int_{0}^{\infty} (\partial_{\tau}(s \tilde{u}_{nl})|_{\tau = 0})^{2} ds \lesssim \int_{0}^{\infty} e^{2s} (\int_{1}^{\cosh s} (e^{s} - t) u^{3}(t, e^{s} - t) dt)^{2} ds \\ + \int_{0}^{\infty} e^{-2s} (\int_{1}^{\cosh s} (t - e^{-s}) u^{3}(t, t - e^{-s}) dt)^{2} ds.
\endaligned$$ By H[ö]{}lder’s inequality, since $e^{s} - \cosh s \sim e^{s}$, $$\label{4.44}
\aligned
\int_{0}^{\infty} e^{2s} (\int_{1}^{\cosh s} (e^{s} - t) u^{3}(t, e^{s} - t) dt)^{2} ds \\ \lesssim \int_{0}^{\infty} \int_{1}^{\cosh s} e^{3s} (e^{s} - t)^{2} u^{6}(t, e^{s} - t) dt ds \\ %\lesssim %\int_{0}^{\infty} \int_{1}^{\cosh(s)} e^{s} (e^{s} - t)^{2} u_{nl}^{4}(t, e^{s} - t) dt ds \\
\lesssim \int_{0}^{\infty} \int_{t^{2} - r^{2} \leq 1} u^{6}(t, r) r^{4} dt dr < \infty.
\endaligned$$ The last inequality follows from global well-posedness of $u$, which implies $\| u \|_{L_{t,x}^{4}([1, 3] \times \mathbf{R}^{3})} < \infty$, $(\ref{4.7})$, Strichartz estimates, and the radial Sobolev embedding theorem, which implies $$\label{4.45}
\| |x|^{1/3} u \|_{L_{t,x}^{6}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \| |\nabla|^{1/6} u \|_{L_{t}^{6} L_{x}^{3}(\mathbf{R} \times \mathbf{R}^{3})}.$$ Also by a change of variables and H[ö]{}lder’s inequality, since $(t - e^{-s}) \gtrsim 1$ for $s \geq 1$ and $t \geq 1$, $$\label{4.46}
\aligned
\int_{1}^{\infty} e^{-2s} (\int_{1}^{\cosh s} (t - e^{-s}) u^{3}(t, t - e^{-s}) dt)^{2} ds \\ \lesssim \int_{1}^{\infty} \int_{1}^{\cosh s} e^{-s} (t - e^{-s})^{2} u^{6}(t, e^{s} - t) dt ds \\ %\lesssim \int_{1}^{\infty} \int_{1}^{\cosh(s)} e^{-s} (t - e^{-s})^{2} u_{nl}^{4}(t, t - e^{-s}) dt ds \\
\lesssim \int_{0}^{\infty} \int_{t^{2} - r^{2} \leq 1} u^{6}(t, r) r^{4} dt dr < \infty.
\endaligned$$ Also, by the radial Sobolev embedding theorem and Young’s inequality, $$\label{4.47}
\aligned
\int_{1}^{\infty} e^{-2s} (\int_{1}^{\cosh s} (t - e^{-s}) u^{3}(t, t - e^{-s}) dt)^{2} ds \\ \lesssim \int_{1}^{3} (\int_{t^{2} - r^{2} \leq 1} u(t, r)^{6} r^{2} dr)^{1/2} dt \lesssim \int_{1}^{3} \frac{1}{(t - 1)^{3/4}} dt < \infty.
\endaligned$$ This takes care of the nonlinear Duhamel piece.
Now consider the linear piece. First consider the contribution of $$\label{4.48}
S(t - 1)(u_{0}, 0).$$ Recall that if $w$ solves $(\ref{4.48})$ and $r > t$, $$\label{4.49}
r w(t,r) = \frac{1}{2} [u_{0}(t + r)(t + r) + u_{0}(r - t) (r - t)],$$ so if $u_{1} = 0$, $u_{l} = S(t - 1)(u_{0}, 0)$, and $$\label{4.50}
\aligned
s \tilde{u}_{l}(\tau, s) = e^{\tau} \sinh s \cdot u_{l}(e^{\tau} \cosh s, e^{\tau} \sinh s) \\ = \frac{1}{2} [u_{0}(e^{\tau + s} - 1) (e^{\tau + s} - 1) + u_{0}(1 - e^{\tau - s}) (1 - e^{\tau - s})].
\endaligned$$ Let $\chi \in C_{0}^{\infty}(\mathbf{R})$ be a function satisfying $$\label{4.51}
1 = \sum_{k \geq 0} \chi(s - k),$$ for any $s \in [0, \infty)$, and $\chi(s - k)$ is supported on $(k - 1) \cdot \ln(2) \leq s \leq (k + 1) \cdot \ln(2)$, and split $$\tilde{u}_{l}(\tau, s) = \tilde{u}_{l}^{(1)}(\tau, s) + \tilde{u}_{l}^{(2)}(\tau, s) + \tilde{u}_{l}^{(3)}(\tau,s) + \tilde{u}_{l}^{(4)}(\tau, s) + \tilde{u}_{l}^{(5)}(\tau,s) + \tilde{u}_{l}^{(6)}(\tau, s),$$ where $$\label{4.52}
\aligned
s \tilde{u}_{l}^{(1)}(\tau, s) = \sum_{k \geq 0} \chi(s - k) (P_{\leq -k} u_{0})(e^{\tau + s} - 1) \cdot (e^{\tau + s} - 1), \\
s \tilde{u}_{l}^{(2)}(\tau, s) = P_{\geq 0} \sum_{k \geq 0} \chi(s - k) (P_{> -k} u_{0})(e^{\tau + s} - 1) \cdot (e^{\tau + s} - 1), \\
s \tilde{u}_{l}^{(3)}(\tau, s) = P_{\leq 0} \sum_{k \geq 0} \chi(s - k) (P_{> -k} u_{0})(e^{\tau + s} - 1) \cdot (e^{\tau + s} - 1), \\
s \tilde{u}_{l}^{(4)}(\tau, s) = \sum_{k \geq 0} \chi(s - k) (P_{\leq k} u_{0})(1 - e^{\tau - s}) \cdot (1 - e^{\tau - s}), \\
s \tilde{u}_{l}^{(5)}(\tau, s) = P_{\leq 0} \sum_{k \geq 0} \chi(s - k) (P_{> k} u_{0})(1 - e^{\tau - s}) \cdot (1 - e^{\tau - s}), \\
s \tilde{u}_{l}^{(6)}(\tau, s) = P_{\geq 0} \sum_{k \geq 0} \chi(s - k) (P_{> k} u_{0})(1 - e^{\tau - s}) \cdot (1 - e^{\tau - s}).
\endaligned$$ Taking the derivative, $$\label{4.53}
\aligned
\partial_{\tau} (s \tilde{u}_{l}^{(1)})(\tau, s)|_{\tau = 0} = \sum_{k \geq 0} \chi(s - k) (P_{\leq -k} u_{0}')(e^{s} - 1) \cdot (e^{s} - 1) e^{s} \\
+ \sum_{k \geq 0} \chi(s - k) (P_{\leq -k} u_{0})(e^{s} - 1) \cdot e^{s}.
\endaligned$$ Then by a change of variables, Hardy’s inequality, and Young’s inequality, $$\label{4.54}
\aligned
\| (\ref{4.53}) \|_{L^{2}[0, \infty)} \lesssim (\sum_{k \geq 0} 2^{k} (\sum_{j \leq -k} \| \chi(s - k) (P_{j} \nabla u_{0})(e^{s} - 1) \|_{L^{2}} \\ + \| \chi(s - k) \frac{1}{|x|} (P_{j} u_{0})(e^{s} - 1) \|_{L^{2}})^{2})^{1/2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.
\endaligned$$ The computation of $\partial_{s}(s \tilde{u}_{l}^{(1)}(\tau, s))|_{\tau = 0}$ is similar, except that, in addition, it is necessary to compute $$\label{4.55}
\sum_{k} \| \chi'(s - k) (P_{\leq -k} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{2}}^{2}.$$ Again, by a change of variables, $$\label{4.56}
(\ref{4.55}) \lesssim \sum_{k \geq 0} 2^{k} (\sum_{j \leq -k} \| \chi'(s - k) \frac{1}{|x|} (P_{j} u_{0})(e^{s} - 1) \|_{L^{2}}^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.$$ By the product rule, $$\label{4.57}
s \partial_{s} \tilde{u}_{l}(\tau, s) = \partial_{s}(s \tilde{u}_{l}(\tau, s)) - \tilde{u}_{l}(\tau, s).$$ By the support properties of $\chi(s - k)$ and the Sobolev embedding theorem, $$\label{4.58}
\| \sum_{k \geq 0} \chi(s - k) (P_{\leq -k} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{\infty}} \lesssim \| u_{0} \|_{\dot{H}^{1/2}},$$ and therefore, $$\label{4.59}
\| \frac{1}{s} \sum_{k \geq 2} \chi(s - k) (P_{\leq -k} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{2}([0, \infty)} \lesssim (\int_{1}^{\infty} \frac{1}{s^{2}} ds)^{1/2} \| u_{0} \|_{\dot{H}^{1/2}} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$ Also, by the support properties of $\chi(s - k)$ and $(\ref{4.58})$, $$\label{4.60}
\| \sum_{k = 0,1} \chi(s - k) P_{\leq -k} u_{0}(e^{s} - 1) \cdot \frac{(e^{s} - 1)}{s} \|_{L^{2}([0, \infty)} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$ Therefore, $\tilde{u}_{l}^{(1)}(\tau, s)|_{\tau = 0}$ has finite energy.
Next, for any $k \geq 0$, $j > -k$, by the product rule and change of variables, $$\label{4.61}
\aligned
\| \partial_{\tau}(\chi(s - k) (P_{j} u_{0})(e^{s + \tau} - 1) \cdot (e^{s + \tau} - 1))|_{\tau = 0} \|_{L^{2}([0, \infty)} \\
\lesssim \| \chi(s - k) (P_{j} \nabla u_{0})(e^{s} - 1) \cdot (e^{s} - 1) e^{s} \|_{L^{2}([0, \infty)} \\
+ \| \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot e^{s} \|_{L^{2}([0, \infty)} \\
\lesssim 2^{k/2} \| P_{j} \nabla u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} + 2^{k/2} \| \frac{1}{|x|} P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}.
\endaligned$$ Therefore, if $f \in \dot{H}^{1/2}(\mathbf{R}^{3})$ is a radial function, by Bernstein’s inequality, $$\label{4.62}
\aligned
\int_{0}^{\infty} (P_{l} f(s)) s \cdot \partial_{\tau}(\chi(s - k) (P_{j} u_{0})(e^{s + \tau} - 1) \cdot (e^{s + \tau} - 1))|_{\tau = 0} ds \\
\lesssim \| P_{l} f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} [2^{k/2} \| P_{j} \nabla u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} \\ + 2^{k/2} \| \frac{1}{|x|} P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}].
\endaligned$$ Summing up, by Young’s inequality, Bernstein’s inequality, $$\label{4.63}
\aligned
\sum_{l \geq j + k > 0} \| P_{l} f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} [2^{k/2} \| P_{j} \nabla u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} \\ + 2^{k/2} \| \frac{1}{|x|} P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}] \lesssim \| f \|_{\dot{H}^{1/2}} \| u_{0} \|_{\dot{H}^{1/2}}.
\endaligned$$ Next, by a change of variables, $$\label{4.64}
\| \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{2}([0, \infty)} \lesssim 2^{-k/2} \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}.$$ By the product rule, $$\label{4.65}
\aligned
\partial_{\tau}(\chi(s - k) (P_{j} u_{0})(e^{s + \tau} - 1) \cdot (e^{s + \tau} - 1))|_{\tau = 0} \\ = \partial_{s} (\chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1)) - \chi'(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1).
\endaligned$$ Integrating by parts, $$\label{4.66}
\aligned
\int_{0}^{\infty} (P_{l} f(s)) s \cdot \partial_{s}(\chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1)) ds \\
= -\int_{0}^{\infty} [(P_{l} \nabla f(s)) s + (P_{l} f(s))] \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) ds \\
\lesssim 2^{-k/2} [\| P_{l} \nabla f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} + \| \frac{1}{|x|} P_{l} f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}] \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}.
\endaligned$$ Summing up, $$\label{4.67}
\aligned
\sum_{0 \leq l < j + k} 2^{-k/2} [\| P_{l} \nabla f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} \\ + \| P_{l} f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}] \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} \lesssim \| f \|_{\dot{H}^{1/2}} \| u_{0} \|_{\dot{H}^{1/2}}.
\endaligned$$ Also, $$\label{4.68}
\aligned
\int_{0}^{\infty} (P_{l} f(s))s \cdot \chi'(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) ds \\ \lesssim \| P_{l} f \|_{L^{2}(2^{k - 1} - 1 \leq s \leq 2^{k + 1})} 2^{-k/2} \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq s \leq 2^{k + 1})}.
\endaligned$$ Then by Bernstein’s inequality, $$\label{4.69}
\aligned
\sum_{0 \leq l < j + k} 2^{-k/2} \| P_{l} f \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1}} \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} \lesssim \| f \|_{\dot{H}^{1/2}} \| u_{0} \|_{\dot{H}^{1/2}}.
\endaligned$$ Therefore, $$\label{4.70}
\| \partial_{\tau}(\tilde{u}_{l}^{(2)}(\tau, s))|_{\tau = 0} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$ Also, by the product rule, and a change of variables $$\label{4.71}
\aligned
\| \partial_{s}(\chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1)) \|_{L^{2}} \lesssim 2^{-k/2} \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} \\
+ 2^{k/2} \| (P_{j} \nabla u_{0}) \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})} + 2^{k/2} \| \frac{1}{|x|} (P_{j} u_{0}) \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}.
\endaligned$$ Meanwhile, $$\label{4.71.1}
\| \frac{1}{s} \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{2}} \lesssim 2^{-k/2} \| P_{j} u_{0} \|_{L^{2}(2^{k - 1} - 1 \leq r \leq 2^{k + 1})}.$$ Then by Bernstein’s inequality and Young’s inequality, $$\label{4.72}
\aligned
\sum_{l} \| P_{l}(\sum_{l \leq k + j, k + j > 0} \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1)) \|_{L^{2}}^{2} \\ \lesssim \sum_{l} 2^{l} \sum_{k} (\sum_{l \leq k + j, k + j > 0} \| \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{2}})^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.
\endaligned$$ Also by Bernstein’s inequality and $(\ref{4.71})$, $$\label{4.73}
\aligned
\sum_{l} \| P_{l}(\sum_{0 < k + j < l} \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1)) \|_{L^{2}}^{2} \\ \lesssim \sum_{l} 2^{l} \sum_{k} (\sum_{0 < k + j < l} \| \chi(s - k) (P_{j} u_{0})(e^{s} - 1) \cdot (e^{s} - 1) \|_{L^{2}})^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.
\endaligned$$ Therefore, we have proved, $$\label{4.74}
\| \partial_{\tau}(\tilde{u}_{l}^{(2)}(\tau, s))|_{\tau = 0} \|_{\dot{H}^{-1/2}} + \| \tilde{u}_{l}^{(2)}(\tau, s)|_{\tau = 0} \|_{\dot{H}^{1/2}} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$
Next, following $(\ref{4.67})$–$(\ref{4.70})$ with $P_{l}$, $l \geq 0$ replaced by $P_{\leq 0}$ and $f \in L^{2}(\mathbf{R}^{3})$, $$\label{4.75}
\| \partial_{\tau}(\tilde{u}_{l}^{(3)}(\tau, s))|_{\tau = 0} \|_{L^{2}} + \| \tilde{u}_{l}^{(3)}(\tau, s)|_{\tau = 0} \|_{\dot{H}^{1}} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$
Next consider $\tilde{u}_{l}^{(4)}(\tau, s)$. By the product rule, $$\label{4.76}
\aligned
\partial_{\tau} (s \tilde{u}_{l}^{(4)}(\tau, s))|_{\tau = 0} = - \sum_{k \geq 0} \chi(s - k) (P_{\leq k} \nabla u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) e^{-s} \\
- \sum_{k \leq 0} \chi(s - k) (P_{\leq k} u_{0})(1 - e^{-s}) e^{-s}.
\endaligned$$ Then, by Young’s inequality, $$\label{4.77}
\aligned
\| \partial_{\tau}(s \tilde{u}_{l}^{(4)}(\tau, s))|_{\tau = 0} \|_{L^{2}([0, \infty)} \lesssim \sum_{k \geq 0} 2^{-k} (\sum_{j \leq k} \| \nabla P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})})^{2} \\
+ \sum_{k \geq 0} 2^{-k} (\sum_{j \leq k} \| \frac{1}{|x|} P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})})^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.
\endaligned$$ Also, by the product rule, $$\label{4.78}
\partial_{s}(s \tilde{u}_{l}^{(4)}(\tau, s)) = -\partial_{\tau}(s \tilde{u}_{l}^{(4)}(\tau, s)) + \sum_{k \geq 0} \chi'(s - k) (P_{k} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}).$$ Then by the finite overlapping property of $\chi(s - k)$ and the radial Sobolev embedding theorem, $$\label{4.79}
\| \sum_{k \geq 0} \chi'(s - k) (P_{k} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) \|_{L^{2}([0, \infty)}^{2} \lesssim \sum_{k \geq 0} \| P_{k} u_{0} \|_{\dot{H}^{1/2}}^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.$$ Therefore, $$\label{4.80}
\| \partial_{s}(s \tilde{u}_{l}^{(4)}(\tau, s))|_{\tau = 0} \|_{L^{2}([0, \infty)} + \| \partial_{\tau}(s \tilde{u}_{l}^{(4)}(\tau, s))|_{\tau = 0} \|_{L^{2}([0, \infty)} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$
Next, by a change of variables, $$\label{4.81}
\| \chi(s - k) (P_{j} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) \|_{L^{2}} \lesssim 2^{k/2} \| P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})}.$$ Therefore, by Young’s inequality, $$\label{4.82}
\| s \tilde{u}_{l}^{5}(\tau, s)|_{\tau = 0} \|_{L^{2}([0, \infty)}^{2} \lesssim \sum_{k \geq 0} 2^{k} (\sum_{j > k} \| P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})})^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.$$ Therefore, by the Fourier support of $\tilde{u}_{l}^{(5)}$, $$\label{4.83}
\| \tilde{u}_{l}^{(5)}(\tau, s)|_{\tau = 0} \|_{\dot{H}^{1}(\mathbf{R}^{3})} \lesssim \| u_{0} \|_{\dot{H}^{1/2}(\mathbf{R}^{3})}.$$ Also, if $f \in L^{2}$ and $f$ is supported on $|\xi| \leq 1$, $$\label{4.84}
\aligned
\int_{0}^{\infty} f(s) s \cdot \partial_{\tau}(s \tilde{u}_{l}^{(5)}(\tau, s))|_{\tau = 0} ds = -\int_{0}^{\infty} f(s) s \cdot \partial_{s}(s \tilde{u}_{l}(\tau, s))|_{\tau = 0} ds \\
- \int_{0}^{\infty} f(s) s \cdot \sum_{k \geq 0} \chi'(s - k) (P_{\geq k} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) ds.
\endaligned$$ Integrating by parts, by $(\ref{4.82})$, $$\label{4.85}
-\int_{0}^{\infty} f(s) s \cdot \partial_{s}(s \tilde{u}_{l}^{(5)}(\tau, s))|_{\tau = 0} ds = \int_{0}^{\infty} \partial_{s}(f(s) s) \cdot s \tilde{u}_{l}^{(5)}(\tau, s)|_{\tau = 0} ds \lesssim \| f \|_{L^{2}} \| u_{0} \|_{\dot{H}^{1/2}}.$$ Also, by $(\ref{4.82})$, $$\label{4.86}
\int_{0}^{\infty} f(s) s \cdot \sum_{k \geq 0} \chi'(s - k) (P_{\geq k} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) ds \lesssim \| f \|_{L^{2}} \| u_{0} \|_{\dot{H}^{1/2}}.$$ Therefore, $$\label{4.87}
\| \partial_{\tau}(s \tilde{u}_{l}^{(5)}(\tau, s))|_{\tau = 0} \|_{L^{2}([0, \infty)} + \| \partial_{s}(s \tilde{u}_{l}^{(5)}(\tau, s))|_{\tau = 0} \|_{L^{2}([0, \infty)} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}.$$
Finally, take $\tilde{u}_{l}^{(6)}(\tau, s)$. Take $f \in \dot{H}^{1/2}$ supported in Fourier space on $|\xi| \geq 1$. Then by the product rule and $(\ref{4.82})$, $$\label{4.88}
\aligned
\| \partial_{s}(\chi(s - k) (P_{j} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s})) \|_{L^{2}([0, \infty)} \\ \lesssim 2^{k/2} \| P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})} \\
+ 2^{-k/2} \| P_{j} \nabla u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})} + 2^{-k/2} \| \frac{1}{|x|} P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})}.
\endaligned$$ Also, by $(\ref{4.82})$ and $(\ref{4.81})$, $$\label{4.89}
\| \frac{1}{s} \chi(s - k) (P_{j} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) \|_{L^{2}([0, \infty)} \lesssim 2^{k/2} \| P_{j} u_{0} \|_{L^{2}(1 - 2^{-k - 1} \leq r \leq 1 - 2^{-k + 1})}.$$ Therefore, by Young’s inequality, $$\label{4.90}
\sum_{l < j + k} 2^{l} \sum_{k} (\sum_{j > k} \| \chi(s - k) (P_{j} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) \|_{L^{2}})^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.$$ Also, by Bernstein’s inequality, $$\label{4.91}
\sum_{l \geq j + k} 2^{-l} \sum_{k} (\sum_{j > k} \| \nabla \chi(s - k) (P_{j} u_{0})(1 - e^{-s}) \cdot (1 - e^{-s}) \|_{L^{2}})^{2} \lesssim \| u_{0} \|_{\dot{H}^{1/2}}^{2}.$$ Therefore, we have finally proved that if $u_{1} = 0$, $$\label{4.92}
\tilde{u}_{l}(\tau, s)|_{\tau = 0} \in \dot{H}^{1/2}(\mathbf{R}^{3}) + \dot{H}^{1}(\mathbf{R}^{3}),$$ and $$\label{4.93}
\partial_{\tau}(\tilde{u}_{l}(\tau, s))|_{\tau = 0} \in \dot{H}^{-1/2}(\mathbf{R}^{3}) + L^{2}(\mathbf{R}^{3}).$$
To compute the contribution of $$\label{4.94}
S(t)(0, u_{1})$$ to $\tilde{u}_{l}(\tau, s)$, observe that $$\label{4.95}
\frac{\sin(t \sqrt{-\Delta})}{\sqrt{-\Delta}} f = \partial_{t} (\frac{\cos(t \sqrt{-\Delta})}{\Delta} f).$$ Plugging in the formula for a solution to the wave equation when $r > t$, let $w(t,r) = \cos(t \sqrt{-\Delta}) f$. Then, $$\label{4.96}
\aligned
\partial_{t} (w(t,r)) = \frac{1}{2r} \partial_{t} (f(t + r)(t + r) + f(r - t) (r - t)) \\ = \frac{1}{2r} [f(t + r) + f'(t + r)(t + r) - f(r - t) - f'(r - t) (r - t)].
\endaligned$$ Then decompose $\tilde{u}_{l}(\tau, s) = \tilde{u}_{l}^{(1)}(\tau, s) + \tilde{u}_{l}^{(2)}(\tau, s) + \tilde{u}_{l}^{(3)}(\tau, s)$, where $$\label{4.96.1}
\aligned
s \tilde{u}_{l}^{(1)}(\tau, s) = \frac{1}{2} [f'(e^{\tau + s} - 1) \cdot (e^{\tau + s} - 1) - f'(1 - e^{\tau - s}) \cdot (1 - e^{\tau - s})], \\
s \tilde{u}_{l}^{(2)}(\tau, s) = \frac{1}{2} (1 - \chi(s)) [f(e^{\tau + s} - 1) - f(1 - e^{\tau - s})], \\
s \tilde{u}_{l}^{(3)}(\tau, s) = \frac{1}{2} \chi(s) [f(e^{\tau + s} - 1) - f(1 - e^{\tau - s})].
\endaligned$$
Since $$\label{4.97}
f = \frac{u_{1}}{\Delta} \in \dot{H}^{3/2}(\mathbf{R}^{3}),$$ the contribution of $$\label{4.98}
f'(e^{\tau + s} - 1) \cdot (e^{\tau + s} - 1) - f'(1 - e^{\tau - s}) \cdot (1 - e^{\tau - s}),$$ to $$\label{4.99}
(\tilde{u}_{l}(\tau, s)|_{\tau = 0}, \partial_{\tau} \tilde{u}_{l}(\tau, s)|_{\tau = 0})$$ may be analyzed in exactly the same manner as the contribution of $S(t)(u_{1}, 0)$. Therefore, $$\label{4.99.1}
\tilde{u}_{l}^{(1)}(\tau, s)|_{\tau = 0} \in \dot{H}^{1/2} + \dot{H}^{1},$$ and $$\label{4.99.2}
\partial_{\tau}(\tilde{u}_{l}^{(1)}(\tau, s))|_{\tau = 0} \in \dot{H}^{-1/2} + L^{2}.$$
Next take $\tilde{u}_{l}^{(2)}(\tau, s)$. By a change of variables, $$\label{4.100}
\int_{1}^{\infty} (\partial_{s} f(e^{s} - 1))^{2} ds = \int_{1}^{\infty} (f'(e^{s} - 1) \cdot e^{s})^{2} ds \lesssim \int |f'(r)|^{2} r dr \lesssim \| f \|_{\dot{H}^{3/2}(\mathbf{R}^{3})}^{2},$$ and $$\label{4.101}
\int_{1}^{\infty} (\partial_{s} f(1 - e^{-s}))^{2} ds = \int_{1}^{\infty} (f'(1 - e^{-s}) \cdot e^{-s})^{2} ds \lesssim |f'(r)|^{2} r dr \lesssim \| f \|_{\dot{H}^{3/2}(\mathbf{R}^{3})}^{2}.$$ By an identical calculation, $$\label{4.102}
\int_{1}^{\infty} (\partial_{\tau} f(e^{s + \tau} - 1)|_{\tau = 0})^{2} ds = \int_{1}^{\infty} (f'(e^{s} - 1) \cdot e^{s})^{2} ds \lesssim \int |f'(r)|^{2} r dr \lesssim \| f \|_{\dot{H}^{3/2}(\mathbf{R}^{3})}^{2},$$ and $$\label{4.103}
\int_{1}^{\infty} (\partial_{s} f(1 - e^{\tau - s})|_{\tau = 0})^{2} ds = \int_{1}^{\infty} (f'(1 - e^{-s}) \cdot e^{-s})^{2} ds \lesssim \int |f'(r)|^{2} r dr \lesssim \| f \|_{\dot{H}^{3/2}(\mathbf{R}^{3})}^{2}.$$ Next, by the fundamental theorem of calculus, for $s_{0} \sim 1$, $$\label{4.104}
s_{0} [f(e^{s_{0}} - 1) - f(1 - e^{-s_{0}})]^{2} = s_{0} [\int_{1 - e^{-s_{0}}}^{e^{s_{0}} - 1} f'(r) dr]^{2} \lesssim \int |f'(r)|^{2} r dr \lesssim \| f \|_{\dot{H}^{3/2}}^{2}.$$ Therefore, by $(\ref{4.101})$ and $(\ref{4.102})$, $$\label{4.105}
\| \partial_{\tau}(\tilde{u}_{l}^{(2)}(\tau, s))|_{\tau = 0} \|_{L^{2}} \lesssim \| f \|_{\dot{H}^{3/2}},$$ and $$\label{4.106}
\| \tilde{u}_{l}^{(2)}(0,s) \|_{\dot{H}^{1}} \lesssim \| f \|_{\dot{H}^{3/2}}.$$
Finally, consider $$\label{4.107}
f(e^{\tau + s} - 1) - f(1 - e^{\tau - s}),$$ when $s < 1$. By direct computation, $$\label{4.108}
\partial_{\tau} [f(e^{\tau + s} - 1) - f(1 - e^{\tau - s})]|_{\tau = 0} = f'(e^{s} - 1) \cdot e^{s} + f'(1 - e^{-s}) \cdot e^{-s}.$$ Then for $g \in \dot{H}^{1/2}$, by Hardy’s inequality, $$\label{4.109}
\int f'(e^{s} - 1) \cdot e^{s} \cdot g(s) s ds + \int f'(1 - e^{-s}) \cdot e^{-s} \cdot g(s) s ds \lesssim \| f \|_{\dot{H}^{3/2}} \| g \|_{\dot{H}^{1/2}}.$$ Also, by the fundamental theorem of calculus, $$\label{4.110}
\aligned
f(e^{s} - 1) - f(1 - e^{-s}) = \int_{s - \frac{s^{2}}{2} + \frac{s^{3}}{3!} - ...}^{s + \frac{s^{2}}{2} + \frac{s^{3}}{3!} + ...} f'(r) dr \\
= \int_{0}^{1} f'(s + \theta (\frac{s^{2}}{2} + \frac{s^{3}}{3!} + ...)) \cdot (\frac{s^{2}}{2} + \frac{s^{3}}{3!} + ...) d\theta \\ + \int_{-1}^{0} f'(s + \theta(\frac{s^{2}}{2} - \frac{s^{3}}{3!} + ...) \cdot (\frac{s^{2}}{2} + \frac{s^{3}}{3!} + ...) d\theta.
\endaligned$$ Therefore, since $\chi(s)$ is supported on $s \leq 1$, $$\label{4.111}
\| f(e^{s} - 1) - f(1 - e^{-s}) \|_{\dot{H}^{1/2}} \lesssim \| f \|_{\dot{H}^{3/2}}.$$ This proves that $$\label{4.112}
\| \tilde{u}_{l}^{(3)}(\tau, s)|_{\tau = 0} \|_{\dot{H}^{1/2}} + \| \partial_{\tau} \tilde{u}_{l}^{(3)}(\tau, s)|_{\tau = 0} \|_{\dot{H}^{-1/2}} \lesssim \| f \|_{\dot{H}^{3/2}}.$$ This finally completes the proof of Lemma $\ref{l4.1}$. $\Box$
Profile decomposition
=====================
*Proof of Theorem $\ref{t1.2}$:* This completes the proof that for any $(u_{0}, u_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$, $(\ref{1.1})$ has a global solution that scatters both forward and backward in time. To prove $(\ref{1.5})$, it remains to prove that for a sequence of initial data $(u_{n}^{0}, u_{n}^{1})$ and for any $A < \infty$, $$\label{5.1}
\| u_{0}^{n} \|_{\dot{H}^{1/2}} + \| u_{1}^{n} \|_{\dot{H}^{-1/2}} \leq A,$$ $$\label{5.2}
\| u^{n} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} \leq f(A) < \infty,$$ where $f : [0, \infty) \rightarrow [0, \infty)$, and $u^{n}$ is the solution to $(\ref{1.1})$ with initial data $(u_{0}^{n}, u_{1}^{n})$.
To prove this, make a profile decomposition.
\[t5.1\] Suppose that there is a uniformly bounded, radially symmetric sequence $$\label{5.3}
\| u_{0}^{n} \|_{\dot{H}^{1/2}(\mathbf{R}^{3})} + \| u_{1}^{n} \|_{\dot{H}^{-1/2}(\mathbf{R}^{3})} \leq A < \infty.$$ Then there exists a subsequence, also denoted $(u_{0}^{n}, u_{1}^{n}) \subset \dot{H}^{1/2} \times \dot{H}^{-1/2}$ such that for any $N < \infty$, $$\label{5.4}
S(t)(u_{0}^{n}, u_{1}^{n}) = \sum_{j = 1}^{N} \Gamma_{n}^{j} S(t)(\phi_{0}^{j}, \phi_{1}^{j}) + S(t)(R_{0, n}^{N}, R_{1,n}^{N}),$$ with $$\label{5.5}
\lim_{N \rightarrow \infty} \limsup_{n \rightarrow \infty} \| S(t)(R_{0,n}^{N}, R_{1,n}^{N}) \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} = 0.$$ $\Gamma_{n}^{j} = (\lambda_{n}^{j}, t_{n}^{j})$ belongs to the group $(0, \infty) \times \mathbf{R}$, which acts by $$\label{5.6}
\Gamma_{n}^{j} F(t,x) = \lambda_{n}^{j} F(\lambda_{n}^{j} (t - t_{n}^{j}), \lambda_{n}^{j} x).$$ The $\Gamma_{n}^{j}$ are pairwise orthogonal, that is, for every $j \neq k$, $$\label{5.7}
\lim_{n \rightarrow \infty} \frac{\lambda_{n}^{j}}{\lambda_{n}^{k}} + \frac{\lambda_{n}^{k}}{\lambda_{n}^{j}} + (\lambda_{n}^{j})^{1/2} (\lambda_{n}^{k})^{1/2} |t_{n}^{j} - t_{n}^{k}| = \infty.$$ Furthermore, for every $N \geq 1$, $$\label{5.8}
\aligned
\| (u_{0, n}, u_{1, n}) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}}^{2} = \sum_{j = 1}^{N} \| (\phi_{0}^{j}, \phi_{0}^{k}) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}}^{2} \\ + \| (R_{0, n}^{N}, R_{1, n}^{N}) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}}^{2} + o_{n}(1).
\endaligned$$
Theorem $\ref{t5.1}$ gives the profile decomposition $$\label{5.9}
S(t)(u_{0}^{n}, u_{1}^{n}) = \sum_{j = 1}^{N} S(t - t_{n}^{j}) (\lambda_{n}^{j} \phi_{0}^{j}(\lambda_{n}^{j} x), (\lambda_{n}^{j})^{2} \phi_{1}^{j}(\lambda_{n}^{j} x)) + S(t)(R_{0, n}^{N}, R_{1,n}^{N}).$$ In the course of proving Theorem $\ref{t5.1}$, [@Ramos] proved $$\label{5.10}
S(\lambda_{n}^{j} t_{n}^{j})(\frac{1}{\lambda_{n}^{j}} u_{0}^{n}(\frac{x}{\lambda_{n}^{j}}), \frac{1}{(\lambda_{n}^{j})^{2}} u_{1}^{n}(\frac{x}{\lambda_{n}^{j}})) \rightharpoonup \phi_{0}^{j}(x),$$ weakly in $\dot{H}^{1/2}(\mathbf{R}^{3})$, and $$\label{5.11}
\partial_{t}S(t + \lambda_{n}^{j} t_{n}^{j})(\frac{1}{\lambda_{n}^{j}} u_{0}^{n}(\frac{x}{\lambda_{n}^{j}}), \frac{1}{(\lambda_{n}^{j})^{2}} u_{1}^{n}(\frac{x}{\lambda_{n}^{j}}))|_{t = 0} \rightharpoonup \phi_{1}^{j}(x)$$ weakly in $\dot{H}^{-1/2}(\mathbf{R}^{3})$. Then after passing to a subsequence, $\lambda_{n}^{j} t_{n}^{j}$ converges to some $t^{j}$. Changing $(\phi_{0}^{j}, \phi_{1}^{j})$ to $S(-t^{j})(\phi_{0}^{j}, \phi_{1}^{j})$ and absorbing the error into $(R_{0, n}^{N}, R_{1, n}^{N})$, $$\label{5.10}
(\frac{1}{\lambda_{n}^{j}} u_{0}^{n}(\frac{x}{\lambda_{n}^{j}}), \frac{1}{(\lambda_{n}^{j})^{2}} u_{1}^{n}(\frac{x}{\lambda_{n}^{j}})) \rightharpoonup \phi_{0}^{j}(x),$$ and $$\label{5.11}
\partial_{t}S(t)(\frac{1}{\lambda_{n}^{j}} u_{0}^{n}(\frac{x}{\lambda_{n}^{j}}), \frac{1}{(\lambda_{n}^{j})^{2}} u_{1}^{n}(\frac{x}{\lambda_{n}^{j}}))|_{t = 0} \rightharpoonup \phi_{1}^{j}(x).$$ Then if $u^{j}$ is the solution to $(\ref{1.1})$ with initial data $(\phi_{0}^{j}, \phi_{1}^{j})$, then $$\label{5.12}
\| u^{j} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} \leq M_{j}.$$
Next, suppose that after passing to a subsequence, $\lambda_{n}^{j} t_{n}^{j} \nearrow +\infty$. Theorem $\ref{t4.0}$ also implies that for any $(\phi_{0}, \phi_{1}) \in \dot{H}^{1/2} \times \dot{H}^{-1/2}$, there exists a solution $u$ to $(\ref{1.1})$ that is globally well-posed and scattering, and furthermore, that $u$ scatters to $S(t)(\phi_{0}, \phi_{1})$ as $t \searrow -\infty$. $$\label{5.13}
\lim_{t \rightarrow -\infty} \| u - S(t)(\phi_{0}, \phi_{1}) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}} = 0.$$ Indeed, by Strichartz estimates, the dominated convergence theorem, and small data arguments, for some $T < \infty$ sufficiently large, $(\ref{1.1})$ has a solution $u$ on $(-\infty, -T]$ such that $$\label{5.14}
\| u \|_{L_{t,x}^{4}((-\infty, -T] \times \mathbf{R}^{3})} \lesssim \epsilon, \qquad (u(-T, x), u_{t}(-T, x)) = S(-T)(\phi_{0}, \phi_{1}).$$ and by Strichartz estimates, $$\label{5.15}
\lim_{t \rightarrow +\infty} \| S(t)(u(-t), u_{t}(-t)) - (\phi_{0}, \phi_{1}) \|_{\dot{H}^{1/2} \times \dot{H}^{-1/2}} \lesssim \epsilon^{3}.$$ Then by the inverse function theorem, there exists some $(u_{0}(-T), u_{1}(-T))$ such that $(\ref{1.1})$ has a solution that scatters backward in time to $S(t)(\phi_{0}, \phi_{1})$, and by Theorem $\ref{t4.0}$, this solution must also scatter forward in time. Therefore, $$\label{5.16}
S(-t_{n}^{j})(\lambda_{n}^{j} \phi_{0}^{j}(\lambda_{n}^{j} x), (\lambda_{n}^{j})^{2} \phi_{1}^{j}(\lambda_{n}^{j} x))$$ converges strongly to $$\label{5.17}
(\lambda_{n}^{j} u^{j}(-\lambda_{n}^{j} t_{n}^{j}, \lambda_{n}^{j} x), (\lambda_{n}^{j})^{2} u_{t}^{j}(-\lambda_{n}^{j} t_{n}^{j}, \lambda_{n}^{j} x))$$ in $\dot{H}^{1/2} \times \dot{H}^{-1/2}$, where $u^{j}$ is the solution to $(\ref{1.1})$ that scatters backward in time to $S(t)(\phi_{0}^{j}, \phi_{1}^{j})$, and the remainder may be absorbed into $(R_{0, n}^{N}, R_{1, n}^{N})$. In this case as well, $$\label{5.18}
\| u^{j} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} \leq M_{j} < \infty.$$ The proof for $\lambda_{n}^{j} t_{n}^{j} \searrow -\infty$ is similar.
Also, by $(\ref{5.8})$, there are only finitely many $j$ such that $\| \phi_{0}^{j} \|_{\dot{H}^{1/2}} + \| \phi_{1}^{j} \|_{\dot{H}^{-1/2}} > \epsilon$. For all other $j$, small data arguments imply $$\label{5.19}
\| u^{j} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})} \lesssim \| \phi_{0}^{j} \|_{\dot{H}^{1/2}} + \| \phi_{1}^{j} \|_{\dot{H}^{-1/2}}.$$ Then by the decoupling property $(\ref{5.7})$, $(\ref{5.12})$, $(\ref{5.19})$, and Lemma $\ref{l6.2}$, $$\limsup_{n \nearrow \infty} \| u^{n} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})}^{2} \lesssim \sum_{j} \| u^{j} \|_{L_{t,x}^{4}(\mathbf{R} \times \mathbf{R}^{3})}^{2} < \infty.$$ This proves Theorem $\ref{t1.2}$. $\Box$
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|
---
abstract: 'In this short note we prove that the Farrell-Jones Fibered Isomorphism Conjecture in $L$-theory, after inverting $2$, is true for a group whose some derived subgroup is free.'
address: |
School of Mathematics\
Tata Institute\
Homi Bhabha Road\
Mumbai 400005, India
author:
- 'S. K. Roushon'
date: 'April 17, 2007.'
title: '$L$-theory of groups with unstable derived series'
---
Introduction
============
In [@R] it was shown that the Fibered Isomorphism Conjecture in $L^{-\infty}$-theory (\[[@FJ], 1.7\]), after inverting $2$ (that is for $\ul L^{-\infty}=L^{-\infty}\otimes {\Bbb Z}[\frac{1}{2}]$), is true for a large class (denoted $\F$) of groups including poly-free groups and one-relator groups.
Here we deduce the conjecture, using the results in [@R], for groups whose some derived subgroup is free. See Remark \[remark\] regarding the relevance of this class of groups.
Throughout the article, by ‘group’ we mean ‘countable group’
We prove the following.
\[solvable\] Let $\G$ be a group. Then the Fibered Isomorphism Conjecture of Farrell and Jones for the $\ul L^{-\infty}$-theory is true for the group $\G\wr F$ if it is true for $\G^{(n)}\wr F$ for some $n$, where $F$ is a finite group and $\G^{(n)}$ denotes the $n$-th derived subgroup of $\G$.
In other words, the $\flv$ (or the $\flf$) is true for $\G$ if the $\flv$ (or the $\flf$) is true for $\G^{(n)}$ for some $n$.
For notations and statement of the conjecture see \[[@R], section 2\].
Consider the following exact sequence. $$1\to \G^{(n)}\to \G\to \G/\G^{(n)}\to 1.$$
Note that $\G/\G^{(n)}$ is a solvable group. Hence applying the hypothesis, Corollary \[solvcor\] and \[[@R], $(2)$ of lemma 2.13\] we complete the proof of the Theorem.
\[free\] Let $G$ be a finite index subgroup of a group $\G$. Assume that $G^{(n)}$ is a free group for some $n$. Then the $\flv$ (or the $\flf$) is true for $\G$.
Using Theorem \[mth\] and Lemma \[imp\] we can assume that $\G=G$. Next we only need to recall that by \[[@R], main lemma\] the $\flv$ (or the $\flf$) is true for any free group and then apply Theorem \[solvable\].
\[remark\][Groups whose derived series does not stabilize (or some derived subgroup is free or surjects onto a free group) are of interest in group theory and topology. See [@R1]. In fact in [@R1] we predicted that these kind of groups appear more often than other groups.]{}
Let us recall the following definition from [@R].
\[begin\] (\[[@R], definition 1.1\])
Let $\F$ be the smallest class of groups satisfying the following conditions
- The following groups belong to $\F$.
1\. Finite groups. 2. Finitely generated free groups. 3. Cocompact discrete subgroups of linear Lie groups with finitely many components.
- (Subgroup) If $H<G\in \F$ then $H\in \F$
- (Free product) If $G_1, G_2\in \F$ then $G_1*G_2\in \F$.
- (Direct limit) If $\{G_i\}_{i\in I}$ is a directed sequence of groups with $G_i\in \F$. Then the limit $\lim_{i\in I}G_i\in \F$.
- (Extension) For an exact sequence of groups $1\to K\to G\to N\to 1$, if $K, N\in \F$ then $G\in \F$.
Let $A$ and $B$ be two groups then by definition the wreath product $A\wr B$ is the semidirect product $A^B\rtimes B$ where the action of $B$ on $A^B$ is the regular action. Let ${\cal {VC}}$ and ${\cal {FIN}}$ denote the class of virtually cyclic groups and the class of finite groups respectively.
We proved the following theorem in \[[@R], theorem 1.1\].
\[mth\] (\[[@R], theorem 1.1\]) Let $\Gamma\in \F$. Then the following assembly maps are isomorphisms for all $n$, for any group homomorphism $\phi :G\to
\Gamma\wr F$ and for any finite group $F$. $${\cal H}^G_n(p, {\bf \ul L}^{-\infty}):{\cal
H}^G_n(E_{\phi^*{\cal
{VC}}(\G\wr F)}(G), {\bf \ul L}^{-\infty})\to
{\cal H}^G_n(pt, {\bf \ul L}^{-\infty})\simeq
\ul L_n^{-\infty}({\Bbb
Z}G).$$ $${\cal H}^G_n(p, {\bf \ul L}^{-\infty}):{\cal
H}^G_n(E_{\phi^*{\cal
{FIN}}(\G\wr F)}(G), {\bf \ul L}^{-\infty})\to
{\cal H}^G_n(pt, {\bf \ul L}^{-\infty})\simeq
\ul L_n^{-\infty}({\Bbb
Z}G).$$
In other words the Fibered Isomorphism Conjecture of Farrell and Jones for the $\ul L^{-\infty}$-theory is true for the group $\G\wr F$. Equivalently, the $\flv(\G)$ and the $\flf(\G)$ are satisfied (see \[[@R], definition 2.1\] for notations).
In [@R] we showed that $\F$ contains some well-known classes of groups. Here we see that $\F$ also contains any virtually solvable group.
\[hyper\] $\F$ contains the class of virtually solvable groups.
Let $\G$ be a virtually solvable group. Using the ‘direct limit’ condition in the definition of $\F$ we can assume that $\G$ is finitely generated, for any countable infinitely generated group is a direct limit of finitely generated subgroups. The following Lemma shows that we can also assume that the group $\G$ is solvable.
\[imp\] Let $G$ be a finitely generated group and contains a finite index subgroup $K$. If $K\in \F$ then $G\in \F$.
By taking the intersection of all conjugates of $K$ in $G$ we get a subgroup $K'$ of $G$ which is normal and of finite index in $G$. Therefore, we can use ‘subgroup’ and ‘extension’ conditions in the definition of $\F$ to conclude the proof of the Lemma.
Hence we have $\G$ a finitely generated solvable group. We say that $\G$ is $n$-step solvable if $\G^{(n+1)}=(1)$ and $\G^{(n)}\neq (1)$. The proof is by induction on $n$. Since countable abelian groups belong to $\F$ (see \[[@R], lemma 4.1\]), the induction starts.
So assume that a finitely generated $k$-step solvable group for $k\leq n-1$ belong to $\F$ and $\G$ is $n$-step solvable.
We have the following exact sequence. $$1\to \G^{(n)}\to \G\to \G/\G^{(n)}\to 1.$$
Note that $\G^{(n)}$ is abelian and $\G/\G^{(n)}$ is $(n-1)$-step solvable. Using the ‘extension’ condition and the induction hypothesis we complete the proof.
Applying Theorems \[mth\] and \[hyper\] we get the following.
\[solvcor\] The $\flv$ (or the $\flf$) is true for any virtually solvable group.
[60]{}
F.T. Farrell and L.E. Jones, Isomorphism conjectures in algebraic $K$-theory, , 6 (1993), 249-297.
W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones Conjectures in $K$- and $L$-theory, Handbook of K-theory Volume 2, editors: E.M. Friedlander, D.R. Grayson, (2005) 703-842, Springer.
S.K. Roushon, The isomorphism conjecture in $L$-theory: poly-free groups and one-relator groups, math.KT/0703879.
[to3em]{}, Topology of 3-manifolds and a class of groups II, 10 (2004), Special Issue, 467-485.
|
---
author:
- |
Aaron Harlap[[$^\dag$]{}]{}[^1] Deepak Narayanan[[$^\ddagger$]{}]{}\
\
Amar Phanishayee[[$^\star$]{}]{} Vivek Seshadri[[$^\star$]{}]{} Nikhil Devanur[[$^\star$]{}]{} Greg Ganger[[$^\dag$]{}]{} Phil Gibbons[[$^\dag$]{}]{}\
\
[*[[$^\star$]{}]{}Microsoft Research [[$^\dag$]{}]{}Carnegie Mellon University [[$^\ddagger$]{}]{}Stanford University*]{}
title: '**[[PipeDream]{}: Fast and Efficient Pipeline Parallel DNN Training ]{}**'
---
[^1]: Work started as part of an internship at Microsoft Research.
|
---
author:
- 'H.A.S. Reid'
- 'H. Ratcliffe'
bibliography:
- 't3\_reid\_ratcliffe\_ms1742.bib'
date: 'Received February 2014; accepted March 2014'
title: A Review of Solar Type III Radio Bursts
---
Introduction
============
The Sun is the most efficient and prolific particle accelerator in our solar system. Electrons are regularly accelerated to near-relativistic energies by the unstable magnetic field of the solar atmosphere. Solar flares are the most violent example of such acceleration, with vast amounts of energy released when the solar magnetic field reconfigures to a lower energy state, releasing energy on the order of $10^{32}$ ergs and accelerating up to $10^{36}$ electrons per second in the solar atmosphere [e.g. @Emslie_etal2012].
Solar radio bursts come in a variety of forms, classified by how their frequency changes in time, known as their frequency drift rate. Initially three types of radio emission were named type I, II and III in order of ascending drift frequency [@WildMccready1950], with types IV and V introduced later. Each type has subtypes that further describe the array of complex behaviour these radio bursts display. In this study we are going to concentrate on the most prolific type of solar radio burst, the type III radio burst. These are a common signature of near-relativistic electrons streaming through the background plasma of the solar corona and interplanetary space, offering a means to remotely trace these electrons. Moreover, their dependence on the local plasma conditions means they act as a probe of the solar coronal plasma and the plasma of the solar wind. The example type III burst in Figure \[fig:typeIII\] shows their main features; they are very bright, transient bursts that usually drift from higher to lower frequencies over time
The precise mechanism for the acceleration of solar electrons is still debated but it is generally attributed to the reconfiguration of an unstable coronal magnetic field, resulting in the conversion of free magnetic energy to kinetic energy. This acceleration usually occurs at solar active regions but can occur when the magnetic field in a coronal hole interacts with the surrounding magnetic field (a process known as interchange reconnection). Moreover, type III radio bursts are observed in relation with X-ray bright points [@Kundu_etal1994].
{width="0.59\columnwidth"}
The first theory of type III bursts was described by @GinzburgZhelezniakov1958. They considered the two-stream instability of an electron beam which generates Langmuir (plasma) waves at the local plasma frequency that can be converted into electromagnetic emission. They proposed that scattering by plasma ions would produce radiation at the plasma frequency (the fundamental component), while the coalescence of two Langmuir waves could produce the second harmonic. The theory has been subsequently discussed and refined by many authors , but the basic two-step process, production of Langmuir waves followed by their conversion into EM emission, remains the same. An overview of the dominant processes is shown in Figure \[fig:typeIII\_flow\].
This *plasma emission mechanism* is now the generally accepted model for type III burst generation. Analytical calculations over many years have shown it is certainly capable of explaining the drift, brightness and harmonic structure of bursts. Alternatives, such as strong turbulence effects like Langmuir wave collapse [e.g. @1983SoPh...89..403G]; conversion of Langmuir waves directly into EM waves [e.g. @Huang1998] and emission via the maser mechanism [e.g. @2002ApJ...575.1094W] can explain some of the properties of type IIIs. Where relevant, we assume plasma emission throughout this review.
The purpose of this study is to give an overview on the type III radio burst as a probe of both accelerated electrons and the background plasma it travels through. In Section \[sec:observations\] we give a detailed account of the observational characteristics of type III radio bursts and we describe how they relate to the properties of the generating electron beam. Section \[sec:electrons\] reviews in-situ plasma observations connected with type III bursts. In Section \[sec:theory\] we will cover some of the theoretical work that has been done on electron beams responsible for type III radio bursts. In Section \[sec:em\] we will give an overview of the theory involved with the generation of electromagnetic emission from plasma waves. Finally we look towards the future in Section \[sec:conclusion\] by covering the “state of the art” simulation codes and the next generation radio telescopes.
![A flow diagram indicating the stages in plasma emission in an updated version on the original theory (adapted from [@Melrose2009])[]{data-label="fig:typeIII_flow"}](ms1742_fig2.png){width="0.59\columnwidth"}
Observed properties of type IIIs {#sec:observations}
================================
A solar type III radio burst is a transient burst of radio emission that starts at higher frequencies and drifts to lower frequencies as a function of time. Their durations, frequency extent and even how fast they drift varies from burst to burst. Their size and intensity are different at different frequencies. The nonlinear emission mechanism can lead to substructure in the dynamic spectra like clumpy emission and harmonic pairs. There are a number of variant type III busts: some have positive drift rates in frequency, some have different polarisation and some vary dramatically in frequency and time. Other wavelengths of light can also be observed at the same time as type III bursts.
We review the main properties of the radio emission that we call a type III burst. When relevant we also highlight what information can be deduced from the observations regarding both the exciting electron beam and the background heliospheric plasma. Type III radio bursts have been studied intensely for the last 60 years so it is beyond our scope to mention all works that have been done on the subject. We try to focus on the most recent works and to mention the range and variability of each property.
Frequency and time {#sec:freqtime}
------------------
### Frequency extent
The starting frequency of type III bursts varies dramatically from burst to burst. During large solar flares type III bursts can start at frequencies in the GHz [e.g. @1983ApJ...271..355B; @StaehliBenz1987; @Benz_etal1992; @Melendez_etal1999]. Typically type III bursts will start at 10s or 100s MHz and can start at even lower frequencies. When two components are observed (see Section \[sec:harmStruc\]), the fundamental often starts at lower frequencies than the harmonic, with the ratio of their onset frequencies often as large as $1:3$ or $1:5$ . When type III bursts are observed at high frequencies they are usually in groups corresponding to multiple accelerated electron beams during solar flares. The variation in starting frequency can be attributed to several causes. Bursts at very high frequency (GHz range) are subject to absorption and have to be intense to be visible above the quiet-Sun background. Moreover collisions in the background plasma damp both electrons and plasma waves making it harder to generate radio emission (see Section \[sec:el\_sims\]). Favourable conditions for high frequency type III emission are acceleration regions at high densities that produce intense electron beams. High energy solar flares are the most common cause of such type III bursts.
The stopping frequency of type III bursts is equally as variable. Some type III bursts will only exist at high frequencies above 100 MHz. Such bursts are attributed to electron beams that are confined to the solar corona by a magnetic field with no access to the upper corona or the heliosphere (closed magnetic field). Other type III bursts can make it to kHz frequencies, going down to 10s kHz or below, where 20 kHz corresponds to the plasma frequency near the Earth. Such bursts are considered interplanetary type III bursts. What frequency constitutes an interplanetary type III burst is not defined, as such a classification requires defining when the corona stops and interplanetary space starts. Certainly if a type III burst makes it to frequencies below 1 MHz (roughly distance of 7 $R_\odot$) it should be considered “interplanetary”. Interplanetary type III bursts are usually made up of a number of distinct type III bursts at high frequencies that cannot be differentiated at lower frequencies (see Figure \[fig:typeIII\]). Using spacecraft, @Leblanc_etal1995 [@Leblanc_etal1996] analysed the stopping frequency of type III radio bursts, finding that weaker radio bursts generally had higher stopping frequencies. What process causes the stopping frequency is not clear but @Leblanc_etal1995 hypothesise that is could be due to electron beam dilution and/or background density fluctuations.
### Burst Duration
The duration of individual type III bursts varies inversely as a function of frequency. Velocity dispersion of the electron beam exciter elongates the beam as a function of distance. As such the beam spends more time at lower frequencies. The rise and decay of type III radio emission in the interplanetary medium generally takes the form of a Gaussian total rise time $t_e$ followed by a power-law e-folding decay time $t_d$. The general trend in emission is a shorter rise time $t_e < t_d$.
A statistical study of rise and decay time between 2.8 MHz and 67 kHz was undertaken by @Evans_etal1973. The study found with a least squares fit through the data the relations $t_e=4.0\times 10^8 f^{-1.08}$ and $t_d=2.0\times 10^8 f^{-1.09}$ where $t$ is in seconds and $f$ is in Hz. A similar result was found by @AlvarezHaddock1973b using a number of different studies in the frequency range 200 MHz to 50 kHz. They found a decay time $t_d=10^{7.7} f^{-0.95}$. Rise and decay times that are roughly inversely proportional to frequency ($t_e\propto t_d\propto f^{-1}$) have been found at decimetric and microwave frequencies [@Benz_etal1983; @StaehliBenz1987]. It is important to note here that the rise and decay times are proportional to each other (longer rise times lead to longer decay times). The power-law form of the decay time is currently unexplained as collisional damping of Langmuir waves would lead to a much longer decay time [e.g. @Abrami_etal1990]. There must be another process which accounts for either the spatial damping of Langmuir waves or the suppression of Langmuir waves inducing electromagnetic emission. Harmonic emission is generally more diffuse in time, having a slower rise and decay time than fundamental emission for a given frequency [e.g. @Caroubalos_etal1974]. Given the differences in the emission mechanisms, particularly the potential involvement of rapidly damped ion-sound waves in fundamental emission, this difference is unsurprising.
### Frequency drift rate
The defining property of a type III radio burst in comparison to other solar radio bursts is their high drift rate $df/dt$ (usually measured in MHz s$^{-1}$). Exactly how the frequency drift rate is calculated is subjective with some authors using the drift rate of the type III burst onset and some authors using the drift rate of the type III burst peak flux. One major survey of type III burst drift rate was done by @AlvarezHaddock1973 who used the rise time of type III bursts between 3 MHz and 50 kHz and combined then with eight other studies up to 550 MHz. They reported a least squares straight line fit to the frequency drift rate over all four orders of magnitude such that $df/dt=-0.01f^{1.84}$. More recently a linear dependence ($df/dt\propto f$) was found for the frequency drift rate by @Melnik_etal2011 between the frequencies 30 MHz to 10 MHz for a number of powerful radio burst occurring during a two month period in the solar maximum of 2002. The value of the frequency drift rate was similar at 10 MHz to @AlvarezHaddock1973 but smaller by a factor of two at 30 MHz. Going to higher frequencies, studies by @Aschwanden_etala1995 and @Melendez_etal1999 looked at radio bursts between 100 MHz and 3000 MHz and found drift rates $df/dt=0.1 f^{1.4}$ and $df/dt= 0.09 f^{1.35}$ respectively. Again, these values differ from @AlvarezHaddock1973 but it must be noted here that @AlvarezHaddock1973 did not use very many points greater than 200 MHz in their study. Going to even higher frequencies a recent study by @Ma_etal2012 looked at microwave type III bursts from 625 MHz to 7600 MHz, similar in range to events considered by @StaehliBenz1987. @Ma_etal2012 found a much faster frequency drift rate, $df/dt = -2.6\times 10^{-6}f^{2.7}$ in the range 635 MHz to 1500 MHz. Both studies quote varying values from hundreds to 17000 MHz s$^{-1}$ at higher frequencies.
The drift rate of type III bursts can be used to assume the speed of the exiting electron beam. Assuming a coronal (or interplanetary) density structure to obtain electron density as a function of distance $n_e(r)$, one can find frequency as a function of distance $f(r)$. Assuming either fundamental or harmonic emission (see Section \[sec:harmStruc\]) $df/dt$ can be converted to $dr/dt=v$ as a estimation of the speed of exciting electrons. Derived exciter speeds for type III bursts are usually fractions of the speed of light and can range from $>0.5$ c in the corona [@Poquerusse1994; @Klassen_etal2003] down to much slower velocities nearer the Earth. Exciter speed derived by @AlvarezHaddock1973 are fast, typically being $>0.2$ c. Exciter speeds derived by @Aschwanden_etala1995 and @Melendez_etal1999 are in the regime of 0.14 c (7 keV). These are slower than the exciter speeds of @AlvarezHaddock1973, however, @Melendez_etal1999 specifically state using the maximum of type III emission and not the onset of type III emission. The latter would lead to higher drift rates and might explain some of the discrepancy between the studies. We expect the drift rate to be faster in the deep corona as the frequency of the background plasma is changing much faster with distance.
Flux and size {#sec:fluxsize}
-------------
### Source flux
At a given frequency the distribution of source flux of type III bursts varies over many orders of magnitude. Statistically the distribution can be approximated as a power-law with a spectral index of 1.7 [@Nita_etal2002; @Saint-Hilaire_etal2013]. This spectral index has been found at GHz frequencies and frequencies at 100s MHz respectively. It is the same spectral index that has been observed in X-ray flare energies [@Crosby_etal1993; @Hannah_etal2008]. This spectral index is also seen in self-organised criticality (SOC) studies [e.g. @LuHamilton1991; @Aschwanden2012].
Statistically, the peak source flux of type III bursts increases as a function of decreasing frequency in the corona. This increase goes up to roughly 1 MHz. On average the peak flux then decreases as a function of decreasing frequency [@Weber1978; @Dulk_etal1984; @Dulk_etal1998]. Exactly how the flux of individual type III bursts varies changes depending on the burst. Some bursts have been observed to peak as low as 0.1 MHz while others peak as high as 5 MHz [@Dulk2000]. An example spectrum can be found in @Dulk_etal2001 showing a rise and decay of type III flux over two and a half orders of magnitude, starting around 50 MHz and ending at 20 kHz. Recently @Saint-Hilaire_etal2013 statistically examined type III burst source fluxes at 450 MHz to 150 MHz. By fitting the power-law distributions of source flux they found the normalisation constants varied with frequency as $A\propto f^{-2.9}$. The increase in source flux as a function of decreasing frequency can be attributed to a number of factors including the increasing ease to generate Langmuir waves, the increasing velocity dispersion of the electron beam as a function of distance and the decrease in collisions.
### Source Size {#sec:sourcesize}
The size of type III bursts increases with decreasing frequency. Measurements at various frequencies for a variety of bursts give averages (half widths to 1/e brightness) of 2 arcmins at 432 MHz, 4.5 arcmins at 150 MHz [@Saint-Hilaire_etal2013], 11 arcmin at 80 MHz, 20 arcmin at 43 MHz [@DulkSuzuki1980], 5 degrees at 1 MHz, 50 degrees at 100 kHz [@Steinberg_etal1985] and 1 AU at 20 kHz (1 AU) [@Lin_etal1973]. The electron beam exciter is guided by the solar/interplanetary magnetic field that expands as a function of distance from the Sun. As the background electron density (frequency) decreases as a function of distance from the Sun, type III burst source sizes increase at lower frequencies. The source sizes above 20 MHz are measured through ground based radio interferometers. Radio emission below $10$ MHz does not typically make it through the ionosphere, so source sizes here are measured using the modulation factor of antenna signal due to spacecraft spin.
A comprehensive study of type III radio source sizes at low frequencies is undertaken by @Steinberg_etal1985 who deduces an $f^{-1}$ variation of source angular size with observing frequency at frequencies below 2 MHz. This is directly proportional to the distance from the Sun implying expansion in a fixed cone of $80^o$ with the apex in the active region. However, there was a correlation between the local plasma frequency and the source size. Scattering from density inhomogeneities in the background plasma expands the apparent source size and occurs within 0.2 AU at frequencies $<200$ kHz. This results in electron beams expansion in a fixed cone of $30^o-40^o$ with the apex in the active region. In the corona @Saint-Hilaire_etal2013 also found an approximate $f^{-1}$ variation from the mean observed size between 150 and 432 MHz over 10 years of type III data using the Nançay Radioheliograph. They also fitted the high end of the source size distribution with a power-law in both size and frequency, finding that the normalisation constant varied as $f^{-3.3}$. It is not obvious why the exponent has such a high magnitude but it could be related to multiple sources merging into one at lower frequencies, given the constraint of 10 second data required for such a huge study of radio bursts.
### Brightness temperature
The brightness temperature $T_b$ of a radio source is the temperature which would result in the given brightness if inserted into the Rayleigh-Jeans law. The brightness temperature can be measured with knowledge of the flux density and the source size and is a commonly used metric in radio astronomy. Type III bursts are characterised by their very large brightness temperatures that typically lie within the range $10^6$ K to $10^{12}$ K although it can rise as high as $10^{15}$ K [@SuzukiDulk1985]. For 10 years of Nançay Radioheliograph data, @Saint-Hilaire_etal2013 investigates the histogram of peak brightness temperatures within the range 150 - 450 MHz. They find that the brightness temperature at all 6 frequencies varies as a power-law with spectral index around -1.8 (larger $T_b$ is less likely than smaller $T_b$). The brightness temperatures varied from $10^6$ K to $10^{10}$ K with a few events $>10^{11}$ K. At lower frequencies the trend is continued, with $T_b$ increasing with decreasing frequency up to around 1 MHz and then either decreasing or remaining constant [@Dulk_etal1984]. The decrease of $T_b$ with decreasing frequency after 1 MHz cannot be explained due to increasing source size. There is also a weak anti-correlation between rise times and $T_b$. Fundamental type III emission is thought to produce higher $T_b$ than harmonic emission [@Dulk_etal1984; @Melrose1989], relating to both a larger flux density and a smaller source.
Scattering, polarisation and harmonic structure
-----------------------------------------------
### Scattering
It is generally accepted that the source sizes of EM emission from the corona will be affected by scattering of radiation en route between source and observer. The rapid time variations in emission are inconsistent with a large source, and thus observations of higher frequency radio bursts with sizes of several arcminutes must be due to scattering. Additional motivation comes from the observations summarised by @1994ApJ...426..774B, that no fine structure is observed in sources at scales below $20^{\prime\prime}$ at 1.5 GHz and around $40^{\prime\prime}$ at 300 MHz. Ray-tracing analyses have been used to study the effect in context of type III emission and a comparable analytical approach was developed by . For example @1972PASAu...2...98R show that a point source at 80 MHz would be observed to be as large as 2 arcminutes, and the directivity of the radiation would be greatly reduced, particularly for fundamental emission, which is emitted into a relatively narrow cone.
A directivity study on 10 years of data from 1995 to 2005 was done by @Bonnin_etal2008 using both the Ulysses and WIND spacecraft. They examined about 1000 type III bursts observed in the frequency range of 940 kHz to 80 kHz. They find that the radiation diagram axis shifts significantly to east from the local magnetic field direction, similar to previous observations [@Hoang_etal1997], with this shift increasing for decreasing frequencies. They attribute this to a density compression when the fast and slow solar wind meet, resulting in a transverse density gradient that bends radiation in the eastward direction. They find no significant variation of directivity with solar activity and latitude.
### Harmonic Structure {#sec:harmStruc}
In a large number of observed radio bursts both the fundamental (F) and harmonic (H) components are seen. Bursts above around 100 MHz generally do not show two separate components and the observed component is thought to be the harmonic. Below 100 MHz both fundamental and harmonic emission are frequently seen [e.g. @Wild_etal1954a; @Stewart1974; @DulkSuzuki1980; @SuzukiDulk1985; @RobinsonCairns1994; @RobinsonCairns1998]. Various theoretical estimates suggest that F emission is more common at large distances from the Sun, while the H component dominates closer, i.e. at higher frequencies [@RobinsonCairns1994; @Dulk_etal1998]. For emission at very high frequencies in the GHz and high MHz ranges, the absorption of radiation due to inverse bremsstrahlung becomes important. Under the assumption of a locally exponential density profile the optical depth is easily calculated . The F emission is highly suppressed above around 500 MHz, while the harmonic is only affected above 1 GHz or so [e.g. @RobinsonBenz2000]. In the 100-500 MHz range absorption is unimportant and cannot explain the rarity of F emission.
The H-F ratio, naively expected to be 2:1, actually ranges from 1.6:1 to 2:1 with a mean near 1.8:1 [@Wild_etal1954a; @Stewart1974]. This may be explained by propagation effects altering the initial ratio of 2:1. Simply put, for EM radiation in plasma the group velocity tends to zero as the frequency approaches the plasma frequency whilst for frequencies much larger than $\omega_{pe}$ it tends to $c$. Thus the fundamental will be delayed relative to the harmonic emission from the same location [e.g. @Dennis_etal1984]. On the dynamic spectrum, the F component will be shifted to later times and the H-F ratio is correspondingly reduced. The occasional instance of smaller ratios around 1.5:1 may suggest the two components are not F and H but rather the second and third harmonics [see e.g. @1974SoPh...36..443Z; @2012OAP....25..181B].
### Polarisation {#sec:polarisation}
Type III bursts are usually weakly circularly polarized with H emission having a lower degree of polarisation than F emission [e.g. @Mclean1971; @SuzukiSheridan1977; @DulkSuzuki1980; @SuzukiDulk1985]. @DulkSuzuki1980 made a thorough analysis of polarisation characteristics of 997 bursts finding the average degree of polarisation of F-H pairs were 0.35 and 0.11 respectively while structureless bursts had only a polarization of 0.06. The maximum F polarisation was around 0.6. At high frequencies (164 - 432 MHz) polarisation has been found to peak either before or simultaneously with the peak flux but never after [@Mercier1990]. Polarisation was also found by @Mercier1990 to increase as a function of frequency, ranging from approximately $5~\%$ at 164 MHz to $15~\%$ at 432 MHz, in contrast to the results of @DulkSuzuki1980. Type III bursts with nearly 100 $\%$ polarisation have been observed at microwave frequencies [@Wang_etal2003]. For solar emission, any linear polarization in the source tends to be obliterated over any finite band of frequencies by differential Faraday rotation of the plane of polarisation during passage through the heliosphere [e.g. @SuzukiDulk1985].
Fundamental emission is generally produced very close to the plasma frequency and therefore below the X-mode cutoff. It would thus have polarisation close to 1. Effects such as mode coupling due to magnetic fields [e.g. @1964SvA.....7..485Z], and scattering due to low-frequency waves [@1984SoPh...90..139W; @1989SoPh..119..143M], or kinetic Alfven waves during propagation are certainly able to reduce the degree of polarisation but cannot explain why fully polarised emission is never seen. @1984SoPh...90..139W therefore proposed that the emission was depolarised to some extent within the source region itself, and that this was inherent to the emission process. For harmonic emission, it was initially thought that the Langmuir waves involved would coalesce head on, and therefore and the correction by @1980PASAu...4...50M concluded that harmonic emission must be weakly O-mode polarised. Later work by @1997SoPh..171..393W showed that relaxing the head-on condition allowed stronger polarisation, with less restriction on the participating Langmuir waves.
Type III burst variants
-----------------------
### Reverse and bi-directional type III bursts
Ordinary type III bursts are produced by an electron beam travelling outwards from the Sun, along the magnetic field, and drift from high to low frequency. Electrons travelling downwards into the solar atmosphere instead encounter plasma of increasing density, and thus drift from lower to higher frequencies. These radio bursts usually have a high starting frequency $> 500$ MHz [@IslikerBenz1994; @AschwandenBenz1997; @RobinsonBenz2000] and fast drift rates due to the increased spatial gradient of the high density lower coronal plasma. In some cases a reverse drift and a normal drift burst are produced at the same time, from the same acceleration region, leading to a bi-directional burst. These are relatively uncommon, with the study of @Melendez_etal1999 finding only 5$\%$ of bursts which extended above 1 GHz had both components in contrast to $66\%$ which had a reverse slope component.
The general suggestion to overcome strong inverse bremsstrahlung damping during escape is that the emission arises in dense loops embedded in less dense surrounding plasma [@Benz_etal1992], which strongly increases the escaping fraction of radiation. Even in this case, it is likely to be only the harmonic component which may be observed [@Chen_etal2013b]. Higher background densities increases the collisional rate and reduces the Langmuir wave growth rate. An increasing background plasma density reduces the level of Langmuir waves generated by an electron beam [e.g. @Kontar2001d]. High frequency type IIIs are thus less frequent and dimmer. In the study of @Melendez_etal1999 the peak fluxes were generally 10 sfu.
Recent simulations were carried out by @Li_etal2011 for bi-directional electron beams producing type III radiation. As previously found from observations they find that downward going electron beams produce much less intense radio emission than upwards propagating electron beams. That @Melendez_etal1999 observes some instances of more intense emission in the reverse type III emission points towards an asymmetry for these events in the electron beam properties, like density. @Li_etal2011 also find the background plasma properties are hugely important to allow the emission at higher frequencies to escape the corona.
### Type IIIb bursts {#sec:IIIb}
Frequency fine structures are commonly observed in type III bursts, and generally referred to as a type IIIb burst. Rather than the smooth emission of a “normal” type III, the emission is fragmented and clumpy. In general, it is the F component which shows this structure, and fine structure in the H component in type IIIb bursts is very rarely observed [@DulkSuzuki1980]. Type IIIb bursts consist of a chain of emission bands, sometimes referred to as striae [@DelanoeBoischot1972], each of which shows minimal frequency drift, whereas the chain as a whole drifts like a normal type III. found that the bandwidth of the striae in the 20-80 MHz range was commonly around 60-100 kHz, similar to their frequency separations. More recent observations by @2010AIPC.1206..445M show similar bandwidths for the striae of 50 kHz, but found strong variations in their duration, flux and frequency drift as a function of the source position on the solar disk.
The common belief [@SmithRiddle1975; @Melrose21980; @Melrose1983; @Li_etal2012] is that density inhomogeneities in the background plasma create a clumpy distribution of Langmuir waves and are the cause of this fine structure. Harmonic emission, with its dependence on Langmuir wave backscattering, may be expected to show less modulation in this case. Fundamental emission, with its rapid growth and dependence on rapidly damped ion-sound waves, would be more susceptible. If density fluctuations are the cause then we may infer this is less intense closer to the Sun, leading to the absence of type IIIb’s at higher frequencies.
Fine structure in the form of “sub-bursts”, also in the decimeter range, was reported by e.g. . In this case, the main burst envelope shows weak emission, and superposed on this are more intense stripes, with rapid frequency drifts on the order of 1 MHz s$^{-1}$, comparable to the main burst drift rate of around 3 MHz s$^{-1}$. These cannot be ascribed to density turbulence, and are as yet unexplained.
### Type U and J bursts
{width="0.59\columnwidth"} \[fig:type\_u\]
The frequency drift rate of radio bursts has been observed to change sign during a normal type III burst, taking the shape of an inverted U or J [@MaxwellSwarup1958] (Figure \[fig:type\_u\]). These bursts are believed to be electron streams travelling along magnetic fields confined to the corona [e.g. @KleinAurass1993; @Karlicky_etal1996]. For the J bursts, the radio emission stops when the electron beam reverses direction but with U bursts it continues to higher frequencies. The rate of occurrence is very low and they generally occur in harmonic emission within the range 20-300 MHz, although fundamental emission has been observed [e.g. @LabrumStewart1970; @AurassKlein1997 and references therein]. The rising and descending branches have spatially separated sources [@AurassKlein1997]. Their polarization is usually below 10 $\%$, agreeing with the properties of harmonic emission. Similar to reverse drift bursts their low occurrence could be to do with increased difficulties to generate Langmuir waves in an increasing density gradient.
### Type V bursts
Closely related to the type III burst are type V bursts, classified due to their long durations (minutes) and wide spectra [@Wild_etal1959]. The type V emission appears as a continuation of a type III burst in the dynamic spectra. Type V bursts are important because their explanation has to be consistent with any model of type III bursts. Type V bursts appear at low frequencies below 120 MHz and generally have 1-3 minute durations [@Dulk_etal1980]. The size of type V bursts increases rapidly with decreasing frequency, with full width at 1/e brightness on average 105 arcmins$^2$ at 80 MHz and 300 arcmins$^2$ at 43 MHz [@Robinson1977] similar to type III bursts [@DulkSuzuki1980]. Type V bursts have also been observed to move relative to the disk surface at speeds $\approx 2~\rm{Mm~s}^{-1}$ [@WeissStewart1965]. A similar problem to type IIIs occurs for the decay of type V emission where the characteristic time of collisional damping of Langmuir waves is much larger than the lifetime of type V emission. Type V polarizations are low (usually $< 0.07~\%$) which suggests harmonic emission. However, it is common to find their polarization opposite in the sense of the corresponding type III. @Dulk_etal1980 suggest the most likely reason for this change is due to X-mode rather than O-mode emission. This could be caused by increased isotropy in the Langmuir wave distribution as the condition for O-mode emission is Langmuir waves within 20$^o$ of the magnetic field. Another deviation of type V emission from their associated type III emission is the occurrence of large position differences, sometimes up to 1 $R_s$ [@WeissStewart1965; @Robinson1977]. This is not always observed and the positions of the type III and V can overlap or only be slightly displaced. One explanation of type V emission is low energy electrons travelling along different magnetic field lines or a variation of the beaming of emission changes the position of the centroids [@Dulk_etal1980]. Another explanation given for type V emission is the electron-cyclotron maser instability [@WingleeDulk1986; @Tang_etal2013].
### Type III-l bursts
Associated with coronal mass ejections (CMEs) and solar energetic protons are type III-l bursts [@Cane_etal2002; @MAcDowall_etal2003]. These bursts start at much lower frequencies than normal (hence the name typeIII-l), around 10 MHz and progress to lower frequencies as a function of time. Moreover the bursts usually start delayed with respect to the associated flare and last on average 20 minutes. It has been suggested [@Cane_etal2002] that type III-l bursts are produced by electrons accelerated by reconnection behind fast CMEs. Conflicting accounts were found by @Macdowall_etal2009 [@CliverLing2009] as the former found type III-l events were associated with intense SEP events favouring flare acceleration of the CME, whilst the latter found type III-l events associated with type II bursts favouring shock acceleration of the CME. However, @GopalswamyMakela2010 found that the presence of a type III-l burst does not always signify the presence of solar energetic protons.
### Type III storms
A phenomena called type III storms can occur when type III bursts are observed quasi-continuously over a period of days [@FainbergStone1970a; @FainbergStone1970b; @FainbergStone1971]. These type III bursts are narrow-band and usually occur at low frequencies $< 100$ MHz. Type III storms are usually observed as harmonic emission with a low degree of polarisation. However, fundamental-harmonic pairs are observed in type III storms with the fundamental component having a higher polarisation and is usually fragmented like a type IIIb burst [@DelanoeBoischot1972]. Type III storms have been associated with type I noise storms, a similar phenomena at higher frequencies but with no frequency drift rate [a good review of type I storms can be found by @Kai_etal1985] which in turn are associated with a solar active region [e.g. @Kayser_etal1987; @Gopalswamy2004]. The polarisation degree of type III storms can remain constant during the entire length of the storm. @Reiner_etal2007 analysed storms below 1 MHz and found only a $5\%$ degree of polarisation, different from the higher $25\%$ polarisation degree found for type III storms above 20 MHz by @Kai_etal1985. However, @Reiner_etal2007 found the polarisation peaks near central meridian crossing of the corresponding radio sources and decreases systematically with decreasing frequency. They deduce from these polarisation measurements that the coronal magnetic field (at least above the active regions they study) does not fall off as $1/R^2$ but slower.
{width="0.49\columnwidth"} \[fig:typeIII\_storm\]
Type III storms usually consist of many faint type III bursts. Analysing the flux distribution at two specific frequencies 1.34 MHz and 860 kHz, @Morioka_etal2007 found the occurrence rate of type III storm emission obeyed a power-law distribution with spectral index of -3.7 and -3.6 respectively. Different spectral indices between -2.36 to -2.10 were found by @Eastwood_etal2010 at a number of different frequencies between 3 and 10 MHz. The discrepancy between the two ranges of indices is suggested by @Eastwood_etal2010 to be due to the averaging of a few storms and the use of data binning in @Morioka_etal2007. In comparison the spectral index of flux occurrence rate of type I emission has been found to be -3 [@MercierTrottet1997].
Multi-wavelength studies {#sec:HXRs}
------------------------
### Type IIIs and X-rays
Type III radio bursts are not the only wavelength of light that can help diagnose the properties of accelerated electrons at the Sun. X-ray emission is frequently observed during solar flares and is believed to also be driven by accelerated electrons. Electrons travel into the dense solar chromosphere and thermalise via electron-ion Coulomb collisions. Bremsstrahlung X-rays are emitted [see @Holman_etal2011 as a recent review] and are detected both directly and through X-rays reflected from the solar surface [e.g. @KontarJeffrey2010]. We can use the X-ray signature to deduce the temporal, spatial, and energetic profile of the energised electrons [e.g. @Kontar_etal2011].
The simultaneous observation of hard X-rays (HXR) and metric/decimetric radio emission is commonplace during flares and the relationship between type III bursts and hard X-ray emissions has been studied for many years [see for example @PickVilmer2008 for a review]. An example flare showing simultaneous X-ray and type III emission is shown in Figure \[fig:nrh\_image\]. A recent study by @Benz_etal2005 [@Benz_etal2007] on 201 flares above GOES class C5 found that nearly all flares were associated with coherent radio emission between the range 4 Ghz to 100 MHz. The majority of flares in the study that had coherent radio emission had a type III burst or groups of type III bursts.
There have been many statistical studies between coherent type III radio emission and HXR bursts [e.g. @Kane1972; @Kane1981; @Hamilton_etal1990; @Aschwanden_etala1995; @ArznerBenz2005]. Bursts of type III radio emission have been found to temporally correlate with bursts in HXRs [e.g. @Aschwanden_etalb1995; @ArznerBenz2005]. However, there is often a small delay in the emission [e.g. @Dennis_etal1984; @Aschwanden_etala1995] of 2 seconds or less. A type III/X-ray correlation is systematically more likely when the intensity of the HXR or radio emission increases [@Kane1981; @Hamilton_etal1990]. Such a property could be the reason for the high association rate of $>C5$ GOES class flares found by @Benz_etal2005. A higher peak starting frequency of the type III bursts has also been found more likely in type III/HXR correlated events along with a low spectral index of HXRs (hard spectrum).
{width="0.59\columnwidth"}
Recently, @Reid_etal2014 found a correlation between the starting frequency of type III bursts and the hard X-ray spectral index in flares. The hard X-ray spectral index is proportional to the accelerated electron beam spectral index. A higher spectral index means the electron beam will become unstable faster and is thus able to generate type III radio emission at lower heights and consequently higher starting frequencies. @Reid_etal2011 [@Reid_etal2014] used the connection between type III starting frequency and HXR spectral index, first used by @Kane_etal1982 [@Benz_etal1983], to deduce the spatial characteristics of flare acceleration regions. They found acceleration region starting heights varied between 25 to 200 Mm with a mean of 100 Mm. The acceleration region vertical extent varied between 16 to 2 Mm with a mean of 8 Mm. Some heights are in good agreement with the height found from time-of-flight X-ray analysis [@Aschwanden_etal1998] but others are higher in the corona.
It should also be theoretically possible to observe hard X-rays associated with the type III producing electron beams in the upper corona. The background density is much lower and so emission is expected to be extremely faint [@Saint-Hilaire_etal2009]. That we have not detected such X-ray emission yet, @Saint-Hilaire_etal2009 use to estimate that the number of electrons escaping in type III producing electron beams must be $\leq 10^{34}$ electrons s$^{-1}$ above 10 keV.
The effect of producing Langmuir waves alters the distribution of electrons (see Section \[sec:theory\]). Such an effect has been studied in the context of flare accelerated electron beam propagating down through the corona and into the dense chromosphere to generate X-rays [@EmslieSmith1984; @HamiltonPetrosian1987; @McClements1987b; @McClements1987a; @Hannah_etal2009; @ZharkovaSiversky2011; @HannahKontar2011; @Hannah_etal2013]. If an electron beam is accelerated high enough in the corona it can become unstable to Langmuir wave growth and the spectral index of the electron beam can be modified. The electron spectrum tends to flatten (decrease in magnitude) and the total amount of X-rays emitted is less. For strong plasma inhomogeneity electrons can be re-accelerated to high energies (see Section \[sec:theory\]) and the total amount of X-ray radiation emitted can be more than predicted with a simple collisional model [@Hannah_etal2013].
### Type IIIs and EUV/X-ray jets {#sec:EUV}
The joint observations of type III bursts and soft X-ray (SXR) jets have been studied by a number of different authors [@Aurass_etal1994; @Kundu_etal1995; @Raulin_etal1996]. They found the emission was co-spatial and the centroids of the type III bursts at different frequencies were aligned with the soft X-ray jet indicating that particle acceleration is taking place when coronal jets occur. Jets that are correlated with type III bursts are also observed in EUV [e.g. @Innes_etal2011; @Klassen_etal2011; @Chen_etal2013a; @Chen_etal2013b] and are commonly associated with $^3$He-rich solar energetic electron events [@Wang_etal2006; @Pick_etal2006; @Nitta_etal2008]. The jets in these studies frequently arose from the border of an active region and a coronal hole. A study involving EUV and X-ray jets where simultaneous type III bursts were observed [@Krucker_etal2011] found hard X-ray footpoints that strongly suggest reconnection with a closed and open magnetic field. Hard X-rays have recently been observed to accompany EUV jets [@BainFletcher2009; @Glesener_etal2012] where type III bursts were also present. This further strengthens the case that acceleration of particles is present at the same time that thermal plasma is ejected from the solar corona in the form of a jet. Other evidence is the in-situ detection of electron spikes $<300$ keV [@Klassen_etal2011; @Klassen_etal2012] at 1 AU which are very closely temporarily correlated with type III bursts and EUV jets although there is a slight timing issue (see Section \[sec:electrons\]).
A recent paper by [@Chen_etal2013b] has used the new upgraded EVLA [@Perley_etal2011] to make images of a type III burst at frequencies between 1 GHz and 2 GHz that occurred at the same time as an EUV jet and HXR footpoint observations. From the images they infer the acceleration site of the type III bursts is below 15 Mm in the solar corona and the electron beam travels along bundles of discrete magnetic loops into the high corona. Moreover, by using EUV images they deduce that the coronal loops the type III producing electron beams are travelling along may be cooler and denser than the surrounding plasma. This dramatically helps the radio emission to escape and is similar to the scenario proposed by @Benz_etal1992 for allowing decimetric emission to escape the corona.
In-situ plasma observations {#sec:electrons}
===========================
Electron beams
--------------
The first in-situ observations of energetic particles [@vanAllen1965] opened up the non-electromagnetic window of flare accelerated particle observations. Since then solar energetic electron events have been found to be closely related observationally [e.g. @Ergun_etal1998; @Gosling_etal2003; @Krucker_etal2007] and theoretically to solar type III radio bursts, having a $99\%$ association rate [@Wang_etal2012].
Impulsive electron events often extend to 1 keV [@Lin_etal1996] with some even extending down to the 0.1 to 1 keV energy range [@Gosling_etal2003]. The presence of such low energy electrons favours injection sites high in the corona at altitudes $\geq 0.1 R_{\odot}$ as Coulomb collisions in the low solar atmosphere will cause low energy electrons to lose their energy. Electron time profiles generally show a rapid onset and also near time-of-flight velocity dispersion [e.g. @Lin1985; @Krucker_etal1999; @Krucker_etal2007]. Electrons also have a beamed pitch-angle distribution at lower energies $<20$ keV [e.g. @Lin1990; @Wang_etal2011] but can have very high pitch angle distributions at high energies of 300 keV.
A statistical study of 10 years of solar energetic electron events using the 3D plasma and Energetic Particle (3DP) instrument [@Lin_etal1995] on the WIND spacecraft was carried out [@Wang_etal2012] finding near 1200 events between 1995 and 2005. They found an occurrence rate of solar electron events can be fitted as a power-law at 40 keV and 2.8 keV with a spectral index between 1.08 - 1.63 and 1.02 - 1.38 respectively, that changes every year. This power-law is smaller than the one associated with type III radio bursts (1.7) observed at the high frequencies by @Saint-Hilaire_etal2013. One possible reason could be due to the high electron background flux at low energies [@Wang_etal2012] obscuring events that could steepen the power-law dependence
Solar impulsive electron events detected in-situ generally display broken power-law energy distributions with lower energies having harder spectra. Broken power-law distributions have been observed for some time [e.g. @Wang_etal1971] but their origin has remained ambiguous, being either a signature of the acceleration mechanism or a transport effect. A recent statistical survey was carried out by @Krucker_etal2009 on 62 impulsive events. They found the average break energy was $\approx 60$ keV with averaged power-law indices below and above the break of $\delta_{low} = 1.9 \pm 0.3$ and $\delta_{high} = 3.6 \pm 0.7$ respectively. The power-law indices have an average ratio $\delta_{low}/\delta_{high}$ of 0.54 with a standard deviation of 0.09. The power-law indices also correlate with a coefficient of 0.74. The presence of the broken power-law can be replicated by simulating electron beam propagation to 1 AU and their interactions with Langmuir waves (see Section \[sec:theory\]).
It is often believed that solar energetic electron events propagate scatter-free from the Sun to the Earth [e.g. @Wang_etal2006]. The observed correlation between the spectral indices of energetic electrons at the Sun from X-ray data and the Earth from in-situ data [@Lin1985; @Krucker_etal2007] is often viewed as an additional support for scatter-free transport. However, the injection time of energetic electrons at different energies does not correspond to the inferred injection time by looking at the associated type III bursts [@Krucker_etal1999; @HaggertyRoelof2002; @Cane2003; @Wang_etal2006; @Wang_etal2011; @Kahler_etal2011b; @Klassen_etal2012]. Low energy electrons (e.g. 10 keV) have an early arrival time while high energy electrons (e.g. 100 keV) have a late arrival time. The early arrival time could be due to pitch angle scattering [e.g. @Tan_etal2011] or electron interaction with Langmuir waves [@KontarReid2009]. It is not entirely clear what causes the later arrival time of the high energy electrons although an increased path length greater than 1 AU has been suggested [e.g. @Kahler_etal2011b].
Langmuir waves
--------------
In-situ observations of Langmuir waves associated with type III radio bursts were first taken by @GurnettAnderson1976 [@GurnettAnderson1977] using the Helios spacecraft at around 0.5 AU. They found that the distribution of Langmuir waves is very clumpy in space. Observations at 1 AU came later using the ISEE-3 spacecraft [@Lin_etal1981]. More recent observations of clumpy Langmuir waves show similar properties [e.g. @Kellogg_etal2009] using the time domain sampler in the WAVES instrument onboard the STEREO spacecraft [@Bougeret_etal2008]. Recently @Ergun_etal2008 modelled the localisation of Langmuir wave packets as a superposition of Langmuir eigenmodes and this concept has been worked on by other authors [e.g. @GrahamCairns2013b]. There has been a number of recent works claiming to observe Langmuir wave collapse in the solar wind [e.g. @Thejappa_etal2012; @Thejappa_etal2013]. However, other studies have analysed the same events but using all three components of the electric field [@Graham_etal2012a; @Graham_etal2012b] and find that the electric field is too weak for wave collapse to occur. Recently @Vidojevic_etal2011 [@Vidojevic_etal2012] analysed the statistical distribution of Langmuir waves measured near 1 AU by the Wind spacecraft [@Bougeret_etal1995]. By looking at the Langmuir wave distributions of 36 different electron beams with careful background subtraction they found that the distribution of Langmuir waves are more accurately modelled by a Pearson’s system of distributions [@Pearson1895] and not a log-normal distribution as previously thought.
Solar Wind Conditions
---------------------
The plasma of the solar corona and the solar wind is a non-uniform, turbulent medium with density perturbations at various length scales [see e.g. @BrunoCarbone2005 as a review]. Observations of how this background plasma fluctuates is important for modelling the Langmuir wave generation of electron beams responsible for type III radio bursts.
In-situ measurements have been used to determine the density spectrum near the Earth and between $0.3$ and $1$ AU with [*Helios*]{} [@MarschTu1990]. The spectral slope at frequencies below $10^{-3}$ Hz were found to have a tendency to get smaller the closer the spacecraft got to the Sun in the fast solar wind. These results were further extended by @Woo_etal1995 using Ulysses remote sensing radio measurements for distances $<40~R_s$ which predicted the decrease in r.m.s. deviation of the density turbulence in the fast solar wind at wavenumber $k=1.4\times10^6~\rm{km}^{-1}$. The results for the slow solar wind density turbulence [@MarschTu1990; @Woo_etal1995] showed a constant level around $10~\%$ which was also found in the later study from @Spangler2002. More recently @Chen_etal2012 [@Chen_etal2013c] have examined the slope of the density fluctuations power spectral density for 17 intervals of solar wind. They find a -5/3 power-law below $0.1~k\rho_i$ where $\rho_i$ is the proton gyroradius. Between 0.1 and 1 $k\rho_i$ the spectrum flattens to -1.1 in-line with other observations at 1 AU [e.g. @Celnikier_etal1983; @KelloggHorbury2005] and closer to the Sun [@ColesHarmon1989; @Coles_etal1991]. Between 3 and 15 $k\rho_i$ @Chen_etal2013c report a steepening of the spectral index to -2.75.
Background solar wind electrons are not in thermal equilibrium with a Maxwellian distribution but exist in a quasi-thermal state, with electrons extending to much higher energies [@Lin_etal1972]. Their velocity distribution function at all pitch angles is usually modelled using two convecting bi-Maxwellians, the core and the halo. A skewed distribution also exists in the fast solar wind parallel to the magnetic field direction. Known as the strahl, this high energy tail usually propagates away from the Sun and has a narrow pitch angle distribution between 10-20$^o$ wide. Observations of the background solar wind electrons have shown that a kappa distribution can better model the solar wind [@Maksimovic_etal2005; @Lechat_etal2010]. The kappa distribution more accurately models the electron temperature whilst having fewer free parameters than the sum of two Maxwellians. Recently [@LiCairns2014] have used a kappa distribution when modelling type III radio bursts. They found that a kappa distribution leads to a type III burst that drifts with a faster velocity. The increased level of higher energy background electrons in the kappa distribution reduce the Langmuir waves that are generated by slower velocity electrons. As such we only observe radio emission generated by higher energy electrons (and hence higher velocity) in the beam. The corresponding type III burst has a higher drift rate in frequency space.
Electron beam interaction with Langmuir waves {#sec:theory}
=============================================
The current generally accepted mechanism for type III burst generation is the plasma emission mechanism as described in the introduction. Since the original proposal of @GinzburgZhelezniakov1958, large amounts of work have been invested in the development of this theory. However, due to its complicated nature the exact details are still not fully understood. Many analytical treatments of the various steps exist and these have been supplemented by numerical simulations in order to develop the general picture of type III burst production as presented here.
The generation process is summarised in Figure \[fig:typeIII\_flow\]. In this section we discuss the electron propagation and the resulting Langmuir wave generation. The evolution of the Langmuir waves via non-linear scattering and wave-wave interactions is the topic of Section \[sec:em\] along with the processes that lead to second harmonic emission (at twice the local plasma frequency) and fundamental emission (first harmonic) at the local plasma frequency.
Quasi-linear theory
-------------------
The rapid frequency drift (see Section \[sec:freqtime\]) of type III bursts is caused by the rapid motion of the generating electrons, and can be derived from a prescribed plasma density profile by assuming that the electrons stream freely [e.g. @Karlicky_etal1996; @RobinsonBenz2000; @Ledenev_etal2004]. Alternatively, assuming a constant velocity for the exciting electrons we can derive the density profile of the ambient plasma [e.g. @2009ApJ...706L.265C]. However, the electrons do not stream freely but produce Langmuir waves. Early numerical simulations by e.g. @TakakuraShibahashi1976 [@Takakura1982], showed that Langmuir wave generation must be considered to reproduce the electron behaviour over long distances, and therefore even the basic properties of type III bursts.
The quasi-linear equations describing the interaction of electrons with Langmuir waves were introduced by @Vedenov_etal1962 [@DrummondPines1962]. The 1-D version of these is often employed in relation to type III bursts, as we may assume the generating electrons are closely tied to the solar magnetic field, as has been observed in-situ near the Earth (Section \[sec:electrons\]). Additional effects are seen in 2-d modelling [e.g. @Ziebell_etal2008a; @Ziebell_etal2008b; @Pavan_etal2009a; @Pavan_etal2009b; @Ziebell_etal2011; @Ziebell_etal2012] such as heating of the background plasma [e.g. @Yoon_etal2012; @Rha_etal2013]. However the 1-d dynamics are generally expected to dominate.
Writing the electron distribution function as $f(v,t)$ \[electrons cm$^{-4}$ s\], normalised so that $\int f(v,t) dv =n_e$ with $n_e$ the plasma density in cm$^{-3}$ and the spectral energy density of Langmuir waves as $W(v,t)$ \[ergs cm$^{-2}$\] with the normalisation $\int W(v,t)\, dv=E_L$ the total energy density of the waves in erg cm$^{-3}$ we have
$$\frac{\partial f(v,t)}{\partial t} =
\frac{4\pi^2e^2}{m_e^2}\frac{\partial }{\partial v}\frac{W}{v}\frac{\partial f(v,t)}{\partial v}
\label{eqk1}$$
$$\frac{\partial W}{\partial t} = \frac{\pi \omega_{pe}}{n_e}v^2W\frac{\partial f(v,t)}{\partial v}
\label{eqk2}$$
where $v$ signifies the electron kinetic velocity and the Langmuir wave phase velocity, as these must be equal for interaction to occur. This is equivalent to the Cherenkov resonance condition that the wave wavenumber and frequency $k, \omega$ and the particle velocity $v$ must satisfy $\omega = k v$.
The quasilinear equations are derived from the Vlasov equation by ignoring all other electromagnetic processes present in the plasma such as collisions. Several assumptions are needed, such as the absence of particle trapping, and the non-magnetisation of the electrons, so that their gyroradius is large and their trajectories almost linear. We also need the weak-turbulence condition which states that the energy in Langmuir waves is much less than that in the background plasma. Provided the perturbations created on a particle through wave-particle interactions are small (for example much less than an electron gyroradius in a gyroperiod) the quasilinear equations are valid.
The growth rate of waves is (amongst other things) proportional to $\partial f/\partial v$ (Equation (\[eqk2\])). When the electron distribution has a positive gradient in velocity space we excite waves and vice versa. An intuitive way to this about this is as follows. The energy exchange between the particles and waves occurs for particles with speeds $v$ close to $\omega/k$; particles with $v<\omega/k$ gain energy from the waves while particles with $v>\omega/k$ lose energy from the waves. The waves gain energy if we have more particles satisfying $v>\omega/k$ and the waves lose energy if we have more particles satisfying $v<\omega/k$.
The effect of wave generation on the electron distribution is more than simple energy loss. The right hand side of Equation (\[eqk1\]) is a diffusion equation, with the diffusion operator $D=W/v$. The transfer of energy from electrons to waves and back to the electrons causes the electrons to diffuse in velocity, smoothing out the positive gradient and slowing the energy transfer. The asymptotic (long time) solution to the quasi-linear equations is thus a plateau in velocity space [e.g. @VedenovRyutov1972; @Grognard1985; @Kontar2001a].
Figure \[fig:WPI\] shows an example of the resonant interaction of Langmuir waves with electrons, and the resulting quasi-linear relaxation. The left panel shows the electron distribution consisting of a background Maxwellian plasma (density $n_e$) and an electron beam (density $n_b$) with a density ratio of $n_e/n_b=10^{-4}$. The right panel shows the spectral energy density of Langmuir waves. This initial beam has positive gradient in velocity, and so is unstable to Langmuir wave growth. After a short time (0.15 s) the electrons have excited a high level of Langmuir waves, and the beam has diffused in velocity, becoming significantly wider in velocity space. At a later time (2.5 s) the electron distribution has relaxed to a plateau with zero gradient in velocity space, and the instability has therefore saturated. The level of induced Langmuir waves has grown, as has their width in wavenumber space.
![A demonstration of an electron beam inducing Langmuir waves in the background plasma. Left: the electron distribution function of a background Maxwellian and an electron beam in velocity space (normalised by the thermal velocity $v_{Te}$. The density ratio is $n_b/n_e=10^{-4}$, $v_{Te}=5.5\times10^{8}~\rm{cm~s}^{-1}$. Right: the spectral energy density of Langmuir waves where the x-axis is normalised by the Debye length. The three different colours correspond to different points in time.[]{data-label="fig:WPI"}](ms1742_fig6){width="79.00000%"}
Electron Beam Instability Distance
----------------------------------
The electrons accelerated during a solar flare are generally expected to have a power-law like distribution in velocity with a negative index. Such a distribution will not generate Langmuir waves. However, time-of-flight effects, where fast electrons outpace slower ones, produce a distribution with positive slope which is unstable to Langmuir wave generation. This occurs only after the electrons have travelled a sufficient distance. This instability distance is closely related to the starting frequency of type III bursts. When the instability distance is small the electron beam will produce Langmuir waves (and hence radio emission) close to the dense acceleration site resulting in a high burst starting frequency. When the instability distance is large, the electron beam travels into much lower density plasma before exciting Langmuir waves and so produces type III emission that starts at lower frequencies.
@Reid_etal2011 modelled the distance that an instantaneously injected electron beam would travel for a range of parameters. They found that the size of the acceleration region and the spectral index of the electron beam were the dominating factors in calculating the starting frequency of the type III burst given a fixed height for the acceleration region. Low spectral indices and small acceleration regions reduce the electron beam instability distance. Such a dependency was used by @Reid_etal2011 [@Reid_etal2014] to deduce heights and sizes of flare acceleration regions (see Section \[sec:HXRs\]). If the electrons are injected over a finite time (rather than instantaneously), the starting frequency will be affected [@ReidKontar2013]. The longer the injection time, the further the beam must travel to become unstable.
Electron Beam Persistence
-------------------------
In the early 60s, @Sturrock1964 noted that without something stopping the beam-plasma instability, the rate of Langmuir wave generation by an electron beam in the solar corona was enough that the beam would lose all energy after propagating only metres. However type III burst observations showed that beams could persist to distances of 1 AU. The *beam-plasma structure* was proposed to overcome this: here the electrons propagate accompanied by Langmuir waves which are continually generated at the front of the beam and reabsorbed at the back [e.g. @ZheleznyakovZaitsev1970; @Zaitsev_etal1972; @Melnik1995]. Numerical work by e.g. @TakakuraShibahashi1976 [@MagelssenSmith1977; @Kontar_etal1998; @Melnik_etal1999] confirmed that the beam-plasma structure could resolve the problem.
Other processes can suppress the beam-plasma instability and therefore the Langmuir wave growth. In particular, density fluctuations in the plasma can shift Langmuir waves out of resonance with the electrons [e.g. @SmithSime1979; @Muschietti_etal1985]. Because the growth rate of Langmuir waves is proportional to their current level at a given wavenumber, this can suppress the instability [in line with @Kontar2001d; @Ledenev_etal2004; @Li_etal2006b; @ReidKontar2010]. The density fluctuations can be either wave modes with wavelengths comparable to the Debye length [@Vedenov_etal1967; @GoldmanDubois1982; @Yoon_etal2005], or longer wavelength inhomogeneities [@Ryutov1969; @NishikawaRyutov1976; @SmithSime1979; @Kontar2001c; @Kontar_etal2012]. In the latter case, the Langmuir wave evolution can be described in the WKB approximation by [e.g. @Ryutov1969; @Kontar2001d] $$\frac{\partial W}{\partial t} - \frac{\partial \omega_{pe}}{\partial x}\frac{\partial W}{\partial k} = 0.$$ and as the plasma frequency is $\omega_{pe} \propto \sqrt{n_e}$, even weak density gradients are seen to strongly affect Langmuir waves.
For a propagating beam-plasma structure, density fluctuations will lead to energy loss, because when Langmuir waves are shifted out of resonance with the beam their energy can no longer be reabsorbed. Recent numerical simulations [@KontarReid2009; @ReidKontar2010; @ReidKontar2013] taking into account (amongst other processes) the refraction of Langmuir waves, found that the beam does lose energy but is still able to persist over distances of 1 AU or more. Additionally, a power-law electron spectrum injected at the Sun was found by @KontarReid2009 to be modified by density fluctuations and further from the Sun is better modelled as a broken power law [@ReidKontar2013]. This is due to the combination of Langmuir wave generation and their spectral evolution. Additionally, the spectral evolution of Langmuir waves to small wavenumbers can in some circumstances lead to acceleration of high-velocity electrons in the tail of the electron distribution, because the Langmuir waves can transfer energy from the beam electrons into this tail. This effect has been studied numerically by e.g. @Kontar2001d [@ReidKontar2010; @Ratcliffe_etal2012; @VoshchepynetsKrasnoselskikh2013; @Krafft_etal2013].
Langmuir Wave Clumping {#sec:el_sims}
----------------------
Plasma density fluctuations are also believed to be responsible for the clumpy Langmuir wave distribution observed in-situ near the Earth. Background density fluctuations are known to correlate to the clumpy observations of Langmuir waves in the solar wind [@Robinson_etal1992]. Recent simulations of the effects of density fluctuations on type III emission were carried out by @Li_etal2012. They considered a plasma density profile with several density enhancements and found these produced strong fine structure in both the fundamental and the harmonic component that resemble stria bursts. However, the fundamental component in these simulations is very weak and thus unobservable. The effect of temperature variation has also been investigated both for the electron temperature [@2011ApJ...730...20L] and the ion temperature [@2011ApJ...730...21L]. Variations only in electron temperature [@2011ApJ...730...20L] produced similar results to those presented for density fluctuations although in general the emission was less intense. Interestingly, large fluctuations in the ion temperature [@2011ApJ...730...21L] were able to produce more distinct fluctuations in the fundamental component and flux levels were able to reach 1 sfu. It is desirable to observe fragmented fundamental emission because the polarisation of type IIIb bursts is usually observed to be quite high (see Section \[sec:polarisation\]). The characteristics of the fine structures in these simulations resemble striae and thus density and/or temperature fluctuations are a promising explanation, although they cannot as yet explain the occurrence of normal H and modulated F pairs.
An alternative way to deal with density fluctuations was introduced by @Robinson1992 [@Robinson_etal1992] called ‘Stochastic Growth Theory’ (SGT). The main principle behind SGT is that density fluctuations induce random growth of Langmuir waves. The bursts of Langmuir waves have an amplification factor of $\exp G$ where $G$ is a random function with zero mean value and non-zero standard deviation. This predicts a log-normal statistical distribution of the electric field amplitude. Stochastic Growth Theory has been used with a variety of different simulation techniques [e.g. @RobinsonCairns1998; @CairnsRobinson1999; @Li_etal2006b; @Li_etal2006c] and has produced simulation results that resemble the clumpy observations of Langmuir waves and synthetic spectra of type III bursts.
Whilst quasilinear theory is very good at capturing the large scale effects of the electron beam - Langmuir wave interaction, it struggles to capture the small scale effects of single wave packets. An alternative approach can be used to model these small scales using the Zakharov equations [@Zakharov1972]. This approach models the electric field with an additional source term to describe the beam particles [e.g. @Zaslavsky_etal2010; @Krafft_etal2013]. Modelling in this way is able to replicate wave clumping and thus strongly varying electric fields that has been observed in the solar wind [e.g. @Ergun_etal2008]. It is also able to describe the reflection of Langmuir waves in density cavities where small scale lengths are important and the quasilinear approach is invalid.
Non-linear processes and Langmuir wave evolution {#sec:em}
================================================
As discussed in the introduction, many years of work have been invested in understanding the plasma emission mechanism. However, due to its complex nature, and the many interconnected processes, there are still many unanswered questions. One key feature of the mechanism is the non-linear processes which control the evolution of the Langmuir waves, and their conversion into radio waves. While their details are complicated, the basic equations provide insight into emission properties such as brightness temperature and angular distribution and show us what plasma properties are important (e.g. temperature ratio). In this section we will present the basic theory of these processes. The derivation of the relevant equations is complicated, and is given in detail in the books by @1980MelroseBothVols [@1995lnlp.book.....T] and in many papers. We give here only a short introduction, and then focus on the application of the non-linear equations to the problem of type III burst generation.
The general form of the non-linear equations
--------------------------------------------
The density fluctuations in the previous section were generally considered to have scale lengths larger than the Langmuir wavelength. However, density-perturbing wave modes such as ion-sound waves can have wavelengths comparable to Langmuir waves. In this case, we can describe their effects as wave-wave interactions using a set of non-linear equations. The scattering of Langmuir waves by individual plasma particles is also described using non-linear equations. Both of these processes are vital for the further evolution of beam-generated Langmuir waves.
The key principle is “detailed balance”: in short we have interactions of the form $\sigma \rightleftarrows \sigma' +\sigma''$ where $\sigma, \sigma'$ and $\sigma''$ denote wavemodes, which can be the same, or different, and/or $\sigma +i \rightleftarrows \sigma' +i'$, where $i, i'$ are an initial and scattered plasma ion. For every wave we lose on the left-hand side we must gain a wave on the right-hand side. Using this principle we can write down the rates of change of the “occupation number” of waves in mode $\sigma$. This is given by[^1] $N_\sigma=(2\pi)^3 W_\sigma /\omega_\sigma$ where $W_\sigma$ is the spectral energy density. The probability of an interaction is then calculated from the detailed plasma physics, and requires that we specify the exact modes involved.
Writing $\vec{k}, \omega, \vec{k'},\omega'$ for the wavevector and frequency of mode $\sigma, \sigma'$ respectively, the ion scattering process $\sigma +i \rightleftarrows \sigma' +i'$ is described by the equation $$\label{eq:IS3D} \frac{ d W_\sigma(\vec{k})}{d t}=\int \frac{d \vec{k}'}{(2\pi)^3} w_i^{\sigma\sigma'}(\vec{k}', \vec{k})\left[\frac{\omega}{\omega'}W_{\sigma'}(\vec{k}')-W_\sigma(\vec{k})-\frac{(2\pi)^3}{k_BT_i}\frac{\omega-\omega'}{\omega'}W_\sigma(\vec{k})W_{\sigma'}(\vec{k}')\right]$$ with probability $w_i^{\sigma\sigma'}$ depending on the wave modes involved, and a second equation with same probability and $\sigma$ and $\sigma'$ interchanged. The plasma ion absorbs the momentum change between the initial and final waves, and in the cases considered here this change is sufficiently small that we may neglect the effect on the ions.
For the process, $\sigma \rightleftarrows \sigma' +\sigma''$ the equations take the form
$$\label{eq:3Wv3D1}\frac{d W_\sigma(\vec{k})}{dt}=\omega \int\int d\vec{k}'d\vec{k}'' w_{\sigma\sigma'\sigma''}(\vec{k}, \vec{k}', \vec{k}'')\left[\frac{W_{\sigma'}(\vec{k}')}{\omega'}\frac{W_{\sigma''}(\vec{k}'')}{\omega''}-\frac{W_{\sigma}(\vec{k})}{\omega}\left(\frac{W_{\sigma'}(\vec{k}')}{\omega'}+\frac{W_{\sigma''}(\vec{k}'')}{\omega''}\right)\right]$$
where $w_{\sigma\sigma'\sigma''}(\vec{k}, \vec{k}', \vec{k}'')$ is the emission probability. Equations for the other wave modes are similar, having the same probability but some sign changes.
The growth of mode $\sigma$ occurs due to the product term (the first term in the square brackets) and the decay of this mode depends on both its own energy density and those of the decay products, giving the other two terms. So for example, if we start with a high level of mode $\sigma$ and thermal levels of $\sigma', \sigma''$, that is, the level generated spontaneously by plasma particles, the interaction $\sigma \rightleftarrows \sigma' +\sigma''$ will proceed to the right, generating the modes on the right-hand side, from that on the left hand side. Eventually, we can reach the state where $d W_\sigma /dt=0$ and the interaction is saturated.
Along with Equation (\[eq:3Wv3D1\]), we have some conservation conditions. Energy conservation in this equation may be expressed by the condition $$\label{eq:consOm}\omega^\sigma(\vec{k})=\omega^{\sigma'}(\vec{k}')+\omega^{\sigma''}(\vec{k}'')$$ where $\omega^\sigma$ is the frequency of a wave in mode $\sigma$, and is included to emphasize that we must consider the relevant dispersion relation for the mode in question. Momentum conservation is given by $$\label{eq:consK}\vec{k}= \vec{k}'+ \vec{k}''$$ for $\vec{k}, \vec{k}', \vec{k}''$ the respective wave vectors. For a 3-wave interaction to be possible we must be able to simultaneously solve energy and momentum conservation using the relevant dispersion relations (frequency-wavevector relations), e.g. those for Langmuir and EM waves: $$\label{eq:dispL}\omega_{L}(\vec{k}) \simeq \omega_{pe} + 3v_{Te}^2k^2/(2\omega_{pe}) \,,\; \omega_{EM}(\vec{k})=(\omega_{pe} + c^2k^2)^{1/2}$$ respectively. Thus for example, the interaction $L + s \rightleftarrows L' $ is allowed, but $L + L' \rightleftarrows L''$ is not as the resulting wave vector must be larger than $k_{De}$, and this is not possible for Langmuir waves.
Non-linear Langmuir wave evolution
----------------------------------
For Langmuir wave evolution, the probability used in Equation (\[eq:IS3D\]) for the scattering $L + i \rightleftarrows L' + i'$ is $$\label{eq:cIon} w_i^{L L'}= \frac{\sqrt{\pi} \omega_{pe}^2}{2 n_e v_{Ti} (1+T_e/T_i)^2} \frac{|{\vec{k}} \cdot {\vec{k}'}| }{k k' |\vec{k}-\vec{k}'|} \exp{\left(-\frac{(\omega'-\omega)^2}{2|\vec{k}'-\vec{k}|^2 v_{Ti}^2}\right)}.$$ Due to the dot product, this probability is dipolar, i.e. it is maximised for $\vec{k}'$ either parallel or antiparallel to $\vec{k}$. The exponential term implies $\omega' \simeq \omega$ and so $|\vec{k}'|\simeq |\vec{k}|$. Finally, the appearance of $|\vec{k}-\vec{k}'|$ in both the exponential and main term implies that we mainly get backscattering, so that $\vec{k}' \simeq - \vec{k}$.
In Equation (\[eq:IS3D\]), the first two terms in the square brackets may be called spontaneous scattering, and give approximately linear growth or decay. Conversely, the third term depends on the spectral energy density already present. Because of the factor $\omega-\omega'$, the scattered waves will grow where $\omega' < \omega$, and decay where $\omega' > \omega$. Thus the main growth occurs for $\vec{k}' \simeq - \vec{k}+ \Delta k$ with the small decrement $\Delta k$.
The main three-wave process affecting the Langmuir waves is $L \rightleftarrows L' + s$ where $s$ denotes an ion-sound wave, with the approximate dispersion relation $$\label{eq:disps}\omega = k v_s \, ,\; v_s=\sqrt{\frac{k_B T_e}{M_i} \left(1+\frac{3 T_i}{T_e}\right)},$$ the sound speed. The probability for this process which appears in Equation (\[eq:3Wv3D1\]) is $$w_{LLs}(\vec{k}, \vec{k}', \vec{k}'') =\frac{\pi \omega_{pe}^2}{4 n_e T_e}\left(1+\frac{3T_i}{T_e}\right)\omega_k^s \left(\frac{(\vec{k}'\cdot\vec{k}^{\prime\prime})^2}{k^{\prime 2}k^{\prime\prime
2}}\right)\delta(\omega_{k'}^l-\omega_{k''}^l-\omega_k^s)$$
Solving the energy and momentum conservation equations (Equations (\[eq:consOm\]) and (\[eq:consK\]) simultaneously using the appropriate wave dispersion relations tells us that the product Langmuir wave has a wavevector approximately anti-parallel to the initial one i.e. $\vec{k}'' \simeq -\vec{k}'$. The ion-sound wave must then have $\vec{k} \simeq 2\vec{k}'$. Thus as in the previous case of ion-scattering, we mainly generate backscattered Langmuir waves.
Again, rather than exact backscattering, the final Langmuir wavenumber has a slight decrement, so we have ${k}''=-{k}' +\Delta k $ with $$\label{eq:deltaK} \frac{\Delta k}{k_{De}}=\frac{2}{3}\sqrt{\frac{m_e}{m_i}}\sqrt{1+\frac{3T_i}{T_e}} \sim \frac{1}{30}.$$ Thus repeated Langmuir wave scatterings produce waves at smaller and smaller wavenumbers, and can lead to significant spectral evolution. This can be important for the electromagnetic emission processes as there can be strong constraints on the participating wavenumbers.
A thermal level of ion-sound waves is naturally present in plasma and the high levels of Langmuir waves generated by an electron beam can produce rapid growth of ion-sound waves and scattered Langmuir waves. However, ion-sound waves are subject to Landau damping with damping coefficient $$\gamma_k^s=\sqrt{\frac{\pi}{8}}\omega_k^s\left[ \frac{v_s}{v_{Te}}+\left(\frac{v_s}{v_{Ti}}\right)^3\exp{-\left(\frac{v_s}{v_{Ti}}\right)^2}\right].$$ In plasma with $T_i \simeq T_e$ this is very strong, approaching the ion-sound wave frequency, and thus the three-wave processes become inefficient. Referring to Equation (\[eq:3Wv3D1\]), we see that this is because the positive “driving” term for the decay process is proportional to $W_s$ and with strong damping this quantity remains small.
Harmonic Emission Equations
---------------------------
Because the temperature ratio in the solar corona and wind can vary [e.g. @1998JGR...103.9553N; @1979JGR....84.2029G] from $T_i/T_e \lesssim 0.1$ to $1$ or even 2, the ion-scattering and ion-sound wave decay processes described in the previous section vary in efficiency, with the latter dominating where the ratio is lower, and the former where it is large. However, in both regimes, we can produce a large level of Langmuir waves oppositely directed to the initial beam-generated population, and thus efficiently produce radiation at the second harmonic of the plasma frequency.
The growth rate for harmonic emission by the process $L+L' \rightleftarrows t$ is again given by Equation (\[eq:3Wv3D1\]), with a probability $$\label{eqn:LLTProb}
w^{LLT}(\vec{k}_1, \vec{k}_2,\vec{k}_T)=\pi \omega_{pe} \frac{(k_2^2-k_1^2)^2 (\vec{k}_T\times\vec{k}_1)^2}{16 m_e n_e k_T^{2}k_1^2k_2^2}\delta(\omega_{k_T}^{T}-\omega_{k_1}^L-\omega_{k_2}^L)$$ where we have labelled the participating Langmuir wavevectors as $k_1, k_2$ for clarity.
Using the energy and momentum conservation conditions (Equations (\[eq:consOm\]) and (\[eq:consK\]) and the dispersion relations (Equation (\[eq:dispL\])) we find that the coalescing Langmuir waves must be approximately oppositely directed. This is referred to as the “head-on approximation” (HOA), and allows us to simplify the probability, by replacing $(k_2^2-k_1^2)^2/k_2^2 = (\vec{k}_T \cdot \vec{k}_1)^2/ k_1^2$.
To derive this, we must assume that $k_T \ll k_1, k_2$. For typical beam electrons, with $v \simeq 10 v_{Te}$ the generated Langmuir waves have $k \simeq 0.1 k_{De}$ and from ${(\omega_{pe}^2 +c^2 k_T^2)^{1/2}}\simeq 2 \omega_{pe}$ we have that $k_T \simeq \sqrt{3} \omega_{pe} /c$. The ratio $k_T/k_1$ is then of the order 0.2, which is not very small. Clearly then the coalescence is not perfectly head on.
Some simple calculations using Equation (\[eqn:LLTProb\]) can show that even accounting for the angle between $k_1$ and $k_2$, the most probable angle between $k_1$ and $k_T$ remains close to 45$^o$. More detailed calculations by e.g. used specific angular distributions for the Langmuir waves to find the emission probability with and without applying the head-on approximation. Significant differences were found at small wavenumbers. Thus the detailed study or simulation of harmonic emission is complicated. It must either be treated using a fully 3-dimensional model, or by analytical angle averaging.
There are however two great simplifications which may be made to the equations for harmonic emission without affecting the results obtained. The energy lost from the Langmuir waves is very small, and so can be neglected when considering the Langmuir wave evolution. The high group velocity of the EM waves, given by $$\label{eq:vg} \vec{v}_g = {c^2 \vec{k}}/{\omega}$$ and approximately equal to $c$ for emission at $2 \omega_{pe}$ means the emission rapidly leaves the source region where Langmuir waves are present. Thus the reverse process of $t \rightarrow L + L'$ can often be ignored, leaving only a growth rate of $d W_{harmonic} /dt \propto W_L W_L'$ for $W_L, W_L'$ the beam generated and backscattered Langmuir wave populations. Balancing this emission rate with the propagation losses from the source we can find the approximate temperature where the emission saturates. This depends on the beam and plasma parameters as well as the source model, but is usually many orders of magnitude above thermal and easily able to explain the observed brightness temperatures.
Fundamental Emission Equations
------------------------------
For emission at the fundamental of the plasma frequency, we have two possible processes, analogous to those for Langmuir waves, namely the scattering by plasma ions, and the decay involving ion-sound waves. Ion-scattering was proposed in the original version of the plasma emission mechanism by @GinzburgZhelezniakov1958. The three-wave processes are generally faster, but if the ion-sound wave damping is very strong, they are suppressed. For kHz wavelengths, i.e. plasma emission in the solar wind, often the ion-temperature is low, so the three-wave processes are fast, and as shown by @1993ApJ...416..831T ion-scattering is too slow to explain the observed brightness temperatures. It is however sufficient in the corona, as shown by @2003SoPh..215..335M.
The processes are the scattering of a Langmuir wave into an electromagnetic wave, $L + i \rightleftarrows t + i'$, and two processes involving ion-sound wave interactions, namely $L + s \rightleftarrows t$, $L \rightleftarrows t + s$. The latter was proposed by e.g. @1986ApJ...308..954L as the most likely generating process for ion-sound turbulence in solar wind, but it is now generally thought that Langmuir wave decay $L \rightleftarrows L' + s$ produces the ion-sound waves and these drive the production of fundamental electromagnetic emission [@Melrose1982; @1994ApJ...422..870R].
The probability for ion-scattering which appears in Equation (\[eq:IS3D\]) is $$w_i^{LT}(\vec{k}_T, \vec{k}_L)= \frac{\sqrt{\pi} \omega_{pe}^2}{2 n_e v_{Ti} (1+T_e/T_i)^2}\frac{|\vec{k}_L \times \vec{k}_T|^2}{k_l^2 k_T^2|\vec{k}_T-\vec{k}_L|} \exp{\left(-\frac{(\omega_T-\omega_L)^2}{2|\vec{k}_T-\vec{k_L}|^2 v_{Ti}^2}\right)}$$ identically to Equation (\[eq:cIon\]). The main difference between this and the previous equation for Langmuir wave scattering is the cross product in the probability, which implies the emission peaks for scattering through 90$^o$.
For the three-wave scattering we have two alternatives $$\label{eqn:fundSprob} w^{LST}(\vec{k}, \vec{k}_L,\vec{k}_T)=\frac{\pi \omega_{pe}^3\left(1+\frac{3T_i}{T_e}\right)}{\omega_{k_T}^T 4n_eT_e} \omega_k^S \frac{|\vec{k}_T\times\vec{k}|^2}{k_T^{2}|\vec{k}_L|^2}\delta(\omega_{k_T}^T-\omega^L_{k_L}\mp\omega^S_k)$$ where the minus sign in the delta function is used for the process $L + s \rightleftarrows t$, the plus sign for $L \rightleftarrows t + s$. The only other difference between these two processes is a slight change in the participating wavenumbers, as we must consider $\vec{k}_t = \vec{k_L} \pm \vec{k}_s$ respectively. The three-wave probability also contains a cross product term, and is thus maximum for scattering by 90$^o$. In both cases (three-wave decay and ion-scattering) we have the condition $k_T \ll k_L$.
If we consider a realistic situation in the solar corona or wind, we have a beam of electrons with some small pitch angle spread. The Langmuir waves this produces also have a small angular spread, and this is only increased by the presence of plasma density fluctuations [e.g. @NishikawaRyutov1976], and by the ion-scattering and three-wave decays acting on the waves. The fundamental emission can therefore be rather non-directional when it is emitted. However, from the expression for the group velocity, Equation (\[eq:vg\]) and the dispersion relation, Equation (\[eq:dispL\]), it is evident that emission close to the plasma frequency cannot propagate into plasma of increasing density. Ray tracing analyses by @Li_etal2006c showed, using typical type III source parameters that fundamental emission emitted into the forwards hemisphere at any angle is rapidly redirected to propagate along the magnetic field direction.
Conclusions {#sec:conclusion}
===========
It is our opinion that solar radio physics is beginning to enter a new era. Regarding type IIIs, new technology is allowing simulations to generate synthetic dynamic spectra for comparison to observations to probe relevant electron beam and background plasma properties. At the same time new technology is giving birth to the next generation of radio telescope. Interferometers with large baselines and huge collecting areas are being trained on the Sun o produce images of radio bursts over many frequencies with impressive angular resolution. We conclude this article by summarising the future of type III radio burst analysis, both numerically and observationally.
“State of the Art” simulations
------------------------------
While initially plasma emission work was mainly theoretical, simulations have always played a key role. Early work by e.g. @TakakuraShibahashi1976 [@Takakura1982] allowed the simulation of bursts at a few wavelengths, and was key to the adoption of the ion-sound wave dependent model.
Only recently have large-scale kinetic simulations become possible, as computing power is a strong limitation. The series of papers by @2008JGRA..11306104L [@2008JGRA..11306105L; @2009JGRA..11402104L] were the first to trace an electron beam from the injection site into the corona and solar wind, and calculate the resulting radio emission fully numerically, rather than by analytical estimates. These simulations have more recently been extended to cover a wider frequency range, and used to explore the effects of the background plasma on the emission. In the solar wind, observations often show a kappa-distribution rather than a Maxwellian background plasma, and this contains more electrons of higher velocity. A similar power-law electron injection forms a higher velocity beam due to time-of-flight effects, and the emission may be expected to show a faster drift. This is confirmed by the simulations of @LiCairns2014, although these consider only instantaneous electron injection.
Ongoing work by the authors aims to combine the large-scale simulations of e.g. @ReidKontar2013 with a model for plasma radio emission used in a different context in . This aims to explore the effects of plasma density fluctuations on the emission.
On the other hand, the fully numerical approach is not the only possibility. The simulations of @RobinsonCairns1998 [@CairnsRobinson1999; @Li_etal2006b; @Li_etal2006c] used a combination of numerics and analytical estimates from Stochastic Growth Theory to reproduce emission. The relevance of SGT remains in question, but the results are useful nonetheless. The localisation of Langmuir waves into discrete clumps, or wave packets, is investigated using the Zakharov equations in e.g. @Zaslavsky_etal2010 [@Krafft_etal2013].
Continued work on the propagation of electrons and their Langmuir wave generation and non-linear evolution is also essential. Langmuir waves can help to explain the observed non-Maxwellian thermal electron distribution in the solar wind [e.g. @2012SSRv..173..459Y]. They may affect the hard-X ray emission from downward electron beams in the corona [e.g. @Hannah_etal2013]. Plasma emission has even been suggested as a source of sub-THz radio emission [@2013AstL...39..650Z].
Next generation radio observations
----------------------------------
There are many solar telescopes all around the world that are able to observe the dynamic spectra of type III radio bursts, too many to mention individually. On the global scale we would like to mention the e-Callisto project [@Benz_etal2009b], an international network of solar radio spectrometers that has more than 66 instruments in more than 35 locations with users from more than 92 countries.
Regarding imaging, the most notable dedicated solar interferometer for type III radio bursts is the Nançay Radioheliograph [@KerdraonDelouis1997] (NRH), based in France, that has been making dedicated solar images since the 1960s and not images between 150 MHz and 450 MHz
The next generation radio observatories being developed around the world are taking observational solar radio astronomy into a revolutionary new phase. Large baseline interferometers are making high resolution imaging spectroscopy observations of type III bursts. This will enable us to address fundamental questions about the energy release site in solar flares and the transport of energetic electrons through the heliosphere. In Europe there is the LOw Frequency ARray [@vanHaarlem_etal2013] (LOFAR), a network of observatories spread across Europe (Netherlands, UK, Germany, France and Sweden). LOFAR operates between the frequencies of 10 MHz and 250 MHz and is providing interferometric imaging with 10s arcsec resolutions. In North America there is the recently upgraded Expanded Very Large Array [@Perley_etal2011] (EVLA) that operates between 1 GHz and 50 GHz and has started solar observing at the end of 2011. In Western Australia there is the Murchison Widefield Array (MWA) [@Lonsdale_etal2009; @Bowman_etal2013] that is a low-frequency radio telescope operating between 80 MHz and 300 MHz, is one of the precursors to the Square Kilometer Array (SKA) and has recently started to observe the Sun [@Oberoi_etal2011].
Other next generation radio telescopes are on the horizon and will soon be ready for solar observations. We now cover them (in no particular order). In China the Chinese Spectral Radio Heliograph (CSRH) [@Yan_etal2009] will be operational in 2014 and will produce dedicated solar imaging between 400 MHz and 15 GHz, an ideal frequency range for type III bursts in the deep corona. In Russia there is the Siberian Solar Radio Telescope (SSRT) that is being upgraded to create interferometric images of the Sun at GHz frequencies. In India an upgrade is currently under way to the GMRT Giant Metrewave Radio Telescope (GMRT) to operate between the frequencies between 50 MHz and 1500 MHz. In America, improvements to the existing array will create the Expanded Owens Valley Solar Array (EOVSA) that will operate between the frequency range 1 GHz to 18 GHz. In South America we will have the Brazilian Decimeric Array that will operate between the frequencies of 1 GHz and 6 GHz. In New Mexico there is the Long Wavelength Array (LWA) [@Lazio_etal2010; @Taylor_etal2012] that will operate at the low frequencies between 10 - 88 MHz.
Extending into space, the upcoming ESA Solar Orbiter mission and NASA Solar Probe Plus mission are journeying close to the Sun to 10s solar radii, and should launch in 2017 and 2018 respectively. Both missions will be armed with in-situ plasma measuring devices to explore the radial dependence of type III radio signals and the plasma properties of electron beams and the ambient solar wind.
This work is supported by a SUPA Advanced Fellowship (Hamish Reid), the European Research Council under the SeismoSun Research Project No. 321141 (Heather Ratcliffe), and the Marie Curie PIRSESGA- 2011-295272 RadioSun project.
[^1]: This definition varies slightly between authors. In @1980MelroseBothVols a factor $\hbar$ appears explicitly, and the $(2\pi)^3$ is omitted with corresponding changes in the quoted probabilities. @1995lnlp.book.....T uses the given definition.
|
---
author:
- |
\
Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki, 305-8571, Japan\
E-mail:
title: Renormalization factor of four fermi operators with clover fermion and Iwasaki gauge action
---
Introduction
============
Calculation of weak matrix elements of phenomenological interest is one of major application of lattice QCD. A calculation of four quark hadron matrix elements with the Wilson fermion encounters an obstacle since unwanted mixing is introduced through quantum correction with operators that have wrong chirality.
One of the solution is to make use of the parity odd operator. By using discrete symmetries of the parity, the charge conjugation and flavor exchanging transformations it was shown [@Donini:1999sf] that the parity odd four quark operator has no extra mixing with wrong operators even without chiral symmetry. One of application of this virtue may be a calculation of the $K\to\pi\pi$ decay amplitude with the Wilson fermion.
An improvement with the clover term is indivisible for the Wilson fermion. The RG improved gauge action of Iwasaki type has a good property at lattice spacing around $a^{-1}\sim2$ GeV imitating that in the continuum. It is plausible to use a combination of the Iwasaki gauge action and the improved Wilson fermion with clover term for our numerical simulation. Unfortunately renormalization factors of the $\Delta S=1$ four quark operators are not available for this combination of action except for $\Delta S=2$ part [@Constantinou:2010zs]. A purpose of this report is to give the renormalization factor of four quark operators perturbatively which contribute to the $K\to\pi\pi$ decay.
Four quark operators {#sec:4fermi}
====================
We adopt the Iwasaki gauge action and the improved Wilson fermion action with the clover term. The Feynman rules for this action is given in Ref. [@Aoki:1998ar]. We shall adopt the Feynman gauge and set the Wilson parameter $r=1$ in the following.
We shall evaluate the renormalization factor of the following ten operators $$\begin{aligned}
&&
Q^{(2n-1)}=
\left({{\overline{s}}}d\right)_L\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}q\right)_L,
\;
Q^{(2n)}=\left({{\overline{s}}}\times d\right)_L
\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}\times q\right)_L,
\;
(n=1,2,5),
\\&&
Q^{(2n-1)}=
\left({{\overline{s}}}d\right)_L\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}q\right)_R,
\;
Q^{(2n)}=\left({{\overline{s}}}\times d\right)_L
\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}\times q\right)_R,
\;
(n=3,4),
\\&&
\alpha^{(1)}_q=\left(1,0,0\right),
\quad
\alpha^{(2)}_q=\alpha^{(3)}_q=\left(1,1,1\right),
\quad
\alpha^{(4)}_q=\alpha^{(5)}_q=\left(1,-\frac{1}{2},-\frac{1}{2}\right)\end{aligned}$$ where $$\begin{aligned}
\left({{\overline{s}}}d\right)_{R/L}={{\overline{s}}}\gamma_\mu\left({1\pm\gamma_5}\right)d\end{aligned}$$ and $\times$ means a following contraction of the color indices $$\begin{aligned}
Q^{(2)}=\left({{\overline{s}}}\times d\right)_L\left({{\overline{u}}}\times u\right)_L
=\left({{\overline{s}}}_a d_b\right)_L\left({{\overline{u}}}_b u_a\right)_L.\end{aligned}$$
We are interested in the parity odd part only, which contribute to the $K\to\pi\pi$ decay amplitude $$\begin{aligned}
&&
Q^{(2n-1)}_{VA+AV}=-Q^{(2n-1)}_{VA}-Q^{(2n-1)}_{AV},
\quad
Q^{(2n)}_{VA+AV}=-Q^{(2n)}_{VA}-Q^{(2n)}_{AV},
\quad (n=1, 2, 5),
\\&&
Q^{(2n-1)}_{VA-AV}=Q^{(2n-1)}_{VA}-Q^{(2n-1)}_{AV},
\quad
Q^{(2n)}_{VA-AV}=Q^{(2n)}_{VA}-Q^{(2n)}_{AV},
\quad (n=3, 4),
\\&&
Q^{(2n-1)}_{VA}=
\left({{\overline{s}}}d\right)_V\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}q\right)_A,
\quad
Q^{(2n-1)}_{AV}=
\left({{\overline{s}}}d\right)_A\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}q\right)_V,
\\&&
Q^{(2n)}_{VA}=
\left({{\overline{s}}}\times d\right)_V
\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}\times q\right)_A,
\quad
Q^{(2n)}_{AV}=
\left({{\overline{s}}}\times d\right)_A
\sum_{q=u,d,s}\alpha^{(n)}_q\left({{\overline{q}}}\times q\right)_V,\end{aligned}$$ where current-current vertex means $$\begin{aligned}
\left({{\overline{s}}}d\right)_V\left({{\overline{q}}}q\right)_A
=\left({{\overline{s}}}\gamma_\mu d\right)\left({{\overline{q}}}\gamma_\mu\gamma_5q\right).\end{aligned}$$
Renormalization factor in ${\overline{\rm MS}}$ scheme
======================================================
We renormalized the lattice bare operators $Q^{(k)}_{\rm lat}$ to get the renormalized operator $Q^{(k)}_{{\overline{\rm MS}}}$. We adopt the ${\overline{\rm MS}}$ scheme with DRED or NDR. We notice there are two kinds of one loop corrections to the operators. One is given by gluon exchanging diagrams given in Ref. [@Constantinou:2010zs; @Martinelli:1983ac] for $\Delta S=2$ operator and the other is the penguin diagrams given in Ref. [@Bernard:1987rw] for $\Delta S=1$ operators.
The renormalization of the operator is given by $$\begin{aligned}
Q^{(i)}_{{\overline{\rm MS}}}=Z_{ij}^gQ^{(j)}_{\rm lat}
+Z_i^{\rm pen}Q^{\rm pen}_{\rm lat}
+Z_i^{\rm sub}O^{\rm sub}_{\rm lat}\end{aligned}$$ where $Q^{(j)}_{\rm lat}$ is the four quark operators on the lattice, $Q^{\rm pen}_{\rm lat}$ is the QCD penguin operator and $O^{\rm sub}_{\rm lat}$ is a lower dimensional operator to be subtracted. $Z_{ij}^g$ comes from gluon exchanging diagrams. $Z_i^{\rm pen}$ is contribution from the penguin diagram.
Gluon exchanging diagrams
-------------------------
For gluon exchanging diagram the one loop contributions are evaluated in terms of those to the quark bilinear operators by using the Fierz rearrangement and the charge conjugation [@Martinelli:1983ac]. Summing up contributions from three types of diagrams [@Martinelli:1983ac] the one loop correction to the four quark operators is given in a form $$\begin{aligned}
Q^{(i)}_{\rm one-loop}=T^{\rm lat}_{ij}Q^{(j)}_{\rm tree},\end{aligned}$$ where $Q^{(j)}_{\rm tree}=Q^{(j)}_{VA\pm AV}$ is a tree level operator. The correction factors are already evaluated for the improved action in Ref. [@Constantinou:2010zs] and is given as follows for our notation of the four quark operators $$\begin{aligned}
T^{\rm lat}_{11}&=&
T^{\rm lat}_{22}=T^{\rm lat}_{33}=T^{\rm lat}_{44}
=T^{\rm lat}_{99}=T^{\rm lat}_{10,10}
{\nonumber}\\&=&
\frac{g^2}{16\pi^2}
\left(-\frac{N^2+2}{N}\ln\left(\lambda a\right)^2
+\frac{N^2-2}{2N}\left(V_V+V_A\right)+\frac{1}{2N}\left(V_S+V_P\right)\right),
\\
T^{\rm lat}_{55}&=&
T^{\rm lat}_{77}
=\frac{g^2}{16\pi^2}
\left(-\frac{N^2-4}{N}\ln\left(\lambda a\right)^2
+\frac{N}{2}\left(V_V+V_A\right)
-\frac{1}{2N}\left(+V_S+V_P\right)
\right),
\\
T^{\rm lat}_{66}&=&
T^{\rm lat}_{88}
=\frac{g^2}{16\pi^2}
\left(-4\frac{N^2-1}{N}\ln\left(\lambda a\right)^2
+\frac{N^2-1}{2N}\left(V_S+V_P\right)\right),
\\
T^{\rm lat}_{12}&=&
T^{\rm lat}_{21}=T^{\rm lat}_{34}=T^{\rm lat}_{43}
=T^{\rm lat}_{9,10}=T^{\rm lat}_{10,9}
=\frac{g^2}{16\pi^2}\frac{1}{2}
\left(6\ln\left(\lambda a\right)^2
+V_V+V_A-V_S-V_P\right),
\\
T^{\rm lat}_{56}&=&
T^{\rm lat}_{78}
=\frac{g^2}{16\pi^2}\frac{1}{2}\left(
-6\ln\left(\lambda a\right)^2
-V_V-V_A+V_S+V_P\right),\end{aligned}$$ where $\lambda$ is a gluon mass introduced for an infra red regularization and the number of color is $N=3$. $V_\Gamma$ is a finite part in one loop correction to the bilinear operator, which is evaluated in Ref. [@Aoki:1998ar] for various gauge actions.
The renormalization factor is given by taking a ratio of quantum corrections with that in the ${\overline{\rm MS}}$ scheme multiplied with the quark wave function renormalization factor $Z_2$ $$\begin{aligned}
&&
Z^g_{ii}(\mu a)=
\frac{\left(Z_2^{{\overline{\rm MS}}}\right)^2\left(1+T_{ii}^{{\overline{\rm MS}}}\right)}
{\left(Z_2^{\rm lat}\right)^2\left(1+T_{ii}^{\rm lat}\right)},
\\&&
Z^g_{ij}(\mu a)=T_{ij}^{{\overline{\rm MS}}}-T_{ij}
\quad (i \neq j).\end{aligned}$$ The correction factor in the DRED ${\overline{\rm MS}}$ scheme is given by $$\begin{aligned}
&&
T^{{\overline{\rm MS}}}_{11}=T^{{\overline{\rm MS}}}_{22}
=T^{{\overline{\rm MS}}}_{33}=T^{{\overline{\rm MS}}}_{44}
=T^{{\overline{\rm MS}}}_{99}=T^{{\overline{\rm MS}}}_{10,10}
=\left(\frac{N^2+2}{N}\right)V^{{\overline{\rm MS}}},
\\&&
T^{{\overline{\rm MS}}}_{12}=T^{{\overline{\rm MS}}}_{21}
=T^{{\overline{\rm MS}}}_{34}=T^{{\overline{\rm MS}}}_{43}
=T^{{\overline{\rm MS}}}_{9,10}=T^{{\overline{\rm MS}}}_{10,9}
=-3V^{{\overline{\rm MS}}},
\\&&
T^{{\overline{\rm MS}}}_{55}=T^{{\overline{\rm MS}}}_{77}
=\left(\frac{N^2-4}{N}\right)V^{{\overline{\rm MS}}},
\\&&
T^{{\overline{\rm MS}}}_{56}=T^{{\overline{\rm MS}}}_{78}=3V^{{\overline{\rm MS}}},
\\&&
T^{{\overline{\rm MS}}}_{66}=T^{{\overline{\rm MS}}}_{88}
=4\frac{N^2-1}{N}V^{{\overline{\rm MS}}},
\\&&
V^{{\overline{\rm MS}}} = \frac{g^2}{16\pi^2}
\left(\log\left(\frac{\mu^2}{\lambda^2}\right)+1\right).\end{aligned}$$ The same infra red regularization with the gluon mass should be adopted. The quark wave function renormalization factor $Z_2$ is given in Ref. [@Aoki:1998ar].
Substituting the above results we have $$\begin{aligned}
Z^g_{11}(\mu a)&=&
Z^g_{22}(\mu a)=Z^g_{33}(\mu a)=Z^g_{44}(\mu a)=Z^g_{99}(\mu a)
=Z^g_{10,10}(\mu a)
{\nonumber}\\&=&
1+\frac{g^2}{16\pi^2}\left(\frac{3}{N}\ln\left(\mu a\right)^2+z^g_{11}\right),
\\
Z^g_{55}(\mu a)&=&Z^g_{77}(\mu a)
=1+\frac{g^2}{16\pi^2}\left(-\frac{3}{N}\ln\left(\mu a\right)^2+z^g_{55}\right),
\\
Z^g_{66}(\mu a)&=&Z^g_{88}(\mu a)
=1+\frac{g^2}{16\pi^2}\left(\frac{3\left(N^2-1\right)}{N}\ln\left(\mu a\right)^2
+z^g_{66}\right),
\\
Z^g_{12}(\mu a)&=&
Z^g_{21}(\mu a)=Z^g_{34}(\mu a)=Z^g_{43}(\mu a)=Z^g_{9,10}(\mu a)
=Z^g_{10,9}(\mu a)
{\nonumber}\\&=&
\frac{g^2}{16\pi^2}\left(-3\ln\left(\mu a\right)^2+z^g_{12}\right),
\\
Z^g_{56}(\mu a)&=&Z^g_{78}(\mu a)
=\frac{g^2}{16\pi^2}\left(3\ln\left(\mu a\right)^2+z^g_{56}\right),
\\
Z^g_{65}(\mu a)&=&Z^g_{87}(\mu a)=\frac{g^2}{16\pi^2}z^g_{65}=0.\end{aligned}$$ The numerical value of the finite part is given in table \[tab:gluon-exchanging\] for $N=3$ as an expansion in $c_{\rm SW}$ $$\begin{aligned}
z^g_{ij}=z^{g(0)}_{ij}+c_{\rm SW}z^{g(1)}_{ij}+c_{\rm SW}^2z^{g(2)}_{ij}.\end{aligned}$$ The finite part for the NDR scheme is given in table \[tab:gluon-exchanging-NDR\]. We need to subtract the evanescent operators in the ${\overline{\rm MS}}$ scheme, which comes from a difference of dimensionality from four for gamma matrices in operator vertex.
--------- ------- ------- --------- ------- ------- --------- -------- --------
(0) (1) (2) (0) (1) (2) (0) (1) (2)
-23.596 3.119 2.268 -25.183 5.420 2.923 -18.041 -4.933 -0.020
--------- ------- ------- --------- ------- ------- --------- -------- --------
: Finite part $z^g_{ij}$ of the renormalization factor from gluon exchanging diagrams in the DRED scheme. []{data-label="tab:gluon-exchanging"}
-------- ------- -------- ------- -------- -------
(0) (1) (2) (0) (1) (2)
-2.381 0.451 -2.020 2.381 -0.451 2.020
-------- ------- -------- ------- -------- -------
: Finite part $z^g_{ij}$ of the renormalization factor from gluon exchanging diagrams in the DRED scheme. []{data-label="tab:gluon-exchanging"}
[$z^{g(0)}_{11}$]{} [$z^{g(0)}_{55}$]{} [$z^{g(0)}_{66}$]{} [$z^{g(0)}_{12}$]{} [$z^{g(0)}_{56}$]{} [$z^{g(0)}_{65}$]{}
--------------------- --------------------- --------------------- --------------------- --------------------- ---------------------
$-24.096$ $-25.350$ $-19.708$ $ -4.881$ $ -1.120$ $-3$
: Finite part $z^{g(0)}_{ij}$ of the renormalization factor from gluon exchanging diagrams in the NDR scheme. $c_{SW}$ dependent terms are the same as that in the DRED scheme $(n=1,2)$.[]{data-label="tab:gluon-exchanging-NDR"}
Penguin diagrams
----------------
Contribution from the penguin diagram is evaluated with the same procedure as in Ref. [@Bernard:1987rw] and the one loop correction to the four quark operators is given in a form $$\begin{aligned}
Q^{(i)}_{\rm one-loop}=\left(T^{\rm pen}_{i}\right)^{\rm lat}
Q^{\rm pen}_{\rm tree},\end{aligned}$$ where $Q^{\rm pen}_{\rm tree}$ is the penguin operator at tree level $$\begin{aligned}
Q^{\rm pen}=\left(Q^{(4)}_{VA+AV}+Q^{(6)}_{VA-AV}\right)
-\frac{1}{N}\left(Q^{(3)}_{VA+AV}+Q^{(5)}_{VA-AV}\right).\end{aligned}$$
The correction factor is given by $$\begin{aligned}
&&
\left(T^{\rm pen}_{i}\right)^{\rm lat}=\frac{g^2}{16\pi^2}\frac{C(Q_i)}{3}
\left(\ln{a^2p^2}+V_{\rm pen}^{\rm lat}\right)\end{aligned}$$ with operator dependent factor $$\begin{aligned}
&&
C\left(Q_1\right)=0,
\quad
C\left(Q_2\right)=1,
\quad
C\left(Q_3\right)=2,
\\&&
C\left(Q_4\right)=C\left(Q_6\right)=\sum_{q=u,d,s}=N_f,
\\&&
C\left(Q_5\right)=C\left(Q_7\right)=0,
\\&&
C\left(Q_8\right)=C\left(Q_{10}\right)=\sum_{q=u,d,s}\alpha_q=N_u-\frac{N_d}{2},
\\&&
C\left(Q_9\right)=-1.\end{aligned}$$ $p$ is a momentum of intermediate gluon propagator given in terms of external quark momentum, for which we set the on-shell condition. The finite part is expanded as $$\begin{aligned}
V_{\rm pen}^{\rm lat}=-1.7128+c_{\rm SW}\left(-1.0878\right).\end{aligned}$$
The correction factor in the ${\overline{\rm MS}}$ scheme is given in a similar form $$\begin{aligned}
&&
Q^{(i)}_{\rm one-loop}=\left(T^{\rm pen}_{i}\right)^{{\overline{\rm MS}}}
Q^{\rm pen}_{\rm tree},
\\&&
\left(T^{\rm pen}_{i}\right)^{{\overline{\rm MS}}}
=\frac{g^2}{16\pi^2}\frac{C(Q_i)}{3}
\left(\ln\left(\frac{p^2}{\mu^2}\right)-\frac{5}{3}-c\left(Q_i\right)\right)\end{aligned}$$ With the same infra red regulator $p$. The scheme dependent finite term is given by $$\begin{aligned}
&&
c^{\rm (NDR)}\left(Q_2\right)=c^{\rm (NDR)}\left(Q_{2n-1}\right)=-1,
\quad
c^{\rm (NDR)}\left(Q_{2n}\right)=0,
\quad(n\ge2)
\\&&
c^{\rm (DRED)}\left(Q_2\right)=c^{\rm (DRED)}\left(Q_{2n-1}\right)
=c^{\rm (DRED)}\left(Q_{2n}\right)=\frac{1}{4},
\quad(n\ge2).\end{aligned}$$
Combining these two contributions the renormalization factor for the penguin operator is given by $$\begin{aligned}
&&
Z_i^{\rm pen}=
\left(T^{\rm pen}_{i}\right)^{{\overline{\rm MS}}}
-\left(T^{\rm pen}_{i}\right)^{\rm lat}
=\frac{g^2}{16\pi^{2}}\frac{C(Q_i)}{3}\left(-\ln a^2\mu^2
+z_i^{\rm pen}\right),
\\&&
z_i^{\rm pen}=-V_{\rm pen}^{\rm lat}-\frac{5}{3}-c_i.\end{aligned}$$ Numerical value of the finite part is given in table \[tab:penguin\].
$z^{\rm pen}_i({\rm DRED})^{(0)}$ $z^{\rm pen}_2({\rm NDR})^{(0)}$ $z^{\rm pen}_{2n-1}({\rm NDR})^{(0)}$ $z^{\rm pen}_{2n}({\rm NDR})^{(0)}$ $(z^{\rm pen}_i)^{(1)}$
----------------------------------- ---------------------------------- --------------------------------------- ------------------------------------- -------------------------
$-0.2039$ $1.0462$ $1.0462$ $0.0461$ $1.0878$
: Finite part of the renormalization factor from the penguin diagram. Coefficients of the term $c_{SW}^k (k=0,1)$ are given in the column marked as $(k)$.[]{data-label="tab:penguin"}
Mixing with lower dimensional operator
--------------------------------------
We shall evaluate the amputated quark bilinear vertex function given by $$\begin{aligned}
I_{k;XY}^{\rm (sub)}&=&
{\left\langle Q^{(k)}_{XY} s_{a\alpha}(-p){{\overline{d}}}_{b\beta}(p) \right\rangle}_{\rm 1PI}.\end{aligned}$$ We consider a leading contribution to the vertex at tree level, which introduces mixing with lower dimensional operators.
We immediately get $$\begin{aligned}
&&
I_{2n-1;VA}^{\rm (sub)}=-I_{2n-1;AV}^{\rm (sub)}=
\alpha^{(n)}_d\delta_{ab}\left(\gamma_5\right)_{\alpha\beta}
\left(I^{\rm (sub)}(m_d)-I^{\rm (sub)}(m_s)\right),
\\&&
I_{2n;VA}^{\rm (sub)}=-I_{2n;AV}^{\rm (sub)}=
N\alpha^{(n)}_d\delta_{ab}\left(\gamma_5\right)_{\alpha\beta}
\left(I^{\rm (sub)}(m_d)-I^{\rm (sub)}(m_s)\right),
\\&&
I^{\rm(sub)}(am)=\int\frac{d^4l}{(2\pi)^4}
\frac{4W(l,am)}{\sin^2l + W(l,am)^2},
\\&&
W(l,am)=am+\sum_{\mu}\left(1-\cos l_\mu\right).
\label{eqn:wilson-term}\end{aligned}$$ which may be evaluated with an expansion in the quark mass $$\begin{aligned}
I^{\rm (sub)}(m)&=&
\frac{1}{a^2}m\frac{d}{d(am)}I^{\rm (sub)}(0)
+\frac{1}{a}m^2\frac{1}{2}\frac{d^2}{d(am)^2}I^{\rm (sub)}(0)
+m^3\frac{1}{6}\frac{d^3}{d(am)^3}I^{\rm (sub)}(0)
+{\cal O}(a).
{\nonumber}\\\end{aligned}$$ The numerical value is given by $$\begin{aligned}
&&
\frac{d}{d(am)}I^{\rm (sub)}(0)=\frac{1}{16\pi^2}\left(-21.466\right),
\\&&
\frac{d^2}{d(am)^2}I^{\rm (sub)}(0)
=\frac{1}{16\pi^2}\left(-14.92\right).
\label{eqn:tree-level}\end{aligned}$$
This contribution introduces a mixing with the lower dimensional bilinear operator $({{\overline{s}}}\gamma_5d)$ multiplied with a mass difference $(m_d-m_s)$. As it is clear from [(\[eqn:wilson-term\])]{} this is due to the chiral symmetry breaking effect in the Wilson fermion. It may be better not to expand in quark mass since the coefficient [(\[eqn:tree-level\])]{} is rather large and ${d^3}/{d(am)^3}I^{\rm (sub)}(0)$ term has an infra red divergence at $m=0$. The subtraction factor is given by $$\begin{aligned}
&&
Z_{2n-1}^{\rm (sub)}=
-2\alpha^{(n)}_d\left(I^{\rm (sub)}(m_d)-I^{\rm (sub)}(m_s)\right),
\quad(n=3,4),
\\&&
Z_{2n}^{\rm (sub)}=
-2N\alpha^{(n)}_d\left(I^{\rm (sub)}(m_d)-I^{\rm (sub)}(m_s)\right),
\quad(n=3,4),
\\&&
Z_{2n-1}^{\rm (sub)}=Z_{2n}^{\rm (sub)}=0,\quad
(n=1,2,5).\end{aligned}$$
Conclusion {#sec:concl}
==========
In this report we have calculated the one-loop contributions for the renormalization factors of parity odd four-quark operators, which contribute to the $K\to\pi\pi$ decay amplitude, in the improved Wilson fermion with clover term and the Iwasaki gauge action. The operators are multiplicatively renormalizable without any mixing with wrong operators that have different chiral structures except for the lower dimensional operator.
Acknowledgment {#acknowledgment .unnumbered}
==============
This work is done for a collaboration with K. -I. Ishikawa, N. Ishizuka, A. Ukawa and T. Yoshié. This work is supported in part by Grants-in-Aid of the Ministry of Education (Nos. 22540265, 23105701).
[99]{} A. Donini, V. Gimenez, G. Martinelli, M. Talevi and A. Vladikas, Eur. Phys. J. C [**10**]{} (1999) 121 \[arXiv:hep-lat/9902030\]. M. Constantinou, P. Dimopoulos, R. Frezzotti, V. Lubicz, H. Panagopoulos, A. Skouroupathis and F. Stylianou, Phys. Rev. D [**83**]{} (2011) 074503 \[arXiv:1011.6059 \[hep-lat\]\]. S. Aoki, K. i. Nagai, Y. Taniguchi and A. Ukawa, Phys. Rev. D [**58**]{}, 074505 (1998) \[arXiv:hep-lat/9802034\]. G. Martinelli, Phys. Lett. [**B141** ]{} (1984) 395.
C. W. Bernard, A. Soni, T. Draper, Phys. Rev. [**D36** ]{} (1987) 3224.
|
---
abstract: 'For the system $$-\D U_i+ U_i=U_i^3-\b U_i\sum_{j\neq i}U_j^2,\qquad i=1,\dots,k,$$ (with $k\geq3$) we prove the existence, for $\b$ large, of positive radial solutions on $\R^N$. We show that, as $\b\to+\infty$, the profile of each component $U_i$ separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing–sign solutions of the scalar equation $-\D W+ W=W^3$. Within an Hartree–Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the $k$–mixtures of Bose–Einstein condensates.'
author:
- |
Susanna Terracini\
Dipartimento di Matematica e Applicazioni\
Università degli Studi di Milano-Bicocca\
Via Bicocca degli Arcimboldi, 8\
20126 Milano, Italy\
`[email protected]`
- |
Gianmaria Verzini\
Dipartimento di Matematica\
Politecnico di Milano\
Piazza Leonardo da Vinci, 32\
20133 Milano, Italy\
`[email protected]`
title: 'Multipulse phases in $k$–mixtures of Bose–Einstein condensates [^1]'
---
Introduction
============
In this paper we seek radial solutions to the system of elliptic equations $$\label{eq:sys}
\left\{
\begin{array}{l}
-\D U_i+ U_i=U_i^3-\b U_i\sum_{j\neq i}U_j^2,\qquad i=1,\dots,k\smallskip\\
U_i\in H^1(\R^N),\quad U_i>0,
\end{array}
\right.$$ with $N=2,3$, $k\geq 3$, and $\b$ (positive and) large, in connection with the changing–sign solutions of the scalar equation $$\label{eq:sing}
-\D W+W=W^3,\qquad W\in H^1(\R^N).$$ It is well known (see, for instance, [@s; @jk]) that this equation admits infinitely many nodal solutions. More precisely, following Bartsch and Willem [@bw], for any $h\in\N$ equation possesses radial solutions with exactly $h-1$ changes of sign, that is $h$ nodal components (“bumps”), with a variational characterization.\
cm 1.0
0.02 0 ()
(-0.6 0) t:V l:0.3 w:0.15 (13 0)
(0 -2) (0 3) t:V
0.04
0.5 () (0 2.5) (1.1 2.5)(1.1 -1.8)(2.2 -1.8) (3.3 -1.8)(3.4 1.3)(4.5 1.3) (5.6 1.3)(5.7 -0.9)(7 -0.9) (8.5 -0.9)(8.5 0.6)(9.5 0.6) (10.5 0.6)(10 0.2)(12.5 0.1) h:C v:C (12.6 -0.4) [$|x|$]{} (4.2 1.6) [$W$]{} (12 2.5) [$h=5$]{}
In the recent paper [@ww], Wei and Weth have shown that, in the case of $k=2$ components, there are solutions $(U_1,U_2)$ such that the difference $U_1-U_2$, for large values of $\b$, approaches some sign–changing solution $W$ of . Hence, one can prescribe the limit shape of $U_1$ and $U_2$ as $W^+$ and $W^-$: this means that each $U_i$ can be seen as the sum of pulses, each converging to one of the bumps of $|W|$.
cm 1.0
0.02 0 ()
(-0.6 0) t:V l:0.3 w:0.15 (13 0)
(0 -0.6) (0 3) t:V 0.02 0 ()
(5.35 2.4) t:V l:0.2 w:0.1 (5 1.6)
(7.75 2) t:V l:0.2 w:0.1 (7.6 1.2) 0.04 0.7 () (0 2.6) (1.1 2.6)(0.6 0.2)(2.2 0.2) (4 0.2)(3.4 1.6)(4.5 1.6) (5.6 1.6)(5.7 0.15)(7 0.15) (9 0.15)(8.5 0.8)(9.5 0.8) (10.5 0.8)(10 0.25)(12.5 0.15) 0.3 (0.1 0.1) (0 0.1) (1.9 0.1)(1.1 2)(2.2 2) (3.3 2)(2.6 0.15)(4.1 0.15) (6.3 0.15)(5.7 1)(7 1) (8.5 1)(8.5 0.26)(9.5 0.18) (10.5 0.1)(10 0.2)(12.5 0.05) h:C v:C (12.6 -0.4) [$|x|$]{} (5.5 2.6) [$U_1\sim W^+$]{} (7.9 2.2) [$U_2\sim W^-$]{}
In the present paper we extend this result to the case of an arbitrary number of components $k\geq3$, proving the existence of solutions to with the property that, for $\b$ large, each component $U_i$ is near the sum of some non–consecutive bumps of $|W|$ (see Theorems \[teo:main\] and \[teo:main2\]).
cm 1.0
0.02 0 ()
(-0.6 0) t:V l:0.3 w:0.15 (13 0)
(0 -0.6) (0 3) t:V
0.04 0.85 () (0 2.4) (1.3 2.4)(0.6 0.15)(2.2 0.15) (4 0.15)(3.4 1.7)(4.5 1.7) (5.6 1.7)(5.7 0.35)(7.2 0.25) (7.7 0.15)(10 0.05)(12.5 0.04) 0.15 (0.1 0.1) (0 0.2) (1.9 0.2)(1.1 2)(2.2 2) (3.8 2)(2.6 0.15)(6.1 0.15) (9.3 0.15)(8 0.8)(9.5 0.8) (10.5 0.8)(10 0.25)(12.5 0.15) 0.5 (0.2 0.1) (0 0.1) (3.9 0.1)(4.5 0.1)(5.5 0.3) (6 0.4)(6 1)(7 1) (8.5 1)(8.5 0.26)(9.5 0.18) (10.5 0.1)(10 0.2)(12.5 0.09) h:C v:C (12.6 -0.4) [$|x|$]{} (1.5 2.7) [$U_1\sim w_1+w_3$]{} (4.5 2.2) [$U_2\sim w_2+w_5$]{} (8.4 1.9) [$U_3\sim w_4$]{} 0.02 0 ()
(1 2.5) t:V l:0.2 w:0.1 (0.7 2.2)
(4 2) t:V l:0.2 w:0.1 (3.2 1.8)
(8.3 1.7) t:V l:0.2 w:0.1 (8 1)
Furthermore, we can prescribe the correspondence between such bumps of $|W|$ and the index $i$ of the component $U_i$ (see Example \[example\]). This, compared with the case $k=2$, provides a much richer structure of the solution set for . This goal will be achieved by a suitable construction inspired by the extended Nehari method (see [@neh]) developed in [@ctv2].
System arises in the study of solitary wave solutions of systems of $k\geq3$ coupled nonlinear Schrödinger equations, known in the literature as Gross–Pitaevskii equations: $$\left\{
\begin{array}{l}
\displaystyle -\imath \partial_t(\phi_i)=\Delta \phi_i-V_i(x)\phi_i+ \mu_i|\phi_i|^2\phi_i-
\sum_{j\neq i}\b_{ij}|\phi_j|^2\phi_i,\qquad i=1,\dots,k\smallskip\\
\phi_i \in H^1(\R^N;\C),\qquad N=1,2,3.
\end{array}
\right.$$ This system has been proposed as a mathematical model for multispecies Bose–Einstein condensation in $k$ different hyperfine spin states (see [@clll] and references therein); such a condensation has been experimentally observed in the triplet states (see [@nature]). Here the complex valued functions $\phi_i$’s are the wave functions of the $i$–th condensate, the functions $V_i$’s represent the trapping magnetic potentials, and the positive constants $\mu_i$’s and $\b_{ij}$’s are the intraspecies and the interspecies scattering lengths, respectively. With this choice the interactions between like particles are attractive, while the interactions between the unlike ones are repulsive; we shall assume that $\b_{ij}=\b_{ji}$, which gives the system a gradient structure. To obtain solitary wave solutions we set $$\phi_i(t,x)=e^{-\imath\lambda_i t}U_i(x),$$ obtaining that the real functions $U_i$’s satisfy $$\label{eq:sys_comp}
\left\{
\begin{array}{l}
\displaystyle -\Delta U_i+\left[V_i(x)+\lambda_i\right]U_i= \mu_iU_i^3-
\sum_{j\neq i}\b_{ij}U_j^2U_i,\qquad i=1,\dots,k\smallskip\\
U_i \in H^1(\R^N).
\end{array}
\right.$$ For the sake of simplicity we assume $V_i(x)\equiv0$, $\lambda_i=\mu_i=1$ and $\b_{ij}=\b$, for every $i$ and $j$, and $N=2,3$, even though our method works also in more general cases, see Remark \[rem:finale\] at the end of the paper. With this choice, system becomes system .
For a fixed $k$, as the interspecific competition goes to infinity, the wave amplitudes $U_i$’s segregate, that is, their supports tend to be disjoint. This phenomenon, called “phase separation”, has been studied, starting from [@ctv; @ctv2], in the case of $\mu_i>0$ and in [@clll] in the case $\mu_i<0$, for least energy solutions in non necessarily symmetric bounded domains. Of course, the number of connected domains of segregation is at least the number of different phases surviving in the limit. For the minimal solutions, the limiting states have *connected* supports[^2]. This is not necessarily the case for solutions which are not characterized as ground states. This is indeed what we show in the present paper, proving the existence of solutions converging to limiting states which supports have a large number of connected components. In this way we obtain a large number of connected domains of segregation with a few phases. Taking the limiting supports as unknown, this can be seen as a free boundary problem. The local analysis of the interfaces and the asymptotic analysis, as the interspecific scattering length grows to infinity has been carried in [@ctv2] for the minimal solutions.
In the recent literature, systems of type have been the object of an intensive research also in different ranges of the interaction parameters, for their possible applications to a number of other physical models, such as the study of incoherent solutions in nonlinear optics. We refer the reader to the recent papers [@ac; @mmp; @dww; @lw2; @ww2] mainly dealing with systems of two equations. For the general $k$–systems we refer to [@lw; @si] and the references therein.
Preliminaries and main results {#sec:basic}
==============================
In the absence of a magnetic trapping potential we shall work in the Sobolev space of radial functions $\spc$, endowed with the standard norm $\|U\|^2=\int_{\R^N}|\nabla U_i|^2+U_i^2\,dx$; it is well known that such functions are continuous everywhere but the origin, thus we are allowed to evaluate them pointwise. As $N=2,3$ implies that $p=4$ is a subcritical exponent, the (compact) embedding of $\spc$ in $L^4(\R^N)$ (see [@s]) will be available:
\[lem:strauss\] If $U\in\spc$ then $\int_{\R^N}U^4\,dx\leq C_S^4 \|U\|^4$, and the immersion $\spc\hookrightarrow L^4(\R^N)$ is compact for $N=2,3$.
We search for solutions of as critical points of the related energy functional $$J_\b(U_1,\dots,U_k)=\sum_{i=1}^k\left[\frac12\|U_i\|^2-\frac14\int_{\R^N}
U_i^4\,dx\right]+\frac\b4\sum_{{i,j=1 \atop i\neq
j}}^k\int_{\R^N}U_i^2U_j^2\,dx$$ (we will always omit the dependence on $\b$ when no confusion arises). In the same way we associate with equation the corresponding functional $$J^*(W)=\frac12\|W\|^2-\frac14\int_{\R^N} W^4\,dx.$$ Let $h\in\N$ be fixed. We introduce the set of the nodal components of radial functions having (at least) $h-1$ ordered zeroes as $$\Xh^*=\left\{(w_1,\dots w_h)\in\left(\spc\right)^h:\,
\begin{array}{cl}
\text{for every }l=1,\dots,h\text{ it holds } w_l\geq0,\,w_l\not\equiv0 \text{ and}\smallskip\\
w_l(x_0)>0 \implies w_p(x)=0 \left\{
\begin{array}{l}
\forall|x|\geq|x_0|\text{ if }p<l\smallskip\\
\forall|x|\leq|x_0|\text{ if }p>l.
\end{array}
\right.
\end{array}
\right\}.$$ We will often write $W=(w_1,\dots w_h)$. By definition, if $W\in\Xh^*$, then $w_l\cdot w_p=0$ a.e. when $l\neq p$. More precisely, the sets $\{w_l>0\}$ are contained in disjoint annuli[^3] and, for $l<p$, the annulus containing $\{w_l>0\}$ is closer to the origin than the one containing $\{w_p>0\}$. As a consequence, we have $J^*(\sum_lw_l)=J^*(\sum_l(-1)^lw_l)=\sum_lJ^*(w_l)$.
We are interested in solutions of with $h$ nodal regions. The *Nehari manifold* related to this problem is defined as $$\Neh^*=\left\{W\in \Xh^*:\,J^*(w_l)=\sup_{\l>0}J^*(\l
w_l)\right\}=\left\{W\in \Xh^*:\,\|w_l\|^2=\int_{\R^N}
w_l^4\,dx\right\}.$$ As a matter of fact one has
\[prop:sing\] Let $$c_\infty=\inf_{W\in\Neh^*}J^*(W)=\inf_{W\in\Xh^*}\sup_{\l_l>0}J^*\left(\sum_{l=1}^h\l_lw_l\right).$$ Then the set $$\Kah=\left\{W\in\Neh^*:\,J^*(W)=c_\infty\right\}$$ is non empty and compact, and, for every $W\in\Kah$, the functions $$\pm\sum_{l=1}^h(-1)^hw_l\text{ solve \eqref{eq:sing}.}\footnote{As a
consequence $\supp w_l$ is an annulus for every $l$, and $\supp
W=\R^N$.}$$ Moreover there exist two constants $0<C_1<C_2$ such that, for every $W\in\Kah$ and for every $l$ it holds $$C^2_1\leq\|w_l\|^2=\int_{\R^N}w_l^4\,dx\leq C^2_2.$$
For the proof of this result, very well known in the literature, we refer to [@bw].
Now, let us consider system . Roughly speaking, we want to construct solutions of in the following way: each $U_i>0$ is the sum of pulses $u_{im}$, where each $u_{im}$ is near some $w_l$ for an appropriate $W\in\Kah$. Maybe an example will make the situation more clear.
\[example\] Let $h=5$ and $k=3$. A possible setting is to search for solutions $U_1=u_{11}+u_{12}$, $U_2=u_{21}+u_{22}$, $U_3=u_{31}$, in such a way that, for some $W\in\Kah$, (for instance) $u_{11}$ is near $w_1$, $u_{21}$ is near $w_2$, $u_{12}$ is near $w_3$, $u_{31}$ is near $w_4$, and $u_{22}$ is near $w_5$. The only rule we want to respect is that two consecutive pulses $w_l$ and $w_{l+1}$ must belong to different components $U_i$ and $U_j$ (see the last figure in the introduction).
The general situation can be treated as follows. Let $h\geq k$ and consider any surjective map $$\s:\{1,\dots,h\}\to\{1,\dots,k\}\quad\text{ such that
}\quad\s(l+1)\neq\s(l)\text{ for }l=1,\dots,h-1$$ (a map that associates each pulse of an element of $\Kah$ to a component $U_i$). The numbers $h_i=\#\s^{-1}(i)$ (the number of pulses associated to the $i$–th component) are such that $h_i\geq1$ and $\sum_1^kh_i=h$. This means that we can (uniquely) define a bijective map onto the set of double indexes $$\tilde\s:\{1,\dots,h\}\to\bigcup_{i=1}^k\{(i,m):\,m=1,\dots,h_i\}$$ where the first index of $\tilde\s$ is given by $\s$, and the second is increasing (when the first is fixed). In this setting, Example \[example\] can be read as $\tilde\s(1)=(1,1)$, $\tilde\s(2)=(2,1)$, $\tilde\s(3)=(1,2)$, $\tilde\s(4)=(3,1)$, $\tilde\s(5)=(2,2)$.[^4].
According to the previous notation we define, for $\eps\leq1$, $$\Xh_\ep=\left\{(u_{11},\dots,u_{kh_k})\in\left(\spc\right)^h:\,\begin{array}{c}u_{im}\geq0
\text{ and there exists }W\in\Kah\text{ such that}\\
\displaystyle\sum_{i=1}^k \sum_{m=1}^{h_i}
\left\|u_{im}-w_{\tilde\s^{-1}(im)}\right\|^2<\ep^2\end{array}\right\},$$ and $U_i=\sum_{m=1}^{h_i}u_{im}$. Sometimes we will use the distance $$d^2_{\tilde
\sigma}\left((u_{11},\dots,u_{kh_k}),W\right)=\sum_{i=1}^k
\sum_{m=1}^{h_i}\left\|u_{im}-w_{\tilde\s^{-1}(im)}\right\|^2$$
\[lem:base\] Using Proposition \[prop:sing\] it is easy to see that
1. $\Xh_\ep$ is contained in an $\ep$–neighborhood of $\Kah$, in the sense of $d_{\tilde\sigma}$; $\Kah\subset\Xh_\ep$ (understanding the identification $w_l=u_{\tilde\s(im)}$);
2. there exist constants $C_1$, $C_2$ not depending on $\b$ and $\ep<1$, such that $0<C_1\leq\|u_{im}\|\leq C_2$, $0<C_1^2\leq\int_{\R^N}u^4_{im}\,dx\leq
C_2^2$
3. $m\neq n$ implies $\int_{\R^N}\nabla u_{im} \cdot \nabla u_{in}<C\ep$, $\int_{\R^N}u_{im}u_{in}<C\ep$, $\int_{\R^N}U_i^2u_{im}u_{in}<C\ep$.
A first important result we want to give, that is underlying the spirit of this whole paper, is the following: in the classical Nehari’s method described above, it is not necessary to perform the min–max procedure on pulses with disjoined support, but we can “mix up”, even tough not too much, the pulses with non adjacent supports.
\[prop:nehari\_mixed\] There exist $\ep_0\leq1$ such that for every $0<\ep<\ep_0$ the following hold. If $(v_{11},\dots,v_{kh_k})\in\Xh_\ep$ is such that $$V_i\cdot V_j=0\quad\text{almost everywhere, for every }i,j,$$ (but $v_{im}\cdot v_{in}$ is not necessarily null) then $$\sup_{\l_{im}>0}J^*\left(\sum_{i,m}\l_{im}v_{im}\right)\geq
c_\infty.$$
To prove the result, we will construct a $h$–tuple $(\tilde
w_1,\dots,\tilde w_h)\in\Neh^*$ such that, for a suitable choice of the positive numbers $\tilde\lambda_{im}$’s, it holds $$J^*\left(\sum\tilde\l_{im}v_{im}\right)=J^*\left(\sum\tilde
w_{\tilde\sigma^{-1}(im)}\right).$$ As a first step, we have to select the (disjointed) supports of such $\tilde w_l$. To do that, using the definition of $\Xh_\ep$, we choose a $W\in\Kah$ such that $\sum_{i,m}
\|v_{im}-w_{\tilde\s^{-1}(im)}\|^2<\ep^2$. When $\ep$ is small we can find $h$ positive radii $r_1,\dots,r_h$ such that, for $|x_l|=r_l$, $v_{\tilde\s(l)}(x_l)w_l(x_l)>0$.[^5] Using this fact we can construct the open connected annuli $A_1,\dots,A_h$ in such a way that
- $|x_l|=r_l\text{ implies }x_l\in A_l$,
- $\bigcup_{l=1}^h\overline{A_l}=\R^N$ and $A_l\cap A_p=\emptyset\text{
when }p\neq l$,
- $\mathrm{supp}V_i\subset\bigcup_{m=1}^{h_i}\overline{A_{\tilde\s^{-1}(im)}}$ for every $i$
(recall that, by assumption, $\mathrm{int}(\mathrm{supp} V_i) \cap
\mathrm{int}(\mathrm{supp} V_j)=\emptyset$). By construction, we obtain that obviously $\supp w_l\cap I_l\neq\emptyset$, while, by connectedness, $$\supp w_{\tilde\s^{-1}(im)}\cap A_{\tilde\s^{-1}(in)}=\emptyset
\text{ when }n\neq m.$$ In particular this last fact implies that, for $n\neq
m$,[^6] $$\label{eq:norme_ristrette}
\left\|\left.v_{im}\right|_{A_{\tilde\s^{-1}(in)}}\right\|^2\leq\ep^2\text{
and hence
}\left\|\left.v_{im}\right|_{A_{\tilde\s^{-1}(im)}}\right\|^2\geq
C_1^2-(h_i-1)\ep^2$$ (with $C_1$ as in Remark \[lem:base\]). Now, depending on the positive parameters $\l_{im}$’s, let us define the functions $\tilde
v_{im}$’s as $$\tilde v_{im}=\left.\left(\sum_{n=1}^{h_i}
\lambda_{in}v_{in}\right)\right|_{A_{\tilde\s^{-1}(im)}}.$$ By construction we have that $\tilde v_{im}\cdot\tilde
v_{jn}\equiv0$ for every choice of the $\l_{im}$’s. We claim the existence of $\tilde\l_{im}$’s such that the corresponding $\tilde
v_{im}$’s satisfy $$F_{im}(\tilde\l_{11},\dots,\tilde\l_{kh_k})=\|\tilde
v_{im}\|^2-\int_{\R^N}\tilde v_{im}^4\,dx=0\quad\text{ for every
}(i,m).$$ This will imply that, writing $\tilde w_{l}=\tilde v_{\tilde\s(l)}$, the $h$–tuple $(w_1,\dots,w_h)$ belongs to $\Neh^*$. Since $J^*\left(\sum_{i,m}\tilde v_{im}\right) = J^*\left(\sum_{i,m}\tilde
\l_{im} v_{im}\right)$, this will conclude the proof of the lemma. In order to prove the claim we will use ($k$ times) a classic result by Miranda concerning the zeroes of maps from a rectangle of $\R^{h_i}$ into $\R^{h_i}$ (see [@mir]), proving the existence, when $\ep$ is sufficiently small, of constants $0<t\leq T$ such that, for every $(i,m)$, $$\label{eq:miranda}
\l_{im}=T,\,t\leq\l_{in}\leq T \implies F_{im}<0,\qquad
\l_{im}=t,\,t\leq\l_{in}\leq T \implies F_{im}>0;$$ from this and Miranda’s Theorem the claim will follow. Let then $(i,m)$ be fixed, $\l_{im}=T$ and, for $n\neq m$, $0\leq \l_{in}\leq
T$. Exploiting Remark \[lem:base\], equation , and the Sobolev embedding of $\spc$ in $L^4(\R^N)$, we obtain[^7] $$\left\|\tilde
v_{im}\right\|=\lambda_{im}\left\|\left.v_{im}\right|_{A_l}+\sum_{n\neq
m}\frac{\l_{in}}{\l_{im}}\left.v_{in}\right|_{A_l}\right\|\leq
T\left(C_2+(h_i-1)\ep\right)$$ and $$\int_{\R^N}\tilde v_{im}^4\,dx\geq \l_{im}^4\int_{A_l} v_{im}^4\,dx=
\l_{im}^4\left(\int_{\R^N} v_{im}^4\,dx-\sum_{p\neq l}\int_{A_p}
v_{im}^4\,dx\right)\geq T^4\left(C_1-(h_i-1)C^4_S\ep^4\right).$$ Choosing $\ep_0$ in such a way that, for $\ep<\ep_0$, the last term is positive, we obtain an inequality of the form $$F_{im}\leq a T^2-bT^4<0\quad\text{if }T\text{ is fixed sufficiently
large,}$$ and the first part of is proved. On the other hand, let now $T$ be fixed as above, $\l_{im}=t$, and, for $n\neq
m$, $t\leq \l_{in}\leq T$. Using again the Sobolev embedding, we have $$F_{im}=\|\tilde v_{im}\|^2-\int_{\R^N}\tilde v_{im}^4\,dx\geq
\|\tilde v_{im}\|^2 - C_S^4\|\tilde v_{im}\|^4=\|\tilde
v_{im}\|^2\left(1-C_S^4\|\tilde v_{im}\|^2\right).$$ Then we simply have to prove that, when $t$ is sufficiently small, $\|\tilde v_{im}\|<1/C^2_S$. As before, by Remark \[lem:base\] and equation , we obtain $$\left\|\tilde
v_{im}\right\|=\left\|\left.\lambda_{im}v_{im}\right|_{A_l}+\sum_{n\neq
m}\left.\l_{in}v_{in}\right|_{A_l}\right\|\leq tC_2+(h_i-1)\ep T.$$ Hence we can choose $t$ and $\ep_0$ sufficiently small in such a way that, for every $\ep<\ep_0$, $\|\tilde v_{im}\|<1/C^2_S$. As we just observed, this implies the second part of , concluding the proof of the proposition.
Since by Remark \[lem:base\] we have $\Neh^*\subset\Xh_\ep$ for every $\ep$, using the previous proposition we obtain the following equivalent characterizations of $c_\infty$: $$\begin{split}
c_\infty
&= \inf_{\begin{array}{c}
w_l\not\equiv0\\
w_l\cdot w_p\equiv0
\end{array}}
\sup_{\l_l>0}J^*\left(\sum_l\l_lw_l\right)\smallskip\\
&= \inf_{\begin{array}{c}
(v_{im})\in\Xh_\ep\\
V_i\cdot V_j\equiv0
\end{array}}
\sup_{\l_{im}>0}J^*\left(\sum_{i,m}\l_{im}v_{im}\right)\smallskip\\
&= \inf_{\begin{array}{c}
(v_{im})\in\Xh_\ep\\
V_i\cdot V_j\equiv0
\end{array}}
\sup_{\l_{im}>0}J_\b\left(\sum_{m}\l_{1m}v_{1m},\dots,\sum_{m}\l_{km}v_{km}\right).
\end{split}$$
In the same spirit of the previous corollary, for $i=1,\dots,k$ and $m=1,\dots,h_i$, let $\l_{im}\geq0$ and $u_{im}\in\spc\setminus\{0\}$. We write $$\begin{array}{l}
\L_i=(\l_{i1},\dots,\l_{ih_i}),\qquad U_i=\sum_m u_{im},\qquad \L_iU_i=\sum_m\l_{im}u_{im},\smallskip\\
\Phi_\b(\L_1,\dots,\L_k)=J_\b(\L_1U_1,\dots,\L_kU_k),
\end{array}$$ in such a way that $\Phi$ is a $C^2$–function. Moreover, let $$M_\b(u_{11},\dots,u_{kh_k})=\sup_{\l_{im}>0}\Phi_\b(\L_1,\dots,\L_k)$$ and finally $$\label{eq:c_beta}
c_{\ep,\b}=\inf_{\Xh_\ep}M_\b.$$ Our main results are the following.
\[teo:main\] There exist $\bar\ep>0$ and $\bar\b=\bar\b(\bar\ep)$ such that, if $0<\ep<\bar\ep$ and $\b>\bar\b$, then $c_{\ep,\b}$ is a critical value for $J_\b$, corresponding to a solution of belonging to $\Xh_\ep$.
\[teo:main2\] Let $0<\ep<\bar\ep$ (as in the previous theorem) be fixed and $(\b_s)_{s\in\N}$ be such that $\b_s\to+\infty$. Finally, let $(U_1^s,\dots,U_k^s)$ be any solution of at level $c_{\ep,\b}$ and belonging to $\Xh_\ep$. Then, up to subsequences, $$d_{\tilde\sigma}\left((U_1^s,\dots,U_k^s),\Kah\right)\to0$$ as $s\to+\infty$.
Estimates for any $\beta$ and $\ep$ small
=========================================
Let us start with some estimates on $c_{\ep,\b}$.
\[lem:c\_b-bdd\] When $\ep$ is fixed, $c_{\ep,\b}$ is non decreasing in $\b$, and $c_{\ep,\b}\leq c_\infty$.
First of all, if $\b_1<\b_2$, then for any $(u_{11},\dots,u_{kh_k})\in\Xh_\ep$ we have $J_{\b_1}(\L_1U_1,\dots,\L_kU_k)< J_{\b_2}(\L_1U_1,\dots,\L_kU_k)$, and we can pass to the inf–sup obtaining $c_{\ep,\b_1}\leq
c_{\ep,\b_2}$. Now, let $\b$ be fixed. By definition we have $$c_\infty=\inf_{\Xh_0}M_\b=\inf\left\{M_\b(u_{11},\dots,u_{kh_k}):\,\text{there
exists } W\in\Xh^*\text{ such that
}u_{im}=w_{\tilde\s^{-1}(im)}\right\}.$$ Indeed, in such situation $w_lw_p=0$ for $l\neq p$ and thus, letting $\mu_l=\l_{\tilde\s(l)}$, we obtain $J_\b(\L_1U_1,\dots,\L_kU_k)=J^*(\sum\mu_lw_l)$. To conclude we simply observe that $$\Xh_0\subset\Xh_\ep\qquad\text{ for every }\ep.\qedhere$$
\[coro:tilde\_X\] Let $$\tilde\Xh_\ep=\left\{(u_{11},\dots,u_{kh_k})\in\Xh_\ep:\,M_\b(u_{11},\dots,u_{kh_k})<
c_\infty+\min\left(1,\frac{1}{\b}\right)\right\}.$$ Then $$c_{\ep,\b}=\inf_{\tilde\Xh_\ep}M_\b.$$
From now on, we will restrict our attention to the elements of $\tilde\Xh_\ep$. We remark that $\tilde\Xh_\ep$ depends on $\b$: actually, if, for some $i\neq j$, $U_iU_j\neq0$ on a set of positive measure, then the corresponding $(u_{11},\dots,u_{kh_k})$ may not belong to $\tilde\Xh_\ep$ if $\b$ is sufficiently large. Nevertheless, all the results we will prove in this section will depend only on $\ep$, and not on $\b$.
\[lem:achieve\_M\] Let $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$. Then $M_\b(u_{11},\dots,u_{kh_k})$ is positive and achieved: $$0<M_\b(u_{11},\dots,u_{kh_k})=\Phi_\b(\bar\L_1,\dots,\bar\L_k),$$ with $$\nabla\Phi_\b(\bar\L_1,\dots,\bar\L_k)\cdot(\bar\L_1,\dots,\bar\L_k)=0.$$
We drop the dependence on $\b$. We observe that $\Phi$ is the sum of two polynomials, which are homogenous of degree two and four, respectively: $$\Phi(\L_1,\dots,\L_k)=\frac12P_2(\L_1,\dots,\L_k)+\frac14P_4(\L_1,\dots,\L_k)$$ where $P_2=\sum_{i,m}\|\l_{im}u_{im}\|^2$. Therefore, for $t\geq0$, $$\Phi(t\L_1,\dots,t\L_k)=\frac12t^2P_2(\L_1,\dots,\L_k)+\frac14t^4P_4(\L_1,\dots,\L_k),$$ Since $u_{im}\not\equiv0$ for every $i$, $m$, if some $\l_{im}$ is different from 0 then $P_2>0$. Thus, for $t$ small, $0<\Phi(t\L_1,\dots,t\L_k)\leq M$.\
As a consequence, we can write $$M=\sup\left\{\Phi(t\L_1,\dots,t\L_k):\,t>0,\,\sum_{i=1}^k|\L_i|^2=1\right\}.$$ On one hand, we have $$\max_{\sum|\L_i|=1}P_2(\L_1,\dots,\L_k)=a>0.$$ On the other hand, since $M<c_\infty+1/\b$, then $$\max_{\sum|\L_i|=1}P_4(\L_1,\dots,\L_k)=-b<0$$ (otherwise we would have $M=+\infty$). But then $$\Phi(t\L_1,\dots,t\L_k)\leq\frac{a}{2}t^2-\frac{b}{4}t^4<0\qquad\text{when
} t^2>\frac{2a}{b},$$ thus $\Phi$ is negative outside a compact set, and $M$ is achieved by some $(\bar\L_1,\dots,\bar\L_k)$.\
Finally, since this is a maximum for $\l_{im}\geq0$, $\bar\l_{im}>0$ implies $\partial_{\l_{im}}\Phi(\bar\L_1,\dots,\bar\L_k)=0$, therefore $$\partial_{\l_{im}}\Phi(\bar\L_1,\dots,\bar\L_k)\cdot\bar\l_{im}=0\quad\text{ for every
}(i,m).
\qedhere$$
Thus, if $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$, then $M_\b$ is a maximum. We want to prove that, when $\ep$ is small (not depending on $\b$), the maximum point is uniquely defined, and it smoothly depends on $(u_{11},\dots,u_{kh_k})$. As a first step, we provide some uniform estimates for its coordinates.
\[lem:R\] There exists $R>0$, not depending on[^8] $\ep\leq\ep_0$ and $\b$, such that, for every $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$,
1. $\nabla\Phi(\bar\L_1,\dots,\bar\L_k)\cdot(\bar\L_1,\dots,\bar\L_k)=0$ implies $\sum_i|\bar\L_i|^2< R^2$;
2. $\sum_i|\L_i|^2= R^2$ implies $\nabla\Phi(\L_1,\dots,\L_k)\cdot(\L_1,\dots,\L_k)<0$.
Since $\Phi\leq M$, using the notations of the proof of the previous lemma we can write $$\Phi(\bar\L_1,\dots,\bar\L_k)=\frac12P_2(\bar\L_1,\dots,\bar\L_k)+
\frac14P_4(\bar\L_1,\dots,\bar\L_k)< c_\infty+1$$ (recall Corollary \[coro:tilde\_X\]) and $$\nabla\Phi(\bar\L_1,\dots,\bar\L_k)\cdot(\bar\L_1,\dots,\bar\L_k)=
P_2(\bar\L_1,\dots,\bar\L_k)+P_4(\bar\L_1,\dots,\bar\L_k)=0,$$ providing $$\label{eq:bdd_norm}
P_2(\bar\L_1,\dots,\bar\L_k)=\sum_{i=1}^k\|\bar\L_iU_i\|^2<4(c_\infty+1).$$ Since every $u_{im}$ is non negative, we have $\|\bar\L_iU_i\|^2\geq\sum_m\bar\l_{im}^2\|u_{im}\|^2$. But we know (Remark \[lem:base\]) that each $\|u_{im}\|$ is bounded from below, providing $$\sum_{i=1}^k|\bar\L_i|^2<\frac{4(c_\infty+1)}{C_1^2}=R^2.$$ Now let $(\L_1,\dots,\L_k)$ be fixed with $\sum_i|\L_i|^2= R^2$. For $t>0$ we write $$f(t)=\nabla\Phi(t\L_1,\dots,t\L_k)\cdot(t\L_1,\dots,t\L_k)=
t^2P_2(\L_1,\dots,\L_k)+t^4P_4(\L_1,\dots,\L_k),$$ and we know, from the discussion above, that $f(\bar t)=0$ implies $\bar t<1$. Recalling that $P_4$ must be negative (otherwise $M=+\infty$) we deduce that $f(1)<0$, concluding the proof.
\[rem:bdd\_three\_addends\] As a consequence of the previous proof (and of Lemma \[lem:achieve\_M\]) we have that, if $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$ and $M_\b(u_{11},\dots,u_{kh_k})=\Phi_\b(\bar\L_1,\dots,\bar\L_k)$, then the three quantities $$\sum_{i=1}^k\|\bar\L_iU_i\|^2,\qquad \sum_{i=1}^k\int_{\R^N}
\left(\bar\L_iU_i\right)^4\,dx,\qquad\b\sum_{{i,j=1 \atop i\neq
j}}^k\int_{\R^N}\left(\bar\L_iU_i\right)^2\left(\bar\L_jU_j\right)^2\,dx,$$ are bounded not depending on $\b$: the first by equation ; the second by the first bound and by the continuous immersion of $\spc$ in $L^4(\R^N)$; the third by the previous bounds and the fact that $M_\b< c_\infty+1$.
\[lem:lambda>1/2\] There exists $0<\ep_1\leq\ep_0$ (not depending on $\beta$) such that if $\ep<\ep_1$, $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$, and $\Phi_\b(\bar\L_1,\dots,\bar\L_k)=M_\b(u_{11},\dots,u_{kh_k})$ then $$\bar\l_{im}>\frac12\qquad\text{for every }(i,m).$$ In particular, since $\bar\l_{im}>0$, $\nabla\Phi_\b(\bar\L_1,\dots,\bar\L_k)=0$.
By the previous lemma we know that $0\leq\bar\l_{im}\leq R$. We choose $(i,m)$ and, for any $(j,n)\neq(i,m)$, we fix $0\leq\l_{jn}\leq R$. We will prove that (for $\ep$ sufficiently small) $$\label{eq:convex}
\Phi|_{\l_{im}=0}<\Phi|_{\l_{im}=1/2}, \qquad\text{ and }\qquad
0\leq\l_{im}\leq1/2 \implies
\partial^2_{\l^2_{im}}\Phi(\L_1,\dots\L_k)>0,$$ and the result will follow. Let $V=\sum_{n\neq m}\l_{in}u_{in}$. Since all the $\l_{in}$’s are bounded, by Remark \[lem:base\] we know that $$\left\langle V,u_{im} \right\rangle=o(1),\quad\int_{\R^N}V^p
u_{im}^{q}\,dx=o(1),\qquad\text{ as }\ep\to0.$$ Moreover, according to the definition of $\Xh_\ep$, let $l=\tilde\s^{-1}(im)$ and $w_l$ be such that $\|u_{im}-w_l\|<\ep$, with $\|w_l\|^2=\int_{\R^N}w_l^4\,dx$. On one hand, we have (as $\ep\to0$) $$\begin{split}
\Phi|_{\l_{im}=1/2}-\Phi|_{\l_{im}=0}&\geq\frac12\left[\left\|V+\frac12u_{im}
\right\|^2-\left\|V\right\|^2\right] - \frac14\int_{\R^N}\left[\left(V+
\frac12u_{im}\right)^4-\left(V\right)^4\right]\,dx\\
&= \frac12\left\|\frac12u_{im}\right\|^2-\frac14\int_{\R^N}\left(
\frac12u_{im}\right)^4\,dx+o(1)\\
&= \frac12\left\|\frac12w_l\right\|^2-\frac14\int_{\R^N}\left(
\frac12w_l\right)^4\,dx+o(1)\\
&=\frac{7}{64}\|w_l\|^2+o(1)>0.
\end{split}$$ On the other hand, with similar calculations, we obtain $$\label{eq:grad_Phi}
\partial_{\l_{im}}\Phi = \left\langle\L_{i}U_i,u_{im}\right\rangle -
\int_{\R^N}(\L_{i}U_i)^3u_{im}\,dx+\b\int_{\R^N}(\L_iU_i)u_{im}\sum_{j\neq
i}(\L_jU_j)^2\,dx$$ and $$\label{eq:der_sec_Phi}
\begin{split}
\partial^2_{\l^2_{im}}\Phi(\L_1,\dots\L_k)&=\|u_{im}\|^2 -
3\int_{\R^N}(\L_{i}U_i)^2u_{im}^2\,dx+\b\int_{\R^N}u_{im}^2\sum_{j\neq
i}(\L_jU_j)^2\,dx\\
&\geq \|w_l\|^2 - 3\int_{\R^N}\l_{im}^2w_l^2\,dx+o(1)\\
&= (1-3\l_{im}^2)\|w_l\|^2+o(1)>0
\end{split}$$ since $\l_{im}\leq1/2$.
\[rem:1/2\] As a byproduct of the previous proof (equation ), we have that, if $\ep<\ep_1$, $(u_{11},\dots,u_{kh_k})\in
\tilde\Xh_\ep$, and $0\leq\l_{jn}\leq R$, then $$\partial_{\l_{im}}\Phi(\L_1,\dots\L_k)|_{\l_{im}=1/2}>0.$$
\[lem:pos\_def\_hess\] There exists $0<\eps_2\leq\eps_1$ (not depending on $\b$) such that if $\ep<\ep_2$, $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$ and $\nabla\Phi(\bar\L_1,\dots,\bar\L_k)=0$ with $\bar\l_{im}>1/2$ then the Hessian matrix $$D^2\Phi(\bar\L_1,\dots,\bar\L_k)\text{ is negative definite.}$$
From we obtain $$\begin{split}
\partial^2_{\l_{im}\l_{jn}}\Phi &= \b\int_{\R^N}2(\L_iU_i) (\L_jU_j) u_{im} u_{jn}\,dx \qquad
\text{if }j\neq i\\
\partial^2_{\l_{im}\l_{in}}\Phi &= \left\langle u_{im},u_{in}\right\rangle - 3\int_{\R^N}
(\L_{i}U_i)^2u_{im}u_{in}\,dx+\b\int_{\R^N} u_{im}u_{in}\sum_{j\neq i}(\L_jU_j)^2\,dx
\end{split}$$ (we computed $\partial^2_{\l^2_{im}}\Phi$ in ). For $m\neq n$ we write $$M^i_{mn}=\left\langle u_{im},u_{in}\right\rangle
-\int_{\R^N}(\bar\L_iU_i)^2 u_{im}u_{in}\,dx$$ in such a way that $$\left\langle\bar\L_i U_i,u_{im}\right\rangle
-\int_{\R^N}(\bar\L_iU_i)^3u_{im}\,dx= \bar\l_{im}\left[\left\|
u_{im}\right\|^2 - \int_{\R^N}(\bar\L_i U_i)^2 u_{im}^2\,dx
+\sum_{n\neq m}\frac{\bar\l_{in}}{\bar\l_{im}}M^i_{mn}\right].$$ Since $\nabla\Phi(\bar\L_1,\dots,\bar\L_k)=0$ we have, for every $i$, $m$, $$\int_{\R^N}(\bar\L_iU_i)^2u_{im}^2= \|u_{im}\|^2+ \sum_{n\neq
m}\frac{\bar\l_{in}}{\bar\l_{im}}M^i_{mn}+
\frac{\b}{\bar\l_{im}}\int_{\R^N}(\bar\L_iU_i)u_{im}\sum_{j\neq
i}(\bar\L_jU_j)^2\,dx.$$ Substituting we obtain $$\label{eq:hess1}
\partial^2_{\l_{im}\l_{im}}\Phi(\bar\L_1,\dots,\bar\L_k)=
-2\underbrace{\|u_{im}\|^2}_{(A)}
-\underbrace{3\sum_{n\neq m}\frac{\bar\l_{in}}{\bar\l_{im}}M^i_{mn}}_{(B)}
+\b\int_{\R^N}\underbrace{u_{im}\left(u_{im}-\frac{3}{\bar\l_{im}}\bar\L_iU_i\right)
\sum_{j\neq i}(\bar\L_jU_j)^2}_{(C)}\,dx,$$ and, for $n\neq m$, $$\label{eq:hess2}
\partial^2_{\l_{im}\l_{in}}\Phi(\bar\L_1,\dots,\bar\L_k) =
\underbrace{M^i_{mn}-2\int_{\R^N}(\bar\L_iU_i)^2u_{im}u_{in}\,dx}_{(B)}
+\b\int_{\R^N}\underbrace{u_{im}u_{in}\sum_{j\neq i}(\bar\L_jU_j)^2}_{(C)}\,dx.$$ As a consequence we can split $D^2\Phi$ as $$D^2\Phi(\bar\L_1,\dots,\bar\L_k)=-2A+B+\b\int_{\R_N} C(x)\,dx,$$ where each of the matrices $A$, $B$, and $C$ contains the corresponding terms in and , and $C$ also contains the terms appearing in $\partial^2_{\l_{im}
\l_{jn}}\Phi$, $i\neq j$. First of all, using Remark \[lem:base\] and the boundedness of the $\bar\l_{im}$’s, we observe that $A$ is diagonal and strictly positive definite, independent of $\eps$, while $B$ is arbitrary small as $\eps$ goes to zero (not depending on $\b$). We will show that $C(x)$ is negative semidefinite for every $x$: this will conclude the proof.
To do that, we will only use that $u_{im}\geq0$ and $\sum_m\bar\l_{im}u_{im}=\bar\L_iU_i$, therefore, without loss of generality, we can put $\bar\l_{im}=1$ for every $(i,m)$[^9]. The matrix $C(x)$ can be written as the sum of matrices $C_{ij}(x)$ where only two components, say $U_i$ and $U_j$, interact. Such matrices, for $x$ fixed, contain many null blocks, corresponding both to the interaction with the other components $U_p$, $p\neq i,j$, and to the pulses of $U_i$ and $U_j$ vanishing in $x$. All those null blocks do not incide on the semidefiniteness of $C_{ij}$; up to the null terms, $C_{ij}$ writes like $$\left(
\begin{array}{cc}
\begin{array}{ccc}
U_j^2(u_{i1}-3U_i)u_{i1} & \cdots & U_j^2u_{i1}u_{ih_i} \\
\vdots & \ddots & \vdots \\
U_j^2u_{ih_i}u_{i1} & \cdots & U_j^2(u_{ih_i}-3U_i)u_{ih_i}
\end{array}
\vline&
\begin{array}{ccc}
2U_iU_ju_{i1}u_{j1} & \cdots & 2U_iU_ju_{i1}u_{jh_j} \\
\vdots & \ddots & \vdots \\
2U_iU_ju_{ih_i}u_{j1} & \cdots & 2U_iU_ju_{ih_i}u_{jh_j}
\end{array}
\\ \hline\hfill
\begin{array}{ccc}
2U_iU_ju_{i1}u_{j1} & \cdots & 2U_iU_ju_{ih_i}u_{j1} \\
\vdots & \ddots & \vdots \\
2U_iU_ju_{jh_j}u_{i1} & \cdots & 2U_iU_ju_{ih_i}u_{jh_j}
\end{array}
\hfill\vline&
\begin{array}{ccc}
U_i^2(u_{j1}-3U_j)u_{j1} & \cdots & U_i^2u_{j1}u_{jh_j} \\
\vdots & \ddots & \vdots \\
U_i^2u_{jh_j}u_{j1} & \cdots & U_i^2(u_{jh_j}-3U_j)u_{jh_j}
\end{array}
\end{array}
\right)$$ (where every term is strictly positive), which has the same signature than $$\left(
\begin{array}{cc}
\begin{array}{ccc}
1-3U_i/u_{i1} & \cdots & 1 \\
\vdots & \ddots & \vdots \\
1 & \cdots & 1-3U_i/u_{ih_i}
\end{array}
\vline& 2\\
\hline\smallskip
\hfill 2
\hfill\vline&
\begin{array}{ccc}
1-3U_j/u_{j1} & \cdots & 1 \\
\vdots & \ddots & \vdots \\
1 & \cdots & 1-3U_j/u_{jh_j}
\end{array}
\end{array}
\right)$$ (we mean that in the two blocks every term is equal to 2). The last matrix can be seen as the sum $$2\left(
\begin{array}{cc}
-1\ \vline& 1\\
\hline\smallskip
\hfill 1\
\hfill\vline&-1
\end{array}
\right) + 3\left(
\begin{array}{cc}
\begin{array}{ccc}
1-1/\a_1 & \cdots & 1 \\
\vdots & \ddots & \vdots \\
1 & \cdots & 1-1/\a_{h_i}
\end{array}
\vline& 0\\
\hline\smallskip
\hfill 0
\hfill\vline&
\begin{array}{ccc}
1-1/\b_1 & \cdots & 1 \\
\vdots & \ddots & \vdots \\
1 & \cdots & 1-1/\b_{h_j}
\end{array}
\end{array}
\right),$$ where $\a_m=u_{im}/U_i$, $\b_m=u_{jm}/U_j$, in such a way that $\sum\a_m=\sum\b_m=1$. It is easy to see that the first addend is negative semidefinite, so the last thing we have to prove is that $\sum_m\a_m=1$, $\a_m>0$, implies that $$D= \left(
\begin{array}{ccc}
1-1/\a_1 & \cdots & 1 \\
\vdots & \ddots & \vdots \\
1 & \cdots & 1-1/\a_{h}
\end{array}
\right)$$ is negative semidefinite. Let $\xi=(\xi_1,\dots,\xi_h)\in\R^h$. Then it is easy to prove that $$\sum_{m=1}^h\frac{\xi_m^2}{\a_m^2}\geq h|\xi|^2,\qquad \text{ thus
}\qquad D\xi\cdot\xi\leq\left(\sum_{m=1}^h\xi_m\right)^2-h|\xi|^2,$$ that is trivially non positive for every $h$ and $\xi$.
\[prop:unique\_lambda\] Let $\ep<\ep_2$ in such a way that all the previous results hold. Then, for every $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$ there exists one and only one choice $$\bar\L_i=\bar\L_i(u_{11},\dots,u_{kh_k})$$ such that $$J_\b\left(\bar\L_1(u_{11},\dots,u_{kh_k})U_1,\dots,\bar\L_k(u_{11},
\dots,u_{kh_k})U_k\right)=M_\b(u_{11},\dots,u_{kh_k}).$$ Moreover, each $\bar\L_i$ is well defined and of class $C^1$ on a neighborhood $N(\tilde\Xh_\ep)$ of $\tilde\Xh_\ep$.
To start with we will show, via a topological degree argument, that $(\bar\L_1,\dots,\bar\L_k)$ is uniquely defined on $\tilde\Xh_\ep$. Indeed, consider the set $$D=\left\{(\L_1,\dots,\L_k)\in\R^h:\,\sum_{i}|\L_i|^2<R^2,\,\l_{im}>1/2\right\}.$$ By Lemma \[lem:R\] and Remark \[rem:1/2\] we know that $\nabla\Phi$ points inward on $\partial D$. As a consequence $-\nabla\Phi$ is homotopically equivalent to a translation of the identity map, and $$\deg\left(\nabla\Phi,0,D\right)=(-1)^h.$$ On the other hand, such degree must be equal to the sum of the local degrees of all the critical points of $\Phi$ in $D$: since these points are all non degenerate maxima (by Lemma \[lem:pos\_def\_hess\]), and they have local degree $(-1)^h$, we conclude that there is only one critical point of $\Phi$ in $D$, and it must be the global maximum point. Therefore the maps $\bar\L_i(u_{11},\dots,u_{kh_k})$ are well defined in $\tilde\Xh_\ep$. Moreover, they are implicitly defined by $$\nabla\Phi(\bar\L_1,\dots,\bar\L_k)=0,$$ thus to conclude we can apply the Implicit Function Theorem in a neighborhood of any point of $\tilde\Xh_\ep$: indeed, $\nabla\Phi$ is a $C^1$ map (both in the $\l$–variables and in the $u$–variables); moreover, its differential with respect to the $\l$–variables is invertible by Lemma \[lem:pos\_def\_hess\] (it is simply $D^2\Phi$).
We observe that, even if $(u_{11},\dots,u_{kh_k})$ belongs to $\tilde\Xh_\ep$, nevertheless this might not be true for the corresponding $(\bar\l_{11}u_{11},\dots,\bar\l_{kh_k}u_{kh_k})$. At this point we can only state a weaker property for those elements of $\tilde\Xh_\ep$ with the corresponding $U_i$’s having disjoint supports.
\[lem:restrict\_ep\_false\] Let $\ep<\ep_2$ in such a way that all the previous results hold, and $(v_{11},\dots,v_{kh_k})\in\tilde\Xh_\ep$ be such that $$V_i\cdot V_j=0\quad\text{almost everywhere, for every }i,j.$$ Then there exists $\delta=\delta(\ep)$ (not depending on $\b$) such that $(\bar\l_{11}v_{11},\dots,\bar\l_{kh_k}v_{kh_k})\in\tilde\Xh_{\delta}$.[^10] Moreover, $\d$ goes to 0 as $\ep$ does.
To start with, we observe that $v_{im}\geq0$ implies $\bar\l_{im}v_{im}\geq0$, and that $M_\b(u_{11},\dots,u_{kh_k})=M_\b(\bar\l_{11}u_{11},\dots,\bar\l_{kh_k}u_{kh_k})$. As a consequence, if we prove that $$\sum_{i,m}\|v_{im}-w_{\tilde\s^{-1}(im)}\|^2<\ep^2\quad\implies\quad
\sum_{i,m}\|\bar\l_{im}v_{im}-w_{\tilde\s^{-1}(im)}\|^2<\delta^2,$$ with $\d$ vanishing when $\ep$ does, we have finished. By assumption we have that $\b\int(\bar\L_iV_i)v_{im}(\bar\L_jV_j)^2=0$ for every choice of the indexes, so that the functions $\bar\l$ are implicitly defined by $$\left\langle\sum_n\bar\l_{in}v_{in},v_{im}\right\rangle-\int_{\R^N}\left(\sum_n
\bar\l_{in}v_{in}\right)^3v_{im}\,dx=0.$$ On the other hand, by the definition of $\Kah$, we know that $$\left\langle\sum_n\bar\l_{in}w_{\tilde\s^{-1}(in)},w_{\tilde\s^{-1}(im)}
\right\rangle-\int_{\R^N}\left(\sum_n\bar\l_{in}w_{\tilde\s^{-1}(in)}
\right)^3w_{\tilde\s^{-1}(im)}\,dx=0\,\iff\,\bar\l_{in}=1\text{ for
every }n.$$ But then our claim directly follows from the Implicit Function Theorem.
Estimates for $\ep$ fixed and $\b$ large
========================================
From now on we choose $\bar\ep>0$ in such a way that, for every $\ep<\bar\ep$, it holds $$\ep<\ep_2\quad\text{and}\quad\d<\ep_2$$ with $\ep_2$ as in Proposition \[prop:unique\_lambda\] and $\d=\d(\ep)$ as in Lemma \[lem:restrict\_ep\_false\]. As we said, $\bar\ep$ do not depend on $\b$. In the following $\ep$ and $\d$ are considered fixed as above.
Since in the following we will let $\b$ move, we observe that, as we already remarked, the set $\tilde\Xh_\ep=\tilde\Xh_{\ep,\b}$ also depends on $\b$, since the functions inside satisfy $M_\b<
c_\infty+1/\b$.
If $\b_1\leq\b_2$, then for any $(u_{11},\dots,u_{kh_k})\in\Xh_\ep$ and any choice of the $\L_i$’s, it holds $J_{\b_1}(\L_1U_1,\dots,\L_kU_k)\leq
J_{\b_2}(\L_1U_1,\dots,\L_kU_k)$. Passing to the supremum we obtain $$\b_1\leq\b_2 \implies
\tilde\Xh_{\ep,\b_2}\subset\tilde\Xh_{\ep,\b_1}.$$
In the following we will deal with sequence of $h$–tuples in $\tilde\Xh_\ep$, with increasing $\b$. For this reason, we start this section with a general result about some convergence property for such sequences.
\[lem:weak\_closure\_infty\] Let the sequence $\b_s\to+\infty$, $s\in\N$, and let us consider a sequence of $h$–tuples $$(u^s_{11},\dots,u^s_{kh_k})\in \tilde\Xh_{\ep,\b_s}.$$ Then, up to a subsequence, $u^s_{im}\to u^*_{im}$, strongly in $\spc$, for every $(i,m)$. Moreover $$U^*_i\cdot U^*_j\equiv0\text{ for }i\neq j,\quad\text{ and
}(u^*_{11},\dots,u^*_{kh_k})\in \tilde\Xh_{\ep,\b}\text{ for every
}\b.$$ Finally, writing $\bar\l^*_{im}=\bar\l_{im}(u^*_{11},\dots,
u^*_{kh_k})$, we have that $\bar\l_{im}^s\to\bar\l_{im}^*$ and $$J_{\b}(\L^*_1U_1^*,\dots,\L^*_kU_k^*)=c_\infty\quad\text{ for every
}\b.$$
By assumption we can find a sequence $(w^s_{1},\dots,w^s_{l})\in\Kah$ such that $\sum_{i,m}\|u^s_{im}-w^s_{l}\|^2<\ep^2$, where $l=\tilde\s^{-1}(im)$. Since $\Kah$ is compact, we have that, up to a subsequence, $w^s_l\to w_l^*$ strongly, and $u^s_{im}\rightharpoonup u_{im}^*$ weakly in $\spc$ (since they are bounded, independently on $\b$ (see Lemma \[lem:R\]), also each $\bar\l_{im}^\b$ converges to some number). By the compact immersion of $\spc$ in $L^4(\R^N)$, we deduce that $u^s_{im}\to u_{im}^*$ strongly in $L^4(\R^N)$. We know by Remark \[rem:bdd\_three\_addends\] that $$\b_s\sum_{{i,j=1 \atop i\neq
j}}^k\int_{\R^N}\left(\bar\L^s_iU^s_i\right)^2\left(\bar\L^s_jU^s_j\right)^2\,dx\leq
C$$ not depending on $\b$. Using the strong $L^4$–convergence and Lemma \[lem:lambda>1/2\] we conclude that $U^*_i\cdot U^*_j\equiv0\text{
for }i\neq j$. To prove that $(u_{11}^*,\dots,u_{kh_k}^*)\in\tilde\Xh_\ep$ we observe that:
- $u_{im}^*\geq0$ by the strong $L^4$–convergence;
- $\sum_{i,m}\|u_{im}^*-w_{l}^*\|^2<\ep^2$ by weak lower semicontinuity of $\|\cdot\|$;
- finally, for every choice of the $\l_{im}$’s, we have that $\liminf\|\L_iU_i^s\|\geq\|\L_iU_i^*\|$, $\lim\int(\L_iU_i^s)^4=
\int(\L_iU_i^*)^4$ and $$\liminf
\b_s\int_{\R^N}\left(\bar\L_iU^s_i\right)^2\left(\bar\L_jU^s_j\right)^2\,dx\geq0
=
\b\int_{\R^N}\left(\bar\L_iU^*_i\right)^2\left(\bar\L_jU^*_j\right)^2\,dx
\quad\text{ for every }\b,$$ providing, for every $\b$, $J_\b(\L_{i}U_{i}^*) \leq\liminf
J_{\b_s}(\L_{i}U_{i}^s)< c_\infty+1/\b_s$, that implies $$M_{\b}(u_{11}^*,\dots,u_{kh_k}^*)\leq c_\infty<c_\infty+\min(1,1/\b).$$
Thus $(u_{11}^*,\dots,u_{kh_k}^*)\in\tilde\Xh_{\ep,\b}$ and we can write $\bar\l_{im}^*=\bar\l_{im}(u_{11}^*,\dots,u_{kh_k}^*)$. Now, since $U_i^*\cdot U_j^*\equiv0$, by Proposition \[prop:nehari\_mixed\] we know that $$J_{\b}(\bar\L^*_{1}U_{1}^*,\dots,\bar\L^*_{k}U_{k}^*)=
\sup_{\l_{im}>0}J^*\left(\sum_{i,m}\l_{im}v_{im}\right)\geq
c_\infty.$$ On the other hand, for what we said, $$c_\infty+1/\b_s\geq
J_{\b_s}(\bar\L^s_{1}U_{1}^s,\dots,\bar\L^s_{k}U_{k}^s)\geq
J_{\b_s}(\bar\L^*_{1}U_{1}^s,\dots,\bar\L^*_{k}U_{k}^s)\geq
J_\b(\bar\L^*_{1}U_{1}^*,\dots,\bar\L^*_{k}U_{k}^*)+o(1),$$ where the second inequality is strict if and only if $\bar\l_{im}^\b\not\to\bar\l_{im}^*$, and the third is strict if and only if $u_{im}^\b\not\to u_{im}^*$. Comparing the last two equations, we obtain that $$J(\bar\L^*_{1}U_{1}^*,\dots,\bar\L^*_{k}U_{k}^*)= c_\infty,\quad
\bar\l_{im}^\b\to\bar\l_{im}^*,\quad\text{ and }u_{im}^\b\to
u_{im}^*\text{ strongly,}$$ concluding the proof.
Now we want to show that, if $\b$ is sufficiently large, then the result of Lemma \[lem:restrict\_ep\_false\] holds on the whole $\tilde\Xh_\ep$, without restrictions.
\[lem:restrict\_ep\_true\] There exists $\b_1$ such that if $\beta>\b_1$ then $$(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep\quad\implies\quad
(\bar\l_{11}u_{11},\dots,\bar\l_{kh_k}u_{kh_k})\in\tilde\Xh_\delta.$$
As in Lemma \[lem:restrict\_ep\_false\], since $M_\b(u_{11},\dots,u_{kh_k})=M_\b(\bar\l_{11}u_{11},\dots,\bar\l_{kh_k}u_{kh_k})$, and $u_{im}\geq0$ implies $\bar\l_{im}u_{im}\geq0$, we only have to prove that, when $\b$ is sufficiently large, $$\sum_{i,m}\|u_{im}-w_{l}\|^2<\ep^2\quad\implies\quad
\sum_{i,m}\|\bar\l_{im}u_{im}-w_{l}\|^2<\delta^2,$$ where $l=\tilde\s^{-1}(im)$. By contradiction, let $\b_s\to+\infty$ and $(u^s_{11},\dots,u^s_{kh_k})\in\tilde\Xh_\ep$ be such that $\sum_{i,m}\|u^s_{im}-w^s_{l}\|^2<\ep^2$, and $\sum_{i,m}\|\bar\l^s_{im}u^s_{im}-w_{l}\|^2\geq\delta^2$ for any $(w_1,\dots,w_h)\in\Kah$. Using Lemma \[lem:weak\_closure\_infty\] we have that $u^s_{im}\to u^*_{im}$, $\bar\l_{im}^s\to
\bar\l_{im}^*$, in such a way that $$(u^*_{11},\dots,u^*_{kh_k})\in\tilde\Xh_\ep,\quad U^*_i\cdot
U^*_j\equiv0,\quad\text{and
}\sum_{i,m}\|\bar\l^*_{im}u^*_{im}-w_{l}\|^2\geq\d,$$ for every $(w_1,\dots,w_h)\in\Kah$. But this is in contradiction with Lemma \[lem:restrict\_ep\_false\].
By the previous lemma we have that, for every $(u_{11},\dots,u_{kh_k})\in\tilde\Xh_\ep$, the corresponding maximum point $(\bar\l_{11}u_{11},\dots,\bar\l_{kh_k}u_{kh_k})$ belongs to $\tilde\Xh_\d$ and $\bar\l_{im}(\bar\l_{11}u_{11},\dots,
\bar\l_{kh_k}u_{kh_k})=1$ for every $(i,m)$. Motivated by this fact we define $$\Neh_\b=\left\{(u_{11},\dots,u_{kh_k})\in\tilde\Xh_{\d,\b}:\,
\bar\l_{im}(u_{11},\dots,u_{kh_k})=1\text{ for every }(i,m)\right\},$$ immediately obtaining that, on $\Neh_\b$, $M_\b\equiv J_\b$ and $$\label{eq:c_beta_con_Nehari}
c_{\ep,\b}\geq\inf_{\Neh_\b}J_\b(U_1,\dots,U_k).$$ As a matter of fact, if $\b$ is sufficiently large, also the opposite inequality holds.
\[lem:excision\] There exists $\b_2\geq\b_1$ such that, if $\b>\b_2$ then $$\Neh_\b\subset\tilde\Xh_{\ep/2}.$$
We argue again by contradiction. Let (up to a subsequence) $\b_s\to+\infty$, $(u^s_{11},\dots,u^s_{kh_k}) \in \Neh_\b$ (and hence $\bar\l_{im}^s=1$) be such that $$\label{eq:out_of_eps/2}
\sum_{i,m}\|u^s_{im}-w_{l}\|^2\geq\frac{\ep^2}{4}\quad\text{ for
every }W\in\Kah.$$ Using Lemma \[lem:weak\_closure\_infty\], we have that $u^s_{im}\to
u_{im}^*$ strongly in $\spc$ and $\bar\l_{im}^*=1$, for every $(i,m)$. As a consequence, defining $w^*_{l}=u^*_{\tilde\sigma(l)}$, we obtain that $J^*(w^*_l)=\sup_\l J^*(\l w^*_l)$ and $J(\sum_l
w_l^*)=c_\infty$. Therefore $(w^*_1,\dots,w^*_h)\in\Kah$, and, obviously $\sum_{i,m}\|u^*_{im}-w^*_{l}\|^2=0$. But this, using strong convergence in , provides a contradiction.
\[rem:equiv\_char\] Taking into account , and the previous lemma (beside the inclusion $\tilde\Xh_{\ep/2}\subset\tilde\Xh_\ep$) we obtain $$c_{\ep,\b}=\inf_{\tilde\Xh_\ep}M_\b\geq\inf_{\Neh_\b}J_\b(U_1,\dots,U_k)\geq
\inf_{\tilde\Xh_{\ep/2}}M_\b\geq\inf_{\tilde\Xh_\ep}M_\b=c_{\ep,\b},$$ obtaining three equivalent characterizations of $c_{\ep,\b}$.
Proof of the main results
=========================
In order to prove our main results, we present an useful abstract lemma.
\[lem:variational\_lemma\] Let $H$ be an Hilbert space, $d$ an integer, $I\in C^2(H^d;\R)$, $\Xh\subset H^d$, and $N(\Xh)$ an open neighborhood of $\Xh$. Let us assume that:
1. $d$ functionals $\bar\l_i\in C^1(N(\Xh);\R)$, $i=1,\dots,d$, are uniquely defined, in such a way that $\bar\l_i>0$ for every $i$ and $$\sup_{\l_i>0}I(\l_1x_1,\dots,\l_dx_d)=I(\bar\l_1(x_1,\dots,x_d)x_1,\dots,\bar\l_d(x_1,\dots,x_d)x_d);$$
2. $(\bar x_1,\dots,\bar x_d)\in\Xh$ is such that $\bar\l_i(\bar x_1,\dots,\bar x_d)=1$ for every $i$, $$\inf_{\Xh}\sup_{\l_i>0}I(\l_1x_1,\dots,\l_dx_d)=I(\bar x_1,\dots,\bar
x_d),$$ and the $d\times d$ matrix $H=\left(\partial^2_{x_ix_j}I(\bar x_1,\dots,\bar x_d)[\bar x_i,\bar x_j]
\right)_{i,j=1,\dots,d}$ is invertible;
3. $(y_1,\dots,y_d)\in H^d$ is such that, for some $\bar t>0$ and $0<\d<1$, $$(s_1\bar x_1+ty_1,\dots,s_d\bar x_d+ty_d)\in\Xh\qquad\text{as }\quad0\leq
t\leq \bar t,\,1-\d\leq s_i\leq1+\d.$$
Then $$\nabla I(\bar x_1,\dots,\bar x_d)\cdot(y_1,\dots,y_d)\geq0.$$
For easier notation we set $\bar x=(\bar x_1,\dots,\bar x_d)$. Since $(s_1\bar x_1+ty_1,\dots,s_d\bar x_d+ty_d)\in\Xh$ we can substitute it in each $\bar\l_i$. To start with, we want to apply the Implicit Function Theorem to $$F(s_1,\dots,s_d,t)= \left(
\begin{array}{c}
\bar\l_1(s_1\bar x_1+ty_1,\dots,s_d\bar x_d+ty_d) \\
\vdots \\
\bar\l_d(s_1\bar x_1+ty_1,\dots,s_d\bar x_d+ty_d) \\
\end{array}
\right) = \left(
\begin{array}{c}
1 \\
\vdots \\
1 \\
\end{array}
\right),$$ in order to write $s_i=s_i(t)$, where $s_i$ is $C^1$, for every $i$. By assumption $F$ is $C^1$ and $F(1,\dots,1,0)=(1,\dots,1)$, thus we only have to prove that the $d\times d$ jacobian matrix $$A=\partial_{(s_1,\dots,s_d)}F(1,\dots,1,0)= \left(
\partial_{x_i}\bar\l_j(\bar x)[\bar x_i]
\right)_{i,j=1,\dots,d} \qquad\text{is invertible}.$$ Therefore let us assume, by contradiction, the existence of a vector $$\label{eq:s_impl_def}
v=(v_1,\dots,v_d)\in\R^d\setminus\{0\}\text{ such that }Av=0.$$ Let us now consider the function $\Phi(\l_1,\dots,\l_d)=I(\l_1x_1,\dots\l_dx_d)$; by definition, the point $(\bar\l_1(x_1,\dots,x_d),\dots\bar\l_d(x_1,\dots,x_d))$ is a free maximum of $\Phi$, and hence $\nabla\Phi(\bar\l_1,\dots,\bar\l_d)=(0,\dots,0)$, that is $$\partial_{x_i}I(\l_1(x_1,\dots,x_d)x_1,\dots,\l_d(x_1,\dots,x_d)x_d)[x_i]=0
\qquad\text{for every }i$$ (in particular, $\partial_{x_i}I(\bar x)[\bar x_i]=0$ for every $i$). We can differentiate the previous equation with respect to $x_j$, obtaining, for every $(z_1,\dots,z_d)\in H^d$, $$\partial^2_{x_ix_j}I(\l_1x_1,\dots,\l_dx_d)
[x_i,\l_jz_j]+\sum_{n=1}^d\partial^2_{x_ix_n}
I(\l_1x_1,\dots,\l_dx_d)[x_i,x_n] \cdot\partial_{x_j}\bar\l_n[z_j]=0
\qquad\text{for }j\neq i$$ and $$\begin{gathered}
\partial^2_{x_ix_i}I(\l_1x_1,\dots,\l_dx_d)
[x_i,\l_iz_i]+\sum_{n=1}^d\partial^2_{x_ix_n}
I(\l_1x_1,\dots,\l_dx_d)[x_i,x_n]
\cdot\partial_{x_i}\bar\l_n[z_i]+\\
+\partial_{x_i}I(\l_1x_1,\dots,\l_dx_d)[z_i]=0.\end{gathered}$$ We can substitute $x_i=\bar x_i$, $z_i=v_i\bar x_i$, and $\bar\l_i=1$ in the previous equations. Recalling that $\partial_{x_i}I(\bar x)[\bar x_i]=0$ for every $i$ we obtain, for every $i$ and $j$ (not necessarily different), $$\sum_{n=1}^d\partial^2_{x_ix_n} I(\bar x)[\bar x_i,\bar x_n]
\cdot\partial_{x_j}\bar\l_n(\bar x)[\bar x_j]\cdot
v_j=-v_j\partial^2_{x_ix_j}I(\bar x) [\bar x_i,\bar x_j].$$ Summing up on $j$, and recalling the definitions of $A$, $v$ (equation ) and $H$, (second assumption of the lemma) we have $$HAv=-Hv,$$ providing a contradiction with the invertibility of $H$.
Hence we obtain the existence of the $C^1$–functions $s_i=s_i(t)$ (for $t$ sufficiently small) such that $\l_i(s_1\bar
x_1+ty_1,\dots,s_d\bar x_d+ty_d)=1$. Let us consider the function $$\varphi(t)=I(s_1(t)\bar x_1+ty_1,\dots,s_d(t)\bar x_d+ty_d).$$ By construction $\varphi$ is $C^1$ and $\varphi(t)\geq0$ for $t\geq0$. We obtain that $$0\leq\varphi'(0)=\sum_{i=1}^d\partial_{x_i}I(\bar x)[s'_i(t)\bar
x_i+y_i]=\nabla I(\bar
x)\cdot(y_1,\dots,y_d)+\sum_{i=1}^ds'_i(t)\partial_{x_i}I(\bar
x)[\bar x_i],$$ and the result follows recalling again that $\partial_{x_i}I(\bar
x)[\bar x_i]=0$ for every $i$.
Now we are finally ready to prove our main results.
Let $\bar\ep$ as above and $\bar\b=\b_2$, in such a way that, for any fixed $\ep<\bar\ep$ and $\b>\bar\b$, all the previous results hold. By Remark \[rem:equiv\_char\], for every integer $s$ we have an element $(u^s_{11},\dots,u^s_{kh_k})\in\Neh_\b$ such that $$c_{\ep,\b}\leq J_\b(U_1^s,\dots,U_k^s)\leq c_{\ep,\b}+\frac{1}{s}.$$ We are in a situation very similar to that in Lemma \[lem:weak\_closure\_infty\] (much easier, in fact, since now $\b$ is fixed). Following the same scheme, one can easily prove that $u^s_{im}\to u_{im}^*$ strongly in $\spc$, with $$(u^*_{11},\dots,u^*_{kh_k})\in\Neh_\b, \quad
J_\b(U^*_1,\dots,U^*_k)=c_{\ep,\b}.$$ Moreover, by Lemma \[lem:excision\], the minimum point is $\ep/2$–near an element of $\Kah$.
It remains to prove that each $U^*_i$ is strictly positive and that $(U^*_1,\dots,U^*_k)$ solves . To do this we will apply Lemma \[lem:variational\_lemma\], letting $H=\spc$, $d=h$, $\Xh=\tilde\Xh_\ep$, $I(u_{11},\dots,u_{kh_k})=
J_\b(U_1,\dots,U_k)$, and $(\bar x_1,\dots,\bar
x_h)=(u^*_{11},\dots,u^*_{kh_k})$. Assumptions 1. and 2. in Lemma \[lem:variational\_lemma\] are satisfied by construction, therefore we have only to choose a variation $(y_1,\dots,y_h)$ and to check assumption 3.: $$\text{``}P=(s_{11}u^*_{11}+t y_1,\dots, s_{kh_k}u^*_{kh_k}+t
y_h)\in\tilde\Xh_\ep\text{ when }t>0\text{ is
small and each }s_{im}\text{ is near }1\text{''}.$$ Under these assumptions on $t$ and $s_{im}$, it is immediate to see that $P$ is $\ep$–near to the same element of $\Kah$ to which $(u^*_{11},\dots,u^*_{kh_k})$ is $\ep/2$–near; moreover, by continuity, $M_\b(P)=J_\b(\bar\L(P)P)<c_{\ep,\b}+1/\b$. Recalling the definition of $\tilde X_\ep$ (Corollary \[coro:tilde\_X\]) we have that assumption 3. is fulfilled whenever each component of $P$ is non negative.
First let us prove that each $U_i$ is strictly positive. Assume not, there exists $x_0\in\R^N$ with, say, $U_1(x_0)=0$. Since $U_1\not\equiv0$ we can easily construct an open, relatively compact annulus $A\ni x_0$ such that $U_1\leq1/2$ on $A$ and $U_1\not\equiv0$ on $\partial A$. For any radial $\f\in
C^{\infty}_0(A)$, $\f\geq0$, we choose the variation $y_1=\f$, $y_l=0$ for $l>1$. Clearly each component of $P$ is non negative, thus Lemma \[lem:variational\_lemma\] implies $$\begin{split}
0&\leq \nabla
I(u^*_{11},\dots,u^*_{kh_k})\cdot(\f,0,\dots,0)=\partial_{u_{11}}
I(u^*_{11},\dots,u^*_{kh_k})[\f]=\\
&=\int_{A}\left[\nabla
U_1\cdot\nabla\f+U_1\left(1-U_1^2+\b\sum_{j\neq
1}U_j^2\right)\f\right]\,dx\\
&=\int_{A}\left[\nabla
U_1\cdot\nabla\f+a(x)U_1\f\,dx\right]\qquad\text{for every radial
}\f\in C^{\infty}_0(A).
\end{split}$$ But then, since $a(x)\geq3/4>0$ on $A$, and $U_1\not\equiv0$ on $\partial A$, the strong maximum principle implies $U_1>0$ on $A$, a contradiction.
Now let us prove that $(U^*_1,\dots,U^*_k)$ solves . Again, assume by contradiction that, for instance, $U_1$ does not satisfy the corresponding equation. Then there exists one radial $\f\in C^\infty_0(\R^N)$, not necessarily positive, such that (up to a change of sign) $$\int_{\R^N}\left(\nabla U_1\cdot\nabla\f+U_1\f-U_1^3\f+\b
U_1\sum_{j\neq 1}U_j^2\f \right)\,dx<0.$$ Moreover we can choose $\f$ with support arbitrarily small. Since $U_1$ is strictly positive, we can then assume that one of its pulse, say $u_{11}$, is strictly positive on the support of $\f$. But then, for $t$ small, also $s_{11}u^*_{11}+t\f$ is positive, therefore Lemma \[lem:variational\_lemma\] (with $y_1=\f$, $y_l=0$ for $l>1$) implies $$0\leq \nabla
I(u^*_{11},\dots,u^*_{kh_k})\cdot(\f,0,\dots,0)=\int_{\R^N}\left(\nabla
U_1\cdot\nabla\f+U_1\f-U_1^3\f+\b U_1\sum_{j\neq 1}U_j^2\f
\right)\,dx,$$ a contradiction.
The proof readily follows by proving that
for every $0<\nu<1$, if $\beta$ is sufficiently large, then $\Neh_\b\subset\tilde\Xh_{\nu\ep}$.
But this can be done following the line of the proof of Lemma \[lem:excision\].
\[rem:finale\] We proved the main result in the simplest case of system . Now we suggest how to modify this scheme in order to treat the general case of system . The main difference is that the role of the associated limiting equation is now played by the minimization problem $$\min_{\Xh^*}\sum_{l=1}^h\dfrac{\int_{\R^N}|\nabla
w_l|^2+\left(V_{\sigma(l)}(x)+\l_{\sigma(l)}\right)w_l^2\,dx}{
\left(\int_{\R^N}\mu_{\sigma(l)}w_l^4\,dx\right)^{1/2}},$$ where now the constants $\mu_i$’s and $\l_i$’s are allowed to take different values, and also the potentials $V_i$’s, with the only constraint that each Schrödinger operator $$-\Delta +V_i(x)+\l_i$$ must be positive. In such a situation, the above minimization problem is always solvable and we call $\Kah$ its solution set. With these changes , in dimensions two and three, Theorem \[teo:main\] and all its proof remain the same.
When $\lim_{|x|\to+\infty}V_i(x)=+\infty$ we can lower the dimension to cover also the dimension $N=1$. In such a case we must change the definition of the Hilbert space we work in choosing for each $i$ the different norm $$\|U_i\|^2=\int_{\R^N}|\nabla
U_i|^2+\left(V_i(x)+\l_i\right)U_i^2\,dx,$$ with the advantage that the embedding in $L^4$ is now compact also in dimension $N=1$.
Finally, let us mention that we can also allow bounded radially symmetric domains instead of $\R^N$, and some non–cubic nonlinearities, provided they are subcritical.
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[^1]: Work partially supported by MIUR, Project “Metodi Variazionali ed Equazioni Differenziali Non Lineari”
[^2]: This is rigorously proven in [@ctv2], while it results from numerical evidence in [@clll].
[^3]: Here and in the following by annuli we mean also balls or exteriors of balls.
[^4]: For easier notation we will write $u_{im}$ instead of $u_{(i,m)}$; from now on we will use the letters $i$, $j$ for the first index and the letters $m$, $n$ for the second, while $l$ is reserved for the components of $W$.
[^5]: Otherwise we would obtain, for some $l$, $v_{\tilde\s(l)}w_l\equiv0$, and hence $\ep^2>\|v_{\tilde\s(l)}-w_l\|^2=\|v_{\tilde\s(l)}\|^2+\|w_l\|^2\geq
2C^2_1$ by Remark \[lem:base\].
[^6]: Since $v_{im}$ vanishes on $\partial I_l$ for every choice of the indexes, $v_{im}|_{I_l}$ belongs to $\spc$ and then, when $\tilde\s(l)\neq(i,m)$, $\ep^2>\|(v_{im}-w_{\tilde\s^{-1}(im)})|_{I_l}\|^2=\|v_{im}|_{I_l}\|^2$.
[^7]: For easier notation we write $l=\tilde\s^{-1}(im)$.
[^8]: Here $\ep_0$ is as in Proposition \[prop:nehari\_mixed\].
[^9]: replacing $\bar\l_{im}u_{im}$ with $u_{im}$ and $\bar\L_iU_i$ with $U_i$.
[^10]: Here and in the following $\bar\l_{im}=\bar\l_{im}(v_{11},\dots,v_{kh_k})$, according to Proposition \[prop:unique\_lambda\].
|
---
abstract: 'We present an Automatic Relevance Determination prior Bayesian Neural Network(BNN-ARD) weight l2-norm measure as a feature importance statistic for the model-x knockoff filter. We show on both simulated data and the Norwegian wind farm dataset that the proposed feature importance statistic yields statistically significant improvements relative to similar feature importance measures in both variable selection power and predictive performance on a real world dataset.'
author:
- |
\
\
\
University of Johannesburg, South Africa
bibliography:
- 'references.bib'
title: An Automatic Relevance Determination Prior Bayesian Neural Network for Controlled Variable Selection
---
Introduction {#sec:intro}
============
Central to many machine learning tasks is a feature selection problem where candidate features are considered such that they maximise the quality and strength of the signal they give about the dynamics of the system or process being modelled[@lu2018deeppink; @candes2018panning]. This will typically require expert judgement or computationally expensive iterative variable selection methods[@candes2018panning].
The knockoff filter of [@candes2018panning] provides a framework for performing variable selection while controlling the false discovery rate (FDR) of null features. Model-X knockoffs work by creating ’knockoff’ variables that imitate the dependency structure within the original candidate feature space but critically with no relationship with the target variable [@barber2015controlling]. These knockoff features can thus be used as variable selection controls for their corresponding candidate features. If a candidate feature on the basis of a specified feature importance metric is less important than its corresponding knockoff then it follows that the candidate feature is irrelevant. Various feature importance metrics have been suggested for determining the relative importance between candidate features and their knockoffs. These have included LASSO coefficients[@barber2015controlling; @candes2018panning], filtering neural network layers [@lu2018deeppink] and permutation based methods[@gimenez2018knockoffs].
In this paper we integrate weight l2-norm feature importance statistics derived from Automatic Relevance Determination(ARD) prior Bayesian Neural Networks(BNNs) into the knockoff filter framework. We show that the proposed feature importance statistics obtain the desired FDR control and display outperformance in statistical power to other nonlinear feature importance measures.
Background {#knocffs}
==========
The modelling setting we consider is one where we have independent and identically distributed observations of candidate feature vectors $(X_{i1}...,X_{ip})\in {R}^p$ and a corresponding scalar $Y_i$, $i=1,....,n$. We assume that Y depends on a subset $\mathcal{H}_0$ of the complete set of candidate features $\mathcal{S}_0$ and is conditionally independent of the features in complement of $\mathcal{H}_0$ given the features in $\mathcal{H}_0$.
The controlled variable selection problem is one that aims to discover as many of the features in $\mathcal{H}_0$ as possible while controlling the FDR which is defined as follows for a set of selected features $\hat{\mathcal{S}}$[@gimenez2018knockoffs]: $$\text{FDR} = \mathbb{E}[\text{FDP}] \quad \text{with} \quad \text{FDP} = \frac{|\hat{\mathcal{S}} \cap\mathcal{H}_0|}{|\mathcal{S}\vee 1|}$$
The recently proposed model-x knockoffs filter [@candes2018panning] has been widely employed as a mechanism for achieving FDR control below some desired threshold $q$.
\[def:sample\] A model-x knockoff for a $p$-dimensional random variable vector ${X}$ is a $\textit{p}$-dimensional random variable vector $\widetilde{X}$ that satisfies the following properties for any subset $\mathcal{S}\in (1,...,p)$:
1. $\big(X,\widetilde{X}\big)_{\text{swap(s)}}\,{\buildrel d \over =}\big(X,\widetilde{X}\big)$ where $\text{swap(s)}$ is an operation swapping corresponding entries of $X$ and $\widetilde{X}$ for each $j\in \mathcal{S}$
2. $\widetilde{X}{\rotatebox[origin=c]{90}{$\models$}}Y|X$ i.e. the response Y is conditionally independent of the knockoff features given the candidate features.
In special cases where features are Gaussian $X \sim \mathcal{N}\big(0,\Sigma\big)$, model-x knockoffs can be constructed by solving constrained optimisation problems for matching second order moments. Once the model-x knockoffs are constructed feature importance statistics $Z_j$ and $\widetilde{Z}_j$ for $X_j$ and $\widetilde{X}_j$ with $j\in (1,...,p)$. $W_j=f(Z_j,\widetilde{Z}_j)$ ard then defined as distance metric between feature importance of a candidate feature $X_j$ and its knockoff $\widetilde{X}_j$ this is typically the l1-norm. Given the relative feature importance statistics $W_j$, the $|W_j|$ are sorted in descending order and we then select the features where $|w_j|$ exceed the threshold $T$ which is defined for a target FDR $q$ as : $$\label{thresh}
T = min\bigg\{t\in \mathcal{W},\frac{1+|\{j:W_j\leq-t\}|}{1\vee|\{j:W_j\geq-t\}|}\leq q\bigg\}$$
Automatic Relevance Determination and Weight l2-norm Feature importance Statistics {#ARD}
==================================================================================
An ARD prior for a BNN is one where weights associated with each input feature connection to the first hidden layer belong to a distinct class and have unique precision parameters $\alpha_c$ for each input. The posterior distribution will then be as follows[@mackay1995probable]:
$$P(w|D,\alpha,\beta)=\frac{1}{Z(\mathbf{\alpha},\beta)}\exp\bigg(\beta E_D+\sum_{c}{\alpha_c} E_{W_C}\bigg)$$
Where $\alpha_c E_W$ is the kernel of the prior distribution on the weights , $\beta E_D$ is the kernel of the data likelihood and $Z(\mathbf{\alpha},\beta)$ is the normalisation constant. The precision hyperparameters for each group of weights are estimated by evidence maximisation or Markov chain Monte Carlo[@mackay1995probable; @neal2012bayesian]. Irrelevant features will have high values of the regularisation parameter meaning that their weights will be forced to decay to values close to zero.
In order to manage the scaling of the regularisation parameters we then consider the l2-norm of the resulting weights as a measure of relative feature importance. We infer the importance of a feature $I(f)$ using the l2-norm all the weights connecting a specific input to the first hidden layer units i.e. $$\textit{I}(f)=\sum_{w\in f}{w_{ij}^2}$$
We incorporate this feature importance measure in the model-x knockoff filter by setting $I(f)$ = $Z_f$.
Experiments
===========
Simulation Studies
------------------
We use simulated data to compare the variable selection performance of our proposed BNN-ARD weight l2-norm feature importance statistic with other non-linear feature importance statistics. We simulate non-linear data from the data model; $ Y = \frac{\big(\mathbf{X}^T\mathbf{\beta}\big)^3}{2} + \varepsilon$. Where Y is a vector of the response, $\mathbf{\beta}$ is a p dimensional vector of coefficients, $\mathbf{X}\in \mathbb{R}^{n\times p}$ is a feature matrix sampled independently from $X \sim \mathcal{N}\big(0,\Sigma\big)$ where elements of the co-variance matrix are set to $\big(\Sigma\big)_{jk}=(\rho^{|j-k|})_{jk}$ with $j,k\in[1,p]$ and $\rho=0.5$.
We randomly set the amplitudes of a subset of relevant features $\mathcal{H}_0$ of 10 randomly selected elements to a value of $3.5$. The total number of candidate features in $\mathcal{S}_0$, $p$ was set to 100. We the compare knockoff filters with the following feature importance statistics: BNN-ARD weight l2-norm; Multi-layer Perceptron(MLP) weight l2-norm ;Random forest(RF) mean decrease in accuracy[@breiman2001random]. We repeat the experiment with 100 random initialisations and report on the mean power and FDR as the target FDR varies.
Real Data Experiments
---------------------
We further assess the relative performance of the knockoff filters based on the various feature importance statistics on the Norwegian wind farm dataset. The dataset consists of 7384 records covering the period from January 2014 to December 2016[@mbuvha2017bayesian]. The eleven candidate features include the wind farm online capacity, one and two hour lagged historical power production values as well as numerical weather prediction(NWP) estimates of humidity, temperature and wind speed.
We compare the performance of the various feature importance statistics based on the testing set Root Mean Square(RMSE) of an MLP trained on features selected by each of the respective feature importance statistics.
Results and Discussion
======================
Figure \[fig:sim\] shows the mean power and empirical FDR across various target FDR levels of the 100 simulations. The results show that the BNN-ARD l2-norm knockoff filter displays persistently higher power in recovering non-null features, relative to the MLP l2-norm knockoff filter and the RF knockoff filter respectively. The corresponding empirical FDR plot is relatively linear below the 45 degree line meaning that all knockoff filters maintained the theoretical Target FDR. A non-parametric Kruskal Wallis[@mckight2010kruskal] test for statistical significance of the differences in power yields a strongly significant p-value of $9.45e-18$. A further Bonferroni test[@bonfurronie] on the pair wise differences in power show statistical significance at a 10 percent level between the BNN-ARD l2-norm and MLP l2-norm filters and at the 5 percent level for both neural network based filters relative to the RF filter.
 \[fig:sim\]
Table \[tab:wind\] shows the mean testing RMSE on the Norwegian wind farm dataset. The results show that the mean testing RMSE based in features selected by the BNN-ARD l2-norm filter is lower than that of the MLP l2-norm filter with statistical significance at FDR target levels greater than 0.25. The features selected by both filters at their respective minimum testing RMSE displayed an 88% overlap. Features such as the third order lag in power production, the second order lag in plant availability and the NWP temperature forecast were irrelevant. The RF filter results are unavailable as the filter yielded empty sets for FDR targets below 0.5. These results show congruence with work previously done on the same dataset by [@mbuvha2017bayesian].
------------------------------ --------- --------- --------- --------- ---------
Feature Importance Statistic
0.2 0.25 0.3 0.4 0.5
MLP-l2 Norm 7257.59 6744.63 7427.79 4398.25 3229.72
ARD-l2 Norm 7325.57 6124.28 5173.02 3717.99 2991.72
------------------------------ --------- --------- --------- --------- ---------
: The mean RMSE based on the different knockoff feature importance statistics after 30 random initialisation across various target FDRs on the Norwegian wind farm dataset[]{data-label="tab:wind"}
|
---
author:
- 'A.N.Panov [^1]'
title: The orbit method for unipotent groups over finite field
---
According to A.A.Kirillov’s orbit method there exists one to one correspondence between the irreducible representations of an arbitrary connected, simply connected Lie group and its coadjoint orbits. This correspondence between orbits and representations makes possible to solve problems of the representation theory in terms of coadjoint orbits. The orbit method initiate many papers beginning from 1962. It turns out that the ideas of the orbit method are useful for the large classes of Lie groups (see [@V1; @V2]), and also for some matrix groups, defined over finite field.
In our paper we obtain formula for multiplicities of certain representations of unipotent groups over finite field in terms of coadjoint orbits (see theorem \[tt\] and corollaries). For reader’s convenience we formulate and prove the maim statements of the orbit method over finite field (see [@Kzh]).
Let $K ={{\Bbb F}}_q$ be a finite field of characteristic $p$ having $q=p^m$ elements. Let ${{\mathfrak g}}$ be a subalgebra of the Lie algebra ${{{\mathfrak u}{\mathfrak t}}}(N,K)$, consisting of all upper triangular matrices with zeros on the diagonal. Suppose that $p$ is large enough to determine the exponential $\exp(x)$ map on ${{\mathfrak g}}$. For instance, let $p{\geqslant}N$. Then the exponential map is a bijection of the Lie algebra ${{\mathfrak g}}$ onto the subgroup $G=\exp({{\mathfrak g}})$ of the unitriangular group ${{{\mathrm U}{\mathrm T}}}(N,K)$. One can define the adjoint representation of the group $G$ on ${{\mathfrak g}}$ by the formula ${{\mathrm{Ad}}}_g(x) = gxg^{-1}$.
Denote by ${{\mathfrak g}}^*$ the conjugate space of ${{\mathfrak g}}$. One can define the coadjoint representation of the group $G$ in ${{\mathfrak g}}^*$ by the formula ${{\mathrm{Ad}}}^*_g{\lambda}(x) = {\lambda}({{\mathrm{Ad}}}_g^{-1} x)$.
Note that if ${{\mathfrak g}}^{\lambda}$ is a stabilazer of ${\lambda}\in{{\mathfrak g}}^*$, then the subgroup $G^{\lambda}=\exp({{\mathfrak g}}^{\lambda})$ is a stabilizer of ${\lambda}$ in $G$. One can calculate the number of elements $|\Omega|$ of the orbit $\Omega={{\mathrm{Ad}}}_G^*({\lambda})$ by the formula $$|\Omega| = \frac{|G|}{|G^{\lambda}|} = \frac{|{{\mathfrak g}}|}{|{{\mathfrak g}}^{\lambda}|} = q^{\dim
{{\mathfrak g}}- \dim {{\mathfrak g}}^{\lambda}} = q^{\frac{1}{2}\dim\Omega}.$$ [**Definition **]{}[****]{}. A subalgebra ${{\mathfrak p}}$ of ${{\mathfrak g}}$ is a polarization of ${\lambda}\in
{{\mathfrak g}}^*$, if ${{\mathfrak p}}$ is a maximal isotropic subspace for the skew symmetric bilinear form $B_{\lambda}(x,y) = {\lambda}([x,y])$ on ${{\mathfrak g}}$. Recall that the subspace ${{\mathfrak p}}$ is isotropic, if $B_{\lambda}(x,y) =0$ for any $x,y\in{{\mathfrak p}}$.
Note that any polarization contains the stabilizer ${{\mathfrak g}}^{\lambda}$, because ${{\mathfrak p}}+{{\mathfrak g}}^{\lambda}$ is an isotropic subspace.\
[**Proposition **]{}[****]{}. Any linear form ${\lambda}$ on a nilpotent Lie algebra ${{\mathfrak g}}$ has a polarization.\
[**Proof**]{}. We shall prove using induction method for the dimension of the Lie algebra ${{\mathfrak g}}$. The statement if obvious for one dimensional Lie algebras, since in this case Lie algebra is a polarization. Assume that the statement is proved for all Lie algebras of dimension $<\dim({{\mathfrak g}})$. We are going to prove the statement for $\dim({{\mathfrak g}})$.
If the dimension of a center ${{\mathfrak z}}$ the Lie algebra ${{\mathfrak g}}$ greater than one, then one can prove existence of polarization applying induction assumption for the factor algebra of ${{\mathfrak g}}$ with respect to the ideal ${{\mathrm{Ker}}}({\lambda}|_{{\mathfrak z}})$. Similarly, for the case $\dim({{\mathfrak z}})=1$, ${\lambda}|_{{\mathfrak z}}= 0$.
Let ${{\mathfrak z}}=Kz$ and ${\lambda}(z)\ne 0$. Consider the two dimensional ideal $Ky+Kz$, containing ${{\mathfrak z}}$. There exists a character ${\alpha}$ of the Lie algebra ${{\mathfrak g}}$ such that ${{\mathrm{ad}}}_u(y) = {\alpha}(u)z$ for any $u\in{{\mathfrak g}}$. The kernel ${{\mathfrak g}}_0$ of the character ${\alpha}$ is an ideal of codimension one in ${{\mathfrak g}}$. There exists an element $x\in{{\mathfrak g}}$ such that ${{\mathfrak g}}= Kx+{{\mathfrak g}}_0$ and $[x,y]=z$.
Denote by ${\lambda}_0$ the restriction of ${\lambda}$ on ${{\mathfrak g}}_0$. According the induction assumption ${\lambda}_0$ has a polarization ${{\mathfrak p}}_0$ in ${{\mathfrak g}}_0$. Let us prove that ${{\mathfrak p}}_0$ is also a polarization for ${\lambda}$ in ${{\mathfrak g}}$. Really, ${{\mathfrak p}}_0$ is a subalgebra and a maximal isotropic subspace in ${{\mathfrak g}}_0$; we will show that ${{\mathfrak p}}_0$ is a maximal isotropic subspace in ${{\mathfrak g}}$. Suppose that one can extent ${{\mathfrak p}}_0$ to an isotropic subspace adding the element $x+u_0$, where $u_0\in{{\mathfrak g}}_0$. Note that $z,y$ belong to the stabilizer ${{\mathfrak g}}_0^{{\lambda}_0}\subset {{\mathfrak p}}_0$. Then $0 ={\lambda}([x+u_0,y]) = {\lambda}([x,y]) = {\lambda}(z) \ne 0$. A contradiction. $\Box$\
[**Proposition **]{}[****]{}\[pp\]. Let ${{\mathfrak p}}$ be a polarization of ${\lambda}\in{{\mathfrak g}}^*$, $P=\exp({{\mathfrak p}})$, $\Omega({\lambda})$ be the coadjoint orbit of ${\lambda}$, $\pi$ be the natural projection of ${{\mathfrak g}}^*$ onto ${{\mathfrak p}}^*$, $L^{\lambda}= \pi^{-1}\pi({\lambda})$. Then\
1) $\dim {{\mathfrak p}}= \frac{1}{2}\left( \dim {{\mathfrak g}}+ \dim{{\mathfrak g}}^{\lambda}\right)$;\
2) $|L^{\lambda}| = \sqrt{|\Omega({\lambda})|}$;\
2) $L^{\lambda}= {{\mathrm{Ad}}}^*_P {\lambda}$, in particular $ L^{\lambda}\subset \Omega({\lambda})$.\
[**Proof**]{}. The statement 1) follows from the formula of dimension of a maximal isotropic subspace for the skew symmetric bilinear form $B_{\lambda}(x,y)$. From 1) we obtain $${{\mathrm{codim}}}\,{{\mathfrak p}}= \frac{1}{2}\left( \dim {{\mathfrak g}}- \dim {{\mathfrak g}}^{\lambda}\right) = \frac{1}{2}\dim \Omega({\lambda}).$$ This implies the statement 2): $$|L^{\lambda}| = q^{{{\mathrm{codim}}}\,{{\mathfrak p}}} = q^{\frac{1}{2}\dim
\Omega({\lambda})} = \sqrt{|\Omega({\lambda})|}.$$
Since ${\lambda}([x,y])=0$ for any $x,y\in{{\mathfrak p}}$, we have ${{\mathrm{ad}}}_{{\mathfrak p}}^* {\lambda}(y)
= 0$ for any $y\in{{\mathfrak p}}$. Then ${{\mathrm{Ad}}}^*_P{\lambda}(y) = {\lambda}(y)$ for any $y\in {{\mathfrak p}}$. This is equivalent to $${{\mathrm{Ad}}}^*_P{\lambda}\subset L^{\lambda}.$$
The equality $ {{\mathrm{Ad}}}^*_P{\lambda}= L^{\lambda}$ is true, since this subsets have equal number of elements: $$|{{\mathrm{Ad}}}^*_P{\lambda}| = \frac{|P|}{|G^{\lambda}|} = q^{\dim {{\mathfrak p}}- \dim {{\mathfrak g}}^{\lambda}} = q^{\frac{1}{2}\dim
\Omega({\lambda})} = |L^{\lambda}|.\,
\Box$$
Fix a non trivial character $e^x: K\to{{\Bbb C}}^*$. We have $$\label{exp}
\sum_{t\in{{\Bbb F}}_q} e^{{\alpha}t} = \left\{\begin{array}{l}
q,~\mbox{если}~~ {\alpha}=0,\\
0,~\mbox{если}~~ {\alpha}\ne 0.\end{array}\right.$$ The equality (\[exp\]) is easy to prove: the image of homomorphism $e^x$ is a subgroup of ${{\Bbb C}}^*$; if ${\alpha}\ne 0$, then this subgroup is nontrivial and coincides with the subgroup of all roots of some order $m\ne 1$ of unity; the sum of all roots of order $m\ne 1$ of unity equals to zero.
Restriction of ${\lambda}$ on its polarization ${{\mathfrak p}}$ defines a character (one dimensional representation) $\xi$ of the group $P=\exp({{\mathfrak p}})$ by the formula $$\xi_{\lambda}(\exp(x)) = e^{{\lambda}(x)}.$$ Consider the induced representation $$T^{\lambda}= {{\mathrm{ind}}}(\xi_{\lambda}, P, G).$$ Denote by $\chi_{\lambda}(g) = {{\mathrm{Tr}}}~ T^{\lambda}(g)$ the character of representation $T^{\lambda}$.\
[**Theorem **]{}[****]{}\[chiq\]. $$\chi_{\lambda}(g) = \frac{1}{\sqrt{|\Omega|}}
\sum_{\mu\in\Omega({\lambda})} e^{\mu(\ln(g))}$$ [**Proof**]{}. Extent $\xi_{\lambda}$ from $P$ to $G$ by the formula $$\tilde{\xi}_{\lambda}(u) = \left\{ \begin{array}{cl} \xi_{\lambda}(u)&, ~~ \mbox{if}
~~ u\in P,\\
0&, ~ ~\mbox{if} ~~ u\notin P.\end{array}\right.$$ Formula (\[exp\]) implies $$\tilde{\xi}_{\lambda}(u) = \frac{1}{|L^{\lambda}|}\sum_{\mu\in L^{\lambda}}
\xi_\mu(u) = \frac{1}{|L^{\lambda}|}\sum_{p\in P} \xi_{{{\mathrm{Ad}}}^*_p({\lambda})}(u).$$
Choose the system of representatives $\{g_i:~ i=\overline{1, k}\}$ of the classes $G/P$. Using the well known formula for induced characters (see [@KR chapter 6]), we obtain $$\chi_{\lambda}(u) = \sum_{g^{-1}_iug_i\in P}\tilde{\xi}_{\lambda}(g^{-1}_iug_i) =
\frac{1}{|L^{\lambda}|} \sum_{i=1, p\in P}^k \xi_{{{\mathrm{Ad}}}_p^*{\lambda}}
(g^{-1}_iug_i) = \frac{1}{|L^{\lambda}|} \sum_{i=1, p\in P}^k
\xi_{{{\mathrm{Ad}}}_{g_ip}^*{\lambda}}(u),$$ Finally, $$\chi_{\lambda}(u) = \frac{1}{|L^{\lambda}|} \sum_{g\in G} \xi_{{{\mathrm{Ad}}}_g^*{\lambda}}(u)
= \frac{1}{\sqrt{|\Omega|}} \sum_{\mu\in\Omega({\lambda})}
e^{\mu(\ln(u))}. \Box$$ [**Theorem **]{}[****]{}\[t\].\
1) $ \dim T^{\lambda}= q^{\frac{1}{2}\dim \Omega({\lambda})} = \sqrt{|\Omega|}$.\
2) The representation $T^{\lambda}$ does not depend on the choice of polarization.\
3) The representation $T^{\lambda}$ is irreducible.\
4) Representations $T^{\lambda}$ and $T^{{\lambda}'}$ are equivalent if and only if ${\lambda}$ and ${\lambda}'$ belong to the same coadjoint orbit.\
5) For any irreducible representation $T$ of the group $G$ there exists ${\lambda}\in{{\mathfrak g}}^*$ such that the representation $T$ is equivalent to $T^{\lambda}$.\
[**Proof**]{}. The statement 1) follows from $$\dim T^{\lambda}= \dim
({{\mathrm{ind}}}(\xi_{\lambda}, P,G)) = q^{{{\mathrm{codim}}}\,{{\mathfrak p}}}.$$ The statement 2) is a corollary of the theorem \[chiq\].
Let us show that the system of characters $\{\chi_{\lambda}\}$, where ${\lambda}$ is running through some system of representatives of the coadjoint orbits, is orthonormal. Let $\Omega$, $\Omega'$ be two coadjoint orbits and ${\lambda}$, ${\lambda}'$ be representatives of this orbits. Then $$(\chi_{\lambda}, \chi_{{\lambda}'}) = \frac{1}{|G|}\sum_{u\in
G}\chi_{\lambda}(u)\overline{\chi_{{\lambda}'}(u)}= \frac{1}{|G|}\cdot
\frac{1}{\sqrt{|\Omega|\cdot|\Omega'|}}\cdot \sum_{\mu\in\Omega,
\mu'\in\Omega', u\in G}\xi_\mu(u)\overline{\xi_{\mu'}(u)}=$$ $$\frac{1}{|G|}\cdot \frac{1}{\sqrt{|\Omega|\cdot|\Omega'|}}\cdot
\sum_{\mu\in\Omega, \mu'\in\Omega', u\in G}e^{(\mu-\mu')\ln(u)}.$$ Applying $$\sum_{x\in{{\mathfrak g}}}e^{\eta(x)}=\left\{\begin{array}{cl}
|G|&, ~~\mbox{if}~~ \eta=0,\\
0&, ~~\mbox{if}~~ \eta\ne 0 \end{array}\right.,$$ we obtain that, if $\Omega \ne \Omega'$, then $(\chi_{\lambda}, \chi_{{\lambda}'})=0$.
In the case $\Omega=\Omega'$, we have got $$(\chi_{\lambda}, \chi_{\lambda}) = \frac{1}{|G|\cdot |\Omega|}
\sum_{\mu, \mu'\in\Omega}\sum_{x\in{{\mathfrak g}}}e^{(\mu-\mu')x} =
\frac{1}{|G|\cdot |\Omega|}\cdot |\Omega| \cdot |G| = 1.$$ This proves 3) and 4).
We shall use notation $T^\Omega$ for the class of equivalent representations $T^{\lambda}$, where ${\lambda}\in\Omega$. To prove statement 5) we verify that the sum of squares of dimensions of irreducible representations $\{ T^\Omega: ~ \Omega\in{{\mathfrak g}}^*/G\}$ equals to the number of elements of the group:\
$$\sum_{\Omega\in{{\mathfrak g}}^*/G} \left(\dim T^\Omega\right)^2 =
\sum_{\Omega\in{{\mathfrak g}}^*/G} \left( \sqrt{|\Omega|} \right)^2 =
\sum_{\Omega\in{{\mathfrak g}}^*/G}|\Omega| = |{{\mathfrak g}}^*| = |G|.$$ This proves 5).$\Box$
[**Lemma **]{}[****]{}\[lsa\]. Let ${{\mathfrak g}}$ be a nilpotent Lie algebra, ${{\mathfrak g}}_0$ be a subalgebra of ${{\mathfrak g}}$ of codimension one. Then ${{\mathfrak g}}_0$ is an ideal of ${{\mathfrak g}}$.\
[**Proof**]{}. Suppose the contrary. Then $[{{\mathfrak g}},{{\mathfrak g}}_0]\ne{{\mathfrak g}}_0$; the exist the elements $y\in{{\mathfrak g}}_0$, $x\notin{{\mathfrak g}}_0$ such that $[x,y] = {\alpha}x
\bmod{{\mathfrak g}}_0$, ${\alpha}\ne 0$. The subalgebra ${{\mathfrak g}}_0$ is invariant with respect to ${{\mathrm{ad}}}_{y_0}$. Since ${{\mathfrak g}}= kx\oplus{{\mathfrak g}}_0$, the operator ${{\mathrm{ad}}}_{y_0}$ is not nilpotent in ${{\mathfrak g}}/{{\mathfrak g}}_0$; this contradicts to assumption that the Lie algebra ${{\mathfrak g}}$ is nilpotent. $\Box$
[**Lemma **]{}[****]{}\[lto\]. Let ${{\mathfrak g}}$, ${{\mathfrak g}}_0$ be as in lemma \[lsa\], $\pi$ is a projection ${{\mathfrak g}}^*\to{{\mathfrak g}}_0^*$; ${\lambda}_0\in{{\mathfrak g}}_0^*$, $\omega = {{\mathrm{Ad}}}_{G_0}^*({\lambda}_0)$, ${{\mathfrak g}}^{{\lambda}_0}
=\{ x\in{{\mathfrak g}}:~ {\lambda}_0([x,{{\mathfrak g}}_0)] = 0\}$.\
1) Let the subalgebra $ {{\mathfrak g}}^{{\lambda}_0}$ belong to ${{\mathfrak g}}_0$. Then\
1a) $\pi^{-1}({\lambda}_0)$ lie in the same содержится ${{\mathrm{Ad}}}^*_G$-orbit $\Omega$;\
1b) $\dim \Omega = \dim \omega +2$ (i.e. $|\Omega| =
q^2|\omega|$) .\
2) Let the subalgebra $ {{\mathfrak g}}^{{\lambda}_0}$ do not lie in ${{\mathfrak g}}_0$. Then for any ${{\mathrm{Ad}}}_G^*$-orbit $\Omega$, which has nonempty intersection with $\pi^{-1}({\lambda}_0)$, the projection $\pi$ establishes one to one correspondence between $\Omega$ and $\omega$; in particular, $\dim \Omega = \dim \omega$.\
[**Proof**]{}.\
1) Suppose that the subalgebra $ {{\mathfrak g}}^{{\lambda}_0}$ belongs to ${{\mathfrak g}}_0$. Since $[{{\mathfrak g}},{{\mathfrak g}}]\subset{{\mathfrak g}}_0$, the formula $B_0(x,y) = {\lambda}_0([x,y])$ defines a skew symmetric bilinear form on the Lie algebra ${{\mathfrak g}}$. The kernel $V$ of the bilinear form $B_0$ coincides with ${{\mathfrak g}}^{{\lambda}_0}$ and belongs to ${{\mathfrak g}}_0$. The kernel $V_0$ of the restriction $B_0$ on ${{\mathfrak g}}_0$ coincides with ${{\mathfrak g}}_0^{{\lambda}_0}$.
Let us prove that there exists a pair of elements $u\in {{\mathfrak g}}\setminus{{\mathfrak g}}_0$ and $v\in V_0$, such that $B_0(u,v)=1$. Really, decompose ${{\mathfrak g}}_0 =
L_0\oplus V_0$, where $L_0$ is a subspace, with the property that the restriction of bilinear form $B_0$ on $L_0$ is nondegenerate. Choose an arbitrary element $u'\in{{\mathfrak g}}\setminus {{\mathfrak g}}_0$ and consider the linear form $B_0(u',\cdot)$ on $L_0$. There exists $x_0\in L_0$ such that $
B_0(u',\cdot) = B_0(x_0,\cdot)$ on $L_0$. The element $u=u'-x_0$ satisfies $B_0(u,L_0)=0$. Since $u\notin V$, there exists $v\in
V_0$ such that $B_0(u,v)=1$.
By direct calculations, we verify that for any ${\lambda}\in\pi^{-1}({\lambda}_0)$ the following equalities are valid $$\label{first}
\left\{\begin{array}{l}{{\mathrm{Ad}}}^*_{\exp(tv)}{\lambda}(u) = {\lambda}(u)+t,\\
{{\mathrm{Ad}}}^*_{\exp(tv)}{\lambda}(y) = {\lambda}(y)
~~\mbox{for~~ any}~~~ y\in{{\mathfrak g}}_0.
\end{array}\right.$$ This implies the statement 1a).
The orbit $\Omega$ is a union $$\label{omegaf}\Omega = \bigcup_{t\in K}\pi^{-1}(\omega_t),$$ where $\omega_t = {{\mathrm{Ad}}}^*_{\exp(tu)}\omega$ is a coadjoint orbit in ${{\mathfrak g}}_0^*$. Let us show that the orbits $\omega_t$ are pairwise different. Really, if not, there exists $t'\ne t''\in K$ and $g'_0,g_0''\in G_0$ such that $${{\mathrm{Ad}}}^*_{\exp(t'u)}{{\mathrm{Ad}}}^*_{g_0'}{\lambda}_0 =
{{\mathrm{Ad}}}^*_{\exp(t''u)}{{\mathrm{Ad}}}^*_{g''_0}{\lambda}_0.$$ Then $${{\mathrm{Ad}}}^*_{\exp(tu)}{{\mathrm{Ad}}}^*_{g_0}{\lambda}_0 = {\lambda}_0,$$ where $t=t'-t''\in K^*$ and $g_0$ is an element of $G_0$. Then the stabilizer $\exp({{\mathfrak g}}^{{\lambda}_0})$ does not belong to $G_0$; this contradicts to assumption of the item 1). Using (\[omegaf\]), we obtain $|\Omega| = q^2|\omega|$. Therefore $\dim \Omega = \dim
\omega + 2$. This proves 1b).
Turn to proof of the item 2). Suppose that the subalgebra $
{{\mathfrak g}}^{{\lambda}_0}$ does not belong to ${{\mathfrak g}}_0$. For an arbitrary nonzero element $x$ of $ {{\mathfrak g}}^{{\lambda}_0}\setminus {{\mathfrak g}}_0$ we have decomposition ${{\mathfrak g}}= Kx\oplus{{\mathfrak g}}_0$. The group $G$ is a semidirect product $G=G_0X$, where $X=\{ \exp(tx):~ t\in K\}$.
Let ${\lambda}\in\pi^{-1}({\lambda}_0)$. The equality $${\lambda}([x,{{\mathfrak g}}]) = {\lambda}([x,Kx\oplus{{\mathfrak g}}_0]) = {\lambda}_0([x,{{\mathfrak g}}_0]) =0$$ implies that $x$ belongs to ${{\mathfrak g}}^{\lambda}$. The subgroup $X$ lies in the stabilizer of ${\lambda}$. The orbit $\Omega({\lambda})$ coincides with ${{\mathrm{Ad}}}^*_{G_0}({\lambda})$. Since the projection $\pi:{{\mathfrak g}}^*\to{{\mathfrak g}}_0^*$ is invariant with respect to ${{\mathrm{Ad}}}^*_{G_0}$, the map $\pi$ project $\Omega({\lambda})$ onto $\omega$.
It remains to show that $$\label{ee} \Omega({\lambda}) \cap\pi^{-1}({\lambda}_0) =
\{{\lambda}\}$$ for any ${\lambda}\in\pi^{1}({\lambda}_0)$. Suppose that ${\lambda}'={{\mathrm{Ad}}}^*_g{\lambda}$ and ${\lambda},{\lambda}'\in\pi^{-1}({\lambda}_0)$. Since $G
= G_0X$ and $X\in G^{\lambda}$, we verify that ${\lambda}' = {{\mathrm{Ad}}}^*_{g_0}{\lambda}$ for some $g_0\in G_0$. As ${\lambda},{\lambda}'\in\pi^{-1}({\lambda}_0)$, the element $g_0$ lies in stabilizer $G_0^{{\lambda}_0}$. Then $g_0 = \exp(y_0)$ for some $y_0\in {{\mathfrak g}}_0^{{\lambda}_0}$. We obtain $$\label{ll} {\lambda}'(x) = {\lambda}({{\mathrm{Ad}}}^*_{\exp(-y_0)}x) = {\lambda}(x)
- {\lambda}({{\mathrm{ad}}}_{y_0}x) + \sum_{k{\geqslant}2}\frac{(-1)^k}{k!}{\lambda}_0({{\mathrm{ad}}}_{y_0}^kx).$$ Since $x\in {{\mathfrak g}}^{\lambda}$, we have $ {\lambda}({{\mathrm{ad}}}_{y_0}x) =0$. As $y_0\in
{{\mathfrak g}}_0^{{\lambda}_0}$, we have ${\lambda}_0({{\mathrm{ad}}}_{y_0}^kx)=0$ for any $k {\geqslant}2$. Substituting into (\[ll\]), we obtain ${\lambda}'(x) = {\lambda}(x)$. Using ${\lambda},
{\lambda}'\in\pi^{-1}({\lambda}_0)$, we conclude ${\lambda}= {\lambda}'$. $\Box$\
[**Lemma **]{}[****]{}\[ss\]. For any subalgebra ${{\mathfrak h}}$ of a nilpotent Lie algebra ${{\mathfrak g}}$ there exists a chain of subalgebras ${{\mathfrak g}}={{\mathfrak g}}_0
\supset{{\mathfrak g}}_1\supset\ldots\supset{{\mathfrak g}}_k={{\mathfrak h}}$ such that ${{\mathfrak g}}_{i+1}$ is an ideal of codimension one in ${{\mathfrak g}}_{i}$ for any $1{\leqslant}i{\leqslant}k-1$.\
[**Proof**]{}. We use the induction method for $\dim {{\mathfrak g}}$. For $\dim {{\mathfrak g}}=1$ the statement is obvious. Assume that the statement is true for $\dim {{\mathfrak g}}= n-1$; let us prove it for $n$. The nilpotent Lie algebra ${{\mathfrak g}}$ has a nonzero central element $z$. Consider projection $\phi:{{\mathfrak g}}\to{\overline{{{\mathfrak g}}}}= {{\mathfrak g}}/Kz$. The image ${\overline{{{\mathfrak h}}}}$ is a subalgebra in ${\overline{{{\mathfrak g}}}}$. As $\dim{\overline{{{\mathfrak g}}}}<n$, according to induction assumption, there exists a chain of subalgebras ${\overline{{{\mathfrak g}}}}={\overline{{{\mathfrak g}}}}_0
\supset{\overline{{{\mathfrak g}}}}_1\supset\ldots\supset{\overline{{{\mathfrak g}}}}_k={\overline{{{\mathfrak h}}}}$, where ${\overline{{{\mathfrak g}}}}_i$ is an ideal of codimension one in ${\overline{{{\mathfrak g}}}}_{i+1}$ for any $1{\leqslant}i{\leqslant}k-1$. Denote ${{\mathfrak g}}_i = \phi^{-1}({\overline{{{\mathfrak g}}}}_i)$. If $z\in{{\mathfrak h}}$, then ${{\mathfrak g}}_k
= {{\mathfrak h}}$; this completes construction of the chain of subalgebras. If $z\notin{{\mathfrak h}}$, then ${{\mathfrak g}}_k = {{\mathfrak h}}+Kz$. It remains to put ${{\mathfrak g}}_{k+1}$ equal to ${{\mathfrak h}}$. $\Box$\
[**Theorem **]{}[****]{}\[tt\]. Let $G=\exp({{\mathfrak g}})$ be an unipotent group over the finite field $K$, ${{\mathfrak h}}$ be a subalgebra of ${{\mathfrak g}}$, $H=exp({{\mathfrak h}})$. Let $\Omega$ (resp. $\omega$) be a coadjoint orbit in ${{\mathfrak g}}^*$ (resp. ${{\mathfrak h}}^*$), $T^\Omega$ and $t^\omega$ be corresponding irreducible representations of $G$ and $H$, $\pi$ be the natural projection ${{\mathfrak g}}^*$ onto ${{\mathfrak h}}^*$. Denote $m(\omega, \Omega) =
\mathrm{mult} (T^\Omega,
\mathrm{ind}(t^\omega,G)) = \mathrm{mult} (t^\omega,
\mathrm{res}(T^\Omega,H))$. Then $$m(\omega, \Omega)
=\frac{|\pi^{-1}(\omega)\cap\Omega|}{\sqrt{|\omega|\cdot |\Omega|}}.$$ [**Proof**]{}. Introduce notations $$P =
|\pi^{-1}(\omega)\cap \Omega|,\quad\quad Q = \sqrt{|\omega|\cdot
|\Omega|},\quad\quad M = {{\mathrm{mult}}}\left(T^\Omega, {{\mathrm{ind}}}(t^\omega,
G)\right).$$
We shall prove that $M=P/Q$ using the induction method with respect to ${{\mathrm{codim}}}({{\mathfrak h}},{{\mathfrak g}})$. If ${{\mathrm{codim}}}({{\mathfrak h}},{{\mathfrak g}})=0$, then $\Omega
= \omega$ and hence $P = |\Omega|$, $Q = |\Omega|$, $M=1$; this proves the equality $M=P/Q$.
Assume that the equality is proved for ${{\mathrm{codim}}}({{\mathfrak h}},{{\mathfrak g}})< k$; let us prove for ${{\mathrm{codim}}}({{\mathfrak h}},{{\mathfrak g}})=k$. The lemma \[ss\] implies that there exists a subalgebra ${{\mathfrak g}}_1$ obeying the conditions ${{\mathfrak g}}\supset{{\mathfrak g}}_1\supset{{\mathfrak h}}$, ${{\mathrm{codim}}}({{\mathfrak h}},{{\mathfrak g}}_1)=1$. Choose ${\lambda}_0\in\omega$. The natural projections $\pi:{{\mathfrak g}}^*\to{{\mathfrak h}}^*$, $\pi_1:{{\mathfrak g}}^*\to{{\mathfrak g}}_1^*$, $\pi_0:{{\mathfrak g}}_1^*\to{{\mathfrak h}}^*$ satisfy $\pi =
\pi_0\pi_1$. For ${{\mathfrak g}}_1^{{\lambda}_0} =\{ x\in{{\mathfrak g}}_1:~ {\lambda}_0[x,{{\mathfrak h}}] = 0 \}$ only two cases are possible: ${{\mathfrak g}}_1^{{\lambda}_0}\subset {{\mathfrak h}}$, or ${{\mathfrak g}}_1^{{\lambda}_0}\not\subset {{\mathfrak h}}$.\
1) Case ${{\mathfrak g}}_1^{{\lambda}_0}\subset {{\mathfrak h}}$. Following lemma \[lto\], $\pi_0^{-1}(\omega)$ belongs to the same coadjoint orbit $\Omega_1\subset {{\mathfrak g}}_1^*$ of the group $G_1 =\exp({{\mathfrak g}}_1)$. Therefore, $\dim \Omega_1 = \dim \omega +2$ and $$\label{dd}
|\Omega_1| = q^2 |\omega| .$$
A polarization ${{\mathfrak p}}_0$ for ${\lambda}_0$ in ${{\mathfrak h}}$ is also a polarization for any ${\lambda}_1\in\pi_0^{-1}$ in ${{\mathfrak g}}_1$. Really, ${{\mathfrak p}}_0$ is an isotropic subspace in ${{\mathfrak g}}_1$ and $${{\mathrm{codim}}}({{\mathfrak p}}_0,{{\mathfrak g}}_1) = {{\mathrm{codim}}}({{\mathfrak p}}_0,{{\mathfrak h}}) +1 =\frac{1}{2}(\dim \omega
+2) =\frac{1}{2} \dim \Omega_1.$$ The induced representation ${{\mathrm{ind}}}(t^\omega, G_1)$ is irreducible and coincides with $T^{\Omega_1}$, $$\label{ii} {{\mathrm{ind}}}(T^{\Omega_1},G) =
{{\mathrm{ind}}}(t^{\omega},G).$$ According to the induction assumption, $$\label{mff}
{{\mathrm{mult}}}\left(T^\Omega, {{\mathrm{ind}}}(T^{\Omega_1}, G)\right) = \frac{
|\pi^{-1}(\Omega_1)\cap \Omega|}{\sqrt{|\Omega_1|\cdot |\Omega|}}.$$ Applying the formula (\[ii\]), we obtain
$$M = \frac{
|\pi^{-1}(\Omega_1)\cap \Omega|}{\sqrt{|\Omega_1|\cdot |\Omega|}}.$$
Using (\[dd\]), we conclude
$$M = \frac{ q P}{\sqrt{q^2|\omega|\cdot |\Omega|}} =\frac{P}{Q}.$$
2\) Case ${{\mathfrak g}}_1^{{\lambda}_0}\not\subset {{\mathfrak h}}$. By the formula (\[ee\]) we obtain $$\pi^{-1}(\omega) = \bigcup_{{\lambda}_1\in\pi^{-1}_0({\lambda}_0)}
\pi_1^{-1}(\Omega_1({\lambda}_1)),$$ where $\Omega_1({\lambda}_1)$ is an orbit of ${\lambda}_1\in{{\mathfrak g}}^*$ with respect to ${{\mathrm{Ad}}}^*_{G_1}$. Appling the induction assumption, we obtain $$P = |\pi^{-1}(\omega)\cap \Omega| =
\sum_{{\lambda}_1\in\pi^{-1}_0({\lambda}_0)}|\pi^{-1}(\Omega_1({\lambda}_1))\cap
\Omega| =$$ $$\sum_{{\lambda}_1\in\pi^{-1}_0({\lambda}_0)}
\sqrt{|\Omega_1({\lambda}_1)|\cdot|\Omega|} \quad {{\mathrm{mult}}}(T^\Omega,
{{\mathrm{ind}}}(T^{\Omega_1({\lambda}_1)},G)).$$ Since $|\Omega_1({\lambda}_1)| =
|\omega|$, we have $$\label{PPP}
P = Q \sum_{{\lambda}_1\in\pi^{-1}_0({\lambda}_0)}{{\mathrm{mult}}}(T^\Omega,
{{\mathrm{ind}}}(T^{\Omega_1({\lambda}_1)},G)).$$ From the other hand, for any polarization ${{\mathfrak p}}_0$ of ${\lambda}_0$ the subalgebra $${{\mathfrak p}}={{\mathfrak g}}_1^{{\lambda}_0} +{{\mathfrak p}}_0$$ is a polarization for any ${\lambda}_1\in\pi^{-1}_0({\lambda}_0)$. Representation ${{\mathrm{ind}}}(\xi_{{\lambda}_0},P_0,P)$ is a direct sum of one dimensional representations $\xi_{{\lambda}_1}$, where ${\lambda}_1\in\pi^{-1}_0({\lambda}_0)$. Therefore, $ {{\mathrm{ind}}}(t^\omega,G)$ is a direct sum of representations ${{\mathrm{ind}}}(T^{\Omega_1({\lambda}_1)},G)$ where ${\lambda}_1\in\pi^{-1}_0({\lambda}_0)$. We obtain $$\label{mmm} M = {{\mathrm{mult}}}\left(T^\Omega, {{\mathrm{ind}}}(t^\omega,
G)\right) = \sum_{{\lambda}_1\in\pi^{-1}_0({\lambda}_0)}{{\mathrm{mult}}}(T^\Omega,
{{\mathrm{ind}}}(T^{\Omega_1({\lambda}_1)},G)).$$ Substituting (\[mmm\]) in (\[PPP\]), we verify $P=QM$. $\Box$\
[**Corollary ** ]{}[****]{}. The irreducible representation $T^\Omega$ occurs in decomposition of ${{\mathrm{ind}}}(t^\omega, G)$ if and only if the orbit $\Omega$ has an nonempty intersection with $\pi^{-1}(\omega)$.\
[**Corollary ** ]{}[****]{}. The irreducible representation $t^\omega$ occurs in decomposition of the restriction of representation $T^\Omega$ on the subgroup $H$ if and only if the orbit $\omega$ lies in $\pi(\Omega)$.\
[**Corollary ** ]{}[****]{}. Let $\Omega$, $\Omega_1$, $\Omega_2$ be coadjoint orbits in ${{\mathfrak g}}^*$. Denote by $|M|$ the number of elements in the subset $$M = \{({\lambda}_1,{\lambda}_2):~~ {\lambda}_1\in\Omega_1,~ {\lambda}_2\in\Omega_2,~
{\lambda}_1+{\lambda}_2\in\Omega \}.$$ Then $${{\mathrm{mult}}}(T^\Omega, T^{\Omega_1}\otimes T^{\Omega_2}) = \frac{
|M|}{\sqrt{|\Omega|\cdot |\Omega_1|\cdot |\Omega_2|}}.$$
[**Proof**]{}. We apply theorem \[tt\] to the group $G\times G$, its coadjoint orbit $\Omega_1\times \Omega_2$, subgroup $H=\{(g,g):~~ g\in G\}$ and its orbit $\omega =\{({\lambda},{\lambda}):~~
{\lambda}\in\Omega\}$. $\Box$
[99]{}
A.A.Kirillov, [*Unitary representations of nilpotent Lie groups*]{}, Usp.Math.Nauk, 1962, [**17**]{}, No. 4, 57-110 (rus).
A.A.Kirillov, Lectures in the orbit method, Graduate Studes in Math., [**64**]{}, 2002. D.Vogan Jr., [*The orbit method and unitary representations for reductive Lie groups*]{}, Algebraic and and analytic methods in representation theory, Sonderborg, 1994, 243-339.
D.Vogan Jr., [*The method of coadjoint orbits for real reductive groups*]{}, Representation theory of Lie groups, 1998, 179-238.
D.Kazhdan, [*Proof of Springer’s Hypothesis*]{}, Israel J.Math., vol. 28, no.4, 1977, 272-286.
Ch.W.Curtis, I.Reiner, Representations theory of finite groups and associative algebras, Interscience Publishes, New York, 1962. М.,Наука, 1969.
[^1]: The paper is supported by the RFBR-grants 12-01-00070, 12-01-00137, 13-01-97000-Volga region-a
|
---
abstract: 'We report time-resolved measurements of current-induced reversal of a free magnetic layer in Py/Cu/Py elliptical nanopillars at temperatures $T$ = 4.2 K to 160 K. Comparison of the data to Landau-Lifshitz-Gilbert macrospin simulations of the free layer switching yields numerical values for the spin torque and the Gilbert damping parameters as functions of $T$. The damping is strongly $T$-dependent, which we attribute to the antiferromagnetic pinning behavior of a thin permalloy oxide layer around the perimeter of the free layer. This adventitious antiferromagnetic pinning layer can have a major impact on spin torque phenomena.'
author:
- 'N. C. Emley'
- 'I. N. Krivorotov'
- 'A. G. F. Garcia'
- 'O. Ozatay'
- 'J. C. Sankey'
- 'D. C. Ralph'
- 'R. A. Buhrman'
bibliography:
- 'Switching\_Speed\_PRL.bib'
title: 'Time-Resolved Spin Torque Switching and Enhanced Damping in Py/Cu/Py Spin-Valve Nanopillars'
---
Experiments [@katine-2000PRL; @urazhdin-2003PRL; @kiselev-2003Nature; @grollier-2000APL] have shown that a spin-polarized current passed through a nanomagnet can excite a dynamic response as the result of a spin torque applied by the conduction electrons [@slonczewski-1996JMMM; @berger-1996PRB]. The potential for technological impact of this spin transfer (ST) effect has inspired research in DC current-induced microwave oscillations [@kiselev-2003Nature; @rippard-2005PRL] and hysteretic switching [@katine-2000PRL; @urazhdin-2003PRL; @grollier-2000APL] in current perpendicular to the plane (CPP) nanopillars and nanoconstrictions. Typically, ST switching data is obtained through the use of slow current ramp rates ($\sim$1 mA/s), but fast pulses ($\sim$10$^{10}$ mA/s) access the regime where thermal activation of the moment over a current-dependent barrier [@li-zhang-2004PRB; @krivorotov-2004PRL] does not play a major role in the switching process. This spin torque-driven regime [@koch-2004PRL] is advantageous for the quantitative examination of the spin torque parameters due to the computational accessibility of numerically integrating the Landau-Lifshitz-Gilbert (LLG) equation for short durations.
Here we report time-resolved measurements of the spin torque-driven switching event in Cu 100/Py 20/Cu 6/Py 2/Cu 2/Pt 30 (in nm, Py = Ni$_{81}$Fe$_{19}$) CPP spin-valve nanopillar structures at bath temperatures $T$ = 4.2 K to 160 K. We compare our experimental results with LLG simulations in the macrospin approximation and find good agreement between simulation and measurement. This both confirms the applicability of the macrospin approximation in the spin torque-driven regime and facilitates the quantitative determination of $T$-dependent spin torque and magnetic damping parameters. At higher $T$ we find that the strength of the spin torque exerted per unit current is in reasonable numerical accord with recent model calculations, and that the damping parameter $\alpha_{0}$ for the nanomagnet excitations is both anomalously high, as suggested by previous pulsed current measurements [@braganca-2005APL], and $T$-dependent. The strong $T$ variation of $\alpha_{0}$, in conjunction with anomalous behavior of the nanomagnet switching fields $H_{S,i}$($T$) in some devices, points to the presence of an adventitious antiferromagnetic oxide layer around the perimeter of the nanomagnet that has a major effect on the nanomagnet dynamics driven by a spin torque.
![(a) Slow ramp rate spin torque switching of a nanomagnet as measured by the GMR effect for sample 1, a 60$\times$190 nm ellipse, at $T$ = 40 K and $H_{\textrm{app}}$ = 385 Oe, which opposes the dipole field so that $H_{\textrm{app}} + H_{\textrm{dip}} \approx 0$. Arrows indicate the scan direction. (b) Pulsed spin torque P to AP switching measured for the same sample at $I$ = 1.07 mA ($\square$) and 2.13 mA ($\blacksquare$). The data (symbols) have been normalized to $M_{x}$ = $\pm$1 for simple comparison with the simulated macrospin switching (lines). Pulse shape distortions are due to the setup (see ref. \[13\]).](Figure1.eps){width="8.5cm"}
The nanopillar devices employed in this study were fabricated using a process described elsewhere [@emley-2004APL]. A slow ramp rate ST scan is shown in Fig. 1(a) for sample 1, a 60$\times$190 nm ellipse. In Fig. 1(b) we show parallel (P) to anti-parallel (AP) switching events for sample 1, averaged over 10,000 switches, taken at pulsed current amplitudes $I$ = 1.07 mA and 2.13 mA at $T$ = 40 K as open and solid squares, respectively. The measured signal is a time-resolved voltage drop $|I\cdot\Delta R|$ from the giant magnetoresistance (GMR) of the sample as the free layer switches from P to AP orientation, where $\Delta R
\equiv R_{x}$(AP)$ - R_{x}$(P) and $R_{x}$ is the 4-point device resistance. The data have been normalized to $M_{x} = +/- 1$ (minimum resistance / maximum resistance) for simple comparison with simulated switching events, described below, which are shown as solid and dashed lines. The abrupt ($\sim$200 ps) jump from $M_{x}$ = -1 to 1 at time = 0 is not a switching event but is simply the rising edge of the current pulse. The more gradual transition between P ($M_{x}$ = 1) and AP (return to $M_{x}$ = -1) is the envelope coming from averaging over thousands of individual switching events, each of which follows a trajectory determined by initial conditions that are randomized by the stochastic thermal fluctuations of the free layer. We define the switching time $t_{switch}$ as the time elapsed between 50% of the signal rise and 50% of the signal drop as indicated in Fig. 1(b) [@jitter].
To obtain a quantitative understanding of the ST switching, we have simulated the nanomagnet dynamics by numerical integration of the LLG equation in the macrospin approximation with the inclusion of a Slonczewski-type spin torque term. $$\begin{aligned}
\frac{d\hat{m}}{dt} = \gamma[\hat{m} \times (\vec{H}_{\textrm{eff}} + \vec{H}_{\textrm{Lang}}(T^{\prime})) -
\alpha(\theta) \hat{m} \times (\hat{m} \times (\vec{H}_{\textrm{eff}}\\ + \vec{H}_{\textrm{Lang}}(T^{\prime})))
- \frac{I\hbar g(\theta)}{e M_{s}(T^{\prime})(area\cdot d)\sin\theta} \hat{m}\times\hat{p}\times\hat{m}]\end{aligned}$$ Here $\gamma$ is the gyromagnetic ratio, $\hat{m}$ is the unit directional vector of the free layer macrospin, $\hat{p}$ is the spin polarization axis, $\theta$ is the in-plane angle between them, $g(\theta)$ is the spin torque function, $M_{s}(T)$ is the free layer magnetization, as measured separately for a continuous 2 nm Py film in a Cu/Py/Cu trilayer that was exposed to the same heat treatments as the nanopillars, $d$ is the nanomagnet thickness, $area = \frac{\pi}{4}ab$ is its lateral area with dimensions $a$ and $b$ that are estimated by OOMMF micromagnetic simulations [@OOMMF] (see below), and $\vec{H}_{\textrm{eff}}$ is the sum of external $\vec{H}_{\textrm{ext}}$, in-plane anisotropy $\vec{H}_{\textrm{K}}$, and out-of-plane anisotropy $\vec{H}_{\perp}$ fields. $\vec{H}_{\textrm{ext}}$ is the sum of the magnetostatic dipole field from the fixed layer $\vec{H}_{\textrm{dip}}$ and the applied field $\vec{H}_{\textrm{app}}$ from the electromagnet, which is adjusted to compensate for $\vec{H}_{\textrm{dip}}$ so $\vec{H}_{\textrm{ext}} \approx 0$.
The initial conditions of the simulation were set by $\theta_{i}$ = $\theta_{0}$ + $\theta_{\textrm{mis}}$ + $\theta_{\textrm{rand}}(T)$, where $\theta_{0}$ = 0$^{\circ}$ for P to AP and 180$^{\circ}$ for AP to P switching and $\theta_{\textrm{mis}}$ represents any systematic angular misalignment between free and fixed layer moments due to the setup and was generally set to 0. The random angle $\theta_{\textrm{rand}}(T)$ is treated as a Gaussian with a standard deviation $\sqrt{k_{B}T/2E_{0}(T)}$ where $E_{0}(T)=E_{0}(4.2 \textrm{
K})[M_{s}(T)/M_{s}(4.2 \textrm{ K})]^{2}$ is the uniaxial anisotropy energy. $E_{0}$(4.2 K), $a$, and $b$ are estimated from $T$ = 0, 2D OOMMF simulations of Py elliptical disks having $H_{\textrm{K}}$ and $\Delta R$ values similar to those measured at 4.2 K. The lateral area is estimated to 12% uncertainty with this method, nearly a factor of 2 better than the inherent shape variation among otherwise identical elliptical patterns due to lithographic fluctuations. Ohmic heating effects during the current pulse are taken into account by locally raising the temperature of the device to $T^{\prime}=\sqrt{T^{2} + 10.23(\textrm{K/mV})^{2}(R_{x}(T)\cdot
I)^{2}}$ [@krivorotov-2004PRL]. A Langevin field $\vec{H}_{\textrm{Lang}}(T^{\prime})$ accounts for thermal fluctuations during the dynamic trajectory, fluctuating randomly in 3-dimensions with a standard deviation $\sqrt{2\alpha_{0}k_{B}T^{\prime}\mu_{0}/\gamma M_{s}(T^{\prime})(area\cdot d)\Delta t}$ where $\Delta t$ = 1 ps is the time step [@li-zhang-2004PRB; @koch-2004PRL; @russek-2005PRB]. Gilbert damping is assigned an angular dependence $\alpha(\theta) = \alpha_{0}[1 - \nu \sin^{2}\theta/(1 - \nu^{2} \cos^{2}\theta)]$, where $\nu$ = 0.33 for Py/Cu/Py nanopillars [@tserkovnyak-2003PRB], but the addition of this angle-dependent damping term had only a small effect on the simulation results.
{width="8.5cm"}
The spin torque function is approximated by $g(\theta) = A \sin\theta/(1 + B \cos\theta)$ where $A$ and $B$ are phenomenological parameters [@braganca-2005APL; @slonczewski-2002JMMM; @xiao-2004PRB]. In our simulations we use $\alpha_{0}$, $A$, and $B$ as $T$-dependent fitting parameters to match the simulated with the measured values of 1/$t_{switch}$ versus $I$ for each $T$, where we allow $\alpha_{0}$ to be different for the two switching directions. In Fig. 1(b) we plot the average of 2000 simulated P to AP switching events at $T$ = 40 K alongside the normalized data for sample 1 with the best fit simulation yielding $A$ = 0.5, $B$ = 0.11, and $\alpha_{0}$ = 0.048. Since the current step in the simulation turns on instantaneously, an average pulse half rise time of 112 ps, measured from data such as those in Fig. 1(b), has been added to all simulated $t_{switch}$. We plot measured 1/$t_{switch}$ versus $I$ for AP to P and P to AP switching at $T$ = 160 K, 40 K, and 4.2 K for sample 2, an 80$\times$180 nm ellipse, together with best fit simulations, all of which are averages over 2000 events, in Fig. 2(a) and 2(b), respectively. Simulations out to long switching times (1/$t_{switch} < 0.1$ ns$^{-1}$) allow for good estimates of the 1/$t_{switch} \rightarrow 0$ intercepts $I_{c0}^{\pm}(T)$, which are the critical currents (+ = P to AP) defining the onset of spin torque-driven switching. These should depend on the spin torque and damping parameters as $I_{c0}^{\pm}(T) \propto \alpha_{0}M_{s}^{2}(T)$ [@koch-2004PRL]. A striking result from these measurements is the strong $T$-dependence of $I_{c0}^{\pm}(T)/M_{s}^{2}(T)$ (Fig. 2(a) inset), which varies by more than 60% over the entire $T$ range, where the upturns at low $T$ indicate a strong dependence of damping, spin torque, or both.
In Fig. 3 the best fit values for $\alpha_{0}$, $A$, and $B$ (assuming $\theta_{\textrm{mis}}$ = 0$^{\circ}$) are plotted as functions of $T$ for sample 2. Uncertainties in the fit parameters, $\Delta\alpha_{0}$ = 0.0035, $\Delta A$ = 0.025, and $\Delta B$ = 0.045, are found through an exploration of parameter space about the best fit values. Accounting for these $T$-dependences, the theoretical prediction of $I_{c0}^{\pm}(T) \propto
\alpha_{0}^{\pm}(T)M_{s}^{2}(T)(1 \pm B(T))/A(T)$ agrees with the measurement to within 10% over the entire range of $T$. All four devices that were extensively studied show an amplitude and $T$-dependence of $\alpha_{0}$ very similar to that of Fig. 3(a); a gradual but significant increase with decreasing $T$ below 160 K, above which the devices are thermally unstable, followed by a stronger increase starting below 60 K - 40 K where the best fit values of $\alpha_{0}$ also suggest differences between the two switching directions. For $T$ $<$ 60 K, the trends in the $T$-dependence of the spin torque parameters $A$ and $B$ vary from sample to sample, but for $T$ $>$ 60 K both consistently show a very mild dependence on $T$ as illustrated in Fig. 3(b). For the four samples studied in detail we found that at 40 K $A$ ranged from 0.5 to 0.68 and $B$ varied from 0.11 to 0.35. In general we also found that $A$ would decrease by 10 or 20% in going from 40 K to 160 K while $B$ would typically vary by 10% or less. These values of $A$ and $B$ and the variation with $T$ $>$ 60 K can be compared with the results of a two-channel model [@valet-fert-1993PRB] with which the measured GMR parameters, $R(T)$ and $\Delta R(T)$, can be used to predict the spin torque parameters [@garcia-unpublished]. This model predicts $A$ = 0.52, $B$ = 0.36 at 40 K, with $A$ decreasing to 0.47 at 160 K and $B$ remaining essentially constant. This is in reasonable accord with the data, given the experimental uncertainties in nanomagnet size and alignment.
{width="8.5cm"}
We attribute the significant $T$-dependence of $\alpha_{0}$ to the presence of a weak antiferromagnetic (AF) layer on the sidewalls of the nanopillar. Although no such AF layer was deliberately deposited, the exposure of the nanopillars to air after ion mill definition undoubtedly oxidized the sidewalls, thus allowing for AF Py oxide to form and weakly exchange bias the ferromagnetic layers. An example of direct evidence for this adventitious exchange biasing is shown in Fig. 4, where switching fields $H_{S1}$ and $H_{S2}$, defined in the inset, from 20 field scans at each $T$, are plotted from 4.2 K to 160 K for sample 3, an 80$\times$180 nm ellipse, a previously unmeasured device cooled in $H_{\textrm{app}} = 0$. Note that $H_{S1}$ varies more rapidly with $T$ than $H_{S2}$, which is indicative of an exchange bias that strengthens with decreasing $T$ (particularly rapidly below 40 K) and promotes AP alignment, i.e. a bias set by the dipole field from the fixed layer. Another key point illustrated in Fig. 4 is the large variation that develops in $H_{S1}$, and to a lesser extent in $H_{S2}$, upon multiple minor loop scans after the device is cooled to low $T$. Initially, the device switches repeatedly with nearly the same switching fields, but after six or seven magnetic reversals the switching fields begin to fluctuate greatly from reversal to reversal, indicating stochastic variations in the net strength of the oxide pinning field. While the effects of the random pinning field are particularly pronounced at 4.2 K they are observed up to 160 K, indicating that some degree of magnetic ordering within the AF persists over this entire $T$ range and also that each reversal of the free layer nanomagnet has an irreversible perturbing effect on the magnetic structure of the AF oxide. It is important to note that the slow ramp rate current-driven switching events at low $T$ for these devices show good reproducibility, with little variation from one sweep to another, as should be the case because ST switching currents are less sensitive to field variations than are the switching fields. The strength of this low $T$ AF exchange biasing varies from device to device, with some samples showing no random variations in $H_{S,i}$. We do not believe that fluctuations of this sort have affected any previously-published conclusions from our group. Nevertheless, the fluctuations visible in some samples indicate clearly the presence of an AF layer that should influence the properties of all nanopillar ST devices.
Exchange biasing in AF/Py films has been demonstrated to dissipate dynamic magnetic energy through a two-magnon scattering process arising from local variations in the interfacial exchange coupling [@mcmichael-1998JAP; @rezende-2001PRB; @weber-2005APL]. Over the course of the pulsed $I$ measurements, the free layer is switched hundreds of millions of times, which the $H_{S,i}$ data indicate should result in the AF layer being on average magnetically ordered but with a finer, more randomized local magnetic structure that leads to strong damping. The rapid increase in damping, observed over the same low $T$ range where both the unidirectional AF pinning field and the critical currents $I_{c0}^{\pm}(T)$ also increase rapidly, is attributed to an increasing portion of the AF oxide layer becoming blocked, thereby simultaneously increasing the amount of interfacial exchange coupling variation seen by the free layer nanomagnet as it moves in its dynamic switching trajectory, consistent with the two-magnon model. The process of inducing randomization in the AF by the nanomagnet reversal itself may also lead to enhanced damping. The unidirectional pinning field is present over the entire $T$ range, with diminishing amplitude with increasing $T$. As the AF grains become unblocked, an additional damping mechanism becomes possible if these grains can undergo reversal on the same time scale that the free layer traces out its dynamic switching trajectory, which can result in domain drag or the “slow relaxer" dissipation process [@mcmichael-2000JAP]. This effect could make a significant contribution to the greater than intrinsic damping that persists to higher $T$.
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Most ST device fabrication processes currently employed expose the sides of the free layer nanomagnet to some level of an oxidizing ambient at some point, either during or after processing and to our knowledge there have been no reports of actively protecting the sidewalls from oxidation. We suggest that the native AF oxide layer that forms can have substantial, previously under-appreciated consequences for the ST behavior, leading to a substantially enhanced damping parameter which directly increases the critical currents for switching. The presence of this AF perimeter layer may also alter the boundary conditions that should be employed in micromagnetic modeling of the free layer nanomagnet behavior and affect the dynamical modes of ST-driven precession. We are currently investigating whether this AF perimeter layer can account, at least in part, for the narrower than predicted ST-induced microwave oscillator linewidths that have been observed in similar nanomagnets at low $T$ [@sankey-2005condmat].
In summary, we have performed time-resolved measurements of the spin torque-driven switching of a Py nanomagnet at $T$ = 4.2 K to 160 K. LLG macrospin simulations are in close quantitative agreement with the ST switching events, yielding values of the parameterized spin torque function $g(\theta) = A \sin\theta/(1 + B \cos\theta)$ and the damping parameter. We find $\alpha_{0}$ to be high, $>$ 0.03, and strongly $T$-dependent, which we attribute to the AF pinning behavior of a thin Py oxide layer on the sidewall of the nanomagnet. The values of $A$ and $B$ are in fair numerical agreement with the spin torque calculated from the two-channel model using the measured magnetoresistance values of the nanopillar spin-valve. There is, however, considerable device-to-device variation in the spin torque asymmetry parameter $B$, which we tentatively attribute to the variable nature of the AF perimeter layer. The presence of an AF oxide layer can have a major effect on the nanomagnet dynamics. Controlling this layer will be important in optimizing spin torque-driven behavior.
This research was supported by ARO - DAAD19-01-1-0541, and by NSF through the NSEC support of the Cornell Center for Nanoscale Systems. Additional support was provided by NSF through use of the facilities of the Cornell Nanoscale Facility - NNIN and the facilities of the Cornell MRSEC.
|
---
abstract: |
We study the obstacle problem for the fractional Laplacian with drift, $\min\left\{{(-\Delta)^s}u + b \cdot \nabla u,\,u -\varphi\right\} = 0$ in ${\mathbb{R}}^n$, in the critical regime $s = \frac{1}{2}$.
Our main result establishes the $C^{1,\alpha}$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form $$u(x)-\varphi(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tilde\gamma(x_0)} + o\left(|x-x_0|^{1+\tilde\gamma(x_0)+\sigma}\right),$$ $$\tilde{\gamma}(x_0) = \frac{1}{2}+\frac{1}{\pi} \arctan (b\cdot e),$$ where $e \in {\mathbb{S}}^{n-1}$ is the normal vector to the free boundary, $\sigma >0$, and $c_0> 0$.
We also establish an analogous result for more general nonlocal operators of order 1. In this case, the exponent $\tilde\gamma(x_0)$ also depends on the operator.
address:
- 'ETH Zürich, Department of Mathematics, Raemistrasse 101, 8092 Zürich, Switzerland'
- 'University of Texas at Austin, Department of Mathematics, 2515 Speedway, TX 78712 Austin, USA'
author:
- 'Xavier Fernández-Real'
- 'Xavier Ros-Oton'
title: |
The obstacle problem for\
the fractional Laplacian with critical drift
---
[^1]
Introduction
============
We consider the obstacle problem for the fractional Laplacian with drift, $$\label{eq.pbintro}
\min\big\{ {(-\Delta)^s}u+ b\cdot \nabla u,\,u -\varphi \big\} = 0\quad\textrm{in}\quad{\mathbb{R}}^n,$$ where $b\in {\mathbb{R}}^n$, and $\varphi: {\mathbb{R}}^n \to {\mathbb{R}}$ is a smooth obstacle.
Problem appears when considering optimal stopping problems for Lévy processes with jumps. In particular, this kind of obstacle problems are used to model prices of (perpetual) American options; see for example [@CF11; @BFR15] and references therein for more details. See also [@Sal12] and [@KKP16] for further references and motivation on the fractional obstacle problem.
We study the regularity of solutions and the corresponding free boundaries for problem . Note that the value of $s\in (0,1)$ plays an essential role. Indeed, if $s > \frac{1}{2}$, then the gradient term is of lower order with respect to ${(-\Delta)^s}$, and thus one expects solutions to behave as in the case $b \equiv 0$. When $s < \frac{1}{2}$ the leading term is $b\cdot\nabla u$ and thus one does not expect regularity results for . Finally, in the borderline case $s = \frac{1}{2}$ there is an interplay between $b\cdot \nabla u$ and $(-\Delta)^{1/2}$, and one may still expect some regularity, but it becomes a delicate issue.
In this work we study this critical regime, $s = \frac{1}{2}$. As explained in detail below, we establish the $C^{1,\alpha}$ regularity of the free boundary near regular points, with a fine description of the solution at such points.
It is important to remark that, when $s=\frac12$, problem is equivalent to the *thin* obstacle problem in ${\mathbb{R}}^{n+1}_+$ with an *oblique* derivative condition on $\{x_{n+1}=0\}$. Thus, our results yield in particular the regularity of the free boundary for such problem, too.
Known results
-------------
The regularity of solutions and free boundaries for was first studied in [@Sil07; @CSS08] when $b = 0$. In [@CSS08], Caffarelli, Salsa, and Silvestre established the optimal $C^{1,s}$ regularity for the solutions and $C^{1,\alpha}$ regularity of the free boundary around regular points. More precisely, they proved that given any free boundary point $x_0\in {\partial}\{u = \varphi\}$, then
(i) either $$0< cr^{1+s} \leq \sup_{B_r(x_0)}(u-\varphi ) \leq Cr^{1+s}$$
(ii) or $$0\leq \sup_{B_r(x_0)} (u-\varphi) \leq Cr^2.$$
The set of points satisfying (i) is called the set of *regular points*, and it was proved in [@CSS08] that this set is open and $C^{1,\alpha}$.
Later, the singular set — those points at which the contact set has zero density — was studied in [@GP09] in the case $s = \frac{1}{2}$. More recently, the regular set was proved to be $C^{\infty}$ in [@JN16; @KRS16]; see also [@KPS15; @DS16]. The complete structure of the free boundary was described in [@BFR15] under the assumption $\Delta \varphi \leq 0$. Finally, the results of [@CSS08] have been extended to a wide class of nonlocal elliptic operators in [@CRS16].
All the previous results are for the case $b = 0 $. For the obstacle problem with drift , Petrosyan and Pop proved in [@PP15] the optimal $C^{1,s}$ regularity of solutions in the case $s > \frac{1}{2}$. This result was obtained by means of an Almgren-type monotonicity formula, treating the drift as a lower order term. In [@GP16], the same authors together with Garofalo and Smit Vega García establish $C^{1,\alpha}$ regularity for the free boundary around regular points, again in the case $ s> \frac{1}{2}$. They do so by means of a Weiss-type monotonicity formula and an epiperimetric inequality. The assumption $s > \frac{1}{2}$ is essential in both works in order to treat the gradient as a lower order term.
Main result
-----------
We study the obstacle problem with critical drift $$\label{eq.obstpb_st}
\begin{array}{rcl}
\min\big\{ (-\Delta)^{1/2}u+ b\cdot \nabla u,\,u -\varphi \big\} & = & 0~~\textrm{ in }~~{\mathbb{R}}^n, \\ \lim_{|x|\to \infty} u (x)& =& 0.
\end{array}$$ Here $b$ is a fixed vector in ${\mathbb{R}}^n$, and the obstacle $\varphi$ is assumed to satisfy $$\label{eq.obst}
\varphi \textrm{ is bounded},~\varphi \in C^{2,1}({\mathbb{R}}^n), \textrm{ and } \{\varphi > 0\}\Subset {\mathbb{R}}^n.$$ The solution to can be constructed as the smallest supersolution above the obstacle and vanishing at infinity.
Our main result reads as follows.
\[thm.1\] Let $u$ be the solution to , with $\varphi$ satisfying , and $b\in {\mathbb{R}}^n$.
Let $x_0\in {\partial}\{u = \varphi\}$ be any free boundary point. Then we have the following dichotomy:
1. either $$0< cr^{1+\tilde\gamma(x_0)} \leq \sup_{B_r(x_0)} (u-\varphi) \leq Cr^{1+\tilde\gamma(x_0)},~~\quad \quad\tilde\gamma(x_0)\in (0,1),$$ for all $r\in (0,1)$,
2. or $$~~~~~~~~~~~~~~~~~~~0\leq \sup_{B_r(x_0)} (u-\varphi) \leq C_\varepsilon r^{2-\varepsilon}\quad\quad\textrm{for all } \varepsilon > 0,~r \in (0,1).$$
Moreover, the subset of the free boundary satisfying ${\rm (i)}$ is relatively open and is locally $C^{1,\alpha}$ for some $\alpha > 0$.
Furthermore, $\tilde\gamma(x_0)$ is given by $$\label{eq.tildegammafls}
\tilde\gamma(x_0) = \frac{1}{2}+\frac{1}{\pi} \arctan \big(b\cdot\nu(x_0)\big),$$ where $\nu(x_0)$ denotes the unit normal vector to the free boundary at $x_0$ pointing towards $\{u > \varphi\}$. Finally, for every point $x_0$ satisfying ${\rm (i)}$ we have the expansion $$\label{eq.expansion}
u(x)-\varphi(x) = c_0\Big((x-x_0)\cdot\nu(x_0)\Big)_+^{1+\tilde\gamma(x_0)} + o\left(\left|x-x_0\right|^{1+\tilde\gamma(x_0)+\sigma}\right)$$ for some $\sigma > 0$, and $c_0 > 0$. The constants $\sigma$ and $\alpha$ depend only on $n$ and $\|b\|$.
We think it is quite interesting that the growth around free boundary points (and thus, the regularity of the solution) depends on the orientation of the normal vector with respect to the free boundary. To our knowledge, this is the first example of an obstacle-type problem in which this happens.
The previous theorem implies that the solution is $C^{1,\gamma_b}$ at every free boundary point $x_0$, with $$\label{eq.gammab}
\gamma_b := \frac{1}{2}- \frac{1}{\pi}\arctan(\|b\|).$$ Nonetheless, the constants may depend on the point $x_0$ considered, so that if we want a uniform regularity estimate for $u$ we actually have the following corollary. It establishes almost optimal regularity of solutions.
\[cor.1\] Let $u$ be the solution to for a given obstacle $\varphi$ of the form , and a given $b\in {\mathbb{R}}^n$. Let $\gamma_b$ given by . Then, for any $\varepsilon > 0$ we have $$\|u\|_{C^{1,\gamma_b - \varepsilon}({\mathbb{R}}^n)}\leq C_\varepsilon,$$ where $C_\varepsilon$ is a constant depending only on $n$, $\|b\|$, $\varepsilon$, and $\|\varphi\|_{C^{2,1}({\mathbb{R}}^n)}$.
In order to prove Theorem \[thm.1\] we proceed as follows. First, we classify convex global solutions to the obstacle problem by following the ideas in [@CRS16]. Then, we show the Lipschitz regularity of the free boundary at regular points, and using the results in [@RS16b] we find that the free boundary is actually $C^{1,\alpha}$. Finally, to prove - we need to establish fine regularity estimates up to the boundary in $C^{1,\alpha}$ domains. This is done by constructing appropriate barriers and a blow-up argument in the spirit of [@RS16]. Notice that, since we do not have any monotonicity formula for problem , our proofs are completely different from those in [@PP15; @GP16].
More general nonlocal operators of order 1 with drift
-----------------------------------------------------
We will show an analogous result for more general nonlocal operators of the form $$\label{eq.L}
L u(x) = \int_{{\mathbb{R}}^n} \left(\frac{u(x+y)+u(x-y)}{2}-u(x) \right)\frac{\mu(y/|y|)}{|y|^{n+1}} dy,$$ with $$\label{eq.L.cond}
\mu \in L^\infty({\mathbb{S}}^{n-1}) ~~ \textrm{ satisfying }~~ \mu (\theta) =\mu (-\theta) ~~\textrm{and}~~ 0<\lambda \leq \mu \leq \Lambda.$$ The constants $\lambda$ and $\Lambda$ are the ellipticity constants. Notice that the operators $L$ we are considering are of order $1$.
The obstacle problem in this case is, then, $$\label{eq.obstpb}
\begin{array}{rcl}
\min\big\{-Lu+ b\cdot \nabla u, u -\varphi \big\}& = & 0~~\textrm{ in }~~{\mathbb{R}}^n, \\ \lim_{|x|\to \infty} u (x)& =& 0.
\end{array}$$
Our main result reads as follows.
\[thm.2\] Let $L$ be an operator of the form -. Let $u$ be the solution to , with $\varphi$ satisfying , and $b\in {\mathbb{R}}^n$.
Let $x_0$ be any free boundary point, $x_0\in {\partial}\{u = \varphi\}$. Then we have the following dichotomy:
1. either $$0< cr^{1+\tilde\gamma(x_0)} \leq \sup_{B_r(x_0)} (u-\varphi) \leq Cr^{1+\tilde\gamma(x_0)},~~\quad \quad\tilde\gamma(x_0)\in (0,1),$$ for all $r\in(0,1)$.
2. or $$~~~~~~~~~~~~~~~~~~~0\leq \sup_{B_r(x_0)} (u-\varphi) \leq C_\varepsilon r^{2-\varepsilon}\quad\quad\textrm{for all }\varepsilon > 0,~r\in(0,1).$$
Moreover, the subset of the free boundary satisfying ${\rm (i)}$ is relatively open and is locally $C^{1,\alpha}$ for some $\alpha > 0$.
Furthermore, the value of $\tilde\gamma(x_0)$ is given by $$\label{eq.tildegamma}
\tilde\gamma(x_0) = \frac{1}{2}+\frac{1}{\pi} \arctan \left(\frac{b\cdot\nu(x_0)}{\chi(\nu(x_0))}\right),$$ where $\nu(x_0)$ denotes the unit normal vector to the free boundary at $x_0$ pointing towards $\{u > \varphi\}$, and $$\label{eq.chi}
\chi(e) = \frac{\pi}{2}\int_{{\mathbb{S}}^{n-1}} |\theta\cdot e| \mu(\theta)d\theta~~\quad\textrm{for}\quad e\in {\mathbb{S}}^{n-1}.$$ Finally, for any point $x_0$ satisfying ${\rm (i)}$ we have the expansion $$u(x)-\varphi(x) = c_0\Big((x-x_0)\cdot\nu(x_0)\Big)_+^{1+\tilde\gamma(x_0)} + o\left(\left|x-x_0\right|^{1+\tilde\gamma(x_0)+\sigma}\right)$$ for some $\sigma > 0$, and $c_0 > 0$. The constants $\sigma$ and $\alpha$ depend only on $n$, the ellipticity constants, and $\|b\|$.
This result extends Theorem \[thm.1\], and the dependence on the operator $L$ is reflected in . For the fractional Laplacian we have $\chi \equiv 1$, and thus becomes .
We will also prove an analogous result to Corollary \[cor.1\] regarding the almost optimal regularity of solutions; see Corollary \[cor.2\].
Structure of the work
---------------------
We will focus on the proof of Theorem \[thm.2\], from which in particular will follow Theorem \[thm.1\]. The paper is organised as follows.
In Section \[sec.2\] we introduce the notation and give some preliminary results regarding nonlocal elliptic problems with drift. In Section \[sec.3\] we establish $C^{1,\tau}$ estimates for solutions to the obstacle problem with critical drift. In Section \[sec.4\] we classify convex global solutions to the problem. In Section \[sec.5\] we introduce the notion of regular points and we prove that blow-ups of solutions around such points converge to convex global solutions. In Section \[sec.6\] we prove $C^{1,\alpha}$ regularity of the free boundary around regular points. In Section \[sec.7\] we establish estimates up to the boundary for the Dirichlet problem with drift in $C^{1,\alpha}$ domains, in particular, finding an expansion of solutions around points of the boundary. In Section \[sec.8\] we combine the results from Sections \[sec.6\] and \[sec.7\] to prove Theorems \[thm.1\] and \[thm.2\]. Finally, in Section \[sec.9\], we establish a non-degeneracy property at all points of the free boundary when the obstacle is concave near the coincidence set.
Notation and preliminaries {#sec.2}
==========================
We begin our work with a section of notation and preliminaries. Here, we recall some known results regarding nonlocal operators with drift, and we also find a 1-dimensional solution.
Throughout the work we will use the following function in order to avoid a heavy reading, $\gamma : {\mathbb{R}}\to (0,1)$, given by $$\label{eq.gamma}
\gamma(t) := \frac{1}{2} + \frac{1}{\pi} \arctan \left(t\right).$$
We next introduce some known results regarding the elliptic problem with drift that will be used. The first one is the following interior estimate.
\[prop.intest\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ solve $$(-L+b\cdot \nabla) u = f,\quad \textrm{in} \quad B_1,$$ for some $f$. Then, if $f\in L^\infty(B_1)$, and for any $\varepsilon > 0$, $$[u]_{C^{1-\varepsilon}(B_{1/2})} \leq C \left(\|f\|_{L^\infty(B_1)} + \|u\|_{L^\infty(B_1)} + \int_{{\mathbb{R}}^n}\frac{|u(y)|}{1+|y|^{n+1}} dy\right),$$ where $C$ depends only on $n$, $\varepsilon$, the ellipticity constants, and $\|b\|$.
The proof of Proposition is given in [@Ser15] in case $b = 0$ (in the much more general context of fully nonlinear equations). The proof of [@Ser15] uses the main result in [@CL14]. The proof of Proposition \[prop.intest\] follows simply by replacing the use of the result [@CL14] in [@Ser15] by [@SS16 Theorem 7.2] or [@CD16 Corollary 7.1].
We also need the following boundary Harnack inequality from [@RS16b].
\[thm.bdharnack\] Let $U\subset{\mathbb{R}}^n$ be an open set, let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$.
Let $u_1, u_2 \in C(B_1)$ be viscosity solutions to $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla) u_i & = & 0 & \quad \textrm{in}\quad U\cap B_1\\
u_i & =& 0 & \quad \textrm{in}\quad B_1\setminus U,\\
\end{array}\right.,\quad i = 1,2,$$ and such that $$u_i \geq 0 \quad \textrm{in}\quad {\mathbb{R}}^n,\quad\quad \int_{{\mathbb{R}}^n} \frac{u_i(y)}{1+|y|^{n+1}} dy = 1,\quad i = 1,2.$$
Then, $$0< c u_2 \leq u_1 \leq Cu_2\quad \textrm{in} \quad U\cap B_{1/2},$$ for some constants $c$ and $C$ depending only on $n$, $\|b\|$, $U$, and the ellipticity constants.
We will also need the following result.
\[thm.bdharnack2\] Let $U\subset{\mathbb{R}}^n$ be a Lipschitz set, let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$.
Let $u_1, u_2 \in C(B_1)$ be viscosity solutions to $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla) u_i & = & g_i & \quad \textrm{in}\quad U\cap B_1\\
u_i & =& 0 & \quad \textrm{in}\quad B_1\setminus U,\\
\end{array}\right.,\quad i = 1,2,$$ for some functions $g_i \in L^\infty(U\cap B_1)$, $i = 1,2$. Assume also that $$u_i \geq 0 \quad \textrm{in}\quad {\mathbb{R}}^n,\quad\quad \int_{{\mathbb{R}}^n} \frac{u_i(y)}{1+|y|^{n+1}} dy = 1,\quad i = 1,2.$$
Then, there exists $\delta > 0$ depending only on $n$, $U$, the ellipticity constants, and $\|b\|$ such that, if $$\|g_i\|_{L^\infty(U\cap B_1)} \leq \delta \quad \textrm{in}\quad U\cap B_1,\quad \quad\quad \quad\quad \quad i = 1,2,$$ then $$\left\|\frac{u_1}{u_2}\right\|_{C^{\sigma}(U\cap B_{1/2})} \leq C,$$ for some constants $\sigma$ and $C$ depending only on $n$, $U$, the ellipticity constants, and $\|b\|$.
Finally, to conclude this section we study how 1-dimensional powers behave with respect to the operator, and in particular, we find a 1-dimensional solution to the problem. This solution is the same as the one that appears as a travelling wave solution in the parabolic fractional obstacle problem for $s = \frac{1}{2}$; see [@CF11 Remark 3.7].
\[prop.1DL\] Let $b\in {\mathbb{R}}$, and let $u\in C({\mathbb{R}})$ be defined by $$u (x) := (x_+)^\beta,$$ for $\beta \in (0,1)$. Then $u$ satisfies $$\begin{split}
(-\Delta)^{1/2} u + bu' = \beta\big(b\sin(\beta\pi)+\cos(\beta\pi)\big)(x_+)^{\beta-1}\quad \textrm{in}\quad {\mathbb{R}}_+,\\ u \equiv 0\quad \textrm{in}\quad {\mathbb{R}}_-.
\end{split}$$
In particular, let us define $$u_0 (x) := C(x_+)^{\gamma(b)},$$ where $$\gamma(t) := \frac{1}{2} + \frac{1}{\pi} \arctan \left(t\right) \in (0,1).$$ Then, $u_0$ satisfies $$\begin{split}
(-\Delta)^{1/2} u_0 + bu_0' = 0\quad \textrm{in}\quad {\mathbb{R}}_+,\\ u_0 \equiv 0\quad \textrm{in}\quad {\mathbb{R}}_-,
\end{split}$$ i.e., $u_0$ is a solution to the 1-dimensional non-local elliptic problem with critical drift and with zero Dirichlet conditions in ${\mathbb{R}}_-$.
Define the harmonic extension to ${\mathbb{R}}^2_+$, $\bar u = \bar u(x,y)$, via the Poisson kernel, so that $\bar u(x, 0) = u(x)$, and $-{\partial}_y \bar u(x, 0) = (-\Delta)^{1/2} u(x)$. We have that $\bar u$ solves, $$\label{eq.expb}
\left\{\begin{array}{rcll}
\Delta \bar u& = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^2\cap \{y > 0\}\\
\bar u & = & 0 & \quad \textrm{in} \quad \{x\leq 0\}\cap \{y = 0\}.\\
\end{array}\right.$$
For simplicity, define the reflected function $w(x, y) = \bar u(-x,y)$, and let us consider that, by separation of variables in polar coordinates, $w(r, \theta) = g(r)h(\theta)$, for $r\geq 0$, $\theta\in [0,\pi]$ (we use the standard variables, $x = r\cos\theta$, $y = r\sin\theta$). Notice that we are considering homogeneous solutions, so that $g(r) = r^\beta$. Then, from we get $$\label{eq.expb2}
\left\{\begin{array}{rcll}
g'' h +r^{-1}g'h + r^{-2}gh'' & = & 0 & \quad \textrm{in}\quad \{r> 0\}\cap \{\theta \in (0,\pi)\}\\
h(0) & = & 0 & \\
\end{array}\right.$$ from which arise that $w$ can be expressed as $$w(r, \theta) = r^\beta \sin(\beta\theta).$$
Now notice that, for $r > 0$, $$((-\Delta)^{1/2} u + bu')(r) = (r^{-1}{\partial}_{\theta} +b{\partial}_r)w(r, \theta)\bigr|_{\theta = \pi} = \beta\left(b\sin(\beta\pi)+\cos(\beta\pi)\right)r^{\beta-1}.$$
Solving for $\beta$ we obtain that it is a solution for $\beta = \gamma(b)$. Moreover, notice that for $\beta <\gamma(b)$ it is a supersolution, and for $\beta >\gamma(b)$ a subsolution.
$C^{1,\tau}$ regularity of solutions {#sec.3}
====================================
In this section we prove $C^{1, \tau}$ regularity of solutions to the obstacle problem with critical drift. For this, we use the method in [@CRS16 Section 2].
Throughout this section we can consider the wider class of nonlocal operators $$\label{eq.L.2}
L u(x) = \int_{{\mathbb{R}}^n} \left(\frac{u(x+y)+u(x-y)}{2}-u(x) \right)\frac{a(y)}{|y|^{n+1}} dy,$$ with $$\label{eq.L.3}
a \in L^\infty ({\mathbb{R}}^n) ~~\textrm{ satisfying }~~ a (y) = a(-y) ~~\textrm{ and }~~ \lambda \leq a \leq \Lambda,$$ so that we are dropping the homogeneity condition of the kernel.
\[lem.basic\] Let $L$ be an operator of the form - and let $b\in {\mathbb{R}}^n$. Let $\varphi$ be any obstacle satisfying , and let $u$ be a solution to . Then,
(a) $u$ is semiconvex, with $$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{\partial}_{ee} u \geq - \|\varphi\|_{C^{1,1}({\mathbb{R}}^n)} ~~\textrm{ for all }~~ e \in {\mathbb{S}}^{n-1}.$$
(b) $u$ is bounded, with $$\|u\|_{L^\infty({\mathbb{R}}^n)} \leq \|\varphi\|_{L^\infty({\mathbb{R}}^n)}.$$
(c) $u$ is Lipschitz, with $$\|u\|_{{\rm Lip}({\mathbb{R}}^n)} \leq \|\varphi\|_{{\rm Lip}({\mathbb{R}}^n)}.$$
The proof is exactly the same as in [@CRS16 Lemma 2.1], since the operator $-L+b\cdot \nabla$ still has maximum principle and is translation invariant.
We next prove the lemma that will yield the $C^{1,\tau}$ regularity of solutions.
\[lem.deltatau\] There exist constants $\tau > 0$ and $\delta > 0$ such that the following statement holds true.
Let $L$ be and operator of the form -, let $b\in {\mathbb{R}}^n$, and let $u\in {\rm Lip}({\mathbb{R}}^n)$ be a solution to $$\begin{array}{rcll}
u &\geq &0& \textrm{in} ~~{\mathbb{R}}^n\\
{\partial}_{ee} u &\geq &-\delta & \textrm{in} ~~B_2~~ \textrm{ for all } e\in {\mathbb{S}}^{n-1}\\
(-L+b\cdot\nabla)(u - u(\cdot - h))&\leq & \delta |h|& \textrm{in} ~~\{u > 0\}\cap B_2~~ \textrm{ for all } h\in {\mathbb{R}}^n,\\
& & &\textrm{in the viscosity sense}. \\
\end{array}$$ satisfying the growth condition $$\sup_{B_R} |\nabla u | \leq R^\tau ~\textrm{ for }~ R\geq 1.$$ Assume that $u(0) = 0$. Then, $$|\nabla u ( x) | \leq 2 |x|^\tau.$$
The constants $\tau$ and $\delta$ depend only on $n$, the ellipticity constants and $\|b\|$.
The proof is very similar to that of [@CRS16 Lemma 2.3].
Define $$\theta(r) := \sup_{\bar r\ge r} \left\{ (\bar r)^{-\tau}\sup_{B_{\bar r}} |\nabla u|\right\}$$
Note that, by the growth control on the gradient, $\theta(r) \leq 1$ for $r \geq 1$. Note also that $\theta$ is nonincreasing by definition.
To get the desired result, it is enough to prove $\theta(r) \leq 2$ for all $r\in (0,1)$. Assume by contradiction that $\theta(r)> 2$ for some $r\in(0,1)$, so that from the definition of $\theta$, there will be some $\bar r\in(r,1)$ such that $$(\bar r)^{-\tau}\sup_{B_{\bar r}} |\nabla u| \geq (1-\varepsilon) \theta(r) \geq (1-\varepsilon) \theta(\bar r) \geq \frac{3}{2},$$ for some small $\varepsilon>0$ to be chosen later.
We now define $$\bar u(x) := \frac{u(\bar r x)}{\theta(\bar r) (\bar r)^{1+\tau}},$$ and $$\begin{aligned}
L_{\bar r} w(x) := \int_{{\mathbb{R}}^n} \left(\frac{w(x+y)+w(x-y)}{2}-w(x) \right)\frac{a(\bar ry)}{|y|^{n+1}} dy\end{aligned}$$ Notice that $L_{\bar r}$ is still of the form -.
The rescaled function satisfies $$\begin{array}{rcll}
\bar u&\geq&0\quad &\textrm{in}\ {\mathbb{R}}^n \\
D^2 \bar u&\geq& -(\bar r)^{2-1-\tau}\delta {\rm Id} \geq -\delta {\rm Id} \quad &\textrm{in}\ B_{2/\bar r} \supset B_2 \\
(-L_{\bar r}+b\cdot \nabla) (\bar u- \bar u(\cdot-\bar h))&\leq& (\bar r)^{-\tau}\delta |\bar r\bar h| \leq \delta |\bar h| \quad &\mbox{in } \{\bar u>0\}\cap B_{2} \\
& & & \mbox{for all }h\in{\mathbb{R}}^n,
\end{array}$$ Moreover, by definition of $\theta$ and $\bar r$, the rescaled function $\bar u$ also satisfies $$\label{eq.growthc11}
1-\varepsilon \le \sup_{|\bar h|\le 1/4} \sup_{B_1} \frac{\bar u- \bar u(\cdot-\bar h)}{|\bar h|} \quad \mbox{and}\quad
\sup_{|\bar h|\le 1/4} \sup_{B_R} \frac{\bar u-\bar u(\cdot-\bar h)}{|\bar h|} \le (R+1/4)^\tau$$ for all $R\ge 1$.
Let $\eta\in C^2_c(B_{3/2})$ with $\eta\equiv 1$ in $B_1$, $\eta \leq 1$ in $B_{3/2}$. Then, $$\sup_{|\bar h|\leq 1/4} \sup_{{B_{3/2}}} \left(\frac{\bar u- \bar u(\cdot-\bar h)}{|\bar h|} + 3\varepsilon\eta\right)\geq 1+2\varepsilon.$$ Fix $h_0\in B_{1/4}$ such that $$t_0 := \max_{\overline{B_{3/2}}} \left(\frac{\bar u - \bar u(\cdot- h_0)}{| h_0|} +3\varepsilon\eta\right)\geq 1+\varepsilon.$$ and let $x_0\in \overline{B_{3/2}}$ be such that $$\label{eq.etatouches}
\frac{\bar u(x_0)- \bar u(x_0- h_0)}{| h_0|} + 3\varepsilon \eta(x_0) = t_0 .$$
Let us denote $$v(x) : = \frac{\bar u(x) - \bar u(x - h_0)}{|h_0|}.$$ Then, we have $$v + 3\varepsilon \eta \leq v(x_0) +3\varepsilon\eta(x_0) = t_0 \quad \textrm{ in }\quad \overline{B_{3/2}}.$$ Moreover, if $\tau$ is taken small enough then $$\sup_{B_4} v \leq (4+1/4)^\tau<1+\varepsilon\leq t_0,$$ so that in particular $x_0$ is in the interior of $B_{3/2}$, and $$\label{eq.etatouches2}
v +3\varepsilon\eta\le t_0\quad \mbox{in }\overline{B_3}.$$ Note also that $x_0\in \{\bar u>0\}$ since otherwise $\bar u(x_0)-\bar u(x_0-h_0)$ would be a nonpositive number.
We now evaluate the equation for $v$ at $x_0$ to obtain a contradiction. To do so, recall that $D^2\bar u \geq -\delta {\rm Id}$ in $B_2$, $\bar u \geq 0$ in ${\mathbb{R}}^n$, and $\bar u(0)=0$. It follows that, for $z\in B_2$ and $t'\in(0,1)$, $$\bar u(t'z) \leq t' \bar u(z)+ (1-t') \bar u(0) + \frac {\delta |z|^2} 2 t'(1-t') \leq \bar u(z) + \frac {\delta |z|^2} 2 t'(1-t')$$ and thus, for $t\in(0,1)$, setting $z =x(1+ t/|x|)$ and $t' = 1/(1+t/|x|)$ we obtain, for $x\in B_1$, $$\bar u(x) -\bar u\left( x + t\frac{x}{|x|}\right) \le \frac \delta 2 (|x|+t)^2 \frac{t/|x|}{(1+t/|x|)^2 } = \frac {\delta |x| t} 2 \le \delta t .$$
Therefore, denoting $e = h_0 /|h_0|$, $t = |h_0|\le 1$ and using that by , if $\tau$ small enough, $$\|\bar u\|_{\rm Lip(B_1)} \le \frac{4}{3},$$ we obtain $$\label{eq.conebd}
\begin{split}
v(x)= \frac{\bar u(x)- \bar u(x- te)}{t}
&\le \frac{\bar u(x)- \bar u(x- te)}{t} + \frac{\bar u\left( x + t\frac{x}{|x|}\right) -\bar u(x) }{t} + \delta
\\
&\le \frac{\bar u\left( x + t\frac{x}{|x|}\right)- \bar u(x- te)}{t} + \delta
\\
&\le \frac{4}{3}\left| e +\frac{x}{|x|} \right|+\delta \,\le\, \frac{1} 4
\end{split}$$ in $\mathcal C_e\cap B_1$ provided $\delta$ is taken smaller than $1/12$; where $\mathcal C_e$ is the cone, $$\mathcal C_e : = \left\{x\,:\, \left|e+ \frac{x}{|x|}\right| \le \frac 18 \right\}.$$
On the other hand, we know that $$\label{eq.boundvxy}
v(x_0 + y) - v(x_0) \leq 3\varepsilon\big(\eta(x_0) - \eta(x_0+y)\big) ~~\textrm{in}~~ B_3.$$
This allows us to define $$\phi(x_0+y) =
\left\{ \begin{array}{rl}
v(x_0) + 3\varepsilon\big(\eta(x_0) - \eta(x_0+y)\big) &\textrm{in }B_{1/8}\\
v(x_0 + y)&\textrm{otherwise}.\\
\end{array}\right.$$
Notice that $\phi$ is regular around $x_0$ and that $\phi \geq v$ everywhere, and recall that $(- L_{\bar r}+b\cdot \nabla) v (x_0) \leq \delta$ in the viscosity sense. Therefore, we have $$\label{eq.deltaineq}
- {L}_{\bar r} \phi(x_0)-C\|b\|\varepsilon \leq (- L_{\bar r}+b\cdot \nabla) \phi(x_0) \leq \delta.$$
Now, using $$1-2\varepsilon\leq v(x_0)\leq 1+\varepsilon,$$ and defining $$\delta \phi(x, y) := \frac{\phi(x+y)+\phi(x-y)}{2}-\phi(x),$$ we can bound $\delta \phi(x_0, y)$ as $$\delta \phi(x_0, y) \leq \left\{\begin{array}{ll}
C\varepsilon |y|^2& \quad\textrm{in}\quad B_2\\[0.3cm]
(|y|+2)^\tau-1+2\varepsilon&\quad\textrm{in}\quad {\mathbb{R}}^n\setminus B_1\\[0.3cm]
-3/8+C\varepsilon&\quad\textrm{in}\quad (-x_0 + \mathcal C_e\cap B_1)\setminus B_{1/4}.
\end{array}\right.$$
The first inequality follows because around $x_0$ and from we have the bound $\delta\phi(x_0,y)\leq \frac{3}{2}\varepsilon \left(2\eta(x_0) - \eta(x_0+y)-\eta(x_0-y)\right)$ and $\eta$ is a $C^2$ function. The second inequality follows from , and using that $\frac{1}{2}\left(|x_0+y|+\frac{1}{4}\right)^\tau + \frac{1}{2}\left(|x_0-y|+\frac{1}{4}\right)^\tau \leq (|y|+2)^\tau$. For the third inequality, notice that $$\begin{aligned}
\delta \phi(x_0, y) & = \frac{v(x_0+y)-v(x_0)}{2} + \frac{v(x_0-y)-v(x_0)}{2}\\
& \leq \frac{1}{8} -\frac{1}{2} + \epsilon + C\varepsilon \leq -\frac{3}{8} + C\varepsilon\quad \textrm{ in } \quad \mathcal (-x_0 + C_e\cap B_1)\setminus B_{1/4},\end{aligned}$$ where we have used to bound the first term and to bound the second one. The constant $C$ depends only on the $\eta$, so it is independent of everything else.
We then find $$\begin{aligned}
{L}_{\bar r} \phi (x_0) \leq &~ \Lambda \int_{B_1} C\varepsilon |y|^2 |y|^{-n-1}dy + \Lambda\int_{{\mathbb{R}}^n\setminus B_1} \bigl\{(|y|+2)^\tau-1+2\varepsilon\bigr\} |y|^{-n-1}dy \\
& + \lambda \int_{(-x_0+\mathcal C_e\cap B_1)\setminus B_{1/4}} \left( -\frac{3}{8} + C\varepsilon \right)|y|^{-n-1}dy \\
\leq & ~ C\varepsilon + C\int_{{\mathbb{R}}^n\setminus B_{1/2}} \bigl\{(|y|+2)^\tau-1\bigr\} |y|^{-n-1}dy - c,\end{aligned}$$ with $c>0$ independent of $\delta$ and $\tau$ (for $\varepsilon$ small).
Thus, combining with we get $$\label{eq.cdelta}
c -C\left( (\|b\| + 1) \varepsilon + \int_{{\mathbb{R}}^n\setminus B_{1/2}} \frac{(|y|+2)^\tau-1}{|y|^{n+1}}dy \right) \leq -C\|b\|\varepsilon - \tilde{L}_{\bar r} \phi(x_0) \leq \delta.$$ If $\varepsilon$ and $\tau$ are taken small enough so that the left-hand side in is greater than $c/2$, we get a contradiction for $\delta \leq c/4$.
The following proposition implies that the solution to the obstacle problem is $C^{1,\tau}$ for some $\tau > 0$.
\[prop.reg.u\] Let $L$ be any operator of the form -, let $b\in {\mathbb{R}}^n$, and let $u\in{\rm Lip}({\mathbb{R}}^n)$ with $u(0) = 0$ be any function satisfying, for all $h\in{\mathbb{R}}^n$ and $e\in {\mathbb{S}}^{n-1}$, and for some $\varepsilon > 0$, $$\begin{array}{rcll}
u&\geq&0\quad &\textrm{in}\ {\mathbb{R}}^n \\
\partial_{ee} u&\geq& -K\quad &\textrm{in}\ B_2 \\
(-L+b\cdot\nabla) (u-u(\cdot-h))& \leq & K|h|\quad &\textrm{in}\ \{u>0\}\cap B_2\\
|\nabla u|&\leq& K(1+|x|^{1-\varepsilon}) \quad &\textrm{in}\ {\mathbb{R}}^n.
\end{array}$$ Then, there exists a small constant $\tau>0$ such that $$\|u\|_{C^{1,\tau}(B_{1/2})}\leq CK.$$ The constants $\tau$ and $C$ depend only on $n$, $\|b\|$, $\varepsilon$, and the ellipticity constants.
The proof is standard and it is exactly the same as the proof of [@CRS16 Proposition 2.4] by means of Lemma \[lem.deltatau\].
Classification of convex global solutions {#sec.4}
=========================================
In this section we prove the following theorem, that classifies all convex global solutions to the obstacle problem with critical drift.
\[thm.clas\] Let $L$ be an operator of the form -. Let $\Omega\subset{\mathbb{R}}^n$ be a closed convex set, with $0\in \Omega$. Let $u\in C^1({\mathbb{R}}^n)$ a function satisfying, for all $h\in {\mathbb{R}}^n$, $$\label{eq.clas}
\left\{\begin{array}{rcll}
(-L+b\cdot\nabla) (\nabla u) & = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n\setminus \Omega\\
(-L+b\cdot\nabla) (u - u(\cdot - h)) & \leq & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n \setminus \Omega\\
D^2 u & \geq & 0 & \quad \textrm{in} \quad {\mathbb{R}}^n\\
u & = & 0 & \quad \textrm{in} \quad \Omega\\
u & \geq & 0 & \quad \textrm{in} \quad {\mathbb{R}}^n.\\
\end{array}\right.$$ Assume also the following growth control satisfied by $u$, $$\label{eq.clas2}
\|\nabla u \|_{L^\infty(B_R)} \leq R^{1-\varepsilon}\quad \textrm{ for all }\quad R \geq 1,$$ for some $\varepsilon > 0$. Then, either $u \equiv 0$, or $$\label{eq.clas3}
\Omega = \{e\cdot x \leq 0 \} \quad \textrm{and}\quad u(x) = C(e\cdot x)_+^{1+\gamma(b\cdot e/\chi(e))},$$ for some $e\in {\mathbb{S}}^{n-1}$ and $C > 0$. The value of $\chi(e)$ is given by with the kernel $\mu$ of $L$, and $\gamma$ is given by .
We start by proving the following proposition.
\[prop.bdharnack\] Let $\Sigma$ be a non-empty closed convex cone, and let $L$ be an operator of the form -. Let $u_1$ and $u_2$ be two non-negative continuous functions satisfying $$\int_{{\mathbb{R}}^n} \frac{u_i(y)}{1+|y|^{n+1}} dy < \infty,\quad i = 1,2.$$
Assume, also, that they are viscosity solutions to $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla) u_i & = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n\setminus \Sigma\\
u_i & =& 0 & \quad \textrm{in}\quad \Sigma\\
u_i & > & 0 & \quad \textrm{in} \quad {\mathbb{R}}^n\setminus \Sigma.
\end{array}\right.$$ Then, $$u_1 \equiv Ku_2 \quad \textrm{in}\quad {\mathbb{R}}^n,$$ for some constant $K$.
The proof is the same as the proof of [@CRS16 Theorem 3.1], using the boundary Harnack inequality in Theorem \[thm.bdharnack\].
Suppose, without loss of generality, that $\Sigma \subsetneq {\mathbb{R}}^n$. Take $P$ a point with $|P| = 1$ and $B_r(P)\subset{\mathbb{R}}^n\setminus\Sigma$ for some $r > 0$, and assume that $u_i (P) = 1$. We want to prove $u_1 \equiv u_2$.
Define, given $R \geq 1$, $$\bar u_i(x) = \frac{u_i(Rx)}{C_i},$$ with $C_i$ such that $\int_{{\mathbb{R}}^n} \bar u_i(y)(1+|y|)^{-n-1} dy = 1$. Thus, by Theorem \[thm.bdharnack\] there exists some $c > 0$ such that $$\label{eq.comparable}
\bar u_1 \geq c \bar u_2 \quad\textrm{and}\quad \bar u_2 \geq c \bar u_1\quad\textrm{in}\quad B_{1/2}.$$
In particular, $\bar u_1 (P/R)$ and $\bar u_2 (P/R)$ are comparable, so that $C_1$ and $C_2$ are comparable. Thus, from , $$u_1 \geq c u_2 \quad\textrm{and}\quad u_2 \geq c u_1\quad\textrm{in}\quad B_{R/2},$$ for any $R\geq 1$, so that the previous inequalities are true in ${\mathbb{R}}^n$.
Now take $$\bar c := \sup\{c > 0 : u_1 \geq cu_2 \quad\textrm{in}\quad {\mathbb{R}}^n\} < \infty.$$
Define $$v = u_1 - \bar c u_2 \geq 0.$$ Either $v \equiv 0$ in ${\mathbb{R}}^n$ or $v >0$ in ${\mathbb{R}}^n\setminus\Sigma$ by the strong maximum principle. If $v \equiv 0$ we are done, because in this case $\bar c = 1$ due to the fact that $u_1(P) = u_2(P) = 1$.
Let us assume then that $v >0$ in ${\mathbb{R}}^n\setminus\Sigma$. Apply the first part of the proof to $v/v(P)$ and $u_2$ to deduce that, for some $\delta > 0$, $v > \delta u_2$. This contradicts the definition of $\bar c$, so $v \equiv 0$ as we wanted.
We can now prove the classification of convex global solutions in Theorem \[thm.clas\]
First, by the same blow-down argument in [@CRS16 Theorem 4.1], we can restrict ourselves to the case in which $\Omega = \Sigma$ for $\Sigma$ a closed convex cone in ${\mathbb{R}}^n$ with vertex at 0.
We now split the proof into two cases:
[*Case 1:*]{} When $\Sigma$ has non empty interior there are $n$ linearly independent unitary vectors $e_i$ such that $-e_i\in \Sigma$. Define $$v_i := {\partial}_{e_i} u,$$ and note that, since $D^2 u \geq 0$ and $-e_i\in \Sigma = \{u = 0\}$, we have $$\label{eq.clas4}
\left\{\begin{array}{rcll}
(-L+b\cdot\nabla) v_i & = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n\setminus \Sigma\\
v_i& = & 0 & \quad \textrm{in}\quad \Sigma\\
v_i& \geq & 0 & \quad \textrm{in} \quad {\mathbb{R}}^n.\\
\end{array}\right.$$
From Proposition \[prop.bdharnack\], we must have $v_i = a_i v_k$ for some $1 \leq k \leq n$, $a_i\in {\mathbb{R}}$, and for all $i = 1,\dots,n$, so that ${\partial}_{e_i - a_ie_k}u \equiv 0$ in ${\mathbb{R}}^n$ for all $i \neq k$. Thus, there exists a non-negative function $\phi: {\mathbb{R}}\to {\mathbb{R}}$, $\phi \in C^1$, such that $u = \phi(e\cdot x)$ for some $e\in {\mathbb{S}}^{n-1}$; so that, since $0\in {\partial}\Sigma$, $\Sigma = \{e\cdot x \leq 0\}$.
Notice that $\phi'\geq 0$ solves $(-L+(b\cdot e){\partial})(\phi') = 0$ in ${\mathbb{R}}_+$ and $\phi' \equiv 0$ in ${\mathbb{R}}_-$, with the growth $\phi' (t)\leq C(1+t^{{1-\varepsilon}})$. From [@RS14 Lemma 2.1], we have $$(\chi(e) (-\Delta)^{1/2}+(b\cdot e){\partial})(\phi') = 0 \quad\textrm{in}\quad {\mathbb{R}}_+,$$ where $\chi(e)$ is given by . Now, a non-negative solution to the previous equation is given by Proposition \[prop.1DL\]. Such solution is unique up to a multiplicative constant thanks to Proposition \[prop.bdharnack\]. Indeed, notice that the hypotheses of the lemma are fulfilled due to the growth control of $\phi'$ and the fact that $\phi'\geq 0$. Thus, we obtain $$\phi(t) = (t_+)^{1+\gamma(b\cdot e)/\chi(e)}\quad\textrm{for}\quad t \in {\mathbb{R}},$$ where $\gamma$ and $\chi$ are given by and respectively.
[*Case 2:*]{} If $\Sigma$ has empty interior then by convexity it must be contained in some hyperplane $H= \{x\cdot e = 0\}$. From Proposition \[prop.reg.u\], rescaling, $$[\nabla u]_{C^\tau(B_R)}\leq C(R),$$ for some constant $C(R)$ depending on $R$; and for any $R \geq 1$. In particular, for any $h\in {\mathbb{R}}^n$, if we define $$v(x) = u(x)- u(x-h)\quad\textrm{for}\quad x\in {\mathbb{R}}^n,$$ then $v\in C^{1,\tau}_{{\rm loc}}({\mathbb{R}}^n)$. This implies that $(-L+b\cdot\nabla)v\in C^{\tau}_{{\rm loc}}({\mathbb{R}}^n)$, but we already knew that $(-L+b\cdot\nabla)v = 0$ in ${\mathbb{R}}^n\setminus H$, so we must have $$(-L+b\cdot\nabla)v = 0\quad\textrm{in}\quad{\mathbb{R}}^n.$$ Now, from the interior estimates in Proposition \[prop.intest\] rescaled on balls $B_R$ we have $$R^{1-\varepsilon/2}[v]_{C^{1-\varepsilon/2}(B_{R/2})} \leq C\left(\|v\|_{L^\infty(B_R)} + \int_{{\mathbb{R}}^n}\frac{|v(Ry)|}{1+|y|^{n+1}}dy\right).$$ On the other hand, from the growth control on the gradient, we have $$\|v\|_{L^\infty(B_R)} \leq |h|R^{1-\varepsilon}.$$ Putting the last two expressions together we reach $$[v]_{C^{1-\varepsilon/2}(B_{R/2})} \leq \frac{C|h|}{R^{\varepsilon/2}}.$$
Now let $R\to \infty$ to obtain that $v$ must be constant for all $h$. That means that $u$ is affine, but $u(0) = 0$ and $u\geq 0$ in ${\mathbb{R}}^n$, so $u\equiv 0$.
Blow-ups at regular points {#sec.5}
==========================
By subtracting the obstacle if necessary and dividing by $C\|\varphi\|_{C^{2,1}({\mathbb{R}}^n)}$, we can assume that we are dealing with the following problem, $$\label{eq.pb}
\left\{\begin{array}{rcll}
u & \geq & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n\\
(-L+b\cdot\nabla) u & \leq & f & \quad \textrm{in}\quad {\mathbb{R}}^n \\
(-L+b\cdot\nabla) u & = & f & \quad \textrm{in}\quad \{u>0\} \\
D^2u & \geq & -{\rm Id} & \quad \textrm{in} \quad {\mathbb{R}}^n.\\
\end{array}\right.$$ Moreover, dividing by a bigger constant if necessary, we can also assume that $$\label{eq.pb2}
\|f\|_{C^1({\mathbb{R}}^n)} \leq 1,$$ and that $$\label{eq.pb3}
\|u\|_{C^{1,\tau}({\mathbb{R}}^n)}\leq 1.$$ The validity of the last expression and the constant $\tau$ come from Proposition \[prop.reg.u\] and Lemma \[lem.basic\].
Let us now introduce the notion of *regular* free boundary point.
We say that $x_0 \in {\partial}\{u > 0\}$ is a *regular* free boundary point with exponent $\varepsilon$ if $$\limsup_{r\downarrow 0} \frac{\| u\|_{L^\infty(B_r(x_0))}}{r^{2-\varepsilon}} = \infty$$ for some $\varepsilon > 0$.
The following proposition states that an appropriate blow up sequence of the solution around a regular free boundary point converges in $C^1$ norm to a convex global solution.
\[prop.regpt\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be a solution to --. Assume that $0$ is a regular free boundary point with exponent $\varepsilon$.
Then, given $\delta > 0$, $R_0 \geq 1$, there exists $r > 0$ such that the rescaled function $$v(x):= \frac{u(rx)}{r\|\nabla u\|_{L^\infty(B_r)}}$$ satisfies $$\|\nabla v \|_{L^\infty(B_R)} \leq 2R^{1-\varepsilon}\quad \textrm{for all}\quad R\geq1,$$ $$\big|(-L+b\cdot\nabla)(\nabla v)\big|\leq \delta\quad \textrm{in}\quad \{v>0\},$$ and $$|v-u_0| + |\nabla v - \nabla u_0|\leq \delta\quad\textrm{in}\quad B_{R_0},$$ for some $u_0$ of the form and with $\|\nabla u_0\|_{L^\infty(B_1)} = 1$.
Before proving the previous proposition, let us prove the following lemma.
\[lem.regpt\] Assume $u \in C^1(B_1)$ satisfies $\|\nabla u\|_{L^\infty({\mathbb{R}}^n)} = 1$, $u(0) = 0$, and $$\sup_{\rho \leq r} \frac{\|u\|_{L^\infty(B_r)}}{r^{2-\varepsilon}} \to \infty\quad \textrm{as} \quad \rho \downarrow 0.$$ Then, there exists a sequence $r_k\downarrow 0$ such that $\|\nabla u\|_{L^\infty(B_{r_k})} \geq \frac{1}{2}r_k^{1-\varepsilon}$, and for which the rescaled functions $$u_k(x) = \frac{u(r_k x)}{r_k \|\nabla u \|_{L^\infty(B_{r_k})}}$$ satisfy $$|\nabla u_k(x)| \leq 2(1+|x|^{1-\varepsilon})\quad\textrm{in}\quad {\mathbb{R}}^n.$$
Define $$\theta(\rho) := \sup_{r \geq \rho} \frac{\|\nabla u\|_{L^\infty(B_{r})}
}{r^{1-\varepsilon}}.$$ Notice that, since $u(0) = 0$, we have $$\frac{\|u\|_{L^\infty(B_{r})}}{r^{2- \varepsilon}}\leq \frac{\|\nabla u\|_{L^\infty(B_{r})}}{r^{1-\varepsilon}}.$$ Therefore, $\theta(\rho) \to \infty$ as $\rho \downarrow 0$, and notice also that $\theta$ is non-increasing.
Now, for every $k\in {\mathbb{N}}$, there is some $r_k\geq \frac{1}{k}$ such that $$\label{eq.theta}
r_k^{\varepsilon-1}\|\nabla u \|_{L^\infty(B_{r_k})} \geq \frac{1}{2}\theta\left(1/k\right) \geq \frac{1}{2} \theta(r_k).$$
Since $\|\nabla u\|_{L^\infty({\mathbb{R}}^n)} = 1$, then $$r_k^{\varepsilon-1} \geq \frac{1}{2}\theta(1/k)\to \infty \quad\textrm{as}\quad k \to \infty,$$ so that $r_k \to 0$ as $k\to \infty$. We also have $\theta(r_k)\geq 1$, and therefore $\|\nabla u\|_{L^\infty(B_{r_k})} \geq \frac{1}{2}r_k^{1-\varepsilon}$.
Finally, from the definition of $\theta$ and , and for any $R \geq 1$, we have $$\|\nabla u_k\|_{L^\infty(B_R)} = \frac{\|\nabla u\|_{L^\infty(B_{r_kR})}}{\|\nabla u\|_{L^\infty(B_{r_k})}} \leq \frac{\theta(r_kR)(r_kR)^{1-\varepsilon}}{\frac{1}{2}(r_k)^{1-\varepsilon}\theta(r_k)} \leq 2R^{1-\varepsilon},$$ which follows from the monotonicity of $\theta$.
We can now prove Proposition \[prop.regpt\], which follows taking the sequence of rescalings given by Lemma \[lem.regpt\] together with a compactness argument.
Let $r_k\downarrow 0$ be the sequence given by Lemma \[lem.regpt\]. Therefore, the functions $$v_k(x) = \frac{u(r_k x)}{r_k\|\nabla u\|_{L^\infty(B_{r_k})}}$$ satisfy $$\|\nabla v_k\|_{L^\infty(B_R)} \leq 2R^{1-\varepsilon}\quad\textrm{for all}\quad R\geq 1,$$ and $$\|\nabla v_k\|_{L^\infty(B_1)} = 1,\quad v_k(0) = 0.$$
Moreover, $$D^2 v_k = \frac{r_k}{\|\nabla u\|_{L^\infty(B_{r_k})}} D^2 u \geq -\frac{r_k}{\|\nabla u\|_{L^\infty(B_{r_k})}} {\rm Id},$$ and, in $\{v_k > 0\}$, $$\begin{aligned}
\big|(-L+b\cdot\nabla)(\nabla v_k)\big| & = \frac{r_k}{\|\nabla u\|_{L^\infty(B_{r_k})}} \big|(-L+b\cdot\nabla)(\nabla u)\big| \\
& \leq \frac{r_k}{\|\nabla u\|_{L^\infty(B_{r_k})}} \|\nabla f\|_{L^\infty} \leq \frac{r_k}{\|\nabla u\|_{L^\infty(B_{r_k})}}.\end{aligned}$$
Notice that, from and with the notation from the proof of Lemma \[lem.regpt\], $$\frac{1}{\eta_k} := \frac{\|\nabla u\|_{L^\infty(B_{r_k})}}{r_k} \geq \frac{\theta(r_k)}{2r^{\varepsilon}_k} \to \infty,\quad\textrm{as}\quad r_k\downarrow 0.$$ Thus, in all we have a sequence $v_k$ such that $v_k\in C^1$, $v_k(0) = 0$, and $$\|\nabla v_k \|_{L^\infty(B_R)} \leq 2R^{1-\varepsilon}\quad \textrm{for all}\quad R\geq1,$$ $$\big|(-L+b\cdot\nabla)(\nabla v_k)\big|\leq \eta_k\quad \textrm{in}\quad \{v_k>0\},$$ $$D^2 v_k \geq -\eta_k {\rm Id},$$ with $\eta_k\downarrow 0$. From the estimates in Proposition \[prop.reg.u\], $$\|\nabla v_k\|_{C^{\tau}(B_R)} \leq C(R)\quad\textrm{for all}\quad R \geq 1,$$ for some constant depending on $R$, $C(R)$. Thus, up to taking a subsequence, $v_k$ converges in $C^1_{\rm loc}({\mathbb{R}}^n)$ to some $v_\infty$ which by stability of viscosity solutions is a convex global solution to the obstacle problem fulfilling .
By the classification theorem, Theorem \[thm.clas\], $v_\infty$ must be of the form . Taking limits $$\|\nabla v_\infty\|_{L^\infty(B_1)} = 1$$ and $v_\infty(0) = 0$. Now the result follows because $\eta_k\downarrow 0$ and $v_k$ converge in $C^1_{\rm loc}({\mathbb{R}}^n)$ to $v_\infty$.
$C^{1,\alpha}$ regularity of the free boundary around regular points {#sec.6}
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In this section we prove $C^{1,\alpha}$ regularity of the free boundary around regular points.
We begin by proving the Lipschitz regularity of the free boundary, as stated in the following proposition.
\[prop.fblip\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be a solution to --. Assume that $0$ is a regular free boundary point.
Then, there exists a vector $e\in {\mathbb{S}}^{n-1}$ such that for any $\ell > 0$, there exists an $r > 0$ and a Lipschitz function $g:{\mathbb{R}}^{n-1}\to {\mathbb{R}}$ such that $$\{u > 0\}\cap B_r = \big\{y_n> g(y_1,\dots,y_{n-1})\big\}\cap B_r,$$ where $y = Rx$ is a change of coordinates given by a rotation $R$ with $Re = e_n$, and $g$ fulfils $$\|g\|_{{\rm Lip} (B_r)} \leq \ell.$$ Moreover, ${\partial}_{e'} u \geq 0$ in $B_{r}$ for all $e'\cdot e \geq \frac{\ell}{\sqrt{1+\ell^2}}$.
The following lemma will be needed in the proof, and it is analogous to [@CRS16 Lemma 6.2].
\[lem.lipreg\] There exists $\eta = \eta(n,\Lambda,\lambda, \|b\|)$ such that the following statement holds.
Let $L$ be an operator of the form -, and let $b\in{\mathbb{R}}^n$. Let $E\subset B_1$ be relatively closed, and assume that, in the viscosity sense, $w\in C(B_1)$ satisfies $$\left\{\begin{array}{rcll}
(-L+b\cdot \nabla)w & \geq & -\eta & \quad \textrm{in}\quad B_1\setminus E\\
w & = & 0 & \quad \textrm{in}\quad E\cup({\mathbb{R}}^n\setminus B_2) \\
w & \geq & -\eta & \quad \textrm{in}\quad B_2 \setminus E,\\
\end{array}\right.$$ and $$\int_{B_1} w_+ \geq 1.$$
Then, $w$ is non-negative in $B_{1/2}$, i.e., $$w \geq 0\quad\textrm{in}\quad B_{1/2}.$$
Let us argue by contradiction, and suppose that the statement does not hold for any $\eta > 0$. Define $\psi\in C_c^2(B_{3/4})$ be a radial function with $\psi \geq 0$, $\psi \equiv 1$ in $B_{1/2}$ and with $|\nabla \psi|\leq C(n)$. Let $$\psi_t(x) := - \eta -t+\eta\psi(x).$$
If $w$ attains negative values on $B_{1/2}$, then there exists some $t_0 > 0$ and $z\in B_{3/4}$ such that $\psi_{t_0}$ touches $w$ from below at $z$, i.e. $\psi_{t_0} \leq w$ everywhere and $\psi_{t_0}(z) = w(z) < 0$. Let $\delta > 0$ be such that $w < 0$ in $B_\delta(z)$ (recall $w$ continuous). Let us now define $$\bar w(x) := \left\{\begin{array}{ll}
w(x) & \quad \textrm{if}\quad x\in {\mathbb{R}}^n\setminus B_\delta(z)\\
\psi_{t_0}(x) & \quad \textrm{if}\quad x\in B_\delta(z).\\
\end{array}\right.$$
Notice that $\bar w$ is $C^2$ around $z$, and is such that $\bar w \leq w$. By definition of viscosity supersolution, we have $$(-L+b\cdot\nabla)\bar w(z) \geq -\eta.$$
On the one hand, this implies $$(-L+b\cdot\nabla)(\bar w-\psi_{t_0})(z) \geq -C\eta,$$ for some $C$ depending on $n$, the ellipticity constants, and $\|b\|$. On the other hand, we can evaluate $\bar w - \psi_{t_0}$ classically at $z$, $$\begin{aligned}
(-L+& b\cdot\nabla)(\bar w-\psi_{t_0})(z) = -L(\bar w-\psi_{t_0})(z) \\
& \leq -\lambda \int_{{\mathbb{R}}^n} (\bar w - \psi_{t_0})(z+y)|y|^{-n-1} dy \leq -c(n) \lambda \int_{B_1\setminus B_{\delta}(z)} (\bar w - \psi_{t_0}) dy\\
&\leq -c(n)\lambda \int_{B_1} w^+ dy \leq -c(n)\lambda.\end{aligned}$$ We used here that $(\bar w - \psi_{t_0}) \chi_{B_1\setminus B_{\delta}(z)} \geq w^+$ in $B_1$.
In all, for $\eta$ small enough depending only on $n$, the ellipticity constants, and $\|b\|$, we reach a contradiction.
With the previous lemma and the results from the previous section, we can now prove Proposition \[prop.fblip\].
Let $\delta > 0$ and $R_0$ to be chosen, and consider the rescaled function from Proposition \[prop.regpt\], $$v(x) = \frac{u(rx)}{r\|\nabla u\|_{L^\infty(B_r)}}.$$
Thanks to Proposition \[prop.regpt\], there exists some $e\in {\mathbb{S}}^{n-1}$ such that $$\left|\nabla v - (x\cdot e)^{\gamma(b\cdot e/\chi(e))}_+ e\right| \leq \delta\quad \textrm{in}\quad B_{R_0}.$$ Recall $\gamma$ and $\chi$ are given by -.
Now let $e'\in {\mathbb{S}}^{n-1}$ be such that (assuming $\ell \leq 1$) $$e'\cdot e \geq \frac{\ell}{\sqrt{1+\ell^2}} \geq \frac{\ell}{2}.$$
Notice that $$\nabla v\cdot e' \geq \frac{\ell}{2} (x\cdot e)^{\gamma(b\cdot e/\chi(e))}_+ - \delta \quad\textrm{in}\quad B_{R_0},$$ and $$\big|(-L+b\cdot\nabla)(\nabla v \cdot e')\big| \leq \delta \quad \textrm{in}\quad \{v > 0\}.$$
Define $$w = \frac{C_1}{\ell} (\nabla v\cdot e')\chi_{B_2},$$ for some $C_1$ such that $$\int_{B_1} w^+ \geq 1.$$
Notice that, if $\delta$ is small enough, then $C_1$ depends only on $n$, $\ell$, $\|b\|$, and the ellipticity constants.
Let us call $E = \{v = 0\}$. If $R_0$ is large enough, depending only on $n$, $\ell$, $\varepsilon$, $\|b\|$, $\delta$, and the ellipticity constants, then $w$ satisfies $$\label{eq.w}
\left\{\begin{array}{ll}
(-L+b\cdot \nabla)w \geq -\frac{CC_1}{\ell}\delta \geq -\eta & \quad \textrm{in}\quad B_1\setminus E\\
w = 0 & \quad \textrm{in}\quad E\cup({\mathbb{R}}^n\setminus B_2) \\
w \geq -\frac{C_1}{\ell}\delta \geq -\eta & \quad \textrm{in}\quad B_2 \setminus E.\\
\end{array}\right.$$ We are using here that, for $x\in B_1\setminus E$, $$\begin{aligned}
(-L+b\cdot \nabla)w(x) & \geq -\frac{C_1}{\ell} \delta- (-L+b\cdot \nabla)\left(\frac{C_1}{\ell} (\nabla v\cdot e')\chi_{B_2^c}\right)(x)\\
& \geq -\frac{C_1}{\ell} \delta + \frac{C_1}{\ell} L (\nabla v\cdot e')\chi_{B_2^c}(x)\\
& \geq -\frac{C_1}{\ell} \delta + \lambda \frac{C_1}{\ell}\int_{B_{R_0-1}} \frac{(\nabla v\cdot e')\chi_{B_2^c}(x+y) + (\nabla v\cdot e')\chi_{B_2^c}(x-y)}{2|y|^{n+1}}\\
&~~~~~~~~~+ \lambda \frac{C_1}{\ell}\int_{B_{R_0-1}^c} \frac{(\nabla v\cdot e')\chi_{B_2^c}(x+y) + (\nabla v\cdot e')\chi_{B_2^c}(x-y)}{2|y|^{n+1}}\\
& \geq -\frac{C_1}{\ell} \delta - \lambda\frac{C_1}{\ell} \hat C\delta - \hat c \geq -\frac{CC_1}{\ell}\delta,\end{aligned}$$ where $R_0$ is chosen large enough so that $\hat c$ can be comparable to the other terms (which can be done, thanks to the fact that $\nabla v$ grows as $R^{1-\varepsilon}$). Notice that $C$ depends only on $\lambda$ and $n$.
In all, we can choose $\delta$ small enough so that $$\frac{CC_1}{\ell}\delta \leq \eta$$ for the constant $\eta$ given in Lemma \[lem.lipreg\].
Therefore, applying Lemma \[lem.lipreg\] to the function $w$ we get that $$w\geq 0\quad\textrm{in}\quad B_{1/2},$$ or equivalently, $${\partial}_{e'} u \geq 0\quad\textrm{in}\quad B_{r/2},$$ for all $e'\in {\mathbb{S}}^{n-1}$ such that $e'\cdot e \geq \frac{\ell}{\sqrt{1+\ell^2}}$. This implies that ${\partial}\{u>0\}$ is Lipschitz in $B_r$, with Lipschitz constant smaller than $\ell$.
Finally, combining Proposition \[prop.fblip\] with the boundary regularity result in Theorem \[thm.bdharnack2\] we show that the free boundary is $C^{1,\alpha}$ around regular points.
\[prop.C1sigma\] Let $L$ be an operator of the form -, and let $b\in{\mathbb{R}}^n$. Let $u$ be a solution to --. Assume that $x_0$ is a regular free boundary point.
Then, there exists $r > 0$ such that the free boundary is $C^{1,\alpha}$ in $B_r(x_0)$ for some $\alpha > 0$ depending only on $n$, $\|b\|$, and the ellipticity constants.
Without loss of generality assume $x_0 = 0$ and that $\nu(0) = e_n$, where $\nu(0)$ denotes the normal vector to the free boundary at 0 pointing towards $\{u > 0\}$.
By Proposition \[prop.fblip\], we already know the free boundary is Lipschitz around 0, with Lipschitz constant 1 in a ball $B_\rho$. Let $v_1 = \frac{1}{\sqrt{2}}\left({\partial}_i u + {\partial}_n u\right)$ for any fixed $i\in\{1,\dots,n-1\}$, and let $v_2 = {\partial}_n u$. We first show that for some $r > 0$ and $\alpha > 0$, $$\label{eq.harnv1v2}
\left\|\frac{v_1}{v_2}\right\|_{C^\alpha\left(\{u > 0\}\cap B_r \right)} = \frac{1}{\sqrt{2}}\left\|1+\frac{{\partial}_i u }{{\partial}_n u}\right\|_{C^\alpha\left(\{u > 0\}\cap B_r \right)} \leq C.$$
Define $w$ as in the proof of Proposition \[prop.fblip\], i.e., $w = C_1 (\nabla v\cdot e')\chi_{B_2}$, where $v$ is the rescaling given by Proposition \[prop.regpt\], and $e'$ is such that $e'\cdot e \geq \frac{\ell}{2}$ (choose $\ell = 1$ for example).
From the proof of Proposition \[prop.fblip\] we know that $w \geq 0 $ in $B_{1/2}$ (if, using the same notation, $R_0$ is large enough and $\delta$ is small enough; i.e., the rescaling defining $v$ is appropriately chosen). Now define $$\tilde{w} = C_1(\nabla v\cdot e')_+$$ and notice that $$\big|(-L+b\cdot \nabla)\tilde{w}\big| \leq \eta \quad \textrm{in}\quad B_{1/4}\setminus \{v = 0\}$$ for some $\eta > 0$ that can be made arbitrarily small by choosing the appropriate (small) $\delta > 0$ and (large) $R_0$ in the rescaling given by Proposition \[prop.regpt\]. The previous inequality follows from the fact that $(\nabla v \cdot e')_- \leq \delta$ in $B_{R_0}$, $(\nabla v \cdot e')_- \leq 2 \left(1+ |x|^{1-\varepsilon}\right)$ in $B_{R_0^c}$, and $ (\nabla v \cdot e')_-\equiv 0$ in $B_{1/2}$.
Let $e_{in} := \frac{1}{\sqrt{2}}\left(e_i+e_n\right)$, and define $w_1= C\left(\nabla v\cdot e_{in}\right)_+$ and $w_2 = C(\nabla v\cdot e_n)_+$ (taking $e' = e_{in}$ and $e' = e_n$). Now notice that $w_1$ and $w_2$ fulfil the hypotheses of the boundary regularity result in Theorem \[thm.bdharnack2\], and $w_1 = C(\nabla v\cdot e_{in})$ and $w_2 = C(\nabla v\cdot e_{n})$ in $B_{1/2}$. Thus, applying Theorem \[thm.bdharnack2\] to $w_1$ and $w_2$ we obtain that there exists some $\alpha > 0$ such that $$\left\|\frac{w_1}{w_2}\right\|_{C^\alpha(\{v > 0\} \cap B_{1/8})} \leq C.$$ Going back to the rescalings defining $\tilde w$ we reach that for some $r> 0$, holds.
Once we have the procedure is standard. Notice that the components of the normal vector to the level sets $\{ u = t\}$ for $t > 0$ can be written as $$\nu^i(x) = \frac{{\partial}_i u}{|\nabla u|}(x) = \frac{{\partial}_i u/{\partial}_n u}{\left(\sum_{j = 1}^{n-1} \left({\partial}_j u/{\partial}_n u\right)^2 + 1\right)^{1/2}},$$ $$\nu^n(x) = \frac{{\partial}_n u}{|\nabla u|}(x) = \frac{1}{\left(\sum_{j = 1}^{n-1} \left({\partial}_j u/{\partial}_n u\right)^2 + 1\right)^{1/2}},$$ for $u(x) = t>0$. In particular, from the regularity of ${\partial}_i u / {\partial}_n u$ given by , we obtain $\nu$ is $C^\alpha$ on these level sets; that is, $|\nu(x) - \nu(y) | \leq C|x-y|^\alpha$ whenever $x, y \in \{u = t\}\cap B_r$. Now let $t\downarrow 0$ and we are done.
Estimates in $C^{1,\alpha}$ domains {#sec.7}
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Once we know that the free boundary is $C^{1,\alpha}$ around regular points, we need to find the expansion of the solution around such points. To do so, we establish fine boundary regularity estimates for solutions to elliptic problem with critical drift in arbitrary $C^{1,\alpha}$ domains. That is the aim of this section.
The main result of this section is the following, for the Dirichlet problem with the operator $-L+b\cdot\nabla$ in $C^{1,\alpha}$ domains. We will use it on the derivatives of the solution to the obstacle problem.
\[thm.expansion\_\] Let $L$ be an operator of the form -, let $b\in {\mathbb{R}}^n$ and let $\Omega$ be a $C^{1,\alpha}$ domain.
Let $f\in L^\infty (\Omega\cap B_1)$, and suppose $u\in L^\infty({\mathbb{R}}^n)$ satisfies $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )u& = & f & \quad \textrm{in}\quad \Omega\cap B_1\\
u& = &0 & \quad \textrm{in}\quad B_1\setminus \Omega. \\
\end{array}\right.$$
Then, for each boundary point $x_0\in B_{1/2}\cap {\partial}\Omega$, there exists a constant $Q$ with $|Q|\leq C\left(\|u\|_{L^\infty({\mathbb{R}}^n)} + \|f\|_{L^\infty(\Omega\cap B_1)}\right)$ such that for all $x\in B_1$ $$\left| u(x) - Q \big((x-x_0)\cdot \nu(x_0) \big)_+^{\tilde\gamma(x_0)} \right| \leq C\left(\|u\|_{L^\infty({\mathbb{R}}^n)} + \|f\|_{L^\infty(\Omega\cap B_1)}\right)|x-x_0|^{\tilde\gamma(x_0)+\sigma},$$ where $\sigma > 0$ and $\nu(x_0)$ is the normal unit vector to ${\partial}\Omega$ at $x_0$ pointing towards the interior of $\Omega$, and $\tilde\gamma(x_0)$ is defined in . The constant $C$ depends only on $n$, $\alpha$, $\Omega$, the ellipticity constants, and $\|b\|$; and the constant $\sigma$ depends only on $n$, $\alpha$, the ellipticity constants, and $\|b\|$.
To prove Theorem \[thm.expansion\_\] we will need several ingredients.
A supersolution and a subsolution
---------------------------------
In this section we denote $$d(x):= {\rm dist}(x, {\mathbb{R}}^n\setminus \Omega).$$ We will also use the following.
\[defi.rendist\] Given a $C^{1,\alpha}$ domain $\Omega$, we consider $\varrho$ a regularised distance function to $C^{1,\alpha}$; i.e., a function that satisfies $$\tilde{K}^{-1} d \leq \varrho \leq \tilde{K} d,$$ $$\|\varrho\|_{C^{1,\alpha}(\Omega)} \leq \tilde{K}\quad \textrm{and}\quad |D^2\varrho|\leq \tilde{K}d^{\alpha - 1},$$ where the constant $\tilde{K}$ depends only on $\alpha$ and the domain $\Omega$.
The existence of such regularised distance was discussed, for example, in [@RS15 Remark 2.2].
We next construct a supersolution, needed in our proof of Theorem \[thm.expansion\_\].
\[prop.supersolution\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $\Omega$ be a $C^{1,\alpha}$ domain for some $\alpha > 0$, and suppose $0\in {\partial}\Omega$.
Let $\nu :{\partial}\Omega\to {\mathbb{S}}^{n-1}$ be the outer normal vector at the points of the boundary of $\Omega$, let $\gamma$ be defined by , and $\chi$ by . Let us also define $$\gamma_0 := \gamma\left(\frac{b\cdot \nu(0)}{\chi(\nu(0))}\right),$$ and $$\label{eq.etanu}
\eta_\nu := \inf \left\{\eta \geq 0 : \gamma\left(\frac{b\cdot \nu(x)}{\chi(\nu(x))}\right) \geq \gamma_0 - \eta\quad \forall x\in {\partial}\Omega\cap B_1 \right\}.$$
Let $\phi := \varrho^\kappa$ for a fixed $0<\kappa<\gamma_0-2\eta_\nu$, and where $\varrho$ is the regularised distance given by Definition \[defi.rendist\]. Then, there exist $\delta > 0$ and $\hat C>0$ such that $$\left\{\begin{array}{rcll}
\hat C(-L+b\cdot\nabla )\phi & \geq& 1 & \quad \textrm{in}\quad B_{1/2}\cap \{x : 0<d(x) \leq \delta\}\\
\hat C\phi&\geq &1 & \quad \textrm{in}\quad B_{1/2}\cap \{x : d(x) \geq \delta\}. \\
\end{array}\right.$$ The constants $\delta$ and $\hat C$ depend only on $n$, $\Omega$, $\kappa$, the ellipticity constants, and $\|b\|$.
Pick any $x_0\in B_{1/2}\cap \{x: d(x) \leq \delta\}$, and define $$l_0 (x) = \big(\varrho(x_0) + \nabla \varrho (x_0)\cdot (x-x_0)\big)_+.$$
Notice that, whenever $l_0 > 0$, if we define $\hat\varrho_0 := \frac{\nabla \varrho(x_0)}{|\nabla \varrho(x_0)|}$ and $z = \hat\varrho_0\cdot x$ then $$\begin{aligned}
(-L+b\cdot\nabla)l_0^\kappa(x) & = \left(\chi(\hat\varrho_0)(-\Delta)^{1/2} + \left(b \cdot \hat\varrho_0\right){\partial}\right)\big(|\nabla\varrho(x_0)|z + c_0\big)^{\kappa}_+ \\
& = |\varrho(x_0)| \chi(\hat\varrho_0) c\big(\kappa, b \cdot \hat\varrho_0/\chi(\hat\varrho_0)\big)\big(|\nabla\varrho(x_0)|z+c_0\big)^{\kappa-1}_+,\end{aligned}$$ where $c_0 = \varrho(x_0) -\nabla \varrho(x_0)\cdot x_0$, and $c(\kappa, b \cdot\hat\varrho_0/\chi(\hat\varrho_0))$ is the constant arising from Proposition \[prop.1DL\]. We want to check that this constant is positive, which is equivalent to saying (again, from Proposition \[prop.1DL\]) that $$\kappa < \gamma\left(\frac{b\cdot \hat\varrho_0}{\chi(\hat\varrho_0)}\right).$$
To see this, it is enough to check that $$\gamma_0 - 2\eta_\nu\leq \gamma\left(\frac{b\cdot \hat\varrho_0}{\chi(\hat\varrho_0)}\right),$$ which will be true for some small $\delta > 0$ and for any $x_0 \in B_{1/2}\cap \{x: d(x) \leq \delta\} $ if $$\lim_{\delta \downarrow 0} \inf_{\substack{y\in B_{1/2} \\ 0<d(y) \leq \delta}}\sup_{x\in {\partial}\Omega\cap B_{3/4}} \frac{\nabla \varrho(y)}{|\nabla \varrho(y)|}\cdot \nu(x) = 1,$$ i.e., $\nabla \varrho$ normalised is close to some unit normal vector to the boundary as $\delta$ goes to zero (notice that $\gamma$ and $\chi$ are continuous). But this is true since $\varrho$ is a $C^{1,\alpha}$ function, so in particular, its gradient is continuous, and the boundary is a level set of $\varrho$; i.e., $\nabla \varrho(y) = |\nabla \varrho (y)|\nu(y)$ for any $y$ on the boundary. It is important to remark that the modulus of continuity of $\nabla \varrho$ depends only on $\Omega$.
Now notice that $$\label{eq.eqx0}
l_0 (x_0) = \varrho(x_0) \quad \quad \nabla l_0 (x_0) = \nabla \varrho (x_0).$$ Let $\tilde{\varrho}$ be a $C^{1,\alpha}({\mathbb{R}}^n)$ extension of $\varrho$ to the whole ${\mathbb{R}}^n$ with $\varrho \leq 0$ in ${\mathbb{R}}^n \setminus \Omega$. Then we have $$\big|\varrho(x_0) +\nabla \varrho(x_0)\cdot y - \tilde{\varrho}(x_0 + y)\big|\leq C|y|^{1+\alpha}.$$ By using that $|a_+ - b_+| \leq |a-b|$ we find $$\big|l_0(x_0 + y) - \varrho(x_0 + y)\big|\leq C|y|^{1+\alpha}.$$
Now, also using that $|a^t - b^t| \leq |a-b|(a^{t-1} + b^{t-1})$ for $a,b \geq 0$, $|a^t - b^t| \leq C|a-b|^t$, and saying $d_0 = d(x_0)$ we get $$\label{eq.supereq}
|\phi - l_0^\kappa|(x_0+y) \leq \left\{\begin{array}{ll}
Cd_0^{\kappa-1}|y|^{1+\alpha} & \quad \textrm{for}\quad y\in B_{d_0/(\tilde{K}+1)}\\
C|y|^{(1+\alpha)\,\kappa}& \quad \textrm{for}\quad y\in B_{1}\setminus B_{d_0/(\tilde{K}+1)} \\
C|y|^{\kappa} &\quad \textrm{for}\quad y\in {\mathbb{R}}^n\setminus B_{1}.\\
\end{array}\right.$$ We have used here that, in $B_{d_0/(\tilde{K}+1)}$, $l_0^{\kappa-1} \leq Cd_0^{\kappa-1}$ and $\varrho^{\kappa-1} \leq Cd_0^{\kappa-1}$. Here, $\tilde{K}$ denotes the constant given in Definition \[defi.rendist\]. Putting all together $$\begin{aligned}
(-L+& b\cdot\nabla)\phi(x_0) =\\
&= (-L+ b\cdot\nabla)(\phi-l_0^\kappa)(x_0)+ (-L+b\cdot\nabla)l_0^\kappa(x_0)\\
& \geq L (l_0^\kappa-\phi)(x_0) + c(\kappa) d_0^{\kappa -1}\\
& = \int_{{\mathbb{S}}^{n-1}} \int_{0}^\infty \left( (l_0^\kappa-\phi)(x_0+ r\theta) + (l_0^\kappa-\phi)(x_0 - r\theta) \right)\frac{dr}{r^{2}}d\mu(\theta) + c(\kappa) d_0^{\kappa -1}\\
& \geq -C\left(\int_0^{d_0/(\tilde{K}+1)} \frac{d_0^{\kappa-1}r^{1+\alpha}}{r^2}dr + \int_{d_0/(\tilde K +1)}^{1} \frac{r^{(1+\alpha)\,\kappa}}{r^2}dr + \int_{1}^{\infty} \frac{r^{\kappa}}{r^2}dr \right) +c(\kappa) \rho^{\kappa -1}\\
& \geq -Cd_0^{\kappa - 1+\alpha}-Cd_0^{(1+\alpha)\,\kappa - 1}+c(\kappa) d_0^{\kappa -1}.\end{aligned}$$
Notice that the right-hand side tends to $+\infty$ as $\delta \downarrow 0$ independently of the $x_0$ chosen. Thus, we can choose $\delta$ small enough so that the right-hand side is greater than 1. Then, by choosing $\hat C \geq 1$ such that $\hat C \phi\geq 1$ in $B_{1/2}\cap \{x: d(x) > \delta\}$ we are done.
We can similarly find a subsolution for the problem. It will be used in the next section.
\[lem.subsolution\] Let $L$ be an operator of the form -, and let $b\in{\mathbb{R}}^n$. Let $\Omega$ be a $C^{1,\alpha}$ domain for some $\alpha > 0$, and suppose $0\in {\partial}\Omega$.
Let $\nu :{\partial}\Omega\to {\mathbb{S}}^{n-1}$ be the outer normal vector at the points of the boundary of $\Omega$, let $\gamma$ be defined by , and $\chi$ by . Let us also define $$\gamma_0 := \gamma\left(\frac{b\cdot \nu(0)}{\chi(\nu(0))}\right),$$ and $$\label{eq.etanu_2}
\eta_\nu^{(2)} := \inf \left\{\eta \geq 0 : \gamma\left(\frac{b\cdot \nu(x)}{\chi(\nu(x))}\right) \leq \gamma_0 + \eta\quad \forall x\in {\partial}\Omega\cap B_1 \right\}.$$
Let $\phi := \varrho^{\kappa_2}$ for any fixed $1>\kappa_2>\gamma_0+2\eta_\nu^{(2)}$. Then, there exist $\delta > 0$ and $\hat C>0$ such that $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )\phi & \leq& -1 & \quad \textrm{in}\quad B_{1/2}\cap \{x : 0<d(x) \leq \delta\}\\
\phi&\leq &\hat C & \quad \textrm{in}\quad B_{1/2}\cap \{x : d(x) > \delta\}. \\
\end{array}\right.$$ The constants $\delta$ and $\hat C$ depend only on $n$, $\Omega$, $\kappa_2$, the ellipticity constants, and $\|b\|$.
The proof follows by the same steps as the proof of Proposition \[prop.supersolution\]. Using the same notation, one just needs to notice that when evaluating $$(-L+b\cdot\nabla)l_0^{\kappa_2}(x) = c\big(\kappa_2, b \cdot \hat\varrho_0/\chi(\hat\varrho_0)\big)\big(|\nabla\varrho(x_0)|z+c_0\big)^{\kappa_2-1}_+,$$ now the constant $c(\kappa_2)$ is negative (independently of the $\kappa_2$ chosen, as before). Thus, $$(-L+ b\cdot\nabla)\phi(x_0)\leq Cd_0^{\kappa_2 - 1+\alpha}+Cd_0^{(1+\alpha)\,\kappa_2 - 1}+c(\kappa) d_0^{\kappa_2 -1},$$ for negative $c(\kappa_2)$, so that if $d_0$ is small enough we obtain the desired result.
Hölder continuity up to the boundary in $C^{1,\alpha}$ domains
--------------------------------------------------------------
The aim of this subsection is to prove Proposition \[prop.boundestimatesg\] below. Before doing that, let us introduce a definition.
\[defi.split\] We say that $\Gamma\subset{\mathbb{R}}^n$ is a $C^{1,\alpha}$ graph splitting $B_1$ into $U^+$ and $U^-$ if there exists some $f_\Gamma\in C^{1,\alpha}({\mathbb{R}}^{n-1})$ such that
1. $\Gamma := \{(x', f_\gamma(x'))\cap B_1 \textrm{ for } x'\in {\mathbb{R}}^{n-1}\}$;
2. $U^+ := \{(x',x_n)\in B_1 : x_n > f_\Gamma(x')\}$;
3. $U^- := \{(x',x_n)\in B_1 : x_n < f_\Gamma(x')\}$.
Under these circumstances, we refer to the $C^{1,\alpha}$ norm of $\Gamma$ as $\|f_\Gamma\|_{C^{1,\alpha}(D')}$, where $D':= \{x'\in {\mathbb{R}}^n : (x',f_\Gamma(x'))\in B_1\}$.
\[prop.boundestimatesg\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $\Gamma$ be a $C^{1,\alpha}$ graph splitting $B_1$ into $U^+$ and $U^-$, according to Definition \[defi.split\], and suppose $0\in \Gamma$.
Let $f\in L^\infty (U^+)$, let $g\in C^\beta(\overline{U_-})$, and suppose $u\in C(\overline{B_1})$ satisfying the growth condition $|u(x) |\leq M (1+|x|)^\Upsilon$ in ${\mathbb{R}}^n$ for some $\Upsilon < 1$. Assume also that $u$ satisfies in the viscosity sense $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )u& = & f & \quad \textrm{in}\quad U^+\\
u& = &g & \quad \textrm{in}\quad U^-. \\
\end{array}\right.$$
Then there exists some $\sigma > 0$ such that $u\in C^\sigma(\overline{B_{1/2}})$ with $$\|u\|_{C^\sigma(B_{1/2})} \leq C\big(\|u\|_{L^\infty(B_1)} + \|g\|_{C^\beta(U^-)} + \|f\|_{L^\infty(U^+)} + M\big).$$ The constants $C$ and $\sigma$ depend only on $n$, $\alpha$, the $C^{1,\alpha}$ norm of $\Gamma$, $\Upsilon$, the ellipticity constants, and $\|b\|$.
Let $\tilde{u} = u \chi_{B_1}$ so that $(-L+b\cdot \nabla)\tilde{u} = f + L (u\chi_{B_1^c}) =: \tilde{f}$ in $U^+\cap B_{3/4}$, and $\tilde{u} = g$ in $U^-$. Note that $\|\tilde{f}\|_{L^\infty(U^+ \cap B_{3/4})} \leq C(\|f\|_{L^\infty(U^+)} + M) =: C_0$ for some constant $C$ depending only on $n$, $\Upsilon$, and the ellipticity constants.
We begin by proving that for some small $\epsilon > 0$, and for some $C$, we have $$\label{eq.utildg}
\|\tilde{u} - g(z)\|_{L^\infty(B_r(z))} \leq Cr^\epsilon\quad \textrm{for all}\quad r\in (0,1),\quad\textrm{and for all}\quad z\in \Gamma\cap B_{1/2},$$ where $\epsilon> 0$ and $C$ depend only on $n$, $C_0$, $\|u\|_{L^\infty(B_1)}$, $\|g\|_{C^\beta(U^-)}$, the ellipticity constants, and $\|b\|$.
Let us define a $C^{1,\alpha}$ domain that will be used in this proof, analogous to a fixed ball if the surface $\Gamma$ was $C^{1,1}$.
Thus, we define $P$ as a fixed $C^{1,\alpha}$ bounded convex domain with diameter 1 that coincides with $\{x = (x_1,\dots,x_n)\in {\mathbb{R}}^n: x_n \geq |(x_1,\dots,x_{n-1})|^{1+\alpha} \}$ in $B_{1/2}$. Let $y_P$ be a fixed point inside the domain, which will be treated as the *center*. Let us call $P_R$ the rescaled version of such domain with diameter $R$ and *center* $y_{P_R}$, and let us define $$P_R^{(\delta)} := \{x\in {\mathbb{R}}^n : {\rm dist}(x, P_R) \leq\delta\}.$$ As an abuse of notation we will also call $P_R$ any rotated and translated version that will be given by the context.
Note that, since $\Gamma$ is $C^{1,\alpha}$, there exists some $\rho_0 \in (0,1)$ depending on the $C^{1,\alpha}$ norm of $\Gamma$ such that any point $z \in \Gamma\cap B_{1/2}$ can be touched by some $P_{\rho_0}$ rotated and translated correspondingly and contained completely in $U^-$.
Let us now consider the supersolution given by Proposition \[prop.supersolution\] with respect to the domain ${\mathbb{R}}^n\setminus P$.
That is, there is some function $\phi_P$ such that, for some constants $\delta > 0$ and $C$ fixed, $$\label{eq.phiP}
\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )\phi_P & \geq& 1 & \quad \textrm{in}\quad P^{(\delta)}\setminus P\\
\phi_P &\geq &1 & \quad \textrm{in}\quad {\mathbb{R}}^n \setminus P^{(\delta)} \\
\phi_P &= &0& \quad \textrm{in}\quad P \\
\phi_P &\leq &Cd^{\kappa}& \quad \textrm{in}\quad {\mathbb{R}}^n,\\
\end{array}\right.$$ where $d = {\rm dist} (x, P)$ and $0<\kappa <\min\left\{\gamma\left(\frac{b'\cdot e}{\chi(e)}\right) : \|b'\| = \|b\|, e\in {\mathbb{S}}^{n-1}\right\}$ can also be fixed — recall that $\gamma$ and $\chi$ are given by -.
![[]{data-label="fig.draw"}](draw3.pdf)
Let $P'$ be a rotated version of $P$, and let $\phi_{P'}$ be the corresponding rotated supersolution. Notice that we can assume that $\phi_{P'}$ also fulfils (with $P'$ instead of $P$), since while the operator $(-L+b\cdot \nabla)$ is not rotation invariant, only an extra positive constant arises depending on the ellipticity constants and $\|b\|$.
Given a rotated, scaled and translated version of the domain $P$, $P_R$, we will denote the corresponding supersolution (the rotated, scaled and translated version of $\phi_P$) by $\phi_{P_R}$.
Let now $z\in \Gamma\cap B_{1/2}$. For any $R\in (0,\rho_0)$ there exists some rescaled, rotated and translated domain $P_R\subset U^-$ touching $\Gamma$ at $z$. Recall that $y_{P_R}$ is the *center* of the domain $P_R$, so that in particular $|z-y_{P_R}| = C_P R$ for some constant $C_P$ that only depends on the domain $P$ chosen ($C_P\in (0,1)$ because the domain $P_R$ has diameter $R$). See Figure \[fig.draw\] for a representation of this situation.
Recall that $\phi_{P_R}$ is the supersolution corresponding to the domain $P_R$, with the $\delta$ given by Proposition \[prop.supersolution\] (now, when rescaling, $\delta$ becomes $R\delta$). Define the function $$\psi(x) = g(y_{P_R}) + \|g\|_{C^\beta(U^-)} \big((1+\delta)R\big)^\beta + \big(C_0 + \|u\|_{L^\infty(B_1)}\big) \phi_{P_R}.$$
Note that $\psi$ is above $\tilde{u}$ in $U^-\cap P^{(R\delta)}_R$, since $\tilde{u} = g$ there and the distance from $y_{P_R}$ to any other point in $P^{(R\delta)}_R$ is at most $(1+\delta)R$.
On the other hand, in $P^{(R\delta)}_R\setminus P_R$ we have $(-L+b\cdot \nabla)\psi \geq (C_0 + \|u\|_{L^\infty(B_1)}) R^{-1} \geq C_0 \geq (-L + b\cdot\nabla) \tilde{u}$ since $R\leq \rho_0 < 1$; and outside $P^{(R\delta)}_R$ we have $\tilde{u} \leq \psi$. In all, $\tilde u \leq \psi$ everywhere by the maximum principle, and thus for any $ R\in (0,\rho_0)$ $$\tilde u (x) - g(z) \leq C\big(R^{\beta} + (r/R)^\kappa\big)\quad \textrm{for all} \quad x\in B_r(z)\quad\textrm{and for all}\quad r\in (0,R\delta),$$ for some constant $C$ that depends only on $n$, $C_0$, $\|u\|_{L^\infty(B_1)}$, $\|g\|_{C^\beta(U^-)}$, the ellipticity constants, and $\|b\|$. If $R$ is small enough we can take $r = R^2$, and repeat this reasoning upside down to get that $$\|\tilde{u} - g(z)\|_{L^\infty(B_r(z))} \leq C\left(r^{\beta/2} + r^{\kappa/2}\right) \leq Cr^\epsilon\quad\textrm{for all } \quad r\in (0,\delta^{2}),$$ for $\epsilon = \min\left\{\frac{\beta}{2},\frac{\kappa}{2}\right\}$. This yields the result by taking a larger $C$ if necessary.
Now let $x, y\in B_{1/2}$, and let $r = |x-y|$. We will show $$|u(x)-u(y)| \leq Cr^\sigma,$$ for some $\sigma > 0$. If $x, y \in U^-$ we are done by the regularity of $g$. If $x\in U^+$, $y\in U^-$, we can take $z$ in the segment between $x$ and $y$, on the boundary $\Gamma$, and compare $x$ and $y$ to $z$, so that it is enough to consider $x, y\in U^+$.
Let $R = {\rm dist}(x, \Gamma) \geq {\rm dist}(y, \Gamma)$, and suppose $x_0,y_0\in \Gamma$ are such that ${\rm dist}(x, \Gamma) = {\rm dist}(x, x_0)$ and ${\rm dist}(y, \Gamma) = {\rm dist}(y, y_0)$. By interior estimates for the problem (see Proposition \[prop.intest\]), $$\label{eq.intest2}
[u]_{C^\epsilon(B_{R/2}(x))} \leq CR^{-\epsilon}.$$
Let $r < 1$, and let us separate two different cases
1. Suppose $r \geq R^2/2$. Then, using and the regularity of $g$ we obtain $$\begin{aligned}
|u(x)-u(y)| & \leq |u(x)-u(x_0)|+|u(x_0)-u(y_0)|+|u(y_0)-u(y)|\\
& \leq CR^{\epsilon} + C(2R+r)^\beta \\
& \leq C(r^{\epsilon/2} + r^{\beta/2}) \leq Cr^{\epsilon/2}.\end{aligned}$$
2. Assume $r \leq R^2/2$, so that $y\in B_{R/2}(x)$. Thus, using , $$|u(x)-u(y)|\leq CR^{-\epsilon} r^\epsilon \leq Cr^{\epsilon/2}.$$
In all, we have found $u\in C^{\sigma}(B_{1/2})$ for $\sigma = \epsilon/2$.
When $U$ is $C^\infty$, the above Hölder estimate follows from the results in [@S94], [@CD01]. We thank G. Grubb for pointing this out to us.
A Liouville theorem
-------------------
We next prove a Liouville-type theorem in the half-space for non-local operators with critical drift, that will be used to prove Theorem \[thm.expansion\_\].
\[thm.liouv\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be any weak solution to $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )u & =& 0& \quad \textrm{in}\quad {\mathbb{R}}^n_+\\
u & = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n_-. \\
\end{array}\right.$$
Assume also that for some $\varepsilon > 0$ and some constant $C$, $u$ satisfies $$\|u\|_{L^\infty(B_R)} \leq C R^{1-\varepsilon}\quad\textrm{for all}\quad R \geq 1.$$
Then, $$u(x) = C(x_n)_+^{\gamma(b_n/\chi)},$$ for some $C > 0$, and where $b_n$ is the $n$-th component of $b$. The constant $\chi$ is defined by $\chi = \chi(e_n)$ where $\chi(e)$ is given by , and $\gamma$ is given by .
Before proving the Liouville theorem, let us prove it in the 1-dimensional case.
Notice that from Proposition \[prop.bdharnack\] it already follows that any non-negative solution must be either $u\equiv 0$ or the one found in Proposition \[prop.1DL\]. Here, however, we need the same result for solutions that may change sign.
\[prop.1DL\_2\] Let $b \in {\mathbb{R}}$, and let $u\in C({\mathbb{R}})$ be a function satisfying $$(-\Delta)^{1/2} u + bu' = 0\quad \textrm{in}\quad {\mathbb{R}}_+,\quad \quad u \equiv 0\quad \textrm{in}\quad {\mathbb{R}}_-,$$ and $|u(x)| \leq C (1+|x|^{1-\varepsilon})$ for some $\varepsilon>0$. Then, $$u (x) = C_0(x_+)^{\gamma(b)},$$ where $\gamma$ is given by .
We first claim that $$\label{eq.claimb}
\left\|u/(x_+)^{\gamma(b)}\right\|_{C^{\sigma}([0,1])}\leq C$$ for some $\sigma > 0$.
Indeed, let $$w = \chi_{[0,2]} u + \kappa \chi_{[3/2,2]},$$ and recall that, for some $\hat C$, $$\|u\|_{L^\infty([0,R])} \leq \hat C R^{1-\varepsilon}.$$
Notice that $w(0) = 0$, and that $w \leq C_0 (x)_+^{\gamma(b)}$ for $x \geq 1$, if $C_0$ is big enough depending only on $\kappa$ and $\hat C$. Choose $\kappa$ so that $(-\Delta)^{1/2} w \leq 0$ in $[0,1]$ so that by the maximum principle $ u = w \leq C_0 (x)_+^{\gamma(b)}$ in $[0,1]$. Doing the same for $-u$ we reach that $$|u| \leq C_0 (x)_+^{\gamma(b)}\quad \textrm{for} \quad x\in [0,1].$$
Define now $\tilde u = u \chi_{(0,m)}+ M(x_+)^{\gamma(b)}$, where $M = M(m)$ is such that $\tilde u \geq 0$ in $(0,m)$. Notice that $\tilde u$ solves an equation of the form $(-\Delta)^{1/2} \tilde u + b\tilde u' = f_m(x)$ in $(0,1)$ for some bounded $f_m$ with $\|f_m\|_{L^\infty(0,1)} \downarrow 0$ as $m \to \infty$. We can now apply Theorem \[thm.bdharnack2\] with $\tilde u$ and $(x_+)^{\gamma(b)}$ to get that for some large enough $m$, $$\left\|\tilde u/(x_+)^{\gamma(b)}\right\|_{C^{\sigma}([0,1])}\leq C,$$ for some $\sigma > 0$. Thus, we get .
Define $v = u - k(x_+)^{\gamma(b)}$, where $k = \lim_{x \downarrow 0} \frac{u(x)}{(x_+)^{\gamma(b)}}$. Then we have $$\label{eq.vnearinf}
|v(x)| \leq C|x|^{1-\varepsilon}\quad \textrm{for} \quad x \geq 1,$$ $$\label{eq.vnear0}
|v(x)| \leq C|x|^{\gamma(b) + \sigma}\quad \textrm{for} \quad x \in [0,2],$$ and we can assume, without loss of generality, that $1-\varepsilon > \gamma(b)+\sigma$. Combining this with the interior estimates from Proposition \[prop.intest\] we obtain $v\in C^{\gamma(b) + \sigma}([0,1])$. Indeed, take $x, y \in [0,1]$, $x < y$. Let $r = y-x$ and $R = |y|$. Now separate two cases
1. If $2r \geq R$, by $$\begin{aligned}
|v(x)-v(y)| & \leq |v(x)| + |v(y)| \leq C(|x|^{\gamma(b)+\sigma} + |y|^{\gamma(b)+\sigma})\\
& \leq C \big((R-r)^{\gamma(b)+\sigma}+R^{\gamma(b)+\sigma}\big) \leq C r^{\gamma(b)+\sigma}.\end{aligned}$$
2. If $2r < R$, then $x, y \in (y-R/2, y+R/2)$. By rescaling the estimates from Proposition \[prop.intest\] and using $$R^{\gamma(b)+\sigma}[v]_{C^{\gamma(b)+\sigma}\left(y-\frac{R}{2}, y+\frac{R}{2}\right)} \leq C\left(\|v\|_{L^\infty\left(y-R, y+R\right) }+R^{1-\varepsilon}\right).$$
Now, from $$\|v\|_{L^\infty\left(y-R, y+R\right) }\leq CR^{\gamma(b)+\sigma},$$ so that $$[v]_{C^{\gamma(b)+\sigma}\left(y-\frac{R}{2}, y+\frac{R}{2}\right)} \leq C.$$
This implies $$\|v\|_{C^{\gamma(b)+\sigma}([0,1])} \leq C,$$ as desired.
Now, we claim that using the interior estimates from Proposition \[prop.intest\] we obtain $$\label{eq.vprime1}
|v'(x)|\leq C|x|^{-\varepsilon} \quad \textrm{for} \quad x\geq 1,$$ and $$\label{eq.vprime2}
|v'(x)| \leq C|x|^{\gamma(b) + \sigma- 1} \quad \textrm{for} \quad x\in[0,1].$$
Let us show that these last inequalities hold. The first one, , follows using that $|v(x)| \leq C(1 + |x|^{1-\varepsilon})$, and that - combined with the rescaled interior estimates in Proposition \[prop.intest\] yield $$\label{eq.vprime0}
[v]_{C^{\gamma(b)+\sigma}(R, 2R)} \leq CR^{1-\varepsilon-\gamma(b)-\sigma}\quad\textrm{for}\quad R \geq 1.$$
Indeed, take $0<\alpha < \gamma(b)+\sigma$, and any $h\in {\mathbb{R}}$ with $|h| \leq R/2$. Then by interior estimates applied to the incremental quotients, $$\left[\frac{v(x+h)-v(x)}{|h|^{\gamma(b)+\sigma}}\right]_{C^{1-\alpha}(R,2R)} \leq CR^{\alpha-\varepsilon-\gamma(b)-\sigma} \quad\textrm{for}\quad R \geq 1,$$ with $C$ independent of the $h$ chosen. In particular, this yields $$[v']_{C^{\gamma(b)+\sigma-\alpha}(R,2R)} \leq CR^{\alpha-\varepsilon-\gamma(b)-\sigma}\quad\textrm{for}\quad R \geq 1.$$ The inequality in now follows comparing the value of $v'(2^k)$ for any $k\in {\mathbb{N}}$ with $v'(1)$ dyadically.
For the second inequality, , we proceed similarly. Take $0<\alpha< \gamma(b)+\sigma$, and for any $R > 0$ fixed take $|h|\leq R/2$ and notice that $$\label{eq.vprime3}
\left[\frac{v(x+h)-v(x)}{|h|^{\gamma(b)+\sigma}}\right]_{C^{1-\alpha}(R,2R)} \leq CR^{\alpha-1}\quad\textrm{for}\quad 0<R<1,$$ with $C$ independent of $h$. This follows from the interior estimates in Proposition \[prop.intest\] and the growth of $\frac{v(x+h)-v(x)}{|h|^{\gamma(b)+\sigma}}$ given by . As before, this implies $$[v']_{C^{\gamma(b)+\sigma-\alpha}(R,2R)} \leq CR^{\alpha-1}\quad\textrm{for}\quad 0<R<1.$$ Finally, the inequality follows comparing the value of $v'(2^{-k})$ with $v'(1)$ dyadically. Thus, and are proved.
Define now the function $$\psi_A(x) = A\left((x_+)^{\gamma(b)} + (x_+)^{\gamma(b) - 1}\right),$$ and notice that $\psi_A$ and $v'$ solve $$\label{eq.psia1}
(-\Delta)^{1/2} \psi_A + b\psi_A' = 0\quad \textrm{in} \quad x> 0,$$ $$\label{eq.psia2}
(-\Delta)^{1/2} v' + b(v')' = 0\quad \textrm{in} \quad x> 0.$$
We have that $\psi_A > v'$ in $\{ x > 0\}$ for some large enough $A$, thanks to the growth of $v'$ in -. Choose the smallest nonnegative $A$ such that $\psi_A\geq v'$. Then, by the growth at zero and infinity of both $v'$ and $\psi_A$ they touch at some point in $(0,\infty)$. Moreover, if $A>0$, then we must have $\psi_A \not\equiv v'$.
Let $x_0 > 0$ be a point where $\psi_A(x_0) = v'(x_0)$. Notice that $\psi_A - v'$ is a non-negative (and non-zero) function with a minimum at $x_0$. Thus, $$\big((-\Delta)^{1/2} (\psi_A - v') + b(\psi_A - v')'\big)(x_0) = (-\Delta)^{1/2} (\psi_A - v')(x_0) < 0,$$ which contradicts the fact that both $\psi_A$ and $v'$ are solutions to the problem, -. Thus, there is no positive $A$ such that $\psi_A$ and $v'$ touch at at least one point, so we must have $v' \leq 0$. Doing the same from below we reach $v' \geq 0$, and therefore $v'\equiv 0$. Hence, since $u(0) = 0$ we find $v \equiv 0$. In particular, this means that $$u = k(x_+)^{\gamma(b)},$$ as desired.
We can now prove the Liouville theorem.
Let us first see that the solution is 1-dimensional in the direction $e_n$.
Given $\rho \geq 1$, define $$v_\rho(x) = \rho^{-\varepsilon+1}u(\rho x).$$
Notice that $$\|v_\rho\|_{L^\infty(B_R)} = \rho^{-\varepsilon+1}\|u(\rho\cdot)\|_{L^\infty(B_R)} = \rho^{-\varepsilon+1}\|u\|_{L^\infty(B_{\rho R})} \leq CR^{1-\varepsilon}.$$ Moreover, by the homogeneity of $(-L+b\cdot\nabla)$, $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )v_\rho & =& 0& \quad \textrm{in}\quad {\mathbb{R}}^n_+\\
v_\rho & = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n_-. \\
\end{array}\right.$$
Define now $\tilde{v}_\rho = v_{\rho}\chi_{B_2}$, so that $\tilde{v}_\rho \in L^\infty({\mathbb{R}}^n)$. We now have $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )\tilde v_\rho & =& g_\rho& \quad \textrm{in}\quad B_1^+\\
\tilde v_\rho & = & 0 & \quad \textrm{in}\quad B_1^-, \\
\end{array}\right.$$ for some $g_\rho$ with $\|g_\rho\|_{L^\infty(B_1^+)}\leq C_0$ with $C_0$ independent of $\rho$. Indeed, $$(-L+b\cdot\nabla )\tilde v_\rho = (-L+b\cdot\nabla )( v_\rho - v_\rho\chi_{B_2^c}) = L (v_\rho\chi_{B_2^c}) \leq C_0\quad \textrm{in}\quad B_1^+,$$ where the last inequality follows thanks to the uniform growth control on $v_\rho$.
Now, by Proposition \[prop.boundestimatesg\], $$\|v_\rho\|_{C^\sigma(B_{1/2})} = \|\tilde{v}_\rho\|_{C^\sigma(B_{1/2})} \leq C,$$ from which $$\label{eq.123}
[u]_{C^\sigma(B_{\rho/2})} = \rho^{-\sigma} [u(\rho\cdot)]_{C^{\sigma}(B_{1/2})} = \rho^{-\sigma+1-\varepsilon} [v_\rho]_{C^\sigma(B_{1/2})} \leq C\rho^{-\sigma+1-\varepsilon}.$$
Now, given $e\in {\mathbb{S}}^{n-1}$ with $e_n = 0$, and for any $h > 0$, define $$w (x) = \frac{u(x+e h)-u(x)}{h^\sigma}.$$ By , $$\|w\|_{L^\infty(B_R)} \leq CR^{-\sigma+1-\varepsilon}\quad\textrm{for all}\quad R \geq 1.$$ We also have $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )w & =& 0& \quad \textrm{in}\quad {\mathbb{R}}^n_+\\
w & = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n_-, \\
\end{array}\right.$$ thanks to the fact that $e$ does not have component in the $n$-th direction, $e_n = 0$.
Repeat the previous argument applied to $w$ instead of $u$, to get $$[w]_{C^\sigma(B_R)} \leq CR^{-2\sigma+1-\varepsilon}\quad \textrm{for all} \quad R\geq 1.$$
Repeating iteratively we get that, for $m = \lfloor \frac{1-\varepsilon}{\sigma}+1 \rfloor$, then $$[w_m]_{C^\sigma(B_R)} \leq CR^{-m\sigma+1-\varepsilon}\quad \textrm{for all} \quad R\geq 1,$$ where $w_m$ is an incremental quotient of order $m$ of $u$. Letting $R\to \infty$ we observe that $w_m \equiv 0$.
Since $w_m$ is any incremental quotient of order $m$, this means that for any fixed $x$, $q_x(y') := u(x+(y',0))$ for $y'\in {\mathbb{R}}^{n-1}$ is a polynomial of order $m-1$ in the $y'$ variables. However, from the growth condition on $u$ the polynomial must grow less than linearly at infinity, and therefore it is constant. This means that for any $x$, $u(x+eh) = u(x)$ for all $h \in {\mathbb{R}}$ and for all $e\in {\mathbb{S}}^{n-1}$ with $e_n= 0$; i.e., $u(x) = u(x_n)$, as we wanted to see.
Now we can proceed as in the proof of the classification theorem, Theorem \[thm.clas\], and use the classification of 1-dimensional solutions from Proposition \[prop.1DL\_2\].
Proof of Theorem \[thm.expansion\_\]
------------------------------------
We now prove the following result, which will directly yield Theorem \[thm.expansion\_\]. For this, we combine the ideas in [@RS16] with Propositions \[prop.boundestimatesg\] and \[prop.1DL\_2\].
\[prop.expansion\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $\Gamma$ be a $C^{1,\alpha}$ graph splitting $B_1$ into $U^+$ and $U^-$ (see Definition \[defi.split\]), and suppose $0\in \Gamma$ and that $\nu(0) = e_n$, where $\nu(0)$ is the normal vector to $\Gamma$ at 0 pointing towards $U^+$.
Let $f\in L^\infty (U^+)$, and suppose $u\in L^\infty({\mathbb{R}}^n)$ satisfies $$\left\{\begin{array}{rcll}
(-L+b\cdot\nabla )u& = & f & \quad \textrm{in}\quad U^+\\
u& = &0 & \quad \textrm{in}\quad U^-. \\
\end{array}\right.$$
Let us denote $\gamma := \gamma\left(\frac{b\cdot \nu(0)}{\chi(\nu(0))}\right) = \gamma(b_n/\chi(e_n))$ and $\chi = \chi(e_n)$ as defined in -, and suppose that $\gamma\in \left[\gamma_0,\gamma_0\left(1+\frac{\alpha}{8}\right)\right]$ for some $\gamma_0 \in(0,1)$ such that $\gamma_0\left(1+\frac{\alpha}{4}\right)< 1$. Suppose also that $\eta_\nu$ as defined in satisfies $\eta_\nu \leq \frac{\alpha\gamma_0}{64}$, and let $\Upsilon = \gamma_0\left(1+\frac{\alpha}{4}\right)$.
Then, there exists $Q$ with $|Q|\leq C\left(\|u\|_{L^\infty({\mathbb{R}}^n)} + \|f\|_{L^\infty(U^+)}\right)$ such that $$\big| u(x) - Q (x_n)_+^{\gamma} \big| \leq C\left(\|u\|_{L^\infty({\mathbb{R}}^n)} + \|f\|_{L^\infty(U^+)}\right)|x|^\Upsilon
\quad \textrm{for all}\quad x \in B_1,$$ where the constant $C$ depends only on $n$, $\alpha$, the $C^{1,\alpha}$ norm of $\Gamma$, $\gamma_0$, the ellipticity constants, and $\|b\|$.
Before proving the previous result let us state a useful lemma. It can be found in [@RS16 Lemma 5.3].
\[lem.RS16\] Let $1>\Upsilon > \beta_0\geq \beta$ and $\nu \in {\mathbb{S}}^{n-1}$ some unit vector. Let $u\in C(B_1)$ and define $$\phi_r (x) := Q_*(r) (x\cdot \nu)^\beta_+,$$ where $$Q_*(r) := {\rm arg~min}_{Q\in {\mathbb{R}}} \int_{B_r} \big(u(x) - Q(x\cdot \nu)^\beta_+\big)^2dx = \frac{\int_{B_r} u(x) (x\cdot \nu)^\beta_+ dx}{\int_{B_r} (x\cdot \nu)^{2\beta}_+ dx}.$$
Assume that for all $r \in (0,1)$ we have $$\|u - \phi_r\|_{L^\infty(B_r)} \leq C_0r^\Upsilon.$$ Then, there is $Q\in {\mathbb{R}}$ with $|Q| \leq C(C_0 + \|u\|_{L^\infty(B_1)})$ such that $$\|u- Q(x\cdot \nu)^\beta_+\|_{L^\infty(B_r)}\leq C C_0 r^\Upsilon$$ for some constant $C$ depending only on $\Upsilon$ and $\beta_0$.
We can now prove Proposition \[prop.expansion\].
Let us argue by contradiction. Suppose that there are sequences $\Gamma_i$, $U^+_i$, $U^-_i$, $L_i$, $b_i$, $u_i$, and $f_i$ that satisfy the assumptions
1. $\Gamma_i$ is a $C^{1,\alpha}$ graph with bounded $C^{1,\alpha}$ norm independently of $i$, splitting $B_1$ into $U^+_i$ and $U^-_i$ with $0\in \Gamma_i$ and with $e_n$ being the normal vector at 0 pointing towards $U^+_i$.
2. $L_i$ are of the form -, and $\|b_i\|=\|b\|$;
3. For each $\Gamma_i$, the corresponding $\eta_\nu$ as defined in fulfils $\eta_\nu \leq (\alpha\gamma_0)/64$;
4. $\|u_i\|_{L^\infty({\mathbb{R}}^n)} + \|f_i\|_{L^\infty(U^+)} = 1$;
5. $u_i$ solves $(-L_i + b_i\cdot \nabla) u_i = f_i$ in $U^+_i$, $u_i = 0$ in $U^-_i$;
6. If we define $\gamma_i := \gamma(b_i\cdot e_n / \chi_i)$ with $\gamma$ as in and $\chi_i = \chi_i(e_n)$ as in with the operator $L_i$, then $\gamma_i \in [\gamma_0,\gamma_0(1+\alpha/8)]$;
but they are such that for all $C > 0$ there exists some $i$ such that there is no constant $Q$ satisfying $$\big| u_i(x) - Q (x_n )_+^{\gamma_i} \big| \leq C|x|^\Upsilon
\quad \textrm{for all}\quad x \in B_1.$$\
[*Step 1: Construction and properties of the blow up sequence.*]{}
Let us denote $$\phi_{i,r} := Q_{i}(r) (x_n)^{\gamma_i}_+,$$ where $$Q_i(r) := {\rm arg~min}_{Q\in {\mathbb{R}}} \int_{B_r} (u_i(x) - Q(x_n)^{\gamma_i}_+)^2dx = \frac{\int_{B_r} u_i(x) (x_n)^{\gamma_i}_+ dx}{\int_{B_r} (x_n)^{2\gamma_i}_+ dx}.$$ From Lemma \[lem.RS16\] with $\beta = \gamma_i$ and $\beta_0 = \gamma_0(1+ \alpha/8)$ we have that $$\sup_i \sup_{r>0} \left\{ r^{-\Upsilon} \|u_i - \phi_{i,r}\|_{L^\infty(B_r)} \right\}= \infty.$$
Define the monotone function $$\theta(r) := \sup_i \sup_{r' > r} \left\{ (r')^{-\Upsilon} \|u_i - \phi_{i,r'}\|_{L^\infty(B_{r'})} \right\}.$$
Note that for $r > 0$, $\theta(r) < \infty$, and $\theta(r) \to \infty$ as $r \downarrow 0$. Now take a sequences $r_m\downarrow 0$ and $i_m$ such that $$(r_m)^{-\Upsilon} \|u_{i_m} - \phi_{i_m, r_m}\|_{L^\infty(B_{r_m})} \geq \frac{\theta(r_m)}{2},$$ and denote $\phi_m = \phi_{i_m, r_m}$.
Consider now $$v_m (x) = \frac{u_{i_m} (r_m x) - \phi_m (r_m x)}{r_m^\Upsilon \theta(r_m)}.$$
By definition of $\phi_m$ we have the orthogonality condition for all $m \geq 1$, $$\label{eq.orth}
\int_{B_1} v_m(x) (x_n)^{\gamma_i}_+ dx = 0.$$
Note that also from the choice of $r_m$ we have a nondegeneracy condition for $v_m$, $$\label{eq.nondeg}
\|v_m\|_{L^\infty(B_1)} \geq \frac{1}{2}.$$
From the definition of $\phi_{i, r}$, $\phi_{i, 2r} - \phi_{i, r} = \big(Q_i (2r) - Q_i (r) \big) (x_n)^{\gamma_i}_+$ so that $$\begin{aligned}
|Q_i(2r) - Q_i(r)| r^{\gamma_i} & = \|\phi_{i, 2r} - \phi_{i, r}\|_{L^\infty(B_r)} \\
& \leq \|\phi_{i, 2r} - u\|_{L^\infty(B_{2r})}+ \|\phi_{i, r} - u\|_{L^\infty(B_r)}\leq Cr^\Upsilon \theta(r).\\\end{aligned}$$
Proceeding inductively, if $R = 2^N$, then $$\label{eq.longeq}
\begin{split}
\frac{r^{\gamma_i - \Upsilon} |Q_i(Rr)- Q_i(r)|}{\theta(r)}& \leq \sum_{j = 0}^{N-1} 2^{j(\Upsilon - \gamma_i)} \frac{{(2^jr)}^{\gamma_i - \Upsilon} |Q_i(2^{j+1}r)- Q_i(2^jr)|}{\theta(r)} \\
& \leq C \sum_{j = 0}^{N-1} 2^{j(\Upsilon - \gamma_i)} \frac{\theta(2^j r)}{\theta(r)} \leq C 2^{N(\Upsilon - \gamma_i)} = CR^{\Upsilon -\gamma_i}.
\end{split}$$
Thus, we obtain a bound on the growth control of $v_m$ given by $$\label{eq.growthctrl}
\|v_m\|_{L^\infty(B_R)} \leq CR^\Upsilon\quad \textrm{for all} \quad R \geq 1.$$
Indeed, $$\begin{aligned}
\|v_m\|_{L^\infty(B_R)} & = \frac{1}{\theta(r_m) r_m^\Upsilon} \|u_i - Q_{i_m}(r_m) (x_n)^{\gamma_i}_+\|_{L^\infty(Rr_m)}\\
& \leq \frac{1}{\theta(r_m) r_m^\Upsilon} \|u_i - Q_{i_m}(Rr_m) (x_n)^{\gamma_i}_+\|_{L^\infty(Rr_m)} +\\
&~~~~~~~~~~~~~~~~+\frac{1}{\theta(r_m) r_m^\Upsilon} |Q_{i_m}(Rr_m) - Q_{i_m}(r_m) | (Rr_m)^{\gamma_i}\\
& \leq \frac{R^\Upsilon \theta(Rr_m)}{\theta(r_m)} + CR^\Upsilon,\end{aligned}$$ and the result follows from the monotonicity of $\theta$.
Notice also that the previous computation in also gives a bound for $Q_i(r)$ given by $$\label{eq.qbound}
|Q_i(r)| \leq C\theta(r),$$ which follows by putting $R = r^{-1}$.
[*Step 2: Convergence of the blow up sequence.*]{}
In this second step we show that $v_m$ converges locally uniformly in ${\mathbb{R}}^n$ to some function $v$ satisfying $$\label{eq.v}
\left\{\begin{array}{rcll}
(-\tilde{L}+ \tilde{b}\cdot\nabla )v& = & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n_+\\
v& = &0 & \quad \textrm{in}\quad {\mathbb{R}}^n_-, \\
\end{array}\right.$$ for some operator $\tilde{L}$ of the form -, $\|\tilde{b}\|= \|b\|$.
To do so, define $$U^+_{R, m} := B_R \cap \left( r_m^{-1}U^+_{i_m}\right) \cap \{x_n > 0\},$$ and suppose that it is well defined by assuming $m$ is large enough so that $Rr_m < 1/2$.
Notice that in $U^+_{R,m}$, $v_m$ satisfies an elliptic equation with drift, $$(-L_{i_m} + b_{i_m}\cdot\nabla) v_m (x) = \frac{r_m}{r_m^\Upsilon\theta(r_m)} f_{i_m} (r_m x)\quad \textrm{in}\quad U^+_{R,m},$$ since we know that $(-L_i+b_i\cdot\nabla) \phi_m = 0$ in $\{x_n > 0\}$. In particular, since $\Upsilon < 1$, the right-hand side converges uniformly to 0 as $r_m \downarrow 0$.
We will now show that $$\label{eq.uimphi}
\|u_{i_m} - \phi_m\|_{L^\infty(B_r \cap (U^-_{i_m} \cup {\mathbb{R}}^n_-)} \leq C\theta(r_m) r^{(1+\alpha)\kappa}\quad\textrm{for all} \quad r < 1/4,$$ and where the constant $C$ is independent of $m$, and $\kappa := \gamma_0\left(1-\frac{\alpha}{16}\right)$. Notice that $\kappa < \gamma_0 - 2\eta_\nu$, so that we can use the supersolution from Proposition \[prop.supersolution\] to get $$|u_{i_m}| \leq C\left({\rm dist}(x, U^-)\right)^{\kappa},$$ with $C$ depending only on $n$, the $C^{1,\alpha}$ norm of $\Gamma$, $\alpha$, the ellipticity constants, and $\|b\|$. On the other hand, by definition of $\phi_m$, $$|\phi_m(x)|\leq C Q_{i_m}(r_m) \left({\rm dist}(x, {\mathbb{R}}^n_-)\right)^{\gamma_i}\leq C\theta(r_m) \left({\rm dist}(x, {\mathbb{R}}^n_-)\right)^{\kappa}\quad \textrm{for all}\quad x\in B_1,$$ where we used . Finally, since the domain is $C^{1,\alpha}$, we have that $${\rm dist}(x, U^-_{i_m}) \leq Cr^{1+\alpha},\quad {\rm dist}(x, {\mathbb{R}}^n_-) \leq Cr^{1+\alpha}\quad\textrm{in}\quad B_r \cap (U^-_{i_m} \cup {\mathbb{R}}^n_-),$$ where the constant $C$ depends only on the $C^{1,\alpha}$ norm of the domain $U^+_{i_m}$, and therefore, it is independent of $m$. Thus, combining the last two expressions we get .
Now, from Proposition \[prop.boundestimatesg\] we have $$\|u_{i_m}\|_{C^\sigma(B_{1/8})} \leq C,$$ uniformly in $m$, for some $\sigma\in (0,\gamma_0)$.
From the regularity of $\phi_m$ this yields, in particular, $$\label{eq.interp1}
\|u_{i_m} - \phi_m\|_{C^\sigma\left(B_r \cap (U^-\cup {\mathbb{R}}^n_-)\right)} \leq C\theta(r_m),$$ where we have used again the bound .
Thus, interpolating and there exists some $\sigma_0 < \sigma$ (depending on $\sigma$, $\gamma_0$, and $\alpha$) such that $$\|u_{i_m} - \phi_m\|_{C^{\sigma_0} (B_r \cap (U^-_{i_m} \cup {\mathbb{R}}^n_-))} \leq C\theta(r_m)r^\Upsilon.$$ Notice that we can do so because $\Upsilon < \kappa(1+\alpha)$. Scaling the previous expression we obtain $$\label{eq.interp2}
\|v_m\|_{C^{\sigma_0} (B_R\setminus U^+_{R,m})} \leq C(R)\quad\textrm{for all }m\textrm{ with } Rr_m< 1/4,$$ for some constant $C(R)$ that depends on $R$, but is independent of $m$.
We now want to apply Proposition \[prop.boundestimatesg\] to $v_m$, rescaled to balls $B_R$. Recall that $$(-L_{i_m} + b_{i_m}\cdot\nabla) v_m (x) = \frac{r_m}{r_m^\Upsilon\theta(r_m)} f_{i_m} (r_m x)\quad \textrm{in}\quad U^+_{R,m},$$ and $v_m$ is $C^{\sigma_0}$ outside $U^+_{R,m}$ by . Notice also that the boundary ${\partial}U^+_{R,m}$ has $C^{1,\alpha}$ norm smaller than the $C^{1,\alpha}$ norm of $\Gamma$ thanks to the fact that we are rescaling with smaller $r_m$ and $Rr_m < 1/4$. Thus, Proposition \[prop.boundestimatesg\] can be applied and we obtain that there exists some $\sigma'>0$ small such that $$\|v_m\|_{C^{\sigma'}(B_{R/2})} \leq C(R)\quad\textrm{for }m\textrm{ with}\quad Rr_m < 1/4.$$ we have again that the constant $C(R)$ depends on $R$, but is independent of $m$; i.e, we have reached a uniform $C^{\sigma'}$ bound on $v_m$ over compact subsets.
Thus, up to taking a subsequence, $v_m$ converge locally uniformly to some $v$.\
[*Step 3: Contradiction.*]{} Up to taking a subsequence if necessary, $L_{i_m}$ converges weakly to some operator $\tilde{L}$ of the form -, and $b_{i_m}$ converges to some $\tilde{b}$ with $\|\tilde{b}\|=\|b\|$. Notice that, in particular, this means that $\gamma_i$ converges to some $\gamma_*\in [\gamma_0,\gamma_0(1+\alpha/8)]$, and $\gamma_*= \gamma(\tilde{b}\cdot e_n/\tilde{\chi})$, where $\tilde{\chi} = \tilde{\chi}(e_n)$ is the associated constant defined as in with the operator $\tilde{L}$.
On the other hand, the domains $U^+_{i_m}$ converge uniformly to ${\mathbb{R}}^n_+$ over compact subsets by construction. Thus, passing all this to the limit, we reach that $v$ satisfies .
Now, passing the growth control to the limit, we reach $$\|v\|_{L^\infty(B_R)}\leq CR^\Upsilon\quad\textrm{for all}\quad R \geq 1,$$ so that we can apply the Liouville theorem in the half space, Theorem \[thm.liouv\], to get $$v(x) = C(x_n)_+^{{\gamma}_*}.$$
Passing to the limit and using this last expression, we obtain $v \equiv 0$. However, by passing to the limit we get $$\|v\|_{L^\infty(B_1)} \geq \frac{1}{2},$$ a contradiction.
The result follows from Proposition \[prop.expansion\] applied to small enough balls so that the condition on $\eta_\nu$ is fulfilled. Notice that the constant $\sigma$ cannot go to 0, because $\tilde\gamma(x_0)$ cannot be made arbitrarily small for a given $L$ and $b$.
Proof of Theorems \[thm.1\] and \[thm.2\] {#sec.8}
=========================================
In this section, we will prove Theorems \[thm.1\] and \[thm.2\]. We already know that if $x_0$ is a regular free boundary point, then the free boundary is $C^{1,\alpha}$ in a neighbourhood. Next, using the results of the previous section, we show that the regular set is open, and that at any regular free boundary point we have below.
\[prop.regopen\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be a solution to --.
Then the set of regular free boundary points is relatively open. Moreover, around each regular point $x_0$ $$\label{eq.ur}
0<c r^{1+\tilde\gamma(x_0)} \leq \sup_{B_r(x_0)} u \leq Cr^{1+\tilde\gamma(x_0)}\quad\textrm{for all}\quad r\leq 1,$$ for some positive constants $c$ and $C$ depending only on $n$, $\|b\|$, and the ellipticity constants. Here, $\tilde\gamma(x_0)$ is given by with $\nu(x_0)$ being the normal vector to the free boundary at $x_0$ pointing towards $\{u > 0\}$.
Suppose without loss of generality that $x_0 = 0$ and $\nu(x_0) = e_n$. The free boundary, $\Gamma$, is $C^{1,\alpha}$ in $B_{r_0}$ for some $\alpha, r_0 > 0$ by Proposition \[prop.C1sigma\]. Apply now Theorem \[thm.expansion\_\] to the partial derivative ${\partial}_n u $ around points $z\in B_{r_0/2}\cap \Gamma$. We obtain $$\label{eq.sigmaappears}
\left|{\partial}_n u (x) - Q(z) \big((x-z)\cdot\nu(z)\big)^{\tilde\gamma(z)}_+\right| \leq C|x-z|^{\tilde\gamma(z) + \sigma},$$ for some $\sigma > 0$, and some constant $C$ independent of $z$.\
[*Step 1: Q is continuous and positive at the origin.*]{} Let us first check that $Q$ is a continuous function on the free boundary at $0$. Indeed, suppose it is not continuous, so that there exists a sequence $z_k \to 0$ on the free boundary such that $\lim_{k\to\infty} Q(z_k) = \bar Q \neq Q(0)$. Then, we have $$\left|{\partial}_n u (x) - Q(z_k) \big((x-z_k)\cdot\nu(z_k)\big)^{\tilde\gamma(z_k)}_+\right| \leq C|x-z_k|^{\tilde\gamma(z_k) +\sigma}.$$
Thus, taking limits as $k \to \infty$, for any fixed $x$, we obtain $$\left|{\partial}_n u (x) - \bar Q (x_n)^{\tilde\gamma(0)}_+\right| \leq C|x|^{\tilde\gamma(0)+\sigma}.$$ We have used here that $\nu$ and $\tilde\gamma$ are continuous. On the other hand, we had $$\left|{\partial}_n u (x) - Q(0) (x_n)^{\tilde\gamma(0)}_+\right| \leq C|x|^{\tilde\gamma(0)+\sigma},$$ so that $$|\bar Q - Q(0)| (x_n)^{\tilde\gamma(0)}_+ \leq C|x|^{\tilde\gamma(0)+\sigma}.$$ Now take $x = (0,t)\in {\mathbb{R}}^{n-1}\times{\mathbb{R}}$ for $t\in{\mathbb{R}}^+$ and let $t\to 0$. It follows $\bar Q = Q(0)$, a contradiction; i.e., $Q$ is continuous at 0.
We now prove that $Q(0) > 0$ (notice that we already know that $Q(0) \geq 0$ because $u \geq 0$). To do so, we proceed by creating an appropriate subsolution using Lemma \[lem.subsolution\].
First of all, consider a fixed bounded strictly convex $C^{1,\alpha}$ domain $P\subset\{u > 0\}$ touching the free boundary at 0, similar to the domains considered in the proof of Proposition \[prop.boundestimatesg\]. Suppose that $P$ has diameter less than 1, and take an $h > 0$ such that, if we denote $\nu_P(z)$ the normal vector to ${\partial}P$ pointing towards the interior of $P$ at $z\in {\partial}P$, then $$\tilde{\gamma}_h := \max\left\{\gamma\left(\frac{b\cdot\nu_P(z)}{\chi(\nu_P(z))}\right) \quad \textrm{for}\quad z\in {\partial}P\cap \{x_n < h\} \right\}\leq \tilde\gamma(0) + \frac{\sigma}{4},$$ where $\sigma$ is the small constant following from Theorem \[thm.expansion\_\] that appears in . Let us call $$\eta_\nu^{(h)} := \tilde{\gamma}_h-\tilde\gamma(0) \geq 0$$ Such $h >0$ exists because $P$ is $C^{1,\alpha}$, and $\gamma$ and $\chi$ are continuous. Take now $\kappa = \tilde\gamma(0) + 3\eta_\nu^{(h)}$, and let $\varrho$ be a regularised distance to ${\mathbb{R}}^n\setminus P$ as in Definition \[defi.rendist\]. In particular, $\varrho \equiv 0$ in ${\mathbb{R}}^n \setminus P$. We will see that $\phi := \varrho^\kappa \leq C{\partial}_n u$ for an appropriate $C$.
By Lemma \[lem.subsolution\] used in $B_h$ we get that for some constant $\delta_0 < h/2$, $$(-L+b\cdot\nabla )\phi \leq -1 \quad \textrm{in}\quad B_{h/2}\cap \{x : 0<d(x, {\mathbb{R}}^n\setminus P) \leq \delta_0\}.$$
Now, since $P$ is strictly convex, we have that there exists some $\delta_P$ with $0< \delta_P\leq \delta_0$ such that $$(-L+b\cdot\nabla )\phi \leq -1 \quad \textrm{in}\quad \{0<x_n <\delta_P\}\cap P.$$
Now consider $v_r$ as the one defined in Proposition \[prop.regpt\] (there it is called $v$), $$v_r(x) = \frac{u(rx)}{r\|\nabla u \|_{L^\infty(B_r)}}.$$
By the same reasoning as in the proof of Proposition \[prop.fblip\] rescaling to a larger ball we have that $$\tilde{w}_r = C_1 ({\partial}_n v_r)\chi_{B_2} \geq 0$$ for $r$ small enough.
From Proposition \[prop.regpt\] we can choose $r$ small enough so that for some positive constant $c$, $$\tilde{w}_{r} > c > 0\quad \textrm{in}\quad P\cap \{x_n \geq \delta_P\}.$$
Moreover, also proceeding as in the proof of Proposition \[prop.fblip\], $(-L+b\cdot \nabla)\tilde{w}_r > -\eta$ in $B_{1}\cap \{v_r > 0\}$ for some arbitrarily small constant $\eta$, making $r$ even smaller if necessary. Thus, we can assume $$(-L+b\cdot \nabla)\tilde{w}_r > -\frac{\tilde{c}}{2}\quad\textrm{in}\quad B_{1}\cap \{v_r > 0\},$$ for some $0<\tilde{c}< c$ to be chosen later.
Now compare the functions $\phi$ and $\tilde{c}^{-1}\tilde{w}_r$. Notice that in ${\mathbb{R}}^n\setminus P$, $\tilde{w}_r \geq \phi \equiv 0$. In $P\cap \{x_n \geq \delta_P\}$, $\tilde{c}$ can be chosen small enough depending on $\delta_P$ and $P$ so that $\tilde{c}^{-1} \tilde{w}_r \geq \phi$ there, because $\tilde{w}_{r} > c > 0$ in $P\cap \{x_n \geq \delta_P\}$. Finally, $$(-L+b\cdot\nabla )\phi \leq (-L+b\cdot\nabla )\tilde{w}_r \quad \textrm{in}\quad \{0<x_n <\delta_P\}\cap P.$$
Thus, by the maximum principle, for this particular $r$ fixed we have that $\tilde{w}_r \geq \tilde{c}\phi$. Going back to the definition of $\tilde{w}_r$, this means that for some $\rho$ and $c$ positive constants $${\partial}_n u(te_n) \geq c \varrho(te_n)\quad \textrm{for}\quad 0<t<\rho.$$ For $\rho$ small enough, $\varrho$ is comparable to $(x_n)_+^\kappa$ along the segment $te_n$, so that we actually have $$\label{eq.orddn}
{\partial}_n u(te_n) \geq c t^\kappa\quad \textrm{for}\quad 0<t<\rho.$$ Now, if $Q(0) = 0$ then $$|{\partial}_n u (x) |\leq C|x|^{\tilde\gamma(0) + \sigma}.$$ Since $\kappa < \tilde\gamma(0)+\sigma$ we get a contradiction with . Thus, $Q(0) > 0$.\
[*Step 2: Conclusion of the proof.*]{} For $z\in \Gamma\cap B_r$ for $r$ small enough we have that $Q(z) > 0$, because $Q$ is continuous and $Q(0) > 0$. In particular, $$\left|{\partial}_n u (x) - Q(z) \big((x-z)\cdot\nu(z)\big)^{\tilde\gamma(z)}_+\right| \leq C|x-z|^{\tilde\gamma(z)+\sigma}.$$
By taking $x = z+te_n$ for $t > 0$ we get $$\left|{\partial}_n u (z+te_n) - Q(z) \big(\nu_n(z) t\big)^{\tilde\gamma(z)}_+\right| \leq Ct^{\tilde\gamma(z)+\sigma}.$$
Integrating with respect to $t$ from $0$ to $t'<1$, using that ${\partial}_nu (z) = 0$ and $\nu_n(z) > 1/2$ for $r$ small enough and recalling that $Q(z)> 0$, we get $$u (z+t'e_n) \geq ct'^{1+\tilde\gamma(z)} > 0 ,$$ so that in particular, $z$ is a regular point; i.e., the set of regular points is relatively open. Doing the same for $z = 0$ we get one of the inequalities from , $$\label{eq.ur2}
\sup_{B_r} u \geq c r^{1+\tilde\gamma(0)} >0 \quad\textrm{for all}\quad r\leq 1.$$
On the other hand, we can also find the expansion at 0 for ${\partial}_i u$ for any $i\in \{1,\dots,n\}$, $$\left|{\partial}_i u (x) - Q_i (x_n)^{\tilde\gamma(0)}_+\right| \leq C|x|^{\tilde\gamma(0)+\sigma}.$$ Therefore, $$|\nabla u (x)|\leq C\left( |x|^{\tilde\gamma(0)} + |x|^{\tilde\gamma(0)+\sigma}\right).$$ Integrating, and using $\nabla u(0) = 0$ $$u (x)\leq C\left( |x|^{1+\tilde\gamma(0)} + |x|^{1+\tilde\gamma(0)+\sigma}\right),$$ i.e., $$\sup_{B_r} u \leq Cr^{1+\tilde\gamma(0)}\quad\textrm{for all}\quad r\leq 1.$$ Thus, combined with , this proves .
\[prop.regopen2\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be a solution to -- and let $x_0$ be a free boundary regular point. Then $$u(x) = c_0\big((x-x_0)\cdot \nu(x_0)\big)^{1+\tilde\gamma(x_0)}_+ + o\left(|x-x_0|^{1+\tilde\gamma(x_0)+ \sigma}\right)$$ with $c_0 > 0$ and for some $\sigma > 0$. Here $\tilde\gamma(x_0)$ is given by , with $\nu(x_0)$ being the normal vector to the free boundary at $0$ pointing towards $\{u > 0\}$; and $\sigma$ depends only on $n$, the ellipticity constants, and $\|b\|$.
Assume that $x_0 = 0$ and $\nu(x_0) = e_n$. From the expansions in the proof of Proposition \[prop.regopen\] we have $$\label{eq.intt}
{\partial}_i u(x) = Q_i(x_n)^{\tilde\gamma(0)}_+ + o\left(|x|^{\tilde\gamma(0)+\sigma}\right),$$ for some $Q_i$, with $Q_n > 0$, and $\sigma > 0$. Now, let $x = (x',x_n)$, with $x' \in {\mathbb{R}}^{n-1}$ and $x_n \in {\mathbb{R}}$. Integrating the expression in the segment with endpoints $0$ and $(x',0)$ we get $$u(x',0) = o\left(|x|^{1+\tilde\gamma(0)+\sigma}\right).$$ Then, integrating in the segment with endpoints $(x',0)$ and $(x',x_n)$ we find $$u(x',x_n) = \frac{Q_n}{1+\tilde\gamma(0)}(x_n)^{1+\tilde\gamma(0)}_+ + o\left(|x|^{1+\tilde\gamma(0)+\sigma}\right).$$ Thus, is proved.
We finally can put all elements together to prove our main results, Theorems \[thm.1\] and \[thm.2\].
After subtracting the obstacle and dividing by a constant, we can assume $u$ is a solution to --. Then the result we want is a combination of Propositions \[prop.C1sigma\], \[prop.regopen\], and \[prop.regopen2\].
It is a particular case of Theorem \[thm.2\]; we only need to check that $\chi \equiv 1$. For this, notice that the kernel is constant and given by $\mu(\theta) = c_{n,1/2}$, where the constant $c_{n,s}$ is the one appearing in the definition of fractional Laplacian, $$c_{n,s} := \left(\int_{{\mathbb{R}}^n} \frac{1-\cos(x_1)}{|x|^{n+2s}}dx\right)^{-1};$$ see for example [@DPV12]. Thus, the value of $\chi$ for $(-\Delta)^{1/2}$ is $$\chi(e) = \frac{\pi c_{n,1/2}}{2}\int_{{\mathbb{S}}^{n-1}} |\theta\cdot e| d\theta.$$
Notice that, by changing variables to polar coordinates, $$\begin{aligned}
c_{n,1/2}^{-1} = \int_{{\mathbb{R}}^n} \frac{1-\cos(x_1)}{|x|^{n+1}}dx = \int_{{\mathbb{S}}^{n-1}}\int_0^\infty \frac{1-\cos(r\theta_1)}{r^2} dr d\theta = \frac{\pi}{2}\int_{{\mathbb{S}}^{n-1}}|\theta_1|d\theta,\end{aligned}$$ where we have used that $\int_0^\infty (1-\cos(t))t^{-2} dt = \pi/2$. This immediately yields that $\chi \equiv 1$ for $(-\Delta)^{1/2}$, as desired.
We next prove the almost optimal regularity of solutions. Given an operator $L$ of the form -, the associated $\chi$ defined as in , and $b\in {\mathbb{R}}^n$, we define $$\label{eq.mingamma}
\gamma^-_{L,b} := \inf_{e\in {\mathbb{S}}^{n-1}} \gamma\left(\frac{b\cdot e}{\chi(e)}\right),$$ where $\gamma$ is given by . Notice that $\gamma_{L,b}^- \in (0,1/2]$.
\[prop.almostoptimal\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be a solution to --. Then, for any $\varepsilon > 0$, $$\|u\|_{C^{1,\gamma^-_{L,b}-\varepsilon}({\mathbb{R}}^n)} \leq C_\varepsilon,$$ where the constant $C_\varepsilon$ depends only on $n$, $L$, $b$, and $\varepsilon$. The constant $\gamma^-_{L,b}$ is given by .
In order to prove the bound we first check the growth of the solution at the free boundary, and then we combine it with interior estimates.
For simplicity, we will denote $\gamma_\varepsilon = \gamma_{L,b}^- - \varepsilon$.
[*Step 1: Growth at the free boundary.*]{} We first prove that, if 0 is a free boundary point, then $$\label{eq.firststep}
\sup_{r > 0} \frac{\|\nabla u\|_{L^\infty(B_r)}}{r^{\gamma_\varepsilon}} \leq C,$$ for some constant $C$ depending only on $n$, $L$, $b$, and $\varepsilon$.
We proceed by contradiction, using a compactness argument. Suppose that it is not true, so that there exists a sequence of functions $u_k$, $f_k$, with $\|u_k\|_{C^{1,\tau}}\leq 1$ for some $\tau > 0$ fixed and $\|f_k\|_{C^1({\mathbb{R}}^n)}\leq 1$, such that $$\left\{\begin{array}{rcll}
u_k & \geq & 0 & \quad \textrm{in}\quad {\mathbb{R}}^n\\
(-L+b\cdot\nabla) u_k & \leq & f_k & \quad \textrm{in}\quad {\mathbb{R}}^n \\
(-L+b\cdot\nabla) u_k & = & f_k & \quad \textrm{in}\quad \{u_k>0\} \\
D^2u_k & \geq & -1 & \quad \textrm{in} \quad {\mathbb{R}}^n,\\
\end{array}\right.$$ but $u_k$ are such that $$\theta(r) := \sup_{i}\, \sup_{r' > r}\,\,(r')^{-\gamma_\varepsilon} \|\nabla u_k\|_{L^\infty(B_{r'})} \to \infty \quad\textrm{as}\quad r\downarrow 0.$$
Notice that for $r > 0$, $\theta(r) < \infty$ and that $\theta$ is a monotone function, with $\theta(r) \to \infty$ as $r \downarrow 0$. Now take sequences $r_m \downarrow 0$ and $i_m$ such that $$r_m^{-\gamma_\varepsilon}\|\nabla u_{i_m}\| \geq \frac{\theta(r_m)}{2},$$ and define the functions $$v_m(x) := \frac{u_{i_m}(r_m x)}{r_m^{1+\gamma_\varepsilon}\theta(r_m)}.$$
Notice that $$\label{eq.nondegfin}
\|\nabla v_m\|_{L^\infty(B_1)} \geq \frac{1}{2},$$ and $$\label{eq.clasfin}
D^2 v_m \geq -\frac{r_m^{1-\gamma_\varepsilon}}{\theta(r_m)}\quad\textrm{in}\quad {\mathbb{R}}^n,\quad\quad |(L+b\nabla)(\nabla v_m)| \leq \frac{r_m^{1-\gamma_\varepsilon}}{\theta(r_m)}\quad\textrm{in}\quad \{v_m > 0\}.$$
On the other hand, $$\label{eq.grofin}
\|\nabla v_m\|_{L^\infty(B_R)} = \frac{\|\nabla u_{i_m}\|_{L^\infty(B_{Rr_m})}}{r_m^{\gamma_\varepsilon}\theta(r_m)} \leq R^{\gamma_\varepsilon} \frac{\theta(Rr_m)}{\theta(r_m)} \leq R^{\gamma_\varepsilon}\quad\textrm{for}\quad R \geq 1.$$
Therefore, noticing that $r_m^{1-\gamma_\varepsilon}/\theta(r_m) \to 0$ as $m\to \infty$, we can apply Proposition \[prop.reg.u\] to deduce that, for some $\tau > 0$ independent of $m$, $$\|v_m\|_{C^{1,\tau}(B_R)}\leq C(R),$$ for some constant depending on $R$, $C(R)$. Let us take limits as $m\to \infty$. By Arzelà-Ascoli, $v_m$ converges, up to taking a subsequence, in $C^{1}_{\rm loc}({\mathbb{R}}^n)$ to some $v_\infty$. By taking to the limit the properties - we reach that $v_\infty$ should be a convex global solution. By the classification theorem, Theorem \[thm.clas\], we have that either $v\equiv 0$ $$v_\infty(x) = C(e\cdot x)_+^{1+\gamma(b\cdot e/\chi(e))}\quad\textrm{for some}\quad e\in {\mathbb{S}}^{n-1},$$ where $\gamma$ and $\chi$ are given by -. Notice, however, that taking to the limit, $v_\infty$ grows at most like $\gamma_\varepsilon$, and by definition $\gamma(b\cdot e/\chi(e)) > \gamma_\varepsilon$. Therefore, we must have $v_\infty \equiv 0$. But this is a contradiction with in the limit. Therefore, we have proved .
[*Step 2: Conclusion.*]{} Let us combine the previous growth with interior estimates to obtain the desired result.
Let $x, y\in {\mathbb{R}}^n$, let $r = |x-y|$ and $R = {\rm dist}(x, \{u = 0\})$. We want to prove that for some constant $C_\varepsilon$ then $$|\nabla u(x)-\nabla u(y)|\leq Cr^{\gamma_\varepsilon}.$$
Without loss of generality and by the growth found in the first step we can assume that $x, y\in \{u > 0\}$. Let $\bar x\in {\partial}\{u = 0\}$ be such that ${\rm dist}(\bar x, x) = R$. We separate two cases:
1. If $4r > R$, $$\begin{aligned}
|\nabla u (x)-\nabla u(y)|& \leq |\nabla u (x)-\nabla u(\bar x)|+|\nabla u (\bar x)-\nabla u(y)| \\
& \leq C\big(R^{\gamma_\varepsilon} + (R+r)^{\gamma_\varepsilon}\big) \leq Cr^{\gamma_\varepsilon},\end{aligned}$$ where we have used the growth found in Step 1.
2. If $4r \leq R$, then $x, y \in B_{R/2}(x)$, and $B_R(x)\subset \{u > 0\}$. Notice that we have $$(-L+b\cdot\nabla)(\nabla u) = \nabla f\quad\textrm{in}\quad B_R(x).$$ From the interior estimates in Proposition \[prop.intest\] rescaled, we have $$R^{\gamma_\varepsilon} [\nabla u]_{C^{\gamma_\varepsilon} (B_{R/2}(x))} \leq C\left(R \|\nabla f\|_{L^\infty(B_R(x))} + \|\nabla u\|_{L^\infty(B_R(x))}+ \int_{{\mathbb{R}}^n} \frac{|\nabla u (Rx)|}{1+|x|^{n+1}}\right).$$
Now notice that thanks to the growth found in Step 1 we have, on the one hand, $$\|\nabla u\|_{L^\infty(B_R(x))} \leq CR^{\gamma_\varepsilon},$$ and on the other hand, $$\int_{{\mathbb{R}}^n} \frac{|\nabla u (Rx)|}{1+|x|^{n+1}} \leq R^{\gamma_\varepsilon} \int_{{\mathbb{R}}^n} \frac{|x|^{\gamma_\varepsilon}}{1+|x|^{n+1}} = CR^{\gamma_\varepsilon},$$ so that putting all together and using $\|\nabla f\|_{L^\infty({\mathbb{R}}^n)} \leq 1$, it yields, $$[\nabla u]_{C^{\gamma_\varepsilon} (B_{R/2}(x))} \leq C\left(1+R^{1-\gamma_\varepsilon}\right).$$ Thus, if $R \leq 4$ we are done. Now suppose $R > 4$. If $r < 1$, by applying interior estimates to $B_1(x)$ we are done. If $r \geq 1$, we are also done, because $|\nabla u(x)-\nabla u(y)|\leq 2\|\nabla u\|_{L^\infty({\mathbb{R}}^n)} \leq C$.
Thus, we have reached the desired result.
As a consequence, we have the following immediate corollary.
\[cor.2\] Let $L$ be an operator of the form -, and let $b\in {\mathbb{R}}^n$. Let $u$ be the solution to for a given obstacle $\varphi$ of the form . Then, for any $\varepsilon > 0$, $$\|u\|_{C^{1,\gamma_{L,b}^- - \varepsilon}({\mathbb{R}}^n)}\leq C_\varepsilon,$$ where $C_\varepsilon$ depends only on $n$, $L$, $b$, $\varepsilon$, and $\|\varphi\|_{C^{2,1}({\mathbb{R}}^n)}$. The constant $\gamma_{L,b}^-$ is given by .
After subtracting the obstacle and dividing by an appropriate constant, we can apply Proposition \[prop.almostoptimal\] and the result follows.
Finally, we prove Corollary \[cor.1\].
After subtracting the obstacle and dividing by a constant, we get that this result is a particular case of Proposition \[prop.almostoptimal\], but the constant $C_\varepsilon$ depends on $b$ and not only on $\|b\|$.
To prove that $C_\varepsilon$ actually depends on $\|b\|$, the proof of Proposition \[prop.almostoptimal\] can be rewritten by taking also sequences of vectors $b_k\in {\mathbb{R}}^n$ with $\|b_k\| = \|b\|$; by compactness, up to a subsequence they converge to some $\tilde b$ with $\|\tilde b\| = \|b\|$ and the rest of the proof is the same.
A nondegeneracy property {#sec.9}
========================
In the obstacle problem for the fractional Laplacian (without drift), in [@BFR15], Barrios, Figalli and the second author proved a non-degeneracy condition at all free boundary points for obstacles satisfying $\Delta \varphi \leq 0$. From this, and by means of a Monneau-type monotonicity formula, they establish a global regularity result for the free boundary.
In the obstacle problem with critical drift for the fractional Laplacian we can actually find a non-degeneracy result analogous to the one found in [@BFR15]. In this case, however, we cannot establish regularity of the singular set, since we do not have (and do not expect) any monotonicity formula for this problem.
\[prop.nondeg\] Let $b \in {\mathbb{R}}^n$, and suppose that $\varphi\in C^{1,1}({\mathbb{R}}^n)$. Assume that $\varphi$ is concave in $\{\varphi > 0\}$ or, more generally, that $$(\Delta+{\partial}_{bb}^2)\,\varphi \leq 0 \quad\textrm{in} \quad\{\varphi > 0\},\quad\varnothing \neq \{\varphi > 0\} \Subset {\mathbb{R}}^n.$$ Let $u$ be a solution to the obstacle problem . Then, there exist constants $c, r_0 > 0$ such that for any $x_0$ a free boundary point then $$\sup_{B_r(x_0)} (u-\varphi) \geq c r^2\quad\textrm{for all}\quad 0 < r < r_0.$$
Let $w := \big( (-\Delta)^{1/2} + b\cdot \nabla\big) u$, so that $w \geq 0$. If $w\equiv 0$, by the interior estimates rescaled, and using that $u$ is globally bounded, we reach $u$ is constant. From $\lim_{|x|\to \infty} u(x) = 0$ we would get $u \equiv 0$, but this is a contradiction with $\varnothing \neq \{\varphi > 0\}$. Thus, $w\not\equiv 0$.
Notice, however, that $w\equiv 0$ in $\{u > \varphi\}$. In particular, given $\bar x\in \{u > \varphi\}$, then $\nabla w(\bar x) = 0$ and $w$ has a global minimum at $\bar x$, so that $$\big((-\Delta)^{1/2} - b\cdot\nabla\big) w (\bar x) = (-\Delta)^{1/2} w (\bar x) < 0.$$
Now, noticing that $\{\varphi > 0\}\Subset{\mathbb{R}}^n$, we get that by compactness there are some $\bar c, \bar r> 0$ such that for any $\bar x\in \{u > \varphi\}$ with ${\rm dist}(\bar x, \{u = \varphi\}) \leq \bar r$ then $$\big((-\Delta)^{1/2} - b\cdot\nabla\big) w (\bar x) \leq -\bar c < 0.$$
Now, since $\big((-\Delta)^{1/2} +b\cdot \nabla\big) u = w$ in ${\mathbb{R}}^n$ and from the semigroup property of the fractional Laplacian, $$-\Delta u - b_ib_j{\partial}_{ij}u = \big((-\Delta)^{1/2} - b\cdot\nabla\big) w \leq -\bar c\quad\textrm{in}\quad\bar U,$$ where $\bar U := \{u > \varphi\}\cap \{{\rm dist}(\cdot, \{u = \varphi\}) \leq \bar r \}$. Note that the operator $\Delta + b_ib_j{\partial}_{ij}$ is uniformly elliptic, with ellipticity constants 1 and $1+\|b\|^2$.
Since $u > 0$ on the contact set, by compactness there exists some $h > 0$ such that $\varphi \geq h$ in $\{u = \varphi\}$. By continuity, there exists some $0<r_0 < \bar r/2$ such that $$\varphi > 0 \quad\textrm{in}\quad U_0 := \{u > \varphi\}\cap \{{\rm dist}(\cdot, \{u = \varphi\}) \leq 2r_0 \}.$$
Now let $\bar x \in U_0$ with ${\rm dist}(\bar x, \{u = \varphi\}) \leq r_0$, and consider $r\in (0,r_0)$. From the condition on $\varphi$, $(\Delta+{\partial}_{bb}^2) \varphi \leq 0 \textrm{ in } \{\varphi > 0\}$, we get that if $\bar u := u - \varphi$ then $$(\Delta+{\partial}_{bb}^2)\,\bar u \geq \bar c>0\quad\textrm{in}\quad \{\bar u > 0\}\cap B_r(\bar x)\subset U_0.$$ Therefore, if we define $$v := \bar u - \frac{\bar c}{2(n+\|b\|^2)}|x-\bar x|^2\quad\textrm{in}\quad \{\bar u > 0\}\cap B_r(\bar x),$$ then $$(\Delta+{\partial}_{bb}^2) v \geq 0.$$
By the maximum principle, if $\Omega_r := \{\bar u > 0\}\cap B_r(\bar x)$ then $$0< \bar u (x_1) \leq \sup_{\Omega_r}\,v = \sup_{{\partial}\Omega_r}\,v.$$ Since $v < 0$ in ${\partial}\{\bar u > 0\} \cap B_r(\bar x)$, $$0 < \sup_{\{\bar u > 0\} \cap {\partial}B_r(\bar x)} v \leq \sup_{{\partial}B_r(\bar x)}\bar u - c r^2,$$ where $c = \frac{\bar c}{2(n+\|b\|^2)}$. Therefore, $c$ is independent of $\bar x$, and we can let $\bar x \to x_0$, to obtain the desired result.
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[^1]: The first author is supported by ERC grant “Regularity and Stability in Partial Differential Equations (RSPDE)” and a fellowship from “Obra Social la Caixa”. The second author is supported by NSF Grant DMS-1565186 and by MINECO Grant MTM-2014-52402-C3-1-P (Spain).
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abstract: 'Let $(M^n, g)$ be a compact Kähler manifold with nonpositive bisectional curvature, we show that a finite cover is biholomorphic and isometric to a flat torus bundle over a compact Kähler manifold $N^k$ with $c_1 < 0$. Moreover, $k$ is the Kodaira dimension of $M$ which is also the maximal rank of the Ricci tensor of $g$ . This confirms a conjecture of Yau. We also prove a global splitting result under some conditions on immersed complex submanifolds.'
address: |
Department of Mathematics\
University of Minnesota\
Minneapolis, MN 55455
author:
- Gang Liu
title: '**Compact Kähler manifolds with nonpositive bisectional curvature**'
---
**[Introduction]{}**
====================
The uniformization theorem of Riemann surfaces says the sign of curvature could determine the conformal structure in some sense. Explicitly, if the curvature is positive, it is covered by either $\mathbb{P}^1$ or $\mathbb{C}$. On the other hand, if the curvature is less than a negative constant, it is covered by the unit disk $\mathbb{D}^2$.
It is natural to wonder whether there are generalizations in higher dimensions. For the compact case, the famous Frankel conjecture says if a compact Kähler manifold has positive holomorphic bisectional curvature, then it is biholomorphic to $\mathbb{CP}^n$. This conjecture was solved by Mori [@[Mo]] and Siu-Yau [@[SiY]] independently. In fact Mori proved the stronger Hartshorne conjecture. Later, Mok [@[Mok]] solved the generalized Frankel conjecture, the result says that, if a compact Kähler manifold has nonnegative holomorphic bisectional curvature, then the universal cover is isometric-biholomorphic to $(\mathbb{C}^k, g_0)\times (\mathbb{P}^{n_1}, \theta_1)\times\cdot\cdot\cdot\times(\mathbb{P}^{n_l}, \theta_l)\times(M_1, g_1)\times\cdot\cdot\cdot\times(M_i, g_i)$, where $g_0$ is flat; $\theta_k$ are metrics on $\mathbb{P}^{n_k}$ with nonnegative holomorphic bisectional curvature; $(M_j, g_j)$ are compact irreducible Hermitian symmetric spaces.
If the curvature is negative, the current knowledge is much less satisfactory. For example, a famous conjecture of Yau says if a simply connected complete Kähler manifold has sectional curvature between two negative constants, then it is a bounded domain. So far, it is not even known whether there exists a nontrivial bounded holomorphic function on such manifolds.
As in the Riemannian case, it is often important to understand the difference between the negative curved case and nonpositive case. The former tends to be hyperbolic in some sense, while the latter usually possesses some rigidity properties. For compact Kähler manifolds with nonpositive holomorphic bisectional curvature, there is a conjecture of Yau:
Let $M^n$ be a compact Kähler manifold with nonpositive holomorphic bisectional curvature. Then there exists a finite cover $M'$ of $M$ such that $M'$ is a holomorphic and metric fibre bundle over a compact Kähler manifold $N$ with nonpositive bisectional curvature and $c_1(N) < 0$, and the fibre is a flat complex torus.
Recall that the fiber bundle $M\to N$ is called a metric bundle, if for any $p\in N$, there is some neighborhood $p\in U \subset N$ such that the bundle over $U$ is isometric to the product of the fiber and $U$. In [@[Y]], Yau proved the following
\[thm2\] Let $M$ be a compact complex submanifold of a complex torus $T^n$. Then $M$ is a torus bundle over a complex submanifold $N$ in $T^n$, such that the induced Kähler metric on $N$ has negative definite Ricci tensor in an open dense set of $N$.
Since complex submanifolds in $T^n$ has nonpositive holomorphic bisectional curvature, Yau’s theorem confirms the conjecture when $M$ is a complex submanifold of $T^n$. Zheng [@[Z]] proved this conjecture under the extra assumption $M$ has nonpositive sectional curvature and the metric is real analytic. In [@[WZ]], Wu and Zheng proved this conjecture by only assuming that the metric is real analytic. They first proved a local splitting by a careful study of the foliation at the points where the ricci curvature has the maximal rank. By real analyticity, the foliation could be extended to the whole manifold. In this note we confirm the conjecture above.
\[thm1\] Let $(M^n, g)$ be a compact Kähler manifold with nonpositive holomorphic bisectional curvature. Then there exists a finite cover $\hat{M}$ of $M$ such that $\hat{M}$ is a holomorphic and metric fibre bundle over a compact Kähler manifold $N^k$ with nonpositive bisectional curvature and $c_1(N) < 0$, and the fibre is a flat complex torus $T$. Furthermore, $\hat{M}$ is diffeomorphic to $T\times N$. Finally, let $r$ be the maximal rank of the Ricci curvature of $g$, then $r = k$.
$dim(N)=Kod(M)$, the Kodaira dimension of $M$.
We also prove the following
\[thm5\] Let $M^n$ be a compact Kähler manifold with nonpositive holomorphic bisectional curvature. Suppose $N^k \subset M$ is a complete(compact or noncompact) immersed complex submanifold of $M$ which is flat and totally geodesic. If in addition, $Ric(M)|_{TN} = 0$, then $M$ splits globally, i.e, the universal cover $\tilde{M}$ is isometric and biholomorphic to $\mathbb{C}^k \times T^{n-k}$ where $T^{n-k}$ is a complete Kähler manifold of dimension $n-k$.
All conditions in theorem $\ref{thm5}$ are “local" around $N$, except that the holomorphic bisectional curvature on $M$ is nonpositive, thus it might be interesting to see that local conditions imply global splitting. Theorem \[thm5\] also holds if we assume the manifold has nonnegative bisectional curvature. We can also weaken the condition by assuming that $M$ is complete with bounded curvature. The condition that $Ric$ vanishes along the tangent of $N$ is necessary. For instance, if $M$ is a compact locally symmetric Hermitian space with rank greater than $1$ covered by an irreducible bounded symmetric domain, then there is a totally geodesic, flat complex submanifold immersed in $M$, however, $M$ does not split.
In [@[WZ]], Wu and Zheng studied the foliation given by the kernel of the Ricci tensor at the points where Ric has the maximal rank. In [@[F]], Ferus showed that the leaves are complete. The following corollary can be regarded as the converse in some sense.
\[cor3\] Let $M^n$ be a compact Kähler manifold with nonpositive bisectional curvature. Define $U(i) = \{x\in M| rank(Ric(x)) = i\}$. Let $p$ be an interior point of $U(i)$, then there is a foliation near $p$ by the kernel of the Ricci curvature. If the leaf through $p$ extends to a complete leaf $L\subset U(i)$ which is the kernel of the Ricci curvature, then $i$ is the maximal rank of the Ricci curvature.
Next we discuss two applications of theorem \[thm1\]. The existence of canonical metric is a central topic in Kähler geometry. Yau [@[Y]] solved the famous Calabi conjecture. He proved that any Kähler manifold with $c_1 < 0$ or $c_1 = 0$ admits a unique Kähler-Einstein metric. Aubin [@[Au]] also obtained the proof when $c_1 < 0$. If a Kähler manifold has nonpositive holomorphic bisectional curvature, it is natural to ask whether there exists canonical metrics.
\[cor1\] Let $(M^n, g)$ be a compact Kähler manifold with nonpositive holomorphic bisectional curvature. Then the manifold admits a canonical metric which is locally a product of a flat metric and a Kähler-Einstein metric with negative scalar curvature. More precisely, the manifold is locally biholomorphic and isometric to $(D^{n-k}, g_1) \times (U^k, g_2)$, where $k = Kod(M)$ and $(D^{n-k}, g_1)$ is a flat complex Euclidean ball with a small radius and $(U^k, g_2)$ is an small ball with Kähler-Einstein metric such that $Ric(g_2)=-g_2$.
According to theorem \[thm1\], there exists a finite cover $\hat{M}$ of $M$ such that there exists a flat fibration $T^{n-k}\to \hat{M}\to N$. The universal cover $\tilde{M}$ is biholomorphic to $\mathbb{C}^{n-k}\times \tilde{N}$ where $\tilde{N}\to N$ is the universal covering. Since $c_1(N) < 0$, $N$ admits a unique Kähler-Einstein metric $g_2$, thus $\tilde{N}$ admits a complete Kähler-Einstein metric with negative scalar curvature. Any element $a\in \pi_1(M)$ induces a deck transformation $f$ on $\tilde{M}$ which descends to a biholomorphism of $\tilde{N}$. By Yau’s Schwartz lemma [@[Y1]], the Kähler-Einstein metric on $\tilde{N}$ is unique. Thus $f$ preserves the Kähler-Einstein metric $g_2$ on $\tilde{N}$. Therefore, the product metric $\mathbb{C}^{n-k}\times (\tilde{N}, g_2)$ descends to a metric to $M$ which is canonical.
We have to lift the metric to the universal cover. Since $\hat{M}$ is not necessarily a regular covering of $M$, there might be no deck transformation on $\hat{M}$.
It is also interesting to analyze the long time behavior of the normalized Kähler-Ricci flow $$\frac{\partial g_{i\overline{j}}}{\partial t} = -R_{i\overline{j}}-g_{i\overline{j}}$$ on such manifolds. Cao [@[C]] proved that if a manifold $(M, \omega)$ has $c_1 < 0$ or $c_1 = 0$ and $c_1 = \lambda [\omega]$, then the Kähler-Ricci flow will converge to the unique Kähler-Einstein metric. Tsuji [@[Ts]] and Tian-Zhang [@[TZ]] proved that if a Kähler manifold has $c_1\leq 0$, then the normalized Kähler-Ricci flow has long time existence. In [@[ST1]], Song and Tian considered the normalized Kähler-Ricci flow on an elliptic surface $f: X\to\Sigma$ where some of the fibers may be singular. It was shown that the solution of the flow converges to a generalized Kähler-Einstein metric. This result was generalized in [@[ST2]] to the fibration $f: X \to X_{can}$ where $X$ is a nonsingular algebraic variety with semi-ample canonical bundle and $X_{can}$ is its canonical model . We have a result with the similar spirit below,
Let $M^n$ be a compact Kähler manifold with nonpositive bisectional curvature, then the for any initial Kähler metric $g(0)$ on $M$, the normalized Kähler-Ricci flow will converge in $C^{\infty}$ to the Kähler-Einstein metric which is a factor in the canonical metric of $M^n$ in corollary \[cor1\].
Taking $\hat{M}$ in theorem \[thm1\], we consider the normalized Kähler-Ricci flow on $\hat{M}$ which is diffeomorphic to $T\times N$. Recall a theorem of M. Gill [@[G]] which generalizes a theorem in [@[SW]] by Song and Weinkove,
\[thm3\] Let $X = Y\times T$ where $Y$ is a Kähler manifold with negative first Chern class and $T$ be a complex torus. Let $\omega(t)$ be the normalized Kähler-Ricci flow on $X$ with any initial metric $\omega(0)$, then $\omega(t)$ converges to $\pi^*(\omega_Y)$ in $C^{\infty}(X, \omega_0)$ sense as $t\to \infty$ where $\pi: X\to Y$ is the projection and $\omega_Y$ is the Kähler-Einstein metric on $Y$.
Note that $\hat{M}$ is not necessarily biholomorphic to $T\times N$. However, $\hat{M}$ is locally biholomorphic to $T\times U$ where $U$ is an open set in $N$, thus there is a flat metric $\omega_T$ on the fibre independent of the projection to $N$. Then one can check that the proof of theorem \[thm3\] in [@[G]] works for this case without any modification.
The proof of theorem \[thm1\] uses Hamilton’s Ricci flow [@[H1]] and Hamilton’s maximum principle for tensors([@[H2]][@[CL]][@[BW]]), together with some argument in [@[WZ]] by Wu and Zheng. We will use the invariant convex set constructed in [@[BW]] by Böhm and Wilking. The key point is to prove that there exists a small $\epsilon > 0$ such that the after the Ricci flow, $Ric(g_t) \leq 0$ for all $0 < t < \epsilon$(note that the holomorphic bisectional curvature is not necessarily nonpositive for small $t$). The final assertion $rank(Ric(g_0)) = k$ will follow from argument of Yu [@[Yu]].
There is a general philosophy that the Ricci flow makes the curvature towards positive, e.g, Hamilton-Ivey pinching estimate [@[H3]][@[I]]. So it might be interesting to see that in our case, at least in a short time, the Ricci curvature remains nonpositive.
Acknowledgements {#acknowledgements .unnumbered}
================
The author would like to express his deep gratitude to his advisor, Professor Jiaping Wang, for his kind help and useful suggestions. He also thanks Professor Fangyang Zheng for his interest in this note. Special thanks also go to Guoyi Xu, Bo Yang and Yuan Yuan for their helpful comments.
**[The proof of theorem \[thm1\]]{}**
=====================================
Let $g(t)$ be the solution to the Ricci flow equation $\frac{\partial g(t)}{\partial t} = -2Ric(g(t))$ with $g(0) = g$. Following Böhm and Wilking in [@[BW]], we shall construct a family of convex sets $V_t$ which are invariant under parallel transport such that the curvature tensor of $g(t)$ lies inside $V_t$ for small $t$.
\[prop\] Let $V_t$ be a family of Kähler algebraic curvature operators satisfying the following conditions:
(1). $Ric(\alpha, \overline{\alpha}) \leq 0$ for any $\alpha \in T^{1,0}M$.
(2). $|R_{x\overline xu\overline v}|^2 \leq (1+tK_1)Ric(u, \overline u)Ric(v, \overline v)$ for any $x , u , v \in T^{1,0}M$ and $|x|_{g(t)} = 1$.
(3).$||R|| \leq K_2 + tK_3$.
Then for suitable positive constants $K_1, K_2, K_3$, there exists a $\epsilon > 0$ such that the $V_t$ is invariant under the Ricci flow for $0 \leq t < \epsilon$. Here $R$ stands for the curvature operator.
First, we prove $V_t$ is a convex for each $t$. It is easy to see that condition (1) and (3) defines a convex set. For condition (2), suppose $R, S$ are two tensors satisfying (1), (2), then for any $0 \leq \lambda \leq 1$, define $$T = \lambda R + (1-\lambda)S.$$ $$\begin{aligned}
|T_{x\overline xu\overline v}|^2 &= |\lambda R_{x\overline xu\overline v} + (1-\lambda)S_{x\overline xu\overline v}|^2 \\&\leq (1 + tK_1)| \lambda \sqrt{Ric_R(u, \overline u)Ric_R(v, \overline v)} + (1-\lambda)\sqrt{Ric_S(u, \overline u)Ric_S(v, \overline v)}|^2 \\&\leq (1 + tK_1)(\lambda Ric_R(u, \overline u) + (1-\lambda) Ric_S(u, \overline u))(\lambda Ric_R(v, \overline v) + (1-\lambda) Ric_S(v, \overline v))\\&=(1+tK_1)Ric_T(u, \overline{u})Ric_T(v, \overline{v}).
\end{aligned}$$ Therefore, $V_t$ is convex.
Now let us check that when $t = 0$, the curvature tensor $R_0$ of $(M^n, g)$ is in $V_0$. If we choose $K_2$ very large, then (1) and (3) hold. To check (2), we notice that for fixed $x$, $R_{x\overline{x}p\overline{q}}$ is a Hermitian form. Let $e_i$ be the eigenvectors where $i = 1, 2,..n$ and $$R_{x\overline{x}e_i\overline{e_j}} = \delta_{ij}\lambda_i$$ where $\lambda_i$ are all nonpositive. Suppose $u = \sum\limits_{i=1}^{n}u_ie_i, v = \sum\limits_{i=1}^{n}v_ie_i$, then $$\begin{aligned}
|R_{x\overline{x}u\overline{v}}|^2 &= |\sum\limits_{i=1}^{n}u_i\overline{v_i}\lambda_i|^2 \\&\leq (\sum\limits_{i=1}^{n}|u_i\sqrt{-\lambda_i}|^2)(\sum\limits_{i=1}^{n}|\overline{v_i}\sqrt{-\lambda_i}|^2) \\&= R_{x\overline{x}u\overline{u}}R_{x\overline{x}v\overline{v}} \\&\leq Ric(u, \overline{u})Ric(v, \overline{v}).
\end{aligned}$$
Let us state Hamilton’s maximum principle for tensors. Let $M^n$ be a closed oriented manifold with a smooth family of Riemannian metric $g(t)$, $t \in [0, T]$. Let $V\rightarrow M$ be a real vector bundle with a time dependent metric $h$ and $\Gamma(V)$ be the vector space of smooth sections on $V$. Let $\nabla^L_t$ denote the corresponding Levi-Civita connection on $(M, g(t))$. Furthermore, let $\nabla_t$ denote a time dependent metric connection on $V$. For a section $R\in \Gamma(V)$, define a new section $\Delta_t R\in \Gamma(V)$ as follows. For $p\in M$ choose an orthonormal basis of $V_p$(the fiber of $V$ at $p$) and extend it along the radial geodesics in $(M, g(t))$ emanating from $p$ by parallel transport of $\nabla_t$ to an orthonormal basis $X_1(q)$, ..., $X_d(q)$ of $V_q$ for all $q$ in a small neighborhood of $p$. If $f_i$ satisfies $R = \sum\limits_{i=1}^d f_iX_i$, then $$(\Delta_tR)(p) = \sum\limits_{i=1}^{d}(\Delta_tf_i)X_i(p)$$ where $\Delta_t$ is the Beltrami Laplacian on functions.
Suppose that a time dependent section $R(\cdot, t)\in \Gamma(V)$ satisfies the parabolic equation $$\label{1}
\frac{\partial R(p, t)}{\partial t} = (\Delta_tR)(p, t) + f(R(p, t))$$ where $f: V\rightarrow V$ is a local Lipschitz map mapping each fibre $V_q$ to itself. Roughly speaking, Hamilton’s maximum principle says that the dynamics of the parabolic equation (\[1\]) is controlled by the ordinary differential equation $$\label{2}
\frac{d R}{d t} = f(R(p, t)).$$
More precisely, we have the following version of Hamilton’s maximum principle in [@[BW]] and [@[CL]]:
\[thm2\] For $t\in [0, \delta]$, let $C(t)\subseteq V$ be a closed subset, depending continuously on $t$. Suppose that each of the sets $C(t)$ is invariant under parallel transport, fiberwise convex and that the family of $C(t)(0\leq t\leq \delta)$ is invariant under the ordinary differential equation (\[2\]). Then for any solution $R(p, t)\in \Gamma(V)$ on $M\times [0, \delta]$ of parabolic equation (\[1\]) with $R(\cdot, 0)\in C(0)$, we have $R(\cdot, t)\in C(t)$ for all $t\in [0, \delta]$.
Let us go back to the proof of proposition \[prop\]. In view of theorem \[thm2\], we just need to prove that $V(t)$ is invariant under the ODE equation of the curvature operator, i.e, we drop the Laplacian in the evolution equation of the curvature operator. For any $R(0) \in V_0$, we consider perturbation $R_{\lambda}(0) = R(0) - \lambda R'$ for the initial condition of the ODE, where $\lambda$ is a small positive number and $R'$ is the curvature tensor with holomorphic sectional curvature $1$. For simplicity, when $\lambda$ is fixed, we use $R$ to denote the solution to the ODE with initial condition $R_\lambda(0)$.
There exist positive constants $\epsilon, A, K_1, K_2, K_3$ which are independent of $\lambda$ such that $\epsilon K_1 \leq 1$ and for any $t \in [0, \epsilon]$, the solution $R$ satisfies
(1’). $Ric(\alpha, \overline{\alpha}) \leq -\frac{\lambda}{2}e^{-At}$ for any $e_\alpha \in T^{1,0}M$ and $|e_\alpha|_{g(t)} = 1$.
(2’). $|R_{x\overline xu\overline v}|^2 \leq (1+tK_1)Ric(u, \overline u)Ric(v, \overline v)$ for any $x , u , v \in T^{1,0}M$ and $|x|_{g(t)} = 1$.
(3’). $||R|| \leq K_2 + tK_3$.
We can find $B > 0$ such that $||R|| \leq B$ for all small $t$ and $\lambda$. Take $K_2 = B$. If $K_3$ is big enough, then (3’) will be preserved for small $t$ and $\lambda$.
\[cl1\] If $R$ satisfies (1’), (2’) and (3’) of the Lemma at time $t$, then there exists $C > 0$ depending only on the bound of the curvature tensor such that $|R_{i\overline{j}k\overline{l}}| \leq C\sqrt{-Ric(i, \overline{i})}$ and $|R_{i\overline{j}k\overline{l}}| \leq C\sqrt{Ric(i, \overline{i})Ric(j, \overline{j})}$ at time t for any $e_i, e_j, e_k, e_j \in T^{1, 0}M$ and that the length is $1$ in $g(t)$.
The proof follows if we polarize the curvature tensor.
In the following, $C$ will denote a positive constant which depends only on the bound of the curvature tensor. $R$ satisfies the ODE $$\frac{d}{dt}R_{i\overline{j}k\overline{l}} = \sum R_{i\overline{j}**}R_{****} + \sum R_{i***}R_{*\overline{j}**}$$ where $*$ are indices. By Claim \[cl1\], we have $$|\frac{d}{dt}R_{i\overline{j}k\overline{l}}| \leq C\sqrt{Ric(i, \overline{i})Ric(j, \overline{j})}$$ It is easy to see that (1’), (2’) and (3’) in the Lemma hold for $t = 0$. If the Lemma is not true, let $t_0$ be the first time so that the Lemma fails. There are two possibilities:
\(i) (1’) does not hold in $[0, t_1)$ for any $t_1 > t_0$.
\(ii) (2’) does not hold in $[0, t_1)$ for any $t_1 > t_0$.
In case (i), after a slight computation, Claim \[cl1\] implies $$\frac{d}{dt}(\frac{Ric(\alpha, \overline\alpha)}{g(t)(\alpha, \overline\alpha)}) \leq -CRic(\alpha, \overline\alpha)$$ for $|\alpha|_{g(t)} = 1$. If $A > 2C$, this contradicts (i).
For case (ii), Claim \[cl1\] gives $$\frac{d}{dt} ((1+tK_1)Ric(u, \overline u)Ric(v, \overline v) - \frac{|R_{x\overline xu\overline v}|^2}{g(t)(x, \overline x)})\geq
(K_1-C)Ric(u, \overline u)Ric(v, \overline v)>0$$ if $|x|_{g(t)} = 1, K_1 > 2C+10, t_0 < \epsilon < \frac{1}{2K_1}$. This contradicts (ii). The Lemma is thus proved.
Proposition \[prop\] follows if we let $\lambda \to 0$ in the Lemma.
By theorem \[thm2\], $Ric(g(t)) \leq 0$ for small $t>0$. If $Ric < 0$ for some small $t > 0$, then $c_1(M) < 0$. Otherwise, the rank of the Ricci curvature is less than $n$ for some $t > 0$. We shall show that the rank of $Ric_t$ is constant and the null space is parallel.
We use the arguments in [@[BW]](page 676-677). Consider $$\frac{\partial Ric(v, \overline v)}{\partial t} = \Delta_t Ric_{v\overline v} + \sum R_{v\overline v **}R_{****} + \sum R_{v****}R_{*\overline v**}.$$ Define $\tilde{Ric}_t = e^{Ht} Ric_t$. By Proposition \[prop\], if $H$ is large, then $$\label{4}
\frac{\partial\tilde{Ric}_{v\overline v}}{\partial t} \leq \Delta_t \tilde{Ric}_{v\overline v}.$$
Now we show that the rank of $\tilde{Ric}$ is constant for small $t>0$. Let $0\geq \mu_1\geq \mu_2\geq....\geq\mu_n$ denote the eigenvalues of $\tilde{Ric}$ and let $$\sigma_l=\mu_1+\mu_2+....+\mu_l.$$ Fix $p\in M$ and let $e_1(t_0), e_2(t_0),..., e_l(t_0)$ be an orthogonal basis of $T^{1, 0}_p(M)$ such that $\sigma_l(t_0) = \sum\limits_{i=1}^{l}\tilde{Ric}_{t_0}(e_i(t_0), \overline{e_i(t_0)})$. Now $$\begin{aligned}
\sigma'_l(t_0): &= \lim\limits_{t\nearrow t_0}\sup \frac{\sigma_l(t_0)-\sigma_l(t)}{t_0-t}\\&
\leq\frac{d}{dt}|_{t=t_0}\sum\limits_{i=1}^l\tilde{Ric}_t(e_i(t_0), \overline{e_i(t_0)})\\&\leq \sum\limits_{i=1}^l\Delta\tilde{Ric}_{t_0}(e_i(t_0), \overline{e_i(t_0)})
\\&\leq\Delta\sigma_l\end{aligned}$$ Thus $$\frac{\partial \sigma_l}{\partial t} \leq \Delta\sigma_l$$ in the support function sense. By the strong maximum principle, either $\sigma_l < 0$ for all small $t>0$ or $\sigma_l \equiv 0$. This proves that $\tilde{Ric}$ has constant rank for small $t > 0$.
Let $v(t) \in T^{1, 0}M$ be a smooth vector field on $M$ depending smoothly on $t$ such that $\tilde{Ric}_t(v, \overline v) = 0$. Since $\tilde{Ric} \leq 0$, from (\[4\]), $$0 = (\frac{\partial}{\partial t}\tilde{Ric})(v, \overline v)\leq \sum\limits_{i=1}^n\tilde{Ric}(\nabla_{e_i}
v, \overline{\nabla_{e_i}v})$$ where $e_i\in T^{1, 0}M$ is a local unitary frame on $M$. This shows that the rank of $Ric_t$ is constant and the null space of $Ric_t$ is parallel. Therefore, $(M, g(t))$ splits locally for all small $t > 0$. Therefore, for metric $g(0)$, the universal cover $\tilde{M}$ is biholomorphic and isometric to $\mathbb{C}^k \times Y^{n-k}$ with the product metric. Note that the Ricci flow on $M$ preserves the local product structure, and for $\epsilon > t > 0$, the Ricci curvature on $Y$ is strictly negative.
The rest proof of Theorem \[thm1\] uses the argument of Wu and Zheng [@[WZ]]. For reader’s convenience, we recall some details here. Denote by $\Gamma$ the deck transformation group. For each $0 \leq t < \epsilon$, denote by $I_1, I_2(t)$ the group of holomorphic isometries of $\mathbb{C}^{k}$ and $Y^{n-k}$ at time $t$. Any $f \in \Gamma$ induces a biholomorphism and isometry on $\mathbb{C}^k \times Y^{n-k}$ for any $0 \leq t < \epsilon$. Therefore $f = (f_1, f_2)$, where $f_1 \in I_1, f_2 \in \cap_{0 \leq t < \epsilon}I_2(t)$. Denote by $p_i: \Gamma \rightarrow I_i$ the projection map, and by $\Gamma_i = p_i(\Gamma)$ the image groups for $i = 1, 2$. Below are two key claims in [@[WZ]]:
\[cl2\] The group $\Gamma_2$ is discrete.
\[cl3\] There exists a finite index subgroup of $\Gamma' \subseteq \Gamma$ such that $\Gamma'_2$ acts freely on $Y$, and $\Gamma'_1$ contains translation only. Here $\Gamma'_i = p_i(\Gamma'), i = 1, 2$.
Wu and Zheng proved the two claims by using ideas in Eberlein [@[E1]][@[E2]] and Nadel [@[N]]. For our case, Claim \[cl2\] follows by applying Wu and Zheng’s argument to $g(t)$ for small $t > 0$(note that in this case $Ric(Y) < 0$). For Claim \[cl3\], Wu and Zheng’s proof can be carried out without any modification.
By Claim \[cl2\] and Claim \[cl3\], we have a finite covering $M' =\tilde{M}/\Gamma'$ over $M$, and a holomorphic surjection $q: M'\rightarrow N$ induced by the projection from $\tilde{M}$ to $Y$. Here $N = Y/\Gamma'_2$ is a compact Kähler manifold. $q$ makes $M'$ a holomorphic fibre bundle over $N$ with fibre being complex torus. $M'$ is also isometric to a flat torus bundle over $N$. By using the same argument in [@[WZ]], Theorem $E$, we can choose $M'$ to be diffeomorphic to $T\times N$.
Finally, we will use the argument in [@[Yu]] to show that the maximal rank of the Ricci curvature of $g$ coincides with the dimension of $N$. Recall corollary C in [@[WZ]]:
\[thm4\] If $M^n$ is a compact Kähler manifold with nonpositive bisectional curvature which has Ricci rank $r < n$, then the open set U in which the Ricci tensor has maximum rank $r$ in the universal cover $\tilde{M}$ is locally holomorphically isometric to $L_a\times Y_a$, where $L_a$ is a complete flat Kähler manifold, and $Y_a$ is a Kähler manifold with nonpositive bisectional curvature and negative Ricci curvature.
Let $f$ be the homomorphic embedding $L_a \to \tilde{M}$ given in the theorem above. By the evolution equation of the Kähler-Ricci flow, $$\label{5}
\frac{\partial}{\partial t}Ric = \sqrt{-1}\partial\overline\partial R$$ where $R$ is the scalar curvature and $Ric = R_{i\overline{j}}dz^i\wedge dz^{\overline{j}}$. Let $p$ be any point in $f_*L_a$. Pulling back $(\ref{5})$ to $L_a$ by $f$ and integrating on the interval $[0, \epsilon]$, we find that for $e_i, e_j \in T^{1,0}L_a$, $$\label{6}
0 \geq f^*Ric_{i\overline{j}}(g(\epsilon))-f^*Ric_{i\overline{j}}(g(0)) = \sqrt{-1}\partial_i\partial_{\overline{j}}\int\limits_0^\epsilon R(p, t)dt,$$ since $Ric(g(\epsilon)) \leq 0$ and $Ric_{i\overline{j}}(g(0)) = 0$ for $e_i, e_j \in T^{1,0}L_a$. $(\ref{6})$ implies that $-\int\limits_0^\epsilon R(x)dt$ is a bounded plurisubharmonic function on $L_a$. Since $L_a$ is flat, the function must be a constant. Therefore $R_{i\overline j}(g(\epsilon)) = 0$ for any $e_i, e_j \in T^{1,0}L_a$. This implies that $r = dim(N)$.
The proof of Theorem \[thm1\] is complete.
The analogous result of Proposition \[prop\] is true for the Riemannian case, i.e, if a compact manifold has nonpositive sectional curvature, then after the Ricci flow, in a short time, the Ricci curvature will be nonpositive.
**[The proof of theorem \[thm5\]]{}**
=====================================
First we run the Kähler-Ricci flow, then by the arguments in section $2$, the Ricci curvature will be nonpositive after a short time. Since $N$ is an immersed totally geodesic flat complex submanifold of $M$ and $Ric(M)|_{TN} = 0$, the last part of the proof in section $2$ applies, e.g, equation $(\ref{6})$. Therefore, $\tilde{M}$ has a flat factor $\mathbb{C}^k$.
Proof of corollary $3$: Let $r$ be the maximal rank of the Ricci curvature of $M$. By using the same proof of theorem $\ref{thm4}$ in [@[WZ]] , we can show that $L$ is an immersed totally geodesic flat complex submanifold of $M$(just observe that near $L$, the rank of Ricci curvature is locally maximal). By theorem $\ref{thm5}$. $i = rank(Ric(g(0), U(i))) \geq rank(Ric(g(\epsilon), U(i))) = rank(Ric(g(\epsilon))) = r$. The proof of corollary $3$ is complete.
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abstract: |
We consider Kirchhoff equations with a small parameter ${\varepsilon}$ such as $${\varepsilon}{u_{{\varepsilon}}}''(t)+(1+t)^{-p}{u_{{\varepsilon}}}'(t)+
{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2\gamma}}A{u_{{\varepsilon}}}(t)=0.$$ We prove the existence of global solutions when ${\varepsilon}$ is small with respect to the size of initial data, for all $0\leq p \leq
1$ and $\gamma \geq 1$. Then we provide global-in-time error estimates on ${u_{{\varepsilon}}}- u$ where $u$ is the solution of the parabolic problem obtained setting formally ${\varepsilon}= 0$ in the previous equation.
[**Mathematics Subject Classification 2000 (MSC2000):**]{} 35B25, 35L70, 35B40.
[**Key words:**]{} hyperbolic-parabolic singular perturbation, Kirchhoff equations, weak dissipation, quasilinear hyperbolic equations, degenerate hyperbolic equations.
author:
- |
Marina Ghisi\
[Università degli Studi di Pisa]{}\
[Dipartimento di Matematica “Leonida Tonelli”]{}\
[ PISA (Italy)]{}\
[e-mail: `[email protected]`]{}
title: 'Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations with weak dissipation'
---
Introduction
============
Let $H$ be a real Hilbert space. For every $x$ and $y$ in $H$, $|x|$ denotes the norm of $x$, and $\langle x,y\rangle$ denotes the scalar product of $x$ and $y$. Let $A$ be a self-adjoint linear operator on $H$ with dense domain $D(A)$. We assume that $A$ is nonnegative, namely $\langle Ax,x\rangle\geq 0$ for every $x\in D(A)$, so that for every $\alpha\geq 0$ the power $A^{\alpha}x$ is defined provided that $x$ lies in a suitable domain $D(A^{\alpha})$.
For every ${\varepsilon}>0$ we consider the Cauchy problem $${\varepsilon}{u_{{\varepsilon}}}''(t)+(1+t)^{-p}{u_{{\varepsilon}}}'(t)+
{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2\gamma}}A{u_{{\varepsilon}}}(t)=0,
\label{pbm:h-eq}$$ $${u_{{\varepsilon}}}(0)=u_0,\hspace{3em}{u_{{\varepsilon}}}'(0)=u_1.
\label{pbm:h-data}$$ Equation (\[pbm:h-eq\]) is the prototype of all degenerate Kirchhoff equations with weak dissipation, such as $${\varepsilon}{u_{{\varepsilon}}}''(t)+(1+t)^{-p}{u_{{\varepsilon}}}'(t)+
m({|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}})A{u_{{\varepsilon}}}(t)=0 \hspace{2em}
\forall t\geq 0,
\label{pbm:h-eq-gen}$$ where $m:[0,+\infty[\to [0,+\infty[$ is a given function which is always assumed to be of class $C^{1}$. It is well known that (\[pbm:h-eq-gen\]) is the abstract setting of a quasilinear nonlocal partial differential equation of hyperbolic type which was proposed as a model for small vibrations of strings and membranes.
Let us start by recalling some general terminology. Equation (\[pbm:h-eq-gen\]) is called *nondegenerate* (or strictly hyperbolic) when $$\mu:=\inf_{\sigma\geq 0}m(\sigma)>0,$$ and *mildly degenerate* when $\mu=0$ but $m({|A^{1/2}u_{0}|^{2}})\neq 0$. In the special case of equation (\[pbm:h-eq\]) this assumption reduces to $$A^{1/2}u_{0}\neq 0.
\label{hp:mdg}$$ Concerning the dissipation term, we have *constant dissipation* when $p=0$ and *weak dissipation* when $p> 0$. Finally, the operator $A$ is called *coercive* when $$\nu:=\inf\left\{\frac{\langle Ax,x\rangle}{|x|^{2}}:x\in D(A),\
x\neq 0\right\}>0,$$ and *noncoercive* when $\nu=0$.
In the following we recall briefly what is a singular perturbation problem and the “state of the art”. For a more complete discussion on this argument we refer to the survey [@trieste] and to the references contained therein. Moreover we concentrate mostly on equation (\[pbm:h-eq\]) recalling only a few facts on the general equation (\[pbm:h-eq-gen\]).
The singular perturbation problem in its generality consists in proving the convergence of solutions of (\[pbm:h-eq\]), (\[pbm:h-data\]) to solutions of the first order problem $$u'(t)+ (1+t)^{p}{{|A^{1/2}u(t)|}^{2\gamma}}Au(t)=0, \hspace{2em}
u(0)=u_{0},
\label{pbm:par}$$ obtained setting formally ${\varepsilon}=0$ in (\[pbm:h-eq\]), and omitting the second initial condition in (\[pbm:h-data\]). In the concrete case, equation in (\[pbm:par\]) is a partial differential equation of parabolic type. With a little abuse of notation in the following we refer to hyperbolic and parabolic problems (or behavior) also in the abstract setting of equations (\[pbm:h-eq\]) and (\[pbm:par\]).
Following the approach introduced by J. L.Lions [@lions] in the linear case, one defines the corrector ${\theta_{{\varepsilon}}}(t)$ as the solution of the second order *linear* problem $${\varepsilon}{\theta_{{\varepsilon}}}''(t)+(1+t)^{-p}{\theta_{{\varepsilon}}}'(t)=0 \hspace{2em}
\forall t\geq 0,
\label{theta-eq}$$ $${\theta_{{\varepsilon}}}(0)=0,\hspace{2em}{\theta_{{\varepsilon}}}'(0)=u_1+
{{|A^{1/2}u_{0}|}^{2\gamma}}Au_{0}=:w_{0}.
\label{theta-data}$$ It is easy to see that ${\theta_{{\varepsilon}}}'(0)={u_{{\varepsilon}}}'(0)-u'(0)$, hence this corrector keeps into account the boundary layer due to the loss of one initial condition. Finally one defines ${r_{{\varepsilon}}}(t)$ and ${\rho_{{\varepsilon}}}(t)$ in such a way that $${u_{{\varepsilon}}}(t)=u(t)+{\theta_{{\varepsilon}}}(t)+{r_{{\varepsilon}}}(t)=u(t)+{\rho_{{\varepsilon}}}(t)\quad\quad\forall
t\geq 0.
\label{defn:rep}$$ With these notations, the singular perturbation problem consists in proving that ${r_{{\varepsilon}}}(t)\to 0$ or ${\rho_{{\varepsilon}}}(t)\to 0$ in some sense as ${\varepsilon}\to 0^{+}$. In particular *time-independent* estimates on ${\rho_{{\varepsilon}}}(t)$ or ${r_{{\varepsilon}}}(t)$ as ${\varepsilon}\to 0^{+}$ are called *global error estimates*.
In this paper we restrict ourself to the so called parabolic regime, namely to the case where $0\leq p \leq 1$. The reason is that equations (\[pbm:h-eq\]) and (\[pbm:h-eq-gen\]) have a different behavior when $p\leq 1$ or $p>
1$. This is true also in the linear nondegenerate case. Let us indeed consider equation $$au''(t)+\frac{b}{(1+t)^{p}}u'(t)+cAu(t)=0,
\label{eq:wirth}$$ where $a$, $b$, $c$ are positive parameters, and $p\geq 0$. This equation was investigated by T. Yamazaki [@yamazaki-lin] and J.Wirth [@wirth]. They proved that (\[eq:wirth\]) has both parabolic and hyperbolic features, and which nature prevails depends on $p$. When $p<1$ the equation has parabolic behavior, in the sense that all its solutions decay to 0 as $t\to +\infty$ as solutions of the parabolic equation with $a=0$. When $p>1$ the same equation has hyperbolic behavior, meaning that every solution is asymptotic to a suitable solution of the non-dissipative equation with $b=0$ (and in particular all non-zero solutions do not decay to zero). In the critical case $p=1$ the nature of the problem depends on $b/a$, with the parabolic behavior prevailing as soon as the ratio is large enough. In and [@jde2] it was proved that also in the case of Kirchhoff equation we have always hyperbolic behavior when $p>1$, meaning that non-zero global solutions (provided that they exist) cannot decay to 0. On the other hand, solutions of the limit parabolic problem decay to zero also for $p>1$, faster and faster as $p$ grows.
The study of the singular perturbation problem has generated a considerable literature in particular regarding the preliminary problem of the existence of global solutions for (\[pbm:h-eq\]) (or (\[pbm:h-eq-gen\])). Despite of this, existence of global solutions without smallness assumptions on ${\varepsilon}$ is a widely open question. The existence of global solutions for (\[pbm:h-eq\]), (\[pbm:h-data\]) in the case of a constant dissipation ($p=0$) and $\gamma \geq 1$ when ${\varepsilon}$ is small and (\[hp:mdg\]) holds true, was established by K. Nishihara and Y. Yamada [@ny] (see also E. De Brito [@debrito1] and Y. Yamada [@yamada] for the nondegenerate case, [@gg:k-dissipative] for the general case and [@ghisi2] for the case $\gamma < 1$). Moreover optimal and ${\varepsilon}$-independent decay estimates were obtained in [@gg:k-decay] (and by T. Mizumachi ([@mizu-ade; @mizu-nc]) and K. Ono ([@ono-kyushu; @ono-aa]) when $\gamma = 1$).
When $0\leq p \leq 1$ the existence of global solutions, always for ${\varepsilon}$ small, in the nondegenerate case was proved in recent years by M. Nakao and J.Bae [@nakao], by T.Yamazaki [@yamazaki-wd; @yamazaki-cwd], and in [@gg:w-ndg].
The first result for (\[pbm:h-eq\]) when $p> 0$ was obtained by K.Ono [@ono-wd]. In the special case $\gamma=1$ he proved that a global solution exists provided that ${\varepsilon}$ is small and $p\in[0,1/3]$. Then for ten years there were no significant progresses. The reason of the slow progress in this field is hardly surprising. In the weakly dissipative case existence and decay estimates have to be proved in the same time. The better are the decay estimates, the stronger is the existence result. This is due to the competition between the smallness of the dissipation term and the one of the nonlinear term. Both of them decay to zero at infinity and it seems fundamental to understand which of them prevails. Ten years ago decay estimates for degenerate equations were far from being optimal, but for the special case $\gamma=1$. In [@gg:k-decay] a new method for obtaining optimal decay estimates was introduced and this allowed a substantial progress. In particular in [@jde2] the following result has been proved.
\[[@jde2]\] \[A\] *[ Let $(u_{0},u_{1})\in D(A)\times D(A^{1/2})$. If the operator $A$ is coercive and $0\leq
p \leq 1$, $\gamma > 0$, for ${\varepsilon}$ small the mildly degenerate problem (\[pbm:h-eq\]), (\[pbm:h-data\]) has a unique global solution such that $$\frac{C_{1}}{(1+t)^{(p+1)/\gamma}}\leq
{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}\leq
\frac{C_{2}}{(1+t)^{(p+1)/\gamma}}
\quad\quad\forall t\geq 0,$$ $$\frac{C_{1}}{(1+t)^{(p+1)/\gamma}}\leq
|A{u_{{\varepsilon}}}(t)|^{2}\leq
\frac{C_{2}}{(1+t)^{(p+1)/\gamma}}
\quad\quad\forall t\geq 0,$$ $$|{u_{{\varepsilon}}}'(t)|^{2}\leq
\frac{C_{2}}{(1+t)^{2+(p+1)/\gamma}}
\quad\quad\forall t\geq 0.$$ If the operator $A$ is only non negative, $\gamma \geq 1$ and $$0\leq p\leq \frac{\gamma^{2}+1}{\gamma^{2}+2\gamma-1},
\label{hp:pnc}$$ for ${\varepsilon}$ small the mildly degenerate problem (\[pbm:h-eq\]), (\[pbm:h-data\]) has a unique global solution such that $$\frac{C_{1}}{(1+t)^{(p+1)/\gamma}}\leq
{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}\leq
\frac{C_{2}}{(1+t)^{(p+1)/(\gamma+1)}}
\quad\quad\forall t\geq 0,$$ $$|A{u_{{\varepsilon}}}(t)|^{2}\leq
\frac{C_{2}}{(1+t)^{(p+1)/\gamma}}
\quad\quad\forall t\geq 0,$$ $$|{u_{{\varepsilon}}}'(t)|^{2}\leq
\frac{C_{2}}{(1+t)^{[2\gamma^{2}+(1-p)\gamma+p+1]/
(\gamma^{2}+\gamma)}}
\quad\quad\forall t\geq 0.$$]{}*
In the case of noncoercive operators this result is not optimal because of (\[hp:pnc\]). This gap is due to the fact that in this second case the estimate on ${|A^{1/2}{u_{{\varepsilon}}}|^{2}}$ is worse. This problem is in some sense unavoidable. Indeed, also in the case of linear equations, small eigenvalues can make worse the decay of solutions. Despite of this, the first result in this paper fills up the gap. The key technical point is that in the noncoercive case a better decay of $|A{u_{{\varepsilon}}}|^{2}$ compensates a worse decay of ${|A^{1/2}{u_{{\varepsilon}}}|^{2}}$ and the two decay rates are strictly related (see Proposition \[priori\]). This unexpected decay property requires some new and subtle estimates. Such estimates improve the decay rates also when (\[hp:pnc\]) is satisfied.
Once we know that a global solution of (\[pbm:h-eq\]) exists, we can focus on the singular perturbation problem. This question was solved in the nondegenerate case. In that case decay - error estimates were proved, that consist in estimating in the same time the behavior of ${u_{{\varepsilon}}}(t)-
u(t)$ as $t\to+\infty$ and as ${\varepsilon}\to 0^{+}$ (see H. Hashimoto and T.Yamazaki [@yamazaki], T.Yamazaki [@yamazaki-wd; @yamazaki-cwd] and [@gg:w-ndg]). On the contrary the singular perturbation problem is still quite open in the degenerate case. With respect to global in time error estimates we indeed know only the following partial result in the constant dissipative case (see [@gg:k-PS] where however more general nonlinearities are considered).
*[ \[thm:error\] If we assume that $p=0$, $\gamma \geq 1$, $(u_{0},u_{1})\in{D(A^{3/2})}\times{D(A^{1/2})}$, then there exists a constant $C$ such that for every ${\varepsilon}$ small we have that $$|{\rho_{{\varepsilon}}}(t)|^{2}+{\varepsilon}|A^{1/2}{\rho_{{\varepsilon}}}(t)|^{2}\leq C{\varepsilon}^{2}
\quad\quad
\forall t\geq 0,$$ $$\int_{0}^{+\infty}
|{r_{{\varepsilon}}}'(t)|^{2}\,dt\leq C{\varepsilon}.$$]{}*
This result is far from being optimal. First of all only the convergence rate of $|{\rho_{{\varepsilon}}}(t)|^{2}$ is optimal, while with these regularity assumptions on the initial data one can expect an optimal convergence rate also for ${|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}$ (see [@gg:l-cattaneo]). Moreover it is limited to equations with constant dissipation. The second result of this paper fills up completely this gap with respect to error estimates in the case of coercive operators or when $p$ verifies (\[hp:pnc\]) (hence always in the case of a constant dissipation) and provides global, but not optimal, error estimates in the remaining cases. Also to prove this second result are fundamental decay estimates as accurate as possible. Conversely it is still open the problem of decay-error estimates in all degenerate cases.
This paper is organized as follows. In Section \[stat\] we state precisely our results. Section \[proofs\] is devoted to the proofs, and it is divided into several parts. In particular to begin with in Section \[sec:ode\] we state and prove some general lemmata, then in Section \[parab\] we consider the parabolic problem (\[pbm:par\]) and finally in Sections \[ex\] and \[error\] we prove the results.
Statements {#stat}
==========
The first result we state concerns the existence of global solutions for (\[pbm:h-eq\]) and their decay properties.
\[th:ex\] Let us assume that $0 \leq p \leq 1$, $\gamma \geq 1$ and $A$ be a nonnegative operator. Let us assume that $(u_{0}, u_{1}) \in
D(A)\times D(A^{1/2})$ satisfy (\[hp:mdg\]).
Then there exists ${\varepsilon}_{0}>0$ such that for every ${\varepsilon}\in(0,{\varepsilon}_{0})$ problem (\[pbm:h-eq\]), (\[pbm:h-data\]) has a unique global solution $${u_{{\varepsilon}}}\in C^{2}([0,+\infty[;H)\cap C^{1}([0,+\infty[;{D(A^{1/2})}) \cap
C^{0}([0,+\infty[;{D(A)}).$$ Moreover there exist positive constants $C_{1}$ and $C_{2}$ such that $$|{u_{{\varepsilon}}}(t)|^{2} \leq C_{1} \quad\quad\forall t\geq 0;
\label{D0}$$ $$\frac{C_{1}}{(1+t)^{(p+1)/\gamma}}\leq
{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}\leq
\frac{C_{2}}{(1+t)^{(p+1)/(\gamma+1)}}
\quad\quad\forall t\geq 0;
\label{D1}$$ $${|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma-1)} |A{u_{{\varepsilon}}}(t)|^{2}\leq
\frac{C_{2}}{(1+t)^{(p+1)}}
\quad\quad\forall t\geq 0;
\label{D2}$$ $$|{u_{{\varepsilon}}}'(t)|^{2}\leq
\frac{C_{2} {|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}{(1+t)^{1-p}}
\quad\quad\forall t\geq 0;
\label{D3}$$ $$\int_{0}^{+\infty}|{u_{{\varepsilon}}}'(t)|^{2} (1+t)\, dt \leq
C_{2}.
\label{D4}$$
*Inequality (\[D2\]) as far we know is new and it is the core of the existence theorem. This estimate says that in the case when the operator $A$ is noncoercive it is of course possible that ${|A^{1/2}{u_{{\varepsilon}}}|}$ decays slow, but in this case $|A{u_{{\varepsilon}}}|$ decays stronger than in the coercive case.*
*When the operator $A$ is noncoercive, $\gamma = 1$ is the only case in which this result is contained in Theorem \[A\] since in the other cases $(\gamma^{2}+1)/(\gamma^{2} + 2\gamma -
1) < 1$ and moreover there is always an improvement on the decay rates (see (\[D2\]) and (\[D3\])).*
The next result regards error estimates. It is divided into two parts. The first one concerns all non negative operators and all $0\leq p \leq 1$ and the exponent of ${\varepsilon}$ in the estimates is not optimal. The second one gives optimal estimates but with some restrictions on the operator or on the admissible values of $p$.
\[thm:dg-error\] Let us assume that $0 \leq p \leq 1$, $\gamma \geq 1$ and $A$ be a nonnegative operator. Let ${u_{{\varepsilon}}}(t)$ be the solution of equation (\[pbm:h-eq\]) with initial data $(u_{0},u_{1})\in D(A^{3/2})\times{D(A^{1/2})}$ satisfying (\[hp:mdg\]). Let $u(t)$ be the solution of the corresponding first order problem (\[pbm:par\]), and let ${r_{{\varepsilon}}}(t)$ and ${\rho_{{\varepsilon}}}(t)$ be defined by (\[defn:rep\]).
Then we have the following conclusions.
1. There exists a constant $C_{3}$ such that for every ${\varepsilon}$ small enough we have that $$|{\rho_{{\varepsilon}}}(t)|^{2}+|A^{1/2}{\rho_{{\varepsilon}}}(t)|^{2} + \int_{0}^{t}
\frac{|{r_{{\varepsilon}}}'(s)|^{2}}{{|A^{1/2}u(s)|}^{2\gamma}}\frac{1}{(1+s)^{p}}ds
\leq C_{3}{\varepsilon}\quad\quad
\forall t\geq 0.
\label{H1}$$
2. If in addition we assume that $$\mbox{$A$ is coercive or } \; 0 \leq p \leq
\frac{\gamma^{2}+1}{\gamma^{2}+2\gamma - 1}
\label{H1cip}$$ then there exists a constant $C_{4}$ such that for every ${\varepsilon}$ small enough we have that $$|{\rho_{{\varepsilon}}}(t)|^{2}+|A^{1/2}{\rho_{{\varepsilon}}}(t)|^{2} + \int_{0}^{t}
\frac{|{r_{{\varepsilon}}}'(s)|^{2}}{{|A^{1/2}u(s)|}^{2\gamma}}
\frac{1}{(1+s)^{p}}ds \leq C_{4}{\varepsilon}^{2}
\quad\quad
\forall t\geq 0.
\label{H1c}$$
*When the initial data are more regular it is of course possible to achieve an estimate on $A{\rho_{{\varepsilon}}}$ like the ones in (\[H1\]) and (\[H1c\]). Moreover in this case one can get also estimates on ${r_{{\varepsilon}}}'$ exactly as in [@gg:k-PS], [@gg:w-ndg] (see also [@trieste]) We do not give here the precise statements and proofs since they only lengthen the paper without introducing new ideas.*
*In the integrals in (\[H1\]), (\[H1c\]) it appears the coefficient ${|A^{1/2}u|}^{-2\gamma}$. When $A$ is a coercive operator we can replace this term with ${|A^{1/2}{u_{{\varepsilon}}}|}^{-2\gamma}$ or $(1+t)^{p+1}$, indeed they all have the same behavior. On the contrary when $A$ is noncoercive the use of ${|A^{1/2}u|}^{-2\gamma}$ seems compulsory.*
Proofs
======
Proofs are organized as follows. First of all in Section \[sec:ode\] we state and prove some general lemmata that do not concern directly the Kirchhoff equation. In Section \[parab\] then we recollect all the properties of the solutions of (\[pbm:par\]) we need. Finally in Section \[ex\] we prove Theorem \[th:ex\] and in Section \[error\] we prove Theorem \[thm:dg-error\].
Basic Lemmata {#sec:ode}
-------------
Numerous variants of the following comparison result have already been used in [@ghisi2; @gg:k-dissipative; @gg:k-decay; @gg:w-ndg; @jde2] and we refer to these ones for the proof.
\[lemma:ode\] Let $T>0$, and let $f:[0,T[\to[0,+\infty[$ be a function of class $C^{1}$. Let $\phi:[0,T[\to[0,+\infty[$ be a continuous function. Then the following implications hold true.
1. \[ODEA1\] Let us assume that there exists a constant $a\geq 0$ such that $$f'(t)\leq
-\phi(t)\sqrt{f(t)}(\sqrt{f(t)}-a)
\hspace{2em}
\forall t\in [0,T[;$$ then we have that $$f(t)\leq \max\{f(0), a^{2}\}
\hspace{2em}
\forall t\in [0,T[.$$
2. \[ODEA2\] Let us assume that there exists a constant $a\geq 0$ such that $$f'(t)\leq
-\phi(t)f(t)(f(t)-a)
\hspace{2em}
\forall t\in [0,T[;$$ then we have that $$f(t)\leq \max\{f(0), a\}
\hspace{2em}
\forall t\in [0,T[.$$
A proof of the next comparison result is contained in [@jde2] (Lemma 3.2).
\[lemma:ode-vs\] Let $w:[0,+\infty[\to[0,+\infty[$ be a function of class $C^{1}$ with $w(0)>0$. Let $a>0$ be a positive constant.
Then the following implications hold true.
1. \[ODEB2\] If $w$ satisfies the differential inequality $$w'(t)\leq
-a(1+t)^{p}\left[w(t)\right]^{1+\gamma}
\quad\quad\forall t\in[0,+\infty[,$$ then for some constant $\gamma_{1}$ we have the following estimate $$w(t)\leq
\frac{\gamma_{1}}{(1+t)^{(p+1)/\gamma}}
\quad\quad\forall t\in[0,+\infty[.$$
2. \[ODEB1\]If $w$ satisfies the differential inequality $$w'(t)\geq
-a(1+t)^{p}\left[w(t)\right]^{1+\gamma}
\quad\quad\forall t\in[0,+\infty[,$$ then for some constant $\gamma_{2}$ we have the following estimate $$w(t)\geq
\frac{\gamma_{2}}{(1+t)^{(p+1)/\gamma}}
\quad\quad\forall t\in[0,+\infty[.$$
Let us now state and prove the third lemma.
\[lemma:ode-GF\] Let $F,\, G:[0,T[\to[0,+\infty[$ be functions of class $C^{1}$. Let $\varphi:[0,T[\to]0,+\infty[$ be a continuous function and $a>0$, $b \geq 0,\, c \geq 0$ be real numbers. Let us assume that in $[0,T[$ the following inequality holds true: $$(F+G)'(t) \leq - \varphi(t) (F(t) + a (G(t))^{2} -bG(t) -
c(G(t))^{3/2}).
\label{0-ode}$$ Let us set $\sigma_{0}:= (c + \sqrt{ab})/a$. Then we get $$G(t) + F(t) \leq \sigma_{0}^{2}(1 + b + c\sigma_{0}) + F(0) + G(0)
+ 1, \quad\quad \forall t\in[0,T[.
\label{1-ode}$$
[[Proof.]{}$\;$]{}Let us set $$S := \sup\{t< T: \:\mbox{ in $[0,t]$ the inequality (\ref{1-ode})
holds true}\}.$$ It is obvious that $S > 0$. We want to prove that $S = T$. Let us assume by contradiction that $S < T$. Therefore in $[0,S[$ the inequality (\[1-ode\]) holds true, moreover $$G(S) + F(S) = \sigma_{0}^{2}(1 + b + c\sigma_{0}) + F(0) +
G(0) + 1
\label{3-ode}$$ and $$(G+F)'(S) \geq 0.
\label{4-ode}$$ If $G(S) > \sigma_{0}^{2}$ then $$a (G(S))^{2}
-bG(S) - c(G(S))^{3/2} = G(S)(aG(S) - b - \sqrt{G(S)} ) > 0.
\label{4b-ode}$$ Indeed let us set $y =\sqrt{G(S)}$, then $$a y^{2} - cy - b > 0 \hspace{1em} \mbox{if}
\hspace{1em} y > \frac{c + \sqrt{c^{2}+ 4ab}}{2a} =: \sigma_{1}$$ and by assumption $\sigma_{1} \leq \sigma_{0} < y$. Plugging (\[4b-ode\]) in (\[0-ode\]) we hence arrive at $$(G+F)'(S) < 0$$ in contrast with (\[4-ode\]).
Let us now assume that $G(S) \leq \sigma_{0}^{2}$. Hence from (\[3-ode\]) we get: $$\begin{aligned}
& F(S) - bG(S) - c(G(S))^{3/2} = & \\
&\sigma_{0}^{2}(1 + b + c\sigma_{0}) + F(0) +
G(0) + 1 - (b+1)G(S) - c(G(S))^{3/2} & \\
& > \sigma_{0}^{2}(1 + b + c\sigma_{0}) - (b+1)G(S) - c(G(S))^{3/2} & \\
& \geq \sigma_{0}^{2}(1 + b + c\sigma_{0}) -
(b+1)\sigma_{0}^{2} - c\sigma_{0}^{3} = 0.&
\end{aligned}$$ Hence by (\[0-ode\]) we obtain once again $$(G+F)'(S) < 0$$ in contrast with (\[4-ode\]). [[10000]{}]{}
The following lemma is essential in the proof of error estimates.
\[mprop\] Let us assume that $m:[0,+\infty[\rightarrow[0,+\infty[$ is a nondecreasing function. Then for all $x,\, y \in D(A)$ we get $$\langle m({|A^{1/2}x|^{2}})Ax - m({|A^{1/2}y|^{2}})Ay, x-y\rangle \geq
\frac{1}{2}\left[m({|A^{1/2}x|^{2}}) + m({|A^{1/2}y|^{2}})\right]{|A^{1/2}(x-y)|^{2}}.$$
[[Proof.]{}$\;$]{}Let us set $$m_{x}:= m({|A^{1/2}x|^{2}}), \quad\quad m_{y}:= m({|A^{1/2}y|^{2}}).$$ Thus an elemental calculation gives: $$\begin{aligned}
& \displaystyle \langle m_{x}Ax - m_{y}Ay, x-y\rangle =
m_{x}{|A^{1/2}x|^{2}} + m_{y}{|A^{1/2}y|^{2}} -(m_{x} +
m_{y})\langle Ax, y\rangle & \\
&=\displaystyle \frac{1}{2}(m_{x} +
m_{y})({|A^{1/2}x|^{2}} + {|A^{1/2}y|^{2}}) -
\displaystyle
\frac{1}{2}\cdot 2(m_{x} + m_{y})\langle Ax, y\rangle + & \\
& \displaystyle+\frac{1}{2}(m_{x} - m_{y})({|A^{1/2}x|^{2}} - {|A^{1/2}y|^{2}}) & \\
& = \displaystyle \frac{1}{2}(m_{x} +
m_{y}){|A^{1/2}(x-y)|^{2}} +
\displaystyle \frac{1}{2}(m_{x} - m_{y})({|A^{1/2}x|^{2}} - {|A^{1/2}y|^{2}}) &
\\
&\geq \displaystyle \frac{1}{2}(m_{x} +
m_{y}){|A^{1/2}(x-y)|^{2}}; &
\end{aligned}$$ where in the last step we exploit that $m$ is nondecreasing, hence $$(m(\alpha) - m(\beta))(\alpha - \beta) \geq 0 \hspace{1em}
\forall \alpha,\, \beta \geq 0.$$ [[10000]{}]{}
The last lemma concerns the integrability properties of the corrector ${\theta_{{\varepsilon}}}$.
\[thetalemma\] Let $0\leq p \leq 1$ and let ${\theta_{{\varepsilon}}}$ be the solution of (\[theta-eq\]), (\[theta-data\]). Let $\delta \geq 0$ and let us assume that ${\varepsilon}<(2+2\delta)^{-1}$. Then there exists a constant $C_{\delta}$ independent from ${\varepsilon}$ and from the initial data such that if $w_{0} \in D(A^{j/2})$ therefore we have $$\int_{0}^{+\infty}(1+t)^{\delta}|A^{j/2}{\theta_{{\varepsilon}}}'(t)| \, dt \leq
C_{\delta}|A^{j/2}w_{0}|{\varepsilon}.$$
[[Proof.]{}$\;$]{} Let us define $$I:= \int_{0}^{+\infty}(1+t)^{\delta}|A^{j/2}{\theta_{{\varepsilon}}}'(t)| \, dt.$$ If $p=1$ then ${\theta_{{\varepsilon}}}'(t) = w_{0}(1+t)^{-1/{\varepsilon}}$ hence thesis follows from $$I =
|A^{j/2}w_{0}|\frac{{\varepsilon}}{1 - (\delta +1){\varepsilon}}.$$ Let us now assume that $p<1$. In such a case we have $${\theta_{{\varepsilon}}}'(t) =
w_{0}\exp\left(-\frac{1}{{\varepsilon}}\frac{1}{
1-p}((1+t)^{1-p} - 1)\right).$$ If we set $$\phi(t) := \min\{t,t^{1-p}\}$$ then it is easy to prove that there exists a constant $\beta_{0}> 0$ such that $$\frac{1}{1-p}((1+t)^{1-p} - 1)\geq \beta_{0}\phi(t).$$
In particular we obtain $$\begin{aligned}
I & \leq &
|A^{j/2}w_{0}|
\int_{0}^{+\infty}(1+t)^{\delta}e^{ -\frac{1}{{\varepsilon}}\beta_{0}\phi(t)}dt
\\
& = & |A^{j/2}w_{0}|\left(
\int_{0}^{1}(1+t)^{\delta}e^{-\frac{t}{{\varepsilon}}\beta_{0}}dt +
\int_{1}^{+\infty}(1+t)^{\delta}e^{-\frac{t^{1-p}}{{\varepsilon}}\beta_{0}}dt\right).
\end{aligned}$$ Let us set ${\varepsilon}s = t$, hence $$\begin{aligned}
I & \leq &
|A^{j/2}w_{0}| {\varepsilon}\left(
\int_{0}^{+\infty}(1+{\varepsilon}s)^{\delta}e^{-\beta_{0}s}ds +
\int_{0}^{+\infty}(1+{\varepsilon}s)^{\delta}e^{-\frac{s^{1-p}}{{\varepsilon}^{p}}\beta_{0}}ds\right)
\\
& \leq & |A^{j/2}w_{0}|{\varepsilon}\left(
\int_{0}^{+\infty}(1+s)^{\delta}e^{-\beta_{0}s}ds +
\int_{0}^{+\infty}(1+s)^{\delta}e^{-\beta_{0}s^{1-p}}ds\right)\\
& = & |A^{j/2}w_{0}|{\varepsilon}C_{\delta}.
\end{aligned}$$ [[10000]{}]{}
The First order problem {#parab}
-----------------------
Theory of parabolic equations of Kirchhoff type is quite well established. These equations appeared for the first time in the pioneering paper [@bernstein] by S. Bernstein and then were considered by many authors (see [@bw; @miletta; @k-par] and [@trieste] for the details). In fact the following result holds true.
\[thp\] Let $A$ be a nonnegative operator, let $0\leq p \leq 1$ and $\gamma \geq 1$. Let $u_{0}\in{D(A)}$.
Then problem (\[pbm:par\]) has a unique global solution $$u\in C^{1}\left([0,+\infty[;H\right)\cap
C^{0}\left([0,+\infty[;{D(A)}\right).$$
If in addition $A^{1/2}u_{0}\neq 0$ then the solution is non-stationary, i.e. ${|A^{1/2}u(t)|^{2}}\neq 0$ for all $t\geq 0$ and $u\in
C^{\infty}\left(]0,+\infty[;D(A^{\alpha})\right)$ for every $\alpha\geq
0$.
In the proposition below we collect all the properties of the solutions of (\[pbm:par\]) we need in proof of error estimates. Only some of these properties require $u_{0} \in {D(A^{3/2})}$. Nevertheless this is an assumption of Theorem \[thm:dg-error\] hence we do not specify in what cases it is in fact necessary or not.
Let $u_{0}\in{D(A^{3/2})}$ and let us assume that all conditions of Theorem \[thp\] are verified. Let $u$ be the global solution of (\[pbm:par\]).
Then the following statements hold true.
- The solution $u$ verifies the standard estimates below: $$\frac{|A^{(k+1)/2}u(t)|^{2}}{|A^{k/2}u(t)|^{2}} \leq
\frac{|A^{(k+1)/2}u_{0}|^{2}}{|A^{k/2}u_{0}|^{2}},
\quad \quad k=1,\, 2, \quad \quad \forall t\geq 0;
\label{dv0}$$ $$\frac{1}{2}|u(t)|^{2}
+\int_{0}^{t}{|A^{1/2}u(s)|}^{2(\gamma+1)}(1+s)^{p} ds =
\frac{1}{2}|u_{0}|^{2}, \quad \quad \forall t\geq 0.
\label{dv1}$$
- The solution $u$ has these decay properties: $$\frac{\gamma_{3}}{(1+t)^{(p+1)/\gamma}}\leq {|A^{1/2}u(t)|^{2}}
\leq \frac{\gamma_{4}}{(1+t)^{(p+1)/(\gamma+1)}},
\quad \quad \forall t\geq 0;
\label{dv2}$$ $${|A^{1/2}u(t)|}^{2(\gamma-1)}|Au(t)|^{2} \leq
\frac{\gamma_{4}}{(1+t)^{p+1}}, \quad \quad \forall t\geq 0.
\label{dv4}$$ If moreover $A$ is a coercive operator then $${|A^{1/2}u(t)|^{2}} \leq
\frac{\gamma_{4}}{(1+t)^{(p+1)/\gamma}},
\quad \quad \forall t\geq 0.
\label{dv3}$$
- The following integrals are bounded: $$\int_{0}^{+\infty}|u'(t)|^{2}(1+t)^{p} dt =
\int_{0}^{+\infty}{|A^{1/2}u(t)|}^{4\gamma}|Au(t)|^{2}(1+t)^{3p}
dt \leq \gamma_{5};
\label{dv5}$$ $$\int_{0}^{+\infty}{|A^{1/2}u(t)|}^{6\gamma}|A^{3/2}u(t)|^{2}(1+t)^{5p}
dt \leq \gamma_{5};
\label{dv6}$$ $$\int_{0}^{+\infty}\left[{|A^{1/2}u(t)|}^{8\gamma}(1+t)^{7p}+
{|A^{1/2}u(t)|}^{6\gamma}(1+t)^{5p}\right]|A^{2}u(t)|^{2} dt \leq
\gamma_{5}.
\label{dv7}$$
[[Proof.]{}$\;$]{}From now in most of the proofs we omit the dependence of $u$ from $t$. Moreover often we use that $0\leq p \leq 1$ but for shortness sake we do not recall it more. Furthermore we use that in $]0,+\infty[$ the solution $u$ is as regular as we want.
#####
It is enough to remark that $$\left(\frac{|A^{(k+1)/2}u|^{2}}{|A^{k/2}u|^{2}}
\right)' = -2(1+t)^{p}\frac{|A^{1/2}u|^{2\gamma}}{|A^{k/2}u|^{4}}
(|A^{(k+2)/2}u|^{2}|A^{k/2}u|^{2} - |A^{(k+1)/2}u|^{4} ) \leq 0,$$ where in the last inequality we exploit that $$|A^{(k+1)/2}u|^{2} =
\langle A^{(k+2)/2}u,A^{k/2}u\rangle \leq |A^{(k+2)/2}u||A^{k/2}u|.
\label{rappA}$$
#####
It suffices to integrate in $[0,t]$ the equality: $$\left(\frac{1}{2}|u|^{2} \right)' +{|A^{1/2}u|}^{2(\gamma+1)}(1+t)^{p} = 0.$$
#####
Using (\[dv0\]) with $k = 1$ we have $$({|A^{1/2}u|^{2}})'= - 2(1+t)^{p}|Au|^{2}{|A^{1/2}u|}^{2\gamma} \geq
-2(1+t)^{p}
\frac{|A u_{0}|^{2}}{|A^{1/2}u_{0}|^{2}}{|A^{1/2}u|}^{2(\gamma+1)} .$$ Therefore estimate form below follows from Statement (\[ODEB1\]) in Lemma \[lemma:ode-vs\].
Let us now remark that $$\left((1+t)^{p+1}\frac{{|A^{1/2}u|}^{2(\gamma +
1)}}{2(\gamma+1)}\right)'+ (1+t)^{2p+1}{|A^{1/2}u|}^{4\gamma}|Au|^{2} =
\frac{p+1}{2(\gamma+1)}(1+t)^{p}{|A^{1/2}u|}^{2(\gamma + 1)}.
\label{decv0}$$ Integrating in $[0,t]$ and using (\[dv1\]) we get: $$(1+t)^{p+1}\frac{{|A^{1/2}u(t)|}^{2(\gamma + 1)}}{2(\gamma+1)}+
\int_{0}^{t}(1+s)^{2p+1}{|A^{1/2}u(s)|}^{4\gamma}|Au(s)|^{2}ds \leq
|u_{0}|^{2} + {|A^{1/2}u_{0}|}^{2(\gamma+1)}.
\label{decv}$$ From this inequality we gain directly the estimate from above in (\[dv2\]).
#####
Let us define $$G(t) = (1+t)^{p+1}{|A^{1/2}u(t)|}^{2(\gamma-1)}|Au(t)|^{2}.$$ We have to prove that $G$ is bounded. Taking the time’s derivative of $G$ we obtain $$G' = \frac{-G}{1+t}\left[2{|A^{1/2}u|}^{2\gamma}
\frac{|A^{3/2}u|^{2}}{|Au|^{2}}(1+t)^{p+1}+2(\gamma - 1)G
-(p+1)\right].$$ Now let us distinguish two cases. If $\gamma > 1$ we have: $$G' \leq\frac{-G}{1+t}\left[2(\gamma - 1)G -(p+1)\right]$$ then thesis follows from Statement (\[ODEA2\]) in Lemma \[lemma:ode\].
Instead if $\gamma = 1$ using (\[rappA\]) with $k=1$ we get $$G'\leq\frac{-G}{1+t}\left[2|Au|^{2}(1+t)^{p+1} -(p+1)\right] =
\frac{-G}{1+t}\left[2G -(p+1)\right],$$ hence we conclude as in the previous case.
#####
Since $\langle Au, u \rangle \geq \nu |u|^{2}$ then $$({|A^{1/2}u|^{2}})'= -
2(1+t)^{p}|Au|^{2}{|A^{1/2}u|}^{2\gamma} \leq -2\nu (1+t)^{p}
{|A^{1/2}u|}^{2(\gamma+1)} .$$ Hence it suffices to apply Statement (\[ODEB2\]) in Lemma \[lemma:ode-vs\].
#####
Since $3p\leq 2p+1$ it is a consequence of (\[decv\]).
#####
A simple computation gives: $$\begin{aligned}
& \displaystyle\left(\frac{1}{2}{|A^{1/2}u|}^{4\gamma}|Au|^{2}(1+t)^{4p}\right)' +
{|A^{1/2}u|}^{6\gamma}|A^{3/2}u|^{2}(1+t)^{5p} = & \\
& 2p{|A^{1/2}u|}^{4\gamma}|Au|^{2}(1+t)^{3p}
-2\gamma{|A^{1/2}u|}^{6\gamma-2}|Au|^{4}(1+t)^{5p} \leq
2{|A^{1/2}u|}^{4\gamma}|Au|^{2}(1+t)^{2p+1}.&\end{aligned}$$ Hence thesis follows integrating in $[0,t]$ and using (\[decv\]).
#####
As in the previous case we have $$\left(\frac{1}{2}{|A^{1/2}u|}^{6\gamma}|A^{3/2}u|^{2}(1+t)^{6p}\right)'
+ {|A^{1/2}u|}^{8\gamma}|A^{2}u|^{2}(1+t)^{7p} \leq
3{|A^{1/2}u|}^{6\gamma}|A^{3/2}u|^{2}(1+t)^{5p}.
\label{Iv1}$$ Moreover from (\[dv0\]) with $k=2$ we get $$\begin{aligned}
&
\displaystyle \left(\frac{1}{2}{|A^{1/2}u|}^{4\gamma}|A^{3/2}u|^{2}(1+t)^{4p}\right)'
+ {|A^{1/2}u|}^{6\gamma}|A^{2}u|^{2}(1+t)^{5p} \leq &\nonumber \\
& 2p{|A^{1/2}u|}^{4\gamma}|A^{3/2}u|^{2}(1+t)^{3p} \leq
2\displaystyle \frac{|A^{3/2}u_{0}|^{2}}{|Au_{0}|^{2}}{|A^{1/2}u|}^{4\gamma}|Au|^{2}(1+t)^{3p}.&
\label{Iv2}\end{aligned}$$ Summing up (\[Iv1\]) and (\[Iv2\]), integrating in $[0,t]$ and using (\[dv6\]) and (\[dv5\]) we end up with (\[dv7\]).
Proof of Theorem \[th:ex\] {#ex}
--------------------------
As in the previous section in most of the proofs we omit the dependence of ${u_{{\varepsilon}}}$ from $t$ and we do not recall more that $0\leq p \leq 1$. We divide the proof into three parts. In the first one we state and prove the energy estimates we need, then we prove the existence of global solutions and finally we give the decay estimates.
### Basic energy estimates
In this section we prove some estimates that involve the following energies: $$\begin{aligned}
Q_{{\varepsilon}}(t) & = &
\frac{|{u_{{\varepsilon}}}'(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{1-p};
\label{defQ} \\
D_{{\varepsilon}}(t) & = & {\varepsilon}\frac{\langle {u_{{\varepsilon}}}'(t),
{u_{{\varepsilon}}}''(t)\rangle}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{2p+1} + \int_{0}^{t}
\frac{{|A^{1/2}{u_{{\varepsilon}}}'(s)|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|^{2}}}(1+s)^{2p+1}ds; \label{defD} \\
R_{{\varepsilon}}(t) & = & \left[{\varepsilon}\frac{|{u_{{\varepsilon}}}''(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}} +
\frac{{|A^{1/2}{u_{{\varepsilon}}}'(t)|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}}\right](1+t)^{2(p+1)};
\label{defR} \\
H_{{\varepsilon}}(t) & = & \left[{\varepsilon}\frac{{|A^{1/2}{u_{{\varepsilon}}}'(t)|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}} +
{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma-1)}|A{u_{{\varepsilon}}}(t)|^{2}\right](1+t)^{p+1}.
\label{defH} \end{aligned}$$
Let us moreover set: $$h_{1}:= 4(|u_{1}|^{2}
+{|A^{1/2}u_{0}|}^{4\gamma}|Au_{0}|^{2}){|A^{1/2}u_{0}|}^{-2(\gamma+1)}, \hspace{1em}h_{2}:= (\gamma - 1)(\sqrt{h_{1}} +1) + \sqrt{\gamma -
1},$$ $$L_{1}:= \left\{
\begin{array}{ll}
(3 + 2h_{2}(\sqrt{h_{1}} +1))h_{2}^{2}(\gamma - 1)^{-2} +
H_{1}(0)+1&
\hspace{1em} \mbox{if $\gamma > 1$} \\
\\
36 + 2 {|A^{1/2}u_{1}|^{2}}{|A^{1/2}u_{0}|}^{-2}+ 2|Au_{0}|^{2} + 2^{-1}|\langle
Au_{0}, u_{1}\rangle|{|A^{1/2}u_{0}|}^{-2} & \hspace{1em} \mbox{if
$\gamma = 1$}.
\end{array}\right.$$ In the following proposition we recollect all the estimates on (\[defQ\]) trough (\[defH\]) we need.
\[priori\] Let us assume that all the hypotheses of Theorem \[th:ex\] are verified. Then there exists ${\varepsilon}_{0}$ with the following property. If ${\varepsilon}\in]0,{\varepsilon}_{0}]$, $S > 0$ and $${u_{{\varepsilon}}}\in C^{2}([0,S[;H)\cap C^{1}([0,S[;{D(A^{1/2})}) \cap
C^{0}([0,S[;{D(A)})$$ is a solution of (\[pbm:h-eq\]), (\[pbm:h-data\]) such that $${|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}> 0 \hspace{1em}
\forall t\in[0,S[,
\label{fond1}$$ $$\frac{|\langle {u_{{\varepsilon}}}'(t),
A{u_{{\varepsilon}}}(t)\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}} \leq
\frac{K_{0}}{1+t}, \hspace{1em}
{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma-1)}|A{u_{{\varepsilon}}}(t)|^{2} \leq
\frac{K_{1}}{(1+t)^{p+1}}, \;\; \forall t\in[0,S[,
\label{fond}$$ then there exists a positive constant $L_{3}$ independent from ${\varepsilon}$ and $S$ such that for every $t\in[0,S[$: $$Q_{{\varepsilon}}(t) \leq \max\{4K_{1}, Q_{1}(0)\}=: L_{2}; \label{SQ}$$ $$D_{{\varepsilon}}(t) \leq D_{{\varepsilon}}(0) + 2 L_{2}(3+2K_{0})
(1+t)^{p+1} +\frac{1}{8(K_{0}+1)} \int_{0}^{t}
\frac{|{u_{{\varepsilon}}}''(s)|^{2} (1+s)^{2p+1}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}}
ds; \label{SD}$$ $$\left[{\varepsilon}\frac{|{u_{{\varepsilon}}}''(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}} +
\frac{{|A^{1/2}{u_{{\varepsilon}}}'(t)|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}}\right](1+t)^{p+1}=
\frac{R_{{\varepsilon}}(t)}{(1+t)^{p+1}} \leq L_{3}+ 2R_{{\varepsilon}}(0);
\label{SR}$$ $$H_{{\varepsilon}}(t) \leq L_{1}.
\label{SH}$$
[[Proof.]{}$\;$]{}Let us set: $$h_{3}:=
4(4\gamma^{2}K_{0}^{2}K_{1} + L_{2}),\hspace{1em}
h_{4}:=h_{3}+ 8(K_{0}+1)(3+2K_{0})L_{2},
\label{defsigma1}$$ $$L_{3}:= 2\left[h_{4} + \frac{L_{2}}{2} + 4
\frac{|u_{1}|}{{|A^{1/2}u_{0}|}^{2(\gamma+1)}}(|u_{1}| +
{|A^{1/2}u_{0}|}^{2\gamma}|Au_{0}|)(K_{0}+1)\right].
\label{defL2}$$
Now let us assume that ${\varepsilon}_{0}$ verifies the following inequalities: $$8{\varepsilon}_{0}(2+(\gamma + 1)K_{0} )\leq 1, \quad\quad
16{\varepsilon}_{0}(K_{0}+1)^{2}\leq 1,
\label{1-ep}$$ $$2{\varepsilon}_{0}(K_{0}+1)(1 + (3 + 2(\gamma + 1)K_{0})^{2})\leq 1/8,
\label{2-ep}$$ $$\sqrt{{\varepsilon}_{0}}\left(\sqrt{L_{3}} +\sqrt{2}
\frac{{|A^{1/2}u_{1}|}}{{|A^{1/2}u_{0}|}}\right) \leq 1.
\label{3-ep}$$
Let us now compute the time’s derivatives of the energies (\[defQ\]) through (\[defH\]). After some computation we find that:
$$Q_{{\varepsilon}}'= Q_{{\varepsilon}}\left(\frac{(1-p)}{1+t}
-2(\gamma+1)\frac{\langle A{u_{{\varepsilon}}}, {u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}
-\frac{2}{{\varepsilon}}\frac{1}{(1+t)^{p}}\right)
-\frac{2}{{\varepsilon}}(1+t)^{1-p}\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}.
\label{derQ}$$
Let us set: $$\begin{aligned}
\varphi_{1}(t) &:= & {\varepsilon}\frac{|{u_{{\varepsilon}}}''(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{2p+1} +\\
& & + {\varepsilon}\left[ \frac{\langle {u_{{\varepsilon}}}'(t),
{u_{{\varepsilon}}}''(t)\rangle}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{2p}\left(2p+1
-2(\gamma+1)\frac{\langle A{u_{{\varepsilon}}}(t),
{u_{{\varepsilon}}}'(t)\rangle}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}}(1+t)\right)\right],\end{aligned}$$ $$\varphi_{2}(t) :=- 2\gamma\left(\frac{\langle A{u_{{\varepsilon}}}(t),
{u_{{\varepsilon}}}'(t)\rangle}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}}\right)^{2}(1+t)^{2p+1},$$ $$\varphi_{3}(t):= - \frac{\langle {u_{{\varepsilon}}}'(t),
{u_{{\varepsilon}}}''(t)\rangle}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{p+1},$$ $$\varphi_{4}(t):= p \frac{|{u_{{\varepsilon}}}'(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}
(1+t)^{p};$$ thus $$D_{{\varepsilon}}'
= \varphi_{1} + \varphi_{2} + \varphi_{3} +
\varphi_{4}.
\label{derD}$$ Moreover $$\begin{aligned}
R_{{\varepsilon}}' & = & 2(1+t)^{2(p+1)}\left(-2\gamma \frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|}^{4}}\langle {u_{{\varepsilon}}}'', A{u_{{\varepsilon}}}\rangle -
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}})|}^{2(\gamma+1)}}\frac{1}{(1+t)^{p}}\right)+
\nonumber \\
& & +2(1+t)^{2(p+1)}\left( p\frac{\langle {u_{{\varepsilon}}}',
{u_{{\varepsilon}}}''\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}\frac{1}{(1+t)^{p+1}}-
\frac{\langle A{u_{{\varepsilon}}}, {u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}
\frac{{|A^{1/2}{u_{{\varepsilon}}}'|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} \right) + \nonumber \\
& & - 2(\gamma+1){\varepsilon}(1+t)^{2(p+1)}\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}
+2(p+1)\frac{R_{{\varepsilon}}}{1+t};
\label{derR}\end{aligned}$$ $$\begin{aligned}
H_{{\varepsilon}}' & = & \frac{p+1}{1+t}H_{{\varepsilon}} -2 (1+t)^{p+1}
\frac{{|A^{1/2}{u_{{\varepsilon}}}'|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}\left(\frac{1}{(1+t)^{p}} +
{\varepsilon}\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} \right)+
\nonumber \\
& & +2(\gamma - 1)(1+t)^{p+1}\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} {|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}|A{u_{{\varepsilon}}}|^{2}.
\label{derH}\end{aligned}$$ We are now ready to prove (\[SQ\]) trough (\[SH\]).
#####
Thanks to (\[derQ\]) we have $$Q_{{\varepsilon}}'\leq
-\frac{1}{{\varepsilon}}Q_{{\varepsilon}}\left(\frac{2}{(1+t)^{p}}-\frac{{\varepsilon}(1-p)}{1+t} +
2{\varepsilon}(\gamma+1)\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}\right) +
\frac{2}{{\varepsilon}}(1+t)^{1-p}\frac{|{u_{{\varepsilon}}}'||A{u_{{\varepsilon}}}|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}.$$ Moreover by (\[fond\]) and (\[1-ep\]) we get $$\begin{aligned}
\frac{2}{(1+t)^{p}}-\frac{{\varepsilon}(1-p)}{1+t} +
2{\varepsilon}(\gamma+1)\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} & \geq & \frac{1}{(1+t)^{p}}(2 - {\varepsilon}-2{\varepsilon}(\gamma+1)K_{0}) \\
& \geq & \frac{7}{4}\frac{1}{(1+t)^{p}} \geq \frac{1}{(1+t)^{p}}.\end{aligned}$$ Hence, using (\[fond\]) once again, we obtain $$\begin{aligned}
Q_{{\varepsilon}}' & \leq & -\frac{1}{{\varepsilon}}\frac{1}{(1+t)^{p}}Q_{{\varepsilon}} +
\frac{2}{{\varepsilon}}\frac{1}{(1+t)^{p}}\sqrt{Q_{{\varepsilon}}}|A{u_{{\varepsilon}}}|{|A^{1/2}{u_{{\varepsilon}}}|}^{\gamma -
1}(1+t)^{(p+1)/2} \\
& \leq & -\frac{1}{{\varepsilon}}\frac{1}{(1+t)^{p}}\sqrt{Q_{{\varepsilon}}} (\sqrt{Q_{{\varepsilon}}}
-2\sqrt{K_{1}}).\end{aligned}$$ Therefore, since $Q_{{\varepsilon}}(0) = Q_{1}(0)$, thesis follows from Statement (\[ODEA1\]) in Lemma \[lemma:ode\].
#####
Thanks to (\[SQ\]), for all $\alpha(t)> 0$ it holds true that in $[0,S[$: $$\begin{aligned}
\frac{|\langle {u_{{\varepsilon}}}'(t),
{u_{{\varepsilon}}}''(t)\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}} & \leq &
\frac{1}{2}\alpha(t)
\frac{|{u_{{\varepsilon}}}''(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}
+ \frac{1}{2\alpha(t)}
\frac{|{u_{{\varepsilon}}}'(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}
\nonumber \\
& \leq & \frac{1}{2}\alpha(t)
\frac{|{u_{{\varepsilon}}}''(t)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}} +
\frac{1}{2\alpha(t)}\frac{L_{2}}{(1+t)^{1-p}}.
\label{SIn}\end{aligned}$$
#####
Let us estimate separately the terms in (\[derD\]).
Using (\[fond\]) and (\[SIn\]) with $\alpha(t) = (1+t)(3 +
2(\gamma+1)K_{0})$ we obtain $$\begin{aligned}
|\varphi_{1}| & \leq &
{\varepsilon}\left[\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1} +
\frac{|\langle {u_{{\varepsilon}}}',
{u_{{\varepsilon}}}''\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}} (3 +
2(\gamma+1)K_{0})(1+t)^{2p}\right] \\
& \leq &
{\varepsilon}\left[\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1}
\left(1+ \frac{1}{2}(3+2(\gamma+1)K_{0})^{2}\right) +
\frac{1}{2}\frac{L_{2}}{(1+t)^{2-3p}}\right]. \end{aligned}$$ Thus from the smallness assumption (\[2-ep\]) we get $$|\varphi_{1}|\leq \frac{1}{16(K_{0}+1)}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1}+
\frac{1}{2}L_{2}(1+t)^{p}.
\label{Sphi1}$$ From (\[SIn\]) with $\alpha(t) = (1+t)^{p}(8(K_{0}+1))^{-1}$ we have $$|\varphi_{3}| \leq \frac{1}{16(K_{0}+1)}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1}
+ 4(K_{0}+1)L_{2}(1+t)^{p}.
\label{Sphi3}$$ Moreover from (\[SQ\]) we get $$|\varphi_{4}| \leq \frac{L_{2}}{(1+t)^{1-p}}(1+t)^{p} \leq
L_{2}(1+t)^{p}.
\label{Sphi4}$$ Finally replacing (\[Sphi1\]), (\[Sphi3\]), (\[Sphi4\]) in (\[derD\]), since $\varphi_{2}\leq 0$ we obtain $$D_{{\varepsilon}}' \leq \frac{1}{8(K_{0}+1)}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1} +
\left(\frac{11}{2} + 4K_{0}\right)L_{2}(1+t)^{p}.$$ Hence (\[SD\]) follows from a simple integration.
#####
Firstly let us estimate some of the terms in (\[derR\]).
Thanks to (\[fond\]) we have $$\begin{aligned}
2\gamma \frac{|\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} \frac{|\langle {u_{{\varepsilon}}}'',
A{u_{{\varepsilon}}}\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}& \leq & 2\gamma \frac{K_{0}}{1+t}
\frac{|{u_{{\varepsilon}}}''|}{{|A^{1/2}{u_{{\varepsilon}}}|}^{(\gamma+1)}}|A{u_{{\varepsilon}}}|{|A^{1/2}{u_{{\varepsilon}}}|}^{\gamma-1}
\nonumber \\
& \leq &
\frac{1}{8}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}\frac{1}{(1+t)^{p}}+
\frac{8\gamma^{2}K_{0}^{2}}{(1+t)^{2-p}}|A{u_{{\varepsilon}}}|^{2}
{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}
\nonumber\\
& \leq & \frac{1}{8}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}\frac{1}{(1+t)^{p}}+
\frac{8\gamma^{2}K_{0}^{2}K_{1}}{(1+t)^{3}}.
\label{1-SR}
\end{aligned}$$ Moreover from (\[SIn\]) with $\alpha(t) = (1+t)/4$ we achieve $$\frac{p}{(1+t)^{p+1}} \frac{|\langle {u_{{\varepsilon}}}',
{u_{{\varepsilon}}}''\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}} \leq
\frac{1}{8}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}\frac{1}{(1+t)^{p}}+
\frac{2L_{2}}{(1+t)^{3}}.
\label{2-SR}$$ Using once again (\[fond\]) we have $$\frac{|\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}\frac{{|A^{1/2}{u_{{\varepsilon}}}'|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}
\leq \frac{K_{0}}{1+t}\frac{{|A^{1/2}{u_{{\varepsilon}}}'|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}.
\label{4-SR}$$ Finally from (\[fond\]) and (\[1-ep\]) we get also $$\begin{aligned}
{\varepsilon}(\gamma+1)\frac{|\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}
& \leq & {\varepsilon}\frac{(\gamma + 1)K_{0}}{1+t}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}
\nonumber \\
& \leq & \frac{1}{8}
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}\frac{1}{(1+t)^{p}}.
\label{3-SR}
\end{aligned}$$
Replacing (\[1-SR\]), (\[2-SR\]), (\[4-SR\]), (\[3-SR\]) in (\[derR\]) and using (\[1-ep\]) and (\[defsigma1\]) we thus obtain $$\begin{aligned}
R_{{\varepsilon}}' & \leq & -
\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}
(1+t)^{p+2}\left(\frac{5}{4}-\frac{2{\varepsilon}(p+1)}{(1+t)^{1-p}}\right)
+
\\
& & +2(K_{0}+p+1)\frac{{|A^{1/2}{u_{{\varepsilon}}}'|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}(1+t)^{2p+1} +
\\
& &
+2(8\gamma^{2}K_{0}^{2}K_{1}+2L_{2})\frac{(1+t)^{p}}{(1+t)^{1-p}} \\
& \leq & -\frac{|{u_{{\varepsilon}}}''|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}
(1+t)^{p+2} + 4(K_{0}+1)\frac{{|A^{1/2}{u_{{\varepsilon}}}'|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}(1+t)^{2p+1}
+ h_{3}(1+t)^{p}.
\end{aligned}$$ Then integrating in $[0,t]$ and using (\[SD\]), since $2p+1 \leq p+2$ we find $$\begin{aligned}
& R_{{\varepsilon}}(t) + \displaystyle \int_{0}^{t}
\frac{|{u_{{\varepsilon}}}''(s)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}}
(1+s)^{p+2}ds
& \nonumber\\
& \leq 4(K_{0}+1)
\displaystyle \int_{0}^{t}\frac{{|A^{1/2}{u_{{\varepsilon}}}'(s)|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|^{2}}}(1+s)^{2p+1} ds
+ \frac{h_{3}}{1+p}(1+t)^{p+1} + R_{{\varepsilon}}(0) & \nonumber\\
& \leq -4(K_{0}+1)\left[{\varepsilon}\displaystyle\frac{\langle {u_{{\varepsilon}}}'(t),
{u_{{\varepsilon}}}''(t)\rangle}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{2p+1} - D_{{\varepsilon}}(0)\right] +
& \nonumber\\
& +\displaystyle \frac{1}{2}\displaystyle \int_{0}^{t}
\frac{|{u_{{\varepsilon}}}''(s)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}}
(1+s)^{p+2}ds+ & \nonumber \\
& \displaystyle + 8(K_{0}+1)(3+2K_{0})L_{2}(1+t)^{p+1}+
\frac{h_{3}}{1+p}(1+t)^{p+1} + R_{{\varepsilon}}(0). \nonumber \label{stimaR}\end{aligned}$$ From this inequality and (\[defsigma1\]) it follows that: $$\begin{aligned}
& R_{{\varepsilon}}(t) + \displaystyle \frac{1}{2}\int_{0}^{t}
\frac{|{u_{{\varepsilon}}}''(s)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}}
(1+s)^{p+2}ds \leq R_{{\varepsilon}}(0) + 4(K_{0}+1)|D_{{\varepsilon}}(0)| +&
\nonumber \\
& +4(K_{0}+1){\varepsilon}\displaystyle\frac{|\langle {u_{{\varepsilon}}}'(t),
{u_{{\varepsilon}}}''(t)\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}(1+t)^{2p+1} +
h_{4}(1+t)^{p+1}.&
\label{5-SR}\end{aligned}$$ Let us now estimate the terms in the right hand side. Let us remark that from (\[SIn\]) with $\alpha(t) =
4{\varepsilon}(K_{0}+1)(1+t)$ we get $$4(K_{0}+1){\varepsilon}\displaystyle\frac{|\langle {u_{{\varepsilon}}}',
{u_{{\varepsilon}}}''\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1} \leq
8{\varepsilon}^{2}(K_{0}+1)^{2}\frac{|{u_{{\varepsilon}}}''|^{2}(1+t)^{2p+2}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}
+ \frac{L_{2}}{2}(1+t)^{3p-1}.$$ Moreover $3p - 1 \leq 2p\leq p+1$, hence using also (\[1-ep\]) we deduce $$4(K_{0}+1){\varepsilon}\displaystyle\frac{|\langle {u_{{\varepsilon}}}',
{u_{{\varepsilon}}}''\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}}(1+t)^{2p+1} \leq
\frac{1}{2}R_{{\varepsilon}} + \frac{1}{2}L_{2}(1+t)^{p+1}.
\label{6-SR}$$ Furthermore we have $$\begin{aligned}
|D_{{\varepsilon}}(0)| & \leq & {\varepsilon}|u''(0)|\frac{|u_{1}|}{{|A^{1/2}u_{0}|}^{2(\gamma+1)}} =
|u_{1}+{|A^{1/2}u_{0}|}^{2\gamma}Au_{0}|\frac{|u_{1}|}{{|A^{1/2}u_{0}|}^{2(\gamma+1)}}
\nonumber \\
& \leq & (|u_{1}| +{|A^{1/2}u_{0}|}^{2\gamma}|Au_{0}|)
\frac{|u_{1}|}{{|A^{1/2}u_{0}|}^{2(\gamma+1)}}.
\label{7-SR}\end{aligned}$$ Plugging (\[6-SR\]) and (\[7-SR\]) in (\[5-SR\]) and recalling the definition in (\[defL2\]) we get $$\begin{aligned}
& \displaystyle\frac{1}{2} R_{{\varepsilon}}(t) + \displaystyle \frac{1}{2}\int_{0}^{t}
\frac{|{u_{{\varepsilon}}}''(s)|^{2}}{{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}}
(1+s)^{p+2}ds & \\
&\leq \left( R_{{\varepsilon}}(0) +
4(K_{0}+1)|D_{{\varepsilon}}(0)| + h_{4} +
\displaystyle \frac{1}{2}L_{2}\right)(1+t)^{p+1} & \\
& \leq \left( R_{{\varepsilon}}(0) + \displaystyle \frac{1}{2}L_{3}\right)(1+t)^{p+1}. &\end{aligned}$$ Therefore inequality (\[SR\]) is proved.
#####
If $\gamma = 1$, thesis, eventually for smaller values of ${\varepsilon}$, follows from Theorem 2.2 in [@jde2] (in particular it is a consequence of (3.52)). Then from now let us assume that $\gamma > 1$. Let us set $$F_{{\varepsilon}}(t):= {\varepsilon}\frac{{|A^{1/2}{u_{{\varepsilon}}}'(t)|^{2}}}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}} (1+t)^{p+1},
\hspace{1em}G_{{\varepsilon}}(t):=
{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma-1)}|A{u_{{\varepsilon}}}(t)|^{2} (1+t)^{p+1}$$ so that $H_{{\varepsilon}} = F_{{\varepsilon}} + G_{{\varepsilon}}$. Hence exploiting (\[fond\]) in (\[derH\]) we obtain: $$H_{{\varepsilon}}' \leq -\frac{2}{{\varepsilon}}F_{{\varepsilon}}\left(\frac{1}{(1+t)^{p}}
-{\varepsilon}(K_{0}+p+1)\frac{1}{1+t}
\right) +\frac{p+1}{1+t}G_{{\varepsilon}} + 2(\gamma -1) \frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} G_{{\varepsilon}}.$$ Thanks to (\[1-ep\]) we have $1 - {\varepsilon}(K_{0}+2) \geq 1/2$, therefore we get $$H_{{\varepsilon}}' \leq -\frac{F_{{\varepsilon}}}{{\varepsilon}}\frac{1}{1+t}
+\frac{2}{1+t}G_{{\varepsilon}} + 2(\gamma -1) \frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} G_{{\varepsilon}}.
\label{1-SH}$$ Now let us recall that $${u_{{\varepsilon}}}' = -(1+t)^{p}({\varepsilon}{u_{{\varepsilon}}}'' + {|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma}A{u_{{\varepsilon}}}),$$ hence $$\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}'\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} = -{\varepsilon}\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}''\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}(1+t)^{p} - \frac{1}{1+t}G_{{\varepsilon}}.$$ Plugging this identity in (\[1-SH\]), we arrive at $$H_{{\varepsilon}}'\leq -\frac{1}{1+t}\left(\frac{F_{{\varepsilon}}}{{\varepsilon}}
-2G_{{\varepsilon}} + 2(\gamma - 1)G_{{\varepsilon}}^{2} +2{\varepsilon}(\gamma -
1)(1+t)^{p+1}G_{{\varepsilon}}\frac{\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}''\rangle}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}}\right).
\label{3-SH}$$ Let us estimate the last term in (\[3-SH\]). Using (\[SR\]) we obtain $$\begin{aligned}
2{\varepsilon}(\gamma -
1)(1+t)^{p+1}\frac{| A{u_{{\varepsilon}}}|
|{u_{{\varepsilon}}}''|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} & = & 2(\gamma - 1){\varepsilon}\frac{|{u_{{\varepsilon}}}''|}{{|A^{1/2}{u_{{\varepsilon}}}|}^{\gamma+1}}(1+t)^{(1+p)/2}\sqrt{G_{{\varepsilon}}}
\nonumber \\
& \leq &2(\gamma - 1) \sqrt{{\varepsilon}}\sqrt{G_{{\varepsilon}}}\sqrt{2R_{{\varepsilon}}(0) +
L_{3}}
\nonumber \\
& \leq & 2(\gamma -
1)\sqrt{G_{{\varepsilon}}}\left(\sqrt{2{\varepsilon}R_{{\varepsilon}}(0) }+
\sqrt{{\varepsilon}}\sqrt{L_{3}}\right).
\label{4-SH}
\end{aligned}$$ By the definition of $h_{1}$ moreover it follows that: $$\begin{aligned}
2{\varepsilon}R_{{\varepsilon}}(0) & =
&2{\varepsilon}^{2}\frac{|{u_{{\varepsilon}}}''(0)|^{2}}
{{|A^{1/2}u_{0}|}^{2(\gamma+1)}}+2{\varepsilon}\frac{{|A^{1/2}u_{1}|^{2}}}{{|A^{1/2}u_{0}|^{2}}}
\nonumber \\
& \leq &
4 (|u_{1}|^{2}+{|A^{1/2}u_{0}|}^{4\gamma}|Au_{0}|^{2})
\frac{1}{{|A^{1/2}u_{0}|}^{2(\gamma+1)}}
+2{\varepsilon}\frac{{|A^{1/2}u_{1}|^{2}}}{{|A^{1/2}u_{0}|^{2}}}
\nonumber \\
& = & h_{1} +2{\varepsilon}\frac{{|A^{1/2}u_{1}|^{2}}}{{|A^{1/2}u_{0}|^{2}}}.
\label{5-SH}
\end{aligned}$$ Using (\[3-ep\]) and (\[5-SH\]), from (\[4-SH\]) we get $$2{\varepsilon}(\gamma - 1)(1+t)^{p+1}\frac{|\langle A{u_{{\varepsilon}}},
{u_{{\varepsilon}}}''\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}|^{2}}} \leq 2(\gamma - 1)\sqrt{G_{{\varepsilon}}}
(\sqrt{h_{1}} + 1).$$ Plugging this estimate in (\[3-SH\]), since ${\varepsilon}\leq 1$ we finally achieve $$H_{{\varepsilon}}' \leq -\frac{1}{1+t}\left(F_{{\varepsilon}} + 2(\gamma - 1)G_{{\varepsilon}}^{2}
-2G_{{\varepsilon}}
-2(\gamma-1)G_{{\varepsilon}}^{3/2}(\sqrt{h_{1}} + 1)\right).$$ Since $H_{{\varepsilon}}(0)\leq H_{1}(0)$, inequality (\[SH\]) follows recalling the definition of $L_{1}$ and applying Lemma \[lemma:ode-GF\] with $$a=2(\gamma-1), \hspace{1em} b = 2, \hspace{1em}
c= 2(\gamma -1)(\sqrt{h_{1}} + 1).$$ [[10000]{}]{}
### Existence of global solutions
#####
Problem (\[pbm:h-eq\]), (\[pbm:h-data\]) admits a unique local-in-time solution, and this solution can be continued to a solution defined in a maximal interval $[0,T[$, where either $T=+\infty$, or $$\limsup_{t\to T^{-}}\left(|A^{1/2}{u_{{\varepsilon}}}'(t)|^{2}+
|A{u_{{\varepsilon}}}(t)|^{2}\right)=+\infty,
\label{limsup}$$ or $$\liminf_{t\to T^{-}}|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}=0.
\label{liminf}$$
We omit the proof of this standard result. The interested reader is referred to [@gg:k-dissipative].
#####
Let us recall that problem (\[pbm:h-eq\]), (\[pbm:h-data\]) admits a first order conserved energy, that is $${\varepsilon}|{u_{{\varepsilon}}}'(t)|^{2} + \frac{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}{\gamma+1}
+2\int_{0}^{t}\frac{|{u_{{\varepsilon}}}'(s)|^{2}}{(1+s)^{p}}ds = {\varepsilon}|u_{1}|^{2}
+\frac{{|A^{1/2}u_{0}|}^{2(\gamma+1)}}{\gamma+1}.
\label{conserv}$$ Therefore ${|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}$ is bounded independently from ${\varepsilon}\leq 1$.
#####
We want to apply Proposition \[priori\]. To this end let us set: $$K_{1}:= L_{1}+1+H_{1}(0), \hspace{1em} K_{0}:=\sqrt{\max\{4K_{1},
Q_{1}(0)\}K_{1}} + \frac{|\langle Au_{0},
u_{1}\rangle|}{{|A^{1/2}u_{0}|^{2}}}.
\label{defk}$$ For such choices of $K_{0}$ and $K_{1}$ let us define $$S:=\sup\{\tau\in[0,T[:\; \mbox{(\ref{fond1}), (\ref{fond}) are
verified for all $ t\in [0,\tau]$}\}.$$
Firstly let us remark that since ${|A^{1/2}u_{0}|^{2}} > 0$, for our choices of $K_{0}$ and $K_{1}$ and ${\varepsilon}\leq 1$ we have $S > 0$. From now furthermore we assume that ${\varepsilon}$ verifies the smallness conditions of Proposition \[priori\]. Thus in $[0,S[$ Proposition \[priori\] holds true.
We want to prove that $S = T$.
Let us assume by contradiction that $S < T$. Then by the regularity properties of ${u_{{\varepsilon}}}$ all the estimates in Proposition \[priori\] hold true in $[0,S]$. Moreover at least one of the following is verified: $${|A^{1/2}{u_{{\varepsilon}}}(S)|^{2}} = 0,
\label{exc1}$$ $$\frac{|\langle {u_{{\varepsilon}}}'(S),
A{u_{{\varepsilon}}}(S)\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}(S)|^{2}}} =
\frac{K_{0}}{1+S},
\label{exc2}$$ $${|A^{1/2}{u_{{\varepsilon}}}(S)|}^{2(\gamma-1)}|A{u_{{\varepsilon}}}(S)|^{2} =
\frac{K_{1}}{(1+S)^{p+1}}.
\label{exc3}$$
#####
Let us set $y(t):={|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}$. Hence by (\[fond\]) in $[0,S[$ we have: $$\frac{y'(t)}{y(t)}\geq -\frac{2K_{0}}{1+t}$$ therefore $$y(t) \geq y(0) e^{-2K_{0}\log(1+t)} =
\frac{{|A^{1/2}u_{0}|^{2}}}{(1+t)^{2K_{0}}};
\label{decy}$$ in particular ${|A^{1/2}{u_{{\varepsilon}}}(S)|^{2}} = y(S) > 0$.
#####
From (\[SQ\]) and (\[SH\]), recalling (\[defk\]) and the definition of $L_{2}$ in Proposition \[priori\] we indeed have $$\begin{aligned}
\frac{|\langle {u_{{\varepsilon}}}'(S),
A{u_{{\varepsilon}}}(S)\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}(S)|^{2}}} & \leq & \frac{|{u_{{\varepsilon}}}'(S)|}{{|A^{1/2}{u_{{\varepsilon}}}(S)|}^{\gamma+1}}
|A{u_{{\varepsilon}}}(S)| {|A^{1/2}{u_{{\varepsilon}}}(S)|}^{\gamma-1} \\
& \leq & \frac{\sqrt{L_{2}}}{(1+S)^{(1-p)/2}}
\frac{\sqrt{L_{1}}}{(1+S)^{(1+p)/2}}
=
\frac{\sqrt{L_{2}L_{1}}}{1+S}<\frac{\sqrt{L_{2}K_{1}}}{1+S}
\leq \frac{K_{0}}{1+S}.\end{aligned}$$
#####
This is an immediate consequence of (\[SH\]).
#####
We have proved that if ${\varepsilon}$ is small enough hence $S = T$ and in $[0,T[$ Proposition \[priori\] holds true. We need only to prove that $T = +\infty$. If it is not the case hence (\[limsup\]) or (\[liminf\]) hold true. Nevertheless (\[liminf\]) is excluded by (\[decy\]). Now let us roll out (\[limsup\]). From (\[conserv\]) we know that ${|A^{1/2}{u_{{\varepsilon}}}|^{2}}$ is bounded from above, hence from (\[SH\]) we deduce that ${|A^{1/2}{u_{{\varepsilon}}}'|^{2}}$ is bounded. Moreover thanks to (\[decy\]) ${|A^{1/2}{u_{{\varepsilon}}}|^{2}}$ is bounded also from below, thus using once again (\[SH\]) we get that $|A{u_{{\varepsilon}}}|^{2}$ is bounded. Hence (\[limsup\]) is false.
### Decay estimates
Since now inequalities (\[SH\]) and (\[SQ\]) hold true in $[0,+\infty[$, we have already proved (\[D2\]) and (\[D3\]). The estimate from below in (\[D1\]) is a consequence of Proposition 3.3 (see (3.21)) in [@jde2], because we have proved that $$\frac{|\langle {u_{{\varepsilon}}}'(t),
A{u_{{\varepsilon}}}(t)\rangle|}{{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}} \leq \frac{K_{0}}{1+t}
\hspace{1em} \forall t \geq 0.$$
#####
Let us recall that we have already supposed that the smallness assumption (\[1-ep\]) is verified. We work as in [@jde2], Section 3.4, hence we skip the details. Let us set $$\mathcal{D}_{{\varepsilon}}(t):=
{\varepsilon}(1+t)^{p}\langle{u_{{\varepsilon}}}'(t),{u_{{\varepsilon}}}(t)\rangle + \frac{1}{2}\left(1 -
\frac{ {\varepsilon}p}{(1+t)^{1-p}}\right)|{u_{{\varepsilon}}}(t)|^{2}.$$ Since $$\mathcal{D}_{{\varepsilon}}' = -(1+t)^{p}{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)} +
{\varepsilon}(1+t)^{p}|{u_{{\varepsilon}}}'|^{2} +
{\varepsilon}\frac{p(1-p)}{2}\frac{|{u_{{\varepsilon}}}|^{2}}{(1+t)^{2-p}},$$ a simple integration gives $$\begin{aligned}
\int_{0}^{t}(1+s)^{p}{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}ds & = &
\mathcal{D}_{{\varepsilon}}(0) - \mathcal{D}_{{\varepsilon}}(t) +
{\varepsilon}\int_{0}^{t}(1+s)^{p}|{u_{{\varepsilon}}}'(s)|^{2}ds \\
& & +
{\varepsilon}\frac{p(1-p)}{2}\int_{0}^{t}\frac{|{u_{{\varepsilon}}}(s)|^{2}}{(1+s)^{2-p}}ds.\end{aligned}$$ Moreover, since $1-3{\varepsilon}\geq 1/4$, it holds true that $$- \mathcal{D}_{{\varepsilon}}(t) \leq
\frac{1}{4}{\varepsilon}(1+t)^{p+1}|{u_{{\varepsilon}}}'(t)|^{2} -
\frac{1}{8}|{u_{{\varepsilon}}}(t)|^{2},$$ hence we get: $$\begin{aligned}
& \displaystyle \frac{1}{8}|{u_{{\varepsilon}}}(t)|^{2} +
\int_{0}^{t}(1+s)^{p}{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2(\gamma+1)}ds \leq
|\mathcal{D}_{{\varepsilon}}(0)| +
\frac{1}{4}{\varepsilon}(1+t)^{p+1}|{u_{{\varepsilon}}}'(t)|^{2}+&
\nonumber \\
& + \displaystyle {\varepsilon}\int_{0}^{t}(1+s)|{u_{{\varepsilon}}}'(s)|^{2}ds +
{\varepsilon}\frac{p(1-p)}{2}\int_{0}^{t}\frac{|{u_{{\varepsilon}}}(s)|^{2}}{(1+s)^{2-p}}ds.&
\label{STD}
\end{aligned}$$ Let us now define $$\mathcal{E}_{{\varepsilon}}(t):= \left({\varepsilon}|{u_{{\varepsilon}}}'(t)|^{2} +
\frac{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}{\gamma+1}\right)(1+t)^{p+1}.$$ A simple computation gives $$\mathcal{E}_{{\varepsilon}}' = -(1+t)\left(2 -
\frac{{\varepsilon}(p+1)}{(1+t)^{1-p}} \right)|{u_{{\varepsilon}}}'|^{2}+
\frac{p+1}{\gamma+1}(1+t)^{p}{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma+1)}.$$ Integrating in $[0,t]$ and using (\[STD\]) we arrive at $$\begin{aligned}
& \displaystyle (1+t)^{p+1}\left(1 -
\frac{p+1}{4(\gamma+1)}\right){\varepsilon}|{u_{{\varepsilon}}}'(t)|^{2} +
\frac{{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma+1)}}{\gamma+1}(1+t)^{p+1} \leq
& \\
&\displaystyle \leq \mathcal{E}_{{\varepsilon}}(0) -
\left(2 - {\varepsilon}(p+1) - {\varepsilon}\frac{p+1}{\gamma+1}\right)
\int_{0}^{t}(1+s)|{u_{{\varepsilon}}}'(s)|^{2}ds + &\\
& \displaystyle + \frac{p+1}{\gamma+1}\left( |\mathcal{D}_{{\varepsilon}}(0)| -
\displaystyle \frac{1}{8}|{u_{{\varepsilon}}}(t)|^{2}
+ {\varepsilon}\frac{p(1-p)}{2}\int_{0}^{t}
\frac{|{u_{{\varepsilon}}}(s)|^{2}}{(1+s)^{2-p}}ds \right). &\end{aligned}$$ Since $2 -2{\varepsilon}(1+p)\geq 1$, then it holds true that $$\begin{aligned}
&\displaystyle \frac{1}{2}\mathcal{E}_{{\varepsilon}}(t) + \int_{0}^{t}(1+s)|{u_{{\varepsilon}}}'(s)|^{2}ds
+ \frac{1}{8} \frac{p+1}{\gamma+1}|{u_{{\varepsilon}}}(t)|^{2}
\leq &
\nonumber \\
&\displaystyle \mathcal{E}_{1}(0)+2|\mathcal{D}_{{\varepsilon}}(0)| +
{\varepsilon}\frac{p+1}{\gamma+1}\frac{p(1-p)}{2}\int_{0}^{t}
\frac{|{u_{{\varepsilon}}}(s)|^{2}}{(1+s)^{2-p}}ds. &
\label{1-SE}\end{aligned}$$ In particular we have $$|{u_{{\varepsilon}}}(t)|^{2} \leq
8\frac{\gamma+1}{p+1}(\mathcal{E}_{1}(0)+2|\mathcal{D}_{{\varepsilon}}(0)|)
+ 4{\varepsilon}(1-p)\int_{0}^{t}
\frac{|{u_{{\varepsilon}}}(s)|^{2}}{(1+s)^{2-p}}ds.$$ From the Gronwall’s Lemma we hence get $$|{u_{{\varepsilon}}}(t)|^{2} \leq 16\frac{\gamma+1}{p+1}
(\mathcal{E}_{1}(0)+2|\mathcal{D}_{{\varepsilon}}(0)|),$$ and finally $$(1-p)\int_{0}^{t}
\frac{|{u_{{\varepsilon}}}(s)|^{2}}{(1+s)^{2-p}}ds \leq 16\frac{\gamma+1}{p+1}
(\mathcal{E}_{1}(0)+2|\mathcal{D}_{{\varepsilon}}(0)|).$$ Now we go back to (\[1-SE\]) and from the previous inequality we obtain $$\frac{1}{2}\mathcal{E}_{{\varepsilon}}(t) + \int_{0}^{t}(1+s)|{u_{{\varepsilon}}}'(s)|^{2}ds
+ \frac{1}{8} \frac{p+1}{\gamma+1}|{u_{{\varepsilon}}}(t)|^{2} \leq
9(\mathcal{E}_{1}(0) + |u_{1}||u_{0}| + |u_{0}|^{2}).$$ By this inequality all the estimates we look for immediately follow. [[10000]{}]{}
Proof of Theorem \[thm:dg-error\] {#error}
---------------------------------
Also in this proof in most cases we omit the dependence of $u$, ${u_{{\varepsilon}}}$, ${\rho_{{\varepsilon}}}$, ${r_{{\varepsilon}}}$ and ${\theta_{{\varepsilon}}}$ from t. Moreover from now on we assume that ${\varepsilon}$ verifies the smallness assumptions of Theorem \[th:ex\] in such a way that ${u_{{\varepsilon}}}$ is globally well defined.
Let us recall that ${r_{{\varepsilon}}}$ and ${\rho_{{\varepsilon}}}$ verify the following problems: $$\left\{
\begin{array}{l}
\displaystyle {\varepsilon}{r_{{\varepsilon}}}'' + {|A^{1/2}u|}^{2\gamma}A{\rho_{{\varepsilon}}}+\frac{1}{(1+t)^{p}}{r_{{\varepsilon}}}'= -{\varepsilon}u''+ ({|A^{1/2}u|}^{2\gamma} -
{|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma})A{u_{{\varepsilon}}}\\
{r_{{\varepsilon}}}(0) = {r_{{\varepsilon}}}'(0) =0;
\end{array}\right.
\label{cpr}$$ and $$\left\{
\begin{array}{l}
\displaystyle {\varepsilon}{\rho_{{\varepsilon}}}'' + ({|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma}A{u_{{\varepsilon}}}- {|A^{1/2}u|}^{2\gamma}Au)
+\frac{1}{(1+t)^{p}}{\rho_{{\varepsilon}}}'= -{\varepsilon}u'' \\ {\rho_{{\varepsilon}}}(0) =
0,\;{\rho_{{\varepsilon}}}'(0) = w_{0};
\end{array}\right.
\label{cpro}$$ where $w_{0}$, ${u_{{\varepsilon}}}$ and $u$ are defined in (\[theta-data\]), (\[pbm:h-eq\]) and (\[pbm:par\]) respectively.
### Fundamental energies
Below we define the energies we use in the proof of Theorem \[thm:dg-error\].
Let us set $$\begin{aligned}
D_{\rho}(t)&:= &\int_{0}^{t}\langle {|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2\gamma}A{u_{{\varepsilon}}}(s) -
{|A^{1/2}u(s)|}^{2\gamma}Au(s), {\rho_{{\varepsilon}}}(s)\rangle (1+s)^{p} ds+ \nonumber \\
& & +{\varepsilon}\langle{\rho_{{\varepsilon}}}'(t),{\rho_{{\varepsilon}}}(t)\rangle (1+t)^{p}+
\frac{1}{2}|{\rho_{{\varepsilon}}}(t)|^{2}(1 - {\varepsilon}p(1+t)^{p-1}) ;
\label{Dro}
\end{aligned}$$ $$E_{\rho}(t):= ({\varepsilon}|{r_{{\varepsilon}}}'(t)|^{2}+
{|A^{1/2}u(t)|}^{2\gamma}{|A^{1/2}{\rho_{{\varepsilon}}}(t)|^{2}})(1+t)^{2p};
\label{Ero}$$ $$F_{\rho}(t):= {\varepsilon}\frac{|{r_{{\varepsilon}}}'(t)|^{2}}{{|A^{1/2}u(t)|}^{2\gamma}}+
{|A^{1/2}{\rho_{{\varepsilon}}}(t)|^{2}}.
\label{Fro}$$ In the proposition below we recollect all the inequalities verified by these energies we need.
\[Basic error estimates\]\[energyest\] Let $A$ be a non negative operator and let us assume that $0\leq p
\leq 1$, $\gamma \geq 1$. Moreover let us suppose that $(u_{0},u_{1})\in
D(A^{3/2})\times D(A^{1/2})$. Then there exists ${\varepsilon}_{0}>0$ such that for all ${\varepsilon}\leq {\varepsilon}_{0}$ and for all $t\geq 0$ we have $$\begin{aligned}
&
\displaystyle |{\rho_{{\varepsilon}}}(t)|^{2}+\int_{0}^{t}({|A^{1/2}u(s)|}^{2\gamma}+{|A^{1/2}{u_{{\varepsilon}}}(s)|}^{2\gamma})
{|A^{1/2}{\rho_{{\varepsilon}}}(s)|^{2}}(1+s)^{p}ds \leq & \nonumber \\
&\displaystyle \leq
\gamma_{6}{\varepsilon}^{2}+8{\varepsilon}\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds;&
\label{stDro1}
\end{aligned}$$ $$F_{\rho}(t) +
\int_{0}^{t}\frac{|{r_{{\varepsilon}}}'(s)|^{2}}{{|A^{1/2}u(s)|}^{2\gamma}}
\frac{1}{(1+s)^{p}} ds
\leq \gamma_{7}{\varepsilon}^{2}+\gamma_{8}
{\varepsilon}\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds;
\label{stFro1}$$ if moreover (\[H1cip\]) is verified then $$E_{\rho}(t) + \int_{0}^{t}|{r_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds \leq
\gamma_{9}{\varepsilon};
\label{stEro1}$$ where all constants do not depend on ${\varepsilon}$ and $t$ but only on the initial data.
[[Proof.]{}$\;$]{}To begin with, we compute the time’s derivatives of (\[Ero\]), (\[Fro\]) and (\[Dro\]). Using (\[cpr\]) it is easy to see that $$\begin{aligned}
E_{\rho}' & = & -|{r_{{\varepsilon}}}'|^{2}(1+t)^{p}\left(2 - \frac{2{\varepsilon}p}{(1+t)^{1-p}}\right) +
2p(1+t)^{2p-1}{|A^{1/2}u|}^{2\gamma}{|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}
+
\nonumber \\
& & +2(1+t)^{2p}{|A^{1/2}u|}^{2\gamma}\langle A{\rho_{{\varepsilon}}}, {\theta_{{\varepsilon}}}'\rangle
-2{\varepsilon}\langle u'',{r_{{\varepsilon}}}'\rangle (1+t)^{2p} +
\nonumber \\
& &
+2(1+t)^{2p}({|A^{1/2}u|}^{2\gamma}- {|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma})\langle
A{u_{{\varepsilon}}},{r_{{\varepsilon}}}'\rangle + \nonumber \\
& &
-2\gamma(1+t)^{3p}{|A^{1/2}u|}^{4\gamma-2}|Au|^{2}{|A^{1/2}{\rho_{{\varepsilon}}}|^{2}} \nonumber
\\
& =: & S_{1}+S_{2}+S_{3}+S_{4}+S_{5}+S_{6},
\label{DEro}
\end{aligned}$$ and $$\begin{aligned}
F_{\rho}' & = &
-\frac{|{r_{{\varepsilon}}}'|^{2}}{{|A^{1/2}u|}^{2\gamma}}\left(\frac{2}{(1+t)^{p}}
-2\gamma{\varepsilon}{|A^{1/2}u|}^{2(\gamma-1)}|Au|^{2}(1+t)^{p}\right)
+ 2 \langle A{\rho_{{\varepsilon}}}, {\theta_{{\varepsilon}}}'\rangle +
\nonumber\\
& & +2\frac{{|A^{1/2}u|}^{2\gamma} -
{|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma}}{{|A^{1/2}u|}^{2\gamma}}\langle A{u_{{\varepsilon}}},{r_{{\varepsilon}}}'\rangle
-2{\varepsilon}\langle u'',{r_{{\varepsilon}}}'\rangle\frac{1}{{|A^{1/2}u|}^{2\gamma}}.
\label{DFro}
\end{aligned}$$ Conversely using (\[cpro\]) we have $$D_{\rho}'= {\varepsilon}|{\rho_{{\varepsilon}}}'|^{2}(1+t)^{p} -{\varepsilon}\langle
u'',{\rho_{{\varepsilon}}}\rangle(1+t)^{p} + \frac{p}{2}(1-p){\varepsilon}(1+t)^{p-2}|{\rho_{{\varepsilon}}}|^{2}.
\label{DDro}$$ Moreover from now on let us assume that ${\varepsilon}_{0}$ verifies also these assumptions (recall that we have already supposed that ${\varepsilon}_{0}$ satisfies the smallness conditions in Theorem \[th:ex\]) $$(1+p){\varepsilon}_{0} \leq \frac{1}{4},\hspace{1em} 2\gamma
\gamma_{4}{\varepsilon}_{0} \leq \frac{1}{4}
\label{hep1}$$ where $\gamma_{4}$ is the constant in (\[dv4\]). In the following we denote by $c_{i}$ various constants that depend only on the initial data. Moreover let us set $$\phi_{\rho}(t):= ({|A^{1/2}u|}^{2\gamma}+
{|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma}){|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}.$$
#####
Thanks to Lagrange’s Theorem for all $t\geq 0$ there exists $\xi_{t}$ in the interval with end points ${|A^{1/2}u(t)|^{2}}$ and ${|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}$ such that $$\begin{aligned}
{|A^{1/2}u(t)|}^{2\gamma}- {|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2\gamma} & =
& \gamma \xi_{t}^{\gamma-1}({|A^{1/2}u(t)|^{2}}-{|A^{1/2}{u_{{\varepsilon}}}(t)|^{2}}) \\
& = & -\gamma
\xi_{t}^{\gamma-1}\langle A^{1/2}(u(t)+{u_{{\varepsilon}}}(t)),A^{1/2}{\rho_{{\varepsilon}}}(t)\rangle.
\end{aligned}$$ Since it is clear that $$\xi_{t}^{\gamma-1}
\leq{|A^{1/2}u(t)|}^{2(\gamma-1)}+{|A^{1/2}{u_{{\varepsilon}}}(t)|}^{2(\gamma-1)},$$ then $$\begin{aligned}
&( {|A^{1/2}u|}^{2\gamma}- {|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma})^{2} \leq
\gamma^{2}\xi_{t}^{2(\gamma-1)}{|A^{1/2}(u+{u_{{\varepsilon}}})|^{2}}{|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}&
\nonumber \\
& \leq
2\gamma^{2}({|A^{1/2}u|}^{2(\gamma-1)}+
{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)})^{2}({|A^{1/2}u|^{2}}+{|A^{1/2}{u_{{\varepsilon}}}|^{2}}){|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}&
\nonumber \\
& \leq 6\gamma^{2} ({|A^{1/2}u|}^{2(\gamma-1)}+
{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}) \phi_{\rho}.&
\label{sdiff} \end{aligned}$$ Moreover computing the time’s derivative of (\[pbm:par\]) we get: $$u''= -p(1+t)^{p-1}{|A^{1/2}u|}^{2\gamma}Au +
(1+t)^{2p}({|A^{1/2}u|}^{4\gamma}A^{2}u+ 2\gamma
{|A^{1/2}u|}^{4\gamma - 2}|Au|^{2}Au),
\label{eqv''}$$ thus $$\begin{aligned}
|u''|^{2} & \leq & 3(1+t)^{2(p-1)}{|A^{1/2}u|}^{4\gamma}|Au|^{2} +
3(1+t)^{4p}{|A^{1/2}u|}^{8\gamma}|A^{2}u|^{2} \nonumber \\
& & +12\gamma^{2}(1+t)^{4p}{|A^{1/2}u|}^{8\gamma-4}|Au|^{6}.
\label{v''-1}\end{aligned}$$ From (\[dv4\]) we also deduce that $${|A^{1/2}u|}^{4(\gamma-1)}|Au|^{4} \leq
\frac{\gamma_{4}^{2}}{(1+t)^{2p+2}}.
\label{stvro}$$ We can now estimate the last term in (\[v”-1\]) using (\[stvro\]), so finally we get $$|u''|^{2} \leq c_{1}(1+t)^{2p-2}{|A^{1/2}u|}^{4\gamma}|Au|^{2}
+ 3(1+t)^{4p}{|A^{1/2}u|}^{8\gamma}|A^{2}u|^{2}.
\label{sv''}$$ Now we are ready to prove (\[stDro1\]), (\[stFro1\]), (\[stEro1\]).
#####
Thanks to Lemma \[mprop\] with $m(r) = r^{\gamma}$ we have $$\frac{1}{2}\phi_{\rho}\leq \langle {|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma}A{u_{{\varepsilon}}}-
{|A^{1/2}u|}^{2\gamma}Au, {\rho_{{\varepsilon}}}\rangle$$ hence integrating (\[DDro\]) in $[0,t]$ we get $$\begin{aligned}
& \displaystyle \frac{1}{2}
\int_{0}^{t}\phi_{\rho}(s)(1+s)^{p}ds +
\frac{1}{2}|{\rho_{{\varepsilon}}}(t)|^{2}(1 - {\varepsilon}p(1+t)^{p-1})&
\nonumber \\
& \displaystyle\leq -
{\varepsilon}\langle{\rho_{{\varepsilon}}}'(t),{\rho_{{\varepsilon}}}(t)\rangle (1+t)^{p}
+{\varepsilon}\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p} ds +&
\nonumber \\
& \displaystyle-{\varepsilon}\int_{0}^{t}\langle
u''(s),{\rho_{{\varepsilon}}}(s)\rangle(1+s)^{p}ds + \frac{p}{2}(1-p){\varepsilon}\int_{0}^{t}(1+s)^{p-2}|{\rho_{{\varepsilon}}}(s)|^{2} ds&
\nonumber \\
&\displaystyle =:\psi_{1}+\psi_{2}+\psi_{3}+\psi_{4}. &
\label{stDro1-1}\end{aligned}$$ Let us now estimate $\psi_{1}$, $\psi_{3}$, $\psi_{4}$.
From (\[D1\]) and (\[D3\]) we obtain $$|{u_{{\varepsilon}}}'|^{2} \leq \frac{C_{2}}{(1+t)^{1-p}}
\frac{C_{2}^{\gamma+1}}{(1+t)^{1+p}} =
\frac{C_{2}^{\gamma+2}}{(1+t)^{2}},$$ and from (\[dv4\]), (\[dv2\]) we have $$|u'|^{2}= (1+t)^{2p}{|A^{1/2}u|}^{4\gamma}|Au|^{2} \leq
\gamma_{4}(1+t)^{2p}\frac{{|A^{1/2}u|}^{2(\gamma+1)}}{(1+t)^{p+1}}\leq
\frac{c_{2}}{(1+t)^{2}}.$$ Therefore, recalling that $|{\rho_{{\varepsilon}}}'|^{2}\leq 2
(|{u_{{\varepsilon}}}'|^{2}+|u'|^{2})$ we get $$|\psi_{1}|\leq {\varepsilon}^{2}|{\rho_{{\varepsilon}}}'|^{2}(1+t)^{2p} +
\frac{1}{4}|{\rho_{{\varepsilon}}}|^{2} \leq c_{3}{\varepsilon}^{2}+ \frac{1}{4}|{\rho_{{\varepsilon}}}|^{2}.
\label{spsi1ro}$$ Now let us estimate $\psi_{3}$. From (\[eqv”\]) we deduce that $$\begin{aligned}
\langle u'',{\rho_{{\varepsilon}}}\rangle & = &
-\frac{p}{(1+t)^{1-p}
}{|A^{1/2}u|}^{2\gamma}\langle A^{1/2}u, A^{1/2}{\rho_{{\varepsilon}}}\rangle + (1+t)^{2p} {|A^{1/2}u|}^{4\gamma}\langle
A^{3/2}u, A^{1/2}{\rho_{{\varepsilon}}}\rangle + \\
& & +2\gamma(1+t)^{2p}{|A^{1/2}u|}^{4\gamma-2}|Au|^{2}
\langle A^{1/2}u,A^{1/2}{\rho_{{\varepsilon}}}\rangle,
\end{aligned}$$ hence $$\begin{aligned}
{\varepsilon}|\langle u'',{\rho_{{\varepsilon}}}\rangle |(1+t)^{p} & \leq &
\frac{1}{4}{|A^{1/2}u|}^{2\gamma}{|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}(1+t)^{p} +
3{\varepsilon}^{2}{|A^{1/2}u|}^{2(\gamma+1)} (1+t)^{3p-2} + \\
& & +3{\varepsilon}^{2}(1+t)^{5p}{|A^{1/2}u|}^{6\gamma}|A^{3/2}u|^{2}+ \\
& & +12\gamma^{2}{\varepsilon}^{2}(1+t)^{5p}{|A^{1/2}u|}^{6\gamma-2}|Au|^{4}.
\end{aligned}$$ Using (\[stvro\]) to estimate the last term in the previous inequality, since $3p-2 \leq p$ finally we obtain $$\begin{aligned}
{\varepsilon}|\langle u'',{\rho_{{\varepsilon}}}\rangle |(1+t)^{p} & \leq &
\frac{1}{4}{|A^{1/2}u|}^{2\gamma}{|A^{1/2}{\rho_{{\varepsilon}}}|^{2}}(1+t)^{p} +
c_{4}{\varepsilon}^{2}{|A^{1/2}u|}^{2(\gamma+1)} (1+t)^{p}+
\nonumber \\
& & +
3{\varepsilon}^{2}(1+t)^{5p}{|A^{1/2}u|}^{6\gamma}|A^{3/2}u|^{2}.
\label{spsi3-1}
\end{aligned}$$ From (\[spsi3-1\]), (\[dv1\]), (\[dv6\]) thus we arrive at $$\begin{aligned}
|\psi_{3}| &\leq &
\frac{1}{4}\int_{0}^{t}{|A^{1/2}u(s)|}^{2\gamma}{|A^{1/2}{\rho_{{\varepsilon}}}(s)|^{2}}(1+s)^{p}ds +
c_{5}{\varepsilon}^{2} \nonumber \\
& & \leq \frac{1}{4}\int_{0}^{t}
\phi_{\rho}(s)(1+s)^{p}ds +
c_{5}{\varepsilon}^{2}.
\label{spsi3ro}
\end{aligned}$$ Let us now consider $\psi_{4}$ and prove that $$p(1-p)\int_{0}^{t}(1+s)^{p-2}|{\rho_{{\varepsilon}}}(s)|^{2}ds \leq c_{6}, \quad
\quad \forall t \geq 0.
\label{spsi4ro-1}$$ If $p=1$ thesis is obvious. If $p<1$ it is enough to prove that $|{\rho_{{\varepsilon}}}|^{2}$ is bounded independently from ${\varepsilon}$ and $t$. But this is a straightaway consequence of (\[D0\]) and (\[dv1\]).
Using (\[spsi4ro-1\]) we then obtain $$\psi_{4} \leq c_{6}{\varepsilon}.
\label{spsi4ro}$$ Now we go back to (\[stDro1-1\]), and using (\[spsi1ro\]), (\[spsi3ro\]), (\[spsi4ro\]), since $p{\varepsilon}\leq 1/4$ we achieve $$\displaystyle \frac{1}{4}\int_{0}^{t}\phi_{\rho}(s)(1+s)^{p}ds +
\frac{1}{8}|{\rho_{{\varepsilon}}}(t)|^{2}\leq c_{7}{\varepsilon}^{2}+c_{6}{\varepsilon}+{\varepsilon}\displaystyle \int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds.
\label{stDro}$$ Let us now remark that thanks to (\[D4\]) and (\[dv5\]) we have $$\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds \leq c_{8}.
\label{SIro}$$ Plugging (\[SIro\]) in (\[stDro\]) we gain $$|{\rho_{{\varepsilon}}}|^{2} \leq c_{9}{\varepsilon}.$$ At this point we can improve estimates (\[spsi4ro-1\]) and (\[spsi4ro\]) as below $$p(1-p)\int_{0}^{t}(1+s)^{p-2}|{\rho_{{\varepsilon}}}(s)|^{2}ds \leq c_{10}{\varepsilon},
\hspace{1em} \psi_{4} \leq c_{10}{\varepsilon}^{2}.$$ Using this last estimate in (\[stDro\]) instead of (\[spsi4ro\]) we finally get $$\displaystyle \frac{1}{4}\int_{0}^{t}\phi_{\rho}(s)(1+s)^{p}ds +
\frac{1}{8}|{\rho_{{\varepsilon}}}(t)|^{2}\leq c_{11}{\varepsilon}^{2}+{\varepsilon}\displaystyle \int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds.$$ that is (\[stDro1\]).
#####
From (\[dv4\]) we have $$2\gamma {|A^{1/2}u|}^{2(\gamma-1)}|Au|^{2}(1+t)^{p}\leq
\frac{2\gamma\gamma_{4}}{1+t}\leq
\frac{2\gamma\gamma_{4}}{(1+t)^{p}},$$ hence from (\[hep1\]), (\[DFro\]), (\[sdiff\]) and (\[sv”\]) we obtain $$\begin{aligned}
F_{\rho}' & \leq &
-\frac{7}{4}\frac{|{r_{{\varepsilon}}}'|^{2}}{{|A^{1/2}u|}^{2\gamma}}
\frac{1}{(1+t)^{p}}+
2{|A^{1/2}{\rho_{{\varepsilon}}}|}|A^{1/2}{\theta_{{\varepsilon}}}|+ \frac{1}{2}\frac{|{r_{{\varepsilon}}}'|^{2}}{{|A^{1/2}u|}^{2\gamma}}
\frac{1}{(1+t)^{p}}
+
\nonumber \\
& & +4\frac{({|A^{1/2}u|}^{2\gamma} -
{|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma})^{2}}{{|A^{1/2}u|}^{2\gamma}}|A{u_{{\varepsilon}}}|^{2}(1+t)^{p}+ 4{\varepsilon}^{2}
\frac{|u''|^{2}}{{|A^{1/2}u|}^{2\gamma}}
(1+t)^{p}
\nonumber \\
& \leq & -\frac{5}{4}\frac{|{r_{{\varepsilon}}}'|^{2}}{{|A^{1/2}u|}^{2\gamma}}
\frac{1}{(1+t)^{p}}+
2{|A^{1/2}{\rho_{{\varepsilon}}}|}|A^{1/2}{\theta_{{\varepsilon}}}|+
\nonumber \\
& & +24\gamma^{2}
\frac{|A{u_{{\varepsilon}}}|^{2}({|A^{1/2}u|}^{2(\gamma-1)}+
{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)})}{{|A^{1/2}u|}^{2\gamma}} \phi_{\rho}(1+t)^{p}+
\nonumber \\
& & + 4c_{1}{\varepsilon}^{2}(1+t)^{3p-2}{|A^{1/2}u|}^{2\gamma}|Au|^{2}
+ 12{\varepsilon}^{2}(1+t)^{5p}{|A^{1/2}u|}^{6\gamma}|A^{2}u|^{2}.
\label{sDFro-1}
\end{aligned}$$ Let us now observe that, thanks to (\[D2\]), (\[dv2\]) and (\[D1\]), we have $$\frac{1}{{|A^{1/2}u|}^{2\gamma}} |A{u_{{\varepsilon}}}|^{2}{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}+
\frac{1}{{|A^{1/2}u|}^{2}}|A{u_{{\varepsilon}}}|^{2}{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}
\frac{1}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}}\leq c_{12}.$$ Replacing this inequality in (\[sDFro-1\]) and integrating we get $$\begin{aligned}
&\displaystyle F_{\rho}(t)+ \frac{5}{4}\int_{0}^{t}
\frac{|{r_{{\varepsilon}}}'(s)|^{2}}{{|A^{1/2}u(s)|}^{2\gamma}}
\frac{1}{(1+s)^{p}} ds \leq 2\sup_{0\leq s\leq
t}{|A^{1/2}{\rho_{{\varepsilon}}}(s)|}\int_{0}^{t}|A^{1/2}{\theta_{{\varepsilon}}}(s)|\,ds+ & \\
& \displaystyle +c_{13}\int_{0}^{t}(1+s)^{p} \phi_{\rho}(s) ds+
4c_{1}{\varepsilon}^{2}\int_{0}^{t}(1+s)^{3p-2}{|A^{1/2}u(s)|}^{2\gamma}|Au(s)|^{2}ds +& \\
& \displaystyle
+
12{\varepsilon}^{2}\int_{0}^{t}(1+s)^{5p}{|A^{1/2}u(s)|}^{6\gamma}|A^{2}u(s)|^{2}ds.&
\end{aligned}$$ We can now use Lemma \[thetalemma\] with $\delta = 0$ and $j =
1$, (\[stDro1\]), (\[dv0\]) with $k=1$, (\[dv7\]) and (\[dv1\]), thus we obtain $$\begin{aligned}
&\displaystyle F_{\rho}(t)+ \frac{5}{4}\int_{0}^{t}
\frac{|{r_{{\varepsilon}}}'(s)|^{2}}{{|A^{1/2}u(s)|}^{2\gamma}}
\frac{1}{(1+s)^{p}} ds \leq c_{14}{\varepsilon}\sup_{0\leq s\leq
t}{|A^{1/2}{\rho_{{\varepsilon}}}(s)|} + c_{15}{\varepsilon}^{2}+ & \\
& \displaystyle+c_{16}{\varepsilon}\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds+
4c_{1}{\varepsilon}^{2}\frac{|Au_{0}|^{2}}
{{|A^{1/2}u_{0}|^{2}}}\int_{0}^{t}(1+s)^{p}{|A^{1/2}u(s)|}^{2(\gamma+1)}ds & \\
&\displaystyle \leq \frac{1}{2}\sup_{0\leq s\leq
t}{|A^{1/2}{\rho_{{\varepsilon}}}(s)|}^{2}+c_{17}{\varepsilon}^{2}+
c_{16}{\varepsilon}\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds.&
\end{aligned}$$ Let now $T>0$ and let us take the essup on $0\leq t\leq T$, then we get $$\frac{1}{2}\sup_{0\leq t\leq T} F_{\rho}(t)+
\frac{5}{4}\int_{0}^{T}
\frac{|{r_{{\varepsilon}}}'(s)|^{2}}{{|A^{1/2}u(s)|}^{2\gamma}}
\frac{1}{(1+s)^{p}} ds \leq c_{17}{\varepsilon}^{2}+
c_{16}{\varepsilon}\int_{0}^{T}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds.$$ Since $T$ is arbitrary we have proved (\[stFro1\]).
#####
Let us estimate separately the terms $S_{1},\ldots, S_{5}$ in (\[DEro\]).
Since ${\varepsilon}\leq 1/4$ then $$S_{1}\leq - \frac{3}{2}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p}.
\label{stS1ro}$$ Moreover $$S_{2}\leq 2(1+t)^{p}{{|A^{1/2}u|}^{2\gamma}}{|A^{1/2}{\rho_{{\varepsilon}}}|^{2}} \leq
2(1+t)^{p}\phi_{\rho}.
\label{stS2ro}$$ Since ${|A^{1/2}u|}^{2\gamma}$ is bounded then $$S_{3}\leq c_{18}(1+t)^{2p}{|A^{1/2}{\rho_{{\varepsilon}}}|}{|A^{1/2}{\theta_{{\varepsilon}}}'|}.
\label{stS3ro}$$ From (\[sv”\]) we deduce $$\begin{aligned}
S_{4}&\leq&
\frac{1}{4}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p}+4{\varepsilon}^{2}|u''|^{2}(1+t)^{3p}\nonumber
\\
&\leq & \frac{1}{4}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p}
+4c_{1}{\varepsilon}^{2}(1+t)^{3p}{|A^{1/2}u|}^{4\gamma}|Au|^{2}+
\nonumber \\
& &+
12{\varepsilon}^{2}(1+t)^{7p}{|A^{1/2}u|}^{8\gamma}|A^{2}u|^{2}.
\label{stS4ro}
\end{aligned}$$ Let us now estimate $S_{5}$. From (\[sdiff\]) and (\[D2\]) we get $$\begin{aligned}
& S_{5} \displaystyle \leq \frac{1}{4}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p} +
4(1+t)^{3p}|A{u_{{\varepsilon}}}|^{2}({|A^{1/2}u|}^{2\gamma} -
{|A^{1/2}{u_{{\varepsilon}}}|}^{2\gamma})^{2} &
\nonumber \\
& \displaystyle \leq \frac{1}{4}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p} +
c_{19}(1+t)^{2p}|A{u_{{\varepsilon}}}|^{2}{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}
\left(1+\frac{{|A^{1/2}u|}^{2(\gamma-1)}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}}\right)
\phi_{\rho}(1+t)^{p} &
\nonumber \\
& \displaystyle \leq \frac{1}{4}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p}
+ c_{20}\frac{1}{(1+t)^{1-p}}
\left(1+\frac{{|A^{1/2}u|}^{2(\gamma-1)}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}}\right)
\phi_{\rho}(1+t)^{p}.&
\label{stS5ro-1}
\end{aligned}$$ Now we want to prove that $$\chi:= \frac{1}{(1+t)^{1-p}}\frac{{|A^{1/2}u|}^{2(\gamma-1)}}{{|A^{1/2}{u_{{\varepsilon}}}|}^{2(\gamma-1)}} \leq c_{21}.
\label{stS5ro-2}$$ When $A$ is coercive this is a consequence of (\[D1\]) and (\[dv3\]). On the other hand if $p\leq
(\gamma^{2}+1)/(\gamma^{2}+2\gamma -1)$ then $$\alpha: = \frac{\gamma^{2}
+1-p(\gamma^{2}+2\gamma-1)}{\gamma(\gamma+1)} \geq 0,$$ hence from (\[D1\]) and (\[dv2\]) we deduce: $$\chi
\leq \frac{c_{22}}{(1+t)^{1-p}} \frac{(1+t)^{(p+1)
(\gamma-1)/\gamma}}{(1+t)^{(p+1)(\gamma-1)/(\gamma+1)}}=
\frac{c_{22}}{(1+t)^{\alpha}}\leq c_{22}.$$ At this point from (\[stS5ro-1\]) and (\[stS5ro-2\]) it follows that: $$S_{5}\leq \frac{1}{4}|{r_{{\varepsilon}}}'|^{2}(1+t)^{p}
+
c_{23}\phi_{\rho}(1+t)^{p}.
\label{stS5ro}$$ If we put (\[stS1ro\]), (\[stS2ro\]), (\[stS3ro\]), (\[stS4ro\]), (\[stS5ro\]) in (\[DEro\]), since $S_{6} \leq 0$ we get: $$\begin{aligned}
E_{\rho}' + |{r_{{\varepsilon}}}'|^{2}(1+t)^{p} & \leq &
c_{24}\phi_{\rho}(1+t)^{p}+ c_{18}(1+t)^{2p}{|A^{1/2}{\rho_{{\varepsilon}}}|}{|A^{1/2}{\theta_{{\varepsilon}}}'|} + \\
& & +
4c_{1}{\varepsilon}^{2}(1+t)^{3p}{|A^{1/2}u|}^{4\gamma}|Au|^{2} + 12{\varepsilon}^{2}(1+t)^{7p}{|A^{1/2}u|}^{8\gamma}|A^{2}u|^{2}.
\end{aligned}$$ Integrating in $[0,t]$ and using (\[stDro1\]), (\[dv5\]), (\[dv7\]) we then achieve: $$\begin{aligned}
& \displaystyle E_{\rho}(t) + \int_{0}^{t}|{r_{{\varepsilon}}}'(s)|^{2}(1+s)^{p} ds \leq
c_{25}{\varepsilon}^{2}+c_{26}{\varepsilon}\int_{0}^{t}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds +&
\nonumber \\
& \displaystyle +c_{18}\sup_{0\leq s \leq
t}(1+s)^{p}{|A^{1/2}{\rho_{{\varepsilon}}}(s)|}{|A^{1/2}u(s)|}^{\gamma}
\int_{0}^{t}\frac{{|A^{1/2}{\theta_{{\varepsilon}}}'(s)|}}{{|A^{1/2}u(s)|}^{\gamma}}(1+s)^{p}ds.&
\label{stEro1-1}
\end{aligned}$$ Thanks to (\[dv2\]) and Lemma \[thetalemma\], with $j = 1$ and $\delta = (3p+1)/2$ (note that thanks to (\[hep1\]) all hypotheses of Lemma \[thetalemma\] are verified) we have $$\int_{0}^{t}\frac{{|A^{1/2}{\theta_{{\varepsilon}}}'(s)|}}{{|A^{1/2}u(s)|}^{\gamma}}(1+s)^{p}ds
\leq c_{27}\int_{0}^{t}{|A^{1/2}{\theta_{{\varepsilon}}}'(s)|}(1+s)^{(3p+1)/2}ds
\leq c_{28}{\varepsilon}.$$ Finally plugging this inequality in (\[stEro1-1\]) and using (\[SIro\]) we gain $$\begin{aligned}
E_{\rho}(t) + \int_{0}^{t}|{r_{{\varepsilon}}}'(s)|^{2}(1+s)^{p} ds & \leq
& c_{25}{\varepsilon}^{2}+c_{29}{\varepsilon}+ c_{30}{\varepsilon}\sup_{0\leq s \leq
t}(1+s)^{p}{|A^{1/2}{\rho_{{\varepsilon}}}(s)|}{|A^{1/2}u(s)|}^{\gamma} \\
& & \leq c_{31}{\varepsilon}+ \frac{1}{2}\sup_{0\leq s \leq
t}E_{\rho}(s).
\end{aligned}$$ Let now $T > 0$, if we take the essup for $0\leq t \leq T$ we get $$\frac{1}{2}\sup_{0\leq t \leq
T}E_{\rho}(t) + \int_{0}^{T}|{r_{{\varepsilon}}}'(s)|^{2}(1+s)^{p} ds \leq
c_{31}{\varepsilon}.$$ Since $T$ is arbitrary we have proved (\[stEro1\]). [[10000]{}]{}
### Conclusion
From (\[SIro\]), (\[stFro1\]), (\[stDro1\]) we straight obtain (\[H1\]).
Let us now prove (\[H1c\]). From (\[stEro1\]) and Lemma \[thetalemma\] with $j=0$ and $\delta = p$ we get $$\begin{aligned}
\int_{0}^{+\infty}|{\rho_{{\varepsilon}}}'(s)|^{2}(1+s)^{p} ds & \leq &
2\int_{0}^{+\infty}|{r_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds +
2\int_{0}^{+\infty}|{\theta_{{\varepsilon}}}'(s)|^{2}(1+s)^{p}ds
\nonumber \\
& \leq & 2\gamma_{9}{\varepsilon}+ 2{\varepsilon}C_{p} \sup_{s\geq 0}|{\theta_{{\varepsilon}}}'(s)|
\leq C {\varepsilon},
\label{SIro1}\end{aligned}$$ where $C$ depends only on the data, since $|{\theta_{{\varepsilon}}}'|\leq |w_{0}|$. Finally replacing this estimate in (\[stDro1\]) and (\[stFro1\]) we obtain inequality (\[H1c\]).
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\[NumeroPagine\]
|
---
author:
- 'A. Kochoska'
- 'N. Mowlavi'
- 'A. Prša'
- 'I. Lecoeur-Taïbi'
- 'B. Holl'
- 'L. Rimoldini'
- 'M. Süveges'
- 'L. Eyer'
bibliography:
- 'bibTex.bib'
date: 'Received 25 Oct 2016; accepted 27 Mar 2017'
title: '*Gaia* Eclipsing Binary and Multiple Systems. A study of detectability and classification of eclipsing binaries with *Gaia*'
---
[In the new era of large-scale astronomical surveys, automated methods of analysis and classification of bulk data are a fundamental tool for fast and efficient production of deliverables. This becomes ever more imminent as we enter the [[*Gaia* ]{}]{}era.]{} [We investigate the potential detectability of eclipsing binaries with [[*Gaia* ]{}]{}using a data set of all [[*Kepler* ]{}]{}eclipsing binaries sampled with [[*Gaia* ]{}]{}cadence and folded with the [[*Kepler* ]{}]{}period. The performance of fitting methods is evaluated with comparison to real [[*Kepler* ]{}]{}data parameters and a classification scheme is proposed for the potentially detectable sources based on the geometry of the light curve fits.]{} [The polynomial chain ([*polyfit*]{}) and [[*two-Gaussian* ]{}]{}models are used for light curve fitting of the data set. Classification is performed with a combination of the $t$-SNE ($t$-distrubuted Stochastic Neighbor Embedding) and DBSCAN (Density-Based Spatial Clustering of Applications with Noise) algorithms.]{} [We find that $\sim68\;\%$ of [[*Kepler* ]{}]{}Eclipsing Binary sources are potentially detectable by [[*Gaia* ]{}]{}when folded with the [[*Kepler* ]{}]{}period and propose a classification scheme of the detectable sources based on the morphological type indicative of the light curve, with subclasses that reflect the properties of the fitted model (presence and visibility of eclipses, their width, depth, etc.).]{}
Introduction
============
The onset of large-scale astronomical surveys is producing a steady flow of a large amount of data, which resulted in many ground-breaking discoveries made merely in the past few decades. Among the most important common objects that can be found in these surveys are binary stars, whose greatest contribution to astronomy is the possibility to directly measure stellar properties to an unprecedented level of accuracy.
With the use of a combination of observational techniques, in particular photometry and spectroscopy, we can obtain a full characterization of the system and its separate components: their orbital elements and dynamics, their absolute masses, radii, temperatures, chemical composition, rotation, the presence of other companions or planets etc. Thus, photometric-variability surveys such as *Hipparcos* [@perryman1997], *MOST* [@pribulla2010], *CoRoT* [@corot], *OGLE-III* [@udalski2008], *ASAS* [@pojmanski2002] and *[[*Kepler* ]{}]{}* [@keplermain], have been of utmost importance to the field of binary stars, yielding extensive eclipsing binary star catalogs with data on several tens of thousands of stars. [[*Gaia* ]{}]{} is expected to boost this number by several orders of magnitude — out of a billion observed sources, up to several million are expected to be eclipsing binaries (four million are predicted by @eyer2013, seven million by @zwitter2002 and half a million by @dischler2005). A portion of these (approximately $12\%$; @eyer2013) are expected to be also spectroscopic binaries, which enables precise determination of their masses and radii.
The full physical characterization of spectroscopic eclipsing binary stars is a highly demanding and often incomplete task due to the parameter space degeneracy of the analysis models. A first step towards the characterization of an eclipsing binary is automated analysis of the geometrical parameters of its light curve (eclipse depth, width, separation, amplitude of ellipsoidal variations), which can be related to the physical system parameters such as periodicity, morphology, eccentricity, inclination, temperature ratio, etc. Likewise, the ever-growing data inflow calls for new, automated classification methods. Several machine learning methods, together with Principal Component Analysis [PCA; @pca] and Locally Linear Embedding [LLE; @lle], $t$-Distributed Stochastic Neighbor Embedding [$t$-SNE; @tsne] and Density-Based Spatial Clustering of Applications with Noise [DBSCAN; @dbscan] have been considered and their performance evaluated on data from photometric and spectroscopic surveys [@caballero2008; @kepler3; @galrave; @kepler7; @suveges2016].
In this paper, we propose a combination of the $t$-SNE and DBSCAN algorithms for the purposes of eclipsing binary light curve classification.
The methodology and results presented here are part of a series of exploratory studies undertaken in the framework of the [[*Gaia* ]{}]{}mission for the implementation of an automated pipeline to process eclipsing binary light curves within the [[*Gaia* ]{}]{}Data Processing and Analysis Consortium (eclipsing binary data from [[*Gaia* ]{}]{}are expected to be delivered to the scientific community not earlier than 2019). In this study, we use [[*Kepler* ]{}]{}eclipsing binary light curves sampled with [[*Gaia* ]{}]{}cadence at their respective positions on the sky. We apply two different techniques to characterize the geometry of the folded light curves, and study the efficiency of the $t$-SNE and DBSCAN algorithms to classify the folded light curves. As a by-product of this analysis we obtain an estimate of the [[*Gaia* ]{}]{}recovery rate of Kepler eclipsing binaries, a number of interest to evaluate the eclipsing binary completeness factor expected from the [[*Gaia* ]{}]{}mission.
An overview of the data sets, light curve fitting and classification methods is given in Sect. \[Sect:Data\_and\_analysis\]. Results of the fitting models comparison are given in Sect. \[Sect:Results\], as well as a classification of the whole and of a filtered data set of [[*Gaia*]{}-sampled ]{}light curves. Conclusions of the paper and future prospects are given in Sect. \[Sect:Conclusions\].
Data and analysis {#Sect:Data_and_analysis}
=================
Overview of methodology {#Sect:overview_of_methods}
-----------------------
We use data from the [[*Kepler* ]{}]{}eclipsing binary catalog [@kepler7] sampled with the expected five year [[*Gaia* ]{}]{}cadence to simulate [[*Gaia* ]{}]{}light curves. The [[*Gaia*]{}-sampled ]{}light curves are then folded using [[*Kepler* ]{}]{}orbital periods and reference times of primary minimum (in barycentric Julian date), and fitted with two models. The first model uses polynomial chain fits ([*polyfits*]{}), as described in [@ebaipaper]. The second model, called the [[*two-Gaussian* ]{}]{}model, chooses the best combination of Gaussian functions to describe the presence of eclipses and a cosine function to describe an ellipsoidal-like variability during the inter eclipses, if present. This model is developed within the [[*Gaia* ]{}]{}pipeline to process the light curves of eclipsing binaries (Mowlavi et al., submitted).
The $t$-SNE algorithm requires a set of data computed in the same set of $x$-axis points. For this purpose, we fit the phase-folded data and compute all models in a set of $N$ equidistant phase points. As mentioned above, the orbital periods and reference times of primary minimum are fixed to their [[*Kepler* ]{}]{}values. Further studies will rely on periods provided by the [[*Gaia* ]{}]{}period-search pipeline. The [[*two-Gaussian* ]{}]{}model fits are only folded with the orbital period, while the shifting to phase zero is done with respect to the phase at maximum magnitude of the phase-folded light curve.
Due to [[*Kepler*]{}’s ]{}unprecedented photometric precision and high cadence observations of the original [[*Kepler* ]{}]{}field of view in Cygnus, its light curves are of remarkable quality. [[*Gaia*]{}’s ]{}scanning law, in contrast, observes a given star 67 times on average in a five year mission lifetime (the actual number of observations of a given star depends on its position in the sky and follows [[*Gaia*]{}’s ]{}Nominal Scanning Law; see Sect. \[Sect:gaiasampled\]), which will result in insufficiently sampled light curves for a portion of the eclipsing binary sources. In our [[*Gaia*]{}-sampled ]{}set, the resulting light curves of these sources typically give poor or unrealistic model fits, which can be automatically isolated and removed by the fitting model itself, in the case of [*two-Gaussians*]{}, or the above-mentioned dimensionality reduction and clustering algorithms, in the case of [*polyfits*]{}.
The dimensionality reduction and clustering method is then used to propose a classification scheme of the remaining sources.
Data sets {#Sect:datasets}
---------
### Original [[*Kepler* ]{}]{}light curves
The data set of eclipsing binaries provided in the [[*Kepler* ]{}]{}Eclipsing Binary Catalog [@kepler7] consists of 2876 binary systems, including eclipsing binary and multiple systems, ellipsoidal variables, and eccentric binaries with dynamical distortions, more commonly known as heartbeat stars [@thompson2012]. All eclipsing binary light curves have geometrical light curve parameters (eclipse depths, widths and separation) determined with the [[*polyfit* ]{}]{}model of @ebaipaper. Classification of the eclipsing binary systems is done via LLE [@kepler3], which yields a number between 0 and 1 that corresponds to the morphology of the system: 0 - 0.5 values are predominantly assigned to detached systems, 0.5 - 0.7 to semi-detached systems, 0.7 - 0.8 to contact binaries and 0.8 - 1 to ellipsoidal variables, while heartbeat stars do not have an assigned value. This parameter is denoted as the morphology parameter (hereafter [*LLE morph*]{}) and is used as a reference for evaluation of our classification methods.
### [[*Gaia*]{}-sampled ]{}[[*Kepler* ]{}]{}light curves {#Sect:gaiasampled}
To simulate [[*Gaia* ]{}]{}time-sampling of the light curves we make use of AGISLab , which is able to predict the transit times of specific sky locations based on the programmed scanning laws. Observation times were computed for a nominal five year [[*Gaia* ]{}]{}mission using the Nominal Scanning Law [^1], see e.g. sect. 5.2 of , for the 2876 Kepler Eclipsing binaries at their original sky-positions in the Kepler field. The resulting number of field-of-view transits is between 69 and 105 with an average of 87 neglecting any (potential) dead-time.
With per-target [[*Gaia* ]{}]{}timestamps available from the scanning-law expectations, we phased them according to the respective target ephemerides, and we linearly interpolated [[*Kepler* ]{}]{}light curves at those phases to obtain simulated [[*Gaia* ]{}]{}flux values. We then unfolded the phases back into time space and used those light curves as pseudo-[[*Gaia* ]{}]{}observations.
[[*Kepler* ]{}]{}per-point uncertainties are assigned statistically, based on the crowding metric and catalog magnitude. Considering that all [[*Kepler* ]{}]{}targets are on the bright [[*Gaia* ]{}]{}end, the pseudo-[[*Gaia* ]{}]{}light curves will be overwhelmingly dominated by shot noise [@christiansen2012]. Because of that we do not assign any per-point uncertainty to data and assume that intrinsic light curve variability in [[*Kepler* ]{}]{}data has a global noise level representative of [[*Gaia* ]{}]{}as well. Thus, our analysis is valid for [[*Gaia* ]{}]{}targets in the shot-noise regime, while better noise models would be needed for fainter targets (see @jordi2010). At this time no such comparison data set is available so we retain a simplified treatment of noise.
Fitting models {#Sect:fitting}
--------------
### Polyfit {#Sect:fitting_Polyfit}
The [[*polyfit* ]{}]{}analytical model is a polynomial chain fit to the data [@ebaipaper]. Individual polynomials in the chain are connected at knots, whose placement is determined iteratively, by minimizing the overall $\chi^2$ value of the fit. The chain is required to be connected and smoothly wrapped in phase space, but not necessarily differentiable at the knots. This way the polynomial fits, or [*polyfits*]{}), are able to reproduce the discrete breaks in light curve flux caused by eclipses. The knots are typically positioned at the top of ingress and egress of the primary and secondary eclipses, each pair spanning a polynomial. For characterizing [[*Kepler* ]{}]{}light curves, we use four knots and four quadratic polynomials, following @ebaipaper.
### Two-Gaussian model {#Sect:fitting_TwoGaussian}
The [[*two-Gaussian* ]{}]{}models aim at characterizing the eclipses and tidal-induced ellipsoidal variability of eclipsing binaries using simple mathematical functions that are fitted to their folded light curves (Mowlavi et al, submitted). The geometry of each eclipse is modeled with a base Gaussian function $G_{\mu_i,\,d_i,\,\sigma_i}(\varphi)$ of depth $d_i$ and width $\sigma_i$ located at phase $\varphi=\mu_i$, where the index $i$ denotes the primary ($i=1$) and secondary ($i=2$) eclipse. The base function is mirrored on phase intervals from $\varphi=-2$ to +2 in order to satisfy the boundary conditions of the periodic signal. The tidal-induced ellipsoidal variability, on the other hand, is modeled with a cosine function with a period equal to half the orbital period. In order to avoid an overfit of the data with non-significant components, various models with different combinations of these functions are first fitted to the folded light curves. The models range from a simple constant model to a full two-Gaussian model with ellipsoidal variability. The model with the highest Bayesian Information Criterion is then retained.
The light curve geometries induced by the eclipses and ellipsoidal variability are in reality more complex than what can be modeled with simple Gaussian and cosine functions, and it is not the aim of the [[*two-Gaussian* ]{}]{}model to provide a precise model of eclipsing binaries. However, the [[*two-Gaussian* ]{}]{}model can adequately estimate, in the majority of cases, the eclipse widths and depths, inter-eclipse separation, and ellipsoidal-like variability amplitude, all of which are used in this study to classify eclipsing binaries from their light curve geometries.
Light curve classification {#Sect:classification}
--------------------------
$t$-SNE is a dimensionality reduction algorithm that is steadily gaining popularity in the scientific community due to its capability to overcome the “crowding problem" present in many other dimensionality reduction techniques (e.g. LLE, SNE, Isomap; @tsne) and thus provides a perfect tool for visualizing high-dimensional data based solely on their similarity, without the need to provide additional data attributes. A $t$-SNE visualization of the original [[*Kepler* ]{}]{}data set is available in @kepler7. In this study, we extend this qualitative visualization technique with quantitative classification based on DBSCAN.
### $t$-SNE
The $t$-SNE algorithm is a modified version of the Stochastic Neighbor Embedding technique and has a specific appeal for visualizing data, since it is capable of revealing both global and local structure in terms of clustering data with respect to similarity. In practice, it takes only one input parameter that defines the configuration of the output map. The so-called perplexity ($perp$) parameter [@tsne] is similar to the number of nearest neighbors in other methods, with the difference that it leaves it up to the method to determine the number of nearest neighbors, based on the data density. This in turn means that the data themselves affect the number of nearest neighbors, which might vary from point to point.
$t$-SNE defines data similarities in terms of conditional probabilities in the high-dimensional data space and their low-dimensional projection. Neighbors of a data point in the high-dimensional data space are picked in proportion to their probability density under a Gaussian, whose variance is determined for each point separately, based on the perplexity value. Therefore, the similarity of two data points is equivalent to a conditional probability. In $t$-SNE, the conditional probability is replaced by a joint probability that depends on the number of data points, which ensures that all data points contribute to the cost function by a significant amount, including the outliers. The conditional probability of the corresponding low-dimensional counterparts in SNE is also defined in terms of a Gaussian probability distribution, but $t$-SNE has introduced a symmetrized Student $t$-distribution. This allows for a higher dispersion of data points in the low-dimensional map and overcomes the crowding problem that results from the overlapping of clusters in the embedding, since a moderate distance in the high-dimensional map can be represented well by larger distances in its low-dimensional counterpart [@tsne]. We have chosen values towards the upper recommended values of the perplexity parameter (30–50), which result in maps with well-defined separate clusters than can be efficiently used for visual inspection or quantitative classification. Lower perplexity values produce too many small clusters that do not bear any significant information, while higher values produce embeddings similar to the $perp=50$ value.
### DBSCAN
The $t$-SNE algorithm does not provide means for classification, but merely visualization of data. However, choosing an appropriate value of the perplexity parameter and implementing a two-step projection ($ND\rightarrow3D\rightarrow2D$) often results in spatially well localized projections of data grouped in several clusters, which allows for the implementation of clustering algorithms to find and isolate different groups. In this step of the analysis we have applied the DBSCAN algorithm [@dbscan], which groups points that are closely packed together (points with many nearby neighbors) and marks as outliers points that lie scattered in low-density regions. The result depends on two parameters: the maximum distance ($\varepsilon$) of all points in the same cluster from a core point and the minimum number of points ($MinPts$) required to form a dense region. It starts with an arbitrary starting point that has not been visited and once that point’s $\varepsilon$-neighborhood is retrieved, a cluster is defined if it contains sufficiently many points. Otherwise, the point is labeled as noise, but this point might later be found in a sufficiently sized $\varepsilon$-environment of a different point and be made part of a cluster. The result is then a list of labeled clusters and the objects corresponding to each of them can be readily retrieved and further analyzed.
Results {#Sect:Results}
=======
Light curve fits
----------------
### [[*Kepler* ]{}]{}data {#Sect:ModelComparison_Kepler}
We have studied a subset of 2861 eclipsing binaries from the Kepler Eclising Binaries catalog in order to evaluate the performance of the [[*two-Gaussian* ]{}]{}model and the classification with $t$-SNE and DBSCAN. The subset is formed by excluding light curves of higher hierarchy objects in multiple systems. For systems with two ephemerides in the catalog, we use only the light curve with the shortest period, while for systems with three or four ephemerides, we keep the light curves corresponding to the two shortest periods.
Several examples of [[*Kepler* ]{}]{}light curves and their models fits are given in Fig. \[Fig:lcfits\].
![Several examples of [[*Kepler* ]{}]{}light curves and their respective model fits with [*polyfits*]{} and [*[*two-Gaussians*]{}*]{}. The plots show the observed [[*Kepler* ]{}]{}light curve (grey dots) in normalized [[*Kepler* ]{}]{}($K$) magnitude, [[*polyfit* ]{}]{}model (solid red line) and [[*two-Gaussian* ]{}]{}model (dashed black line). Magnitudes are obtained from the [[*Kepler* ]{}]{}detrended flux and normalized to a reference value of $0$ out of eclipse.[]{data-label="Fig:lcfits"}](kepler_lcs_2-1.png){width="\hsize"}
---------------------------------------------- --------------- ---------------
Primary Secondary
Model ([[*Kepler* ]{}]{}data set) eclipse \[%\] eclipse \[%\]
[*Polyfit*]{} and [[*two-Gaussian* ]{}]{} 95.3 66.8
None 0.0 14.1
Only [*polyfits*]{} 4.7 3.7
Only [*two-Gaussians*]{} 0.0 15.4
Data set ([[*two-Gaussian* ]{}]{}model)
[[*Kepler* ]{}]{}and [[*Gaia*]{}-sampled ]{} 63.8 50.0
None 4.7 16.5
Only [[*Kepler* ]{}]{} 31.4 32.2
Only [[*Gaia*]{}-sampled ]{} 0.10 1.3
---------------------------------------------- --------------- ---------------
: The rate of eclipse identification by the two models on the set of 2861 phase-folded [[*Kepler* ]{}]{}and [[*Gaia*]{}-sampled ]{}light curves.[]{data-label="Tab:identification-rate1"}
The eclipse detection overlap between the two models is given in Table \[Tab:identification-rate1\]. The [[*two-Gaussian* ]{}]{}model fails to detect a primary eclipse for approximately 5% of the light curves. They are shown to correspond to very noise light curves, where an actual eclipse is practically invisible, even if present. The primary eclipse detected by [*polyfits*]{} in these systems is negligible, and we conclude that the failure of the [[*two-Gaussian* ]{}]{}model to detect it is not due to deficiencies of the model, but is governed by the data themselves. There is also a small portion (3.7%) of the light curves where the [[*two-Gaussian* ]{}]{}model fails to detect a secondary eclipse. These are mainly attributed to very narrow secondary eclipses, that do not contain many data points, as well as eccentric binaries, where the initial conditions are not suitable for the models to converge to a Gaussian at the position of the secondary eclipse. A relatively large number of cases ($15.4\%$) have an identified secondary eclipse by the [[*two-Gaussian* ]{}]{}model only, where a secondary Gaussian function has been fitted to the inter-eclipse variability. The differences in the model results are thus mainly a consequence of how each model adapts to the particular structure of the data. Examples of light curves that illustrate these discrepancies are given in Fig. \[fig:discrepant\_lcs\].
![Examples of [[*Kepler* ]{}]{}light curves where [*polyfits*]{} and [*two-Gaussians*]{} give discrepant results. The plots show the observed [[*Kepler* ]{}]{}light curve (grey dots) in normalized [[*Kepler* ]{}]{}($K$) magnitude, [[*polyfit* ]{}]{}model (solid red line) and [[*two-Gaussian* ]{}]{}model (dashed black line). Top left: a light curve where both models agree; top right: no primary eclipse detected by the [[*two-Gaussian* ]{}]{}model; bottom left: no secondary eclipse detected by the [[*two-Gaussian* ]{}]{}model; bottom right: no secondary eclipse detected by [[*polyfit* ]{}]{}.[]{data-label="fig:discrepant_lcs"}](Kepler_all-1.png){width="\hsize"}
For many of the cases the [[*polyfit* ]{}]{}and [[*two-Gaussian* ]{}]{}models look almost identical, but the underlying model parameters can differ greatly, in particular the eclipse widths. [*Polyfits*]{} compute them based on the positions of the knots in the polynomial chain [@ebaipaper], while the eclipse width in the [[*two-Gaussian* ]{}]{}model corresponds to the widths of the Gaussian functions at a reference magnitude equal to 2% of the respective depth. For contact and ellipsoidal systems, which show continuous variability, the polynomial knots are usually positioned midway through the eclipses, resulting in a width of $\Delta\varphi\sim0.25$, while in the [[*two-Gaussian* ]{}]{}model the eclipse width of these objects would typically saturate at $\Delta\varphi\sim0.5$ and is limited to $\Delta\varphi=0.4$. This disparity should be taken into account when comparing values from the [[*two-Gaussian* ]{}]{}model and the [[*polyfit* ]{}]{}values given in the [[*Kepler* ]{}]{}Eclipsing Binaries catalog.
### [[*Gaia*]{}-sampled ]{}data
Unlike [*Kepler*]{}, [[*Gaia* ]{}]{}light curves have sparse (irregular) sampling. This introduces an additional difficulty to the models and is especially notable in [*polyfits*]{}, which in the absence of a well-defined light curve, tend to fit the polynomial chain to the out-of-eclipse noise. The [[*two-Gaussian* ]{}]{}models, on the other hand, will either fit a constant model to the noise or produce a smooth curve resembling a physical light curve. This is a great advantage when dealing with light curves with well-defined eclipses, but noisy inter-eclipses, as well as for discarding cases where the points in an eclipse cannot be used for any quantitative analysis (bottom left panel of Fig. \[Fig:pf\_vs\_2g\_lcs\]).
![Same as Fig. \[Fig:lcfits\], but for [[*Gaia*]{}-sampled ]{}data. Top left: good quality data and both matching fits. Top right: good data, slightly discrepant fits. Bottom left: bad phase coverage in eclipse, [*two-Gaussians*]{} fit a constant model, while [*polyfits*]{} find eclipse. Bottom right: bad quality light curve and corresponding model fits.[]{data-label="Fig:pf_vs_2g_lcs"}](gaia_lcs_2-1.png){width="\hsize"}
The eclipse identification rate with [*two-Gaussian*]{} models on the [[*Gaia*]{}-sampled ]{}data set again shows a loss of about $31.5\%$ of the primary eclipse and an additional $16.8\%$ of secondary eclipse identifications, which is a consequence of the deterioration of light curve quality. This gives an estimate of $\sim32\%$ of Kepler Eclipsing Binaries that will likely not be identified with [[*Gaia* ]{}]{}sampling. A summary of these results is given in Table \[Tab:identification-rate1\].
Light curve classification {#light-curve-classification}
--------------------------
We use a combination of the two techniques to classify a data set of $M$ light curves, based on the per-point similarity of the phase-folded light curve models, computed in $N$ phase points. Our [[*Gaia*]{}-sampled ]{}[[*Kepler* ]{}]{}data set consists of $M=2861$ light curve models computed in $N=1000$ equidistant phase points ranging from $-0.5$ to $0.5$ with imposed periodic boundary conditions. The input is an $M\times N$ array of the light curve model magnitudes. Each row is first rescaled so that its magnitudes span the range of $[0,1]$, via:
$$m_i(\mathrm{rescaled}) = \frac{m_i - \min(m_i)}{\max(m_i-\min(m_i))}$$
where $m_i$ is the array of model magnitudes of the $i$-th source in the input array. This ensures that the mapping is only sensitive to the light curve shape and unaffected by the different primary eclipse depths. The rescaled magnitude array is first mapped to a three-dimensional map with $t$-SNE. The three-dimensional map serves as input to the second step of the mapping, in which a two-dimensional $t$-SNE map is produced. The two-dimensional $t$-SNE map is then scanned by DBSCAN, which defines a set of clusters and labels them, returning the per-point labels in a $M\times1$ array, which can be then assigned to the original input set of light curves. An illustration of the algorithm flow is given in Fig. \[Fig:flowchart\].
![Flowchart of the $t$-SNE+DBSCAN application to a set of $M$ phase-folded light curve models computed in $N$ equidistant phase points.[]{data-label="Fig:flowchart"}](tsne_flowchart_2-1.png){width="0.8\hsize"}
### Polyfit {#Sect:fullDataSet_Polyfits}
{width="0.32\hsize"} {width="0.32\hsize"} {width="0.32\hsize"}
{width="0.32\hsize"} {width="0.32\hsize"} {width="0.32\hsize"}
We use the Barnes-Hut version of $t$-SNE [@VanDerMaaten2014] with $perp_{3D} = 50$ and $perp_{2D} = 45$ and DBSCAN with $\varepsilon = 5$ and $MinPts = 50$ that result in the identification of five [[*polyfit* ]{}]{}clusters. The left panel of Fig. \[fig:pf\_full\_tsne\] shows the two-dimensional map color-coded by the detected DBSCAN classes. A visual inspection of the sources pertaining to each of the classes and their true [[*Kepler* ]{}]{}[[*polyfit* ]{}]{}parameters is used to define the descriptive classification provided in Table \[Tab:dbscan\_stats\]. It shows that the mapping has been driven by the morphology of the systems, ranging from close eclipsing binaries and ellipsoidal variables to detached systems, as well as the relative depth of the eclipses. Phase-folding of the [[*polyfit* ]{}]{}models has been performed with the known value of the zero-time reference point of primary eclipse obtained by [[*Kepler* ]{}]{}light curves, which is why we get a class of sources with a more conspicuous secondary eclipse. Had this value not been provided to the model, classes 3 and 4 would have most likely been merged into one class of light curves with only one conspicuous eclipse, centered at orbital phase $\varphi = 0$. Classes 1 and 2 contain the close binary types (semi-detached –SD–, contact binaries –CB– and ellipsoidal variables –ELV–), while class 0 contains the poor fits, i.e. [*polyfits*]{} which do not resemble a physical light curve.
In addition to the automated approach, we have performed a visual evaluation of each [[*polyfit* ]{}]{}with three flags: good (g), medium (m) and bad (b) quality fit. Out of the 2861 sources, 1308 (46%) were marked as good, 770 (27%) as medium, and only 783 (27%) as bad. Actually, the DBSCAN class 0 contains $\sim93.6\%$ of the visually flagged poor fits, $\sim 6\%$ of the medium quality and only $\sim 0.4\%$ of good quality fits, which shows that class 0 is predominantly composed of the visually marked bad fits. The distribution of the flags over the $t$-SNE projection is given in the middle panel of Fig. \[fig:pf\_full\_tsne\]. The distribution of the [[*Kepler* ]{}]{}morphology parameter derived with LLE (@kepler3; right panel of Fig. \[fig:pf\_full\_tsne\]) shows the continuous transition from detached ($morph=0$) to contact/ellipsoidal ($morph=1$) binaries over the four classes containing good quality fits, while the poor [[*polyfit* ]{}]{}class is shown to be mainly composed of detached systems, whose eclipses are likely not observed in [[*Gaia* ]{}]{}cadences.
We use [*Kepler*]{}-derived parameters and quantities related to the quality of fits to search for indicators of the underlying causes for the bad data quality resulting in poor fits. The histograms of the orbital period, primary eclipse depth and eclipse separation parameters over the different classes given in Fig. \[fig:pfhistograms\] show that class 0 is composed of both long and short period binaries and low to high eclipse depth values, but the low-eclipse depth systems prevail. Based on these parameter distributions, we conclude that class 0 is composed of sources that will likely not show any eclipsing binary features due to the possible miss of eclipse or low eclipse signal buried in the background noise. Classes 3 and 4 show similar parameter distributions, but their light curves have at least one prominent eclipse, thus we can consider those “lucky catch eclipses” that happen in [[*Gaia* ]{}]{}cadences. The wider distributions of eclipse separation in classes 0, 3 and 4 show that they contain eccentric systems, as well as light curves where the secondary eclipse is not present (eclipse separation equal to 1). This similarity of class 0 to classes 3 and 4 further reinforces the necessity of a filtering approach primarily based on the light curve shape, since no threshold can be set to the values of any of the characteristic light curve parameters that would provide a clear distinction between poor and good quality fits.
\[Tab:dbscan\_stats\]
### Two-Gausssian {#Sect:fullDataSet_TwoGaussian}
The [[*two-Gaussian* ]{}]{}model has a built-in mechanism of identifying the light curves which do not show any eclipses, through the choise of a constant as the best-fit model. Out of the 2861 [[*Gaia*]{}-sampled ]{}light curves, 859 or about 30% were fitted with a constant model. 575 or 66% of these light curves also belong to the bad polyfit class, while the remaining 34% primarily correspond to light curves with one or just a few data-points in eclipses, which the [[*two-Gaussian* ]{}]{}model flags as insignificant, while [[*polyfit* ]{}]{}still fits an eclipse. Examples of these light curves are given in Fig. \[fig:gs\_c\_goodpf\].
![Examples of [[*Gaia*]{}-sampled ]{}light curves where [[*polyfit* ]{}]{}fits an eclipse, while [*two-Gaussians*]{} do not. The plots show the observed [[*Kepler* ]{}]{}light curve (grey dots) in normalized [[*Kepler* ]{}]{}($K$) magnitude, [[*polyfit* ]{}]{}model (solid red line) and [[*two-Gaussian* ]{}]{}model (dashed black line).[]{data-label="fig:gs_c_goodpf"}](CgoodPf-1.png){width="\hsize"}
### Filtered data set
The [[*two-Gaussian* ]{}]{}model has a more thorough and consistent approach towards filtering data that is unlikely to pass [[*Gaia*]{}’s ]{}eclipsing binary detection pipeline, thus we remove the 859 light curves fitted with the constant model and propose a classification scheme on the remaining 2002 sources. The application of $t$-SNE+DBSCAN with $perp_{3D}=35$, $perp_{2D}=35$, $\varepsilon=2.6$ and $MinPts=18$ on the [[*two-Gaussian* ]{}]{}models results in nine classes (Fig. \[fig:2g\_tsne\]), marked from 1 to 9, which can be used to define a classification scheme based on the morphology of the systems and geometry of the light curve fits, given in Table \[Tab:2gfiltered\]. Representative light curves of each class are given in Fig. \[fig:2g\_tsne\].
{width="0.48\hsize"} {width="0.42\hsize"}
{width="0.45\hsize"} {width="0.45\hsize"}
class class description subclass class \# subclass description \# of sources %
------- ------------------------------------ ---------- ---------- ------------------------------ --------------- ----
D-1 1 one conspicuous eclipse 429 21
D-2 2 two conspicuous eclipses 105 5
D+SD-1 3 one conspicuous eclipse 178 9
D+SD-2 4 two conspicuous eclipses 180 9
SD+CB semi-detached and contact binaries / 5 208 10
CB contact binaries / 6 115 6
CB+ELV-a 7 eclipse depth ratio $< 1$ 212 11
CB+ELV-b 8 eclipse depth ratio $\sim 1$ 403 20
ELF ellipsoidal fits / 9 172 9
{width="0.32\hsize"} {width="0.32\hsize"} {width="0.32\hsize"}
Six main classes have been defined based on the light curve morphology, ranging from detached (D), detached and semi-detached (D+SD), semi-detached and contact binaries (SD+CB), contact binaries (CB), contact binaries and ellipsoidal variables (CB+ELV) and ellipsoidal fits (ELF). A clear distinction between overlapping morphological types among the different classes (D+SD, SD+CB and CB+ELV) cannot be made because their light curve fits are intrinsically similar and the continuous transition from one morphological type to another is an inherent property of the light curve shapes. A more detailed distinction can be made through the inspection of the individual light curve properties, while in some cases full modeling of the system might be required for the accurate determination of its morphological type. The proposed morphological classes are thus an initial indication of the system morphology based on its light curve fit with the [[*two-Gaussian* ]{}]{}model.
The subclasses of each class are based on the geometrical properties of the light curves. The presence and visibility of eclipses define the subclasses in the detached and semi-detached classes: one conspicuous eclipse (D-1 and D+SD-1) or two conspicuous eclipses (D-2 and D+SD-2), while in the CB+ELV subclasses two eclipses are visible in most of the cases, thus the sub-classification is driven by the eclipse or ellipsoidal variation widths and depths. In contact systems, the eclipse widths and depths can also be used as indicators of the physical system parameters, like filling factor and temperature equilibrium. Wider eclipses, characteristic for classes 7 and 8, point to a larger filling factor, while similar eclipse depths, characteristic for classes 6 and 8, point to a system close to thermal equilibrium (e.g. a W UMa star).
The distribution of the [[*Kepler* ]{}]{}LLE morphology parameter over classes (left panel of Fig. \[fig:2g\_tsne\_dist\] and middle panel of Fig. \[fig:2gshistograms\]), suggests that the proposed classification scheme corresponds to the morphological type of the observed sources determined on the true [[*Kepler* ]{}]{}light curves. The orbital period distribution (left panel of Fig. \[fig:2gshistograms\]) further supports this notion, with transitions from long-periods in the detached to shorter in semi-detached and contact classes.
The distribution of the different [[*two-Gaussian* ]{}]{}model types over the projection (right panel of Fig. \[fig:2g\_tsne\_dist\]) shows that the projection is highly driven by the choice of the fitting model. This is expected since the model defines the light curve geometry, but the different widths and depths of the eclipses lead to mixing of the models in most of the classes, with the exception of class 9, which is composed solely of ellipsoidal fits.
The distribution of the reduced $\chi^2$ value (right panel of Fig. \[fig:2gshistograms\]) is an indicator of the fit quality, which indirectly influences the classification. The classes corresponding to detached systems have both low and high reduced $\chi^2$, due to the small width and contribution of the eclipses to the overall light curve, while as we move towards closer systems with wider and more significant eclipses, the distributions are dominated by lower reduced $\chi^2$ values. This is a valuable indicator of the reliability of light curve parameters provided for each class.
Class 9 shows the most peculiar parameter distributions of all, pertaining to both very low and very high values of the morphology parameter, as well as predominantly higher reduced $\chi^2$ values. These indicate that class 9 is not only composed of ellipsoidal fits which correspond to true ellipsoidal variables, but also of detached systems where an eclipse has not been observed and the cosine function is fitted to the inter-eclipse scatter. This flags class 9 and the model parameters derived for the sources in it as unreliable and subject to further filtering.
Conclusions and future prospects {#Sect:Conclusions}
================================
In this paper, we have presented for the first time a proposed method of automated reduction and classification of [[*Gaia* ]{}]{}eclipsing binary data.
Results from both the analysis of the bad cases identified by the [*polyfits*]{} and the [[*two-Gaussian* ]{}]{}models, and the comparison of eclipse detection rates of the [[*two-Gaussian* ]{}]{}model applied to real [[*Kepler* ]{}]{}and [[*Gaia*]{}-sampled ]{}[[*Kepler* ]{}]{}data, show that about 68% of all eclipsing binaries in the magnitude interval of [[*Kepler* ]{}]{}are detectable by [[*Gaia* ]{}]{}over a five year mission. The orbital parameters and morphologies derived from [[*Kepler* ]{}]{}data show that the 32% non-detectable sources are mainly long-period, detached binaries, with very narrow eclipses that can easily be missed in the $\sim87$ light curve points expected to be observed (on average) in Cygnus by [[*Gaia* ]{}]{}over the five years. Given that the all-sky average number of observations for [[*Gaia* ]{}]{}is $\sim67$, we expect the all-sky detectability to be lower than 68%. The other type of sources less likely to be detected comprises systems with very low eclipse depths that can easily be buried in the data noise.
We investigated the efficiency of the combined use of the $t$-SNE and DBSCAN algorithms to classify eclipsing binary light curves. The application to [[*Kepler* ]{}]{}eclipsing binary light curves sampled at observation times predicted by the [[*Gaia* ]{}]{}scanning law and characterized with the [[*two-Gaussian* ]{}]{}model shows that the method is successful in identifying six broad classes corresponding to the system morphology (detached, detached+semi-detached, semi-detached+contact binaries, contact binaries, contact binaries+ellipsoidal variables and ellipsoidal fits).
An additional sub-classification is introduced based on the properties of the fitted models (presence and visibility of eclipses and eclipse widths) to distinguish them from the physical properties of the observed systems, which may not be correctly evaluated due to the irregular sampling in [[*Gaia* ]{}]{}observations (for example, systems with only one observed eclipse may in reality also have a prominent secondary eclipse that was not observed with [*Gaia*]{}’s scanning law).
The thorough testing, formulation and implementation of automated reduction and classification techniques are of highest priority in the [[*Gaia* ]{}]{}era. One aspect of the method that is still open for adjustment on real [[*Gaia* ]{}]{}data is its performance on data sets of much larger scale than the one used in this study. Several optimization approaches using parametric mapping between the high- and low-dimensional space (i.e. parameteric $t$-SNE) are being considered, as well as a combination of more than one classification approach (see @suveges2016). The final aim of this collaborative effort is to provide [[*Gaia* ]{}]{}data archive users with a clean set of geometrical light curve parameters of eclipsing binaries, an estimate of their credibility, and classification types that would make the selection of a desired data subset as effortless and reliable as possible.
We are grateful to our referee Dr. J. A. Caballero for his constructive input, which substantially improved the quality and presentation of the paper.
A. K. gratefully acknowledges the MSE postdoctoral fellowship of the College of Liberal Arts and Sciences at Villanova University.
[^1]: Alternative scanning laws were used during two months prior to the start of the Nominal Scanning Law (NSL), but those did not intersect with the Kepler field.
|
---
abstract: 'We present an explicit product formula for the spherical functions of the compact Gelfand pairs $(G,K_1)= (SU(p+q), SU(p)\times SU(q))$ with $p\ge 2q$, which can be considered as the elementary spherical functions of one-dimensional $K$-type for the Hermitian symmetric spaces $G/K$ with $K= S(U(p)\times U(q))$. Due to results of Heckman, they can be expressed in terms of Heckman-Opdam Jacobi polynomials of type $BC_q$ with specific half-integer multiplicities. By analytic continuation with respect to the multiplicity parameters we obtain positive product formulas for the extensions of these spherical functions as well as associated compact and commutative hypergroup structures parametrized by real $p\in]2q-1,\infty[$. We also obtain explicit product formulas for the involved continuous two-parameter family of Heckman-Opdam Jacobi polynomials with regular, but not necessarily positive multiplicities. The results of this paper extend well known results for the disk convolutions for $q=1$ to higher rank.'
author:
- |
Margit Rösler\
Institut für Mathematik, Universität Paderborn\
Warburger Strasse 100, D-33098 Paderborn\
[email protected]\
and\
Michael Voit\
Fakultät Mathematik, Technische Universität Dortmund\
Vogelpothsweg 87, D-44221 Dortmund, Germany\
e-mail: [email protected]
title: A multivariate version of the disk convolution
---
Key words: Hypergeometric functions associated with root systems, Heckman-Opdam theory, Jacobi polynomials, disk hypergroups, positive product formulas, compact Grassmann manifolds, spherical functions.
AMS subject classification (2000): 33C67, 43A90, 43A62, 33C80.
Introduction
============
It is well-known that the spherical functions of Riemannian symmetric spaces of compact type can be considered as Heckman-Opdam polynomials, which are the polynomial variants of Heckman-Opdam hypergeometric functions. In particular, the spherical functions of Grassmann manifolds $SU(p+q,
\mathbb F)/S(U(p, \mathbb F)\times U(q,\mathbb F))$ with $p\geq q$ and $\mathbb F=\mathbb R,
\mathbb C, \mathbb H$ can be realized as Heckman-Opdam polynomials of type $BC_q$ with certain mulitplicities given by the root data, namely $$k= (d(p-q)/2, (d-1)/2, d/2) \quad \text{with } \,d= \dim_\mathbb R \mathbb F \in\{1,2,4\},$$ corresponding to the short, long and middle roots, respectively. We refer to [@HS], [@H] and [@O1] for the foundations of Heckman-Opdam theory, and to [@RR] for the compact Grassmann case. Recall that the spherical functions of a Gelfand pair $(G,K)$ can be characterized as the continuous, $K$-biinvariant functions on $G$ satisfying the product formula $$\label{prodformel} \varphi(g)\varphi(h) = \int_K \varphi(gkh)dk\quad (g,h\in G)$$ where $dk$ is the normalized Haar measure on $K.$ In [@RR], the product formula for the spherical functions of $SU(p+q,
\mathbb F)/S(U(p, \mathbb F)\times U(q,\mathbb F)),$ considered as functions on the fundamental alcove $$\label{A_q} A_q = \{t\in \mathbb R^q:\> \pi/2\ge t_1\ge
t_2\ge\ldots\ge t_q\ge0\},$$ was extended by analytic interpolation to all real parameters $p\in]2q-1,\infty[.$ This led to positive product formulas for the associated $BC$-Heckman-Opdam polynomials as well as commutative hypergroups structures on $A_q$ with the Heckman-Opdam polynomials as characters. For a background on hypergroups see [@J], where hypergroups are called convos. For the non-compact Grassmannians over $\mathbb R, \, \mathbb C$ and $\mathbb
H$, similar constructions had previously been carried out in [@R2]. In the case $\mathbb F= \mathbb C$, where the non-compact Grassmannian $SU(p,q)/ S(U(p)\times U(q))$ is a Hermitian symmetric space, the approach of [@R2] was extended to the spherical functions of $U(p,q)/U(p)\times SU(q)$ in [@V9]. Here the analysis was based on the decisive fact that the spherical functions of this (non-symmetric) space are intimately related to the generalized spherical functions of the symmetric space $SU(p,q)/ S(U(p)\times U(q))$ of one-dimensional $K$-type. These in turn can be expressed in terms of Heckman-Opdam hypergeometric functions according to [@HS], Section 5.
In the present paper, we consider the pair $(G,K_1)$ with $G=SU(p+q)$ and $K_1= SU(p)\times SU(q)$ (over $\mathbb F=\mathbb C$), where we assume $p>q$. In this case, $(G,K_1)$ is a Gelfand pair. By a decomposition of Cartan type, we identify the spherical functions of $G/K_1$ as functions on the compact cone $$X_q:=\{(zr_1,r_2, \ldots, r_q): \quad 0\le r_1\le\ldots r_q\le 1, \>
z\in\mathbb T\}\subset \mathbb C\times\mathbb R^{q-1}$$ with the torus $\mathbb T:=\{z\in\mathbb C:\> |z|=1\}$. Similar as in the non-compact case [@V9], the spherical functions of $G/K_1$ can be considered as elementary spherical functions of $K$-type $\chi_l\, ( l\in \mathbb Z$) for the Hermitian symmetric space $G/K$ with $K= S(U(p)\times U(q)).$ According to the results of Heckman [@HS], they can be expressed in terms of of Heckman-Opdam polynomials of type $BC_q$, depending on the integral parameter $p>q$ and the spectral parameter $l\in\mathbb Z$. Under the stronger requirement $p\geq 2q$, we proceed similar as in [@RR] and write down the product formula for the spherical functions of $G/K_1$ as product formulas on $X_q$.
In Section 4 we then extend this formula to a product formula on $X_q$ for a continuous range of parameters $p\in] 2q-1,\infty[$ by means of Carlson’s theorem, a principle of analytic continuation. In particular, we obtain a continuous family of associated commutative hypergroup structures on $X_q$. We determine the dual spaces and Haar measures of these hypergroups. For $q=1$, the space $X_q$ is the complex unit disk, and the associated hypergroups are the well-known disk hypergroups studied in [@AT], [@Ka], and [@BH], where the associated product formulas are based on the work of Koornwinder [@K2].
For each real $p\in] 2q-1,\infty[$, the associated hypergroup structure on $X_q$ contains a compact subgroup isomorphic to the one-dimensional torus $(\mathbb T,\cdot)$. Moreover, the quotient $X_q/\mathbb T$ can be identified with the alcove $A_q$ and carries associated quotient convolution structures; see [@J] and [@V1] for the general background. In this way we in particular recover the above-mentioned hypergroup structures of [@RR] on $A_q$ for $\mathbb F=\mathbb C$. More generally, we obtain in Section 5 explicit continuous product formulas and convolution structures on $A_q$ for all Jacobi polynomials of type $BC_q$ with multiplicities $$k=(k_1,k_2,k_3)= (p-q-l,1/2+l, 1)$$ with $p\in] 2q-1,\infty]$ and $l\in\mathbb R$. Unfortunately, for general $l\ne0$, the positivity of these product formulas remains open.
**Acknowledgement.** It is a pleasure to thank Maarten van Pruijssen for valuable hints concerning Gelfand pairs.
Preliminaries
=============
We start our considerations with the compact Grassmann manifolds $G/K$ over $\mathbb C$, where $G= SU(p+q)$ and $K=S(U(p)\times U(q))$ with $p\geq q\geq 1.$ From the Cartan decomposition of $G $ (see [@RR] or Theorem VII.8.6 of [@Hel]) it follows that a system of representatives of the double coset space $G //K$ is given by the matrices $$a_t =
\begin{pmatrix} I_{p-q}&0&0\\0 & \cos \underline t & -\sin \underline t
\\0 &\sin \underline t & \cos \underline t \end{pmatrix},\quad t\in A_q$$ with $A_q$ as in ; here $\underline t$ is the $q\times q$-diagonal matrix with the components of $t$ as entries, and $\cos\underline t,\, \sin\underline t$ are understood componentwise. We recall from [@RR] how the double coset representatives can be determined explicitly:
For $X\in M_q(\mathbb C)$ denote by $\,\sigma_{sing}(X) =
\sqrt{\text{spec}(X^*\!X)} = (\sigma_1 \ldots, \sigma_q) \in \mathbb R^q$ the vector of singular values of $X$, decreasingly ordered by size. Write $g\in G$ in $(p\times q)$-block notation as $$\label{block} g = \begin{pmatrix} A(g) & B(g)\\
C(g) & D(g)
\end{pmatrix},$$ and suppose that $g \in K b_tK.$ Then $$\label{xspec} t =
\text{arccos}(\sigma_{sing}(D(g))).$$
By [@RR], the spherical functions of $G/K$ are given by Heckman-Opdam polynomials, as follows: Consider $\mathbb R^q$ with the standard inner product $\langle \,.\,,\,.\,\rangle$ and denote by $F_{BC_q}(\lambda,k;t)$ the Heckman-Opdam hypergeometric function associated with the root system $$R=2BC_q =
\{ \pm 2e_i, \pm 4e_i, 1\leq i\leq q\}\cup\{ \pm 2e_i \pm 2e_j: 1\leq i<j\leq q\}\subset\mathbb R^q$$ and with multiplicity parameter $k=(k_\alpha)_{\alpha \in R}$, in the notion of [@R2]. Recall that there exists an open regular set $K^{reg}\subset \mathbb C^3$ such that $F_{BC_q}(\lambda,k;\,.\,)$ exists in a suitable tubular neighbourhood of $\mathbb R^q\subset \mathbb C^q.$ We also write $k=(k_1, k_2, k_3)$ where $k_1, k_2, k_3$ belong to the short, long and middle roots, respectively. Fix the positive subsystem $\,R_+ = \{ 2e_i, 4e_i, 2e_i \pm 2e_j: 1\leq i<j\leq q\}.$ Writing $\alpha^\vee:=2\alpha/\langle\alpha,\alpha\rangle$, the associated set of dominant weights is $$\label{dominantweights} P_+ := \{ \lambda \in \mathbb R^q:
\langle\lambda,\alpha^\vee\rangle \in \mathbb Z_+
\,\,\forall\,\alpha\in R_+\}=\{\lambda\in 2\mathbb Z_+^q:\> \lambda_1\ge
\lambda_2\ge\ldots\ge\lambda_q\}.$$ Notice that our normalization of root sytems and multiplicities is in accordance with [@HS], [@O1] but differs from the “geometric” notion of [@RR], where both are rescaled by a factor $2$.
We consider the renormalized Heckman-Opdam polynomials associated with $R$ and $k$ in trigonometric notion as in [@RR], which are defined by $$\label{HOrenorm} R_\lambda(k;t) = F_{BC_q}(\lambda + \rho(k),k;it), \quad
\lambda \in P_+ \,.$$ Here $$\rho(k) = \frac{1}{2}\sum_{\alpha \in R_+} k_\alpha \alpha$$ and $$c(\lambda,k) = \prod_{\alpha\in R_+}
\frac{\Gamma\bigl(\langle\lambda,\alpha^\vee\rangle +
\frac{1}{2}k_{\alpha/2}\bigr)}{\Gamma \bigl(\langle\lambda,\alpha^\vee\rangle +
\frac{1}{2}k_{\alpha/2} + k_\alpha\bigr)}\cdot \prod_{\alpha\in R_+}
\frac{\Gamma
\bigl(\langle\rho(k),\alpha^\vee\rangle + \frac{1}{2}k_{\alpha/2} +
k_\alpha\bigr)}{\Gamma\bigl(\langle\rho(k),\alpha^\vee\rangle +
\frac{1}{2}k_{\alpha/2}\bigr)}\,$$ is the generalized $c$-function, with the convention $\,k_{\alpha/2} := 0$ if $\,\alpha/2\notin R$. The Heckman-Opdam polynomials are holomorphic on $\mathbb C^q$. Indeed, $F_{BC_q}(\lambda + \rho(k),k;\,.\,)$ is holomorphic on all of $\mathbb C^q$ if and only if $\lambda \in P_+$, see [@HS].
According to Theorem 4.3. of [@RR], the spherical functions of $G/K=SU(p+q)/S(U(p)\times U(q))$ can be considered as trigonometric polynomials on the alcove $A_q\subset \mathbb R^q$ and are given by $$\varphi_\lambda (t) = R_\lambda(k;t), \quad \lambda \in P_+$$ with the multiplicity $$k = (k_1, k_2, k_3) = (p-q, 1/2, 1).$$ For fixed $q\ge1$ and $p\geq 2q$, the product formula (\[prodformel\]) for the $\varphi_\lambda$ was written in [@RR] as a product formula on $A_q$ depending on $p$ in a way which allowed extension to all real parameters $p> 2q-1$ by analytic continuation; see Theorem 4.4. of [@RR].
Spherical functions of $(SU(p+q),
SU(p)\times SU(q))$ and their product formula
=============================================
Let us turn to the pair $(G, K_1):=(SU(p+q), SU(p)\times SU(q)), $ where we assume $p>q$. In this case, $(G,K_1)$ is a Gelfand pair according to the classification of [@Kr]. In this section, we derive an explicit product formula for the spherical functions of $(G, K_1)$. First, we determine a system of representatives for the double coset space $G//K_1$. For this, consider the compact set $$X_q := \{x= (zr_1,r_2,\ldots, r_q): z\in \mathbb T, \,r_i\in \mathbb R \text{
with }0\leq r_1 \leq \ldots \leq r_q \leq 1\}\subseteq \mathbb C\times
\mathbb R^{q-1}.$$ If $q=1$, this is just the closed unit disc in $\mathbb C$. For $q\geq 2$, the set $X_q$ can be interpreted as a cone of real dimension $q+1$ with $X_{q-1}$ as basis set.
For $x=(zr_1, r_2, \ldots, r_q) \in X_q$ we write $$x=[r,z] \,\,\text{ with } \,r= (r_1, \ldots, r_q) .$$ Note that the phase factor $z\in
\mathbb T$ is arbitrary if $r_1 = 0.$ Now consider the alcove $A_q$. We define an equivalence relation on $A_q\times
\mathbb T$ via $$(t,z) \sim (t^\prime, z^\prime ) :\Longleftrightarrow\, t= t^\prime \,\text{
and }\, z = z^\prime \text{ if }\, t_1 = t_1^\prime <\frac{\pi}{2}.$$ Then the mapping $\, A_q\times \mathbb T \,\to X_q\,, \, (t,z) \mapsto [\cos t, z]$ induces a homeomorphism between the quotient space $(A_q\times \mathbb T)/\sim\,$ and the cone $X_q$.
For $z\in \mathbb T$, let $h(z) = \text{diag}(z, 1\ldots, 1) \in
M_q(\mathbb C).$ Then a set of representatives of $G// K_1$ is given by the matrices $b_x, x\in X_q$ with $$\label{b-x-z-def}
b_{x} = \begin{pmatrix} I_{p-q}&0&0\\0 & h(z^{-1})\,\cos\underline t
\,&
-\sin\underline t
\\0 & \sin\underline t & h(z)\cos \underline t\,\end{pmatrix} \text { for }\,x
= [\cos t, z] \,\text{ with } t\in A_q.$$ In particular, the double coset space $G//K_1$ is naturally homeomorphic with the cone $X_q$.
We first check that each double coset has a representative of the stated form. In fact, by the known Cartan decomposition of $G$ with respect to $K$, each $g\in G$ can be written as $$g=\begin{pmatrix} u_1&0\\0&v_1\end{pmatrix} a_t
\begin{pmatrix} u_2&0\\0&v_2\end{pmatrix}$$ with $t\in A_q$ and $u_i\in U(p)$, $v_i\in U(q)$ satisfying $\det(u_i)\cdot \det(v_i) = 1.$ As $U(q) = SU(q) \rtimes H_q$ with $H_q = \{ h(z): z\in \mathbb T\}$, there are $z_1, z_2 \in \mathbb T$ such that $ \widetilde v_1 := v_1h(z_1)^{-1}\in SU(q)$ and $\, \widetilde v_2 :=
h(z_2)^{-1}v_2\in SU(q).$ For $z,w\in \mathbb T,$ define $\, H(w,z)= \text{diag}(w,1, \ldots, 1, z, 1, \ldots, 1)
\in M_{p}(\mathbb C)$ with the entry $z$ in position $p+1$. Put $\,\widetilde u_1 := u_1\cdot H(z_1/z_2,z_2)$ and $\,\widetilde
u_2 := H(z_2/z_1, z_1)\cdot u_2\in U(p).$ Then $\widetilde u_1, \widetilde u_2 \in SU(p)$, and a short calculation in $p\times q$-blocks gives $$\begin{aligned}
g \,=\,& \begin{pmatrix}
u_1 & 0 \\
0 & v_1 \end{pmatrix}
\begin{pmatrix}
I_{p-q}&0&0\\0 & \cos \underline t & -\sin \underline t
\\0 &\sin \underline t & \cos \underline t \,
\end{pmatrix}
\begin{pmatrix} u_2 & 0 \\
0 & v_2
\end{pmatrix} \\
\,=\, & \begin{pmatrix}
\widetilde u_1 & 0 \\
0 & \widetilde v_1 \end{pmatrix}
\begin{pmatrix}
I_{p-q}&0&0\\0 & \,h(z_1)^{-1}h(z_2)^{-1} \cos \underline t\, & \,-\sin
\underline t
\\0 &\,\sin \underline t \,& \, h(z_1)h(z_2)\cos \underline t \,
\end{pmatrix}
\begin{pmatrix} \widetilde u_2 & 0 \\
0 & \widetilde v_2
\end{pmatrix}.\end{aligned}$$ Thus $g\in K_1 b_x K_1$ with $x= [\cos t, z_1z_2]$.
In order to show that the $b_{x}$ are contained in different double cosets for different $x,$ we analyze how the parameter $x$ depends on $g\in G$. Let $$g = \begin{pmatrix} \widetilde u_1 & 0 \\
0 & \widetilde v_1
\end{pmatrix} b_{x}
\begin{pmatrix} \widetilde u_2 & 0 \\
0 & \widetilde v_2
\end{pmatrix}$$ with $\widetilde u_i \in SU(p), \,\widetilde v_i \in SU(q).$ Suppose that $x= [r,z]$ with $r= \cos t, \, t\in A_q$. Using the $(p\times q)$-block notation , we obtain $$D(g) = \widetilde v_1 h(z) \underline r \,\widetilde v_2\,, \quad
\Delta(D(g)) = z\cdot\prod_{i=1}^q r_i\,= \, z\cdot |\Delta(D(g))|,$$ Here and throughout the paper, $\Delta$ denotes the usual determinant of a complex square matrix. Thus $\, r= \cos t\, = \sigma_{sing}(D(g)).$ Further, $z$ is uniquely determined by $g$ exactly if $r_1\not=0$, which is equivalent to $\prod_{i=1}^q r_i \not = 0. $ In this case, $ \, z= \frac{\Delta D(g)}{|\Delta D(g)|}.$ Thus $(t,z)\in A_q\times \mathbb T$ is uniquely determined by $g$ up to the equivalence $\sim$, which proves our statement.
The proof of the above lemma reveals the following equivalence for $g\in G$: $$g\in K_1b_x K_1 \, \Longleftrightarrow \, x = [r,z]\,\text{ with }\, r= \sigma_{sing}(D(g)), \,
z = \arg\Delta(D(g)),$$ with the argument $\,\arg:\mathbb C \to \mathbb T$ defined by $$\arg\,z := \frac{z}{|z|}\, \text{ if } z\not = 0, \,\, \arg\,0\,:= 1.$$
We are now going to write the general product formula (\[prodformel\]) for spherical functions as a product formula on the cone $\,X_q\cong (A_q\times \mathbb T)/\sim\,.$ For $x = [\cos t, z_1], \, x^\prime = [ \cos t^\prime, z^\prime]\in X_q$ and $K_1$-biinvariant $f\in C(G)$ we have to evaluate the integral $$\int_{ K_1} f(b_{x}kb_{x^\prime})\,dk.$$ Write $\,\displaystyle k= \begin{pmatrix} u & 0\\
0 & v \end{pmatrix}\, $ with $u\in SU(p), \, v\in SU(q).$ Then $(p\times q)$-block calculation gives $$b_{x}kb_{x^\prime} =\begin{pmatrix} * & *\\
* & D(b_{x}kb_{x^\prime}) \end{pmatrix}$$ where $$D(b_{x}kb_{x^\prime}) =
(0,\sin\underline t\,)\,u \begin{pmatrix}0\\ -\sin\underline t^\prime\end{pmatrix}
+h(z) \cos \underline t \,v\, h(z^\prime)\cos \underline t^\prime\, .$$ With the block matrix $$\sigma_0 := \begin{pmatrix}0_{(p-q)\times q}\\ I_q\end{pmatrix} \in M_{p,q}(\mathbb C)$$ this can be written as $$\label{D(.)}
D(b_{x} k b_{x^\prime}) = \,-\sin\underline t \,\sigma_0^* u
\sigma_0\sin\underline t^\prime \,+\,
h(z z^\prime)\,\cos \underline t\, v\, \cos \underline t^\prime .$$ Regarding $K_1$-biinvariant functions $f\in C(G)$ as continuous functions on $X_q$, we have $$\int_{K_1} f(b_{x}kb_{x^\prime})dk\,=\,
\int_{ SU(p)\times SU(q)} f\left([\sigma_{sing}(D(b_xkb_{x^\prime})),
\arg\Delta(D(b_{x}k b_{x^\prime})\,]\right)dk$$ with $D(b_{x}kb_{x^\prime})$ from . Notice that $\,\sigma_0^* u \sigma_0\in M_q(\mathbb C)$ is the lower right $q\times q$-block of $u$ and is contained in the closure of the ball $$B_q:=\{w\in M_q(\mathbb C):\> w^*w\le I_q\},$$ where $w^*w\le I_q$ means that $I_q-w^*w$ is positive semidefinite. We now assume that $p\geq 2q$ and reduce the $SU(p)$-integration by means of Lemma 2.1 of [@R2]. Notice first that for continuous $g$ on $\overline B_q$, $$\int _{SU(p)} g(\sigma_0^*u\sigma_0) du = \int _{U(p)} g(\sigma_0^*u\sigma_0) du,$$ where $du$ denotes the normalized Haar measure in each case. Thus Lemma 2.1 of [@R2] gives
$$\begin{aligned}
\int_{K_1} f(b_{x}kb_{x^\prime})dk&\,=\,\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)}
f\bigr(\big[\sigma_{sing}(-\sin\underline t \,w\sin\underline t^\prime +
h(zz^\prime)\cos \underline t\, v\cos \underline t^\prime
), \notag \\
&\arg\Delta(-\sin\underline t \,w \sin\underline t^\prime +
h(zz^\prime)\cos \underline t\, v \cos \underline t^\prime
)\big]\bigr)
\Delta(I_q-w^*w)^{p-2q} dv dw,\end{aligned}$$
where $$\kappa_p=\int_{B_q}\Delta(I_q-w^*w)^{p-2q}\> dw$$ and $dw$ means integration with respect to Lebesgue measure on $B_q$. After the substitution $w\mapsto
h(zz^\prime)w$, we finally arrive at the following
\[prop-group-torus-convo-Xq\] Suppose that $p\ge 2q$. Then the product formula for the spherical functions of the Gelfand pair $(G, K_1)$, considered as functions on the cone $X_q$, can be written as $$\begin{aligned}
\label{group-torus-convo}
\phi([\cos t,z])\phi([\cos t^\prime, z^\prime]) \,=\,
\frac{1}{\kappa_p}&\int_{B_q}\int_{SU(q)}
\phi\bigr(\big[\sigma_{sing}(-\sin\underline t \,w\sin\underline t^\prime +
\,\cos \underline t\,v\cos \underline t^\prime),\notag \\
zz^\prime\cdot \arg\Delta(&-\sin\underline t \,w\sin\underline t^\prime +
\cos \underline t\, v \cos \underline t^\prime)\big]\bigr)\cdot
\Delta(I_q-w^*w)^{p-2q}\, dvdw.\notag\end{aligned}$$
We remark that for $p=2q-1$, a degenerate version of this integral formula may be written down by using the coordinates introduced in Section 3 of [@R1].
We next turn to the classification of the spherical functions of $(G, K_1)$. Note first that $K = K_1\rtimes H$ with $H = \{ H_z, z\in \mathbb T\}$ where $H_z$ is the diagonal matrix with entries $z$ in position $p-q+1$, $1/z$ in position $p+1$ and $1$ else. Let $\chi:K \to \mathbb T$ be the homomorphism with kernel $ K_1$ and $\chi(H_z):= z.$ Then the characters of $K$ are given by the functions $\chi_l(k) = \chi(k)^l, \,l\in \mathbb Z$, and we have the following characterization:
\[euiv-def-spher\] For $\phi\in C(G)$ the following properties are equivalent: =-1pt
1. $\phi$ is $K_1$-spherical, i.e., $K_1$-biinvariant with $\phi(g)\phi(h)=\int_{K_1} \phi(gkh) dk\,$ for all $g,h\in G$.
2. $\phi$ is an elementary spherical function for $(G,K)$ of type $\chi_l$ for some $l\in\mathbb Z$, i.e. $\phi$ is not identical zero and satisfies the twisted product formula $$\label{twist-prod}
\phi(g)\phi(h)=\int_{ K} \phi(gkh)\chi_l(k)dk \quad\text{ for all }\,
g,h\in G.$$
We here adopt the notion of elementary spherical functions of type $\chi_l$ according to [@HO]. Each such function automatically satisfies $\varphi (e) = 1$ as well as the $\chi_l$-bi-coinvariance condition $$\label{twist} \varphi(k_1gk_2) = \chi_l(k_1k_2)^{-1} \cdot \varphi(g) \quad \text {for all }\, g\in G,
k_1, k_2 \in K,$$ see Lemma 3.2. of [@HO]. The discussion of [@HS] is based on a different, but equivalent definition of elementary spherical functions of $K$-type (for the non-compact dual), which requires together with a system of invariant differential operators on sections in an associated homogeneous line bundle, see Definition 5.2.1 of [@HS]. For the equivalence of definitions we refer to Theorem 3.2 of [@S].
The proof of this result, which we expect to be well-known, can be carried out for instance precisely as in Lemma 2.3 of [@V9].
For an arbitrary irreducible Hermitian symmetric space and its compact dual, the elementary spherical functions of type $\chi_l$ can be written as modifications of Heckman-Opdam hypergeometric functions, see Section 5 of [@HS], in particular Theorem 5.2.2 and Corollary 5.2.3., as well as [@HO]. In the compact case, they correspond to the $\chi_l$-spherical representations of $G$ which were classified in [@Sch]. To become explicit in the particular case of our compact symmetric spaces $G/K= SU(p+q)/S(U(p)\times U(q)),$ recall the set $P_+$ of dominant weights from as well as the renormalized Heckman-Opdam polynomials $R_\lambda$ of type $BC_q$ from . According to [@HS], the elementary spherical functions of $(G,K)$ of type $\chi_l$, considered as functions on $A_q$, are given by $$\label{chispher}t\mapsto \prod_{j=1}^q \cos^{|l|}\! t_j\cdot
R_\lambda(k(p,q,l);t), \quad\lambda\in P_+\,,\, l\in\mathbb Z$$ with multiplicity parameters $$k(p,q,l):=(p-q-|l|, 1/2+|l|, 1), \quad l\in \mathbb Z.$$ Notice at this point that for $F_{BC_q}$, the set $$\{ k= (k_1,k_2,k_3): \text{Re}\, k_3 \geq 0, \,\text{Re}(k_1+k_2) \geq 0\}$$ is contained in $K^{reg}$ (see Remark 4.4.3 of [@HS]), and so in particular the multiplicities $k(p,q,l)$ are regular. The associated $\rho$-function is $$\rho(k(p,q,l))=
(p-q +|l|+1)\sum_{j=1}^q e_j\,+ 2 \sum_{j=1}^q (q-j)e_j.$$ We now turn to the spherical functions of $(G,K_1)$, which we again consider as functions on the cone $X_q$.
\[classification-spher-allg\] Let $(G,K) = (SU(p+q),S(U(p)\times U(q)))$ and $K_1 = SU(p)\times SU(q).$ Then the spherical functions of the Gelfand pair $(G, K_1)$, considered as functions on $X_q$, are precisely given by $$\label{def-spherical-jacobi}
\phi_{\lambda,l}^p([\cos t,z])=z^l\cdot\prod_{j=1}^q \cos^{|l|}\! t_j\cdot
R_\lambda(k(p,q,l);t), \quad\lambda\in P_+\,,\, l\in\mathbb Z$$ with the multiplicity $k(p,q,l)=(p-q-|l|,\> \frac12 +|l|,\> 1)\in K^{reg}.$
First observe that $\phi_{\lambda,l}^p$ is indeed well-defined as a function on $X_q$, because the right side is zero if $t_1 = \pi/2$, independently of $z\in \mathbb T.$ Now suppose that $\varphi: G\to \mathbb C$ is spherical for $(G,K_1)$. Then by Lemma \[euiv-def-spher\] it is $\chi_l$-spherical for some $l\in \mathbb Z$. Consider $x=[\cos t, z] \in X_q$ and write $b_x = H_{1/\sqrt{z}}\,a_tH_{1/\sqrt{z}}$ with an arbitrary square root of $z$. Then in view of , $$\varphi(b_x) = z^l \cdot \varphi(a_t),$$ where $\varphi(a_t)$ is of the form . This proves the statement.
\[q1–example-1\] Here $G/K = SU(p+1)/S(U(p)\times U(1))\cong U(p+1)/U(p)$ and $G//K$ is homeomorphic to the unit disk $X_1=\{z\in \mathbb C: |z|\leq 1\}~=~D$. We shall identify the spherical functions $\phi_{\lambda,l}^p $ as the well-known disk polynomials on $D$ introduced in [@K2], which are known to be the spherical functions of $(U(p+1),U(p)).$
We have $R_+ = \{ 2,4\} \subset \mathbb R$ and $P_+=2\mathbb Z_+$. According to the example on p.89f of [@O1], $F_{BC_1}(\lambda,
k;t)$ may be expressed as a $_2F_1$- (Gaussian) hypergeometric function. Consider the renormalized one-dimensional Jacobi polynomials $$\label{classical-jacobi-pol}
R_n^{(\alpha,\beta)}(x)= \>_2F_1(\alpha+\beta+n+1, -n, \alpha+1; (1-x)/2)
\quad(x\in\mathbb R, \> n\in\mathbb Z_+)$$ for $\alpha,\beta>-1.$ Then the Heckman-Opdam polynomials associated with $R=2BC_1$ and multiplicity $k=(k_1,k_2)$ can be written as $$\label{ident-jacobi}
R_{2n}(k;t)=R_n^{(\alpha,\beta)}(\cos 2t)\quad\quad (n\in \mathbb Z_+, \>
t\in[0,\pi/2])$$ with $$\alpha=k_1+k_2-1/2, \quad \beta= k_2-1/2;$$ c.f. equation (5.4) of [@RR], where a different scaling of roots and multiplicites is used. Writing $r=\cos t $, we have $\cos 2t = 2r^2-1$. We thus obtain from Theorem \[classification-spher-allg\] the well-known fact that the spherical functions of $(G,K)$ are given, as functions on $D$, by the so-called disk polynomials $$\label{disk-polynimials}
\widetilde\phi_{n,l}(zr)= \varphi_{2n, l}^p([r,z]) = z^l r^{|l|}\cdot R_n^{(p-1,|l|)}(2r^2-1) \quad\quad
(z\in\mathbb T,\> r\in[0,1])$$ with $l\in\mathbb Z$, $n\in\mathbb Z_+$. Moreover, the product formula of Theorem \[prop-group-torus-convo-Xq\] becomes in this case $$\label{disk-polynimial-product-formulas}
\widetilde\phi_{n,l}(zr)\cdot \widetilde\phi_{n,l}(z^\prime s)=
\frac{1}{\kappa_p}\int_{D}
\widetilde\phi_{n,l}(zz^\prime (rs-w\sqrt{1- r^2} \sqrt{1- s^2} ))\cdot
(1-|w|^2)^{p-2}\> dw$$ with $$\kappa_p=\int_{D}(1-|w|^2)^{p-2}\> dw= \frac{\pi}{p-1}.$$ This formula is well known; see for instance [@AT], [@Ka].
Convolution algebras on the cone $X_q$ for a continuous family of multiplicities
================================================================================
In this section, we extend the product formula for the spherical functions of $(G,K_1)$ in Theorem \[prop-group-torus-convo-Xq\] from integers $p\ge 2q$ to a continuous range of parameters $p\in]2q-1,\infty[$. We show that for each $p\in]2q-1,\infty[$, the corresponding product formula induces a commutative Banach algebra structure on the space of all bounded Borel measures on $X_q$ and an associated commutative hypergroup structure. These hypergroups generalize the known disk hypergroups for $q=1$ which were studied for instance in [@AT], [@Ka]; see also the monograph [@BH].
As ususal, the basis for analytic continuation will be Carlson’s theorem, which we recapitulate for the reader’s convenience from [@Ti]:
\[continuation\] Let $f$ be holomorphic in a neighbourhood of $\{z\in \mathbb C:{\rm Re\>} z \geq 0\}$ satisfying $f(z) =
O\bigl(e^{c|z|}\bigr)$ for some $c<\pi$. If $f(z)=0$ for all nonnegative integers $z$, then $f$ is identically zero on $\{{\rm Re\>} z>0\}$.
It is now straightforward, but a bit nasty in detail to prove that the product formula of Theorem \[prop-group-torus-convo-Xq\] can be extended analytically with respect to the variable $p$. For the necessary exponential bounds, one has to use that the coefficients of the Jacobi polynominals $P_\lambda(k;t) = c(\lambda+\rho(k),k)^{-1}R_\lambda(k;t)$ are rational in the multiplicity $k$, see Par. 11 of [@M]. As the arguments are the same as those in the proof of Theorem 4.1 of [@R2] and Theorem 3.2 of [@V9], we skip the details. We obtain:
\[general-twodim-prod-form\] Let $p\in]2q-1,\infty[$, $\lambda\in P_+$, and $l\in\mathbb Z$. Then the functions $$\phi_{\lambda,l}^p([\cos t,z])=z^l\cdot\prod_{j=1}^q \cos^{|l|} t_j\cdot
R_\lambda(k;t)$$ on $X_q$ with multiplicity $k=(p-q-|l|,\> \frac12 +|l|,\> 1)\in K^{reg}$ satisfy the product formula $$\begin{aligned}
\phi_{\lambda,l}^p([\cos t,z])\cdot\phi_{\lambda,l}^p([\cos t^\prime, z^\prime]) \,=\,
\frac{1}{\kappa_p}&\int_{B_q}\int_{SU(q)}
\phi_{\lambda,l}^p\bigr(\big[\sigma_{sing}(-\sin\underline t \,w\sin\underline t^\prime +
\,\cos \underline t\,v\cos \underline t^\prime),\notag \\
zz^\prime\cdot \arg\Delta(-\sin\underline t &\,w\sin\underline t^\prime +
\cos \underline t\, v \cos \underline t^\prime)\big]\bigr)\cdot
\Delta(I_q-w^*w)^{p-2q}\, dvdw.\notag\end{aligned}$$ for $(t,z),(t^\prime,z^\prime)\in A_q\times\mathbb T$.
The positive product formula in Theorem \[general-twodim-prod-form\] for $p\in]2q-1,\infty[$ leads to a continuous series of probability-preserving commutative convolution algebras on the cone $ X_q$. In fact, similar to the noncompact case [@V9], we obtain commutative hypergroups structures on $X_q$ with the $\phi_{\lambda,l}^p$ ($\lambda\in P_+,
\>l\in\mathbb Z$) as hypergroup characters. To start with, let us briefly recapitulate some notions from hypergroup theory from [@J], [@BH].
A hypergroup is a locally compact Hausdorff space $X$ with a weakly continuous, associative and bilinear convolution $*$ on the Banach space $M_b(X)$ of all bounded regular Borel measures on $X$ such that the following properties hold:
1. For all $x,y\in X$, $\delta_x*\delta_y$ is a compactly supported probability measure on $X$ such that the support $\text{supp}(\delta_x*\delta_y)$ depends continuously on $x,y$ with respect to the so-called Michael topology on the space of compact subsets of $X$ (see [@J] for details).
2. There exists a neutral element $e\in X$ with $\delta_x*\delta_e=
\delta_e*\delta_x=\delta_x$ for all $x\in X$.
3. There exists a continuous involution $x\mapsto\overline x$ on $X$ such that $e\in \text{supp} (\delta_x*\delta_y)$ holds if and only if $y=\overline x$, and such that $(\delta_x*\delta_y)^-= \delta_{\overline y}*\delta_{\overline x}$, where for $\mu\in M_b(X)$, the measure $\mu^-$ denotes the pushforward of $\mu$ under the involution.
Due to weak continuity and bilinearity, the convolution of bounded measures on a hypergroup is uniquely determined by the convolution of point measures. A hypergroup is called commutative if so is the convolution. We recall from [@J] that for a Gelfand pair $(G,K)$, the double coset space $G//K$ carries the structure of a commutative hypergroup in a natural way. For a commutative hypergroup $X$ the dual space is defined by $$\widehat X = \{\varphi\in C_b(X): \,\varphi\not\equiv 0, \, \varphi( x* \overline
y ):=
(\delta_x*\delta_{\overline y})(\varphi) =
\varphi(x)\overline{\varphi(y)} \,\, \forall\, x,y\in X\} .$$ Each commutative hypergroup $(X,*)$ admits a (up to a multiplicative constant unique) Haar measure $\omega_X$, which is characterized by the condition $\omega_X(f)=\omega_X(f_x)$ for all continuous, compactly supported $f\in C_c(X)$ and all $x\in X$, where $f_x$ denotes the translate $f_x(y)=(\delta_y*\delta_x)(f)$.
Now let $p\in ]2q-1,\infty[$. Using the positive product formula of Theorem \[general-twodim-prod-form\], we introduce the convolution of point measures on $X_q$ by $$\begin{aligned}
\label{def-convolution-aq}
(\delta_{[\cos t,z]}*_p \delta_{[\cos t^\prime,z^\prime]})(f):=
& \,\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)}
f\bigl(\big[\sigma_{sing}(-\sin\underline t \,w \sin\underline t^\prime \,+\,
\cos \underline t\, v\,\cos \underline t^\prime
),\notag \\
zz^\prime\cdot \arg\Delta &(-\sin\underline t \,w\sin\underline t^\prime \,+\,
\cos \underline t\, v \cos \underline t^\prime)\big]\bigr)
\Delta(I_q-w^*w)^{p-2q} dv dw\end{aligned}$$ for $f\in C_b(X_q)$.
\[mainhypergroup\] Let $q\ge 1$ be an integer and $p\in]2q-1,\infty[$. Then the convolution $*_p$ defined in (\[def-convolution-aq\]) extends uniquely to a bilinear, weakly continuous, commutative convolution on the Banach space $M_b(X_q)$. This convolution is also associative, and $(X_q, *_{p})$ is a commutative hypergroup with $[(1, \ldots, 1),1]$ as identity and with the involution $\overline{[r,z]}:= [r,\overline z]$. A Haar measure of the hypergroup $(X_q, *_{p})$ is given by $$d\omega_p([r,z]) = \prod_{j=1}^q r_j(1-r_j^2)^{p-q}\cdot\prod_{1\leq i < j\leq q}
(r_i^2 - r_j^2)^2\,drdz$$ with the Lebesgue measure $dr$ on $\mathbb R^q$ and the normalized Haar measure $dz$ on $\mathbb T.$ Finally, the dual space is given by $$(X_q, *_{p})^\wedge = \{\varphi_{\lambda, l}^p: \lambda\in P_+,\, l\in \mathbb Z\}.$$
Note that $\omega_p$ is the pushforward measure under the mapping $(t,z)\mapsto [\cos t, z], \, A_q\times \mathbb T \to X_q\,$ of the measure $$d\widetilde\omega_p(t,z) = \prod_{j=1}^q \cos t_j\sin^{2p-2q+1}t_j
\cdot\prod_{1\leq i < j\leq q} (\cos(2t_i)-\cos(2t_j))^2 dtdz$$ on $A_q\times \mathbb T.$
For the proof of Theorem \[mainhypergroup\], consider the measure $\omega_p$ defined above. We start with the following observation:
\[orthobasis\] The functions $$\varphi_{\lambda,l}^p \quad (\lambda \in P_+, \, l\in \mathbb Z)$$ form an orthogonal basis of $L^2(X_q, \omega_p),$ and their $\mathbb C$-span is dense in the space $C(X_q)$ of continuous functions on $X_q$ with respect to $\|.\|_\infty.$
For $k=k(p,q,l)$, the Heckman-Opdam polynomials $R_\lambda(k;t)$ are orthogonal on the alcove $A_q$ with respect to the weight $$\begin{aligned}
\delta_k(t) = &\,\prod_{\alpha\in R_+} \big\vert e^{i\langle\alpha,t\rangle/2} -
e^{-i\langle\alpha,t\rangle/2}\big\vert^{2k_\alpha} \\
=\, & const\cdot\prod_{j=1}^q \sin^{2p-2q+1}\!t_j\, \cos^{2|l|+1}\! t_j \cdot\prod_{1\leq i<j\leq q} (\cos(2t_i) - \cos(2t_j))^2
\end{aligned}$$ This immediately implies that the functions $(t,z)\mapsto\varphi_{\lambda,l}^p([\cos t,z])$ are orthogonal on $A_q\times \mathbb T$ with respect to $\widetilde\omega_p.$ Let $V$ denote the subspace of $C(X_q)$ spanned by $\{\varphi_{\lambda, l}^p: \lambda\in P_+, \, l\in \mathbb Z\}.$ Clearly, $V$ is an algebra which is stable under complex conjugation. Further, $V$ separates points on $X_q$, because the $R_\lambda(k;\,.\,)$ span the space of Weyl group invariant trigonometric polynomials. Hence by the Stone-Weierstraß theorem, $V$ is dense in $C(X_q)$ with respect to $\|.\|_\infty$. The rest is obvious.
The statements are clear for integer values of $p$, where $*_p$ is just the convolution of the double coset hypergroup $SU(p+q)//SU(p)\times SU(q).$ >From the definition of the convolution for general $p$ we see that $\delta_{[r,z]} *_{p} \delta_{[r^\prime, z^\prime]}$ is a probability measure and that the mapping $([r,z],[r^\prime, z^\prime])\mapsto \delta_{[r,z]} *_{p} \delta_{[r^\prime, z^\prime]}$ is weakly continuous. It is now standard (see [@J]) to extend the convolution of point measures uniquely in a bilinear, weakly continuous way to a probability preserving convolution on $M_b(X_q)$. For commutativity and associativity, it suffices to consider point measures. Let $[r_i, z_i]\in X_q\,,\, 1\leq i\leq 3.$ Then for $f= \varphi_{\lambda,l}^p$ with $\lambda\in P_+$ and $l\in \mathbb Z$ we have $$(\delta_{[r_1,z_1]}*_{p} \delta_{[r_2,z_2]})(f) = f([r_1,z_1]) f([r_2,z_2]) =
(\delta_{[r_2,z_2]}*_{p} \delta_{[r_1,z_1]})(f)$$ and in the same way, $$((\delta_{[r_1,z_1]}*_{p} \delta_{[r_2,z_2]})*_{p}
\delta_{[r_3,z_3]})(f) =
(\delta_{[r_1,z_1]}*_{p} (\delta_{[r_2,z_2]}*_{p} \delta_{[r_3,z_3]}))(f).$$ By Lemma \[orthobasis\], both identities remain valid for arbitrary $f\in C(X_q).$ The remaining hypergroup axioms are immediate from the fact that the supports of convolution products of point measures are independent of $p$. For the statement on the Haar measure, the argumentation is similar to [@RR]. Notice first that by definitition of hypergroup translates, the identity $$\int_{X_q} f_x\, d\omega_p = \int_{X_q} f d\omega_p \quad \text{for all }
x\in X_q.$$ holds for $f= \varphi_{\lambda, l}^ p$ with arbitrary $ \lambda\in P_+, \, l\in \mathbb Z.$ In view of Lemma \[orthobasis\], it extends to arbitrary $f\in C(X_q)$, hence $\omega_p$ is a Haar measure. Finally, it is clear that the $\varphi_{\lambda, l}^p$ are hypergroup characters. There are no further ones, because the characters of a compact hypergroup are orthogonal with respect to its Haar measure.
The hypergroups $(X_q\,, *_{p})$ have a prominent subgroup. For this we recall that a closed, non-empty subset $H\subset X_q$ is a subgroup if for all $x,y\in H$, $\delta_x*_p\delta_{\overline y}$ is a point measure with support in $H$. It is clear from (\[def-convolution-aq\]) that $$H:=\{[1,z] =(z,1,\ldots, 1): z\in\mathbb T\}$$ is a subgroup of $(X_q, *_{p})$ which is isomorphic to the torus group $\mathbb T$.
The cosets $$x*_pH:=\bigcup_{y\in H}{\rm supp}\> (\delta_x*_p\delta_{ y}),
\quad x\in X_q$$ form a disjoint decomposition of $X_q$, and the quotient $$X_q/H:=\{x*_p\!H: \> x\in X_q \}$$ is again a locally compact Hausdorff space with respect to the quotient topology, c.f. Section 10.3. of [@J]. Moreover, $$\label{quotient-allg}
(\delta_{x*_pH}*_p\delta_{y*_pH})(f):=\int_X f(z*_pH)\>
d(\delta_{x}*_p\delta_{y})(z),
\quad x,y\in X_q\, , \, f\in C_b(X_q/H)$$ establishes a well-defined quotient convolution and an associated commutative quotient hypergroup; see [@J], [@V1], and the references given there. We may identify $X_q/H$ topologically with the alcove $A_q$ via $[\cos t,z]*_p\!H \mapsto t.$ It is then immediate from that the quotient convolution on $A_q$ derived from $\ast_p$ is given by $$\begin{aligned}
\label{def-convolution-aqres}
(\delta_{t}&*_p \delta_{t^\prime})(f)=\\
&=\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)}
f\bigl(\arccos(\sigma_{sing}(-\sin\underline t \,w\sin\underline t^\prime \,+\,
\cos \underline t\, v \cos \underline t^\prime
))\bigr) \cdot\Delta(I_q-w^*w)^{p-2q} dv dw\notag\end{aligned}$$ for $t, t^\prime\in A_q$ and $f\in C_b(A_q).$ These are precisely the hypergroup convolutions studied in Section 6 of [@RR]. For integers $p\ge 2q$, this connection just reflects the fact that for $G=SU(p+q), K= S(U(p)\times U(q))$ and $K_1= SU(p)\times SU(q),$ we have $$(G// K_1)/(K//K_1)\simeq G//K$$ as hypergroups. This a fact which holds for general commutative double coset hypergroups, see Theorem 14.3 of [@J].
1\. We mention at this point that the Haar measure of the hypergroup $(X_q, *_p)$, which was obtained in Theorem \[mainhypergroup\] by an orthogonality argument, can alternatively be calculated by using Weil’s integration formula for Haar measures on hypergroups and their quotients (see [@Her], [@V1]). In the same way as in the non-compact case treated in [@V9], the Haar measure can thus be obtained from the known Haar measure of the quotient $(X_q, *_p)/H$. For integers $p\ge 2q$, the hypergroup $(X_q, \ast_p)$ can be identified with the double coset hypergroup $SU(p+q)//SU(p)\times SU(q),$ and its Haar measure therefore coincides by construction (see [@J]) with the pushforward measure of the Haar measure on $SU(p+q)$ under the canonical projection $$SU(p+q)\to SU(p+q)//SU(p)\times SU(q)\simeq X_q\,.$$
2\. Concerning their structure, the hypergroups $(X_q, \ast_p)$ are also closely related to continuous family of commutative hypergroups $(C_q\times \mathbb R, \ast_p)$ with $ p\geq 2q-1$ and the $BC_q$-Weyl chamber $\, C_q = \{(x_1, \ldots, x_q)\in \mathbb R^q: x_1 \geq \ldots \geq x_q\geq 0\}$ which were studied in [@V2]. For integral $p$, $(C_q\times \mathbb R, \ast_p)$ is an orbit hypergroup under the action of $U(p)\times U(q)$ on the Heisenberg group $M_{p,q}(\mathbb C)\times \mathbb R$. The characters are given in terms of multivariable Bessel- and Laguerre functions.
Continuous product formulas for Heckman-Opdam Jacobi polynomials
================================================================
Fix the rank $q\ge1$ and a parameter $p\in ]2q-1,\infty[$. Recall that for $l\in\mathbb Z$ the functions $$\phi_{\lambda,l}^p([\cos t, z]) = z^l\cdot
\prod_{j=1}^q \cos^{|l|}\! t_j\cdot R_\lambda(k(p,q,l);t)$$ satisfy the product formula of Theorem \[general-twodim-prod-form\]. We shall now extend this fomula to exponents $l\in\mathbb R$ via Carlson’s theorem, and write it down as a product formula for the Jacobi polynomials $R_\lambda(k;t)$ with $k=k(p,q,l)$. This will work out smoothly only for non-degenerate arguments $\,t, t^\prime\in A_q$ with $t_1, t_1^\prime \ne \pi/2$. Notice first that for a product formula for the Jacobi polynomials, we may restrict our attention to $l\in[0,\infty[$ as $k(p,q,l)$ depends on $|l|$ only. In the following, we shall use the abbreviation $$d(t,t^\prime;v,w):= -\sin\underline t \,w \sin\underline t^\prime \,+\,\cos \underline t\, v \cos \underline t^\prime\,.$$ The main result of this section is
Let $q\ge1$ be an integer, $p\in ]2q-1,\infty[$, $l\in[0,\infty[$, and $k=k(p,q,l)$. Then for all $\lambda\in P_+$ and $\, t, t^\prime\in A_q$ with $t_1, t_1^\prime\ne \pi/2$, $$\begin{aligned}
\label{prod-formel-allg-Jacobi}
R_\lambda(k; t) R_\lambda(k; t^\prime)\,
=\,\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)} &
R_\lambda\bigl(k;\arccos(\sigma_{sing}(d(t,t^\prime;v,w)))\bigr) \cdot\\
&\cdot {\rm Re}\biggl[\biggl(\frac{\Delta(d(t,t^\prime;v,w))}{\Delta(\cos \underline
t)\cdot
\Delta(\cos \underline t^\prime)}\biggr)^{\!l}\,\biggr]
\cdot\Delta(I_q-w^*w)^{p-2q} dvdw.\notag\end{aligned}$$
For $l\in \mathbb Z_+$, $z=z^\prime =1$ and $t, t^\prime\in A_q$, the product formula in Theorem \[general-twodim-prod-form\] implies $$\begin{aligned}
\label{hilf-prod-f}
&\Bigl(\prod_{j=1}^q \cos t_j \cos t_j^\prime\Bigr)^l \cdot
R_\lambda(k; t) R_\lambda(k; t^\prime)=\\
&=\frac{1}{\kappa_p}\int_{B_q}\int_{SU(q)}\Delta(d(t,t^\prime;v,w))^l \cdot
R_\lambda\bigl(k;\arccos(\sigma_{sing}(d(t,t^\prime;v,w)))\bigr)
\cdot\Delta(I_q-w^*w)^{p-2q} dv dw.
\notag\end{aligned}$$ Our condition on $t,t^\prime$ assures that both sides of (\[hilf-prod-f\]) are analytic in $l$ for $ {\rm
Re} (l)> 0$. Moreover, by Section 11 of [@M], the coefficients of the Jacobi polynomials $R_\lambda(k;t)$ with respect to the exponentials $e^{i\langle\mu,t\rangle}\,(\mu\in P_+)$ are rational in $k$. Therefore both sides of (\[hilf-prod-f\]) satisfy the growth condition of Carlson’s theorem. This implies that (\[hilf-prod-f\]) remains correct for all $l\in[0,\infty[$. As $R_\lambda(k;t)$ is real for all $\lambda\in P_+$ and $t\in A_q$, the claimed product formula now follows from (\[hilf-prod-f\]) by taking real parts.
Contrary to the non-compact case in Section 5 of [@V9], it seems to be difficult in the present setting to derive positivity of the product formula except for the known case $l=0$. This problem already appears in rank one, i.e. for $q=1$. We discuss this case for illustration.
\[1-dim-bsp-fortsetzung\] Let $q=1$, $p\ge 2q-1= 1$, $l\in[0,\infty[$, and $k=k(p,q,l)= (p-1-l, \frac{1}{2}+l).$ Resuming the notions from Example \[q1–example-1\], we have $$\alpha=k_1+k_2-1/2=p-1>0,
\quad
\beta=k_2-1/2=l.$$ Consider the classical (normalized) one-dimensional Jacobi polynomials $R_n^{(\alpha,\beta)}$ with $$R_n^{(\alpha,\beta)}(\cos 2\theta)=R_{2n}(k;\theta),$$ c.f. . With the identity $\cos 2\theta=2\cos^2\theta -1$, product formula (\[prod-formel-allg-Jacobi\]) becomes $$\begin{aligned}
\label{Jacobi_1}
R_n^{(\alpha,\beta)}&(\cos 2\theta) R_n^{(\alpha,\beta)}(\cos 2\theta^\prime)
=\\
=&\frac{\alpha}{\pi} \int_0^1 \int_{-\pi}^{\pi}
R_n^{(\alpha, \beta)} (2|b+are^{i\varphi}|^2 -1) \cdot
\frac{(b+are^{i\varphi})^\beta}{b^\beta} \cdot r(1-r^2)^{\alpha-1} dr d\varphi
\notag\end{aligned}$$ for $\theta,\theta^\prime\in [0,\pi/2[$, with $\,a:= \sin \theta\sin\theta^\prime$ and $b: = \cos\theta\cos\theta^\prime>0$. Following Section 5 of [@K3], we use the change of variables $(r,\phi)\mapsto (t,\psi) $ with $$b+are^{i\varphi} = te^{i\psi} \text{ and }\, t\geq 0.$$ Then $a^2r\> dr\> d\phi= t\>
dt\> d\psi$ and identity becomes, for $0< \theta,\theta^\prime < \pi/2$: $$\begin{aligned}
\label{Jacobi_2}
R_n^{(\alpha,\beta)}&(\cos 2\theta) R_n^{(\alpha,\beta)}(\cos 2\theta^\prime)
=\\
=&\frac{\alpha}{\pi}\cdot\frac{1}{b^\beta a^{2\alpha}}
\int_0^\infty\int_{-\pi}^{\pi} R_n^{(\alpha,\beta)}(2t^2-1)
(te^{i\psi})^\beta
(a^2-b^2-t^2+2bt\cos\psi)_+^{\alpha-1} t\> dt\> d\psi\notag\\
=&\frac{\alpha}{\pi}\cdot\frac{2}{b^\beta a^{2\alpha}}\int_0^1
R_n^{(\alpha,\beta)}(2t^2-1) \Bigl(\int_{0}^{\pi}
e^{i\beta\psi}(a^2-b^2-t^2+2bt\cos\psi)_+^{\alpha-1}d\psi\Bigr)t^{\beta+1} dt.
\notag\end{aligned}$$ Here the notation $$(x)_+^\lambda = \begin{cases} x^\lambda & \text{ if $x>0$,}\\
0 & \text{ if $x\leq 0$}
\end{cases}$$ is used. Notice for the last equality that $\,t=|b+are^{i\phi}|\le a+b\le 1$. Notice also that for $\theta=0$ or $\theta^\prime=0$, the product formula degenerates in a trivial way due to $R_n^{(\alpha,\beta)}(1)=1$.
On the other hand, for $\alpha>\beta$, the Jacobi polynomials satisfy the well-known positive product formula ([@K1]) $$\begin{aligned}
\label{Jacobi_3}
R_n^{(\alpha,\beta)}&(\cos 2\theta)R_n^{(\alpha,\beta)}(\cos 2\theta^\prime) =\\
=&c_{\alpha,\beta}\int_0^1\int_0^\pi
R_n^{(\alpha,\beta)}(2|b+are^{i\varphi}|^2-1)\cdot
(1-r^2)^{\alpha-\beta-1}r^{2\beta+1}\sin^{2\beta}\varphi\, dr d\varphi
%\notag
%\\
%=&\frac{c_{\alpha,\beta}}{2}\int_0^1\int_{-\pi}^\pi
% R_n^{(\alpha,\beta)}(2|b+are^{i\varphi}|^2-1)\cdot
%(1-r^2)^{\alpha-\beta-1}r^{2\beta+1}\sin^{2\beta}\varphi\> dr \> d\varphi
\notag\end{aligned}$$ with some constant $c_{\alpha,\beta}>0$. By the same substitution as above this can be brought into kernel form, $$\begin{aligned}
\label{Jacobi_4}
R_n^{(\alpha,\beta)}&(\cos 2\theta)R_n^{(\alpha,\beta)}(\cos 2\theta^\prime) = \\
&=
\frac{c_{\alpha,\beta}}{a^{2\alpha}}\cdot
\int_0^1 R_n^{(\alpha,\beta)}(2t^2-1) \Bigl(\int_{0}^\pi
(a^2-b^2-t^2+2bt\cos\psi)_+^{\alpha-\beta-1}\sin^{2\beta}\!\psi d\psi\Bigr)
t^{2\beta+1}dt.
\notag\end{aligned}$$ As the integrals in and are identical for all $n$, we conclude that, for all $t\in [0,1]$, $$c_{\alpha,\beta}(tb)^\beta \!
\int_{0}^\pi(a^2-b^2-t^2+2bt\cos\psi)_+^{\alpha-\beta-1}\sin^{2\beta}\!\psi
d\psi \,=\, \frac{2\alpha}{\pi} \int_{0}^\pi e^{i\beta\psi}
(a^2-b^2-t^2+2bt\cos\psi)_+^{\alpha-1}d\psi.$$ This identity seems not obvious, and it would be desirable to have an elementary proof for it which possibly might be extended to the higher rank case.
So far, a positive product formula such as formula (\[Jacobi\_3\]) of Koornwinder seems to be a difficult task in rank $q\geq 2$. However, at least the first step above in the case $q=1$, that is from to , can be partially extended to $q\ge 2$. Indeed, consider the product formula (\[prod-formel-allg-Jacobi\]) for $t,t^\prime\in A_q$. We define the matrices $$a_1:=\sin \underline t\,, \,
a_2:=\sin \underline t^\prime\,,\, b= b(v):= \cos \underline t\, v \cos \underline t^\prime\, \in M_q(\mathbb C)$$ and consider the polar decomposition $\,b-a_1wa_2=: ru\,$ with positive semidefinite $r\in M_q(\mathbb C)$ and $u\in U(q)$. We now carry out the change of variables in two steps. First, for $\,\widetilde w:= a_1wa_2\in M_q(\mathbb C)$, we have $\,d\widetilde w= const\cdot\Delta(a_1a_2)^{2q}\> dw$. Moreover, the polar decomposition $\,b-\widetilde w=\sqrt{r}u\,$ (for non-singular $r$) leads to $\,d\widetilde w= const\cdot drdu,$ where $dr$ means integration with respect to the Lebesgue measure on the cone $\Omega_q$ of positive definite matrices as an open subset of the vector space of all Hermitian matrices, and $du$ is the normalized Haar measure of $U(q)$; see Proposition XVI.2.1 of [@FK]. Formula (\[prod-formel-allg-Jacobi\]) now reads $$\begin{aligned}
\label{prod-formel-allg-Jacobi-kern}
R_\lambda(k; t)R_\lambda(k; t^\prime)= & \notag\\
=\,const\cdot\int_{\Omega_q} &
R_\lambda\bigl(k;\arccos(\sigma_{sing}(\sqrt{r}))\bigr) \cdot
\frac{\Delta(r)^{l/2}}{\Delta(\cos \underline t)^l\Delta(\cos \underline t^\prime)^l
\Delta(\sin \underline t)^{2q}\Delta(\sin \underline t^\prime)^{2q}}\cdot
\notag\\
&\cdot\Bigl(\int_{SU(q)}\int_{U(q)}
\Delta\bigl(H(t,t^\prime,r,u,v)_+\bigr)
\Delta(u)^l\,dv du\Bigr) dr\end{aligned}$$ where $$H(t,t^\prime,r,u,v)=I_q \,-\, a_2^{-1}\bigl(b(v)^*-u^*\sqrt{r}\bigr) a_1^{-2}
\bigl(b(v)-\sqrt{r}u\bigr) a_2^{-1}$$ and the subscript $+$ means that this term is put zero for matrices which are not positive definite. The analysis of the origin of these formulas shows that in the outer integral, $r$ in fact runs through the set $\{r\in\Omega_q:\> I_q-r >0\}$. One may speculate that (\[prod-formel-allg-Jacobi-kern\]) can be brought into a kernel form by replacing the integration over $r\in \Omega_q$ by an integration over the spectrum of $r$ as a subset of $\{\rho\in\mathbb R^q:\> 1 \ge\rho_1\ge\cdots\ge\rho_q\ge 0\}$.
[9999]{}=-3pt H. Annabi, K. Trimeche: Convolution généralisée sur le disque unité. *C.R. Acad. Sci. Ser.* A 278 (1974), 21–24. W.R. Bloom, H. Heyer, Harmonic analysis of probability measures on hypergroups. *De Gruyter Studies in Mathematics* 20, de Gruyter-Verlag Berlin, New York 1995. J. Faraut, A. Korányi, Analysis on symmetric cones. Oxford Science Publications, Clarendon press, Oxford 1994. G. Heckman, Dunkl Operators. Séminaire Bourbaki 828, 1996–97; *Astérisque* 245 (1997), 223–246. G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Part I; Perspectives in Mathematics, vol. 16, Academic Press, California, 1994. S. Helgason, Groups and Geometric Analysis. Mathematical Surveys and Monographs, vol. 83, AMS 2000. P. Hermann, Induced representations and hypergroup homomorphisms. *Monatsh. Math.* 116 (1993), 245–262. V.M. Ho, G. Olafsson, Paley-Wiener theorem for line bundles over compact symmetric spaces. ArXiv:1407.1489v1 (2014). R.I. Jewett, Spaces with an abstract convolution of measures, *Adv. Math.* 18 (1975), 1–101. Y. Kanjin, A convolution measure algebra on the unit disc. *Tohoku Math. J.* 28 (1976), 105–115. T. Koornwinder, The addition formula for Jacobi polynomials I. Summary of Results, *Indag. Math.* 34 (1972), 188–191. T. Koornwinder, The addition formula for Jacobi polynomials II. The Laplace type integral representation and the product formula. Report TW 133/72, Mathematisch Centrum, Amsterdam, 1972. T. Koornwinder, Jacobi Polynomials II. An analytic proof of the product formula. *SIAM J. Math. Anal.* 5 (1974), 125–137. M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen. *Compos. Math.* 38 (1979), 129–153. I.G. Macdonald, Orthogonal polynomials associated with root systems. Séminaire Lotharingien de Combinatoire 45 (2000), Article B45a. E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras. *Acta Math.* 175 (1995), no.1, 75–121. H. Remling, M. Rösler, Convolution algebras for Heckman-Opdam polynomials derived from compact Grassmannians. *J. Approx. Theory* (2014), DOI: 10.1016/j.jat.2014.07.005, arXiv:0911.5123. M. Rösler, Bessel convolutions on matrix cones. *Compos. Math.* 143 (2007), 749–779. M. Rösler, Positive convolution structure for a class of Heckman-Opdam hypergeometric functions of type BC. *J. Funct. Anal.* 258 (2010), 2779–2800. H. Schlichtkrull, One-dimensional $K$-types in finite dimensional representations of semisimple Lie groups: a generalization of Helgason’s theorem. *Math. Scand.* 54 (1984), 279–294. N. Shimeno, The Plancherel formula for spherical functions with one dimensional $K$-type on a simply connected simple Lie group of hermitian type. *J. Funct. Anal.* 121 (1994), 330–388. E.C. Titchmarsh, The Theory of Functions. Oxford Univ. Press, London, 1939. M. Voit, Properties of subhypergroups. *Semigroup Forum* 56 (1998), 373–391. M. Voit, Multidimensional Heisenberg convolutions and product formulas for multivariate Laguerre polynomials. *Coll. Math.* 123 (2011), 149–179. M. Voit, Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC. *J. Lie Theory* 25 (2015), 9–36; arXiv:1310.3075.
|
---
abstract: |
There is a compelling intellectual case for exploring whether purely unitary quantum theory defines a sensible and scientifically adequate theory, as Everett originally proposed. Many different and incompatible attempts to define a coherent Everettian quantum theory have been made over the past fifty years. However, no known version of the theory (unadorned by extra ad hoc postulates) can account for the appearance of probabilities and explain why the theory it was meant to replace, Copenhagen quantum theory, appears to be confirmed, or more generally why our evolutionary history appears to be Born-rule typical.
This article reviews some ingenious and interesting recent attempts in this direction by Wallace, Greaves-Myrvold and others, and explains why they don’t work. An account of one-world randomness, which appears scientifically satisfactory, and has no many-worlds analogue, is proposed. A fundamental obstacle to confirming many-worlds theories is illustrated by considering some toy many-worlds models. These models show that branch weights can exist without having any role in either rational decision-making or theory confirmation, and also that the latter two roles are logically separate.
Wallace’s proposed decision theoretic axioms for rational agents in a multiverse and claimed derivation of the Born rule are examined. It is argued that Wallace’s strategy of axiomatizing a mathematically precise decision theory within a fuzzy Everettian quasiclassical ontology is incoherent. Moreover, Wallace’s axioms are not constitutive of rationality either in Everettian quantum theory or in theories in which branchings and branch weights are precisely defined. In both cases, there exist coherent rational strategies that violate some of the axioms.
author:
- Adrian Kent
title: 'One world versus many: the inadequacy of Everettian accounts of evolution, probability, and scientific confirmation'
---
Introduction
============
Some common ground
------------------
Although I disagree with the Everettian contributors to this volume on some fundamental questions, I think they deserve much credit for developing some creative and interesting ideas and arguments, which have certainly helped advance our understanding of fundamental science. To elaborate on this, let me note some points on which I agree with many Everettians.
First, the Everettian programme had a sensible motivation. Everett asked[@everettone], in effect, whether quantum theory really needs to be framed in such a way that the evolution of the wave function is governed by two different laws: generic unitary evolution together with the projection postulate when measurement takes place. It’s a good question. Even if, rather than the projection postulate, quantum theory came equipped with a precise extra dynamical law implying the postulate as an approximation, it would be natural to ask if we really needed it. As it is, there is quite a compelling case for exploring whether we can make sense of purely unitary quantum theory.
Second, it [*is*]{} a sensible project to try to extract a physical ontology from a unitarily evolving quantum state vector, given a theory of the initial state or initial conditions, a Hilbert space defining a representation of position, momentum and other canonical operators, and a dynamical theory that expresses the Hamiltonian in terms of these operators. Whether the project succeeds in producing an ontology with the properties that Everettians fondly imagine is another question – but certainly one worth discussing. One still, strangely, sometimes hears the argument that it is illegitimate – a basic misunderstanding of quantum theory – even to examine the possibility of giving the state vector a direct physical interpretation. This seems to me simply unimaginative dogma. Everettians are right to insist that their programme should be judged on whether or not it works, not on whether it respects pre-Everettian quantum orthodoxies.
Third, neither the apparently fantastic nature of the Everettian worldview, nor the superficial conflict between postulating multiple independent mutually inaccessible worlds and Occam’s razor, are entirely compelling arguments against the Everett programme. One needs to consider Everettian ideas in the context of other attempts to make sense of quantum theory, and in detail. One of the great intellectual challenges of theoretical physics is to find a mathematically elegant, universally applicable, Lorentz covariant, scientifically adequate version of quantum theory that supplies a well-defined realist ontology. If the Everett programme really could produce a well-defined Lorentz covariant physical ontology that adds little or no arbitrary structure to the mathematics of quantum theory, and that reproduces all the scientific successes of Copenhagen quantum theory within its domain of validity, it would have solved this fundamental problem. Given the present alternatives, we would, I think, at that point, have to consider it seriously as a possible account of reality.
Now, in fact, I think that the Everett programme fails in these ambitions, for reasons explained below. I am also optimistic[@akoneworld] that we can find simpler one-world versions of quantum theory that have all the aforementioned virtues and none of the problems that afflict, and I think ultimately doom, the Everett programme. But I see no way to make either conclusion so transparently true as to eliminate the need for argument.
Fourth, it matters – it is scientifically important to understand – whether the Everettian programme can possibly succeed. If Everettians really could produce a theory of reality with all the proclaimed virtues, it would clearly weaken (though not eliminate) the motivation for other attempts at solving the quantum reality problem – just as finding a consistent quantum theory of gravity would weaken (though not eliminate) the motivation for looking for others. Conversely, if, as I argue, the Everettian programme has fairly definitely failed, then the problem of finding a viable formulation of quantum theory applicable to closed quantum systems looms rather large among the concerns of theoretical physics. The failure of the Everett programme adds to the likelihood that the fundamental problem is not our inability to interpret quantum theory correctly but rather a limitation of quantum theory itself. If so, my guess is that we most likely won’t find an adequate cosmological theory so long as we assume that quantum theory is universally valid – so we should be looking for possible signals of the failure of quantum theory applied to the universe. Likewise, if so, quantum interference quite likely breaks down somewhere between the microscopic and the macroscopic – so we should be working harder to characterize the most promising types of experiment to test this.
Everettian ideas have been around for fifty years, and influential for at least the last thirty. Yet there has never been a consensus among theoretical physicists either that an Everettian account of quantum theory can be made precise and made to work, or that the Everettian programme has been comprehensively refuted. These questions are quite central to the future of theoretical, experimental and observational physics. We need to resolve them and move forward.
Everett’s elusive essence
-------------------------
> “When he died, his heirs found nothing save chaotic manuscripts. His family, as you may be aware, wished to condemn them to the fire; but his executor – a Taoist or Buddhist monk – insisted on their publication.”
>
> “We descendants of Ts’ui Pên,” I replied, “continue to curse that monk. Their publication was senseless. The book is an indeterminate heap of contradictory drafts.”
>
> (Jorge Luis Borges, [*The Garden of Forking Paths*]{} [@borges])
> “ $\ldots$ so crowded with $\ldots$ empty sophistication that it is extremely difficult to perceive the simple errors at the basis. It is like fighting the hydra – cut off one ugly head, and eight formalizations take its place.”
>
> (P.K. Feyerabend, [*How to Defend Society Against Science*]{} [@feyerabend])
After fifty years, there is no well-defined, generally agreed set of assumptions and postulates that together constitute “the Everett interpretation of quantum theory”. Far from it: Everett[@everettone; @everetttwo], DeWitt[@dewitt], Graham[@graham], Hartle[@hartlefreq], Geroch[@geroch], Deutsch[@deutschone], Deutsch[@deutschtwo], Saunders[@saundersvol], Barbour[@barbour] (partly inspired by Bell[@bellmw], though Bell’s aim was not to inspire), Albert-Loewer[@almanyminds], Coleman[@coleman], Lockwood[@lockwood], Wallace[@wallacevolone], Wallace[@wallacevoltwo], Vaidman[@vaidman], Papineau[@papineauvol], Greaves[@greaves], Greaves-Myrvold[@greavesmyrvoldvol], Gell-Mann and Hartle[@hartlevol], Zurek[@zurekvol] and Tegmark[@tegmarkvol], among many others, have offered distinctive and often fundamentally conflicting views on what precisely one needs to assume in order to get the Everett programme off the ground, and what precisely an Everettian (or, some say, post-Everettian) version of quantum theory entails.
I am primarily interested here in contrasting realist “one-world” and “many-worlds” accounts of quantum theory. By [*one-worlders*]{}, I mean those who aim to find a version of quantum theory in which quantum experiments really have only one outcome, we really have only one version of our future selves at any future time, and some intrinsic randomness in nature determines which outcome occurs and which future self is realised from among the range of possibilities defined by the theory. For example, within its domain of validity, the Copenhagen interpretation of quantum theory is a one-world theory. By [*many-worlders*]{}, I mean those who share Everett’s view that a unitarily evolving quantum state vector should be interpreted as directly representing reality, and the future versions of ourselves that observe different outcomes of quantum experiments should be interpreted as equally real.
So, I will not discuss here attempts at “post-Everettian” interpretations like those of Gell-Mann and Hartle[@hartlevol] and Zurek[@zurekvol], which fall into neither camp, and seem – despite much critical probing – unclear on, or uncommitted to taking a stance on, precisely what, if anything, in the theory corresponds to objective external reality. (Extensive critiques of Gell-Mann and Hartle’s approach can however be found elsewhere[@dowkerkentone; @dowkerkenttwo; @kentone; @kenttwo; @kentthree; @kentfour].)
My main focus is on the recent attempts by Wallace[@wallacevolone; @wallacevoltwo], Greaves and Myrvold[@greavesmyrvoldvol], and, to a lesser extent, Papineau[@papineauvol] and Saunders[@saundersvol], to define, analyse and test realist many-worlds interpretations. These authors offer different, and on some points mutually inconsistent, approaches, but nonetheless share enough common perspectives to be considered together. Their papers include some very interesting and creative arguments, which raise important scientific questions. However, I will argue below that none of their approaches produces a scientifically adequate version of quantum theory.
Shadowing these discussions is the spectre of the ’many-minds interpretation’ set out some time ago by Albert and Loewer [@almanyminds]. Essentially everyone, including Albert and Loewer, agrees that the many-minds interpretation, while logically consistent and in accord with the data, is utterly unsatisfactory, since it adds to the Everettian formalism a collection of ad hoc postulates which not only are (even by Everettian standards) fantastic, but also undercut the motivation for taking Everett seriously, namely that it purports to explain how to make sense of quantum theory without adding extra equations or interpretational postulates. So, no one – certainly no one represented in this volume – wants to be a many-minder. And here lies the problem: it seems to me (and to others – see in particular Albert’s contribution[@albertvol] to this volume) that, at various points in their arguments, Saunders, Wallace, Greaves-Myrvold and Papineau tacitly – and, since they reject the many-minds interpretation, illegitimately – appeal to many-minds intuitions. Indeed, at least in the first three cases, it seems to me that if one fleshed their ideas out into a fully coherent and complete interpretation, one would necessarily arrive either at the many-minds interpretation or something even worse. I will elaborate on this below.
Of course, these discussions crucially turn on our understanding of what counts as scientifically adequate. The idea that reality contains many essentially independent quasiclassical worlds corresponding to different possible cosmological and experimental outcomes clearly isn’t, per se, susceptible to logical refutation. That isn’t at issue. The key question, to my mind — and I think modern Everettians, including the authors considered here, generally agree — is whether we can find an appealingly simple version of quantum theory in which a realist many-worlds ontology is essential (i.e. there is no equally simple one-world variant) and which (at minimum) replicates all the scientific successes of one-world quantum theory (i.e. quantum theory including some form of the projection postulate, or some principle from which it can, approximately and within a suitable domain of validity, be derived). I believe that we can’t. In particular, it seems to me the Everettian programme has not produced and cannot produce a scientifically adequate alternative account that reproduces the standard one-world account of probabilistic inferences derived from quantum theory – despite the ingenious recent attempts of contributors to this volume.
Some commentators sympathetic to the Everettian programme[@greavesmyrvoldvol; @papineauvol] argue that a double standard is at work here: that criticisms of the Everettian programme’s attempt to account for the appearance of probability can and should equally well be applied to the standard understanding of the role of probability in one-world versions of quantum theory, and indeed of probabilistic scientific statements in general. To respond to this point, I consider below some fundamental differences between randomness (or apparent randomness) in one-world quantum theory and its purported Everettian analogue, and point out what seem to me irresolvable problems with the latter.
One-world theories and probability {#oneworldprob}
==================================
Copenhagen quantum theory is a one-world version of quantum theory: any given experiment or quantum event has a number of possible outcomes, but only one actual outcome. Some other non-Everettian variants and modifications of quantum theory, such as de Broglie-Bohm theory and dynamical collapse models, similarly randomly select from many possible physical evolutions, and can be (and usually are) interpreted as defining a unique quasiclassical world. The consistent histories approach [@hartlevol], if combined with an (alas unknown) suitable set selection rule, would also lead naturally to a one-world interpretation, in which reality is described by one randomly chosen history from the selected set. And these by now venerable contenders certainly don’t exhaust the possible options.[@akoneworld] My aim here is not to advocate a specific one-world version or variant of quantum theory, or to assess the current state of the art, but rather to compare and contrast one-world and many-world accounts of probability. For that purpose, let us suppose, for the sake of argument, that we have to hand a particular one-world theory that implies that, while the universe could have evolved in a (presumably very large) number of different ways, one quasiclassically evolving world – the one we observe – was randomly selected.
One-world versions of quantum theory, together with hypotheses about the initial conditions and unitary evolution, predict the probabilities of our experimental results and observations. We test the theory and these hypotheses by checking whether the results are of a form we would typically expect given the predicted probabilities. In practice, pretty much everyone agrees on the methodology of theory confirmation, at least sufficiently so that, for example, everyone agrees that, within the domain of validity of Copenhagen quantum theory, the Born rule is very well confirmed statistically. However, there is much less agreement on how, or even whether, we can make sense of fundamentally probabilistic physical theories. What exactly, if anything, does it mean to say that the probability of the universe turning out the way it did was $0.00038$?
Everettian authors have stressed this last point lately. We should not, they argue, apply different standards to one-world and many-worlds quantum theory. If our account of standard probability applied to one-world quantum theory is suspect, or incomplete, or involves ad hoc postulates, we cannot reasonably reject an alternative many-worlds account on the grounds that it runs into difficulties that might, on close analysis, turn out to be precisely analogous.
There are several possible responses for one-worlders here. One response is to try to defend or buttress or further develop frequentism, or another standard account of standard probability. A second is to try to point out some insuperable problems with many-worlds accounts of probability, and thus make the case that, whatever difficulties one-world quantum theory might run into, many-worlds quantum theory cannot possibly be satisfactory. A third is to argue that the difficulties that many-worlders face in dealing with probability are worse than – not, as claimed, precisely analogous to – those faced by one-worlders.
I think the first of these options is worth pursuing. I think too that the second and third lines of argument are valid, and I will develop them later. But, in this section, I want to make a separate point. I want to suggest a non-standard account in which the scientific space usually occupied by one-world probabilistic theories is filled instead by deterministic theories with a large amount of theoretically unspecified data. This allows us to compare, verify and falsify theories, and to recover essentially all of current science, without assigning a fundamental role to probability [*per se*]{}. Convinced believers in a chancy world might regard this as a useful fall-back position pending a fully satisfactory explanation of standard probability. It might, alternatively, be seen as an account with enough attractions of its own that it could be preferable to any standard account involving probability. Either way, it offers a way of making scientific sense of one-world quantum theory that has no many-worlds analogue.
Consider a probabilistic theory $T$, and suppose for simplicity that it predicts a finite set of probabilistic events, labelled by the index $i$, each with finitely many possible outcomes $x^j_i$, labelled by the index $j \in J_i$, for which it predicts nonzero probabilities $p^j_i$. For simplicity, we also suppose for the moment that the possible outcomes for any given event, and their probabilities, are independent of the outcome of any other event. We say two events $i$ and $i'$ are of the same type, according to $T$, if the sets $\{p^j_i
\}$ and $\{ p^j_{i'} \}$ are identical. Let $B = \{ 0 , 1 \}$, $B^* =
\{ \emptyset , 0 , 1 , 00, 01 , \ldots \}$ be the set of finite binary strings, and $B^r$ the set of length $r$ binary strings. Let $n =
\prod_i | J_i | $ be the size of the list of possible sets of outcomes, which we write as $N = \{ 1 , 2 , \ldots , n \}$.
A length $r$ code for the outcomes is any surjective map $C: B^r
\rightarrow N $. Given such a code, we can define an alternative probabilistic theory $T^C$ by stipulating that a binary string $b$ in $B^r$ is randomly chosen from the uniform distribution, and that the outcomes are given by $C(b)$. By taking $r$ sufficiently large, and choosing $C$ so that $ | \{ b : C(b) = i \} | \approx 2^r p(i ) $ for each $i \in N$, we can find theories $T^C$ whose probability assignments are arbitrarily close to those of $T$.
A length $r$ subcode for the outcomes is any map (not necessarily surjective) $C: B^r \rightarrow N$. Again, given a subcode, we can define a probabilistic theory $T^C$ as above: here $T^C$ may assign zero probability to some outcomes for which $T$ assigns non-zero probability.
We can define another type of theory from the triple $( T^C , C , r)$: a theory that simply states that the data will be those predicted by $T^C$ and $C$ given some length $r$ binary string as input, and makes no prediction about the binary string. We call this theory $D(
T^C , C , r)$, using $D$ to emphasize that we now regard the theory as deterministic. One might view the binary string in $D ( T^C , C , r
)$ as playing a role analogous to a constant of nature in a deterministic theory: its value is not fixed by the theory, and can only be determined empirically. In this case, even if the map $C$ is injective, determining the entire string would require observing every random event in the universe.
Now, on the view that there is a unique “correct” fundamentally probabilistic theory of nature $T$, each probabilistic theory of the form $ T^C $ must be either equivalent to $T$ (which is possible only if the probabilities $p^j_i$ are all dyadic), or else incorrect (though possibly a good approximation to $T$). Note though that, given a finite set of data, many other probabilistic theories besides $T$, including some of the form $T^C$, will be consistent with the data. Indeed, we would generally expect some theories $T'$ to fit the data better than $T$, in the sense that the same sets of events are of the same type according to $T$ and $T'$, and the probabilities $p'^j_i$ are closer than $p^j_i$ to the observed relative frequencies for events of the same type. If we nonetheless regard $T$ as likelier to be correct than $T'$, it must be for reasons other than purely empirical – presumably on grounds of elegance or simplicity. And if we maintain that there is a unique correct fundamental theory, it seems to follow that the correct theory is determined by a set of probabilities $\{ p^j_i \}$ not determined by the physical universe (although perhaps very well approximated by relative frequencies of physical events).
Here’s an alternative view. It may be, if not meaningless, then at least unnecessary, to appeal to the idea of a unique correct fundamentally probabilistic theory of nature, or even to define probability as a fundamental physical concept. Instead of considering probabilistic theories $T^C$, we can compare deterministic theories $D ( T^C , C , r )$ against one another and against the data. In evaluating these theories, we use the criteria of simplicity and elegance. These criteria have no precise mathematical definition. They include judgements about the form of $T^C$ and $C$, as well as the parameter $r$ (which is a precise measure of complexity for the part of the theory defined by the unknown binary string). In saying that one theory $D( T^C , C , r )$ is our best current theory – or perhaps that our best descriptions of nature are given by a class of similar such theories – we mean that we can’t presently find a substantially simpler and more elegant theory that fits the data. The stronger meta-theoretic hypothesis that a theory $D( T^C , C , r ) $ is, up to approximate equivalence, [*the*]{} best theory of nature implies that, given all the physical data in the universe, one would not be able to find a simpler, more elegant, compelling theory.
This could be made more quantitative by formalising the discussion within the context of a fixed model of computation, for instance a (classical) Turing machine. (This is why we have chosen to consider theories with unknown binary strings, although of course bases other than binary could also be used.) Here, a theory is a program for generating a mathematical representation of the complete set of physical data. A theory with unknown data is a program that requires an unknown input string of stipulated length. The theory’s simplicity depends, inter alia, on both the length of the program and the length of the required input string. Each of these is a natural simplicity parameter. The halting time of the program is another significant parameter, which gives one way of quantifying the elegance of a theory.
In principle, within a fixed computation model, it’s possible to carry out an exhaustive search of all theories with total length $ \leq L$ that halt after $\leq N$ steps. In principle, thus, given all the physical data, one can test the hypothesis that $D ( T^C, C, r)$ is the best theory among all those whose program and input strings satisfy given length bounds, and which satisfy other stipulated simplicity and elegance constraints, that halt after any given finite time, relative to a fixed computation model.
Thus, instead of talking about a probabilistic physical theory that produces a random set of physical data, we can consider a deterministic physical theory whose definition includes a set of pre-determined but a priori unknown physical data, together with the meta-theoretic hypothesis that this description is essentially algorithmically incompressible. If we learn empirically that the data are in fact significantly compressible, then this hypothesis is refuted, and we may replace the theory by a more economical one.
It should be stressed that these measures of simplicity and elegance are by no means intended to be an exhaustive list. For example, another elegance criterion is given by the principle of scientific induction, which suggests we should prefer a theory that suggests that a hitherto apparently fair coin will continue to be apparently fair over one that suggests that it will henceforth always come up heads, even though the latter theory requires a shorter input string (and so is simpler by one of the above measures).[^1] Comparing scientific theories generally involves a wide and arguable variety of quantitative and qualitative simplicity and elegance criteria, and nothing in this account alters that: the aim here is only to propose a different treatment of apparent randomness when comparing theories.
Example: reinterpreting a fair coin
-----------------------------------
For example, in a universe with an apparently random process that apparently mimics a fair coin and produces a large number $N$ of apparently independent outcomes, our meta-theoretic hypothesis might suggest that we cannot find a simpler correct theory than one that states that the length $N$ binary string is essentially algorithmically incompressible. If, in fact, the string $S$ turns out to consist of $0.01 N$ zeroes and $
0.99 N$ ones, we can certainly generate a more economical theory, and this hypothesis is refuted.
According to the standard account of probabilistic theories, if a probabilistic theory $PT$ says that zeroes and ones are equiprobable and independently generated, the outcome $S$ is extremely improbable, but not logically impossible. The theory $PT$ is thus not logically refuted by the outcome $S$. In practice we would reject it – but, without a fundamentally satisfactory account of probability, it is hard to give a completely satisfactory justification for doing so.
In our alternative account, however, no such problem arises. Our hypothesis predicts that a given physical dataset is essentially incompressible — where “essentially” incorporates some judgements about the tradeoffs between small gains in compression of the dataset and simplicity and elegance in other aspects of the theory. If the dataset turns out to be a string such as $S$ that is significantly compressible, so that we can fit the data by a simpler theory, the hypothesis is falsified and the original theory replaced.
Example: reinterpreting a biased coin
-------------------------------------
Now consider a universe with an apparently random process that apparently mimics a coin with bias $p > \frac{1}{2}$ towards zero and produces $N$ apparently independent outcomes. We can then produce theories that state that the length $N$ binary string is compressible to $H(p) N + o (N)$ bits. For example, a theory which says that the length $N$ string will contain between $p N - 10 \sqrt{N}$ and $p N + 10 \sqrt{N}$ zeroes has the required compression, since we can binary code all such strings in a code of length $H(p) N + o (N)$. Clearly there are many somewhat similar such theories — the string contains between $p N - 9 \sqrt{N}$ and $p N + 9 \sqrt{N}$ zeroes, between $p N - 11 \sqrt{N}$ and $p N + 11 \sqrt{N}$ zeroes, and so on. On this view of scientific accounts of apparently random data, that’s the best one can hope for: generically, no single clearly optimal theory will emerge. However, we can hypothesize that theories of roughly this length are essentially best possible – i.e. that the string cannot be compressed to significantly shorter than $H(p)N$ bits – and [*this*]{} hypothesis is testable and falsifiable.
Again, these theories reproduce deterministically predictions that the standard probabilistic theory says hold with probability very close (but not equal) to one. They exclude some very low probability events which would, if realised, in practice persuade almost everyone that the probabilistic theory was wrong, even though their occurrence is logically consistent with the theory.
Conclusion
----------
According to this account, we should consider one-world quantum theory as a theory which requires a binary string as input, and consider it alongside the meta-theoretic hypotheses that (a) there is no significantly more compressed description of the data obtained from quantum experiments than that given by encoding them in binary, using a coding that would produce an approximately uniform distribution over binary strings if the data were probabilistically generated via the Born rule, (b) the data can indeed be thus described. If one of these hypotheses turns out to be incorrect – if, for example, the data in all Bell experiments consistently show significantly greater violations of the CHSH inequality than quantum theory predicts – then we must find a better theory. Conversely, the theory logically (not merely with high probability) implies that we will see no consistent regularities in our experimental data that would, on the usual account, be highly improbable.
Among the scientific virtues of this account, as I see it, are its explicitness about the provisional nature of our theories, and its undogmatic sidestepping of the problem of giving a fundamental meaning to probability. It recognizes the possibility that random-seeming data may turn out to have a simpler description. It recognizes too that, if we find consistent regularities that a probabilistic theory says are highly improbable, then we should and will feel impelled to produce a better theory. At the same time, it stays silent on the question of whether random-seeming physical data are genuinely randomly generated in some fundamental sense, and hence avoids the need to explain what such an assertion could really mean and how we could be persuaded of its truth.
One-world quantum theory, read in this way, allows us to draw logical inferences about the physical world. It predicts – it is not merely consistent with the fact – that there will be no regularities in the data of a type that would allow for a significantly simpler theoretical description. If that prediction turns out to be wrong, the theory is refuted. Interpreted thus, one-world quantum theory can be read as a well-formulated scientific theory, in a way that allows a straightforward account of scientific confirmation and refutation. If we assume it is correct, we have an explanation for the apparent fact that our evolutionary and experimental histories contain no regularities that would be inexplicably improbable according to the Born rule. To the extent that the project outlined above can be fleshed out and succeeds – and I am optimistic that it can and does – proponents of one-world quantum theory can rest relatively easy on the question of randomness.
Toy many-worlds theories and their uses
=======================================
If we knew of probability theory [*only*]{} through its use in Copenhagen quantum theory – if we had no familiarity with coin tosses, dice rolls, noise, or any other effectively unpredictable classical systems – we would probably be (even more) deeply confused about the nature of both quantum theory and probability. I suspect this is the cause of much of the continuing confusion over many-worlds quantum theory: discussions need simultaneously to grapple with the quite unfamiliar concept of many branching worlds and the specific peculiarities of Everettian quantum theory.
This motivates defining some simpler many-worlds theories. Another reason for doing so is that some key Everettian ideas – for example, Greaves and Myrvold’s attempt [@greavesmyrvoldvol] at an account of many-worlds theory confirmation – can really only sensibly be discussed if we can consider a class of many-worlds theories, not just the single example of Everettian quantum theory. Readers may initially find the form of the following theories a little intellectually unsettling, but I recommend persevering: they shed a great deal of light on Everettian arguments.
Let me stress right away that these are not perfect models for Everettian quantum theory. That is, in fact, part of the point: they allow us to separate out general claims about rationality and theory confirmation in multiverse theories from claims that rely on specific features of quantum theory. In particular, they allow us to see why Greaves-Myrvold’s account of many-world theory confirmation doesn’t work.
Some toy multiverses
--------------------
The following toy multiverses are all classical, in the sense that the state of any branch at any time is defined by a classical physical theory, and they all have a definite branching structure.
Consider, first, the branching multiverse $CBU_1$, which includes conscious inhabitants, and also includes a machine with a red button on it and a tape emerging from it, with a sequence of numbers on it, all in the range $0$ to $(N-1)$. Whenever the red button is pressed in some universe within the multiverse, that universe is deleted, and $N$ successor universes are then created. All the successors are in the same classical state as the original (and so, by hypothesis, all include conscious inhabitants with the same memories as those who have just been deleted), except that a new number has been written onto the end of the tape, with the number $i$ being written in the $i$-th successor universe.
Suppose, too, that the multiverse’s inhabitants believe that something like this is indeed happening. The numbers on the tape play a significant role in their society. In particular, it is quite common to place bets on future numbers, and social mores ensure that such bets are always honoured. Of course, since one’s own universe will be destroyed before the next number is written, placing such a bet means – they correctly believe – redistributing resources amongst one’s successors. Some inhabitants may find reasons for preferring some redistributions over others. We need not discuss yet precisely what these reasons and preferences (both of which may be different for different inhabitants) may be.
It might be helpful to imagine that the universes are being run on a simulator by technologically advanced beings, who simply end one simulation whenever the red button is pressed, and then start simulating the successor universes from the appropriate initial states. We will sometimes assume that the inhabitants, indeed, believe this to be the case.
Suppose, further, that some of the inhabitants of $CBU_1$ have acquired the theoretical idea that the laws of their multiverse might attach [*weights*]{} to branches, i.e. a number $p_i$ is attached to branch $i$, where $p_i \geq 0$ and $\sum_i p_i =1$. They may have different theories about how these weights are defined: for instance, that the weights are always $\{ p_i \}$, that they are always $\{ q_i \}$, that they vary over time according to some rule, and so on. As it happens, though, these theories are all incorrect: there are no weights attached to the branches. To be clear: this is not to say that the branches have equal weight. Nor are they necessarily physically identical aside from the tape numbers. They may perhaps be distinguished by other features: for example, if they are simulations, they may be simulated by different hardware or software. However, any such differences do not yield any natural quantitative definition of branch weights. There is just no fact of the matter about branch weights in this multiverse.
The multiverse $CBU_2$ is similar to $CBU_1$. In this universe, there [*are*]{} indeed numbers attached to the branches, but the way they are attached means that they should (by our lights, and also by the inhabitants’, if only they understood the full picture) have no significance to any decisions the inhabitants make about bets/redistributions. For instance, we could extend the simulation idea, and imagine that the technologically advanced beings simply choose, on whim, to write the number $p_i$ somewhere inconspicuous in the simulation of successor universe $i$, in such a way that it has no effect on the inhabitants, and that it has no other significance.
The multiverse $CBU_3$ is similar to $CBU_2$. However, this time the numbers attached to the branches by the physical theory are attached in such a way that it can be plausibly argued that they [*could*]{} reasonably play a significant role in the decisions the inhabitants make about bets/redistributions. For instance, we could imagine that when the technologically advanced beings create successor universes, they create not just one successor corresponding to each outcome $i$, but a number of distinct successor universes, all identical apart from their outcome values, and the number containing outcome $i$ is proportional to the weight $p_i$. (We assume here the $p_i$ are rational numbers.)
Some possible strategies {#strategies}
------------------------
Consider an inhabitant of any of the above multiverses, who believes that the weight $p_i $ is attached to the outcome $i$. Suppose they are offered a variety of bets that give their successor a good $G_i$ in a universe in which outcome $i$ obtains, and they (the original inhabitant) attach utility $U_i$ to this good. We suppose the $U_i$ are finite real numbers, not necessarily positive (the goods may be bads); and, of course, both $G_i$ and $U_i$ depend on the bet.
How might they proceed to evaluate and rank such bets? [*Weight-sensitive*]{} inhabitants believe that branch weights exist and should play a role in their betting preferences. [*Weight-indifferent*]{} inhabitants may also believe that the physical theory attaches weights, but if so, do not believe they are of any relevance to a rational betting strategy. (Such an inhabitant might, for example, believe that they live in a multiverse like $CBU_2$.) Among their options is to mimic the strategy of a weight-sensitive inhabitant, except that they treat all branch weights as equal. By this means, given any weight-sensitive strategy, we can define a corresponding weight-indifferent strategy. Here are some examples of weight-sensitive strategies:
- The [*mean utilitarian*]{} ranks bets according to the value of $\sum_i p_i U_i$.
- The [*Price-Rawlsian*]{}’s dominant concern[@pricevol; @rawls] is with the welfare of their least satisfied future self. They rank bets first according to $\min ( U_i )$, and then some list of tie-breaking criteria. To be definite, let’s say their next criterion is the value of $ \sum{p_j} $, summed over all $j$ such that $U_j = \min ( U_i )$, followed by $\min ( U_j: U_j \neq \min (U_i ) )$, and so on.
- The [*future self elitist*]{}’s dominant concern is that the best possible version of their future self should be realized somewhere; they have little interest in mediocre future selves, whom they regard as losers. Their bet rankings are thus dominated by $\max ( U_i )$, and they break ties using the mirror image of the Price-Rawlsian’s criteria.
- The [*rivalrous future self elitist*]{} takes things one stage further. Not only do they identify their interests exclusively with those of their best possible future self, but they regard that self as in competition with the others, and feel happiest – all else being equal – if that competition is won by as large a margin as possible. They rank bets first by $\max (U_i )$, then by $ \max ( U_i ) -
\max ( U_j : U_j \neq \max (U_i ) ))$, and so on.
- The [*median utilitarian*]{}’s dominant concern is for median utility. Reordering the index labels so that $U_1 \leq U_2 \leq \ldots \leq U_n$, let $j$ be such that $\sum_{i=1}^{j-1} p_i < \frac{1}{2}$ and $\sum_{i=1}^{j} p_i \geq \frac{1}{2}$: they rank bets first according to the value of $U_j$. (They also have some tie-breaking criteria: one option is to break ties by considering the mean utility.)
- The [*$x$-percentile utilitarian*]{}’s dominant concern is for the utility of the future self ranked at $x$% in the distribution. They proceed like the median utilitarian, with $\frac{1}{2}$ replaced by $\frac{x}{100}$. The Price-Rawlsian, median utilitarian and future self elitist are all special cases.
- The [*future self democrat*]{} believes her preference should be that which would result from a democratic vote among her future selves. Given a finite list of possible bets, for each value of $x$, she asks herself how she would order her preferences among the bets, if she knew that she would become the future self ranked at the $x$-percentile of the elected bet. (The answer might be that her future self’s voting preference would always be dominated by its own welfare under this hypothesis, but it need not: it depends whether she cares about the welfare of contemporaneous versions of herself in other branches.) She then tallies the votes, integrating over $x$ using branch weight measure, and using, say, a single transferable vote system. The winner of the vote is her preferred bet. If the election is tied, she has more than one equally preferred bet.
- An example of a [*future self distribution engineer*]{} is someone who seeks to maximise an expression of the form $$\sum_i f_1 (U_i ) p_i + \sum_{ij} f_2 (U_i , U_j ) p_i p_j + \ldots \, ,$$ where the $U_i$ are the utilities of future branches with weights $p_i$, and $f_n$ is some given joint function of $n$ variables.[^2]
Many-worlds rationality reconsidered in toy models
--------------------------------------------------
> “If you do what you’ve always done, you’ll get what you always got.”
>
> (variously attributed)
According to Wallace [@wallacevoltwo] and Greaves-Myrvold [@greavesmyrvoldvol], we should [*define*]{} rational behaviour in a multiverse via axioms generalizing those proposed by Savage[@savage] in order to justify using the standard calculus of probabilities and utility functions for rational decisions in a single world in which future events are uncertain.
### Savagean rationality in one world
Savage, engagingly and rather admirably, presented his approach to rationality in the presence of (one-world) uncertainty
> “$\ldots$ in a tentative spirit, for I realize that the serious blemishes in it apparent to me are not the only ones that will be discovered by critical readers.” [@savagequote]
Everettian neo-Savageans [@wallacevoltwo; @greavesmyrvoldvol], as I read them, seem rather less self-critical — puzzlingly so, since applying Savagean decision theory to Everettian quantum theory raises many new questions without solving any of the old ones. This raises some general worries, which are developed to some extent elsewhere in this paper, but might also be taken in other directions. First, if Savage’s axioms are, in fact, unable to give a completely satisfactory account of ideal rational behaviour in the presence of one-world uncertainty, it seems very unlikely that a completely satisfactory axiomatic treatment of many-worlds rationality can be produced by generalizing them. Second, giving a satisfactory account of ideal rational behaviour in the presence of one-world uncertainty (or some many-worlds generalization) may in any case not be enough. (For one thing, we are not ideal rational agents. For another, as Albert [@albertvol] has eloquently stressed, there is a crucial difference between showing that one can find a rational justification for behaving as though the world were a certain way and showing that the world actually is that way.) Third, however far Savage can or cannot guide rational agents in one uncertain world, it isn’t obvious that his programme generalizes [*at all*]{} to many-worlds theories in general or to Everettian quantum theory in particular.
### Many-worlds rationality according to Greaves-Myrvold
Let me now focus on Greaves-Myrvold’s axioms, which are intended to apply to general many-worlds theories, and so can straightforwardly be considered within the toy models described above. I will consider later Wallace’s arguments, which are framed for the special case of Everettian quantum theory, and for the moment simply note that their logic suggests the same conclusions here as Greaves-Myrvold’s.
In Greaves-Myrvold’s view, the mean utilitarian’s strategy is rationally justifiable, and the others are branded irrational, since they violate one or more of the axioms. For example, the future self elitist and the Price-Rawlsian violate their continuity postulate, $P6$, the median utilitarian violates $P2$, the future self democrat violates transitivity, $P1a$, and the rivalrous future self elitist the dominance postulate $P3$.
However, the fact is that each of these strategies is well-defined and has a coherent motivation (and many other such examples could also be constructed). To brand them irrational seems to me itself irrational dogma. Even the most contentious case, the rivalrous future self elitist, has a coherent, if ungenerous, philosophy of life in the multiverse and a rational strategy for implementing it. Note too that some of these strategies have arguable theoretical advantages over the mean utilitarian strategy. For instance, one can be an $x$-percentile utilitarian, or a future self democrat (if they are [*purely self-concerned*]{}, in the sense that each future self’s preferences among options are completely determined by the implications for its own welfare), without having to quantify the utility of the possible outcomes: one needs only a preference ordering. This is arguably advantageous, since even if one accepts Savage’s postulates [@savage] and, hence, the conclusion that one’s preferences must be defined by some utility function, it may be difficult or even impractical to compute the relevant function for general outcomes, and yet relatively easy to identify preferences among any finite list of outcomes.
In short, Greaves-Myrvold’s postulates only express in more abstract form a preference for being a mean utilitarian – i.e., for one possible choice among many. Their postulates are plausible possible prescriptions for rational behaviour when considering the welfare of a population of future selves, but also logically inconsistent with other plausible prescriptions. This shouldn’t come as a complete surprise: after all, Arrow’s celebrated impossibility theorem [@arrow] taught us that plausible decision theoretic principles for populations may be inconsistent.
Granted, the one-world counterparts of some of these strategies may look peculiar. But one can consistently accept the many-worlds strategies as rational and reject their one-world counterparts. As Price [@pricevol] has persuasively argued, many-worlds agents can offer reasoned justifications for their strategies that aren’t available to their one-world counterparts. The many-worlds future self elitist knows that his best possible future self will be an [*actual*]{} future self, while his one world counterpart doesn’t. The many-worlds future self democrat knows that there really will be a population of future selves who have preferences among the betting choices, while her one world counterpart knows there won’t be; and so on.
One could, of course, adopt a weaker position. One could take Greaves-Myrvold’s and Wallace’s accounts of rationality as simply suggesting a possible attitude one [*might*]{} adopt to life in an Everettian multiverse, an attitude defined by a set of rules which are consistent and have some pleasant mathematical features but which are not meant to constitute a dogma. On this liberal reading, Greaves-Myrvold’s preferred strategy could be termed “rational”, in the sense of being well-defined and internally consistent, without denying the existence of other equally rational strategies. The problem is that abandoning any claim of uniqueness also removes the purported connection between theoretical reasoning and empirical data, and this is disastrous for the programme of attempting to interpret Everettian quantum theory via decision theory. If Wallace’s arguments are read as suggesting no more than that one can consistently adopt the Born rule if one pleases, it remains a mystery as to how and why we arrived at the Born rule empirically. If Greaves-Myrvold’s arguments are read as merely suggesting a possible attitude one might choose to take about testing and confirming many-worlds theories, one’s left to investigate how many other equally valid attitudes there might be, and whether they mightn’t – disastrously – imply the confirmation of inconsistent theories from the same data.
Rationality and feasibility
---------------------------
Consider now a rather more complicated multiverse, $CBU_4$. Here, the universes are definitely being simulated by technologically advanced beings, and the inhabitants know it. They also know that, after the red button is pressed, there is a list of outcomes $i$, and that the list is indeterminately long (and possibly infinite). They know too that, for each $i$, some number of successor universes containing outcome $i$ will be created. They do not know the number of successors there will be of each type: these vary for each $i$, and vary each time the button is pressed, at the whim of the simulators. What they do know – because, let’s say, the simulators have credibly promised them – is that numbers playing the role of additive weights, following certain rules, will be written inconspicuously into each simulated universe. Thus, if their universe has the number $x$ written in it, and the red button is pressed, and there are $n_i$ successor universes with outcome $i$, these successors will have numbers of the form $ x q^j_i $ written into them, where the label $j$ runs from $1$ to $n_i$, $q^j_i \geq 0$, and $$\label{weightrelation}
\sum_j q^j_i = p_i \, .$$ Here the $p_i$ are known to be constants (i.e. they take the same value each time the button is pressed), with $p_i \geq 0$, and $ \sum_i p_i = 1$. The inhabitants know the values of a finite set of the $p_i$, those with index $i \in I$, whose sum $ \sum_{i \in I} p_i < 1 $.
What the inhabitants would [*like*]{} to do – what they feel rationality would mandate they do if they could – is express betting/distribution preferences that value each successor universe equally. But they can’t – they don’t know how many successors will be created for any given $i$, nor do they know how long the list of possible outcomes $i$ is. Nor can they express betting/distribution preferences that value each outcome equally, regardless of the number of successor universes containing it – again, they don’t know how long the list of possible outcomes is.
What they [*can*]{} do is express betting/distribution preferences, for bets on the known possible outcomes, treating the known values of $p_i$ as probability weights. Doing so is equivalent to treating a successor universe with the number $y$ written into it as having an importance proportional to $y$ – a rule which can be consistently applied, despite their ignorance about the number of successors of each type, because of equation (\[weightrelation\]). So, they have a consistent, feasible strategy available to them. Moreover, if they want to assign a measure of importance to each individual universe, and they want the importance they assign to the set of universes containing outcome $i$ to be independent of the number of such universes, this is the [*only available*]{} rule. Nonetheless, it doesn’t seem to have a fundamental rational justification. The numbers written into the universes happen to follow convenient bookkeeping rules, but they have no significance: there is no fundamental [*reason*]{} to treat the numbers as a measure of importance of their universes.
From this, I think we should conclude two things, to be borne in mind when we come to consider Wallace’s arguments. It can make perfect sense, in a multiverse theory, to say that there exists a rational optimal strategy that is inaccessible to the agents in that multiverse. Conversely, the fact that a strategy is available does not [*per se*]{} make it rationally compelling, even if it is the unique available strategy satisfying some pleasant consistency properties: rational compulsion also needs rational justification, which may or may not exist.
Why many-worlds theory confirmation doesn’t work
================================================
Everettian quantum theory is essentially useless, as a scientific theory, unless it can explain the data that confirm the validity of Copenhagen quantum theory within its domain – unless, for example, it can explain why we should expect to observe the Born rule to have been very well confirmed statistically. Evidently, Everettians cannot give an explanation that says that all observers in the multiverse will observe confirmation of the Born rule, or that very probably all observers will observe confirmation of the Born rule. On the contrary, many observers in an Everettian multiverse will definitely observe convincing [*disconfirmation*]{} of the Born rule. Nor can one look at Everettian quantum theory and conclude that any given observer in the multiverse will probably observe confirmation: the theory has no notion of standard probability available to even make sense of any such claim. And if the theory doesn’t explain the data, the data don’t support the theory.
There seems to be no good way around this, and if so, then that’s the end of Everettian quantum theory as a serious contender: a theory with no predictive power [*should*]{} lose the scientific competition against theories that predict what we actually see. However, Greaves and Myrvold [@greavesmyrvoldvol] have offered an attempt at a solution, by giving a general account purporting to explain why agents who take seriously the possibility of many-worlds theories can use observational data to confirm particular theories and refute others. Their account is illuminating, and raises some very interesting questions about many-worlds theories. Ultimately, though, it seems to me that it does not show, as claimed, the possibility of explaining our observations from a many-worlds theory and thus confirming one many-worlds theory against another. Rather, it highlights some apparently insuperable problems that prevent us from doing so. As Greaves and Myrvold’s arguments are set out in detail elsewhere in this volume, in this discussion I will simply summarize the implications of their confirmation algorithm in toy models, and point out the problems that arise.
The problem of inappropriate self-importance
--------------------------------------------
It suffices to consider very simple many-worlds theories, containing classical branching worlds in which the branches correspond to binary outcomes of definite experiments. Consider thus the [*weightless multiverse*]{}, a many-worlds theory of type $CBU_1$, in which the machine produces only two possible outcomes, writing $0$ or $1$ onto the tape. Recall that in $CBU_1$ there is no fact of the matter about weights attached to the branches containing $0$ outcomes and $1$ outcomes, although the inhabitants think there may be. This is the many-worlds analogue of an indeterministic one-world theory containing a sequence of binary experimental outcomes which are not only not determined but also not governed by any probabilistic law. Suppose now that the inhabitants begin a series of experiments in which they push the red button on the machine a large number, $N$, times, at regular intervals. Suppose too that the inhabitants believe (correctly) that this is a series of independent identical experiments, and moreover – this is not essential, but simplifies the discussion – believe this [*dogmatically*]{}: no pattern in the data will shake their faith. Suppose also that they believe (incorrectly) that their multiverse is governed by a many-worlds theory with unknown weights attached to $0$ and $1$ outcomes, identical in each trial, and seek to discover the (actually nonexistent) values of these weights by following Greaves-Myrvold’s learning algorithm.
After $N$ trials, the multiverse contains $2^N$ branches, corresponding to all $N$ possible binary string outcomes. The inhabitants on a string with $pN$ zero and $(1-p)N$ one outcomes will, with a degree of confidence that tends towards one as $N$ gets large, tend to conclude that the weight $p$ is attached to zero outcome branches and weight $(1-p)$ is attached to one outcome branches. In other words, everyone, no matter what outcome string they see, tends towards complete confidence in the belief that the relative frequencies they observe represent the weights.
Let’s consider further the perspective of inhabitants on a branch with $pN$ zero outcomes and $(1-p)N$ one outcomes. They do not have the delusion that all observed strings have the same relative frequency as theirs: they understand that, given the hypothesis that they live in a multiverse, [*every*]{} binary string, and hence every relative frequency, will have been observed by someone. So how do they conclude that the theory that the weights are $(p, 1-p)$ has nonetheless been confirmed? Because, following Greaves-Myrvold’s reasoning, they have concluded that the weights measure the [*importance*]{} of the branches for theory confirmation. Since they believe they have learned that the weights are $(p,1-p)$, they conclude that a branch with $r$ zeroes and $(N-r)$ ones has importance $p^r (1-p)^{N-r}$. Summing over all the branches with $pN$ zeroes and $(1-p)N$ ones, or very close to those frequencies, thus gives a set of total importance very close to $1$; the remaining branches have total importance very close to $0$. So, on a set of branches that dominates the importance measure, the theory that the weights are (very close to) $(p, 1-p)$ is indeed correct. All is well! By definition, the important branches are the ones that matter for theory confirmation. The theory is indeed confirmed!
The problem, of course, is that this reasoning applies equally well for all the inhabitants, whatever relative frequency $p$ they see on their branch. All of them conclude that their relative frequencies represent (to very good approximation) the branching weights. All of them conclude that their own branches, together with those with identical or similar relative frequencies, are the important ones for theory confirmation. All of them thus happily conclude that their theories have been confirmed. And, recall, all of them are wrong: there are actually no branching weights.
### Comparison with the one-world case
It’s illuminating to compare the case of an inhabitant of the analogous one-world universe, in which pressing the red button produces either a $0$ or a $1$ on the tape but there is no law, either deterministic or probabilistic, governing these outcomes. After $N$ experiments in which he sees $pN$ zeroes and $(1-p)N$ ones, he tends towards confidence in the theory that zeroes have probability $p$ and ones have probability $(1-p)$.
Let us again restrict attention to theories — in this case probabilistic one-world theories — that dogmatically assume the experiments are identical and independent. Among such theories, the selected theory does indeed characterize, better than all its competitors, all the relevant data in the universe — i.e., all the outcomes of the $N$ experiments. Of course, further data could change that conclusion. But, so long as we consider only the relevant data, it’s something of a puzzle to pin down whether it’s wrong to adopt the theory [*pro tem*]{}, and if so precisely why. Is there a physically meaningful sense in which a universe that [*looks*]{} as though it contains data resulting from a sequence of independent identical coin tosses with a probability $p$ of outcome zero is distinct from one that [*does*]{} contain such data? And if so, how precisely should we characterize the distinction?
On the view of physical randomness discussed in section \[oneworldprob\], the answer to the first question is no. In any case, however one answers the questions, it seems that any possible error here must be subtler than and distinct from the error highlighted above in the many-worlds case. In the many-worlds case, recall, all observers are aware that other observers in worlds with other data must exist, but each is led to construct a spurious measure of importance that favours their own observations against the others’, and this leads to an obvious absurdity. In the one-world case, observers treat what actually happened as important, and ignore what didn’t happen: this doesn’t lead to the same difficulty.
### Numbers in the sky
Consider next the [*decorative weight multiverse*]{}, a type $CBU_2$ variant of the weightless universe. This universe has a constant of nature fixed by the technologically advanced beings, a real number $p$, with $0 < p < 1$. As before, whenever the red button is pressed in a simulated universe, that universe is deleted, and successor universes with outcomes $0$ and $1$ written on the tape are initiated. This time, the technologically advanced beings also write the numbers $p$ and $(1-p)$ in an inaccessible part of the skies of the $0$ and $1$ successor universes, respectively. These numbers are visible to the inhabitants, but have no other physical significance.
There is thus a formal sense in which distinct weights are attached to the $0$ and $1$ branches. However, by hypothesis, these weights are decorative: there are no rational grounds for assigning them any fundamental physical meaning or any rôle in constraining rational actions. We can thus run through a discussion of theory confirmation precisely parallel to that for the weightless multiverse.
This illustrates again that the mere fact that Born weights are mathematically defined in Everettian quantum theory does not [*per se*]{} justify assigning them any role in theory confirmation. They could be merely decorative.
Separating caring weights from theory confirmation
--------------------------------------------------
To investigate further it’s helpful to consider branching world models in which there [*are*]{} weights attached to the branches, in such a way that the weights could plausibly be regarded as important for making rational decisions. I want here to tell specific stories about the weights, in order to illustrate a crucial distinction between two possible definitions of importance.
### The replicating multiverse
Consider first the [*replicating multiverse*]{}, a multiverse of type $CBU_3$ with a machine like the one above, in which the branches arise as the result of technologically advanced beings running simulations. Whenever the red button is pressed in a simulated universe, that universe is deleted, and successor universes with outcomes $0$ and $1$ written on the tape are initiated. Suppose, in this case, that each time, the beings create [*three*]{} identical simulations with outcome $0$, and just one with outcome $1$. From the perspective of the inhabitants, there is no way to detect that outcomes $0$ and $1$ are being treated differently, and so they represent them in their theories with one branch each. In fact, though, given this representation, there is an at least arguably natural sense in which they ought to assign to the outcome $0$ branch three times the importance of the outcome $1$ branch: in other words, they ought to assign branch weights $(\frac{3}{4}, \frac{1}{4} )$.
They don’t know this. But suppose, as before, that they believe that there are unknown weights attached to the branches, and follow the Greaves-Myrvold procedure for identifying those weights. What happens now? After $N$ runs of the experiment, there will actually be $4^N$ simulations – although in the inhabitants’ theoretical representation, these are represented by $2^N$ branches. Of the $4^N$ simulations, almost all (for large $N$) will contain close to $\frac{3N}{4}$ zeroes and $\frac{N}{4}$ ones. These simulations will contain inhabitants who, following Greaves-Myrvold, believe they have confirmed that the branch weights (in their own theoretical representation, which remember contains only $2^N$ branches) are very close to $(\frac{3}{4}, \frac{1}{4} )$. They believe too that the weights define an importance measure on the branches: a branch with $r$ zeroes and $(N-r)$ ones has importance (very close to) $ ( \frac{3}{4} )^r ( \frac{1}{4} )^{N-r}$. They thus conclude that their weight assignment will be confirmed on a set of branches whose total importance is close to $1$.
Now, I think I can see how to run some, though not all, of an argument that supports this conclusion. The branch importance measure defined by inhabitants who find relative frequency $\frac{3}{4}$ of zeroes corresponds to the counting measure on simulations. If we could argue, for instance by appealing to symmetry, that each of the $4^N$ simulations is equally important, then this branch importance measure would indeed be justified. If we could also argue, perhaps using some form of anthropic reasoning, that there is an equal chance of finding oneself in any of the $4^N$ simulations, then the chance of finding oneself in a simulation in which one concludes that the branch weights are (very close to) $(\frac{3}{4} , \frac{1}{4} )$ would be very close to one. Turning that around, the theory that the branch weights are $(\frac{3}{4} , \frac{1}{4} )$ would then imply that, with high probability, one should expect to see relative frequency of zeroes close to $\frac{3}{4}$. There would indeed then to be a sense in which the branch weights define which subsets of the branches are important for theory confirmation.
It seems hard to make this argument rigorous. In particular, the notion of “chance of finding oneself” in a particular simulation doesn’t seem easy to define properly. Still, we have an arguably natural measure on simulations, the counting measure, according to which most of the inhabitants will arrive at (close to) the right theory of branch weights. That might perhaps be progress.
### The qualia enhancing multiverse
But consider now the [*qualia enhancing multiverse*]{}, again a multiverse with the same type of machine, in which the branches arise in the way we’ve previously considered, as the result of technologically advanced beings running simulations. Whenever the red button is pressed in a simulated universe, that universe is deleted, and successor universes with outcomes $0$ and $1$ written on the tape are initiated. This time, though, the beings create just one simulation with outcome $0$, and one with outcome $1$, but devise their simulations so that the qualia – the mental sensations – of the inhabitants in the outcome $0$ simulation are three times as intense. As before, from the perspective of the inhabitants, there is no way to detect that outcomes $0$ and $1$ are being treated differently, and so they represent them in their theories with one branch each.
There is, again, an arguably natural sense in which they ought – if they were aware of the rules of their multiverse – to assign to the outcome $0$ branch three times the importance of the outcome $1$ branch: in other words, they ought to assign branch weights $(\frac{3}{4}, \frac{1}{4} )$. Recall, pleasure and pain in outcome $0$ branches have tripled in intensity. The welfare of successors on outcome $0$ branches is felt more intensely, and in that sense it matters more.
Let me deal here with three possible objections:
- It might be argued that qualia enhancement should be analysed differently, as an example of an unannounced alteration in utility functions: the actual payoff of winning a bet with outcome $0$ is three times the expected payoff, since the inhabitants don’t expect any qualia enhancement. Certainly it [*could*]{} be analysed in this way. But this reflects an arbitrary choice that always needs to be made in many-worlds theories. (Precisely the same argument could be made in the case of the replicating multiverse, for example.) The statement that one branch is $N$ times as important as another can always be recast as a statement that utilities on the first branch are rescaled by $N$ relative to those on the second. So, we can legitimately analyse qualia enhancement as an effect altering the relative importance of branches, and it’s interesting to do so, as this lets us test general propositions about the confirmation of theories attaching importance to branches.
- The reader may not believe that there is a sensible account of experience involving qualia, or that intensifying qualia makes any sense. Never mind. It’s just a useful device to make a point about branch measures. It could be formulated in another way: we could suppose that the simulators arrange that all bets have payoffs with three times the expected utility on outcome $0$, while erasing the relevant bits of the inhabitants’ memories so that they’re not aware that the payoff tripled.
- One might also worry that inhabitants in an outcome $0$ branch would notice that the intensity of their qualia has just tripled. For the sake of the argument, we must assume not. Insofar as the notion of qualia enhancement makes sense, this seems reasonable: their memories will triple in intensity along with everything else.
Suppose, once again, that the inhabitants believe that there are unknown weights attached to the branches, and follow the Greaves-Myrvold procedure for identifying those weights. What happens now? After $N$ runs of the experiment, there will be $2^N$ simulations – now correctly represented by $2^N$ branches in the inhabitants’ many-worlds theory. The simulations will contain inhabitants who, following Greaves-Myrvold, believe they have confirmed that the branch weights are very close to $(p, 1-p)$, because their observed relative frequency is $p = r/N$, for each $r$ in the range $0 \leq r \leq N$. They believe that the weights define an importance measure on the branches: a branch with $r$ zeroes and $(N-r)$ ones has importance (very close to) $ ( p )^r (1-p )^{N-r}$. They thus conclude that their weight assignment will be confirmed on a set of branches whose total importance is close to $1$. Now, in one sense, the inhabitants whose observed relative frequency $p = 3/4$ are a special case. Their inferred importance measure equals the natural importance measure defined by qualia intensity. And if we weight the branches by this importance measure, it is the case, by the same calculation as before, that, on a set of branches with total measure close to one, the inhabitants end up with (very close to) the “right” branch weights, $ ( \frac{3}{4}, \frac{1}{4} )$.
But wait! If we count the simulations, the inhabitants who arrive at weights $ ( \frac{3}{4} , \frac{1}{4} )$ are a tiny minority. Almost everyone arrives at the wrong branch weights – and, as in our earlier example, almost everyone arrives at a measure of importance according to which branches with (very close to) their observed relative frequency are the important ones. By the natural simulation counting measure, theory confirmation has spectacularly failed.
What these last two examples show is that there are two distinct senses, which Greaves-Myrvold and Wallace fail to separate, in which a branch weight might possibly be said to be a measure of importance. It could be said to be a “caring measure”, if there is some reason to care differently about the welfare of successors on different branches. And it could be said to be, for want of a better term, an “explanatory counting measure”, if there is some reason to think that we are likelier to [*find ourselves*]{} on some branches rather than others – or some other argument to show that a branching theory which predicts the observed relative frequencies (or other data) on a set of branches of high explanatory counting measure thereby explains them. What we’ve seen is that the first property doesn’t necessarily imply the second, and it’s the second that is needed for an adequate account of branching theory confirmation.
Couldn’t a many-worlds theorist then simply [*postulate*]{} the existence of an explanatory counting measure? (And perhaps also postulate that a caring measure exists and equals the explanatory counting measure?)
A preliminary remark: even postulating a caring measure – which [*has*]{} been proposed[@papineauvol] in the Everettian literature – already seems a very strange manoeuvre. Physical theories can certainly give reasons for rational agents to perform certain actions if they have certain goals. But what’s envisaged here is a theory that [*by fiat*]{} imposes a constraint on rational behaviour. I’m not clear – and at least some Everettians (e.g. [@saundersvol]) seem to share this worry – that this makes any sense, either as an idea about physics, or about rationality.[^3]
In any case, when it comes to postulating an explanatory counting measure, one should be clear: the proposal is that a many-worlds theory [*defines*]{}, by fiat, without any attempt at further justification, whose observations matter and whose may be neglected, when it comes to testing and confirming the theory. The theory defines its own – highly non-standard – criteria for deciding whether or not it is a scientific success.
One could play this sort of game, of course, even in one world. For example, Alice could define a theory that includes – as a postulate, with no further explanation – the principle that everyone who agrees with her observations and her theoretical interpretation is important for theory confirmation, and everyone else is negligible. She could then announce, after checking with the important people, that her theory is confirmed. This would be self-consistent, and maybe politically adept, but it wouldn’t be science.
It’s no more scientifically respectable to declare that we can, without further justification, confirm Everettian quantum theory by neglecting the observations made on selected low Born weight branches. A Pavlovian association of low Born weight with small probability – illegitimately carried over from one world quantum theory – may perhaps lend an aura of greater respectability. But in Everettian quantum theory the Born weight is simply a number attached to branches. It has no intrinsic relevance to theory confirmation, and unless we add further structure to the theory, we cannot justify assigning it any such role.
Note again the contrast here with the one-world case: one-world probabilists do not pick and choose which observations are to be used for theory generation or confirmation.
Many-worlds confirmation: conclusion
------------------------------------
To explain how we could come to confirm Everettian many-worlds quantum theory it is not enough to note that we have Born weights to hand and so can automatically give them a confirmation-theoretic rôle. As the decorative weight multiverse illustrates, branch weights can be simply irrelevant to theory formation and confirmation.
Nor can Wallace’s arguments for treating the Born weights as a caring measure suffice, even if we take Wallace’s result at face value. As the qualia enhancing multiverse illustrates, a caring measure is not necessarily an explanatory counting measure.
Thus, the most sympathetic (though unauthorised) translation of Greaves-Myrvold’s account of many-worlds and confirmation that I can find requires us to add structure that [*justifies*]{} the existence of an explanatory counting measure. This requires interpreting Everettian quantum theory, along with competing many-worlds theories theories, as modelled by versions of the replicating multiverse, with branches constantly being deleted, and successor branches created. We need to postulate that the number of simulations or realisations of a given branch at a given time is proportional to the branch weight, and to assume that it is rational to treat all realisations as equally valuable. We need also to postulate something like an anthropic principle that tells us that, in some sense that needs to be properly defined, the chance of finding ourselves in one of a given class of realisations at a given time is proportional to the number of realisations in the class.
This, if it could be made rigorous, would suggest something resembling the objectively determinist “momentary minds” version of Albert-Loewer’s many-minds interpretation[@almanyminds; @bellmw; @barbour], in which the minds exist only instantaneously, with no continuous identity extending over time. This isn’t a picture I find easy to take seriously. As I read them, none of the Everettian contributors to the present volume would wish to defend this account – and yet it seems very closely aligned with some of their intuitions. Let me close here by inviting readers to see if they can find a better way of rigorously justifying Greaves’ gloss:[@greaves]
> But since we have a measure over our successors, we can, if we find it intuitive, talk of ’how much successor’ sees spin-up. I have a preference for my spin-down successor to receive chocolate, rather than my spin-up successor, because there is more of the former; more of my future lies that way. Thus, I think, Lockwood’s (1996) talk of a ’superpositional dimension’, and/or Vaidman’s (1998, 2001) suggestion that we speak of the amplitude-squared measure as a ’measure of existence’, are somewhat appropriate (although we are not to regard lower-weight successors as less real, for being real is an all-or-nothing affair – we should say instead that there is less of them).
Fuzziness, rationality and decision theory in many worlds
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Two of the most interesting recent developments in the Everettian literature, in my view, have been the attempt to argue for an intrinsically fuzzy emergent quasiclassical ontology[@wallacevolone] and (as already discussed) the attempt to reinterpret Born weights via a many-worlds version of decision theory[@wallacevoltwo]. Interesting, but flawed – each project has deep problems, and they appear to be based on inconsistent premises.
Fuzziness and its limitations
-----------------------------
Granted, as Wallace[@wallacevolone] notes, viable higher level scientific theories can and do, indeed, supervene on more fundamental theories. Objects in those theories need not have any unique and precise definition in terms of fundamental concepts: there is, indeed, no unique, natural, precise, chemical characterisation of a tiger.
Nonetheless, there is a very strong reason for seeking[@akcritique] a precise mathematical formulation of the intuition that many branching worlds emerge from unitary quantum theory – or else a precise mathematical formulation of some other structure consistent with Everettian ideas – namely, that it is not at all clear that, without such a formulation, we have a well-defined scientific theory to discuss. (This, it seems to me, is why both Everettians[@graham; @deutschone] and critics[@bellmw; @almanyminds] have often attempted to find mathematical structures that might explain the notion of branching.) The alternative strategy, proposed by Wallace[@wallacevolone], of trying to interpret the implications of a fundamentally mathematical theory in terms of higher level fuzzily defined constructs carries a very obvious danger — namely, a retreat into vagueness and hand-waving on points where precision really is required. It’s hard to run a serious argument (pro or con), let alone prove a rigorous theorem, if one doesn’t, in the end, know quite what one’s talking about.
Fuzzy minds
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A case in point is Wallace’s appeal to functionalist intuitions in trying to give an account of the mind states of agents in Everettian quantum theory. Readers are, I think, owed a much more precise explanation of what, actually, is supposed to follow from this, since some rather crucial points appear to turn on unspecified details.
For instance, on this account, do distinct mind states necessarily correspond to orthogonal quantum states? If so, wouldn’t this account necessarily supply us with a preferred orthogonal decomposition of the unitarily evolving quantum state? And wouldn’t this, pace Wallace[@wallacevolone], allow a precise definition of a relevant branching structure after all?
Wallace places great emphasis on the lack of a unique natural definition of a quasiclassical branch, and hence the impossibility of agents formulating a rational strategy based on counting distinct future branches. But it’s at least as relevant to examine whether our account of mind states supplies a natural definition of a future self, and whether it might be possible for agents to formulate a rational strategy based on counting distinct future selves? Can’t an agent identify successor selves as distinct if and only if they have distinct mind states, ascribe to distinct successors a branching history corresponding to that recorded in their memories, and use [*those*]{} data to define a rational strategy for taking account of their welfare? (These points are pursued further in appendix \[countdesc\].)
On the other hand, if non-orthogonal quantum states [*could*]{} correspond to distinct mind states, how would we even begin to connect quantum theory with even the appearance of probabilities? Quantum theory gives no general rule to calculate a probability of a transition from an unknown state belonging to one fuzzily defined set of states (corresponding to mind state A) to an unknown state belonging to another (corresponding to mind state B). But that’s what we’d need to calculate, in principle, in order to obtain a number corresponding to the apparent probability of arriving at state B when starting in state A. Maybe one could cook up such a rule, and then explain how the Born rule emerges as an approximation under suitable circumstances – but it’s not obvious how, and this would certainly be going beyond quantum theory as presently understood.
Both options thus lead to serious, perhaps insuperable, difficulties.
Can precise preferences arise in a fuzzy ontology?
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Another very basic worry about Wallace’s programme is its equivocation over mathematical rigour. Everything in Wallace’s ontology that’s relevant to rational decisions — including agents, the quasiclassical branches they inhabit, the branch states, and the branch Born weights, and the distinction between micro-states and macro-states — is intrinsically fuzzily defined[@wallacevolone]. There is, on Wallace’s account, no precise fact of the matter about the different quasiclassical states that would result after a bet on a quantum experiment, nor about the Born weights of the branches corresponding to those quasiclassical states. And this isn’t merely because quantum theory doesn’t supply a unique natural definition of elementary branches and branching events: the [*total*]{} Born weight of all the quasiclassical branches describing a spin-up outcome of a Stern-Gerlach experiment isn’t precisely defined either.
Now, to be sure, the total weight [*is*]{} supposed to be approximately defined. We are supposed, on Wallace’s account, to be able to say that it’s in a range of the form $ R = ( p - \epsilon, p + \epsilon )$, where $\epsilon$ is very small, and $p$ thus represents an approximate total Born weight.[^4] But we’re not supposed to be able, on this account, to reduce $\epsilon$ to zero: below some level of precision, it becomes unavoidably arbitrary, just a matter of taste in your choice of branch definition, whether you take the total weight as $p_1 \in R$ or $p_2 \in R$.
And yet, Wallace’s decision theoretic programme postulates that each rational agent should have a [*precisely specified*]{} and [*complete*]{} preference ordering among a very large class of possible unitary maps that produce different possible future global states. Where could such a preference ordering possibly come from? The ordering is supposed to be agent-dependent. Physics doesn’t equip rational agents with some personal preference ordering on global states: they have to arrive at their preferences by introspection and reasoning. If one accepts Wallace’s conclusions, the only ultimately relevant quantities are branch weights and the agent’s personal utilities for macrostates (whose existence is supposed to follow given the preference ordering axioms). But even a super-agent who finds they can calculate the former and can identify the latter by pure introspection would find these quantities only fuzzily defined – so that, in comparing some pairs $( U_1 , U_2 )$ of actions on a given state $\ket{ \psi }$, however hard they try and however carefully they analyse the alternatives, they wouldn’t be able to identify a reliable preference, [*not*]{} because the resulting global states are precisely equivalent, but because their difference is fuzzily ambiguous. On some views, $U_1 \ket{ \psi }$ would seem very slightly preferable; on others, $U_2 \ket{ \psi }$ would. In Wallace’s notation [@wallacevolone] for preference orderings, neither $U_1 \succeq_{\psi} U_2$ nor $U_2 \succeq_{\psi} U_1$ would hold in all ways of looking at the situation. Nor does it seem legitimate to postulate that $ U_1 \sim_{\psi} U_2 $ must hold in such cases. One can imagine the possibility of a sequence $( U_1 , \ldots , U_n )$ such that no preference can reliably be identified between $U_i \ket {\psi }$ and $U_{i+1} \ket{\psi}$, for $i=1 , \ldots, (n-1)$, but nonetheless setting $U_i \sim_{\psi} U_{i+1}$ violates transitivity, since $U_1 \succ_{\psi} U_n $ [*does*]{} hold no matter what view the agent adopts of the fuzzy facts.
We are not, in any case, super-agents, and can only read Wallace’s arguments as prescriptions for ideal rationality rather than descriptions of our real-world behaviour. None of us in fact has a complete and precise preference ordering among the relevant unitaries. Wallace, in effect, is telling us that we should ideally adjust our reasoning and their behaviour so as to be consistent with some complete preference ordering. But how? There is no natural algorithm available: any choice will involve uncountably many arbitrary decisions on pairs of preferences.[@savageworry] And why? Given that no choice of ordering will have any intelligible justification, even after the entire analysis is complete, how can there be a rational compulsion to make some choice (even if, counterfactually, it were practical)?
Here, it seems to me Wallace’s prescription runs into essentially the same difficulties that he identifies in other ways of thinking about Everettian branching. One [*could*]{}, in principle, find some (perhaps ad hoc) prescription defining a branching structure for the unitarily evolving state vector, and one could then use this structure to define a rational Born-rule-independent strategy based on branch counting. Wallace accepts that such a strategy is not logically inconsistent, but argues that it is likely to be difficult to implement in practice (because defining a precise branching structure is difficult) and hard to justify in principle (because the definition seems to require ad hoc choices). Both objections apply – arguably with at least equal force – to the Wallace programme.
This also reinforces the point that the case for Wallacean rationality cannot possibly rely on the lack of any practical alternative strategy. A very practical alternative is to follow whatever combination of instinct and reasoning evolution provided us before we became aware of Everettian quantum theory. Altering that strategy so as to comply rigorously with Wallace’s axioms isn’t practical; even coming close to doing so may not be. To be persuaded that we ought to try, we would need to be rationally persuaded not only that we should ideally be Wallaceans, but also that there is a practical method which allows us to become closer to being Wallaceans, and that we will be better off if we employ this method.[^5]
In short, given Wallace’s account of a fuzzy ontology, there seems a strong reason to doubt Wallace’s most basic postulate of rationality, $R1$, which states that rational agents have a [*complete*]{} (or [*connected*]{}) preference ordering on the unitary operations available to them at any state $\ket{\psi}$. No actual agent in a fuzzy Everettian ontology will ever be able to arrive at such an ordering in practice. Moreover, even if they had infinite computational power, fixing an ordering would require making a very complicated ad hoc choice which can have no complete rational justification. Yet without $R1$, the purported derivation of the Born rule[@wallacevoltwo] fails at the first step.
It’s not clear to me that there is any fix for this, but let me comment briefly on two possible responses.
First, one might perhaps try weakening the postulate $R1$ to suggest that agents have, or should aspire to have, a preference ordering that [*approximates*]{} a complete ordering, in the hope of then proving that their policy should approximate Born-weighted mean utilitarianism. One problem with this is that one would need first to find and justify a suitable definition of approximation applied to preferences between pairs of unitary operations. As these are unquantified binary relations, it doesn’t seem obvious that any suitable definition exists.
Second, one might consider the desperate resort of [*postulating*]{} a total ordering as part of the physical theory. But even that surely isn’t available here. The orderings, recall, are agent-dependent, and even the most postulate-happy Everettian would surely recoil from requiring that [*fundamental physical laws*]{} specify independently, agent by agent, the preferences of every agent instantiated in nature.
Trying to formulate a rigorous decision theory for preferences in a fuzzy ontology may thus be rather like trying to build a skyscraper on mud.
Circularity of the Wallace programme?
-------------------------------------
Zurek[@zurekvol] flags another worry about the logical relation between the two parts[@wallacevolone; @wallacevoltwo] of Wallace’s programme, namely an apparent circularity. Wallace envisages a fuzzy quasiclassical ontology arising as the result of mathematical regularities observable within components of the unitarily evolving universal wave function. These regularities are supposed, in a realistic cosmological model, to arise through the decoherence of classical variables and to be defined by what Gell-Mann and Hartle term a quasiclassical domain [@hartlevol], in which, for example, operators approximately quantifying local mass densities approximately follow classical equations of motion with probability close to one. Here the probability for a history defined by a sequence of operators $P_1 (t_1) , \ldots , P_n (t_n )$ is given by the decoherence functional $$\Tr ( P_n (t_n ) \ldots P_1 (t_1 ) \rho_{{\rm initial}} P_1 (t_1 ) \ldots
P_n (t_n ) ) \, .$$
In other words, the ontology is [*defined*]{} by applying the Born rule. Even if one could show, as Wallace claims, that agents defined within that ontology are rationally justified in using the Born rule as a calculus for decisions, it would seem incorrect to portray this argument as a [*derivation*]{} of the Born rule within Everettian quantum theory. Wallace’s argument should rather be understood as attempting to show something weaker: that the Born rule re-emerges as output (albeit, to be fair, in an interesting and non-obvious way) if assumed as input. Even if correct, this would leave open the possibility that there are many different consistent and essentially inequivalent ways of defining ontologies that include distinct types of agents for whom different rational decision calculi can be established. It would thus fail to explain whether and (if so) why our own decision calculus should be based on the Born rule. It would also leave open the questions as to whether and (if so) how agents in some consistently defined Everettian ontology can arrive at the rational decision calculus appropriate to their ontology.
Problems with Born-weighted mean utilitarianism
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Wallace[@wallacevoltwo], developing earlier ideas of Deutsch[@deutschtwo], partly in response to criticisms (e.g. Ref. [@barnumetal]) of the latter, then goes on to argue that from a few simple and purportedly natural axioms we can prove that rational agents who believe themselves to be in a universe described by many-worlds quantum theory are rationally required to (a) have a utility function that quantifies the value they assign to possible future quasiclassical events, (b) act so as to maximise their Born-weighted mean utility.
As we just saw, Wallace’s first postulate, $R1$, seems to run into a fundamental obstacle, since neither Born weights nor quasiclassical histories (and thus their utility) are precisely defined in his ontology, and without $R1$ the decision theoretic argument, which, inter alia, implies the existence of a utility function, fails. Moreover, even for an agent who [*has*]{} a utility function applicable to all relevant quasiclassical histories, the strategy of maximising Born-weighted mean utility is not well-defined. For a real world agent in state $\psi$ there will generally be available unitaries $U_1$ and $U_2$ for which it’s a matter of arbitrary definitional choice whether $U_1$ or $U_2$ produces higher Born-weighted mean utility.
There’s a further practical problem, which isn’t apparent in simple models of many-worlds experiments but is a serious worry in realistic applications. To be a rigorous Born-weight mean utilitarian in the real world, one must allow for the possibility of small Born weight branches with extreme negative or positive utility. The mean Born-weighted utility of a bet that, with Born weight close to $1$, involves small utility gains or losses, is radically altered if it also creates a $10^{-25}$ Born weight branch of utility $-10^{30}$. Now, the Deutsch-Wallace-Savage arguments imply no bounds on agents’ utility functions. It seems unlikely that any [*a priori*]{} argument can supply one, since pure rationality imposes no bound on utility functions – and in practice, for example, there seems to be no generally agreed lower bound on the utility cost assigned to the destruction of the Earth or similar catastrophes[@akrisk]. A rigorous real world Born-weight mean utility calculation thus typically requires very careful analysis of small weight branches. In fact, even ensuring that the sum defining the mean utility [*converges*]{} requires careful analysis of small weight branches: consider, for example, the possibility of a set of branches of weight $2^{-n}$ and utility $-3^n$ for all integers $n \geq N$.
Practically speaking, the best that real world agents are likely to be able to do is first simplify their model, by excluding events below some weight threshold, and then estimate a Born-weighted mean utility within that model — with no assurance that the estimated mean utility is close to the true mean utility (if indeed the latter exists).[^6] This needs emphasising, since much of Wallace’s case against alternative rational strategies is based on the claim that they are ill-defined or impractical or both. Actually, as we will see, alternative strategies can sometimes be rather [*better*]{} defined and [*more*]{} practical than Born-weight mean utilitarianism.
Everettian many-worlds rationality reconsidered
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### General remarks on life in a multiverse
It seems [*prima facie*]{} surprising to claim that mathematical analysis could show that Born-weight mean utilitarianism, or any other strategy, is the unique rational way of optimizing the welfare of one’s own, and other people’s, many future selves in a multiverse. After all, human parents are faced with the not entirely disanalogous question of how to take into account the welfare of their genetic descendants in (most of us assume) a single world, and it’s a notoriously complex problem. People generally care not only about their descendants’ present welfare, but also about their expected future welfare after our death. They can, and sometimes do, frame guiding rules of thumb to arbitrate between competing claims on their resources – for instance, to divide their estate equally among their children, or to divide it according to their need. They take into account their children’s relationships with one another, with others, and with society. They tend to care about immediate descendants more than distant ones, in a way that generally follows no well-defined formula. Evolutionarily developed instincts also impel a more general concern for our genes and those of the species. This concern probably cannot be precisely codified, but we can often find principles with which they are roughly aligned and which roughly characterise the behaviour they motivate. For instance, some species’ instinctive behaviour might be roughly modelled as aiming to maximise an individual’s expected number of descendants after $10^2$ years. Some humanists’ aims might be modelled as aiming to maximise the survival probability of the human race (and its genetic successors) over the next $10^9$ years.
Some of these principles require impossible calculations to implement precisely, but can nonetheless legitimately be regarded as rational aims. If we adopt them, we commit ourselves to trying to satisfy them as best we can. In general, they imply conflicting courses of action. No one, I think, would seriously claim that any one of them is uniquely rationally preferable to all the others. We just make decisions as best we can, imperfectly guided by logic, sometimes perhaps trying our best to optimise quantities we know we cannot properly calculate. And we always did: before we were capable of rational reflection, evolution equipped us to muddle through, sometimes following one rule of thumb, sometimes another. That’s life. Why should we expect evolution or rationality to have equipped us any better when faced with the bewilderingly underdetermined imperative to care about our and everyone else’s quantum descendants in a hypothetical multiverse?
### Alternative Born-weight-sensitive strategies
Suppose, for the sake of the discussion, that we can somehow ignore the fuzziness of the ontology. Suppose that we have an agent faced with a finite number of choices $j$, each of which will create quasiclassical branches (although not a unique quasiclassical branching structure) with well-defined utilities $U^j_i$, in such a way that the set $S^j_i$ of branches with the same utility $U^j_i$ has a well-defined total Born weight $p^j_i$, and that the sums $\mu^j = \sum_i p^j_i U^j_i$ are finite.[^7]
Consider again some of the strategies listed in Sec. \[strategies\]. The $x$-percentile utilitarian, for $0<x<100$, always has a well-defined strategy, as does the future self democrat. The future self elitist and Price-Rawlsian’s strategies are defined provided that $\max_{j} \sup_i (U^j_i )$ and $\max_j \inf_i (U^j_i )$, respectively, are defined. These will always hold true if the indexing set $I \ni i$ is finite. They need not hold true if the branch utilities are unbounded above or below (possibilities which are not usually considered by Everettians, and which perhaps might be excluded by assumption, but possibilities nonetheless).
As a practical matter, unless low Born weight extreme utility branches can be excluded, the future self elitist and Price-Rawlsian may have difficulty optimising their strategies, even if an optimal strategy exists, since calculating $\sup_i (U^j_i ) $ or $\inf_i (U^j_i )$ requires analysing low Born weight branches that realise, or converge towards, the extreme utility values. This is also be a problem – which may be easier or harder, depending on the details – for the mean utilitarian. Generically, it should not be a significant problem for the $x$-percentile utilitarian (for most $x$, say $1 < x < 99$), assuming the utility function is generically well-behaved over the range, since the utility at the $x$-th percentile is then relatively insensitive to small perturbations of $x$, and so the calculation is relatively insensitive to the details of low Born weight extreme utility branches. It should also generally not be a problem for a purely self-concerned future self democrat, who would generally hope to be able to attain a majority decision without counting the votes from low Born weight extreme utility branches.[^8]
### Some other strategies
The [*Gell-Mann–Hartle aesthete*]{} fixes a particularly pretty quasiclassical consistent set $S$, which she uses to define a way of counting branches containing her future selves.[^9] Her quantum ontology is Everettian: she agrees that her selected set has no fundamental physical significance. However, she thinks one needs [*some*]{} way of weighting future selves and that this one is as rationally defensible as Born-rule-weighting or any other, and more aesthetically pleasing.
The [*value teleologist*]{} fixes a particular cosmological final density matrix $\rho_f$, whose spectrum does not include zero. In considering whether or not to accept a generalized bet on a quantum experiment, or indeed making any decision dependent on a quantum event, he uses pre- and post-selection, with some standard theory of the initial cosmological conditions defining the initial state $\rho_i$, and with $\rho_f$ defning the final state, in order to calculate the probabilities of the future worlds corresponding to the possible outcomes.[@aharonovetal; @gmhtwotime] He bets as if these were the actual probabilities. This is not because he believes they are – he believes in deterministic unitary quantum mechanics and so doesn’t think probabilities are fundamental, and in any case his physical theory is a standard cosmological theory with initial state $\rho_i$ and no post-selection on $\rho_f$. However, for aesthetic or existential reasons, his interest in future events is conditional on the chosen final state post-selection.
### Wallace’s rationality postulates
We noted already that branch weight, branch macrostate, branch microstate, and reward are all only fuzzily defined in Wallace’s ontology[@wallacevolone], and that this gives strong reason to doubt Wallace’s ordering axiom $R1$. It casts doubt too on whether the availability axioms $A3-A5$ and the rationality axioms $R3$, $R5$ even have a precise definition.
Wallace’s diachronic consistency axiom, $R2$, is violated by the $x$-percentile utilitarian strategy, among others. Now, to be fair, one can find examples where the two conflict which illustrate some motivation for diachronic consistency. Consider the possibility of being offered $N$ dollars per unit time to stand in a radiation field, with a risk $p$ of lethality per unit time. An $x$-percentile utilitarian who considers this offer will generally find their response depends on the timescale over which they regard their decisions as binding: it could seem a good offer considered as valid for the next second, and then good again for each successive second, but a bad offer if they have to make a single decision about whether to accept for the next hour.
Yet, even in this rather unusual example, the motivation for $x$-percentile utilitarians is still clear when $x$ is close to $0$ or $100$, and their actual strategy is intended as a practical approximation to their ideal strategy of Price-Rawlsianism or future self elitism. The Price-Rawlsian will decline unless the total risk is zero; the future self elitist will accept unless the survival probability is zero. Note too that even here $x$-percentile utilitarianism [*is*]{} a well-defined strategy once a timescale for decisions is fixed. In the more normal circumstance of separated discrete decisions, $x$-percentile utilitarianism seems both rationally defensible and practical, which suggests that the diachronic consistency axiom is less rationally compelling than Wallace argues.
Another reason to doubt $R2$, it seems to me, is that, pace Wallace’s comment[@wallacevoltwo] –
> “In the presence of widespread, generic violation of diachronic consistency, agency in the Everett universe is not possible at all.”
– diachronic consistency actually [*is*]{}, strictly speaking, generically violated in real world decisions. A Savagean or Wallacean rational agent, recall, has to be equipped with a utility function as well as a probability measure for outcomes. Rationality is silent on the precise form of the utility function. If we have one, it reflects our current values. These generally change over time, as we do, partly as a result of decisions we have previously taken, whose outcomes affect us in ways we cannot reliably predict beforehand: our own natures are too complex and too opaque to us, and we also change in response to our environment, which is also complex and unpredictable. The best it seems to me that one might hope to say of diachronic consistency in real world decisions is that pretty often, in the short term, it approximately holds — which clearly isn’t a strong enough assumption to prove an interesting decision theoretic representation theorem.
Elga’s proposal that Everettians might have a rational preference for future self diversity[@wallacevoltwoelga] also seems pertinent here, as does the case for rationally preferring future society diversity. (Why [*not*]{} exploit the scope for political compromise by causing society to evolve in different ways along different branches?) In both cases, it seems to me, contra Wallace[@wallacevoltwoelga], diachronic consistency can be rationally violated. I can consistently believe now that it’s a good thing that the global state should include future copies of me as a king and a beggar, while knowing that, if I ever find myself a beggar, I would strive to become a king if I could.[^10] From the perspective of my future beggar self, the unpleasantness of finding that [*he*]{} is the beggar outweighs the satisfaction of knowing that diversity was achieved. From my present perspective, the prospect of diversity nonetheless remains appealing.
As Wallace himself notes earlier in his discussion[@wallacetwodictates]:
> $\ldots$ to make a copy of myself and send him off to do a dangerous or disagreeable task – and $\ldots$ to take actions designed to prevent him shirking that task $\ldots$ is not [*irrational*]{}.
Indeed – and this remains true if the task, for which I have a strong present desire, is to ensure future self diversity. The fact that my future selves will never interact makes no difference to the rational justification.
Turning briefly to other postulates:
- Microstate indifference, $R3$, can be violated by value teleologist strategies, among others.
- Continuity, $R4$, is violated by $x$-percentile utilitarian strategies, among others.[^11]
- Branching indifference, $R5$, is violated by Gell-Mann–Hartle aesthete strategies, among others.
### Summary
Wallace argues that strategies other than mean utilitarianism turn out, on closer inspection, either to be not rigorously defined, completely impractical, or to violate criteria such as diachronic consistency that allegedly define the very essence of rationality. The last two claims — impracticality and violation of rational essentials – surely require mathematical underpinning and justification, if they are to have any possible relevance to what is presented as a rigorous mathematical argument. For example, an account of practicality needs some complexity criteria for rational agent computations: one could then at least discuss the empirical justification for the proposed criteria and whether and when they actually distinguish mean utilitarianism from other strategies. Similarly, if one accepts that diachronic consistency is generically violated and sometimes grossly violated in the real world, an account of its role in decisions needs to quantify and compare the degree of violation implied by different strategies in different circumstances. At present, though, the arguments for diachronic consistency and those concerning practicality rest only on very debatable verbal intuitions.
As for lack of rigorous definition, there seems to be a danger of a double standard, whereby the fuzziness of the ontology is used to point out difficulties for alternative strategies (though in fact it also causes difficulties for mean utilitarianism), while the arguments for mean utilitarianism are justified in the context of toy models in which a precise definition of a branching structure can be found (in which case many strategies other than mean utilitarianism can be precisely defined). The case has not been made that mean utilitarianism is well-defined or practical in Wallace’s fuzzy Everettian ontology, in which the mean utility of a strategy can at best only be fuzzily defined. One can imagine examples in which it either fails to be finite or is impractical to estimate – and it seems hard to exclude the possibility that these features often apply in the real world. One can also easily construct examples in which other strategies are easier either to approximate or to implement precisely.
Wallace’s rationality postulates, likewise, are hard to motivate in Wallace’s fuzzy Everettian ontology, where they are ultimately intended to apply, but where they are difficult, perhaps impossible, to define precisely. They generally appear, in any case, possible but uncompelling guides for rational agents. Where defined and practical, mean utilitarianism is certainly a rationally defensible strategy, with some mathematically convenient properties. But, like other critics[@pricevol; @albertvol], I am far from persuaded that, if I were an Everettian, I should or would be a Born-weighted mean utilitarian.
Against subjective uncertainty
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One of the stranger claims in the recent Everettian literature is the suggestion, first made by Saunders [@saundersprob; @saundersvol; @saunderswallace; @wallaceepiq], that Everettian quantum theory, although deterministic, nonetheless has a natural probabilistic interpretation that can be found not by amending the theory or by adding further postulates, but simply by – somehow – analysing the experience and linguistic usages of agents, that is, creatures like ourselves, in an Everettian universe. In support of this claim are offered highly technical and controversial arguments concerning the philosophy of language. It seems to me simply a mistake, an exercise in wish fulfilment, to think that anything of significance to fundamental physics could turn on such questions, as though waving the magic wand of linguistic philosophy over a unitarily evolving state vector could somehow conjure up a probability measure and a sample space.[^12]
Consider Wallace’s succinct summary[@wallaceepiq] of the argument:
> “\[The argument for subjective uncertainty\] may be summarised as follows: in ordinary, non-branching situations, the fact that I expect to become my future self supervenes on the fact that my future self has the right causal and structural relations to my current self so as to count as my future self. What, then, should I expect when I have two or more such future selves? There are only three possibilities:
>
> 1. I should expect abnormality: some experience which is unlike normal human experience (for instance, I might expect somehow to become both future selves).
>
> 2. I should expect to become one or the other future self.
>
> 3. I should expect nothing: that is, oblivion.
>
> Of these, (3) seems absurd: the existence of either future self would guarantee my future existence, so how can the existence of more such selves be treated as death? (1) is at least coherent – we could imagine some telepathic link between the two selves. However, on any remotely materialist account of the mind this link will have to supervene on some physical interaction between the two copies – an interaction which is not in fact present. This leaves (2) as the only option, and in the absence of some strong criterion as to which copy to regard as “really” me, I will have to treat the question of which future self I become as (subjectively) indeterministic.”
This is a false trichotomy. Consider an (obviously simplified) Everettian description of an experiment in which an agent Alice, initially in brain state $\ket{0}_A$, observes a system in a quantum superposition $\sum_{i=1}^2 c_i \ket{i}_S$, where $\ket{1}_S$ and $\ket{2}_S$ correspond, say, to the up and down states of a spin $1/2$ particle, and becomes entangled: $$\label{spinexpt}
\ket{0}_A \sum_{i=1}^2 c_i \ket{i}_S \rightarrow
\sum_{i=1}^2 c_i \ket{i}_A \ket{i}_S \, .$$ Here $\ket{i}_A$ is Alice’s brain state after observing the system in state $i$, for $i = 1,2$.
Now, as an aside, we actually [*should*]{} take possibility (3) seriously, for two reasons. First, our conclusions ought to be based on empirical evidence rather than prejudice. We do not know that Everettian quantum theory is actually correct; we do not have a good theory of how consciousness is attached to quantum states; we do not know that we or any other agents have ever been in a superposition of macroscopically distinct brain states. We thus do not know whether, if we were able to place an agent in such a superposition, they would experience anything — nor, if so, what. Second, there’s a coherent view of Everettian quantum theory in which we are continually being replaced by multiple copies of future selves. On this view, even if we assume that superposed selves have individual experiences, [*we*]{} will experience nothing in future (though our various future selves will).
The more immediately pertinent point, though, is that if we do take Everettian quantum theory seriously, it says, indeed, that Alice becomes entangled in a macroscopic superposition. A coherent way of describing this, which respects the link between brain states and mind states, is that just as materially she becomes several future selves, her mind becomes several [*disjoint, non-interacting*]{} future minds, with no telepathic link: i.e. option (1) without Wallace’s misleading gloss.
The one description that seems obviously wrong, given the rules of the game Wallace sets out, is option (2): this really [*is*]{} an account of mind that supervenes on something not present in the physics, namely a probabilistic evolution law taking brain state $\ket{0}_A$ to one of the states $\ket{i}_A$.
The dangers of attaching some fuzzy theory of experience to Everettian quantum theory provoke two comments:
First, the fact that we don’t have a good theory of mind, even in classical physics, doesn’t give us a free pass to conclude anything we please. That way lies scientific ruin: [*any*]{} physical theory is consistent with [*any*]{} observations if we can bridge any discrepancy by tacking on arbitrary assumptions about the link between mind states and physics. We should, rather, be all the more cautious and tentative in offering any conclusion.
Second, the fact that at present no theory of mind can be expressed [*purely*]{} mathematically doesn’t remove the obligation to strive to express one’s ideas in mathematics [*as far as possible*]{}. Adorning Everettian quantum theory with extra assumptions expressed in words – for instance, as arguments in linguistic philosophy – without equations doesn’t alter the fact that one’s making extra assumptions: it merely makes them more vaguely expressed.
Consider[^13] Saunders’ exposition[@saundersvol]:
> Consider a simple example. Alice, we suppose, is about to perform a Stern-Gerlach experiment; she understands the structure of the apparatus and the state preparation device, and she is convinced EQM is true. In what sense does she learn, post-branching, something new? The answer is that each Alice, post-branching, learns something new (or is in a position to learn something new) – each will say something (namely, “I know the outcome is spin-up (respectively, spin-down), and not spin-down (respectively, spin-up)”) that Alice prior to branching cannot say. It is true that Alice, prior to branching, knows that this is what each successor will say – but still she herself cannot speak in this way. The implication of this line of thought is that, appearances notwithstanding, prior to branching Alice does not know everything there is to know. What is it she does not know? I say “appearances notwithstanding” for of course in one sense (we may suppose) Alice does know everything there is to know: she knows (we might as well assume) the entire corpus of impersonal, scientific knowledge. But what that does not tell her is just which person she is – or where she is located – in the wave-function of the universe.
But equation (\[spinexpt\]) suggests there is no meaning to this question before the experiment.[^14] Nothing in the mathematics corresponds to “Alice, who will see spin up” or “Alice, who will see spin down”. On the left we have “Alice, before the experiment”; on the right we have “Alice, who has seen spin up” and “Alice, who has seen spin down”. If one wants to postulate an “Alice, who will see spin up”, well, one can – but one should then include her in the mathematics. One could, for instance, start with a postulate of the form:
> “(P) the probability that A’s mind ends up believing that spin is up is $| c_0 |^2$ and the probability that A’s mind ends up believing that spin is down is $ | c_1 |^2$.”
This – Albert and Loewer’s “single mind view” [@almanyminds] – gives only one sentient future Alice. To introduce a collection of present Alices who in future will experience each of the different possible experimental outcome, one could, instead, follow Albert and Loewer in postulating a continuum of Alice minds of which a proportion $| c_0 |^2$ will see spin up. One could, in short, adopt the many-minds interpretation. I am not persuaded that there is a legitimate alternative formulation of Saunders’ account.
Further comments on counting descendants {#countdesc}
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Wallace places great stress on the fact that there is no unique natural definition of a quasiclassical branch in an Everett universe, and so no way of counting the number of branches with any given feature. For example, a naive analysis of a quantum experiment with three possible outcomes might suggest that a single branch, pre-experiment, divides into three, post-experiment. But this, Wallace stresses, neglects the fact that quantum interactions take place very frequently in time and densely in space, outside our control. A careful attempt to quantify quasiclassical branches would show many branches splitting into many more during the lifetime of the experiment; however, there is no unique natural definition that would allow us to pin these numbers down. Hence, it is argued, there is no way of implementing the naive idea of using branch counts to define a rational strategy – an approach which would, if it worked, be a coherent alternative to the Born-rule-dependent strategy, and so refute the claim that the latter is the unique rational strategy.
There is indeed no known natural way of characterising and counting branches. It is worth reconsidering, however, whether there may nonetheless be a natural way for an agent about to observe an experiment to characterise and hence count his descendants, by considering their memory states after the experiment.
Consider first a simple model of an observer, apparatus and quantum state which evolve unitarily during an experiment so that $$| O_0 > | A_0
> | Q_0 > \rightarrow \sum_{j=1}^3 a_j | O_j > | A_j > | Q_j > \, ,$$ where the first state is at time $0$ and the second at time $t$, the state $ | A_j > $ is the apparatus state registering outcome $j$ and $ | O_j > $ is the observer state having observed the apparatus registering outcome $j$. The agent in state $ | O_0 >$ can reasonably say that he will have three successors $ | O_j >$ at time $t$, corresponding to his three future distinct brain states.
Now consider a more detailed model in which we include an environment, and suppose that $$\label{modeltwo} | O_0 > | A_0 > |
Q_0 > | E_0 > \rightarrow \sum_{j=1}^3 \sum_{i=1}^{n_j}
a_{ij} | O_j > | A_{ij} > | Q_{j}> | E_{ij} > \, .$$ Again, the sum over $j$ represents the three possible observer states. The sums over $i$ represent decompositions into orthogonal quasiclassical branches, and the number of terms $n_j$ in each sum depends on an arbitrary choice of definition of quasiclassical branch from among many possible definitions. Since this is supposed to be a model of the same experiment, we have that $ \sum_i | a_{ij} |^2 = | a_j |^2 $ for $j = 1,2,3$. The agent in state $| O_0 >$ has no unique natural way of characterising the branches. However, he could consistently view each of the three components, containing the state $| O_j >$, $j=1,2,3$, as representing precisely one successor.[^15] After all, he is interested in his successors’ welfare, and this is determined by their mind states; at this point in the analysis, at least, it is not affected by the state of the rest of the universe.
It might be objected that we still have not taken sufficiently into account the pervasiveness of environment-induced quantum interactions, which will presumably also be taking place within the agent’s brain during the experiment. A more detailed model still would replace the sums on the right hand side of (\[modeltwo\]) by sums including at least small components of a variety of different agent mind states, corresponding to different brain states that arise through zapping by stray cosmic rays and other quantum effects.
A related objection is that this way of counting successors leads to ambiguities when sequences of experiments are carried out. Suppose that the agent will carry out a second experiment, with two possible outcomes, if he observes outcome $1$ in the first experiment, but not otherwise. After the first experiment but before the second, it appears that he has three successors, whose welfare he should value equally. After the second experiment, it appears that he has four successors, whose welfare he should again value equally. But since two of these descend from one of the original successors, and each “inherit” any resources he “bequeaths” to that successor, the two ways of counting lead to different allocation strategies – i.e. they disagree on which “bets” he should be willing to accept on the experiment.
To these objections, however, the agent could make several responses.
First, that his policy is not incoherent, but merely so far incompletely specified. In the case of the two experiments, he does care equally about the welfare of his three successors between the experiments, and he also cares equally about the welfare of his four successors after the experiments. To formulate a more precise policy, he would need to set out some way of trading off the welfares of different successors at different times: for example, by summing the time-integrated welfares over each distinct life-time segment. That this may become complicated doesn’t imply that the aim is not rational. Indeed, we face very similar problems in worrying about the well-being of our genetic descendants. It is perfectly rational to value the welfare of all of your children equally, and also perfectly rational to value the welfare of all of your as yet unborn grandchildren equally, in both cases ceteris paribus. However, finding a rational asset allocation policy that respects both these preferences may require some further policy decisions, complicated calculations and predictions of uncertain future events.
Second, that he would indeed take into account the welfare of all his successors, including those whose mind states differ because of environmentally induced interactions, if he could. Here again, he can maintain that he has a rational policy in principle, albeit one that he cannot fully implement it in practice because of the impracticality of carrying out the relevant calculations. And again, he can say that the latter caveat does not detract from the rationality of his goal.
Third, that a consistent strategy for weighting his concern for the welfare of successors could be defined by considering the branching structure recorded in their own memory states. In the example above, successors who experience two experiments in succession remember that fact, and this distinguishes them from successors who experience only one experiment. It’s logically consistent – and not obviously any more absurd than any Born-weight-dependent many-worlds strategy – to assign the former caring weight $1/6$ and the latter caring weight $1/3$.
To these responses, Everettians might in turn object that there is no natural way, even in principle, of characterising and counting all the possible mind states of successors of an agent exposed to real-world environmental interactions. But if the Everettian case eventually turns on [*this*]{} point, then the objection to rational strategies based on counting successors ultimately arises from an intrinsic vagueness in the quasifunctionalist theory of mind attached to the quantum formalism by Wallace et al., not, as claimed, from the vagueness in the notion of a quasiclassical branch. It seems a most uncomfortable defence of a purportedly fundamental theory to say that it is not well enough developed for us to be able to assess whether or not one of the key arguments advanced in its favour is valid.
Further Comment on physical laws of rational compulsion {#rationalconstraint}
=======================================================
What could it possibly mean to believe that the [*laws of physics per se rationally compel*]{} a particular behaviour for rational agents in a branching multiverse? The idea here, to be clear, is not merely the truism that the laws of physics imply significant facts about the world which rational agents might, or even must, sensibly take into account. It is that there are basic postulates, on an equal footing with other physical laws, that state by fiat that a particular type of behaviour is rationally compulsory for rational agents. I don’t think I know what this can mean – the idea of such a law isn’t consistent with my understanding of either physics or rationality – but the idea is definitely in play in some discussions of many-worlds theories in this volume. Papineau[@papineauvol] proposes an axiom of this type, and Greaves-Myrvold[@greavesmyrvoldvol] consider how to (purportedly) confirm theories including such axioms. My impression is that many Everettians share Greaves’ view[@greaves] that resorting to such a postulate would be, at least, an adequate fall-back should Wallace’s[@wallacevoltwo] and other arguments not hold up.
Here’s a point that seems not to have been considered in the Everettian literature. If we [*were*]{} to take seriously the idea that physical axioms can rationally compel rational beings to act in a certain way – by fiat, without further justification – then we must also take seriously the possibility that these rationally compelling axioms can take unfamiliar forms. For instance, in our branching universe, there’s no reason to restrict to axioms that require rational preferences to be given by the ordering of values of expressions of the form $ \sum_i p_i U_i $, where $U_i$ is the agent’s utility for the outcome on branch $i$ and the $p_i$ are positive branch weights satisfying the Kolmogorov axioms. There’s nothing [*logically*]{} inconsistent about postulating laws with negative or complex $p_i$, or preference orderings given by general joint functions $ f( { p_i } , { U_i } )$, or indeed any other mathematical structure one cares to dream up.[^16]
Such laws would generally violate Savage’s axioms and perhaps other cherished intuitions about rational behaviour. But once one enters the strange game of postulating [*physical*]{} laws [*defining*]{} rational behaviour in multiverses, one needn’t restrict one’s postulates to intuitions developed in an attempt to provide a foundation for decision theory in a single chancy universe. Everettians who miss this point seem to me like hypothetical seventeenth century theorists who learn Hooke’s law, come up with the idea that one can postulate abstract physical force laws that define forces between objects unmediated by springs, but then still maintain that these laws necessarily have to set force proportional to separation. Their boldness is inconsistently selective: an abstract law need not be constrained by the details of the concrete model that inspired it.
Of course, Everettians who think it makes sense to postulate laws of rational compulsion still have the option of basing their postulates on one-world probability theory, and specifically on optimising Born-rule-weighted average utility. But one needs to be clear that, [*prima facie*]{}, this is an arbitrary choice from a very large range of possibilities. Along with everything else in this peculiar game, that choice seems to lack justification. Moreover, as the above analysis of Greaves-Myrvold’s account of confirmation applied to the weightless universe shows, allowing arbitrary choices of laws of rational compulsion means not only that many mutually inconsistent choices can be postulated in the same multiverse but also that each of them can be, by their own lights, confirmed.
A possible empirical distinction between many-worlds and one-world quantum theory
=================================================================================
Finally, suppose, notwithstanding all the arguments above, that we arrive at an Everettian theory that, while perhaps ad hoc and unattractive, is coherent – for example, some version of the many-minds interpretation[@almanyminds]. It is generally believed that, without very advanced technology which allows the re-interference of macroscopically distinct branches, such a theory will necessarily be empirically indistinguishable from Copenhagen quantum theory.
The following argument against this conclusion relies on anthropic reasoning and also on the hypothesis that species may evolve a consistent preference for or against higher population expectation over higher survival probability. Anthropic reasoning is notoriously tricky to justify, and we may anyway not necessarily have evolved demonstrable consistent preferences one way or the other, so the argument may not necessarily have practical application. Nonetheless, it does show in principle that evolutionary evidence could make many-worlds theories more or less plausible.
Consider a simple model of two species $A$ and $B$, both of which begin with population $P$ and are offered, each year, the option of doing something that depends on a quantum event and carries a $0.5$ probability of extinction and a $0.5$ probability of trebling the species population. Suppose that, if they reject the option, their population remains constant, as it does in between these decisions. Species $A$ is risk-averse, and so always declines the option. Species $B$ is risk-tolerant, and instinctively driven to maximise expected population, and so always accepts.
Now let $N$ be a large integer. After $N$ years, if one-world quantum theory is correct, species $A$ will have population $P$, and species B will have either population $0$ (with probability $ ( 1 - ( \frac{1}{2} )^N )$) or population $3^N$ (with probability $ ( \frac{1}{2} )^N$). In other words, species $B$ will almost surely be extinct. If these are the only two species, and you are alive in the $N$-th year, almost certainly you belong to species $A$.
If many-worlds quantum theory is correct, species $A$ still has population $P$ in all branches. Species $B$ has population $0$ in branches of total Born weight $ ( 1 - ( \frac{1}{2} )^N )$, and population $3^N$ in branches of total Born weight $ ( \frac{1}{2} )^N$. Now, if anthropic reasoning is justifiable here, and you are alive in the $N$-th year, almost certainly you belong to species $B$. (There are $( \frac{3}{2} )^N$ times as many minds belonging to species $B$ as to $A$ after $N$ years.)
In other words, there is a sense in which long-run evolutionary success is defined by different measures in one-world and many-worlds quantum theory. If anthropic reasoning were justifiable, then one could in principle infer whether one-world or many-worlds quantum theory is likelier correct by seeing whether one belongs to a Born-weighted expected population maximising species or to a risk-averse species that seeks to maximise its Born-weighted survival probability. Readers may thus wish to consider whether their species has evolved a coherent strategy of either type.[^17]
I am very grateful to Jonathan Barrett for many thoughtful and constructive comments on and criticisms of a preliminary version of the manuscript, as well as other very valuable conversations, and also to Hilary Greaves and David Wallace for patiently tolerating and helpfully engaging with my critical probing. Thanks too to David Albert, Harvey Brown, Jeremy Butterfield, Chris Fuchs, Lucien Hardy, Graeme Mitchison, Wayne Myrvold, David Papineau, Huw Price, Simon Saunders, Tony Short, John Sipe, Rob Spekkens and Tony Sudbery for some valuable conversations. This research was partially supported by a grant from The Foundational Questions Institute (fqxi.org) and by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
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[^1]: I thank Jonathan Barrett for this point and this example.
[^2]: There are more general possibilities.
[^3]: See Appendix \[rationalconstraint\] for further discussion.
[^4]: That all ambiguities in total weights of quasiclassical outcomes are necessarily very small seems plausible and is what Wallace expects. Given the level of conceptual imprecision in discussing the emergence of quasiclassical structures from unitary quantum theory, though, it is hard to be certain even of this.
[^5]: And to run such an argument, Wallaceans would, inter alia, need to find suitable precise definitions of “closer” and “better off”.
[^6]: Given the fuzzy ontology, we should more accurately say “to any possible assignment of the imprecisely defined value of the true mean utility”. For the sake of readability, we take this qualification as read in what follows.
[^7]: Without these assumptions, the mean utilitarian’s strategy isn’t defined, in which case Wallace’s argument has failed.
[^8]: Of course, in both these last two cases, it [*could*]{} still be a problem if the numbers so conspire.
[^9]: I thank Hans Westman and Ward Struyve for suggesting this example.
[^10]: Or perhaps, considering the uneasiness of crown-wearing heads, vice versa.
[^11]: Wallace[@wallacevoltwo] notes and discusses the special case of the Price-Rawlsian.
[^12]: Incidentally, this issue has also divided Everettians: Papineau[@papineausu] and, at one point, Greaves [@greavessu], have also argued that subjective uncertainty is not to be had in Everettian quantum theory.
[^13]: My remarks here follow Albert’s lucid discussion[@albertvol].
[^14]: The same is true in more realistic models.
[^15]: He is not rationally compelled to accept this view. The point is that in this model successor counting is mathematically well-defined and hence it’s [*possible*]{} to use it to define a rational strategy.
[^16]: Indeed, one could imagine an exotic story about inverted qualia and hence reversing of utilities on some branches, which justifies giving them negative weights. And perhaps, some might argue, the complex quantum amplitudes defining the path integral should be interpreted as directly defining rational constraints on an agent’s preferences for the entire set of paths defining a hypothetical future unitary evolution.
[^17]: Unfortunately, I suspect mine has not.
|
---
abstract: 'It was suggested that light-cone superstring field theory (LCSFT) and matrix string theory (MST) are closely related. Especially the bosonic twist fields and the fermionic spin fields in MST correspond to the string interaction vertices in LCSFT. Since CFT operators are characterized by their OPEs, in our previous work we realized the most important OPE of the twist fields by computing contractions of the interaction vertices using the bosonic cousin of LCSFT. Here using the full LCSFT we generalize our previous work into the realization of OPEs for a vast class of operators.'
author:
- |
[Isao Kishimoto]{}[^1]\
[*High Energy Accelerator Research Organization (KEK),*]{}\
[*Tsukuba 305-0801, Japan*]{}\
[Sanefumi Moriyama]{}[^2]\
[*Graduate School of Mathematics, Nagoya University,*]{}\
[*Nagoya 464-8602, Japan*]{}[^3]\
[*Center for Theoretical Physics, Massachusetts Institute of Technology,*]{}\
[*Cambridge, MA02139, USA*]{}\
date: 'November, 2006'
title: '\'
---
Introduction
============
Search for a fundamental field-theoretical formulation of string theory began in the old days of dual resonance models. However, none of these attempts have been completed yet for the superstring theory.
The light-cone superstring field theory (LCSFT) is the most successful formulation thus far and was first constructed in [@GS; @GSB]. The starting point is the Green-Schwarz action in the light-cone gauge. Out of the action we can construct the free Hamiltonian $H_0$ and two free supercharges $Q_0^{\dot a}, \tilde Q_0^{\dot a}$, which satisfy the supersymmetry algebra: $$\begin{aligned}
\{Q_0^{\dot{a}},Q_0^{\dot{b}}\}
=\{\tilde Q_0^{\dot{a}},\tilde Q_0^{\dot{b}}\}
&=2\delta^{\dot{a}\dot{b}}H_0,
\label{susyalg1}\\
[Q_0^{\dot{a}},H_0]=[\tilde Q_0^{\dot{a}},H_0]&=0.
\label{susyalg2}\end{aligned}$$ The interaction terms are added to these charges so that the total charges satisfy the same supersymmetry algebra order by order. Namely, we construct the interaction terms by replacing the free charges $H_0, Q_0^{\dot a}, \tilde Q_0^{\dot a}$ in , by the full order charges $$\begin{aligned}
H&=H_0+g_sH_1+\cdots,\\
Q^{\dot a}&=Q^{\dot a}_0+g_sQ^{\dot a}_1+\cdots,\\
\tilde{Q}^{\dot a}&=\tilde{Q}^{\dot a}_0+g_s\tilde{Q}^{\dot a}_1+\cdots,\end{aligned}$$ and determining the interaction terms order by order. The result for the first order interaction terms is $$\begin{aligned}
|H_1\rangle_{123}
&=Z^i\bar Z^jv^{ji}(Y)|V\rangle_{123},
\label{SFTH1}\\
|Q_1^{\dot{a}}\rangle_{123}
&=\bar Z^is^{i\dot{a}}(Y)|V\rangle_{123},\\
|\tilde Q_1^{\dot{a}}\rangle_{123}
&=Z^i\tilde s^{i\dot{a}}(Y)|V\rangle_{123}.\end{aligned}$$ Here $|V\rangle_{123}$ is the kinematical three-string interaction vertex constructed by the overlapping condition and $Z^i$ $(\bar Z^i)$ is the holomorphic (anti-holomorphic) part of the bosonic momentum at the interaction point, whose divergence is regularized: $$\begin{aligned}
\Bigl(P^{(1)i}+\frac{1}{2\pi\alpha_1}X^{(1)i\prime}\Bigr)(\sigma_1)
|V\rangle_{123}
\sim\frac{1+ i\epsilon(\sigma_{\rm int}-\sigma_1)}
{4\pi\sqrt{-\alpha_{123}}
\sqrt{|\sigma_{\rm int}-\sigma_1|}}Z^i|V\rangle_{123},
\label{Z}\end{aligned}$$ with $\alpha_r=p^+_{r}$, $\alpha_{123}=\alpha_1\alpha_2\alpha_3$ and $\sigma_{\rm int}$ being the interaction point. We take the range of $\sigma_1$ to be $-\pi\alpha_1\le\sigma_1\le\pi\alpha_1$, which implies $\sigma_{\rm int}=\pi\alpha_1$ for $\sigma_1\sim\pi\alpha_1$ and $\sigma_{\rm int}=-\pi\alpha_1$ for $\sigma_1\sim-\pi\alpha_1$, respectively. $\epsilon(x)$ is the step function and defined by $\epsilon(x)=+1$ ($-1$) for $x>0$ ($x<0$). Similarly, $Y^a$ is the regularization of the fermionic momentum at the interaction point: $$\begin{aligned}
\lambda^{(1)a}(\sigma_1)|V\rangle_{123}
\sim\frac{1}{4\pi\sqrt{-\alpha_{123}}
\sqrt{|\sigma_{\rm int}-\sigma_1|}}Y^a
|V\rangle_{123},
\label{Y}\end{aligned}$$ and the prefactors $v^{ji}(Y)$, $s^{i\dot{a}}(Y)$ and $\tilde{s}^{i\dot{a}}(Y)$ are functions of $Y^a$ and were originally given in complicated forms. Here we find that these functions can be simply put into the hyperbolic functions: $$\begin{aligned}
v^{ji}(Y)&=\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij}
=\delta^{ij}+\frac{1}{2!}({\!\not\hspace{-0.5mm}Y}^2)^{ij}
+\frac{1}{4!}({\!\not\hspace{-0.5mm}Y}^4)^{ij}+\frac{1}{6!}({\!\not\hspace{-0.5mm}Y}^6)^{ij}+\frac{1}{8!}({\!\not\hspace{-0.5mm}Y}^8)^{ij},
\label{v}\\
s^{i\dot a}(Y)&=\sqrt{-\alpha_{123}}\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i}
=\sqrt{-\alpha_{123}}\Bigl[({\!\not\hspace{-0.5mm}Y})^{\dot{a}i}
+\frac{1}{3!}({\!\not\hspace{-0.5mm}Y}^3)^{\dot{a}i}+\frac{1}{5!}({\!\not\hspace{-0.5mm}Y}^5)^{\dot{a}i}
+\frac{1}{7!}({\!\not\hspace{-0.5mm}Y}^7)^{\dot{a}i}\Bigr],
\label{s}\\
\tilde{s}^{i\dot a}(Y)
&=i\sqrt{-\alpha_{123}}\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}}
=i\sqrt{-\alpha_{123}}\Bigl[({\!\not\hspace{-0.5mm}Y})^{i\dot{a}}
+\frac{1}{3!}({\!\not\hspace{-0.5mm}Y}^3)^{i\dot{a}}+\frac{1}{5!}({\!\not\hspace{-0.5mm}Y}^5)^{i\dot{a}}
+\frac{1}{7!}({\!\not\hspace{-0.5mm}Y}^7)^{i\dot{a}}\Bigr],
\label{tildes}\end{aligned}$$ where we have constructed gamma matrices with spinor indices $\hat\gamma^a$ out of the standard gamma matrices with the vector indices $\gamma^i$ using the triality of $SO(8)$ and defined ${\!\not\hspace{-0.5mm}Y}$ as $$\begin{aligned}
{\!\not\hspace{-0.5mm}Y}=\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a\hat{\gamma}{}^a
=\begin{pmatrix}0&{\!\not\hspace{-0.5mm}Y}_{i\dot a}\\
{\!\not\hspace{-0.5mm}Y}_{\dot{a}i}&0\end{pmatrix},
\label{Yslash}\end{aligned}$$ with $\eta^*=e^{-\frac{i\pi}{4}}$. Note that the indices of the functions in – are consistent because cosh is an even function while sinh is an odd one. For more details of the prefactors, see Appendix A. As pointed out in [@GS], these quantities satisfy the following Fourier identities: $$\begin{aligned}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij}
&=\left(\alpha_{123}\over 2\right)^4\int d^8\phi\,
e^{{2\over \alpha_{123}}\phi^aY^a}
\bigl[\cosh\!\not\hspace{-1mm}\phi\bigr]^{ji},
\label{fourier1}\\
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i}
&=-i\left(\alpha_{123}\over 2\right)^4\int d^8\phi\,
e^{{2\over \alpha_{123}}\phi^aY^a}
\bigl[\sinh\!\not\hspace{-1mm}\phi\bigr]^{i\dot{a}},\\
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}}
&=i\left(\alpha_{123}\over 2\right)^4\int d^8\phi\,
e^{{2\over \alpha_{123}}\phi^aY^a}
\bigl[\sinh\!\not\hspace{-1mm}\phi\bigr]^{\dot{a}i},
\label{fourier3}\end{aligned}$$ which will play important roles in our following computation. The program of constructing the interaction terms is successful at the first order, though it is too complicated to proceed to higher orders.
Recently, there is a significant breakthrough in the construction [@DM]. The point is to relate LCSFT to another formulation of the superstring theory known as matrix string theory (MST) [@MSSBS; @DVV]. MST stems from the matrix formulation of light-cone quantization of M-theory [@BFSS] and takes the form of the maximally supersymmetric Yang-Mills theory. To relate MST to the perturbative string, we note that the Yang-Mills coupling constant $g_{\rm YM}$ is related to the string coupling constant $g_{\rm s}$ and the string length $\sqrt{\alpha'}$ by $g_{\rm YM}^{-1}\sim g_{\rm s}\sqrt{\alpha'}$. Hence, the free string limit corresponds to the IR limit and the first order interaction term to the least irrelevant operator. From the requirement of the dimension counting and the locality of the interaction, we expect that the first order interaction term is a dimension three operator constructed essentially out of the bosonic twist fields and the fermionic spin fields. The interaction term of MST is proposed to be [@DVV] $$\begin{aligned}
H_1=\sum_{m,n}\int d\sigma
\bigl([\tau^i\bar{\tau}^j]\Sigma^i\bar{\Sigma}^j\bigr)_{m,n},
\label{MSTH1}\end{aligned}$$ where $\tau^i(z)$ is the excited ${\mathbb Z}_2$ twist field defined as $$\begin{aligned}
\partial X^i(z)\cdot\sigma(0)
\sim\frac{1}{\sqrt{z}}\tau^i(0),
\label{tau}\end{aligned}$$ with $\sigma(z)$ being the elementary ${\mathbb Z}_2$ twist field. Since the holomorphic part of the twist field does not completely decouple from the anti-holomorphic part due to the zero mode contribution, we combine them with the square parenthesis in . $\Sigma^i(z)$ and $\Sigma^{\dot{a}}(z)$ (which will appear later) are the spin fields for the Green-Schwarz fermion $\theta^a(z)$. The indices $m$ and $n$ of the twist fields denote the string bits where the “exchange” interaction takes place.
Now we can find a close analogy between and and between and , if we regard $[\sigma\bar\sigma](z,\bar{z})$ as $|V\rangle_{123}$ and $[\tau^i\bar\tau^j](z,\bar{z})$ as $Z^i\bar{Z}^j|V\rangle_{123}$. Following this analogy between LCSFT and MST, two supercharges of MST were written down explicitly in [@M]: $$\begin{aligned}
Q_1^{\dot{a}}=\sum_{m,n}\int d\sigma
\bigl([\sigma\bar{\tau}^i]\Sigma^{\dot{a}}\bar{\Sigma}^i\bigr)_{m,n},
\quad
\tilde{Q}_1^{\dot{a}}=\sum_{m,n}\int d\sigma
\bigl([\tau^i\bar{\sigma}]\Sigma^i\bar{\Sigma}^{\dot{a}}\bigr)_{m,n},\end{aligned}$$ and the supersymmetry algebra was checked. These arguments of supercharges are consistent with the pioneering argument in [@DVV] and with the relation between LCSFT and MST proposed in [@DM].
After observing the analogy between LCSFT and MST, we would like to establish the correspondence next. Since operators in conformal field theory are characterized by their OPEs, we need to reproduce the OPEs in terms of LCSFT in order to completely confirm the correspondence. In our previous work [@KMT], we assumed the correspondence between the bosonic cousin of LCSFT $$\begin{aligned}
|H_1^{\rm boson}\rangle_{123}=|V^{\rm b}\rangle_{123},
\label{bLCSFT}\end{aligned}$$ and the “bosonic” version of MST [@Rey] $$\begin{aligned}
H_1^{\rm boson}
=\sum_{m,n}\int d\sigma\bigl([\sigma\bar\sigma]\bigr)_{m,n},
\label{bMST}\end{aligned}$$ and reproduced the OPE of the twist field $$\begin{aligned}
[\sigma\bar\sigma](z,\bar z)\cdot[\sigma\bar\sigma](0,0)
\sim\frac{1}{\bigl[|z|^{1/4}(\log|z|)^{1/2}\bigr]^{d-2}},
\label{twosigma}\end{aligned}$$ by computing the corresponding contractions of the string interaction vertices in the bosonic LCSFT. On the LCSFT side, it is well-known that the first order interaction vertex in the bosonic LCSFT takes the same form as that in the full LCSFT, if we drop the prefactors and pick up the bosonic sector in the kinematical interaction vertex $|V\rangle_{123}$. The superscript “b” in denotes that we pick up the bosonic sector. On the MST side, of course there is no nonperturbative justification for the bosonic MST, but the interaction term looks natural considering the correct dimension of the operators $[\sigma\bar\sigma]$ and the correspondence between $[\sigma\bar\sigma](z,\bar{z})$ and $|V\rangle_{123}$.
Since the string interaction vertex describes both the processes of one string splitting into two strings and two strings joining into one string, there are two diagrams which correspond to the OPE . One of them is a tree diagram with two incoming strings and two outgoing strings of the same string lengths, while the other is a one-loop diagram with one incoming string and one outgoing string. (See Fig. 1 and Fig. 2 in the following sections.) For the two realizations of the OPE, the corresponding contractions of the interaction vertices are given by $$\begin{aligned}
&{}_{36}\langle R^{\rm b}|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
|V^{\rm b}\rangle_{123}|V^{\rm b}\rangle_{456},\nonumber\\
&{}_{14}\langle R^{\rm b}|{}_{25}\langle R^{\rm b}|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}
|V^{\rm b}\rangle_{123}|V^{\rm b}\rangle_{456}.
\label{RVV}\end{aligned}$$ We found that both the contractions of the string vertices give exactly the same singular behavior expected from the OPE .
Though the correspondence seems to work well also in the bosonic case, the action still lacks justification. In this paper we would like to return to our original motivation of investigating the correspondence in the supersymmetric case, where we typically have to deal with the prefactors. First, we repeat the computation of the bosonic sector. In order to reproduce the OPE $$\begin{aligned}
[\tau^i\bar\sigma](z,\bar{z})\cdot[\tau^k\bar\sigma](0,0)
\sim\frac{\delta^{ik}}{z^2\bar{z}(\log|z|)^4},\end{aligned}$$ in terms of LCSFT, from the correspondence dictionary we need to consider two contractions: $$\begin{aligned}
&{}_{36}\langle R^{\rm b}|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
Z_{123}^i|V^{\rm b}\rangle_{123}
Z_{456}^k|V^{\rm b}\rangle_{456},\nonumber\\
&{}_{14}\langle R^{\rm b}|{}_{25}\langle R^{\rm b}|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}
Z_{123}^i|V^{\rm b}\rangle_{123}
Z_{456}^k|V^{\rm b}\rangle_{456},
\label{RVVb}\end{aligned}$$ for the tree diagram and the one-loop diagram. After the computation in the bosonic sector, we shall proceed to the fermionic sector. Among various OPEs of the spin fields, we would like to concentrate on the realization of the OPE $$\begin{aligned}
\Sigma^i(z)\bar\Sigma^j(\bar{z})\cdot\Sigma^k(0)\bar\Sigma^l(0)
\sim\frac{\delta^{ik}\delta^{jl}}{|z|^2}.\end{aligned}$$ This OPE corresponds to the fermionic parts of the contractions between two first order Hamiltonians $|H_1\rangle$ in LCSFT: $$\begin{aligned}
&{}_{36}\langle R^{\rm f}|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}|V^{\rm f}\rangle_{123}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{kl}|V^{\rm f}\rangle_{456},\nonumber\\
&{}_{14}\langle R^{\rm f}|{}_{25}\langle R^{\rm f}|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}|V^{\rm f}\rangle_{123}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{kl}|V^{\rm f}\rangle_{456}.
\label{RVVf}\end{aligned}$$ The superscript “f” denotes that we pick up the fermionic sector. Comparing with the previous case of , we note that in the computation of and we have to deal with the prefactors $Z^i$ or $[\cosh{\!\not\hspace{-0.5mm}Y}]^{ij}$.
Since both the reflector and the interaction vertex basically take the Gaussian form, we will utilize the Gaussian convolution formula in our computation. The bosonic case of the Gaussian convolution formula is well known and can be found, for example, in [@KMT]. For the fermionic oscillators $S_m$, $S^{\dagger}_n$ satisfying $\{S_m,S^{\dagger}_n\}=\delta_{m,n}$, the formula reads $$\begin{aligned}
&\langle 0|\exp\biggl({1\over 2}{\boldsymbol{S}}^{\rm T}M{\boldsymbol{S}}
+{\boldsymbol{k}}^{\rm T}{\boldsymbol{S}}\biggr)
\exp\biggl({1\over 2}{\boldsymbol{S}}^{\dagger{\rm T}}N{\boldsymbol{S}}^\dagger
+{\boldsymbol{l}}^{\rm T}{\boldsymbol{S}}^\dagger\biggr)|0\rangle\nonumber\\
&=\sqrt{\det(1+MN)}
\exp\biggl({1\over 2}{\boldsymbol{k}}^{\rm T}N\frac{1}{1+MN}{\boldsymbol{k}}
+{1\over 2}{\boldsymbol{l}}^{\rm T}\frac{1}{1+MN}M{\boldsymbol{l}}
+{\boldsymbol{l}}^{\rm T}\frac{1}{1+MN}{\boldsymbol{k}}\biggr).
\label{gaussian}\end{aligned}$$
To deal with the bosonic prefactor $Z_{123}^iZ_{456}^k$ in , we need to introduce the source term $e^{\beta_{123}^iZ_{123}^i+\beta_{456}^kZ_{456}^k}$, take the derivative with respect to $\beta$ and finally set $\beta=0$. However, the fermionic prefactor $[\cosh{\!\not\hspace{-0.5mm}Y}_{123}]^{ij}[\cosh{\!\not\hspace{-0.5mm}Y}_{456}]^{kl}$ in looks much more complicated than the bosonic one $Z_{123}^iZ_{456}^k$. Fortunately, using the Fourier transformation of the prefactor we can compute the contractions with the simple source term $e^{\frac{2}{\alpha_{123}}(\phi_{123}Y_{123}-\phi_{456}Y_{456})}$ as in the bosonic case and perform the $\phi$ integration afterwards. The surprise is that, although the right hand side of consists of the bilinear terms of the sources ${\boldsymbol{k}}$ and ${\boldsymbol{l}}$, after the computation we find that the bilinear terms of $\phi$ do not appear in the final result. Therefore, in the $\phi$ integration we can apply once again and write down the answer explicitly.
The contents of the present paper are as follows. In the next section, we recapitulate some ingredients of LCSFT, which will be necessary in our computation. After the review we proceed to the computation of the contractions of the string interaction vertices which correspond to the OPEs. We first compute the contractions of the tree diagram in Section 3 in the ordering of the bosonic sector and the fermionic sector. Then we turn to the computation of the one-loop diagram in the same ordering in Section 4. Finally we conclude our paper with some comments. Appendix A is devoted to our new notations of the prefactors. In Appendix B we collect several relations of the Neumann coefficient matrices and prove some preliminary formulas for Appendix C. In Appendix C we evaluate the small intermediate time behavior of some Neumann coefficient matrix products. In Appendix D we collect some formulas for the gamma matrices. The results in the appendices are necessary at each stage of our diagram computation.
LCSFT
=====
In this section we would like to recapitulate some ingredients of LCSFT, which are necessary in the computation of following sections. The first order interaction terms are expressed by the three-string Fock space and given by [@GSB] $$\begin{aligned}
|H_1\rangle_{123}&=Z^i\bar Z^j\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij}
|V\rangle_{123},\\
|Q_1^{\dot a}\rangle_{123}
&=\sqrt{-\alpha_{123}}\bar{Z}^i\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i}
|V\rangle_{123},\\
|\tilde Q_1^{\dot a}\rangle_{123}
&=i\sqrt{-\alpha_{123}}Z^i\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}}
|V\rangle_{123}.\end{aligned}$$ Here the kinematical interaction vertex $|V\rangle_{123}
=|V^{\rm b}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle$, determined by the overlapping conditions of strings, is given by $$\begin{aligned}
&|V^{\rm b}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
=\int\delta^{\rm b}(1,2,3)
e^{E^{\rm b}+\bar{E}^{\rm b}}
|p_1\rangle_1|p_2\rangle_2|p_3\rangle_3,\\
&|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
=\int\delta^{\rm f}(1,2,3)e^{E^{\rm f}}
|\lambda_1\rangle_1|\lambda_2\rangle_2|\lambda_3\rangle_3,\end{aligned}$$ with the zero-mode delta functions being $$\begin{aligned}
&\int\delta^{\rm b}(1,2,3)=\int
\frac{d^8p_1}{(2\pi)^8}\frac{d^8p_2}{(2\pi)^8}\frac{d^8p_3}{(2\pi)^8}
(2\pi)^8\delta^8(p_1+p_2+p_3),\\
&\int\delta^{\rm f}(1,2,3)=\int d^8\lambda_1d^8\lambda_2d^8\lambda_3
\delta^8(\lambda_1+\lambda_2+\lambda_3),\end{aligned}$$ and the oscillator bilinears $E^{\rm b}$ (and its anti-holomorphic cousin $\bar{E}^{\rm b}$) and $E^{\rm f}$ given by $$\begin{aligned}
E^{\rm b}&=\frac{1}{2}
{\boldsymbol{a}}^{(12)\dagger{\rm T}}N^{12,12}{\boldsymbol{a}}^{(12)\dagger}
+{\boldsymbol{a}}^{(12)\dagger{\rm T}}N^{12,3}{\boldsymbol{a}}^{(3)\dagger}
+\frac{1}{2}
{\boldsymbol{a}}^{(3)\dagger{\rm T}}N^{3,3}{\boldsymbol{a}}^{(3)\dagger}\nonumber\\
&\qquad+\bigl({\boldsymbol{N}}^{12{\rm T}}{\boldsymbol{a}}^{(12)\dagger}
+{\boldsymbol{N}}^{3{\rm T}}{\boldsymbol{a}}^{(3)\dagger}\bigr){\mathbb P}_{123}
-\frac{\tau_0}{2\alpha_{123}}{\mathbb P}_{123}^2,\\
E^{\rm f}&=\frac{1}{2}
{\boldsymbol{S}}^{(12)\dagger{\rm T}}[\hat{N}]^{12,12}{\boldsymbol{S}}^{(12)\dagger}
+{\boldsymbol{S}}^{(12)\dagger{\rm T}}[\hat{N}]^{12,3}{\boldsymbol{S}}^{(3)\dagger}
+\frac{1}{2}
{\boldsymbol{S}}^{(3)\dagger{\rm T}}[\hat{N}]^{3,3}{\boldsymbol{S}}^{(3)\dagger}
\nonumber\\&\qquad
-\sqrt{2}\Lambda_{123}
\bigl(\hat{{\boldsymbol{N}}}{}^{12{\rm T}}{\boldsymbol{S}}^{{\rm II}(12)\dagger}
+\hat{{\boldsymbol{N}}}{}^{3{\rm T}}{\boldsymbol{S}}^{{\rm II}(3)\dagger}\bigr).\end{aligned}$$ The explicit forms of the building blocks of the prefactors $Z^i$ and $Y^a$, defined in and , are given by $$\begin{aligned}
&Z^i={\mathbb P}_{123}^i
-\alpha_{123}\sum_{r=1}^3\sum_{n=1}^\infty
\frac{n}{\alpha_r}N_n^ra^{(r)i}_{-n},\\
&Y^a=\Lambda^a_{123}
-\frac{\alpha_{123}}{\sqrt{2}}\sum_{r=1}^3\sum_{n=1}^\infty
\sqrt{\frac{n}{\alpha_r}}N_n^rS^{{\rm I}(r)a}_{-n}.\end{aligned}$$
Our notation for the bosonic part is exactly the same as those in our previous work [@KMT]. Hence, we shall only explain our notation for the fermionic part in detail. We define the zero mode $\Lambda_{123}^a$ to be $\Lambda_{123}^a=\alpha_1\lambda_2^a-\alpha_2\lambda_1^a$, and adopt the matrix notation for the infinite nonzero modes of the fermionic oscillators $S^{{\rm I}(r)a}_m$ and $S^{{\rm II}(r)a}_m$ satisfying $$\begin{aligned}
\{S^{{\rm I}(r)a}_m,S^{{\rm I}(s)b}_n\}
=\delta_{m+n,0}\delta^{r,s}\delta^{a,b},\quad
\{S^{{\rm II}(r)a}_m,S^{{\rm II}(s)b}_n\}
=\delta_{m+n,0}\delta^{r,s}\delta^{a,b},\quad
\{S^{{\rm I}(r)a}_m,S^{{\rm II}(s)b}_n\}=0,\end{aligned}$$ and $S^{{\rm I}(r)a}_m|\lambda_r\rangle_r
=S^{{\rm II}(r)a}_m|\lambda_r\rangle_r=0$ ($m>0$) on the vacuum ket-state $|\lambda_r\rangle_r$. We first rewrite the fermionic oscillators $S^{{\rm I}(r)a}_m$ and $S^{{\rm II}(r)a}_m$ into the vector forms $(m>0)$ $$\begin{aligned}
({\boldsymbol{S}}^{{\rm I}(r)})_m=S^{{\rm I}(r)}_m,\quad
({\boldsymbol{S}}^{{\rm II}(r)})_m=S^{{\rm II}(r)}_m,\quad
({\boldsymbol{S}}^{{\rm I}(r)\dagger})_m=S^{{\rm I}(r)}_{-m},\quad
({\boldsymbol{S}}^{{\rm II}(r)\dagger})_m=S^{{\rm II}(r)}_{-m}.\end{aligned}$$ Then we combine the fermionic oscillators of the incoming/outgoing strings as $$\begin{aligned}
{\boldsymbol{S}}^{{\rm I}(12){\rm T}}=\begin{pmatrix}
{\boldsymbol{S}}^{{\rm I}(1){\rm T}}&{\boldsymbol{S}}^{{\rm I}(2){\rm T}}
\end{pmatrix},\quad
{\boldsymbol{S}}^{{\rm II}(12){\rm T}}=\begin{pmatrix}
{\boldsymbol{S}}^{{\rm II}(1){\rm T}}&{\boldsymbol{S}}^{{\rm II}(2){\rm T}}
\end{pmatrix},\end{aligned}$$ and finally pair up the oscillators with the index I and the oscillators with the index II: $$\begin{aligned}
{\boldsymbol{S}}^{(12){\rm T}}=\begin{pmatrix}
{\boldsymbol{S}}^{{\rm I}(12){\rm T}}&{\boldsymbol{S}}^{{\rm II}(12){\rm T}}
\end{pmatrix},\quad
{\boldsymbol{S}}^{(3){\rm T}}=\begin{pmatrix}
{\boldsymbol{S}}^{{\rm I}(3){\rm T}}&{\boldsymbol{S}}^{{\rm II}(3){\rm T}}
\end{pmatrix}.\end{aligned}$$
Correspondingly, we repeat the same manipulation for the Neumann coefficient matrices. The Neumann coefficient matrices for the fermionic oscillators $\hat N^{r,s}$, $\hat{{\boldsymbol{N}}}{}^{r}$ are constructed out of those for the bosonic oscillators $N^{r,s}$, ${\boldsymbol{N}}^r$ by $$\begin{aligned}
\hat N^{r,s}=(C/\alpha_r)^{\frac{1}{2}}
N^{r,s}(C/\alpha_s)^{-\frac{1}{2}},\quad
\hat{{\boldsymbol{N}}}{}^{r}=(C/\alpha_r)^{\frac{1}{2}}{\boldsymbol{N}}^r.\end{aligned}$$ We combine the Neumann coefficient matrices of the incoming/outgoing strings by $$\begin{aligned}
\hat{N}^{12,12}=\begin{pmatrix}
\hat{N}^{1,1}&\hat{N}^{1,2}\\\hat{N}^{2,1}&\hat{N}^{2,2}
\end{pmatrix},\quad
\hat{N}^{3,12}=\begin{pmatrix}
\hat{N}^{3,1}&\hat{N}^{3,2}\end{pmatrix},\quad
\hat{N}^{12,3}=\begin{pmatrix}
\hat{N}^{1,3}\\\hat{N}^{2,3}\end{pmatrix},\quad
\hat{{\boldsymbol{N}}}{}^{12}=\begin{pmatrix}
\hat{{\boldsymbol{N}}}{}^1\\\hat{{\boldsymbol{N}}}{}^2\end{pmatrix},\end{aligned}$$ and pair the Neumann coefficient matrices for the fermionic oscillators with the index I and those for the fermionic oscillators with the index II: $$\begin{aligned}
[\hat{N}]^{12,12}
\!\!=\!\begin{pmatrix}0&\!\!\!-\hat N^{12,12{\rm T}}\\
\hat N^{12,12}&\!\!\!0\end{pmatrix},\quad
[\hat{N}]^{12,3}
\!\!=\!\begin{pmatrix}0&\!\!\!-\hat N^{3,12{\rm T}}\\
\hat N^{12,3}&\!\!\!0\end{pmatrix},\quad
[\hat{N}]^{3,3}
\!\!=\!\begin{pmatrix}0&\!\!\!-\hat N^{3,3{\rm T}}\\
\hat N^{3,3}&\!\!\!0\end{pmatrix}.
\label{hatN}\end{aligned}$$
Using the matrix notation, the fermionic part of the reflector $\langle R(3,6)|=\langle R^{\rm b}(3,6)|\langle R^{\rm f}(3,6)|$ is expressed as $$\begin{aligned}
\langle R^{\rm f}(3,6)|=\int\delta^{\rm f}(3,6)
{}_3\langle\lambda_3|{}_6\langle\lambda_6|
\exp\bigg(\frac{1}{2}{\boldsymbol{S}}^{(36){\rm T}}R{\boldsymbol{S}}^{(36)}\bigg),\end{aligned}$$ with the vacuum bra-state satisfying ${}_r\langle\lambda_r|S^{{\rm I}(r)}_{-m}
={}_r\langle\lambda_r|S^{{\rm II}(r)}_{-m}=0$ and $\langle\lambda|\lambda'\rangle=\delta^8(\lambda-\lambda')$. Here the fermionic oscillators are defined as ${\boldsymbol{S}}^{(36){\rm T}}=({\boldsymbol{S}}^{(3){\rm T}}\quad{\boldsymbol{S}}^{(6){\rm T}})$ and the matrix $R$ is given by $$\begin{aligned}
R=\begin{pmatrix}0&i[\Sigma^1]\\-i[\Sigma^1]&0\end{pmatrix},\quad
[\Sigma^1]=\begin{pmatrix}0&1\\1&0\end{pmatrix},
\label{Sigma1}\end{aligned}$$ where the matrices $1$ and $0$ in $[\Sigma^1]$ denote the infinite-dimensional unity matrix and the infinite-dimensional zero matrix respectively and are the Neumann coefficient matrices for the oscillators with the indices I and II.
Tree Amplitude
==============
After the recapitulation of LCSFT in the previous section, we would like to proceed to realize various OPEs using the string interaction vertices. As explained in the introduction, we have two ways to realize the OPEs. This section will devote to the calculation of the four-point tree diagram with two incoming strings 1 (with length $\alpha_1(>0)$) and 2 (with length $\alpha_2(>0)$), joining and splitting again into two outgoing strings 4 and 5 of the same length as 1 and 2, respectively. (See Fig. 1.)
\[0.6\][![Four-string tree diagram.[]{data-label="fig:4string"}](tree.eps "fig:")]{}
Bosonic sector
--------------
Let us start with realizing the OPE $$\begin{aligned}
[\tau^i\bar\sigma](z,\bar{z})\cdot[\tau^k\bar\sigma](0)
\sim\frac{\delta^{ik}}{z^2\bar{z}(\log|z|)^4},
\label{twotau}\end{aligned}$$ with the tree diagram. For this purpose, we shall consider the bosonic effective four-string interaction vertex $$\begin{aligned}
&|A^{\rm b}(1,2,4,5)\rangle=\langle R^{\rm b}(3,6)|
e^{-{T\over |\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}\nonumber\\
&\quad\times
Z_{123}^i|V^{\rm b}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
Z_{456}^k|V^{\rm b}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$ Comparing the OPE with , we have an extra factor of $1/z$. Therefore, we expect that the extra prefactors $Z_{123}^iZ_{456}^k$ in the effective four-string vertex $|A^{\rm b}(1,2,4,5)\rangle$ induce an extra $1/T$ factor in the limit $T\to+0$. Here note that we can identify the relative distance in $z$ as the relative distance in $T$.
For the computation of the effective four-string interaction vertex $|A^{\rm b}(1,2,4,5)\rangle$, it is sufficient to consider the generating function $$\begin{aligned}
&|A^{\rm b}_\beta(1,2,4,5)\rangle=\langle R^{\rm b}(3,6)|
e^{-{T\over |\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\nonumber\\
&\quad\times
e^{\beta_{123}^iZ_{123}^i+\beta_{456}^kZ_{456}^k}
|V^{\rm b}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
|V^{\rm b}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$ The computation is almost the same as that performed in [@KMT]. The only difference is the source term, but the effect can be easily absorbed by completing the square. The result is given as $$\begin{aligned}
&|A^{\rm b}_\beta(1,2,4,5)\rangle
=(\det)^{-8}\int\delta^{\rm b}(1,2,4,5)
e^{F^{\rm b}(1,2,4,5)}\nonumber\\
&\quad\times e^{\beta_{123}{\cal Z}_{123}+\beta_{456}{\cal Z}_{456}
+\frac{1}{2}(\beta_{123}^2+\beta_{456}^2)a_2
+\beta_{123}\beta_{456}b_2}
|p_1\rangle_1|p_2\rangle_2|p_4\rangle_4|p_5\rangle_5,
\label{bosonictree}\end{aligned}$$ where various expressions are $$\begin{aligned}
&\det=\det\Bigl[1-\bigl(e^{-\frac{T}{2|\alpha_3|}C}
N^{3,3}e^{-\frac{T}{2|\alpha_3|}C}\bigr)^2\Bigr]
\sim 2^{-\frac{5}{12}}\mu^{\frac{1}{6}}
\biggl[\frac{T}{|\alpha_{123}|^{1/3}}\biggr]^{\frac{1}{4}},\\
&\qquad\mu=\exp\biggl(-\tau_0\sum_{t=1}^3\frac{1}{\alpha_t}\biggr),
\quad\tau_0=\sum_{t=1}^3\alpha_t\log|\alpha_t|,\\
&\int\delta^{\rm b}(1,2,4,5)
=\int\frac{d^8p_1}{(2\pi)^8}\frac{d^8p_2}{(2\pi)^8}
\frac{d^8p_4}{(2\pi)^8}\frac{d^8p_5}{(2\pi)^8}
(2\pi)^8\delta^8(p_1+p_2+p_4+p_5),\\
&\lim_{T\to +0}F^{\rm b}(1,2,4,5)
=-\bigl({\boldsymbol{a}}^{(12)\dagger{\rm T}}{\boldsymbol{a}}^{(45)\dagger}
+\bar{{\boldsymbol{a}}}^{(12)\dagger{\rm T}}\bar{{\boldsymbol{a}}}^{(45)\dagger}\bigr)
-(p_1+p_4)^2\lim_{T\to +0}b
\nonumber\\
&\qquad-(p_1+p_4)\alpha_3{\boldsymbol{N}}^{3{\rm T}}(N^{12,3})^{-1}
\bigl({\boldsymbol{a}}^{(12)\dagger}+{\boldsymbol{a}}^{(45)\dagger}
+\bar{{\boldsymbol{a}}}^{(12)\dagger}+\bar{{\boldsymbol{a}}}^{(45)\dagger}\bigr),
\label{Fb}\\
&b=\alpha_3^2
{\boldsymbol{N}}^{3{\rm T}}\circ\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ{\boldsymbol{N}}^3
\sim 2\log\frac{|\alpha_3|}{T},\label{blog}\end{aligned}$$ which have already appeared in [@KMT]. Here we drop the subscript $T$ in $b_T$ of [@KMT], because all the quantities depend on $T$. The new effect of the source term is taken care of by $$\begin{aligned}
&{\cal Z}_{123}=(1-a_1)\mathbb{P}_{123}-b_1\mathbb{P}_{456}
-{\boldsymbol{a}}^{\rm T}{\boldsymbol{a}}^{(12)\dagger}
+{\boldsymbol{b}}^{\rm T}{\boldsymbol{a}}^{(45)\dagger},
\label{Z123}\\
&{\cal Z}_{456}=-b_1\mathbb{P}_{123}+(1-a_1)\mathbb{P}_{456}
-{\boldsymbol{b}}^{\rm T}{\boldsymbol{a}}^{(12)\dagger}
+{\boldsymbol{a}}^{\rm T}{\boldsymbol{a}}^{(45)\dagger},
\label{Z456}\end{aligned}$$ with various new quantities defined by $$\begin{aligned}
a_1&=\alpha_{123}{\boldsymbol{N}}^{3{\rm T}}(C/\alpha_3)\circ
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
N^{3,3}\circ{\boldsymbol{N}}^3,\\
b_1&=\alpha_{123}{\boldsymbol{N}}^{3{\rm T}}(C/\alpha_3)\circ
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
{\boldsymbol{N}}^3,\\
a_2&=(\alpha_{123})^2{\boldsymbol{N}}^{3{\rm T}}(C/\alpha_3)\circ
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
N^{3,3}\circ(C/\alpha_3){\boldsymbol{N}}^3,\\
b_2&=(\alpha_{123})^2{\boldsymbol{N}}^{3{\rm T}}(C/\alpha_3)\circ
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
(C/\alpha_3){\boldsymbol{N}}^3,\\
{\boldsymbol{a}}^{\rm T}
&=\alpha_{123}\Bigl({\boldsymbol{N}}^{12{\rm T}}(C/\alpha_{12})
+{\boldsymbol{N}}^{3{\rm T}}(C/\alpha_3)\circ
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
N^{3,3}\circ N^{3,12}\Bigr),\\
{\boldsymbol{b}}^{\rm T}&=\alpha_{123}{\boldsymbol{N}}^{3{\rm T}}(C/\alpha_3)\circ
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
N^{3,12}.\end{aligned}$$ Here $\circ$ denotes the matrix multiplication with $e^{-\frac{T}{|\alpha_3|}C}$ inserted and $(1-(N^{3,3})^2_\circ)^{-1}_\circ$ is defined by $$\begin{aligned}
\bigl(1-(N^{3,3})^2_\circ\bigr)^{-1}_\circ
=1+N^{3,3}\circ N^{3,3}\circ
+N^{3,3}\circ N^{3,3}\circ N^{3,3}\circ N^{3,3}\circ+\cdots.\end{aligned}$$ Note that $C/\alpha_{12}$ is the bookkeeping notation for $$\begin{aligned}
C/\alpha_{12}
=\begin{pmatrix}C/\alpha_1&0\\0&C/\alpha_2\end{pmatrix},\end{aligned}$$ and should not be confused with $\alpha_{123}=\alpha_1\alpha_2\alpha_3$. To get back to the effective four-string interaction vertex $|A^{\rm b}(1,2,4,5)\rangle$ we need to take the derivative of the generating function $|A^{\rm b}_\beta(1,2,4,5)\rangle$ and set $\beta=0$ finally: $$\begin{aligned}
|A^{\rm b}(1,2,4,5)\rangle=\frac{\partial}{\partial\beta_{123}^i}
\frac{\partial}{\partial\beta_{456}^k}
|A^{\rm b}_\beta(1,2,4,5)\rangle\biggr|_{\beta=0}
=\bigl(b_2\delta^{ik}+{\cal Z}_{123}^i{\cal Z}_{456}^k\bigr)
|A^{\rm b}_\beta(1,2,4,5)\rangle\biggr|_{\beta=0}.\end{aligned}$$
Let us take the short intermediate time limit $T\to +0$ hereafter to reproduce the OPE . Note that the nonzero-mode part of the reflector already appears correctly in and no extra contributions from the prefactor and are added to the nonzero-mode part because of $\lim_{T\to +0}{\boldsymbol{a}}=\lim_{T\to +0}{\boldsymbol{b}}={\boldsymbol{0}}$ , as proved in Appendix B.2. Hence, we shall concentrate on the zero-mode contribution. Using $$\begin{aligned}
{\cal Z}_{123}^i\sim-{\cal Z}_{456}^i\sim -b_1\alpha_3(p_1+p_4)^i,\end{aligned}$$ which can be shown with $\lim_{T\to +0}(1-a_1-b_1)=0$ , the zero-mode part of the effective interaction vertex $|A^{\rm b}(1,2,4,5)\rangle$ is given as $$\begin{aligned}
|A^{\rm b}(1,2,4,5)\rangle\biggr|_{0}&\sim\bigl(b_2\delta^{ik}
+(b_1\alpha_3)^2(p_1+p_4)^i(p_1+p_4)^k)\bigr)e^{-b(p_1+p_4)^2}
\delta^8(p_1+p_2+p_4+p_5)\nonumber\\
&=\biggl[\biggl(b_2+\frac{(b_1\alpha_3)^2}{2b}\biggr)\delta^{ik}
+\frac{(b_1\alpha_3)^2}{4b^2}\partial_{p_1^i}\partial_{p_1^k}\biggr]
e^{-b(p_1+p_4)^2}\delta^8(p_1+p_2+p_4+p_5)\nonumber\\
&\sim b_2\delta^{ik}\cdot e^{-b(p_1+p_4)^2}\delta^8(p_2+p_5).\end{aligned}$$ In the last line we have picked up the most singular term using the short intermediate time behavior of $b$ , $b_1$ and $b_2$ . The behavior of $b_2$ in the short intermediate time limit $T\to+0$ is $b_2\sim-\alpha_{123}/(2T)$. Since we have an extra $1/T$ factor compared with the case with no prefactors, this result is exactly what we have expected from the OPE . After all the final result is given as $$\begin{aligned}
|A^{\rm b}(1,2,4,5)\rangle
\sim 2^{-\frac{29}{3}}\pi^{-4}\mu^{-\frac{4}{3}}
|\alpha_{123}|^{\frac{5}{3}}
\frac{\delta^{ik}}{T^3}\biggl(\log\frac{T}{|\alpha_3|}\biggr)^{-4}
|R^{\rm b}(1,4)\rangle|R^{\rm b}(2,5)\rangle.
\label{twotautree}\end{aligned}$$
Fermionic sector
----------------
In this subsection we would like to realize the OPE $$\begin{aligned}
\Sigma^i(z)\bar\Sigma^j(\bar z)\cdot\Sigma^k(0)\bar\Sigma^l(0)
\sim\frac{\delta^{ik}\delta^{jl}}{|z|^2},
\label{Sigmaijkl}\end{aligned}$$ in terms of the string interaction vertex. This OPE corresponds to the fermionic part of the contraction between two first order Hamiltonians $H_1\cdot H_1$. Therefore, we shall compute the fermionic effective four-string interaction vertex $$\begin{aligned}
&|A^{\rm f}(1,2,4,5)\rangle=\langle R^{\rm f}(3,6)|
e^{-{T\over |\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}\nonumber\\
&\quad\times\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{kl}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$
Due to the Fourier transformation formula , we can evaluate the effective four-string interaction vertex $|A^{\rm f}(1,2,4,5)\rangle$ by the generating function $$\begin{aligned}
&|A^{\rm f}_\phi(1,2,4,5)\rangle
=\langle R^{\rm f}(3,6)|
e^{-{T\over|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\nonumber\\&\quad\times
e^{\frac{2}{\alpha_{123}}(\phi_{123}Y_{123}-\phi_{456}Y_{456})}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$ In order to calculate the generating function $|A^{\rm f}_\phi(1,2,4,5)\rangle$, we first rewrite the reflector and the interaction vertices into the following expression: $$\begin{aligned}
&\langle R^{\rm f}(3,6)|
e^{-{T\over |\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
=\int\delta^{\rm f}(3,6){}_3\langle\lambda_3|{}_6\langle\lambda_6|
\exp\bigg(\frac{1}{2}{\boldsymbol{S}}^{(36){\rm T}}
\tilde{M}{\boldsymbol{S}}^{(36)}\bigg),\\
&e^{\frac{2}{\alpha_{123}}
(\phi_{123}\Lambda_{123}-\phi_{456}\Lambda_{456})}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle
\nonumber\\&\quad
=\int\delta^{\rm f}(1,2,3)\delta^{\rm f}(4,5,6)
\exp\bigg(\frac{1}{2}{\boldsymbol{S}}^{(36)\dagger{\rm T}}
\tilde{N}{\boldsymbol{S}}^{(36)\dagger}
+\tilde{{\boldsymbol{l}}}{}^{\rm T}{\boldsymbol{S}}^{(36)\dagger}
+\tilde{P}\bigg)
|\lambda_1\rangle_1|\lambda_2\rangle_2
\cdots|\lambda_6\rangle_6,\end{aligned}$$ with ${\boldsymbol{S}}^{(36)}
=\begin{pmatrix}{\boldsymbol{S}}^{(3)}&{\boldsymbol{S}}^{(6)}\end{pmatrix}$, ${\boldsymbol{S}}^{(1245)}
=\begin{pmatrix}{\boldsymbol{S}}^{(12)}&{\boldsymbol{S}}^{(45)}\end{pmatrix}$ and $$\begin{aligned}
&\tilde{M}=e^{-\frac{T}{|\alpha_3|}C}\begin{pmatrix}
0&i[\Sigma^1]\\-i[\Sigma^1]&0\end{pmatrix},\quad
\tilde{N}=\begin{pmatrix}
[\hat{N}]^{3,3}&0\\0&[\hat{N}]^{3,3}\end{pmatrix},\\
&\tilde{{\boldsymbol{l}}}{}^{\rm T}={\boldsymbol{S}}^{(1245)\dagger{\rm T}}
\begin{pmatrix}[\hat{N}]^{3,12{\rm T}}&0\\
0&[\hat{N}]^{3,12{\rm T}}\end{pmatrix}
-\sqrt{2}\begin{pmatrix}
\phi_{123}&\Lambda_{123}&i\phi_{456}&i\Lambda_{456}
\end{pmatrix}\hat{{\boldsymbol{N}}}{}^{3{\rm T}},\\
&\tilde{P}=\frac{1}{2}{\boldsymbol{S}}^{(1245)\dagger{\rm T}}
\begin{pmatrix}[\hat{N}]^{12,12}&0\\0&[\hat{N}]^{12,12}\end{pmatrix}
{\boldsymbol{S}}^{(1245)\dagger}
\nonumber\\&\qquad
-\sqrt{2}\begin{pmatrix}
\phi_{123}&\Lambda_{123}&i\phi_{456}&i\Lambda_{456}
\end{pmatrix}\hat{{\boldsymbol{N}}}{}^{12{\rm T}}{\boldsymbol{S}}^{(1245)\dagger}
+\frac{2}{\alpha_{123}}(\phi_{123}\Lambda_{123}-\phi_{456}\Lambda_{456}).\end{aligned}$$ Then the calculation can be easily done with the help of the fermionic Gaussian convolution formula . The result is given as $$\begin{aligned}
&|A^{\rm f}_\phi(1,2,4,5)\rangle=(\det)^8\int\delta^{\rm f}(1,2,4,5)
e^{\frac{2}{\alpha_{123}}
(\phi_{123}{\cal Y}_{123}-\phi_{456}{\cal Y}_{456})}
e^{F^{\rm f}(1,2,4,5)}
|\lambda_1\rangle_1|\lambda_2\rangle_2
|\lambda_4\rangle_4|\lambda_5\rangle_5,
\label{fermionictree}\end{aligned}$$ with various expressions defined as $$\begin{aligned}
&\int\delta^{\rm f}(1,2,4,5)
=\int d^8\lambda_1d^8\lambda_2d^8\lambda_4d^8\lambda_5
\delta^8(\lambda_1+\lambda_2+\lambda_4+\lambda_5),\\
&F^{\rm f}(1,2,4,5)=\frac{1}{2}{\boldsymbol{S}}^{(1245)\dagger{\rm T}}
M{\boldsymbol{S}}^{(1245)\dagger}
+{\boldsymbol{k}}^{\rm T}{\boldsymbol{S}}^{(1245)\dagger},\\
&{\cal Y}_{123}=(1-a_1)\Lambda_{123}
-b_1\Lambda_{456}
-{1\over\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(12)\dagger{\rm T}}
(C/\alpha_{12})^{-\frac{1}{2}}{\boldsymbol{a}}
+{i\over\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(45)\dagger{\rm T}}
(C/\alpha_{12})^{-{1\over 2}}{\boldsymbol{b}},\\
&{\cal Y}_{456}=-b_1\Lambda_{123}+(1-a_1)\Lambda_{456}
+{1\over\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(12)\dagger{\rm T}}
(C/\alpha_{12})^{-{1\over 2}}{\boldsymbol{b}}
-{i\over\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(45)\dagger{\rm T}}
(C/\alpha_{12})^{-{1\over 2}}{\boldsymbol{a}}.\end{aligned}$$ Here the effective Neumann coefficient matrix $M$ and ${\boldsymbol{k}}^{\rm T}$ are given as $$\begin{aligned}
&M=\begin{pmatrix}
[A]&i[B]\\-i[B]&[A]\end{pmatrix},\quad
[A]=\begin{pmatrix}
0&-A^{\rm T}\\A&0\end{pmatrix},\quad
[B]=\begin{pmatrix}
0&B^{\rm T}\\B&0\end{pmatrix},\\
&{\boldsymbol{k}}^{{\rm T}}=-\sqrt{2}\begin{pmatrix}
0&\Lambda_{123}{\boldsymbol{U}}^{\rm T}
-\alpha_3(\lambda_1+\lambda_4){\boldsymbol{V}}^{\rm T}&
0&i(\Lambda_{456}{\boldsymbol{U}}^{\rm T}
+\alpha_3(\lambda_1+\lambda_4){\boldsymbol{V}}^{\rm T})
\end{pmatrix},\end{aligned}$$ with the building blocks being $$\begin{aligned}
&A=\hat{N}^{12,12}+\hat{N}^{12,3}\circ\hat{N}^{3,3}\circ
\bigl(1-(\hat{N}^{3,3})^2_\circ\bigr)^{-1}_\circ
\hat{N}^{3,12},\\
&B=\hat{N}^{12,3}\circ
\bigl(1-(\hat{N}^{3,3})^2_\circ\bigr)^{-1}_\circ
\hat{N}^{3,12},\\
&{\boldsymbol{U}}=\hat{{\boldsymbol{N}}}^{12}
+\hat{N}^{12,3}\circ\bigl(1-\hat{N}^{3,3}\bigr)^{-1}_\circ
\hat{{\boldsymbol{N}}}^{3},\\
&{\boldsymbol{V}}=\hat{N}^{12,3}\circ
\bigl(1-(\hat{N}^{3,3})^2_\circ\bigr)^{-1}_\circ
\hat{{\boldsymbol{N}}}^{3}.\end{aligned}$$ What is surprising in this result is that although in the bosonic case we find the linear source term $\beta$ induces the squared source term $\beta^2$ in the final result , the squared source term $\phi^2$ is absent in the current final result . The reason is that both the Neumann coefficient matrix in the interaction vertex and that in the reflector connect the oscillators with the index I and the oscillators with the index II, but in the Fourier transformation formula the source $\phi$ only couples to the oscillators with the index I. Due to this fact, we can perform the $\phi$ integration without difficulty. After performing the inverse Fourier transformation for the result of the generating function $|A^{\rm f}_\phi(1,2,4,5)\rangle$ , we find the effective four-string interaction vertex $|A^{\rm f}(1,2,4,5)\rangle$ itself is given by $$\begin{aligned}
&|A^{\rm f}(1,2,4,5)\rangle
=(\det)^8\int\delta^{\rm f}(1,2,4,5)\nonumber\\
&\quad\times\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{ij}
\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{456}\bigr]^{kl}
e^{F^{\rm f}(1,2,4,5)}
|\lambda_1\rangle_1|\lambda_2\rangle_2
|\lambda_4\rangle_4|\lambda_5\rangle_5.\end{aligned}$$
Note that the kinematical overlapping part in the short intermediate time limit $T\to +0$ is given as $$\begin{aligned}
&\lim_{T\to +0}A=0,\quad\lim_{T\to +0}B=1,\quad
\lim_{T\to +0}{\boldsymbol{U}}={\boldsymbol{0}},\\
&\lim_{T\to +0}{\boldsymbol{V}}=\bigl(C/\alpha_{12}\bigr)^{\frac{1}{2}}
\bigl(N^{3,12}\bigr)^{-1}{\boldsymbol{N}}^3
=-\frac{\alpha_3}{2}\bigl(C/\alpha_{12}\bigr)^{\frac{3}{2}}
A^{(12){\rm T}}C^{-1}{\boldsymbol{B}},\end{aligned}$$ which implies $$\begin{aligned}
&\lim_{T\to +0}F^{\rm f}(1,2,4,5)
=i({\boldsymbol{S}}^{{\rm I}(45)\dagger{\rm T}}{\boldsymbol{S}}^{{\rm II}(12)\dagger}
-{\boldsymbol{S}}^{{\rm I}(12)\dagger{\rm T}}{\boldsymbol{S}}^{{\rm II}(45)\dagger})
\nonumber\\
&\qquad-\sqrt{2}\alpha_3(\lambda_1+\lambda_4)
\lim_{T\to +0}{\boldsymbol{V}}^{\rm T}
(i{\boldsymbol{S}}^{{\rm II}(45)\dagger}-{\boldsymbol{S}}^{{\rm II}(12)\dagger}),
\label{Ff}\end{aligned}$$ while the singular behavior of the prefactors are $$\begin{aligned}
{\cal Y}_{123}^a\sim-{\cal Y}_{456}^a\sim{\cal Y}^a,\quad
{\cal Y}^a=b_1\alpha_3(\lambda_1+\lambda_4)^a,\label{calY}\end{aligned}$$ if we use the short intermediate time behavior of $1-a_1$, $b_1$, ${\boldsymbol{a}}$ and ${\boldsymbol{b}}$ in . Since we have normalized $Y$ by $\sqrt{-\alpha_{123}}$ as in , the result of implies $$\begin{aligned}
{\!\not\hspace{-1mm}{\cal Y}}_{456}=\frac{\sqrt{2}}{\sqrt{-\alpha_{456}}}
\eta^*{\cal Y}_{456}^a\hat\gamma^a
\sim\frac{\sqrt{2}}{i\sqrt{-\alpha_{123}}}
\eta^*(-{\cal Y}_{123}^a)\hat\gamma^a
=i{\!\not\hspace{-1mm}{\cal Y}}_{123},\end{aligned}$$ where the phase $i$ induce the effect of the transposition: $$\begin{aligned}
\bigl[\cosh i{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{kl}
=\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{lk}.
\label{coshiY}\end{aligned}$$
To reproduce the tensor structure of the most singular term in the OPE , let us first rewrite the prefactors into $$\begin{aligned}
\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{ij}\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{lk}
=\frac{1}{2^4}\sum_{m=0}^8{(-1)^{\frac{1}{2}m(m-1)}\over m!}
\hat{\gamma}^{c_1\cdots c_m}_{ik}
(\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\hat{\gamma}^{c_1\cdots c_m}\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123})_{lj},
\label{coshcoshfierz}\end{aligned}$$ and then expand the hyperbolic functions into polynomials to study the singular behavior of each term. Here in we have used the Fierz identity $$\begin{aligned}
M_{AB}N_{CD}=\frac{(-1)^{|M||N|}}{2^4}
\sum_{m=0}^8{(-1)^{{1\over 2}m(m-1)}\over m!}
\hat{\gamma}^{c_1\cdots c_m}_{AD}
(N\hat{\gamma}^{c_1\cdots c_m}M)_{CB},
\label{Fierz}\end{aligned}$$ with $\hat{\gamma}^{c_1\cdots c_m}
=\hat{\gamma}^{[c_1}\cdots\hat{\gamma}^{c_m]}$. Note that from the index structure of $\hat{\gamma}^{c_1\cdots c_m}_{ik}$, the summand of is non-vanishing only when $m$ is even.
Since ${\cal Y}$ is the only singularity, the more ${\cal Y}$’s we have, the more singular the expression is. In order to extract the coefficient of the most singular term ($p+q=8$) $$\begin{aligned}
{\cal Y}^{a_1}\cdots{\cal Y}^{a_p}{\cal Y}^{b_1}\cdots{\cal Y}^{b_q}
=\epsilon^{a_1\cdots a_pb_1\cdots b_q}\delta^8({\cal Y}),\quad
\delta^8({\cal Y})={\cal Y}^1\cdots{\cal Y}^8,
\label{delta8}\end{aligned}$$ we note the formulas [@Kugo] $$\begin{aligned}
&\frac{1}{q!}\epsilon^{a_1\cdots a_pb_1\cdots b_q}
\hat\gamma^{b_1\cdots b_q}
=(-1)^{\frac{1}{2}p(p-1)}\hat\gamma^{a_1\cdots a_p}
\hat\gamma_9,
\label{epsilongamma}\\
&\frac{(-1)^{\frac{1}{2}p(p-1)}}{p!}
\hat\gamma^{a_1\cdots a_p}
\hat\gamma^{c_1\cdots c_m}
\hat\gamma^{a_1\cdots a_p}
=d_{p,m}\hat\gamma^{c_1\cdots c_m},
\label{sandwich}\end{aligned}$$ with $\hat{\gamma}_9$ and $d_{p,m}$ defined by $$\begin{aligned}
\hat{\gamma}_9=\hat{\gamma}^1\cdots\hat{\gamma}^8
=\begin{pmatrix}\delta^{ij}&0\\
0&-\delta_{\dot a\dot b}\end{pmatrix},\quad
(1+x)^{8-m}(1-x)^m=\sum_{p=0}^8(-1)^{pm}d_{p,m}x^p.\end{aligned}$$ There are lots of useful formulas of $d_{p,m}$. We collect some of them in Appendix D, which are necessary in this paper. Using the formulas , and , we find the most singular part of the prefactors is given as $$\begin{aligned}
\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{ij}\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{lk}
=16\nu^8
\delta_{ik}\delta_{jl}\delta^8({\cal Y})
+{\cal O}({\cal Y}^6),\quad
\nu=\sqrt{\frac{2}{-\alpha_{123}}}\eta^*.
\label{coshcosh}\end{aligned}$$ This extra fermionic delta function will eliminate the extra term in $F^{\rm f}(1,2,4,5)$ and turn $|A^{\rm f}(1,2,4,5)\rangle$ into two reflectors $$\begin{aligned}
|A^{\rm f}(1,2,4,5)\rangle
\sim 2^{\frac{26}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{ik}\delta_{jl}}{T^2}
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle,
\label{Sigmaijkltree}\end{aligned}$$ as we have expected from the OPE .
Other processes
---------------
Other processes corresponding to the OPEs (up to the numerical factor) $$\begin{aligned}
&\Sigma^{\dot{a}}(z)\bar{\Sigma}^i(\bar{z})
\cdot\Sigma^{\dot{b}}(0)\bar{\Sigma}^j(0)
\sim\frac{\delta^{\dot{a}\dot{b}}\delta^{ij}}{|z|^2},\quad
\Sigma^i(z)\bar{\Sigma}^{\dot{a}}(\bar{z})
\cdot\Sigma^j(0)\bar{\Sigma}^{\dot{b}}(0)
\sim\frac{\delta^{ij}\delta^{\dot{a}\dot{b}}}{|z|^2},
\label{other1}\\
&\Sigma^i(z)\bar{\Sigma}^j(\bar{z})
\cdot\Sigma^{\dot{a}}(0)\bar{\Sigma}^k(0)
\sim\frac{\gamma^i_{a\dot{a}}\delta^{jk}}{\sqrt{z}\bar{z}}
\theta^a(0),\quad
\Sigma^i(z)\bar{\Sigma}^j(\bar{z})
\cdot\Sigma^k(0)\bar{\Sigma}^{\dot{a}}(0)
\sim\frac{\delta^{ik}\gamma^j_{a\dot{a}}}{\sqrt{\bar{z}}z}
\bar\theta^a(0),
\label{other2}\\
&\Sigma^{\dot{a}}(z)\bar{\Sigma}^i(\bar{z})
\cdot\Sigma^j(0)\bar{\Sigma}^{\dot{b}}(0)
\sim\frac{\gamma^j_{a\dot{a}}\gamma^i_{b\dot{b}}}{|z|}
\theta^a(0)\bar\theta^b(0),
\label{other3}\end{aligned}$$ can also be evaluated as in the previous subsection. These OPEs correspond to the fermionic sector of the $Q_1^{\dot{a}}\cdot Q_1^{\dot{b}}$, $\tilde Q_1^{\dot{a}}\cdot\tilde Q_1^{\dot{b}}$, $H_1\cdot Q_1^{\dot{a}}$, $H_1\cdot\tilde Q_1^{\dot{a}}$ and $Q_1^{\dot{a}}\cdot\tilde Q_1^{\dot{b}}$ contractions respectively. Since all of the string interaction vertices have the identical overlapping part $|V^{\rm f}\rangle$, we would like to concentrate on the prefactors hereafter.
The tree diagram corresponding to can be evaluated exactly in the same way except that instead of we use $$\begin{aligned}
\bigl[\sinh i{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{b}j}
=i\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{j\dot{b}},\quad
\bigl[\sinh i{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{j\dot{b}}
=i\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{b}j},\end{aligned}$$ and instead of we use . Since the contribution of the prefactors is given by $$\begin{aligned}
&\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{a}i}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{j\dot{b}}
=-16\nu^8
\delta_{ij}\delta_{\dot a\dot b}\delta^8({\cal Y})
+{\cal O}({\cal Y}^6),
\label{sinhsinh1}\\
&\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{i\dot{a}}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{b}j}
=16\nu^8
\delta_{ij}\delta_{\dot a\dot b}\delta^8({\cal Y})
+{\cal O}({\cal Y}^6),
\label{sinhsinh2}\end{aligned}$$ the effective interaction vertices are computed to be $$\begin{aligned}
&\langle R^{\rm f}(3,6)|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{\dot{a}i}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{\dot{b}j}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
-i2^{\frac{26}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{\dot{a}\dot{b}}\delta_{ij}}{T^2}
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle,
\label{other11tree}\\
&\langle R^{\rm f}(3,6)|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{i\dot{a}}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{j\dot{b}}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
i2^{\frac{26}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{ij}\delta_{\dot{a}\dot{b}}}{T^2}
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle.
\label{other12tree}\end{aligned}$$
To evaluate the tree diagram corresponding to and , a little more effort is required. Since the most singular term $p+q=8$ vanishes in this case, we have to consider terms with $p+q<8$. First we note that a natural generalization of for $p+q<8$ is $$\begin{aligned}
{\cal Y}^{a_1}\cdots{\cal Y}^{a_p}{\cal Y}^{b_1}\cdots{\cal Y}^{b_q}
=\frac{(-1)^{\frac{1}{2}r(r-1)}}{r!}
\epsilon^{d_1\cdots d_ra_1\cdots a_pb_1\cdots b_q}
\frac{\partial}{\partial{\cal Y}^{d_1}}\cdots
\frac{\partial}{\partial{\cal Y}^{d_r}}\delta^8({\cal Y}),
\label{subsingular}\end{aligned}$$ if we define $r=8-(p+q)$. Using and , we find two expressions for each term of the polynomial expansion of the hyperbolic functions: $$\begin{aligned}
\frac{1}{p!q!}({\cal Y}^a\gamma^a)^p
\hat\gamma^{c_1\cdots c_m}({\cal Y}^b\gamma^b)^q
&=\frac{1}{r!}
G^{c_1\cdots c_m,d_1\cdots d_r}_p
\hat\gamma_9
\frac{\partial}{\partial{\cal Y}^{d_1}}\cdots
\frac{\partial}{\partial{\cal Y}^{d_r}}\delta^8({\cal Y})
\nonumber\\
&=\frac{(-1)^m}{r!}
\tilde G^{d_1\cdots d_r,c_1\cdots c_m}_q
\hat\gamma_9
\frac{\partial}{\partial{\cal Y}^{d_1}}\cdots
\frac{\partial}{\partial{\cal Y}^{d_r}}\delta^8({\cal Y}),
\label{GG}\end{aligned}$$ with $G^{c_1\cdots c_m,d_1\cdots d_r}_p$ and $\tilde G^{d_1\cdots d_r,c_1\cdots c_m}_q$ defined as $$\begin{aligned}
&G^{c_1\cdots c_m,d_1\cdots d_r}_p
=\frac{(-1)^{\frac{1}{2}p(p-1)}}{p!}
\hat\gamma^{a_1\cdots a_p}
\hat\gamma^{c_1\cdots c_m}
\hat\gamma^{a_1\cdots a_pd_1\cdots d_r},\\
&\tilde G^{d_1\cdots d_r,c_1\cdots c_m}_q
=\frac{(-1)^{\frac{1}{2}q(q-1)}}{q!}
\hat\gamma^{d_1\cdots d_rb_1\cdots b_q}
\hat\gamma^{c_1\cdots c_m}
\hat\gamma^{b_1\cdots b_q}.\end{aligned}$$
For the computation of the tree diagram corresponding to , we consider the case $p+q=7$ or $r=1$. For this purpose, $G^{c_1\cdots c_m,d}_p$ is evaluated in Appendix D: $$\begin{aligned}
G^{c_1\cdots c_m,d}_p
=\biggl(\sum_{\stackrel{0\le p_0\le p}{p_0\equiv p\mod 2}}d_{p_0,m}\biggr)
\hat\gamma^{c_1\cdots c_m}\hat\gamma^d
-\biggl(\sum_{\stackrel{0\le p_0\le p}{p_0\equiv p-1\mod 2}}d_{p_0,m}\biggr)
\hat\gamma^d\hat\gamma^{c_1\cdots c_m}.
\label{Gvalue}\end{aligned}$$ Using and , we find the contribution of the prefactors is given by $$\begin{aligned}
&\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{ij}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{k\dot{a}}
=-8\nu^7
\delta_{jk}\gamma^i_{a\dot a}
{\partial\over\partial{\cal Y}^a}\delta^8({\cal Y})
+{\cal O}({\cal Y}^5),\label{coshsinh1}\\
&\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{ij}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{a}k}
=8\nu^7
\delta_{ik}\gamma^j_{a\dot a}
{\partial\over\partial{\cal Y}^a}\delta^8({\cal Y})
+{\cal O}({\cal Y}^5).\label{coshsinh2}\end{aligned}$$ To translate these results into the effective interaction vertices, we need a fermionic expansion formula: $$\begin{aligned}
\left[\frac{\partial}{\partial\xi^a}\delta^8(\xi)\right]
\delta^8(\xi+\eta)e^{\xi^c\zeta^c}
=\left[\frac{\partial}{\partial\xi}
-\frac{\partial}{\partial\eta}-\zeta\right]^a
\delta^8(\xi)\delta^8(\eta),
\label{derive}\end{aligned}$$ which can be proved by $$\begin{aligned}
\xi^b{\partial\over\partial\xi^a}\delta^8(\xi)=
\delta^b_a\delta^8(\xi),\quad
\delta^8(\xi+\eta)=\delta^8(\eta)
+\xi^b\frac{\partial}{\partial\eta^b}\delta^8(\eta)
+{\cal O}(\xi^2),\quad
e^{\xi^c\zeta^c}=1+\xi^c\zeta^c+{\cal O}(\xi^2).\end{aligned}$$ After plugging $\xi^a=(\lambda_1+\lambda_4)^a$, $\eta^a=(\lambda_2+\lambda_5)^a$ and $\zeta^a=-\sqrt{2}\alpha_3\lim_{T\to +0}{\boldsymbol{V}}^{\rm T}
(i{\boldsymbol{S}}^{{\rm II}(45)\dagger}-{\boldsymbol{S}}^{{\rm I}(12)\dagger})$ into for our purpose, we find the expression appearing on the right hand side can be rewritten into $$\begin{aligned}
&\biggl[\frac{\partial}{\partial\lambda_2}
-\frac{\partial}{\partial\lambda_1}
-\sqrt{2}\alpha_3\lim_{T\to +0}
{\boldsymbol{V}}^{\rm T}(i{\boldsymbol{S}}^{{\rm II}(45)\dagger}
-{\boldsymbol{S}}^{{\rm II}(12)\dagger})\biggr]^a
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle\nonumber\\
&\quad=\bigl[\vartheta^{(2)}(\sigma_{2,\rm int})
-\vartheta^{(1)}(\sigma_{1,\rm int})\bigr]^a
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle,\end{aligned}$$ with the fermionic coordinate $$\begin{aligned}
\vartheta^{(r)}(\sigma_r)=\vartheta_r
+\sqrt{\frac{2}{\alpha_r}}\sum_{n=1}^\infty
\biggl(\bigl(S_n^{{\rm I}}+S_{-n}^{{\rm II}}\bigr)
\cos\frac{n\sigma_r}{|\alpha_r|}
+\bigl(S_n^{{\rm II}}-S_{-n}^{{\rm I}}\bigr)
\sin\frac{n\sigma_r}{|\alpha_r|}\biggr),\end{aligned}$$ and $\sigma_{1,\rm int}=\pi\alpha_1$, $\sigma_{2,\rm int}=0$. Therefore, the effective string interaction vertices corresponding to are given as $$\begin{aligned}
&\langle R^{\rm f}(3,6)|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{\dot{a}k}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
-\eta^*2^{\frac{20}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{jk}\gamma^i_{a\dot{a}}}{T^{3/2}}
\bigl[\vartheta^{(2)}(\sigma_{\rm int})
-\vartheta^{(1)}(\sigma_{\rm int})\bigr]^a
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle,
\label{other21tree}\\
&\langle R^{\rm f}(3,6)|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{k\dot{a}}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
\eta^*2^{\frac{20}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{ik}\gamma^j_{a\dot{a}}}{T^{3/2}}
\bigl[\vartheta^{(2)}(\sigma_{\rm int})
-\vartheta^{(1)}(\sigma_{\rm int})\bigr]^a
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle.
\label{other22tree}\end{aligned}$$ Here we have abbreviated $\sigma_{1,{\rm int}}$ and $\sigma_{2,{\rm int}}$ as $\sigma_{\rm int}$.
Finally, let us turn to the effective string interaction vertex corresponding to . We can easily show with the help of that $p+q\ge 7$ does not contribute. Hence, let us consider the case of $p+q=6$. Though it is not easy to find the value of $G^{c_1\cdots c_m,d_1d_2}_p$ or $\tilde G^{d_1d_2,c_1\cdots c_m}_q$ separately, we find in Appendix D that the difference can be evaluated as $$\begin{aligned}
G^{c_1\cdots c_m,d_1d_2}_p-\tilde G^{d_1d_2,c_1\cdots c_m}_p
=\biggl(\sum_{\stackrel{0\le p_0\le p}{p_0\equiv p\mod 2}}
d_{p_0,m}\biggr)
[\hat\gamma^{c_1\cdots c_m},\hat\gamma^{d_1d_2}].
\label{Gdiff}\end{aligned}$$ To use this result properly, let us first combine two expressions of into $$\begin{aligned}
&\frac{1}{p!q!}\bigl(({\cal Y}^a\hat\gamma^a)^p
\hat\gamma^{c_1\cdots c_m}({\cal Y}^b\hat\gamma^b)^q
+({\cal Y}^a\hat\gamma^a)^q
\hat\gamma^{c_1\cdots c_m}({\cal Y}^b\hat\gamma^b)^p\bigr)
\nonumber\\
&\quad=\frac{1}{4}\bigl(G^{c_1\cdots c_m,d_1d_2}_p
+(-1)^m\tilde G^{d_1d_2,c_1\cdots c_m}_p
+G^{c_1\cdots c_m,d_1d_2}_{6-p}
+(-1)^m\tilde G^{d_1d_2,c_1\cdots c_m}_{6-p}\bigr)\hat\gamma_9
\frac{\partial}{\partial{\cal Y}^{d_1}}
\frac{\partial}{\partial{\cal Y}^{d_2}}\delta^8({\cal Y}).
\label{Gpm}\end{aligned}$$ Fortunately, since we only want to consider the case with $m$ being odd because of the index structure of the gamma matrices in the summand of , reduces to the difference of $G^{c_1\cdots c_m,d_1d_2}_p$ and $\tilde G^{d_1d_2,c_1\cdots c_m}_p$. Therefore, we can apply the result of directly to find $$\begin{aligned}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{a}i}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{b}j}
\sim\nu^6
(\hat\gamma^c)_{\dot{a}j}([\hat\gamma^c,\hat\gamma^{d_1d_2}])_{\dot{b}i}
\frac{\partial}{\partial{\cal Y}^{d_1}}
\frac{\partial}{\partial{\cal Y}^{d_2}}\delta^8({\cal Y}),\end{aligned}$$ with the help of . Using the formula for the gamma matrices $(\hat\gamma^c)_{\dot{a}j}([\hat\gamma^c,\hat\gamma^{d_1d_2}])_{\dot{b}i}
=4\hat\gamma^{[d_1}_{j\dot{a}}\hat\gamma^{d_2]}_{i\dot{b}}$, we arrive at the result: $$\begin{aligned}
&\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{a}i}
\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}_{123}\bigr]^{\dot{b}j}
=4\nu^6
\gamma^j_{a\dot{a}}\gamma^i_{b\dot{b}}
{\partial\over\partial{\cal Y}^a}
{\partial\over\partial{\cal Y}^b}\delta^8({\cal Y})
+{\cal O}({\cal Y}^4).
\label{sinhsinh}\end{aligned}$$ To translate our result into the effective interaction vertex, we introduce another fermionic expansion formula similar to $$\begin{aligned}
&\left[\frac{\partial}{\partial\xi^a}\frac{\partial}{\partial\xi^b}
\delta^8(\xi)\right]
\delta^8(\xi+\eta)e^{\xi^c\zeta^c}
=\left[\frac{\partial}{\partial\xi}-\frac{\partial}{\partial\eta}
-\zeta\right]^a
\left[\frac{\partial}{\partial\xi}-\frac{\partial}{\partial\eta}
-\zeta\right]^b
\delta^8(\xi)\delta^8(\eta),
\label{derivative}\end{aligned}$$ which can be proved with $$\begin{aligned}
\xi^c\frac{\partial}{\partial\xi^a}\frac{\partial}{\partial\xi^b}
\delta^8(\xi)
=-\delta^c_b\frac{\partial}{\partial\xi^a}\delta^8(\xi)
+\delta^c_a\frac{\partial}{\partial\xi^b}\delta^8(\xi),\quad
\xi^c\xi^d
\frac{\partial}{\partial\xi^a}\frac{\partial}{\partial\xi^b}
\delta^8(\xi)
=-(\delta^c_a\delta^d_b-\delta^c_b\delta^d_a)\delta^8(\xi).\end{aligned}$$ Finally, using the effective string interaction vertex is expressed as $$\begin{aligned}
&\langle R^{\rm f}(3,6)|
e^{-\frac{T}{|\alpha_3|}(L_0^{(3)}+\bar{L}_0^{(3)})}
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{\dot{a}i}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{j\dot{b}}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle
\nonumber\\
&\quad\sim
-2^{\frac{14}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\gamma^j_{a\dot{a}}\gamma^i_{b\dot{b}}}{T}
\bigl[\vartheta^{(2)}(\sigma_{\rm int})
-\vartheta^{(1)}(\sigma_{\rm int})\bigr]^a
\bigl[\vartheta^{(2)}(\sigma_{\rm int})
-\vartheta^{(1)}(\sigma_{\rm int})\bigr]^b
|R^{\rm f}(1,4)\rangle|R^{\rm f}(2,5)\rangle.
\label{other3tree}\end{aligned}$$ This result exactly takes the form expected from .
One-Loop Amplitude
==================
In the previous section we have computed one realization of the OPE via the tree diagram. Here we would like to proceed to the other realization via the one-loop diagram: the incoming string $6$ splits into two short strings and join again into the outgoing string $3$. (See Fig.2.) Since most of the computations are parallel to the previous section, we will be short in the presentation and put the prime $P'$ on every corresponding quantity $P$ in the tree diagram to make the similarity clear.
\[0.6\][![Two-string one-loop diagram.[]{data-label="fig:loop"}](loop.eps "fig:")]{}
Bosonic sector
--------------
We start with the bosonic sector again. We would like to compute the effective two-string interaction vertex $$\begin{aligned}
&|A^{\prime\rm b}(3,6)\rangle
=\langle R^{\rm b}(1,4)|\langle R^{\rm b}(2,5)|
e^{-{T\over\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})}
e^{-{T\over\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\quad\times
Z_{123}^i|V^{\rm b}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
Z_{456}^k|V^{\rm b}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle,\end{aligned}$$ to see whether the prefactor $Z_{123}^iZ_{456}^k$ gives the extra contribution of $1/T$. As in the previous section, let us first consider the generating function $$\begin{aligned}
&|A^{\prime\rm b}_\beta(3,6)\rangle
=\langle R^{\rm b}(1,4)|\langle R^{\rm b}(2,5)|
e^{-{T\over\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})}
e^{-{T\over\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\quad\times e^{\beta_{123}^iZ_{123}^i+\beta_{456}^kZ_{456}^k}
|V^{\rm b}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
|V^{\rm b}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$ The result of the generating function is $$\begin{aligned}
&|A^{\prime\rm b}_\beta(3,6)\rangle
=(\det{}')^{-8}\int\frac{d^8p_1}{(2\pi)^8}
\int\delta^{\rm b}(3,6)e^{F^{\prime{\rm b}}(3,6,p_1)}\nonumber\\
&\quad\times
e^{\beta_{123}{\cal Z}'_{123}+\beta_{456}{\cal Z}'_{456}
+\frac{1}{2}(\beta_{123}^2+\beta_{456}^2)a'_2+\beta_{123}\beta_{456}b'_2}
|p_3\rangle_3|p_6\rangle_6,\end{aligned}$$ with various expressions denoting $$\begin{aligned}
&\det{}'=\det\Bigl[1-\bigl(e^{-\frac{T}{2\alpha_{12}}C}
N^{12,12}e^{-\frac{T}{2\alpha_{12}}C}\bigr)^2\Bigr],\\
&\int\delta^{\rm b}(3,6)
=\int\frac{d^8p_3}{(2\pi)^8}\frac{d^8p_6}{(2\pi)^8}
(2\pi)^8\delta^8(p_3+p_6),\\
&\lim_{T\to +0}F(3,6,p_1)
=-\bigl({\boldsymbol{a}}^{(3)\dagger{\rm T}}{\boldsymbol{a}}^{(6)\dagger}
+\bar{{\boldsymbol{a}}}^{(3)\dagger{\rm T}}\bar{{\boldsymbol{a}}}^{(6)\dagger}\bigr)
+\Bigl(\lim_{T\to +0}c\Bigr)
\biggl[p_1-\frac{\alpha_1}{\alpha_3}p_3\biggr]^2
\nonumber\\&\quad
+\Bigl(\lim_{T\to +0}{\boldsymbol{C}}^{\rm T}\Bigr)
({\boldsymbol{a}}^{(3)\dagger}-{\boldsymbol{a}}^{(6)\dagger}
+\bar{{\boldsymbol{a}}}^{(3)\dagger}-\bar{{\boldsymbol{a}}}^{(6)\dagger})
\biggl[p_1-\frac{\alpha_1}{\alpha_3}p_3\biggr],\\
&{\cal Z}'_{123}=(1-a'_1-b'_1)\mathbb{P}_{123}
-{\boldsymbol{a}}^{\prime\rm T}{\boldsymbol{a}}^{(3)\dagger}
+{\boldsymbol{b}}^{\prime\rm T}{\boldsymbol{a}}^{(6)\dagger},\\
&{\cal Z}'_{456}=(1-a'_1-b'_1)\mathbb{P}_{123}
-{\boldsymbol{b}}^{\prime\rm T}{\boldsymbol{a}}^{(3)\dagger}
+{\boldsymbol{a}}^{\prime\rm T}{\boldsymbol{a}}^{(6)\dagger}.\end{aligned}$$ Here various quantities are given as $$\begin{aligned}
c&=2\alpha_3^2\Bigl(\frac{T/2-\tau_0}{\alpha_{123}}
+{\boldsymbol{N}}^{12{\rm T}}\circ'\bigl(1-N^{12,12}\bigr)^{-1}_{\circ'}
{\boldsymbol{N}}^{12}\Bigr),\\
{\boldsymbol{C}}&=\alpha_3\Bigl({\boldsymbol{N}}^3+N^{3,12}\circ'
\bigl(1-N^{12,12}\bigr)^{-1}_{\circ'}{\boldsymbol{N}}^{12}\Bigr),\\
a'_1&=\alpha_{123}{\boldsymbol{N}}^{12{\rm T}}(C/\alpha_{12})\circ'
\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
N^{12,12}\circ'{\boldsymbol{N}}^{12},\\
b'_1&=\alpha_{123}{\boldsymbol{N}}^{12{\rm T}}(C/\alpha_{12})\circ'
\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
{\boldsymbol{N}}^{12},\\
a'_2&=(\alpha_{123})^2{\boldsymbol{N}}^{12{\rm T}}(C/\alpha_{12})\circ'
\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
N^{12,12}\circ'(C/\alpha_{12}){\boldsymbol{N}}^{12},\\
b'_2&=(\alpha_{123})^2{\boldsymbol{N}}^{12{\rm T}}(C/\alpha_{12})\circ'
\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
(C/\alpha_{12}){\boldsymbol{N}}^{12},\\
{\boldsymbol{a}}^{\prime\rm T}&=\alpha_{123}\Bigl({\boldsymbol{N}}^{3{\rm T}}
(C/\alpha_3)
+{\boldsymbol{N}}^{12{\rm T}}(C/\alpha_{12})\circ'
\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
N^{12,12}\circ'N^{12,3}\Bigr),\\
{\boldsymbol{b}}^{\prime\rm T}&=\alpha_{123}{\boldsymbol{N}}^{12{\rm T}}
(C/\alpha_{12})\circ'
\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}N^{12,3},\end{aligned}$$ with $\circ'$ denoting the matrix multiplication with $e^{-\frac{T}{\alpha_{12}}C}$ inserted. Again, we dropped the subscript $T$ in $c_T$ and ${\boldsymbol{C}}_T$ from our previous paper [@KMT]. To consider the effective two-string interaction vertex $|A^{\prime\rm b}(3,6)\rangle$, we take the derivative of the generating function $|A^{\prime\rm b}_\beta(3,6)\rangle$ as in the previous section: $$\begin{aligned}
|A^{\prime\rm b}(3,6)\rangle
=\frac{\partial}{\partial\beta_{123}^i}
\frac{\partial}{\partial\beta_{456}^k}
|A^{\prime\rm b}_\beta(3,6)\rangle\biggr|_{\beta=0}
=\bigl(b'_2\delta^{ik}
+{\cal Z}_{123}^{\prime i}{\cal Z}_{456}^{\prime k}\bigr)
|A^{\prime\rm b}_\beta(3,6)\rangle\biggr|_{\beta=0},\end{aligned}$$ where in the last expression the factor $(b'_2\delta^{ik}
+{\cal Z}_{123}^{\prime i}{\cal Z}_{456}^{\prime k})$ should be interpreted to be in the $p_1$ integration of $|A^{\prime\rm b}_\beta(3,6)\rangle$.
Let us consider the short intermediate time limit hereafter. In , we prove that $$\begin{aligned}
\lim_{T\to +0}{\boldsymbol{a}}'=\lim_{T\to +0}{\boldsymbol{b}}'
=\frac{\alpha_{123}}{2}
\biggl[\frac{C}{\alpha_3}\biggr]^{-1}\sum_{t=1}^3
A^{(t)}\biggl[\frac{C}{\alpha_t}\biggr]^2{\boldsymbol{N}}^t,\end{aligned}$$ which, as argued around (C.19) in [@GSB], gives the difference of the delta functions of physically the same point and vanishes essentially. Hence, we can concentrate on the zero-mode part again. Since we have $$\begin{aligned}
{\cal Z}_{123}^{\prime i}\sim{\cal Z}_{456}^{\prime i}
\sim-2b'_1\alpha_3\Bigl(p_1-\frac{\alpha_1}{\alpha_3}p_3\Bigr)^i,\end{aligned}$$ because of $\lim_{T\to 0}(1-a'_1+b'_1)=0$ , the contribution of the zero-mode part is given by $$\begin{aligned}
&|A^{\prime\rm b}(3,6)\rangle\biggr|_0
\sim\int\frac{d^8p_1}{(2\pi)^8}\bigl(b'_2\delta^{ik}
+4b_1^{\prime 2}\alpha_3^2
(p_1-{\scriptstyle\frac{\alpha_1}{\alpha_3}}p_3)^i
(p_1-{\scriptstyle\frac{\alpha_1}{\alpha_3}}p_3)^k
\bigr)\nonumber\\
&\qquad\times e^{c\left(p_1-\frac{\alpha_1}{\alpha_3}p_3\right)^2
+{\boldsymbol{C}}^{\rm T}\left({\boldsymbol{a}}^{(3)\dagger}-{\boldsymbol{a}}^{(6)\dagger}
+\bar{{\boldsymbol{a}}}{}^{(3)\dagger}-\bar{{\boldsymbol{a}}}{}^{(6)\dagger}
\right)\left(p_1-\frac{\alpha_1}{\alpha_3}p_3\right)}.\end{aligned}$$ Comparing $b'_2\sim{\cal O}(T^{-1})$ with $b_1^{\prime 2}/c\sim{\cal O}\bigl(T^{-1}(\log T)^{-1}\bigr)$ due to and , we find that the first term is more singular. Because of $\lim_{T\to +0}({\boldsymbol{C}})_m({\boldsymbol{C}})_n/c=0$ [@KMT] (See also Appendix C.3.), the leading contribution is given by $$\begin{aligned}
|A^{\prime\rm b}(3,6)\rangle\biggr|_0
\sim b'_2\delta^{ik}\cdot\frac{1}{2^8\pi^4(-c)^4}.
\label{loopzeromode}\end{aligned}$$ The behavior of $b'_2$ is roughly $b'_2\sim-\alpha_{123}/(2T)$ in the limit $T\to +0$, so we have found the expected singular behavior again: $$\begin{aligned}
|A^{\prime{\rm b}}(3,6)\rangle
\sim 2^{-\frac{29}{3}}\pi^{-4}\mu^{-\frac{4}{3}}
|\alpha_{123}|^{\frac{5}{3}}
\frac{\delta^{ik}}{T^3}\biggl(\log\frac{T}{|\alpha_3|}\biggr)^{-4}
|R^{\rm b}(3,6)\rangle.
\label{twotauloop}\end{aligned}$$
Fermionic sector
----------------
Let us turn to the fermionic sector of the one-loop diagram. Here we would like to compute the effective two-string interaction vertex $$\begin{aligned}
&|A^{\prime\rm f}(3,6)\rangle
=\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-{T\over\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})}
e^{-{T\over\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\quad\times\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{kl}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$ As in the tree diagram let us consider the generating function first: $$\begin{aligned}
&|A^{\prime{\rm f}}_\phi(3,6)\rangle
=\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-{T\over\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})}
e^{-{T\over\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}
\nonumber\\&
\quad\times e^{\frac{2}{\alpha_{123}}
(\phi_{123}Y_{123}-\phi_{456}Y_{456})}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle.\end{aligned}$$ After applying the Gaussian convolution formula , we find $$\begin{aligned}
&|A^{\prime{\rm f}}_\phi(3,6)\rangle=(\det{}')^8\int\delta^{\rm f}(3,6)
\int d^8\lambda_1
e^{\frac{2}{\alpha_{123}}
(\phi_{123}{\cal Y}'_{123}-\phi_{456}{\cal Y}'_{456})}
e^{F^{\prime\rm f}(3,6)}
|\lambda_3\rangle_3|\lambda_6\rangle_6,\end{aligned}$$ with various expressions defined by $$\begin{aligned}
&F^{\prime\rm f}(3,6)
=\frac{1}{2}{\boldsymbol{S}}^{(36)\dagger{\rm T}}M'{\boldsymbol{S}}^{(36)\dagger}
+{\boldsymbol{k}}^{\prime\rm T}{\boldsymbol{S}}^{(36)\dagger},\label{Ffloop}\\
&{\cal Y}'_{123}=(1-a'_1-b'_1)\Lambda_{123}
-\frac{1}{\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(3)\dagger{\rm T}}
(C/\alpha_{3})^{-\frac{1}{2}}{\boldsymbol{a}}'
+\frac{i}{\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(6)\dagger{\rm T}}
(C/\alpha_{3})^{-\frac{1}{2}}{\boldsymbol{b}}',\\
&{\cal Y}'_{456}=(1-a'_1-b'_1)\Lambda_{123}
+{1\over\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(3)\dagger{\rm T}}
(C/\alpha_{3})^{-{1\over 2}}{\boldsymbol{b}}'
-{i\over\sqrt{2}}{\boldsymbol{S}}^{{\rm I}(6)\dagger{\rm T}}
(C/\alpha_{3})^{-{1\over 2}}{\boldsymbol{a}}'.\end{aligned}$$ Here $M'$ and ${\boldsymbol{k}}^{\prime\rm T}$ are given as $$\begin{aligned}
&M'=\begin{pmatrix}
[A']&-i[B']\\i[B']&[A']\end{pmatrix},\quad
[A']=\begin{pmatrix}
0&-A^{\prime\rm T}\\A'&0\end{pmatrix},\quad
[B']=\begin{pmatrix}
0&B^{\prime\rm T}\\B'&0\end{pmatrix},\\
&{\boldsymbol{k}}^{\prime{\rm T}}=-\sqrt{2}\begin{pmatrix}
0&1&0&i\end{pmatrix}
\Lambda_{123}{\boldsymbol{U}}^{\prime\rm T},
\label{kloop}\end{aligned}$$ with the building blocks being $$\begin{aligned}
&A'=\hat{N}^{3,3}+\hat{N}^{3,12}\circ'\hat{N}^{12,12}\circ'
\bigl(1-(\hat{N}^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
\hat{N}^{12,3},\\
&B'=\hat{N}^{3,12}\circ'
\bigl(1-(\hat{N}^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
\hat{N}^{12,3},\\
&{\boldsymbol{U}}'=\hat{{\boldsymbol{N}}}{}^{3}
+\hat{N}^{3,12}\circ'\bigl(1-\hat{N}^{12,12}\bigr)^{-1}_{\circ'}
\hat{{\boldsymbol{N}}}{}^{12}.\label{bsUprime}\end{aligned}$$ Transforming back to the original effective interaction vertex with , we find our result is given as $$\begin{aligned}
&|A^{\prime{\rm f}}(3,6)\rangle=(\det{}')^8\int\delta^{\rm f}(3,6)
\int d^8\lambda_1
\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{123}'\bigr]^{ij}\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}_{456}'\bigr]^{kl}
e^{F^{\prime\rm f}(3,6)}
|\lambda_3\rangle_3|\lambda_6\rangle_6.
\label{lambda1int}\end{aligned}$$
Note that for $T\to +0$, we have $$\begin{aligned}
\lim_{T\to +0}A'=0,\quad\lim_{T\to +0}B'=1,\quad
\lim_{T\to +0}{\boldsymbol{U}}'={\boldsymbol{0}},\end{aligned}$$ which implies $$\begin{aligned}
&\lim_{T\to +0}F^{\prime{\rm f}}(3,6)
=i(-{\boldsymbol{S}}^{{\rm I}(6)\dagger{\rm T}}{\boldsymbol{S}}^{{\rm II}(3)\dagger}
+{\boldsymbol{S}}^{{\rm I}(3)\dagger{\rm T}}{\boldsymbol{S}}^{{\rm II}(6)\dagger}).\end{aligned}$$ Also, due to which essentially means $\lim_{T\to +0}{\boldsymbol{a}}'=\lim_{T\to +0}{\boldsymbol{b}}'={\boldsymbol{0}}$, we have $$\begin{aligned}
{\cal Y}^{\prime a}_{123}\sim{\cal Y}^{\prime a}_{456}
\sim{\cal Y}^{\prime a},\quad
{\cal Y}^{\prime a}
=-2b'_1\alpha_3
\Bigl(\lambda_1-\frac{\alpha_1}{\alpha_3}\lambda_3\Bigr)^a,\end{aligned}$$ where we have used $\Lambda_{123}=\Lambda_{456}=\alpha_3\lambda_1-\alpha_1\lambda_3$. As in the previous section, this relation implies $$\begin{aligned}
{\!\not\hspace{-1mm}{\cal Y}}'_{456}\sim-i{\!\not\hspace{-1mm}{\cal Y}}'_{123},\quad
\bigl[\cosh-i{\!\not\hspace{-1mm}{\cal Y}}'_{123}\bigr]^{kl}
=\bigl[\cosh{\!\not\hspace{-1mm}{\cal Y}}'_{123}\bigr]^{lk}.\end{aligned}$$ Hence, we can repeat the evaluation with the Fierz identity analogous to and find $$\begin{aligned}
|A^{\prime{\rm f}}(3,6)\rangle
\sim 2^{\frac{26}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{ik}\delta_{jl}}
{T^2}|R^{\rm f}(3,6)\rangle,
\label{Sigmaijklloop}\end{aligned}$$ with the use of . This expression of the effective interaction vertex gives the expected results from the OPE .
Other processes
---------------
Similarly, using $$\begin{aligned}
\bigl[\sinh-i{\!\not\hspace{-1mm}{\cal Y}}'_{123}\bigr]^{\dot{b}j}
=-i\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}'_{123}\bigr]^{j\dot{b}},\quad
\bigl[\sinh-i{\!\not\hspace{-1mm}{\cal Y}}'_{123}\bigr]^{j\dot{b}}
=-i\bigl[\sinh{\!\not\hspace{-1mm}{\cal Y}}'_{123}\bigr]^{\dot{b}j},\end{aligned}$$ and the one-loop analogues of and , we can also evaluate the effective interaction vertices $$\begin{aligned}
&\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\qquad\times\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{\dot{a}i}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{\dot{b}j}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
i2^{\frac{26}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{\dot{a}\dot{b}}\delta_{ij}}{T^2}
|R^{\rm f}(3,6)\rangle,
\label{other11loop}\\
&\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\qquad\times\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{i\dot{a}}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{j\dot{b}}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
-i2^{\frac{26}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{ij}\delta_{\dot{a}\dot{b}}}{T^2}
|R^{\rm f}(3,6)\rangle,
\label{other12loop}\end{aligned}$$ which again give the expected results from the OPEs in .
To evaluate the one-loop diagram corresponding to the OPEs in and , we need more efforts. Using the one-loop analogues of , and , we find that the most singular terms in these cases do not have eight enough $\Lambda_{123}$’s to survive the $\lambda_1$ integration in an expression similar to . Therefore, we need to take $\Lambda_{123}$ in ${\boldsymbol{k}}'$ out of the exponential factor $e^{F^{\prime{\rm f}}(3,6)}$ to compensate the $\lambda_1$ integration.
An explicit asymptotic expression of ${\boldsymbol{U}}'$ is necessary, since $\Lambda_{123}$ in ${\boldsymbol{k}}'$ always appears simultaneously with ${\boldsymbol{U}}'$, when we take $\Lambda_{123}$ out of the exponential part $e^{F^{\prime{\rm f}}(3,6)}$ using the formulas $$\begin{aligned}
\biggl[\frac{\partial}{\partial\xi^a}\delta^8(\xi)\biggr]
e^{\xi^c\zeta^c}
=\biggl[\frac{\partial}{\partial\xi}-\zeta\biggr]^a\delta^8(\xi),\quad
\biggl[\frac{\partial}{\partial\xi^a}
\frac{\partial}{\partial\xi^b}\delta^8(\xi)\biggr]
e^{\xi^c\zeta^c}
=\biggl[\frac{\partial}{\partial\xi}-\zeta\biggr]^a
\biggl[\frac{\partial}{\partial\xi}-\zeta\biggr]^b\delta^8(\xi).\end{aligned}$$ Using ${\boldsymbol{a}}'_0$ and ${\boldsymbol{b}}'_0$ with the definition given by and and the asymptotic expression given by and , we find the asymptotic expression of ${\boldsymbol{U}}'$ is $$\begin{aligned}
{\boldsymbol{U}}'=\sqrt{C/\alpha_3}\frac{{\boldsymbol{a}}'_0+{\boldsymbol{b}}'_0}{\alpha_1\alpha_2}
\sim\sqrt{C/\alpha_3}\frac{\pi^2\alpha_1\alpha_2}
{|\alpha_3|^2\log(T/|\alpha_3|)}C{\boldsymbol{B}},\end{aligned}$$ where we have plugged in the value of $g$ which is determined in Appendix C.4.
Our final result can be summarized by the expression of the fermionic momentum acting on the reflector $$\begin{aligned}
\bigl[\lambda^{(3)}(\sigma)+\lambda^{(6)}(\sigma)\bigr]|R(3,6)\rangle
=\frac{1}{\sqrt{2\alpha_3\pi}}\sum_{n=1}^\infty
(S^{{\rm II}(3)}_{-n}+iS^{{\rm II}(6)}_{-n})
\sin\frac{n\sigma}{|\alpha_3|}|R(3,6)\rangle,\end{aligned}$$ with the fermionic momentum given by $$\begin{aligned}
\lambda^{(r)}(\sigma)=\frac{1}{2\pi|\alpha_r|}\biggl[
\lambda_r+\sqrt{\frac{\alpha_r}{2}}\sum_{n=1}^\infty
\biggl((S^{{\rm II}(r)}_n+S^{{\rm I}(r)}_{-n})
\cos\frac{n\sigma}{|\alpha_r|}
+(S^{{\rm II}(r)}_{-n}-S^{{\rm I}(r)}_n)
\sin\frac{n\sigma}{|\alpha_r|}\biggr)
\biggr].\end{aligned}$$ The results are given by $$\begin{aligned}
&\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\qquad\times\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{\dot{a}k}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
\eta^*2^{\frac{20}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{jk}\gamma^i_{a\dot{a}}}{T^{3/2}}
4\pi\bigl[\lambda^{(3)}(\sigma_{3,{\rm int}})
+\lambda^{(6)}(\sigma_{3,{\rm int}})\bigr]^a
|R^{\rm f}(3,6)\rangle,
\label{other21loop}\\
&\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\qquad\times\bigl[\cosh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{ij}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{k\dot{a}}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
-\eta^*2^{\frac{20}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\delta_{ik}\gamma^j_{a\dot{a}}}{T^{3/2}}
4\pi\bigl[\lambda^{(3)}(\sigma_{3,{\rm int}})
+\lambda^{(6)}(\sigma_{3,{\rm int}})\bigr]^a
|R^{\rm f}(3,6)\rangle,
\label{other22loop}\\
&\langle R^{\rm f}(1,4)|\langle R^{\rm f}(2,5)|
e^{-\frac{T}{\alpha_1}(L_0^{(1)}+\bar{L}_0^{(1)})
-\frac{T}{\alpha_2}(L_0^{(2)}+\bar{L}_0^{(2)})}\nonumber\\
&\qquad\times\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{123}\bigr]^{\dot{a}i}
|V^{\rm f}(1_{\alpha_1},2_{\alpha_2},3_{\alpha_3})\rangle
\bigl[\sinh{\!\not\hspace{-0.5mm}Y}_{456}\bigr]^{j\dot{b}}
|V^{\rm f}(4_{-\alpha_1},5_{-\alpha_2},6_{-\alpha_3})\rangle\nonumber\\
&\quad\sim
2^{\frac{14}{3}}\mu^{\frac{4}{3}}|\alpha_{123}|^{-\frac{2}{3}}
\frac{\gamma^j_{a\dot{a}}\gamma^i_{b\dot{b}}}{T}
4\pi\bigl[\lambda^{(3)}(\sigma_{3,{\rm int}})
+\lambda^{(6)}(\sigma_{3,{\rm int}})\bigr]^a
4\pi\bigl[\lambda^{(3)}(\sigma_{3,{\rm int}})
+\lambda^{(6)}(\sigma_{3,{\rm int}})\bigr]^b
|R^{\rm f}(3,6)\rangle,
\label{other3loop}\end{aligned}$$ with $\sigma_{3,{\rm int}}=\pi\alpha_2$, which match exactly with the OPEs and .
Conclusion
==========
In this paper, we have completed our previous attempts of realizing all the OPEs in MST using the interaction vertices in LCSFT. We have found all the diagrams reproduce the correct OPEs and established the correspondence between LCSFT and MST. Especially, we find the OPEs , , , , are realized by the tree diagrams in , , , , , , and the loop diagrams in , , , , , , . It would be interesting to understand the relation between our current computations and those in [@AF] where the Veneziano amplitude was reproduced from MST.
We have to confess that we do not fully understand why the holomorphic quantity $\theta^a(z)$ and the anti-holomorphic quantity $\bar{\theta}^a(\bar{z})$ in and are realized as $\vartheta^{(2)}(\sigma_{{\rm int}})
-\vartheta^{(1)}(\sigma_{{\rm int}})$ in the tree diagrams while as $4\pi\bigl[\lambda^{(3)}(\sigma_{{\rm int}})
+\lambda^{(6)}(\sigma_{{\rm int}})\bigr]$ in the loop diagrams. Roughly speaking, two sets of fermions are separated as holomorphic $\theta^a(z)$ and anti-holomorphic $\bar{\theta}^a(\bar{z})$ in MST while their linear combinations play the role of the fermionic coordinate $\vartheta(\sigma)$ and the fermionic momentum $\lambda(\sigma)$ in LCSFT. But we cannot make the exact correspondence clear.
Aside from the main result, we have several comments. First of all, the notoriously complicated prefactors in LCSFT are put into much simpler expressions –. As in Appendix A, using these expressions, the supersymmetry algebras are shown easily. We hope these expressions will make LCSFT more accessible to non-experts of the subject.
Secondly, we have performed the computation of the tree and one-loop diagrams. At first sight the computation in the fermionic sector seems impossibly complicated. Fortunately, since the result of the generating function does not have the squared term of the source, we can perform the inverse Fourier transformation without difficulty and write down the result explicitly. We hope this fact will enable other important calculations in LCSFT.
Having acquired enough information of the first order interaction term, we would like to turn to the contact terms next. We wish to report progress in this direction in the near future.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank S. Dobashi, S. Fujii, J. Gomis, H. Hata, K. Murakami, S. Rey and S. Teraguchi for valuable discussions and comments. Discussions during the KEK Theory Workshop 2006 and the YITP workshop YITP-W-06-11 are useful. We are grateful to the organizers of these workshops. S.M. would also like to thank Kyoto University and KEK for hospitality where part of this work was done. This work is supported partly by Grant-in-Aid for Young Scientists (\#18740143) from the Japan Ministry of Education, Culture, Sports, Science and Technology, partly by Nishina Memorial Foundation, partly by Inamori Foundation and partly by funds provided by the US Department of Energy (DOE) under cooperative research agreement DE-FG02-05ER41360. The calculations of various Neumann coefficients using [*Mathematica*]{} were partly carried out on sushiki at YITP in Kyoto University.
Prefactors
==========
In this section, we would like to recapitulate the prefactors of the Green-Schwarz-Brink light-cone superstring field theory. The prefactors were thought to be notoriously complicated. We would like to show here that we can simplify the expressions of the prefactors in our new notation. First of all, we note that due to the triality of $SO(8)$ we can construct the gamma matrices with the spinor indices $$\begin{aligned}
\hat{\gamma}{}^a
=\begin{pmatrix}0&\hat{\gamma}^a_{i\dot{a}}\\
\hat{\gamma}^a_{\dot{a}i}&0\end{pmatrix},\end{aligned}$$ by the gamma matrices with the vector indices $\hat{\gamma}^a_{i\dot{a}}=\hat{\gamma}^a_{\dot{a}i}
\equiv\gamma^i_{a\dot{a}}$. Here we have used $i,j,k,\cdots$ to represent the vector indices, $a,b,c,\cdots$ to represent the spinor indices and $\dot{a},\dot{b},\dot{c},\cdots$ to represent the cospinor indices. The new gamma matrices with the spinor indices satisfy the standard anti-commutation relations: $$\begin{aligned}
\hat{\gamma}^a_{i\dot{a}}\hat{\gamma}^b_{\dot{a}j}
+\hat{\gamma}^b_{i\dot{a}}\hat{\gamma}^a_{\dot{a}j}
=2\delta^{ab}\delta_{ij},\quad
\hat{\gamma}^a_{\dot{a}i}\hat{\gamma}^b_{i\dot{b}}
+\hat{\gamma}^b_{\dot{a}i}\hat{\gamma}^a_{i\dot{b}}
=2\delta^{ab}\delta_{\dot{a}\dot{b}}.
\label{spinorgamma}\end{aligned}$$ If we define ${\!\not\hspace{-0.5mm}Y}$ as $$\begin{aligned}
{\!\not\hspace{-0.5mm}Y}=\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a\hat{\gamma}{}^a
=\begin{pmatrix}0&{\!\not\hspace{-0.5mm}Y}_{i\dot a}\\
{\!\not\hspace{-0.5mm}Y}_{\dot{a}i}&0\end{pmatrix},\end{aligned}$$ using the modified gamma matrices $\hat{\gamma}{}^a$, we find the complicated prefactors of Hamiltonian and two supercharges, $v^{ji}(Y)$, $s^{i\dot{a}}(Y)$, $\tilde{s}^{i\dot{a}}(Y)$ as well as the auxiliary quantity $m^{\dot{a}\dot{b}}(Y)$ can be written as $$\begin{aligned}
v^{ji}(Y)=\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij},\quad
&m^{\dot{a}\dot{b}}(Y)=\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}\dot{b}},\\
s^{i\dot a}(Y)
=\sqrt{-\alpha_{123}}\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i},\quad
&\tilde s^{i\dot a}(Y)
=i\sqrt{-\alpha_{123}}\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}},\end{aligned}$$ where the indices of the function are consistent because $\cosh$ is an even function while $\sinh$ is an odd function.
Let us show that the supersymmetry algebra can be proved easily with our new notation hereafter. The $Y^a$ derivative and the $Y^a$ multiplication are paired into two anti-commuting operators $D^a$ and $D^{*a}$. We shall modify the definition of two anti-commuting operators slightly by $$\begin{aligned}
D^a=i\sqrt{\frac{-\alpha_{123}}{2}}D_+^a,\quad
D^{*a}=-\sqrt{\frac{-\alpha_{123}}{2}}D_-^a,\end{aligned}$$ with $$\begin{aligned}
D_\pm^a=\sqrt{\frac{-\alpha_{123}}{2}}\frac{1}{\eta^*}
\frac{\partial}{\partial Y^a}
\pm\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a.\end{aligned}$$ Then we can easily find how the operators $D^a_\pm$ act on $\cosh{\!\not\hspace{-0.5mm}Y}$ and $\sinh{\!\not\hspace{-0.5mm}Y}$. Since the derivative $\partial/\partial Y^a$ can act on any ${\!\not\hspace{-0.5mm}Y}$ in the polynomial expansion of the hyperbolic functions, we need a formula to bring $\hat{\gamma}^a$ to the most left side or the most right side of the expression. By iterative use of , we find $$\begin{aligned}
&{\!\not\hspace{-0.5mm}Y}^k\hat{\gamma}^a-(-1)^k\hat{\gamma}^a{\!\not\hspace{-0.5mm}Y}^k
=(-1)^{k-1}2k\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a{\!\not\hspace{-0.5mm}Y}^{k-1}
=2k{\!\not\hspace{-0.5mm}Y}^{k-1}\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a,\\
&\sqrt{\frac{-\alpha_{123}}{2}}\frac{1}{\eta^*}
\frac{\partial}{\partial Y^a}{\!\not\hspace{-0.5mm}Y}^k
=k\hat{\gamma}^a{\!\not\hspace{-0.5mm}Y}^{k-1}
-k(k-1)\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a{\!\not\hspace{-0.5mm}Y}^{k-2}\nonumber\\
&\qquad\qquad\qquad=(-1)^{k-1}\biggl(k{\!\not\hspace{-0.5mm}Y}^{k-1}\hat{\gamma}^a
-k(k-1){\!\not\hspace{-0.5mm}Y}^{k-2}\sqrt{\frac{2}{-\alpha_{123}}}\eta^*Y^a\biggr),\end{aligned}$$ for $k=1,2,\cdots$. The results of the computation are given as $$\begin{aligned}
&D_+^a\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij}
=\bigl[\hat\gamma^a\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij},\quad
D_-^a\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{ij}
=-\bigl[(\sinh{\!\not\hspace{-0.5mm}Y})\hat\gamma^a\bigr]^{ij},\\
&D_+^a\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}\dot{b}}
=\bigl[\hat\gamma^a\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}\dot{b}},\quad
D_-^a\bigl[\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}\dot{b}}
=-\bigl[(\sinh{\!\not\hspace{-0.5mm}Y})\hat\gamma^a\bigr]^{\dot{a}\dot{b}},\\
&D_+^a\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i}
=\bigl[\hat\gamma^a\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i},\quad
D_-^a\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{\dot{a}i}
=\bigl[(\cosh{\!\not\hspace{-0.5mm}Y})\hat\gamma^a\bigr]^{\dot{a}i},\\
&D_+^a\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}}
=\bigl[\hat\gamma^a\cosh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}},\quad
D_-^a\bigl[\sinh{\!\not\hspace{-0.5mm}Y}\bigr]^{i\dot{a}}
=\bigl[(\cosh{\!\not\hspace{-0.5mm}Y})\hat\gamma^a\bigr]^{i\dot{a}},\end{aligned}$$ which imply $$\begin{aligned}
&D^av^{ij}(Y)=\frac{i}{\sqrt{2}}\gamma^j_{a\dot{a}}s^{i\dot{a}}(Y),\quad
D^{*a}v^{ij}(Y)
=-\frac{i}{\sqrt{2}}\gamma^i_{a\dot{a}}\tilde{s}^{j\dot{a}}(Y),\\
&D^am^{\dot{a}\dot{b}}(Y)
=\frac{1}{\sqrt{2}}\gamma^i_{a\dot{a}}\tilde{s}^{i\dot{b}}(Y),\quad
D^{*a}m^{\dot{a}\dot{b}}(Y)
=\frac{1}{\sqrt{2}}\gamma^i_{a\dot{b}}s^{i\dot{a}}(Y),\\
&D^as^{i\dot{a}}(Y)
=-\frac{i\alpha_{123}}{\sqrt{2}}\gamma^j_{a\dot{a}}v^{ij}(Y),\quad
D^{*a}s^{i\dot{a}}(Y)
=\frac{\alpha_{123}}{\sqrt{2}}
\gamma^i_{a\dot{b}}m^{\dot{a}\dot{b}}(Y),\label{Ds}\\
&D^a\tilde{s}^{i\dot{a}}(Y)
=\frac{\alpha_{123}}{\sqrt{2}}
\gamma^i_{a\dot{b}}m^{\dot{b}\dot{a}}(Y),\quad
D^{*a}\tilde{s}^{i\dot{a}}(Y)
=\frac{i\alpha_{123}}{\sqrt{2}}\gamma^j_{a\dot{a}}v^{ji}(Y).
\label{Dtildes}\end{aligned}$$ Using and we can further show $$\begin{aligned}
&\sqrt{2}\alpha_{123}\delta_{\dot{a}\dot{b}}v^{ij}(Y)
=i\gamma^j_{a\dot{a}}D^as^{i\dot{b}}(Y)
+i\gamma^j_{a\dot{b}}D^as^{i\dot{a}}(Y)
=-i\gamma^i_{a\dot{a}}D^{*a}\tilde{s}^{j\dot{b}}(Y)
-i\gamma^i_{a\dot{b}}D^{*a}\tilde{s}^{j\dot{a}}(Y),\\
&\sqrt{2}\alpha_{123}\delta^{ij}m^{\dot{a}\dot{b}}(Y)
=\gamma^j_{a\dot{a}}D^a\tilde{s}^{i\dot{b}}(Y)
+\gamma^i_{a\dot{a}}D^a\tilde{s}^{j\dot{b}}(Y)
=\gamma^j_{a\dot{b}}D^{*a}s^{i\dot{a}}(Y)
+\gamma^i_{a\dot{b}}D^{*a}s^{j\dot{a}}(Y).\end{aligned}$$ All these formulas are sufficient to prove the supersymmetry algebra.
Neumann coefficient matrices
============================
Convention
----------
We would like to present the definition of various Neumann coefficient matrices in this appendix, in order to fix the convention used in this paper as well as to make preparations for the next appendix.
In [@GS] the overlapping condition was rewritten in terms of the mode expansion and the matrices $A^{(1)}$, $A^{(2)}$, ${\boldsymbol{B}}$ and $C$ were introduced as $$\begin{aligned}
(A^{(1)})_{mn}&=\sqrt{\frac{n}{m}}\frac{(-1)^m}{\pi\alpha_1}
\!\int_0^{\pi\alpha_1}\!\!\!d\sigma\,
2\cos\frac{n\sigma}{\alpha_1}\cos\frac{m\sigma}{\alpha_3}
=\sqrt{\frac{m}{n}}\frac{(-1)^m}{\pi\alpha_3}
\!\int_0^{\pi\alpha_1}\!\!\!d\sigma\,
2\sin\frac{n\sigma}{\alpha_1}\sin\frac{m\sigma}{\alpha_3},\\
(A^{(2)})_{mn}&=\sqrt{\frac{n}{m}}\frac{(-1)^m}{\pi\alpha_2}
\!\int_{\pi\alpha_1}^{\pi(\alpha_1+\alpha_2)}\!\!\!d\sigma\,
2\cos\frac{n(\sigma-\pi\alpha_1)}{\alpha_2}
\cos\frac{m\sigma}{\alpha_3}\nonumber\\
&=\sqrt{\frac{m}{n}}\frac{(-1)^m}{\pi\alpha_3}
\!\int_{\pi\alpha_1}^{\pi(\alpha_1+\alpha_2)}\!\!\!d\sigma\,
2\sin\frac{n(\sigma-\pi\alpha_1)}{\alpha_2}
\sin\frac{m\sigma}{\alpha_3},\\
(\boldsymbol{B})_m&=\frac{2(-1)^{m+1}}{\sqrt{m}\pi\alpha_1\alpha_2}
\!\int_0^{\pi\alpha_1}\!\!\!d\sigma\,\cos\frac{m\sigma}{\alpha_3}
=\frac{2(-1)^{m}}{\sqrt{m}\pi\alpha_1\alpha_2}
\!\int_{\pi\alpha_1}^{\pi(\alpha_1+\alpha_2)}\!\!\!d\sigma\,
\cos\frac{m\sigma}{\alpha_3},\\
(C)_{mn}&=m\delta_{mn}.\end{aligned}$$ In terms of these matrices, the Neumann coefficient matrices are given as $$\begin{aligned}
N^{r,s}=\delta^{rs}-2A^{(r)\rm T}\Gamma^{-1}A^{(s)},\quad
\boldsymbol{N}^r=-A^{(r)\rm T}\Gamma^{-1}\boldsymbol{B},\quad
\Gamma=1+A^{(1)}A^{(1)\rm T}+A^{(2)}A^{(2)\rm T},
\label{Neumann}\end{aligned}$$ if we define $(A^{(3)})_{mn}=\delta_{mn}$ in addition. It was found in [@GS] by explicit computation that these matrices satisfy the relations ($r,s=1,2$) $$\begin{aligned}
&-\frac{\alpha_r}{\alpha_3}
A^{(r){\rm T}}CA^{(s)}
=\delta_{rs}C,\quad
A^{(r){\rm T}}C\boldsymbol{B}=0,\quad
\frac{1}{2}\alpha_1\alpha_2
\boldsymbol{B}^{{\rm T}}C\boldsymbol{B}=1,\quad
-\frac{\alpha_3}{\alpha_r}
A^{(r){\rm T}}\frac{1}{C}A^{(s)}=\delta_{rs}\frac{1}{C},
\nonumber\\
&\qquad\qquad\sum_{t=1}^3\alpha_tA^{(t)}\frac{1}{C}A^{(t){\rm T}}
=\frac{1}{2}\alpha_1\alpha_2\alpha_3
\boldsymbol{B}\boldsymbol{B}^{{\rm T}},\quad
\sum_{t=1}^3\frac{1}{\alpha_t}A^{(t)}CA^{(t){\rm T}}=0.
\label{unitary}\end{aligned}$$ As was pointed out in [@KMT; @GMP3], these relations can simply be interpreted as the unitarity of the overlapping transformation between the incoming and outgoing strings where no information is lost. Due to we can also prove the following relations without difficulty. $$\begin{aligned}
\sum_{t=1}^3N^{r,t}N^{t,s}=\delta_{r,s},\quad
\sum_{t=1}^3N^{r,t}{\boldsymbol{N}}^t=-{\boldsymbol{N}}^r.
\label{NN}\end{aligned}$$ As in [@KMW1], we adopt $$\begin{aligned}
\bigl(A^{(12)}\bigr)^{-1}
=-\bigl(C/\alpha_{12}\bigr)A^{(12){\rm T}}\bigl(C/\alpha_3\bigr)^{-1},
\label{inverse}\end{aligned}$$ to be the inverse of $A^{(12)}=\begin{pmatrix}A^{(1)}&A^{(2)}\end{pmatrix}$, since we can show that it is a right inverse as well as a left inverse by applying .
Tree diagram formulas
---------------------
Here we would like to prove some preliminary formulas $$\begin{aligned}
\lim_{T\to +0}{\boldsymbol{a}}=\lim_{T\to +0}{\boldsymbol{b}}={\boldsymbol{0}},\quad
\lim_{T\to +0}\bigl((1-a_1)-b_1\bigr)=0,
\label{aabb}\end{aligned}$$ which appear in the main text and will also be necessary in the next appendix.
Using we find that $$\begin{aligned}
&\lim_{T\to +0}{\boldsymbol{a}}^{\rm T}=\alpha_{123}
\biggl({\boldsymbol{N}}^{12{\rm T}}\frac{C}{\alpha_{12}}
+{\boldsymbol{N}}^{3{\rm T}}\frac{C}{\alpha_3}
N^{3,3}\bigl(N^{12,3}\bigr)^{-1}\biggr),\\
&\lim_{T\to +0}{\boldsymbol{b}}^{\rm T}=\alpha_{123}
{\boldsymbol{N}}^{3{\rm T}}\frac{C}{\alpha_3}\bigl(N^{12,3}\bigr)^{-1}.\end{aligned}$$ With the help of the expression for $N^{r,s}$ in , we can put the above two expressions into $$\begin{aligned}
&\lim_{T\to +0}{\boldsymbol{a}}^{\rm T}=\alpha_{123}
\biggl({\boldsymbol{N}}^{12{\rm T}}\frac{C}{\alpha_{12}}
-\frac{1}{2}{\boldsymbol{N}}^{3{\rm T}}\frac{C}{\alpha_3}
\Gamma\bigl(A^{(12){\rm T}}\bigr)^{-1}
+{\boldsymbol{N}}^{3{\rm T}}\frac{C}{\alpha_3}
\bigl(A^{(12){\rm T}}\bigr)^{-1}\biggr),\label{alim}\\
&\lim_{T\to +0}{\boldsymbol{b}}^{\rm T}
=-\frac{\alpha_{123}}{2}{\boldsymbol{N}}^{3{\rm T}}\frac{C}{\alpha_3}
\Gamma\bigl(A^{(12){\rm T}}\bigr)^{-1}.\label{blim}\end{aligned}$$ If we plug in the expression for $(A^{(12)})^{-1}$ we find the first term and the last term of cancel each other. Therefore both the expressions and reduce to the same form. Furthermore, if we plug in the expression of $\Gamma$ and $(A^{(12)})^{-1}$ , we obtain $$\begin{aligned}
\lim_{T\to +0}{\boldsymbol{a}}^{\rm T}=\lim_{T\to +0}{\boldsymbol{b}}^{\rm T}
=-\frac{\alpha_{123}}{2}\biggl(
-{\boldsymbol{N}}^{3{\rm T}}A^{(12)}\frac{C}{\alpha_{12}}
+{\boldsymbol{N}}^{3{\rm T}}\frac{C}{\alpha_3}A^{(12)}\biggr).\end{aligned}$$ As in [@GS], from the definition of $\Gamma$ , we can easily compute $\Gamma CA^{(12)}$ using : $$\begin{aligned}
\Gamma CA^{(12)}=CA^{(12)}-\alpha_3A^{(12)}C/\alpha_{12}.\end{aligned}$$ By multiplying ${\boldsymbol{B}}^{\rm T}\Gamma^{-1}$ from the left, we find finally $$\begin{aligned}
\lim_{T\to +0}{\boldsymbol{a}}^{\rm T}=\lim_{T\to +0}{\boldsymbol{b}}^{\rm T}
={\boldsymbol{0}}^{\rm T}.\end{aligned}$$
For the second formula of let us repeat our formal computation: $$\begin{aligned}
\lim_{T\to +0}\bigl((1-a_1)-b_1\bigr)
=1-\frac{\alpha_1\alpha_2}{2}{\boldsymbol{B}}^{\rm T}\frac{1}{\Gamma}C{\boldsymbol{B}}.\end{aligned}$$ Since computation of $\Gamma C{\boldsymbol{B}}$ with and leads to $\Gamma C{\boldsymbol{B}}=C{\boldsymbol{B}}$, we find $$\begin{aligned}
{\boldsymbol{B}}^{\rm T}\frac{1}{\Gamma}C{\boldsymbol{B}}={\boldsymbol{B}}^{\rm T}C{\boldsymbol{B}},\end{aligned}$$ which combined with implies the second formula.
Loop diagram formulas
---------------------
Let us turn to the proof of $$\begin{aligned}
\lim_{T\to +0}{\boldsymbol{a}}^{\prime{\rm T}}
=\lim_{T\to +0}{\boldsymbol{b}}^{\prime{\rm T}}
=\frac{\alpha_{123}}{2}\sum_{t=1}^3{\boldsymbol{N}}^{t{\rm T}}
\biggl[\frac{C}{\alpha_r}\biggr]^2A^{(r){\rm T}}
\biggl[\frac{C}{\alpha_3}\biggr]^{-1}\!\!\!,
\quad\lim_{T\to +0}\bigl((1-a'_1)+b'_1\bigr)=0,
\label{aabbprime}\end{aligned}$$ in this subsection. The proof is parallel to the previous subsection. For the first formula, we find both of the expressions reduce to $$\begin{aligned}
\lim_{T\to +0}{\boldsymbol{a}}^{\prime{\rm T}}
=\lim_{T\to +0}{\boldsymbol{b}}^{\prime{\rm T}}
=-\frac{\alpha_{123}}{2}{\boldsymbol{N}}^{12{\rm T}}
\frac{C}{\alpha_{12}}\bigl(A^{(12)}\bigr)^{-1}\Gamma.\end{aligned}$$ Plugging the expression of $\Gamma$ and $(A^{(12)})^{-1}$, we find the result does not vanish but gives instead this time. The second formula can also be proved similarly.
Small time behavior of the matrix products
==========================================
Tree diagram formulas
---------------------
In this subsection we would like to evaluate $a_1$, $b_1$ and $b_2$, which is necessary for our analysis of small intermediate time behavior of the tree diagram amplitude. Let us define as in [@HIKKO2] $(i,j\geq 0)$ $$\begin{aligned}
&\bar{a}_{i,j}=\alpha_1\alpha_2{\boldsymbol{N}}^{3{\rm T}}
C^i\circ
N^{3,3}\circ\left(1-(N^{3,3})^2_\circ\right)^{-1}_\circ
C^j{\boldsymbol{N}}^3,\label{aij}\\
&\bar{b}_{i,j}=\alpha_1\alpha_2{\boldsymbol{N}}^{3{\rm T}}
C^i\circ
\left(1-(N^{3,3})^2_\circ\right)^{-1}_\circ C^j
{\boldsymbol{N}}^3.\label{bij}\end{aligned}$$ Then, according to [@HIKKO2] we can show that these quantities satisfy the relations $$\begin{aligned}
&|\alpha_3|{\partial\over\partial T}\log\det=-\bar{a}_{1,1},
\label{logdettree}\\
&|\alpha_3|{\partial\over\partial T}\bar{a}_{i,j}
=\bar{b}_{i,1}\bar{b}_{1,j},\label{atree}\\
&|\alpha_3|{\partial\over\partial T}\bar{b}_{i,j}
=\bar{b}_{i,1}\bar{a}_{1,j}-\bar{b}_{i,j+1},\label{btree}\end{aligned}$$ using the decomposition formula [@GS] $$\begin{aligned}
\frac{C}{\alpha_r}N^{r,s}+N^{r,s}\frac{C}{\alpha_s}
=-\alpha_1\alpha_2\alpha_3\frac{C}{\alpha_r}{\boldsymbol{N}}^r
{\boldsymbol{N}}^{s\rm T}\frac{C}{\alpha_s}.
\label{decompose}\end{aligned}$$ Combining with our results from the bosonic case [@KMT] $$\begin{aligned}
\det\sim 2^{-\frac{5}{12}}\mu^{\frac{1}{6}}
\biggl[{T\over|\alpha_{123}|^{1/3}}\biggr]^{\frac{1}{4}},\quad
\bar{b}_{0,0}\sim-2\frac{\alpha_1\alpha_2}{\alpha_3^2}
\log{T\over|\alpha_3|},\end{aligned}$$ we have especially $$\begin{aligned}
&1-a_1=1-\bar{a}_{1,0}\sim
\sqrt{\frac{2\alpha_1\alpha_2}{\alpha_3^2}{|\alpha_3|\over T}},\quad
b_1=\bar{b}_{1,0}\sim
\sqrt{\frac{2\alpha_1\alpha_2}{\alpha_3^2}{|\alpha_3|\over T}},
\label{a_1b_1}\\
&b_2=\alpha_1\alpha_2\bar{b}_{1,1}\sim
-{\alpha_{123}\over 2T}.\label{b_2}\end{aligned}$$
Loop diagram formulas\[sec:loop diagram\]
-----------------------------------------
As the Neumann matrix products and are defined in [@HIKKO2] to analyze the tree diagram amplitude, let us define $$\begin{aligned}
\bar{a}'_{i,j}&=(\alpha_1\alpha_2)^{\frac{i+j+1}{2}}\alpha_3
{\boldsymbol{N}}^{12{\rm T}}
\left[C/\alpha_{12}\right]^i
\circ'N^{12,12}\circ'
\left(1-(N^{12,12)})^2_{\circ'}\right)^{-1}_{\circ'}
\left[C/\alpha_{12}\right]^j
{\boldsymbol{N}}^{12},\\
\bar{b}'_{i,j}&=(\alpha_1\alpha_2)^{\frac{i+j+1}{2}}\alpha_3
{\boldsymbol{N}}^{12{\rm T}}
\left[C/\alpha_{12}\right]^i\circ'
\left(1-(N^{12,12})^2_{\circ'}\right)^{-1}_{\circ'}
\left[C/\alpha_{12}\right]^j{\boldsymbol{N}}^{12},\end{aligned}$$ for the loop diagram amplitude. These quantities satisfy the following identities: $$\begin{aligned}
&\sqrt{\alpha_1\alpha_2}{\partial\over\partial T}\log\det{}'
=-\bar{a}'_{1,1},\label{logdetloop}\\
&\sqrt{\alpha_1\alpha_2}{\partial\over\partial T}\bar{a}'_{i,j}
=\bar{b}'_{i,1}\bar{b}'_{1,j},\label{aloop}\\
&\sqrt{\alpha_1\alpha_2}{\partial\over\partial T}\bar{b}'_{i,j}
=\bar{b}'_{i,1}\bar{a}'_{1,j}-\bar{b}'_{i,j+1},\label{bloop}\end{aligned}$$ which imply especially the following relations: $$\begin{aligned}
&\alpha_1\alpha_2{\partial^2\over\partial T^2}\log\det{}'
=-(\bar{b}'_{1,1})^2\\
&\alpha_1\alpha_2{\partial^2\over\partial T^2}
(\bar{a}'_{0,0}+\bar{b}'_{0,0})
=\bar{b}'_{1,1}\left(\bar{b}'_{1,0}-(1-\bar{a}'_{1,0})\right)^2,\\
&\sqrt{\alpha_1\alpha_2}{\partial\over\partial T}
\left(\bar{b}'_{1,0}-(1-\bar{a}'_{1,0})\right)
=\bar{b}'_{1,1}\left(\bar{b}'_{1,0}-(1-\bar{a}'_{1,0})\right).\end{aligned}$$ Combining with the results from our bosonic analysis [@KMT], $$\begin{aligned}
&\sqrt{-c}\det{}'\sim 2^{\frac{1}{12}}\mu^{\frac{1}{6}}
\left[{T\over|\alpha_{123}|^{1/3}}
\left(\log{T\over|\alpha_3|}\right)^2\right]^{\frac{1}{4}},
\label{cdet}\\
&c=\frac{\alpha_3(T-2\tau_0)}{\alpha_1\alpha_2}
+\frac{2\alpha_3}{\sqrt{\alpha_1\alpha_2}}
(\bar{a}'_{0,0}+\bar{b}'_{0,0}),\end{aligned}$$ we find that the quantities $a'_1=\bar{a}'_{1,0}$, $b'_1=\bar{b}'_{1,0}$ and $b'_2=\sqrt{\alpha_1\alpha_2}\alpha_3\bar{b}'_{1,1}$ appearing in the main text should satisfy $$\begin{aligned}
&\left(b'_2\over\alpha_{123}\right)^2\sim{1\over 4T^2}
\biggl[\,1+2\biggl(\log\frac{T}{|\alpha_3|}\biggr)^{\!-1}
\!\!\!\!\!+2\biggl(\log\frac{T}{|\alpha_3|}\biggr)^{\!-2}\,\biggr]
+{1\over 2}{\partial^2\over\partial T^2}\log(-c),\\
&{\alpha_1\alpha_2\over\alpha_3}{\partial^2\over\partial T^2}c
=2{b'_2\over\alpha_{123}}\bigl(b'_1-(1-a'_1)\bigr)^2,\\
&{\alpha_{123}\over b'_2}{\partial\over\partial T}
\bigl(b'_1-(1-a'_1)\bigr)=b'_1-(1-a'_1).\end{aligned}$$ Solving the asymptotic behavior by first adopting the ansatz of the Laurent expansion of $T$ and then correcting by the Laurent expansion of $\log T$, we find that $$\begin{aligned}
&b'_1-(1-a'_1)\sim\frac{g}{\sqrt{T}}
\biggl(\log{T\over|\alpha_3|}\biggr)^{\!-1},\quad
{b'_2\over\alpha_{123}}\sim-\frac{1}{T}\biggl[\,{1\over 2}
+\biggl(\log{T\over|\alpha_3|}\biggr)^{\!-1}\,\biggr],\quad
\label{b'_2}\\
&{\alpha_1\alpha_2\over\alpha_3}c\sim-g^2
\biggl(\log{T\over|\alpha_3|}\biggr)^{\!-1},
\label{c}\end{aligned}$$ where $g$ is an undetermined constant independent of the intermediate time $T$. Moreover, due to , we find explicitly $$\begin{aligned}
b'_1\sim\frac{g}{2\sqrt{T}}
\biggl(\log{T\over|\alpha_3|}\biggr)^{-1},\quad
1-a'_1\sim-\frac{g}{2\sqrt{T}}
\biggl(\log{T\over|\alpha_3|}\biggr)^{-1}.
\label{a'_1b'_1}\end{aligned}$$ Note that a combination $2b'_1/\sqrt{-c}$ does not depend on the undetermined constant $g$. Combining with we find especially $$\begin{aligned}
\bigl(2b'_1\det{}'\bigr)^8
\sim\frac{2^{\frac{2}{3}}\mu^{\frac{4}{3}}}
{|\alpha_{123}|^{2/3}T^2}
\left(\frac{\alpha_1\alpha_2}{\alpha_3}\right)^4,
\label{bdet}\end{aligned}$$ which appears in the main text.
Some Identities
---------------
In this subsection let us make a small digression to clarify several relations of the Neumann coefficient products. As a result, among others, we will show $$\begin{aligned}
\lim_{T\to+0}\frac{({\boldsymbol{C}})_m({\boldsymbol{C}})_n}{c}=0,
\label{CC/c}\end{aligned}$$ which was conjectured in [@KMT] and is also needed in our computation in . Our result in this subsection will also enable the evaluation of $g$ in the next subsection.
We start with proving $$\begin{aligned}
(1-a'_1)^2-b_1^{\prime 2}=1.
\label{a1b1}\end{aligned}$$ Our strategy is basically the same as the derivation of the differential equations – and –. First of all let us rewrite $a'_1$ as follows: $$\begin{aligned}
a'_1=\frac{\alpha_{123}}{2}{\boldsymbol{N}}^{12\rm T}
\biggl[\frac{C}{\alpha_{12}}\frac{N^{12,12}}{1-(N^{12,12})^2}
+\frac{N^{12,12}}{1-(N^{12,12})^2}\frac{C}{\alpha_{12}}\biggr]
{\boldsymbol{N}}^{12}.
\label{a1sym}\end{aligned}$$ Here (until ) note that $\circ'$ in the multiplication between Neumann matrices is implicit. We omit it shortly just to simplify our notation. The key point is to regard the decomposition formula as an anti-commutation relation between the Neumann coefficient matrix $N^{12,12}$ and $C/\alpha_{12}$ and move $C/\alpha_{12}$ all the way from the right to the left. The quantity in the square bracket of is given as ($C_{12}=C/\alpha_{12}$) $$\begin{aligned}
\bigl[\cdots\bigr]&=C_{12}N^{12,12}+N^{12,12}C_{12}\nonumber\\
&\qquad+C_{12}(N^{12,12})^3+N^{12,12}C_{12}(N^{12,12})^2
-N^{12,12}C_{12}(N^{12,12})^2-(N^{12,12})^2C_{12}N^{12,12}\nonumber\\
&\qquad\qquad+(N^{12,12})^2C_{12}N^{12,12}+(N^{12,12})^3C_{12}\nonumber\\
&\qquad+\cdots,\end{aligned}$$ which can be resumed into $$\begin{aligned}
\bigl[\cdots\bigr]
&=-\alpha_{123}\biggl\{
\frac{1}{1-(N^{12,12})^2}\frac{C}{\alpha_{12}}{\boldsymbol{N}}^{12}
{\boldsymbol{N}}^{12\rm T}\frac{C}{\alpha_{12}}\frac{1}{1-(N^{12,12})^2}
\nonumber\\&\qquad
-\frac{N^{12,12}}{1-(N^{12,12})^2}\frac{C}{\alpha_{12}}{\boldsymbol{N}}^{12}
{\boldsymbol{N}}^{12\rm T}\frac{C}{\alpha_{12}}\frac{N^{12,12}}{1-(N^{12,12})^2}
\biggr\},
\label{resum}\end{aligned}$$ where we have used the decomposition formula . This implies that can be expressed as $$\begin{aligned}
a'_1=\frac{1}{2}\bigl(-(b'_1)^2+(a'_1)^2\bigr),\end{aligned}$$ which is exactly what we want in .
Similarly, if we further define ${\boldsymbol{a}}'_0$ and ${\boldsymbol{b}}'_0$ as $$\begin{aligned}
&{\boldsymbol{a}}'_0=\alpha_1\alpha_2\Bigl({\boldsymbol{N}}^3
+N^{3,12}\circ'\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
N^{12,12}\circ'{\boldsymbol{N}}^{12}\Bigr),\label{a0loop}\\
&{\boldsymbol{b}}'_0=\alpha_1\alpha_2
N^{3,12}\circ'\bigl(1-(N^{12,12})^2_{\circ'}\bigr)^{-1}_{\circ'}
{\boldsymbol{N}}^{12},\label{b0loop}\end{aligned}$$ we can prove the following formulas algebraically, $$\begin{aligned}
&{\boldsymbol{a}}'-C{\boldsymbol{a}}'_0={\boldsymbol{a}}'a'_1-{\boldsymbol{b}}'b'_1,\\
&{\boldsymbol{b}}'+C{\boldsymbol{b}}'_0=-{\boldsymbol{a}}'b'_1+{\boldsymbol{b}}'a'_1.\end{aligned}$$ Using , we can solve these equations for ${\boldsymbol{a}}'$ and ${\boldsymbol{b}}'$: $$\begin{aligned}
&{\boldsymbol{a}}'=(1-a'_1)C{\boldsymbol{a}}'_0+b'_1C{\boldsymbol{b}}'_0,\label{veca1}\\
&{\boldsymbol{b}}'=-b'_1C{\boldsymbol{a}}'_0-(1-a'_1)C{\boldsymbol{b}}'_0.
\label{vecb1}\end{aligned}$$
We have found several algebraical formulas thus far. Let us turn to the proof of the formula . Noting ${\boldsymbol{C}}$ can be expressed as $$\begin{aligned}
{\boldsymbol{C}}=\frac{\alpha_3}{\alpha_1\alpha_2}({\boldsymbol{a}}'_0+{\boldsymbol{b}}'_0),\end{aligned}$$ let us study the short intermediate time behavior of ${\boldsymbol{a}}'_0$ and ${\boldsymbol{b}}'_0$. Our strategy is as follows. We first consider the derivatives of ${\boldsymbol{a}}'_0$ and ${\boldsymbol{b}}'_0$. From the experience of the previous two subsections, we know roughly the results should be given by ${\boldsymbol{a}}'$ and ${\boldsymbol{b}}'$, which are expressed again by ${\boldsymbol{a}}'_0$ and ${\boldsymbol{b}}'_0$ by and . Therefore we can solve the differential equations explicitly.
Similarly to the previous two subsections, we find that $$\begin{aligned}
&\alpha_3\frac{d}{dT}{\boldsymbol{a}}'_0={\boldsymbol{b}}'b'_1
=-(b'_1)^2C{\boldsymbol{a}}'_0-(1-a'_1)b'_1C{\boldsymbol{b}}'_0,\\
&\alpha_3\frac{d}{dT}{\boldsymbol{b}}'_0=-{\boldsymbol{b}}'(1-a'_1)
=(1-a'_1)b'_1C{\boldsymbol{a}}'_0+(1-a'_1)^2C{\boldsymbol{b}}'_0,\end{aligned}$$ where in the last equations we have used and . Plugging the small intermediate time behavior of $1-a'_1$ and $b'_1$ , we find $$\begin{aligned}
&\frac{d}{d\log(T/|\alpha_3|)}{\boldsymbol{a}}'_0
\sim-\frac{g^2}{4\alpha_3\bigl(\log(T/|\alpha_3|)\bigr)^2}
C({\boldsymbol{a}}'_0-{\boldsymbol{b}}'_0),\label{smallveca0}\\
&\frac{d}{d\log(T/|\alpha_3|)}{\boldsymbol{b}}'_0
\sim-\frac{g^2}{4\alpha_3\bigl(\log(T/|\alpha_3|)\bigr)^2}
C({\boldsymbol{a}}'_0-{\boldsymbol{b}}'_0).
\label{smallvecb0}\end{aligned}$$
These differential equations imply that ${\boldsymbol{a}}'_0-{\boldsymbol{b}}'_0$ is $T$-independent at the leading order. Using we find $$\begin{aligned}
&{\boldsymbol{a}}'_0-{\boldsymbol{b}}'_0\sim\alpha_1\alpha_2\frac{2}{1-N^{3,3}}{\boldsymbol{N}}^3
=-\alpha_1\alpha_2{\boldsymbol{B}},
\label{veca0minusb0}\\
&{\boldsymbol{a}}'_0+{\boldsymbol{b}}'_0\sim 0,\end{aligned}$$ in the exact limit of $T\to+0$. Plugging back to and , we find the expression for ${\boldsymbol{a}}'_0$ and ${\boldsymbol{b}}'_0$: $$\begin{aligned}
&{\boldsymbol{a}}'_0
\sim-\frac{\alpha_1\alpha_2}{2}{\boldsymbol{B}}
-\frac{g^2\alpha_1\alpha_2}{4\alpha_3\log(T/|\alpha_3|)}C{\boldsymbol{B}},
\label{a0loopT}\\
&{\boldsymbol{b}}'_0
\sim\frac{\alpha_1\alpha_2}{2}{\boldsymbol{B}}
-\frac{g^2\alpha_1\alpha_2}{4\alpha_3\log(T/|\alpha_3|)}C{\boldsymbol{B}},
\label{b0loopT}\end{aligned}$$ which implies $$\begin{aligned}
{\boldsymbol{C}}\sim-\frac{g^2}{2\log(T/|\alpha_3|)}C{\boldsymbol{B}}.
\label{asympC}\end{aligned}$$ The final result shows .
Evaluation of $g$
-----------------
In order to obtain explicit formulas corresponding to and for the one-loop diagram in LCSFT, we have to evaluate the constant $g$, which appeared in Appendix \[sec:loop diagram\]. Here, we determine it by computing a one-loop diagram with two gravitons inserted in two ways, using respectively bosonic LCSFT and the $\alpha=p^+$ HIKKO string field theory [@alpha=p+], and comparing their results. Note that in [@KMT] we applied the same method for a one-loop diagram with two tachyons inserted to determine $K_T=\mu^4(4\pi)^{-12}|\sqrt{-c}\det'|^{-24}$ or .
Let us consider, in bosonic LCSFT, a contraction of two graviton states $\langle\zeta_r,-k_r^i|=
\zeta^{ij}_r\langle-k_r^i|a_1^{(r)i}\bar{a}_1^{(r)j},~(r=3,6)$ with $k_r^i\zeta^{ij}_r=0,
\zeta^{ij}_r=\zeta^{ji}_r,\zeta^{ij}_r\delta_{ij}=0$ and $|B(3,6)\rangle$, which is given by (34) in [@KMT]. It is evaluated as $$\begin{aligned}
\langle\zeta_3,-k_3^i|\langle\zeta_6,-k_6^i|B(3,6)\rangle
\sim\zeta_3^{ij}\zeta_6^{kl}
\left[\delta^{il}\delta^{jk}\left(-\frac{1}{2c}
\bigl(({\boldsymbol C})_1\bigr)^2\right)^2
+\delta^{jl}\delta^{ik}
\right]K_T(2\pi)^{24}\delta^{24}(k_3+k_6),
\label{eq:2gravB}\end{aligned}$$ for $T\to +0$, where $K_T$ appears similarly to the computation of the one-loop diagram with two tachyons inserted [@KMT]. From , we have $$\begin{aligned}
\label{eq:c12g}
-\frac{1}{2c}
\bigl(({\boldsymbol C})_1\bigr)^2
\sim\frac{g^2}{2\pi^2|\alpha_1\alpha_2/\alpha_3|}
\frac{\sin^2(\pi\alpha_1/|\alpha_3|)}{-\log(T/|\alpha_3|)}\,,\end{aligned}$$ which implies that we can determine $g$ from the evaluation of .
Including the light-cone directions and the level matching projection, the total amplitude for the one-loop diagram with two gravitons is computed as $$\begin{aligned}
T_{36}&=
\langle\zeta_3,-k_3|\langle\zeta_6,-k_6|
\langle R^{{\rm LC}}(2,5)|\langle R^{{\rm LC}}(1,4)|
\Delta_1\Delta_2
|V^{{\rm LC}}(1,2,3)\rangle|V^{{\rm LC}}(4,5,6)\rangle\nonumber\\
&=\int_0^\infty dT\int d\alpha_1
\oint\frac{d\theta_1}{2\pi}\oint\frac{d\theta_2}{2\pi}
(2\pi)^2\delta(k_3^-+k_6^-)\delta(k_3^++k_6^+)
\frac{e^{2Tk_3^-}}{4\pi\alpha_1\alpha_2}
\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times
\langle\zeta_3,-k_3^i|\langle\zeta_6,-k_6^i
|B_{\theta_1,\theta_2}(3,6)\rangle\,,
\label{eq:T36LCSFT}\end{aligned}$$ where $\Delta_r$ is the propagator combined with the level matching projection: $$\begin{aligned}
\Delta_r=\frac{1}{-2p^+_rp^-_r+L_0^{(r)}+\bar L_0^{(r)}}{\cal P}_r
=\int_0^\infty\frac{dT_r}{\alpha_r}\oint\frac{d\theta_r}{2\pi}
e^{-\frac{T_r}{\alpha_r}(-2p^+_rp^-_r+L_0^{(r)}+\bar L_0^{(r)})}
e^{i\theta_r(L_0^{(r)}-\bar L_0^{(r)})},\end{aligned}$$ and $|B_{\theta_1,\theta_2}(3,6)\rangle$ is the effective interaction vertex rotated from the original one $|B(3,6)\rangle$ with $|B_{\theta_1=0,\theta_2=0}(3,6)\rangle=|B(3,6)\rangle$.
The above on-shell amplitude $T_{36}$ can be also obtained in the framework of the $\alpha=p^+$ HIKKO string field theory: $$\begin{aligned}
T_{36}&=
\langle\zeta_3,-k_3|\langle\zeta_6,-k_6|
\langle R^{\alpha=p^+}(2,5)|\langle R^{\alpha=p^+}(1,4)|
\frac{b_0^{(1)}\bar b_0^{(1)}}
{L_0^{{\rm tot}(1)}+\bar L_0^{{\rm tot}(1)}}
{\cal P}_1^{\rm tot}
\frac{b_0^{(2)}\bar b_0^{(2)}}
{L_0^{{\rm tot}(2)}+\bar L_0^{{\rm tot}(2)}}
{\cal P}_2^{\rm tot}\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times
|V^{\alpha=p^+}(1,2,3)\rangle|V^{\alpha=p^+}(4,5,6)\rangle\nonumber\\
&=\int_0^{\infty}dT_1\int_0^{\infty}dT_2
\oint\frac{d\theta_1}{2\pi}\oint\frac{d\theta_2}{2\pi}
\langle\zeta_3,-k_3|\langle\zeta_6,-k_6|
\langle R^{\alpha=p^+}(2,5)|\langle R^{\alpha=p^+}(1,4)|
\frac{1}{\alpha_1\alpha_2}\nonumber\\
&\qquad\times
b_0^{(1)}\bar b_0^{(1)}b_0^{(2)}\bar b_0^{(2)}
e^{-\frac{T_1}{\alpha_1}(L_0^{{\rm tot}(1)}+\bar L_0^{{\rm tot}(1)})
-\frac{T_2}{\alpha_2}(L_0^{{\rm tot}(2)}+\bar L_0^{{\rm tot}(2)})
+i\theta_1(L_0^{{\rm tot}(1)}-\bar L_0^{{\rm tot}(1)})
+i\theta_2(L_0^{{\rm tot}(2)}-\bar L_0^{{\rm tot}(2)})}
\nonumber\\
&\qquad\qquad\qquad\qquad\qquad\qquad\times
|V^{\alpha=p^+}(1,2,3)\rangle|V^{\alpha=p^+}(4,5,6)\rangle,\end{aligned}$$ which includes the ghost part in addition to the light-cone directions. We calculate it using the CFT correlator on the torus $u$-plane ($u\sim u+1\sim u+\tau$): $$\begin{aligned}
T_{36}&=\int_0^{\infty}dT_1\int_0^{\infty}dT_2
\oint\frac{d\theta_1}{2\pi}\oint\frac{d\theta_2}{2\pi}
\bigl\langle
(\alpha_1\alpha_2)^{-1}
b^{(1)}\bar b^{(1)}b^{(2)}\bar b^{(2)}\,
V_{\zeta_6,k_6}(U_6,\bar U_6)
V_{\zeta_3,k_3}(U_3,\bar U_3)
\bigr\rangle_{\tau}\,,
\label{eq:T36corr}\end{aligned}$$ where $b^{(i)}$ is given by a contour integral on $C_i$ in the $u$-plane, which is denoted in Fig. 3 [@KMT] for a pure imaginary $\tau$, and $V_{\zeta,k}(u,\bar u)$ is the graviton vertex: $$\begin{aligned}
b^{(i)}=\int_{C_i}\frac{du}{2\pi i}\alpha_i\frac{du}{d\rho}b(u),\quad
V_{\zeta,k}(u,\bar u)
=\frac{1}{4}\zeta^{ij}c(u)\bar c(\bar u)\!:\!
i\partial X^i(u)i\bar\partial\bar X^j(\bar u)
e^{ik_{\mu}X^{\mu}(u,\bar u)}\!\!:\,.\end{aligned}$$ Note that here we have used the on-shell condition for the graviton vertices: $k_3^{\mu}k_{3\mu}=k_6^{\mu}k_{6\mu}=0$. The light-cone diagram ($\rho$-plane) can be obtained from the torus $u$-plane by the generalized Mandelstam map: $$\begin{aligned}
\label{eq:mandel_gen1}
\rho(u)=|\alpha_3|\left(
\log\frac{\vartheta_1(u-U_6|\tau)}{\vartheta_1(u-U_3|\tau)}
+2\pi i\frac{{\rm Im}(U_3-U_6)}{{\rm Im}\tau}u\right),\end{aligned}$$ with $|\alpha_3|=k_3^+$ and $U_3+U_6=0$. It is related to the parameters on the light-cone diagram as $$\begin{aligned}
&\rho(u+1)-\rho(u)=-2\pi i\alpha_1,
\label{eq:cond_cyc1}
\\
&\rho(u+\tau)-\rho(u)=T_2-T_1-i(\alpha_2\theta_2-\alpha_1\theta_1),
\label{eq:cond_cyc2}
\\
&\rho(u_{-})-\rho(u_{+})=T_2-i\alpha_2\theta_2,
\quad\frac{d\rho(u_{\pm})}{du}=0\,.
\label{eq:cond_intT}\end{aligned}$$ From the explicit computation with the $\alpha=p^+$ prescription, which provides $\delta({\rm Re}(\rho(u+\tau)-\rho(u)))=\delta(T_2-T_1)$ in the integrand, (\[eq:T36corr\]) can be rewritten as $$\begin{aligned}
T_{36}&=\int_0^{\infty}dT\int d\alpha_1
\oint\frac{d\theta_1}{2\pi}
\oint\frac{d\theta_2}{2\pi}
(2\pi)^{26}\delta^{26}(k_6+k_3)
\frac{\alpha_1\alpha_2}{2^{38}\pi^{25}\alpha_3^4}
\nonumber\\
&\qquad\times
\Bigl|g_1'(u_+-U_6|\tau)-g_1'(u_+-U_3|\tau)\Bigr|^{-2}
\Bigl|({\rm Im}\tau)^{\frac{1}{4}}\eta(\tau)\Bigr|^{-48}
\nonumber\\
&\qquad\times
\zeta_3^{ij}\zeta_6^{kl}
\biggl[
\delta^{il}\delta^{jk}\biggl(\frac{\pi}{{\rm Im}\tau}\biggr)^2
+\delta^{jl}\delta^{ik}
\biggl|\frac{\pi}{{\rm Im}\tau}+g_1'(U_6-U_3|\tau)\biggr|^2
\biggr]\,,
\label{eq:T36f}\end{aligned}$$ with $g_1'(\nu|\tau)=\partial^2_{\nu}\log\vartheta_1(\nu|\tau)$. The factors in the second and the third line are functions of two complex parameters $\tau,U_3-U_6$, which are related to 4 real parameters $T(=T_1=T_2),\alpha_1,\theta_1,\theta_2$ by (\[eq:cond\_cyc1\]), (\[eq:cond\_cyc2\]) and (\[eq:cond\_intT\]).
Similarly, for the one-loop amplitude with two tachyons inserted we find[^4] $$\begin{aligned}
S_{36}&=\langle -k_3|\langle -k_6|
\langle R^{\alpha=p^+}(2,5)|\langle R^{\alpha=p^+}(1,4)|
\frac{b_0^{(1)}\bar b_0^{(1)}}{L_0^{(1)}+\bar L_0^{(1)}}
{\cal P}_1^{\rm tot}
\frac{b_0^{(2)}\bar b_0^{(2)}}{L_0^{(2)}+\bar L_0^{(2)}}
{\cal P}_2^{\rm tot}\nonumber\\
&\qquad\times
|V^{\alpha=p^+}(1,2,3)\rangle|V^{\alpha=p^+}(4,5,6)\rangle\nonumber\\
&=\int_0^{\infty}dT\int d\alpha_1
\oint\frac{d\theta_1}{2\pi}\oint\frac{d\theta_2}{2\pi}
(2\pi)^{26}\delta^{26}(k_6+k_3)
\frac{\alpha_1\alpha_2}{2^{38}\pi^{25}\alpha_3^4}
\label{eq:S36}\\
&\qquad\times
\Bigl|g_1'(u_+-U_6|\tau)-g_1'(u_+-U_3|\tau)\Bigr|^{-2}
\Bigl|({\rm Im}\tau)^{\frac{1}{4}}\eta(\tau)\Bigr|^{-48}
\left|e^{\pi\frac{({\rm Im}(U_3-U_6))^2}{{\rm Im}\tau}}\frac{
\partial_{\nu}\vartheta_1(\nu|\tau)|_{\nu=0}}
{\vartheta_1(U_6-U_3|\tau)}\right|^4.\nonumber\end{aligned}$$ For $\theta_1=\theta_2=0,T\to +0$, the extra factor of the integrand in compared to that in can be evaluated as $$\begin{aligned}
&\zeta_3^{ij}\zeta_6^{kl}
\biggl[
\delta^{il}\delta^{jk}\biggl(\frac{\pi}{{\rm Im}\tau}\biggr)^2
+\delta^{jl}\delta^{ik}
\biggl|\frac{\pi}{{\rm Im}\tau}+g_1'(U_6-U_3|\tau)\biggr|^2
\biggr]\times
\left|e^{\pi\frac{({\rm Im}(U_3-U_6))^2}{{\rm Im}\tau}}
\frac{\partial_{\nu}\vartheta_1(\nu|\tau)|_{\nu=0}}
{\vartheta_1(U_6-U_3|\tau)}\right|^{-4}\nonumber\\
&\qquad\qquad
\sim\zeta_3^{ij}\zeta_6^{kl}
\biggl[\delta^{il}\delta^{jk}
\biggl(\frac{\sin^2(\pi\alpha_1/|\alpha_3|)}
{-\log(T/|\alpha_3|)}\biggr)^2
+\delta^{jl}\delta^{ik}
\biggr].\end{aligned}$$ Comparing this factor with and , we finally obtain $$\begin{aligned}
\label{eq:gana}
g=\sqrt{2}\pi|\alpha_1\alpha_2/\alpha_3|^{1/2}.\end{aligned}$$
Formulas for the gamma matrices
===============================
Here we would like to prove the formulas and first. The point is to find a recursion relation. Multiplying by $\hat\gamma^{d_1\cdots d_r}$ and using the gamma matrix product formula (See e.g. [@Fujii]) $$\begin{aligned}
\hat\gamma^{a_1\cdots a_p}\hat\gamma^{b_1\cdots b_q}
=\sum_{k=0}^{\min(p,q)}
\frac{(-1)^{pk-\frac{1}{2}k(k+1)}p!q!}{(p-k)!(q-k)!k!}
\delta^{[a_1}_{[b_1}\cdots\delta^{a_k}_{b_k}
\hat\gamma^{a_{k+1}\cdots a_p]}{}_{b_{k+1}\cdots b_q]},\end{aligned}$$ where $[\cdots]$ denotes the anti-symmetrization of the indices, we find the recursive relations for small $r$: $$\begin{aligned}
&G^{c_1\cdots c_m,d}_p
=d_{p,m}\hat\gamma^{c_1\cdots c_m}\hat\gamma^d
-d_{p-1,m}\hat\gamma^d\hat\gamma^{c_1\cdots c_m}
+G^{c_1\cdots c_m,d}_{p-2},\\
&G^{c_1\cdots c_m,d_1d_2}_p-\tilde G^{d_1d_2,c_1\cdots c_m}_p
=d_{p,m}[\hat\gamma^{c_1\cdots c_m},\hat\gamma^{d_1d_2}]
+G^{c_1\cdots c_m,d_1d_2}_{p-2}-\tilde G^{d_1d_2,c_1\cdots c_m}_{p-2}.\end{aligned}$$ Using these formula recursively, we can prove and without difficulty.
For the explicit computation of $G^{c_1\cdots c_m,d}_p$ and $G^{c_1\cdots c_m,d_1d_2}_p-\tilde G^{d_1d_2,c_1\cdots c_m}_p$, we need several summation formulas of $d_{p,m}$. For this purpose first we note that $d_{p,m}$ has the residue formula: $$\begin{aligned}
d_{p,m}
=(-1)^{pm}\oint_{z=0}{dz\over 2\pi i}z^{-1-p}(1+z)^{8-m}(1-z)^m
=\sum_{s=0}^{\min(p,m)}
{(-1)^s(8-m)!m!\over(p-s)!(8-m-p+s)!s!(m-s)!}.\end{aligned}$$ Using this expression we find we can show the following formulas. $$\begin{aligned}
&\sum_{q=0}^4d_{2q,m}=128(\delta_{m,0}+\delta_{m,8}),
\label{sumd1}\\
&\sum_{q=0}^3d_{2q+1,m}=128(\delta_{m,0}-\delta_{m,8}),
\label{sumd2}\\
&\sum_{q=0}^3\sum_{q_0=0}^qd_{2q_0+1,m}
=320(\delta_{m,0}-\delta_{m,8})-32(\delta_{m,1}-\delta_{m,7}),
\label{sumd3}\\
&\sum_{q=0}^3\sum_{q_0=0}^qd_{2q_0,m}
=256(\delta_{m,0}+\delta_{m,8})+32(\delta_{m,1}+\delta_{m,7}),
\label{sumd4}\\
&\sum_{q=0}^2\sum_{q_0=0}^qd_{2q_0+1,m}
=192(\delta_{m,0}-\delta_{m,8})-32(\delta_{m,1}-\delta_{m,7}).
\label{sumd5}\end{aligned}$$
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G. E. Arutyunov and S. A. Frolov, “Virasoro amplitude from the S(N) R\*\*24 orbifold sigma model,” Theor. Math. Phys. [**114**]{}, 43 (1998) \[arXiv:hep-th/9708129\]. G. E. Arutyunov and S. A. Frolov, “Four graviton scattering amplitude from S(N) R\*\*8 supersymmetric orbifold sigma model,” Nucl. Phys. B [**524**]{}, 159 (1998) \[arXiv:hep-th/9712061\]. J. Gomis, S. Moriyama and J. w. Park, “Open + closed string field theory from gauge fields,” Nucl. Phys. B [**678**]{} (2004) 101 \[arXiv:hep-th/0305264\]. I. Kishimoto, Y. Matsuo and E. Watanabe, “Boundary states as exact solutions of (vacuum) closed string field theory,” Phys. Rev. D [**68**]{} (2003) 126006 \[arXiv:hep-th/0306189\]. E. Cremmer and J. L. Gervais, “Infinite Component Field Theory Of Interacting Relativistic Strings And Dual Theory,” Nucl. Phys. B [**90**]{}, 410 (1975). H. Hata, K. Itoh, T. Kugo, H. Kunitomo and K. Ogawa, “Covariant String Field Theory. 2,” Phys. Rev. D [**35**]{}, 1318 (1987). T. Kugo and B. Zwiebach, “Target space duality as a symmetry of string field theory,” Prog. Theor. Phys. [**87**]{}, 801 (1992) \[arXiv:hep-th/9201040\]. T. Asakawa, T. Kugo and T. Takahashi, “One-loop tachyon amplitude in unoriented open-closed string field theory,” Prog. Theor. Phys. [**102**]{}, 427 (1999) \[arXiv:hep-th/9905043\]. Y. Fujii, “Introduction to the Theory of Supergravity,” ISBN: 4782810121, in Japanese.
[^1]: [[email protected]]{}
[^2]: [[email protected]]{}
[^3]: Permanent address
[^4]: Because we have included the level matching projection which was omitted in [@KMT], we can reproduce a modular invariant measure by computing the Jacobian using , and : $$\begin{aligned}
\int_0^{\infty}dT\int \!d\alpha_1\!
\oint\frac{d\theta_1}{2\pi}\!
\oint\frac{d\theta_2}{2\pi}
\frac{\alpha_1\alpha_2}{\alpha_3^4}(\cdots)
=\frac{1}{2\pi}\!\int\! d^2\tau dx dy
\left|g_1'(u_+-U_6|\tau)-g_1'(u_+-U_3|\tau)\right|^2(\cdots).\end{aligned}$$ Here $x,y$ are real parameters defined by $U_3-U_6=x+y\tau$. However, we have only to use the expression of the integrand for $\theta_1=0,\theta_2=0$, namely without projection, in the evaluation of $K_T~(T\to +0)$ in [@KMT] and $g$ here.
|
---
abstract: 'Meta-learning extracts the common knowledge from learning different tasks and uses it for unseen tasks. It can significantly improve tasks that suffer from insufficient training data, e.g., few-shot learning. In most meta-learning methods, tasks are implicitly related by sharing parameters or optimizer. In this paper, we show that a meta-learner that explicitly relates tasks on a graph describing the relations of their output dimensions (e.g., classes) can significantly improve few-shot learning. The graph’s structure is usually free or cheap to obtain but has rarely been explored in previous works. We develop a novel meta-learner of this type for prototype based classification, in which a prototype is generated for each class, such that the nearest neighbor search among the prototypes produces an accurate classification. The meta-learner, called “Gated Propagation Network (GPN)”, learns to propagate messages between prototypes of different classes on the graph, so that learning the prototype of each class benefits from the data of other related classes. In GPN, an attention mechanism aggregates messages from neighboring classes of each class, with a gate choosing between the aggregated message and the message from the class itself. We train GPN on a sequence of tasks from many-shot to few-shot generated by subgraph sampling. During training, it is able to reuse and update previously achieved prototypes from the memory in a life-long learning cycle. In experiments, under different training-test discrepancy and test task generation settings, GPN outperforms recent meta-learning methods on two benchmark datasets. The code of GPN and dataset generation is available at <https://github.com/liulu112601/Gated-Propagation-Net>.'
author:
- |
Lu Liu$^{1}$, Tianyi Zhou$^{2}$, Guodong Long$^{1}$, Jing Jiang$^{1}$, Chengqi Zhang$^{1}$\
$^{1}$Center for Artificial Intelligence, University of Technology Sydney\
$^{2}$Paul G. Allen School of Computer Science & Engineering, University of Washington\
[[email protected], [email protected], [email protected]]{}\
[[email protected], [email protected]]{}
title: 'Learning to Propagate for Graph Meta-Learning'
---
Introduction
============
The success of machine learning (ML) during the past decade has relied heavily on the rapid growth of computational power, new techniques training larger and more representative neural networks, and critically, the availability of enormous amounts of annotated data. However, new challenges have arisen with the move from cloud computing to edge computing and Internet of Things (IoT), and demands for customized models and local data privacy are increasing, which raise the question: how can a powerful model be trained for a specific user using only a limited number of local data? Meta-learning, or “learning to learn”, can address this few-shot challenge by training a shared meta-learner model on top of distinct learner models for implicitly related tasks. The meta-learner aims to extract the common knowledge of learning different tasks and adapt it to unseen tasks in order to accelerate their learning process and mitigate their lack of training data. Intuitively, it allows new learning tasks to leverage the “experiences” from the learned tasks via the meta-learner, though these tasks do not directly share data or targets. Meta-learning methods have demonstrated clear advantages on few-shot learning problems in recent years. The form of a meta-learner model can be a similarity metric (for K-nearest neighbor (KNN) classification in each task) [@snell2017prototypical], a shared embedding module [@oreshkin2018tadam], an optimization algorithm [@l2o] or parameter initialization [@finn2017model], and so on. If a meta-learner is trained on sufficient and different tasks, it is expected to be generalized to new and unseen tasks drawn from the same distribution as the training tasks. Thereby, different tasks are related via the shared meta-learner model, which implicitly captures the shared knowledge across tasks. However, in a lot of practical applications, the relationships between tasks are known in the form of a graph of their output dimensions, for instance, species in the biology taxonomy, diseases in the classification coding system, and merchandise on an e-commerce website.
![ [**LEFT:**]{} t-SNE [@maaten2008visualizing] visualization of the class prototypes produced by GPN and the associated graph. [**RIGHT:**]{} GPN’s propagation mechanism for one step: for each node, its neighbors pass messages (their prototypes) to it according to attention weight $a$, where a gate further choose to accept the message from the neighbors $g^{+}$ or from the class itself $g^{*}$.[]{data-label="fig:hier-prop"}](resources/hier-prop-wide-v4-sips.pdf){width="\linewidth"}
In this paper, we study the meta-learning for few-shot classification tasks defined on a given graph of classes with mixed granularity, that is, the classes in each task could be an arbitrary combination of classes with different granularity or levels in a hierarchical taxonomy. The tasks can be classification of cat vs mastiff (dog) or an m-vs-rest task, e.g. classification that aims to distinguish among cat, dog and others. In particular, we define the graph with each class as a node and each edge connecting a class to its sub-class (i.e., children class) or parent class. In practice, the graph is usually known in advance or can be easily extracted from a knowledge base, such as the WordNet hierarchy for classes in ImageNet [@imagenet]. Given the graph, each task is associated with a subset of nodes on the graph. Hence, tasks can be related through the paths on the graph that links their nodes even when they share few output classes. In this way, different tasks can share knowledge by message passing on the graph. We develop Gated Propagation Network (GPN) to learn how to pass messages between nodes (i.e., classes) on the graph for more effective few-shot learning and knowledge sharing. We use the setting from [@snell2017prototypical]: given a task, the meta-learner generates a prototype representing each class by using only few-shot training data, and during test a new sample is classified to the class of its nearest prototype. Hence, each node/class is associated with a prototype. Given the graph structure, we let each class send its prototype as a message to its neighbors, while a class received multiple messages needs to combine them with different weights and update its prototype accordingly. GPN learns an attention mechanism to compute the combination weights and a gate to filter the message from different senders (which also includes itself). Both the attention and gate modules are shared across tasks and trained on various few-shot tasks, so they can be generalized to the whole graph and unseen tasks. Inspired by the hippocampal memory replay mechanism in [@bendor2012biasing] and its application in reinforcement learning [@mnih2015human], we also retain a memory pool of prototypes per training class, which can be reused as a backup prototype for classes without training data in future tasks. -1
We evaluate GPN under various settings with different distances between training and test classes, different task generation methods, and with or without class hierarchy information. To study the effects of distance (defined as the number of hops between two classes) between training and test classes, we extract two datasets from *tiered*ImageNet [@ren2018meta]: *tiered*ImageNet-Far and *tiered*ImageNet-Close. To evaluate the model’s generalization performance, test tasks are generated by two subgraph sampling methods, i.e., random sampling and snowball sampling [@goodman1961snowball] (snowball sampling can restrict the distance of the targeted few-shot classes). To study whether/when the graph structure is more helpful, we evaluate GPN with and without using class hierarchy. We show that GPN outperforms four recent few-shot learning methods. We also conduct a thorough analysis of different propagation settings. In addition, the “learning to propagate” mechanism can be potentially generalized to other fields.-1
Related Works
=============
Meta-learning has been proved to be effective on few-shot learning tasks. It trains a meta learner using augmented memory [@santoro2016meta; @kaiser2017learning], metric learning [@vinyals2016matching; @snell2017prototypical; @closerlook] or learnable optimization [@finn2017model]. For example, prototypical network [@snell2017prototypical] applied a distance-based classifier in a trained feature space. We can extend the single prototype per class to an adaptive number of prototypes by infinite mixture model [@allen2019infinite]. The feature space could be further improved by scaling features according to different tasks [@oreshkin2018tadam]. Our method is built on prototypical network and improves the prototype per class by propagation between prototypes of different classes. Our work also relates to memory-based approaches, in which feature-label pairs are selected into memory by dedicated reading and writing mechanisms. In our case, the memory stores prototypes and improves the propagation efficiency. Auxiliary information, such as unlabeled data [@ren2018meta] and weakly-labeled data [@ppn] has been used to embrace the few-shot challenge. In this paper, we improve the quality of prototype per class by sending messages between prototypes on a graph describing the class relationships. Our idea of prototype propagation is inspired by belief propagation [@Pearl82; @Weiss00lbp], message passing and label propagation [@Zhu02labelprop; @Zhou04labelprop]. It is also related to Graph Neural Networks (GNN) [@henaff2015deep; @velivckovic2018graph], which applies convolution/attention iteratively on a graph to achieve node embedding. In contrast, the graph in our paper is a computational graph in which every node is associated with a prototype produced by an CNN rather than a non-parameterized initialization in GNN. Our goal is to obtain a better prototype representation for classes in few-shot classification. Propagation has been applied in few-shot learning for label propagation [@liu2018transductive] in a transductive setting to infer the entire query set from support set at once.-1
Graph Meta-Learning {#sec:graph-meta-learning}
===================
Problem Setup {#sec:mtl-setup}
-------------
[r]{}[0.6]{}
Notation Definition
--------------------------------------- ----------------------------------------------------------
$\mathcal Y$ Ground set of classes for all possible tasks
$\mathcal G=(\mathcal Y, E)$ Category graph with nodes $\mathcal Y$ and edges $E$
$\mathcal N_y$ The set of neighbor classes of $y$ on graph $\mathcal G$
$\mathcal M(\cdot; \Theta)$ A meta-learner model with paramter $\Theta$
$T$ A few-shot classification task
$\mathcal T$ Distribution that each task $T$ is drawn from
$\mathcal Y^T\subseteq \mathcal Y$ The set of output classes in task $T$
$(\vx, y)$ A sample with input data $\vx$ and label $y$
$D^{T}$ Distribution of $(\vx, y)$ in task $T$
$\mP_{y}$ Final output prototype of class $y$
$\mP^{t}_{y}$ Prototype of class $y$ at step $t$
$\mP^{t}_{y\rightarrow y}$ Message sent from class $y$ to itself
$\mP^{t}_{\mathcal N_y\rightarrow y}$ Message sent to class $y$ from its neighbors
We study “graph meta-learning” for few-shot learning tasks, where each task is associated with a prediction space defined by a subset of nodes on a given graph, e.g., 1) for classification tasks: a subset of classes from a hierarchy of classes; 2) for regression tasks: a subset of variables from a graphical model as the prediction targets; or 3) for reinforcement learning tasks: a subset of actions (or a sub-sequence of actions). In real-world problems, the graph is usually free or cheap to achieve and can provide weakly-supervised information for a meta-learner since it relates different tasks’ output spaces via the edges and paths on the graph. However, it has been rarely considered in previous works, most of which relate tasks via shared representation or metric space.
In this paper, we will study graph meta-learning for few-shot classification. In this problem, we are given a graph with nodes as classes and edges connecting each class to its parents and/or children classes, and each task aims to learn a classifier categorizing an input sample into a subset of $N$ classes given only $K$ training samples per class. Comparing to the traditional setting for few-shot classification, the main challenge of graph meta-learning comes from the mixed granularity of the classes, i.e., a task might aim to classify a mixed subset containing both fine and coarse categories. Formally, given a directed acyclic graph (DAG) $\gG=(\mathcal Y, E)$, where $\mathcal Y$ is the ground set of classes for all possible tasks, each node $y\in \mathcal Y$ denotes a class, and each directed edge (or arc) $y_{i}\rightarrow y_{j}\in E$ connects a parent class $y_{i}\in\mathcal Y$ to one of its child classes $y_{j}\in\mathcal Y$ on the graph $\gG$. We assume that each learning task $T$ is defined by a subset of classes $\mathcal Y^T\subseteq \mathcal Y$ drawn from a certain distribution $\mathcal T(\mathcal G)$ defined on the graph, our goal is to learn a meta-learner $\mathcal M(\cdot;\Theta)$ that is parameterized by $\Theta$ and can produce a learner model $\mathcal M(T;\Theta)$ for each task $T$. This problem can then be formulated by the following risk minimization of “learning to learn”: $$\begin{aligned}
\label{equ:opt-obj}
\min_\Theta\mathbb E_{T\sim\mathcal T(\mathcal G)}\left[\mathbb E_{(\vx,y)\sim\mathcal D^T} -\log\Pr(y|\vx; \mathcal{M}(T, \Theta)))\right],\end{aligned}$$ where $\mathcal D^T$ is the distribution of data-label pair $(\vx,y)$ for a task $T$. In few-shot learning, we assume that each task $T$ is an $N$-way-$K$-shot classification over $N$ classes $\mathcal Y^T\subseteq \mathcal Y$, and we only observe $K$ training samples per class. Due to the data deficiency, conventional supervised learning usually fails. We further introduce the form of $\Pr(y|\vx; \mathcal{M}(T; \Theta))$ in Eq. . Inspired by [@snell2017prototypical], each classifier $\mathcal M(T; \Theta)$, as a learner model, is associated with a subset of prototypes $\mP_{\mathcal Y^T}$ where each prototype $\mP_y$ is a representation vector for class $y\in\mathcal Y^T$. Given a sample $\vx$, $\mathcal M(T; \Theta)$ produces the probability of $\vx$ belonging to each class $y\in\mathcal Y^T$ by applying a soft version of KNN: the probability is computed by an RBF Kernel over the Euclidean distances between $f(\vx)$ and prototype $\mP_y$, i.e., $$\begin{aligned}
\label{equ:euc-dis}
\Pr(y|\vx; \mathcal{M}(T; \Theta)) \triangleq \frac{\exp(-\|f(\vx)-\mP_{y})\|^2)}{\sum_{z\in \mathcal Y^T}\exp(-\|f(\vx)-\mP_{z})\|^2)},\end{aligned}$$ where $f(\cdot)$ is a learnable representation model for input $\vx$. The main idea of graph meta-learning is to improve the prototype of each class in $\mP$ by assimilating their neighbors’ prototypes on the graph $\mathcal G$. This can be achieved by allowing classes on the graph to send/receive messages to/from neighbors and modify their prototypes. Intuitively, two classes should have similar prototypes if they are close to each other on the graph. Meanwhile, they should not have exactly the same prototype since it leads to large errors on tasks containing both the two classes. The remaining questions are 1) how to measure the similarity of classes on graph $\mathcal G$? 2) how to relate classes that are not directly connected on $\mathcal G$? 3) how to send messages between classes and how to aggregate the received messages to update prototypes? 4) how to distinguish classes with similar prototypes?
Gated Propagation Network {#sec:mtl-GPN}
-------------------------
[r]{}[0.4]{}
{width="40.00000%"}
We propose [*Gated Propagation Network*]{} (GPN) to address the graph meta-learning problem. GPN is a meta-learning model that learns how to send and aggregate messages between classes in order to generate class prototypes that result in high KNN prediction accuracy across different $N$-way-$K$-shot classification tasks. Technically, we deploy a multi-head dot-product attention mechanism to measure the similarity between each class and its neighbors on the graph, and use the obtained similarities as weights to aggregate the messages (prototypes) from its neighbors. In each head, we apply a gate to determine whether to accept the aggregated messages from the neighbors or the message from itself. We apply the above propagation on all the classes (together with their neighbors) for multiple steps, so we can relate the classes not directly connected in the graph. We can also avoid identical prototypes of different classes due to the capability of rejecting messages from any other classes except the one from the class itself. In particular, given a task $T$ associated with a subset of classes $\mathcal Y^T$ and an $N$-way-$K$-shot training set $\mathcal D^T$. At the very beginning, we compute an initial prototype for each class $y\in\mathcal Y^T$ by averaging over all the $K$-shot samples belonging to class $y$ as in [@snell2017prototypical], i.e., $$\begin{aligned}
\label{equ:proto-avg}
\mP^0_{y}\triangleq\frac{1}{|\{(\vx_i,y_i)\in \mathcal D^T: y_i=y\}|} \sum_{(\vx_i,y_i)\in \mathcal D^T, y_i=y}f(\vx_i).\end{aligned}$$ GPN repeatedly applies the following propagation procedure to update the prototypes in $\mP_{\mathcal Y^T}$ for each class $y\in\mathcal Y^T$. At step-$t$, for each class $y\in\mathcal Y^T$, we firstly compute the aggregated messages from its neighbors $\mathcal N_y$ by a dot-product attention module $a(p, q)$, i.e., $$\begin{aligned}
\label{equ:prop}
\mP^{t+1}_{\mathcal N_y\rightarrow y}\triangleq\sum_{z\in\mathcal N_{y}} a(\mP^{t}_{y}, \mP^{t}_{z}) \times \mP^{t}_{z},~~a(p, q)=\frac{\langle h_{1}(p), h_2(q)\rangle}{\|h_1(p)\| \times \|h_2(q)\|}.\end{aligned}$$ where $h_1(\cdot)$ and $h_2(\cdot)$ are learnable transformations and their parameters $\Theta^{prop}$ are parts of the meta-learner parameters $\Theta$. To avoid the propagation to generate identical prototypes, we allow each class $y$ to send its own last-step prototype $\mP^{t}_{y}$ to itself, i.e., $\mP^{t+1}_{y\rightarrow y}\triangleq \mP^{t}_{y}$. Then we apply a gate $g$ making decisions of whether accepting messages $\mP^{t+1}_{\mathcal N_y\rightarrow y}$ from its neighbors or message $\mP^{t+1}_{y\rightarrow y}$ from itself, i.e.-1 $$\begin{aligned}
\label{equ:head}
\mP^{t+1}_y \triangleq g \mP^{t+1}_{y\rightarrow y} + (1-g) \mP^{t+1}_{\mathcal N_y\rightarrow y},~~g = \frac{ \exp[\gamma\cos(\mP_{y}^{0}, \mP^{t+1}_{y\rightarrow y})] }{ \exp[\gamma\cos(\mP_{y}^{0}, \mP^{t+1}_{y\rightarrow y})] + \exp[\gamma\cos(\mP_{y}^{0}, \mP^{t+1}_{\mathcal N_y\rightarrow y})]},\end{aligned}$$ where $\cos(p,q)$ denotes the cosine similarity between two vectors $p$ and $q$, and $\gamma$ is a temperature hyper-parameter that controls the smoothness of the softmax function. To capture different types of relation and use them jointly for propagation, we apply $k$ modules of the above attentive and gated propagation (Eq. (\[equ:prop\])-Eq. (\[equ:head\])) with untied parameters for $h_{1}(\cdot)$ and $h_{2}(\cdot)$ (as the multi-head attention in [@vaswani2017attention]) and average the outputs of the $k$ “heads”, i.e., $$\begin{aligned}
\label{equ:multi-head}
\mP^{t+1}_y = \frac{1}{k} \sum\nolimits_{i=1}^{k} \mP^{t+1}_y[i],\end{aligned}$$ where $\mP^{t+1}_y[i]$ is the output of the $i$-th head and computed in the same way as $\mP^{t+1}_y$ in Eq. . In GPN, we apply the above procedure to all $y\in\mathcal Y^T$ for $\mathcal T$ steps and the final prototype of class $y$ is given by $$\begin{aligned}
\label{equ:p-final}
\mP_{y}\triangleq\lambda \times \mP^0_{y} + (1-\lambda) \times \mP^{\mathcal T}_{y}.\end{aligned}$$ GPN can be trained in a life-long learning manner that relates tasks learned at different time steps by maintaining a memory of prototypes for all the classes on the graph that have been included in any previous task(s). This is especially helpful to learning the above propagation mechanism, because in practice it is common that many classes $y\in\mathcal Y^T$ do not have any neighbor in $\mathcal Y^T$, i.e., $\mathcal N_y\cap \mathcal Y^T=\emptyset$, so Eq. cannot be applied and the propagation mechanism cannot be effectively trained. However, by initializing the prototypes of these classes to be the ones stored in memory, GPN is capable to apply propagation over all classes in $\mathcal N_y\cup\mathcal Y^T$ and thus relate any two classes on the graph, if there exists a path between them and all the classes on the path have prototypes stored in the memory.
[r]{}[0.52]{}
$\gG=(\gY, E)$, memory update interval $m$, propagation steps $\mathcal T$, total episodes $\tau_{total}$; $\Theta^{cnn}$, $\Theta^{prop}$, $\Theta^{fc}$, $\tau\leftarrow 0$; Update prototypes in memory by Eq. ; Draw $\alpha\sim$Unif$[0,1]$; Train a classifier to update $\Theta^{cnn}$ with loss $\sum
\nolimits_{(\vx,y)\sim \mathcal D} -\log\Pr(y|\vx; \Theta^{cnn}, \Theta^{fc})$; Sample a few-shot task $T$ as in Sec. \[sec:training-strategy\]; Construct MST $\mathcal Y^T_{MST}$ as in Sec. \[sec:training-strategy\]; For $y\in\mathcal Y^T_{MST}$, compute $\mP^0_{y}$ by Eq. if $y \in T$, otherwise fetch $\mP^0_{y}$ from memory; For all $y\in\mathcal Y^T_{MST}$, concurrently update their prototypes $\mP_y^t$ by Eq. -; Compute $\mP_{y}$ for $y\in\mathcal Y^T_{MST}$ by Eq.; Compute $\log\Pr(y|\vx; \Theta^{cnn}, \Theta^{prop})$ by Eq. for all samples $(\vx,y)$ in task $T$; Update $\Theta^{cnn}$ and $\Theta^{prop}$ by minimizing $\sum\limits_{(x,y)\sim \mathcal D^T} -\log\Pr(y|x; \Theta^{cnn}, \Theta^{prop})$;
Training Strategies {#sec:training-strategy}
-------------------
**Generating training tasks by subgraph sampling:** In meta-learning setting, we train GPN as a meta-learner on a set of training tasks. We can generate each task by sampling targeted classes $\mathcal Y^T$ using two possible methods: random sampling and snowball sampling [@goodman1961snowball]. The former randomly samples $N$ classes $\mathcal Y^T$ without using the graph, and they tend to be weakly related if the graph is sparse (which is often true). The latter selects classes sequentially: in each step, it randomly sample classes from the hop-$k_{n}$ neighbors of the previously selected classes, where $k_{n}$ is a hyper-parameter controlling how relative the selected classes are. In practice, we use a hybrid of them to cover more diverse tasks. Note $k_{n}$ also results in a trade-off: the classes selected into $\mathcal Y^T$ are close to each other when $k_n$ is small and they can provide strong training signals to learn the message passing; on the other hand, the tasks become hard because similar classes are easier to be confused.
**Building propagation pathways by maximum spanning tree:** During training, given a task $T$ defined on classes $\mathcal Y^T$, we need to decide the subgraph we apply the propagation procedure to, which can cover the classes $z\notin \mathcal Y^T$ but connected to some class $y\in\mathcal Y^T$ via some paths. Given that we apply $\mathcal T$ steps of propagation, it makes sense to add all the hop-1 to hop-$\mathcal T$ neighbors of every $y\in\mathcal Y^T$ to the subgraph. However, this might result in a large subgraph requiring costly propagation computation. Hence, we further build a maximum spanning tree (MST) [@kruskal1956shortest] (with edge weight defined by cosine similarity between prototypes from memory) $\mathcal Y^T_{MST}$ for the hop-$\mathcal T$ subgraph of $\mathcal Y^T$ as our “propagation pathways”, and we only deply the propagation procedure on the MST $\mathcal Y^T_{MST}$. MST preserves the strongest relations to train the propagation and but significantly saves computations.
**Curriculum learning:** It is easier to train a classifier given sufficient training data than few-shot training data since the former is exposed to more supervised information. Inspired by auxiliary task in co-training [@oreshkin2018tadam], during early episodes of training[^1], with high probability we learn from a traditional supervised learning task by training a linear classifier $\Theta^{fc}$ with input $f(\cdot)$ and update both the classifier and the representation model $f(\cdot)$. We gradually reduce the probability later on by using an annealed probability $0.9^{20 \tau / \tau_{t}}$ so more training will target on few-shot tasks. Another curriculum we find helpful is to gradually reduce $\lambda$ in Eq. (\[equ:p-final\]), since $\mP^0_y$ often works better than $\mP^{\mathcal T}_y$ in earlier episodes but with more training $\mP^{\mathcal T}_y$ becomes more powerful. In particular, we set $\lambda= 1-\tau / \tau_{t}$.
The complete training algorithm for GPN is given in Alg. \[alg:sub-graph-train\]. On image classification, we usually use CNNs for $f(\cdot)$. In GPN, the output of the meta-learner $\mathcal M(T;\Theta)=\{\mP^{y}\}_{y\in\mathcal Y^T}$, i.e., the prototypes of class $y$ achieved in Eq. (\[equ:p-final\]), and the meta-learner parameter $\Theta=\{\Theta^{cnn}, \Theta^{prop}\}$.
Applying a Pre-trained GPN to New Tasks {#sec:gpn-test}
---------------------------------------
The outcomes of GPN training are the parameters $\{\Theta^{cnn}, \Theta^{prop}\}$ defining the GPN model and the prototypes of all the training classes stored in the memory. Given a new task $T$ with target classes $\mathcal Y^T$, we apply the procedure in lines 11-17 of Alg.\[alg:sub-graph-train\] to obtain the prototypes for all the classes in $\mathcal Y^T$ and the prediction probability of any possible test samples for the new task. Note that $\mathcal Y^T_{MST}$ can include training classes, so the test task can benefit from the prototypes of training classes in memory. However, this can directly work only when the graph already contains both the training classes and test classes in $T$. When test classes $\mathcal Y^T$ are not included in the graph, we apply an extra step at the beginning in order to connect test classes in $\mathcal Y^T$ to classes in the graph: we search for each test class’s $k_{c}$ nearest neighbors among all the training prototypes in the space of $\mP_y^0$, and add arcs from the test class to its nearest classes on the graph.
[|l|l|l|l|l|l|l|l|l|l|ll]{} & & &\
& & & & & & &\
\#cls & \#img & \#cls & \#img & & \#cls & \#img & \#cls & \#img & & &\
773 & 100,320 & 315 & 45,640 & 145,960 & 773 & 100,320 & 26 & 12,700& 113,020\
\[table:datasets\]
Experiments {#sec:experiments}
===========
In experiments, we conduct a thorough empirical study of GPN and compare it with several most recent methods for few-shot learning in 8 different settings of graph meta-learning on two datasets we extracted from ImageNet and specifically designed for graph meta-learning. We will briefly introduce the 8 settings below. First, the similarity between test tasks and training tasks may influence the performance of a graph meta-learning method. We can measure the distance/dissimilarity of a test class to a training class by the length (i.e., the number of edges) of the shortest path between them. Intuitively, propagation brings more improvement when the distance is smaller. For example, when test class “laptop” has nearest neighbor “electronic devices” in training classes, the prototype of electronic devices can provide more related information during propagation when generating the prototype for laptop and thus improve the performance. In contrast, if the nearest neighbor is “device”, then the help by doing prototype propagation might be very limited. Hence, we extract two datasets from ImageNet with different distance between test classes and training classes. Second, as we mentioned in Sec. \[sec:gpn-test\], in real-world problems, it is possible that test classes are not included in the graph during training. Hence, we also test GPN in the two scenarios (denoted by GPN+ and GPN) when the test classes have been included in the graph or not. At last, we also evaluate GPN with two different sampling methods as discussed in Sec. \[sec:training-strategy\]. The combination of the above three options finally results in 8 different settings under which we test GPN and/or other baselines.
Datasets {#sec:dataset}
--------
**Importance.** We built two datasets with different distance/dissimilarity between test classes and training classes, i.e., *tiered*ImageNet-Close and *tiered*ImageNet-Far. To the best of our knowledge, they are the first two benchmark datasets that can be used to evaluate graph meta-learning methods for few-shot learning. Their importance are: 1) The proposed datasets (and the method to generate datasets) provide benchmarks for the novel graph meta-learning problem, which is practically important since it uses the normally available graph information to improve the few-shot learning performance, and is a more general challenge since it covers classification tasks of any classes from the graph rather than only the finest ones. 2) On these datasets, we empirically justified that the relation among tasks (reflected by class connections on a graph) is an important and easy-to-reach source of meta-knowledge which can improve meta-learning performance but has not been studied by previous works. 3) The proposed datasets also provide different graph morphology to evaluate the meta knowledge transfer through classes in different methods: Every graph has 13 levels and covers $\sim$ 800 classes/nodes and it is flexible to sample a subgraph or extend to a larger graph using our released code. So we can design more and diverse learning tasks for evaluating meta-learning algorithms.-1
**Details.** The steps for the datasets generation procudure are as follows: 1) Build directed acyclic graph (DAG) from the root node to leaf nodes (a subset of ImageNet classes [@ren2018meta]) according to WordNet. 2) Randomly sample training and test classes on the DAG that satisfy the pre-defined minimum distance conditions between the training classes and test classes. 3) Randomly sample images for every selected class, where the images of a non-leaf class are sampled from their descendant leaf classes, e.g. the animal class has images sampled from dogs, birds, etc., all with only a coarse label “animal”. The two datasets share the same training tasks and we make sure that there is no overlap between training and test classes. Their difference is at the test classes. In *tiered*ImageNet-Close, the minimal distance between each test class to a training class is 1$\sim$4, while the minimal distance goes up to 5$\sim$10 in *tiered*ImageNet-Far. The statistics for *tiered*ImageNet-Close and *tiered*ImageNet-Far are reported in Table \[table:datasets\].
Experiment Setup
----------------
We used $k_{n}=5$ for snowball sampling in Sec. \[sec:training-strategy\]. The training took $\tau_{total}=$350$k$ episodes using Adam [@kingma2015adam] with an initial learning rate of $10^{-3}$ and weight decay $10^{-5}$. We reduced the learning rate by a factor of $0.9\times$ every 10$k$ episodes starting from the 20$k$-th episode. The batch size for the auxiliary task was $128$. For simplicity, the propagation steps $\mathcal T=2$. More steps may result in higher performance with the price of more computations. The interval for memory update is $m=3$ and the the number of heads is $5$ in GPN. For the setting that test class is not included in the original graph, we connect it to the $k_{c}=2$ nearest training classes. We use linear transformation for $g(\cdot)$ and $h(\cdot)$. For fair comparison, we used the same backbone ResNet-08 [@he2016deep] and the same setup of the training tasks, i.e., $N$-way-$K$-shot, for all methods in our experiments. Our model took approximately 27 hours on one TITAN XP for the 5-way-1-shot learning. The computational cost can be reduced by updating the memory less often and applying fewer steps of propagation.
Results
-------
**Selection of baselines.** We chose meta-learning baselines that are mostly related to the idea of metric/prototype learning (Prototypical Net [@snell2017prototypical], PPN [@ppn] and Closer Look [@closerlook]) and prototype propagation/message passing (PPN [@ppn]). We also tried to include the most recent meta-learning methods published in 2019, e.g., Closer Look [@closerlook] and PPN [@ppn].
The results for all the methods on *tiered*ImageNet-Close are shown in Table \[table:exp\_close\_random\] for tasks generated by random sampling, and Table \[table:exp\_close\_snowball\] for tasks generated by snowball sampling. The results on *tiered*ImageNet-Far is shown in Table \[table:exp\_far\_random\] and Table \[table:exp\_far\_snowball\] with the same format. GPN has compelling generalization to new tasks and shows improvements on various datasets with different kinds of tasks. GPN performs better with smaller distance between the training and test classes, and achieves up to $\sim$8% improvement with random sampling and $\sim$6% improvement with snowball sampling compared to baselines. Knowing the connections of test classes to training classes in the graph (GPN+) is more helpful on *tiered*ImageNet-Close, which brings 1$\sim$2% improvement on average compared to the situation without hierarchy information (GPN). The reason is that *tiered*ImageNet-Close contains more important information about class relations that can be captured by GPN+. In contrast, on *tiered*ImageNet-Far, the graph only provides weak/far relationship information, thus GPN+ is not as helpful as it shows on *tiered*ImageNet-Close.
**Model** **5way1shot** **5way5shot** **10way1shot** **10way5shot**
------------------------------------------- --------------------- --------------------- --------------------- --------------------- -- --
Prototypical Net [@snell2017prototypical] 42.87$\pm$1.67% 62.68$\pm$0.99% 30.65$\pm$1.15% 48.64$\pm$0.70%
GNN [@gnn2018few] 42.33$\pm$0.80% 59.17$\pm$0.69% 30.50$\pm$0.57% 44.33$\pm$0.72%
Closer Look [@closerlook] 35.07$\pm$1.53% 47.48$\pm$0.87% 21.58$\pm$0.96% 28.01$\pm$0.40%
PPN [@ppn] 41.60$\pm$1.59% 63.04$\pm$0.97% 28.48$\pm$1.09% 48.66$\pm$0.70%
[GPN]{} 48.37$\pm$1.80% 64.14$\pm$1.00% 33.23$\pm$1.05% 50.50$\pm$0.70%
[GPN+]{} **50.54$\pm$1.67**% **65.74$\pm$0.98**% **34.74$\pm$1.05**% **51.50$\pm$0.70**%
: Validation accuracy (mean$\pm$CI$\%95$) on $600$ test tasks achieved by GPN and baselines on *tiered*ImageNet-[**Close**]{} with few-shot tasks generated by [**random sampling**]{}.
\[table:exp\_close\_random\]
**Model** **5way1shot** **5way5shot** **10way1shot** **10way5shot**
------------------------------------------- --------------------- --------------------- --------------------- --------------------- -- --
Prototypical Net [@snell2017prototypical] 35.27$\pm$1.63% 52.60$\pm$1.17% 26.08$\pm$1.04% 41.48$\pm$0.76%
GNN [@gnn2018few] 36.50$\pm$1.03% 52.33$\pm$0.96% 27.67$\pm$1.01% 40.67$\pm$0.90%
Closer Look [@closerlook] 34.07$\pm$1.63% 47.48$\pm$0.87% 21.02$\pm$0.99% 33.70$\pm$0.44%
PPN [@ppn] 36.50$\pm$1.62% 52.50$\pm$1.12% 27.18$\pm$1.08% 40.97$\pm$0.77%
[GPN]{} 39.56$\pm$1.70% 54.35$\pm$1.11% 27.99$\pm$1.09% 42.50$\pm$0.76%
[GPN+]{} **40.78$\pm$1.76**% **55.47$\pm$1.41**% **29.46$\pm$1.10**% **43.76$\pm$0.74**%
: Validation accuracy (mean$\pm$CI$\%95$) on $600$ test tasks achieved by GPN and baselines on *tiered*ImageNet-[**Close**]{} with few-shot tasks generated by [**snowball sampling**]{}.
\[table:exp\_close\_snowball\]
**Model** **5way1shot** **5way5shot** **10way1shot** **10way5shot**
------------------------------------------- --------------------- --------------------- --------------------- --------------------- -- --
Prototypical Net [@snell2017prototypical] 44.30$\pm$1.63% 61.01$\pm$1.03% 30.63$\pm$1.07% 47.19$\pm$0.68%
GNN [@gnn2018few] 43.67$\pm$0.69% 59.33$\pm$1.04% 30.17$\pm$0.47% 43.00$\pm$0.66%
Closer Look [@closerlook] 42.27$\pm$1.70% 58.78$\pm$0.94% 22.00$\pm$0.99% 32.73$\pm$0.41%
PPN [@ppn] 43.63$\pm$1.59% 60.20$\pm$1.02% 29.55$\pm$1.09% 46.72$\pm$0.66%
[GPN]{} **47.54$\pm$1.68**% **64.20$\pm$1.01**% **31.84$\pm$1.10**% **48.20$\pm$0.69**%
[GPN+]{} **47.49$\pm$1.67**% **64.14$\pm$1.02**% **31.95$\pm$1.15**% **48.65$\pm$0.66**%
: Validation accuracy (mean$\pm$CI$\%95$) on $600$ test tasks achieved by GPN and baselines on *tiered*ImageNet-[**Far**]{} with few-shot tasks generated by [**random sampling**]{}.
\[table:exp\_far\_random\]
**Model** **5way1shot** **5way5shot** **10way1shot** **10way5shot**
------------------------------------------- --------------------- --------------------- --------------------- --------------------- -- --
Prototypical Net [@snell2017prototypical] 43.57$\pm$1.67% 62.35$\pm$1.06% 29.88$\pm$1.11% 46.48$\pm$0.70%
GNN [@gnn2018few] 44.00$\pm$1.36% 62.00$\pm$0.66% 28.50$\pm$0.60% 46.17$\pm$0.74%
Closer Look [@closerlook] 38.37$\pm$1.57% 54.64$\pm$0.85% 30.40$\pm$1.09% 33.72$\pm$0.43%
PPN [@ppn] 42.40$\pm$1.63% 61.37$\pm$1.05% 28.67$\pm$1.01% 46.02$\pm$0.61%
[GPN]{} **47.74$\pm$1.76**% **63.53$\pm$1.03**% **32.94$\pm$1.16**% **47.43$\pm$0.67**%
[GPN+]{} **47.58$\pm$1.70**% **63.74$\pm$0.95**% **32.68$\pm$1.17**% **47.44$\pm$0.71**%
: Validation accuracy (mean$\pm$CI$\%95$) on $600$ test tasks achieved by GPN and baselines on *tiered*ImageNet-[**Far**]{} with few-shot tasks generated by [**snowball sampling**]{}.
\[table:exp\_far\_snowball\]
Visualization of Prototypes Achieved by Propagation
---------------------------------------------------
We visualize the prototypes before (i.e., the ones achieved by Prototypical Networks) and after (GPN) propagation in Figure. \[fig:tsne-cases\]. Propagation tends to reduce the intra-class variance by producing similar prototypes for the same class in different tasks. The importance of reducing intra-class variance in few-shot learning has also been mentioned in [@closerlook; @gidaris2018dynamic]. This result indicates that GPN is more powerful to find the relations between different tasks, which is essential for meta-learning.
![ Prototypes before (top row) and after GPN propagation (bottom row) on *tiered*ImageNet-Close by random sampling for 5-way-1-shot few-shot learning. The prototypes in top row equal to the ones achieved by prototypical network. Different tasks are marked by a different shape ($\circ$/$\times$/$\triangle$), and classes shared by different tasks are highlighted by non-grey colors. It shows that GPN is capable to map the prototypes of the same class in different tasks to the same region. Comparing to the result of prototypical network, GPN is more powerful in relating different tasks. []{data-label="fig:tsne-cases"}](resources/tsne-prop-crop.pdf){width="\linewidth"}
Ablation Study {#sec:ablation}
--------------
In Table \[table:case-study\], we report the performance of many possible variants of GPN. In particular, we change the task generation methods, propagation orders on the graph, training strategies, and attention modules, in order to make sure that the choices we made in the paper are the best for GPN. For task generation, GPN adopts both random and snowball sampling (**SR-S**), which performs better than snowball sampling only (S-S) or random sampling only (R-S). We also compare different choices of propagation directions, i.e., **N$\rightarrow$C** (messages from neighbors, used in the paper), F$\rightarrow$C (messages from parents) and C$\rightarrow$C (messages from children). B$\rightarrow$P follows the ideas of belief propagation [@pearl1982reverend] and applies forward propagation for $\mathcal T$ steps along the hierarchy and then applies backward propagation for $\mathcal T$ steps. M$\rightarrow$P applies one step of forward propagation followed by a backward propagation step and repeat this process for $\mathcal T$ steps. The propagation order introduced in the paper, i.e., N$\rightarrow$C, shows the best performance. It shows that the auxiliary tasks (**AUX**), maximum spanning tree (**MST**) and multi-head (**M-H**) are important reasons for better performance. We compare the multi-head attention (**M-H**) using multiplicative attention (**M-A**) and using additive attention (A-A), and the former has better performance.
-- -- -- -- -- -- -- -- -- -- -- -- -- ---------------------
46.20$\pm$1.70%
49.33$\pm$1.68%
42.60$\pm$1.61%
37.90$\pm$1.50%
47.90$\pm$1.72%
46.90$\pm$1.78%
41.87$\pm$1.72%
45.83$\pm$1.64%
49.40$\pm$1.69%
46.74$\pm$1.71%
**50.54$\pm$1.67**%
-- -- -- -- -- -- -- -- -- -- -- -- -- ---------------------
: Validation accuracy (mean$\pm$CI%95) of GPN variants on *tiered*ImageNet-Close for 5-way-1-shot tasks. Original GPN’s choices are in **bold** fonts. Details of the variants are given in Sec. \[sec:ablation\].
\[table:case-study\]
Acknowledgements {#acknowledgements .unnumbered}
================
This research was funded by the Australian Government through the Australian Research Council (ARC) under grants 1) LP160100630 partnership with Australia Government Department of Health and 2) LP150100671 partnership with Australia Research Alliance for Children and Youth (ARACY) and Global Business College Australia (GBCA). We also acknowledge the support of NVIDIA Corporation and Google Cloud with the donation of GPUs and computation credits.
Visualization Results
=====================
Prototype Hierarchy
-------------------
We show more visualizations for the hierarchy structure of the training prototypes in Figure. \[fig:two-hier\].
![Visualization of the hierarchy structure of subgraphs from the training class prototypes transformed by t-SNE.[]{data-label="fig:two-hier"}](resources/two-hier-app-crop-v2.pdf){width="\linewidth"}
Prototypes Before and After Propagation
---------------------------------------
We show more visualization examples for the comparison of the prototypes learned before (Prototypical Networks) and after propagation (GPN) in Figure. \[fig:tsne-cases-app\].
![Prototypes before and after GPN propagation on *tiered*ImageNet-Close by random sampling for 5-way-1-shot few-shot learning. The prototypes in top row equal to the ones achieved by prototypical network. Different tasks are marked by a different shape ($\circ$/$\times$/$\triangle$), and classes shared by different tasks are highlighted by non-grey colors. It shows that GPN is capable to map the prototypes of the same class in different tasks to the same region. Comparing to the result of prototypical network, GPN is more powerful in relating different tasks.[]{data-label="fig:tsne-cases-app"}](resources/tsne-appendix-all.pdf){width="\linewidth" height="20cm"}
[^1]: We update GPN in each episode $\tau$ on a training task $T$, and train GPN for $\tau_{total}$ episodes.
|
---
abstract: 'Let $G$ be a directed graph with $n$ vertices and $m$ edges, and let $s \in V(G)$ be a designated source vertex. We consider the problem of single source reachability (SSR) from $s$ in presence of failures of edges (or vertices). Formally, a spanning subgraph $H$ of $G$ is a [*$k$-Fault Tolerant Reachability Subgraph ($k$-FTRS)*]{} if it has the following property. For any set $F$ of at most $k$ edges (or vertices) in $G$, and for any vertex $v\in V(G)$, the vertex $v$ is reachable from $s$ in $G-F$ if and only if it is reachable from $s$ in $H - F$. Baswana et.al. \[STOC 2016, SICOMP 2018\] showed that in the setting above, for any positive integer $k$, we can compute a $k$-FTRS with $2^k n$ edges. In this paper, we give a much simpler algorithm for computing a $k$-FTRS, and observe that it extends to higher connectivity as well. Our results follow from a simple application of *important separators*, a well known technique in Parameterized Complexity.'
author:
- 'Daniel Lokshtanov[^1]'
- 'Pranabendu Misra[^2]'
- 'Saket Saurabh[^3]'
- 'Meirav Zehavi[^4]'
bibliography:
- 'references.bib'
title: A Brief Note on Single Source Fault Tolerant Reachability
---
[^1]: University of California Santa Barbara, USA. `[email protected]`
[^2]: Max-Planck Institute for Informatics, Saarbrucken, Germany. `[email protected]`
[^3]: Institute of Mathematical Sciences, Chennai, India. `[email protected]`
[^4]: Ben-Gurion University, Beersheba, Israel. `[email protected]`
|
---
author:
- 'R. Cesaroni'
- 'M.T. Beltrán'
- 'Q. Zhang'
- 'H. Beuther'
- 'C. Fallscheer'
date: 'Received date; accepted date'
title: ' Dissecting a hot molecular core: The case of G31.41+0.31[^1] '
---
Introduction {#sint}
============
High-mass stars are usually defined as those exceeding $\sim$8 , based on the fact that stars above this mass limit do not have a pre-main sequence phase (Palla & Stahler [@past]). This means that accretion is ongoing until the star ignites hydrogen burning and reaches the zero-age main sequence (ZAMS). At this point the strong radiation pressure may halt and even reverse infall and thus stop further growth of the stellar mass. This led to the so-called “radiation pressure problem”. Recent studies have demonstrated that this limitation holds only in spherical symmetry. As first envisioned by Nakano ([@nakano]) and recently demonstrated by Krumholz et al. ([@krum]) and Kuiper et al. ([@kuip]), accretion through a circumstellar disk can explain the formation of stars up to the upper limit of the initial mass function, by allowing part of the photons to escape along the disk axis and boosting the ram pressure of the accreting gas through the small disk solid angle. It also appears that the powerful ionizing fluxes from these OB-type stars are not sufficient to destroy the disk, which eventually turns into an ionized, rotating accretion flow close to the star (Sollins et al. [@sollins]; Keto [@keto07]).
For these reasons it seems established that circumstellar accretion disks play a crucial role in the formation of [*all*]{} stars and not only solar-type stars. This theoretical result contrasts with the limited observational evidence of disks in high-mass (proto)stars. Only in recent years the number of disk candidates associated with luminous young stellar objects (YSOs) has significantly increased, mostly owing to the improvement of (sub)millimeter interferometers in terms of angular resolution and sensitivity. The main problems that one has to face in this type of search are the large distances to the sources (typically a few kpc) and the confusion caused by stellar crowding (OB-type stars form in clusters). These factors may explain the failure to detect disks in O-type stars, as opposed to the number of detections obtained for B-type stars (Cesaroni et al. [@ppv]). In association with the most luminous YSOs one finds only huge ($\la$0.1 pc), massive (a few 100 ) cores, with velocity gradients suggesting rotation. These objects, named “toroids”, are likely non-equilibrium structures, because the ratio between the accretion time scale and the rotation period is shorter than for disks: this implies that the toroid does not have enough time to adjust its structure to the new fresh material falling onto it (Cesaroni et al. [@ppv]; Beltrán et al. [@bel11]).
With this in mind, one can see that understanding the formation of the most massive stars ($>20~\Msun$) may benefit from a detailed investigation of toroids, also because these objects might be hosting true circumstellar disks in their interiors. Moreover, their mere existence may set tighter constraints on theoretical models. Despite the number of candidates, the existence of rotating toroids is still a matter of debate. What is questioned is the nature of the velocity gradient, which is sometimes interpreted as expansion instead of rotation (see e.g. Gibb et al. [@gibb] and Araya et al. [@araya] and references therein), thus suggesting that one might be seeing a compact bipolar outflow rather than a rotating core. In an attempt to distinguish between these two possibilities and more in general to shed light on these intriguing objects, we have focused our attention on one of the best examples: the hot molecular core (HMC) G31.41+0.31 (hereafter G31.41).
This prototypical HMC is located at a kinematic distance of 7.9 kpc and was originally imaged in the high-excitation (4,4) inversion transition of ammonia (Cesaroni et al. [@cesa94]) and in the (6–5) rotational transitions of methyl cyanide (; Cesaroni et al. [@cesag31]). The latter showed the existence of a striking velocity gradient (centered at an LSR velocity of $\sim$96.5 ) across the core in the NE–SW direction, already suggested by the distribution and velocities of OH masers (Gaume & Mutel [@gamu]). Follow-up interferometric observations with better angular resolution and in high-energy tracers have confirmed this result and revealed the presence of deeply embedded YSOs, which in all likelihood explains the temperature increase toward the core center (Beltrán et al. [@bel04], [@bel05], hereafter BEL04 and BEL05; Cesaroni et al. [@cesa10]). The G31.41 HMC is separated by $\sim$5 from an ultracompact (UC) region, and overlaps in projection on a diffuse halo of free-free emission, possibly associated with the UC region itself (see e.g. Fig. 2c of Cesaroni et al. [@cesa98]). The SPITZER/GLIMPSE images (Benjamin et al. [@benj]) show that the HMC lies in a complex pc-scale region where both extended emission and multiple stellar sources are detected. All these facts complicate the interpretation of the HMC structure and its relationship with the molecular surroundings and call for additional high-quality observations.
An important aspect of a proper interpretation of the velocity gradient observed in the HMC is the existence of an associated molecular outflow. A reasonable assumption, suggested by the analogy with low-mass YSOs, is that a disk actively undergoing accretion must be associated with a bipolar jet/outflow expanding along the disk rotation axis. Therefore, a way to distinguish between the two interpretations of the HMC velocity gradient (expansion vs rotation) is to search for a molecular outflow perpendicular to the gradient. Previous interferometric observations in the (1–0) line have suggested the presence of a collimated outflow directed SE–NW (Olmi et al. [@olmi96b]), but the poor uv coverage, especially on the shortest baselines, makes this result questionable. We have therefore conducted new interferometric observations at 1.3 mm with the Submillimeter Array (SMA) in the typical HMC tracer and in standard outflow tracers such as and . In order to be sensitive to extended structures filtered out by the interferometer, we have also mapped the region over $\sim$2with the IRAM 30-m telescope. The observational details are given in Sect. \[sobs\], while the results are illustrated in Sect. \[sres\] and discussed in Sect. \[sdis\]. Finally, the conclusions are drawn in Sect. \[scon\].
Observations and data reduction {#sobs}
===============================
Our observations have been performed with the SMA interferometer and IRAM 30-m telescope. The former was used with the aim to image the and (2–1) lines and the (12–11) transitions, but the 2 GHz bandwidth in both the LSB and USB allowed us to cover a much larger number of lines from a plethora of different molecules. Despite the richness of the spectrum, in this article we will present only the results obtained for the above mentioned lines, which are the best tracers for our purposes. The 30-m telescope was used to fill the zero-spacing of the SMA maps in the and lines and thus help establishing the morphology of the emission at all velocities and on all scales. Technical details of the observations are given in the next two sections.
------- ------------ ----- -----------------
Resolution synth. beam, PA
()
cont. — 7 088 x 072, 566
0.6 90 089 x 075, 528
0.6 90 089 x 075, 528
1.0 60 089 x 073, 534
1.0 50 088 x 075, 527
------- ------------ ----- -----------------
: Parameters of SMA images. The RMS of the lines is estimated in each channel[]{data-label="tobs"}
SMA interferometer
------------------
Observations of G31.41 were carried out with the SMA [^2] (Ho et al. [@ho2004]) in the 230 GHz band in the compact and extended configurations. The correlator was configured to a uniform spectral resolution of 0.41 MHz (0.6 ) over the entire 4 GHz spectral window. With IF frequencies of 4 to 6 GHz, the observations covered the rest frequencies from 219.3–221.3 GHz in the lower side band (LSB), and 229.3–231.3 GHz in the upper side band (USB). The compact configuration data were obtained on 2007 July 09 with eight antennas. With 225 GHz zenith opacity of 0.05 to 0.1, the typical double side-band system temperatures were around 200 K. The very extended data were obtained on 2007 May 21 with seven antennas. At 225 GHz the zenith opacity was about 0.2 and the double side-band system temperatures were around 400 K. For both tracks, we used Vesta and 3C273 for flux and bandpass calibrations. The time-dependent gains were calibrated using 1751+096 and 1830+063. The phase center of the observations was $\alpha$(J2000)=18$^{\rm h}$47$^{\rm m}$34315, $\delta$(J2000)=–0112459. The primary beam of the 6-m antennas is about 55 at the operating frequencies.
The visibility data were calibrated in the IDL superset MIR[^3] and MIRIAD (Sault et al. [@sault]), and were exported to MIRIAD format for imaging. The projected baselines of the combined visibilities from both configurations range from 12 m to 500 m. The continuum data were constructed from line free channels and the continuum was subtracted from the line uv data.
Continuum and channel maps of the and lines were created and cleaned in MIRIAD by weighting the data with “robust=0” to find a compromise between angular resolution and sensitivity to extended structures. The resulting synthesized beam is approximately $0\farcs88\times0\farcs75$, with P.A. 53. The $1\,\sigma$ rms is 5.8 mJy/beam in the continuum, and 89 mJy/beam in the line images with 0.6 spectral resolution. The and (2–1) line data were imaged in a similar fashion, but using natural weighting to enhance extended emission. The resulting synthesized beam is $0\farcs89\times1\farcs73$ with P.A. 53. The $1\,\sigma$ rms in the 1 channel maps is 50 mJy/beam.
The and data were also combined with the 30-m data (see Sect. \[ssd\]) to recover extended structure resolved out by the interferometer. The single-dish maps were Fourier-transformed and then suitably sampled in the uv domain, then merged with the SMA data. Finally natural weighted maps were created using the same procedure as for the non-merged data. The resulting synthesized beam is $2\farcs3\times1\farcs3$ with P.A. 63and the $1\,\sigma$ rms in the 1 channel maps is 66 mJy/beam.
IRAM 30-m telescope {#ssd}
-------------------
Maps of G31.41 with the 30-m telescope were made on November 4th, 2007. The HERA multi-beam receiver was used to cover a region of $2\arcmin\times2\arcmin$ centered on $\alpha$(J2000)=18$^{\rm h}$47$^{\rm m}$343 and $\delta$(J2000)=–0112459, using the on-the-fly-mode. To prevent systematic effects, the region was scanned alternatively along right ascension and declination, and all the data were eventually averaged to obtain the final map. The receiver was tuned to the frequencies of the (2–1) (HERA1) and (2–1) (HERA2) lines, which were hence observed simultaneously. Both lines were covered with the VESPA autocorrelator, with 0.4 spectral resolution. The reference position used for all maps was $\alpha$(J2000)=18$^{\rm h}$45$^{\rm m}$1524 and $\delta$(J2000)=–00555609. This was carefully chosen from CO surveys such as that by Sanders et al. ([@sand]), and checked to be free of emission in the observed lines. Data were reduced with the program CLASS of the GILDAS package[^4] and channel maps with 1 resolution were created.
Results {#sres}
=======
In the following we illustrate the results obtained, first for the typical hot core tracers (continuum, , and lines[^5] ) and then for the outflow tracers ( and isotopomers).
Core tracers {#score}
------------
Compared to previous Plateau de Bure interferometer (PdBI) observations in the same lines (BEL04, BEL05), our SMA observations benefit from a better uv coverage and more circular beam, thus allowing us to attain sub-arcsec resolution in all directions (see Table \[tobs\]; for comparison, the synthesized beam of the PdBI images was 11$\times$05 with PA=189).
To give an idea of the complexity of the line emission from the G31.41 HMC, in Fig. \[fmcnsp\] we show a spectrum covering all (12–11) transitions with $K\le10$, obtained by integrating the emission over the whole core. For an estimate of the and line parameters we refer to the study of BEL05 (see their Table 4). We did not attempt the identification of all lines detected, because this goes beyond the purposes of the present study. However, we note that a number of and transitions are heavily blended with each other and/or with other lines. Only the $K$=2,3,4,8 and the $K$=2,6 components appear free enough of contamination to be used for a detailed investigation. A similar consideration holds for the vibrationally excited ($v_8$=1) lines, shown in the same figure. Here, only the ($K$=3,$l$=–1) and ($K$=6,$l$=1) transitions are clearly detected and sufficiently separated from other lines to be considered for our study. Note that our bandwidth covers all $v_8$=1 lines down to the (11,–1) at 220788.016 MHz, but the lines below 221.19 GHz are too weak or too blended: this is the reason why the (12–11) $v_8$=1 spectrum in Fig. \[fmcnsp\] is displayed only above this frequency.
------------------------------ ---------- ------------- --------- ---------- ----------
$\alpha$ $\delta$ $T_{\rm B}$ $S_\nu$ FWHM $\Theta$
(J2000) (J2000) (K) (Jy) (arcsec) (arcsec)
18$^{\rm h}$47$^{\rm m}$3431 –0112460 68 4.6 11 076
------------------------------ ---------- ------------- --------- ---------- ----------
: Continuum parameters, i.e. peak position, peak brightness temperature in the synthesized beam, integrated flux density, full width at half maximum (FWHM), and deconvolved angular diameter []{data-label="tcont"}
Figure \[fmapsu\] shows continuum and line emission maps obtained by averaging the emission under the line and Table \[tcont\] lists the main parameters of the continuum emission. The most obvious feature is that the maps of all tracers are slightly elongated in the NE–SW direction, suggesting that the distribution of the molecular gas is flattened approximately in the direction defined by the two free-free continuum sources detected by Cesaroni et al. ([@cesa10]).
From the same figure one can also note that the high-energy transitions and the isotopomer emission peak approximately at the same position as the continuum, whereas the maximum of the low-energy $K$=2 component appears to be slightly offset from it. Such an offset is evident for all components with $K\le6$, while the $K$=7 and 8 line maps look more circular and peak at the core center. Nothing can be said about the $K$=9 transition, which is heavily blended with the (2–1) line. A plausible interpretation is that the opacity decreases with increasing excitation energy. In confirmation of that, one can see that this effect is even more prominent in the maps obtained by integrating the line emission over a narrow velocity range around the systemic LSR velocity, i.e. from 95.5 to 97.3 (see Fig. \[fmapsd\]). Now, the $K$=2 emission clearly splits into two peaks located on opposite sides with respect to the continuum and also the corresponding line map shows a similar pattern. We conclude that opacity plays a crucial role in the emission of this HMC and cannot be neglected in the analysis of the line emission.
A qualitative impression of the emission can be obtained from visual inspection of the channel maps shown in Fig. \[fmcnchm\]. In order to improve the signal-to-noise, such maps have been obtained by averaging the emission in the $K$=2, 3, and 4 components. These transitions have been chosen because they do not seem to be affected by significant contamination by other lines and have comparable excitation energies, which suggests that they are likely tracing the same gas.
Clearly, the emission gradually drifts from SW to NE with increasing velocity and outlines an “8-shaped” pattern close to the systemic velocity of $\sim$96.5 . These characteristics resemble those observed in rotating circumstellar disks around low-mass stars (see e.g. Fig. 1 of Simon et al. [@simon]). We will discuss this problem in Sect. \[sotto\]. A more quantitative analysis of the velocity field traced by the emission is attained – in analogy with the previous study by BEL04 – by fitting the $K$=0 to 4 components simultaneously, after fixing their separations in frequency to the laboratory values and forcing the line widths to be equal[^6]. This procedure improves the quality of the fit with respect to fitting a single $K$ line, but we stress that a similar result is obtained by fitting a single Gaussian to any of the unblended $K$ components. A map of the LSR velocity is presented in Fig. \[fvmap\], which confirms the existence of a clear velocity gradient directed approximately NE–SW (P.A.$\simeq$68) and roughly centered on the peak of the continuum emission. A different and more detailed representation of the same velocity gradient is presented in Fig. \[fpvplots\], where the position–velocity plots along the velocity gradient (P.A.=68) and perpendicular to it (P.A.=–22) are shown for three different lines of and . Note that to enhance the signal-to-noise ratio in the plots, we have averaged the emission along the direction perpendicular to the cut along which the position–velocity plot is calculated. As expected, no velocity trend is seen in the plots along P.A.=–22.
It is worth comparing our results to those of BEL04. With respect to their Fig. 2e, our Fig. \[fvmap\] presents a slightly broader velocity range ($\sim$5.5 instead of $\sim$4 ), over a larger region ($\sim$3 instead of $\sim$2). These differences are probably caused by the different angular resolutions and sensitivities. A more evident discrepancy appears in the map of the (12–11) emission, i.e. our Fig. \[fmapsd\] and their Fig. 1. In the former, the $K$=2 emission appears to outline two peaks located approximately to the NW and SE with respect to the core center, whereas in Fig. 1 of BEL04 the peaks lie to the E and W. However, one should keep in mind that this map was obtained integrating the emission under the $K$=0,1, and 2 components. Moreover, we have averaged the emission over a narrow interval around the peak, whereas BEL04 integrated under the whole line profile. Finally, the angular resolution is different for the two data sets. In order to attain a more consistent comparison, we have reconstructed our maps using the same clean beam as BEL04 and integrating over the same frequency interval. The new map (shown in Fig. \[fpdb\]) is now much more similar to that of BEL04, demonstrating how the apparent morphology of the emission may depend significantly on the resolution. Given the larger number of antennas and more circular beam, we believe that our SMA images reproduce the structure of the emission more faithfully than the old maps by BEL04 and BEL05.
Outflow tracers {#soutf}
---------------
One of the purposes of the present study was to compare the structure and kinematics of the HMC with those of a possible bipolar outflow associated with it. The existence of such an outflow had been suggested by Olmi et al. ([@olmi96b]), whose Plateau de Bure images in the (1–0) line seem to reveal two narrow lobes oriented SE–NW (see their Fig. 5). With this in mind, we have imaged G31.41 in the and $J$=2–1 rotational transitions with both the SMA and 30-m IRAM telescope, to recover also the emission filtered out by the interferometer. Figure \[fchm12co\] shows channel maps in the (2–1) line and effectively demonstrates the complexity of the region. Only at high velocities the emission appears quite compact, whereas close to the systemic velocity the structure presents a complicated pattern. In particular, from 95 to 100 the emission is uniformly distributed over an extended region: this means that the line opacity is sufficiently high that one can see only the surface of the cloud at these velocities, consistent with the relatively low values of the brightness temperature (10–15 K).
One arrives at the same conclusion by looking at Fig. \[fcospts\], where a comparison is shown of the single-dish spectra obtained by averaging the emission of three different CO isotopomers over a square region, 30 in size, centered at the HMC position. While the and profiles are double-peaked with a dip at $\sim$97 , the optically thin(ner) (2–1) line (Cesaroni, unpublished data) is Gaussian and peaks right at the velocity of the dip in the other two lines. This is an indication of self-absorption and thus of high optical depth in the and transitions.
An apparent feature of the (2–1) profile is the presence of broad wings. From Fig. \[fchm12co\] one sees that this high-velocity emission originates from 2–3 compact structures located roughly to the E (red-shifted) and W (blue-shifted) of the HMC. Whether this morphology is consistent with that of a bipolar outflow cannot be trivially decided on the basis of the evidence presented so far and requires a detailed analysis and discussion that we postpone to Sect. \[svgr\].
Discussion {#sdis}
==========
The main purpose of the present study is to shed light on the nature of the velocity gradient observed in this HMC (e.g. Fig. \[fvmap\]). As already mentioned, BEL04 and BEL05 favored the rotating toroid scenario. In contrast, other authors (Gibb et al. [@gibb]; Araya et al. [@araya]) preferred the outflow interpretation, thus posing a problem that we wish to address with our new SMA data.
Besides the (possible) existence of rotation and/or outflow, the situation in G31.41 is complicated by the existence of infall, detected by Girart et al. ([@gira]) as an inverse P-Cygni profile of the (7–6) line observed with the SMA. Therefore, before making a comparative discussion of the different models for the velocity gradient, we need to verify to what extent our measurements may be affected by infall.
Evidence for infall
-------------------
As already discussed in Sect. \[soutf\], Fig. \[fcospts\] clearly reveals that the and (2–1) lines are affected by self-absorption. However, only a weak asymmetry is seen in the line profile, with the red-shifted peak being slightly more prominent than the blue-shifted one. If such an asymmetry were caused by infalling gas, the blue-shifted peak should be stronger; hence we can conclude that no obvious evidence of infall is present on the large scale (30 or 1.1 pc) over which the spectra in Fig. \[fcospts\] have been calculated. It thus seems that the infall detected by Girart et al. ([@gira]) occurs close to the HMC. Do we see any evidence of this in the HMC tracers that we observed? All and (12–11) lines shown in Figure \[fmcnsp\] have Gaussian profiles with no hint of (self-)absorption, unlike the (7–6) line observed by Girart and collaborators. However, this is not surprising, because both the excitation energy and critical density of the latter (49 K and $\sim10^7$ ) are less than those of the former transitions (58–926 K and $\ga10^8$ ). This implies that the (7–6) line traces large radii of the HMC that are characterized by a relatively low temperature, and thus absorb the hot continuum photons emitted from the central region of the core. This effect does not apply to the (12–11) transitions, because these arise from a smaller hot shell, whose excitation temperature is much more similar to the brightness temperature of the dust continuum.
Despite the apparent lack of evidence for infall in the transitions, we have attempted a more detailed analysis to see whether some hint of asymmetry was present in the line profiles. The idea is that if the line shape is skewed toward the blue (although very weakly), a Gaussian fit should give a peak velocity less than the systemic velocity. With this in mind, in Fig. \[fperp\] we plot the line first moment (solid curves) as a function of distance from the HMC center along an axis with P.A.=–22, i.e. perpendicular to the direction of the velocity gradient. For the sake of completeness, also the line zero moment (integrated intensity) along the same direction is shown (dashed curves). This is done for the $K$=2,3,4, and 8 components of the ground-state transition and the (3,-1) line of $v_8$=1. Evidently, up to $K$=8 the line velocity attains a minimum value close to the HMC center, whereas this minimum is not seen in the $v_8$=1 line. This proves that the lower energy lines are slightly skewed to the blue toward the bright continuum peak, as expected for weak red-shifted absorption caused by infall. This effect is not seen in the highest energy line, coming from the innermost, hottest layers of the core.
We conclude that some of the (12–11) lines are affected by red-shifted absorption, although in a much less prominent way than the (7–6) line measured by Girart et al. ([@gira]). Note that this absorption causes only a marginal shift in the line velocity, of $\sim$0.5 , and is going to have only negligible effects on our study of the velocity shift of $\sim$5 observed across the HMC.
The NE–SW velocity gradient {#svgr}
---------------------------
The existence of a velocity gradient in the lines is proved beyond any doubt by Figs. \[fmcnchm\], \[fvmap\], and \[fpvplots\]. The high angular resolution allows us to establish that the velocity is shifting smoothly across the core, and one can reasonably exclude that this shift is caused by two unresolved velocity components (sub-cores) with different velocities. The question we intend to address here is whether this velocity gradient is caused by rotation of the HMC about a SE–NW axis, as hypothesized by BEL04, or to expansion in a bipolar outflow oriented NE–SW, as proposed by Gibb et al. ([@gibb]) and Araya et al. ([@araya]). How can one distinguish between these two scenarios? As explained in Sect. \[sint\], it is common belief that disks and outflows are tightly associated, with the latter being ejected along the rotation axis of the former. Therefore, if the velocity gradient in the G31.41 HMC is caused by rotation, one would expect to detect a bipolar outflow perpendicular to it, on a larger scale. In contrast, if the velocity gradient is tracing the “root” of a bipolar outflow, on a larger scale this outflow should become clearly visible along the same direction defined by the (small-scale) velocity gradient. Our combined single dish and interferometric observations in typical outflow tracers such as and should hence be well suited for our purposes, given the sensitivity to both small (a few arcsec) and large (a couple of arcmin) scales.
The results reported in Sect. \[soutf\] are quite ambiguous. The blue- and red-shifted and gas is oriented like the velocity gradient, suggesting that the two are manifestations of the same phenomenon. The existence of a NE–SW bipolarity in the emission can be appreciated in Fig. \[fcorb\], where we show the blue- and red-shifted (2–1) emission in pairs of channels equally offset from the systemic velocity. One is tempted to conclude that is indeed associated with a bipolar outflow, whose densest component is traced by the emission. However, this bipolarity in the CO maps is seen only on a small scale, because no evidence of high-velocity CO emission is found beyond $\sim$5 (i.e. 0.2 pc) from the HMC, implying an unusually small size for a typical outflow associated with a high-mass star-forming region. This casts some doubt on the outflow interpretation. Below we discuss the two hypotheses (outflow and toroid) in detail.
### Bipolar outflow
The outflow hypothesis is supported by a couple of facts. First of all, if the gradient is caused by a rotating toroid, one expects to find a bipolar outflow perpendicular to it, but this is not seen in our maps. Second, the position velocity plot of the emission in the direction of the velocity gradient suggests that the gas velocity is proportional to the distance from the star (as discussed later in Sect. \[storo\]), consistent with the Hubble-law expansion observed in molecular outflows from YSOs.
In order to check the plausibility of the outflow scenario, we have calculated the outflow parameters from the , , , and lines, in the latter two cases using the combined 30-m and SMA data. These are given in Table \[tout\], where we report the mass of the outflow, $M$, the momentum, $P$, the energy, $E$, and the corresponding rates obtained by dividing the previous quantities by the dynamical time scale (see below). In practice, $M=\sum_i m_i$, $P=\sum_i m_i V_i$, and $E=\sum_i (1/2) m_i
V_i^2$, where the sums are extended over only those channels, $i$, falling in the velocity ranges given in the footnotes of Table \[tout\], and $m_i$ is the mass moving with velocity $V_i$ relative to the systemic velocity. This mass is computed by integrating the line emission inside the regions corresponding to the $5\sigma$ level of the blue- and red-shifted emission. Note that the velocity intervals were chosen by inspecting the channel maps and selecting only those channels where the emission was sufficiently strong and at the same time not affected by the missing flux problem close to the systemic velocity. In our calculations we assume a temperature of $\sim$100 K for both molecules, intermediate between the peak brightness temperature (77 K) of the (2–1) line and the rotational temperature (164 K) estimated by BEL05 from the (6–5), (12–11), and (6–5) $v_8$=1 lines (see their Fig. 6). While this temperature may seem too high for the outflow component traced by the CO isotopomers, assuming 50 K instead of 100 K would reduce the outflow parameters by only a factor 0.6. The abundance of and relative to are assumed equal to $10^{-8}$ and $10^{-4}$ respectively (see e.g. Van Dishoek et al. [@vand]), while the isotopic ratios / and / are taken equal to 50 after Wilson & Rood ([@wiro]), for a galactocentric distance of 4.5 kpc. The dynamical time scale of the outflow, $t_{\rm dyn}\simeq4\times10^3$ yr, is calculated from the maximum size ($\sim$0.12 pc) and velocity ($\sim$30 ) of the lobes. Note that the (unknown) inclination of the outflow with respect to the line of sight has not been taken into account in our estimates.
---------------------------------------- -------------------- ------------------ ----------------- -----------------
parameter $^a$ $^b$
$K$=4 $^c$ $K$=2 $^d$
$M (M_\odot)$ 3.8 20 60 290
$P (M_\odot\,\kms)$ 48 230 350 1200
$E (L_\odot\,{\rm yr})$ $5.9\times10^4$ $2.2\times10^5$ $1.7\times10^5$ $4.2\times10^5$
$\dot{M} (M_\odot\,{\rm yr}^{-1})$ $9.5\times10^{-4}$ $5\times10^{-3}$ 0.015 0.07
$\dot{P} (M_\odot\,\kms{\rm yr}^{-1})$ 0.012 0.057 0.087 0.30
$\dot{E} (L_\odot)$ 14.7 54.6 41.8 104
---------------------------------------- -------------------- ------------------ ----------------- -----------------
: Outflow parameters calculated from different lines. The dynamical time scale is $t_{\rm dyn}\simeq4\times10^3$ yr and a gas temperature of 100 K is assumed in the calculations[]{data-label="tout"}
$^a$ blue wing from 78.9 to 89.9 , red wing from 105.9 to 127.9 . Abundance relative to : $10^{-4}$\
$^b$ blue wing from 77.9 to 88.9 , red wing from 103.9 to 112.9 . Abundance relative to : $2\times10^{-6}$\
$^c$ blue wing from 88.9 to 91.9 , red wing from 100.9 to 103.9 . Abundance relative to : $10^{-8}$\
$^d$ blue wing from 90.1 to 92.5 , red wing from 99.1 to 101.5 . Abundance relative to : $2\times10^{-10}$
From Table \[tout\] one notes that the values increase with decreasing abundance of the molecule. This can be explained in terms of decreasing optical depth, because higher opacities lead to an underestimate of the column density and hence of the mass. Indeed, from the ratio between the and emission in the line wings one derives opacities as high as $\sim$30 for . Therefore, the most reliable estimates should be those obtained from . These depend significantly on the abundance, which is known to present considerable variations in molecular clouds. However, the value assumed in our calculations is one of the highest found in the literature and we therefore believe that our estimates are likely to be lower limits. This leads us to another consideration. Our estimates are very close to the largest outflow parameters ever measured (see e.g. López-Sepulcre et al. [@lopsep] and Wu et al. [@wu04]), corresponding to a YSO powering the outflow of at least $\sim10^5~L_\odot$. More precisely, using Wu et al. ([@wu04]) relationship between bolometric luminosity ($L_{\rm bol}$) of the powering source and outflow momentum rate, from the value in Table \[tout\] one obtains $L_{\rm bol}=6\times10^6$ . This is much higher than the luminosity estimated for the HMC ($\sim10^5~L_\odot$) by Osorio et al. ([@osor]) and that obtained from the corresponding IRAS fluxes ($2.6\times10^5$ ; see Cesaroni et al. [@cesa94]). Moreover, the momentum rate of 0.3 yr$^{-1}$ is an order of magnitude higher than that estimated by Cesaroni et al. ([@cesa10]) from VLA observations of the free-free continuum emission ($\sim$0.03–0.06 yr$^{-1}$).
It is also worth noting that the velocity gradient in G31.41 seems to involve the whole core and not only the gas emitting in the line wings. Usually, in outflow sources the emission close to the systemic velocity – i.e. that inside the FWHM of the line – traces the molecular core; in contrast, in G31.41 most of the emission is affected by the velocity gradient, as one can see from the channel maps in Fig. \[fmcnchm\], where only emission over a limited velocity range ($\sim$2–3 ) appears to arise from the central region. As discussed below in Sect. \[sotto\], this situation is reminiscent of that observed in circumstellar disks around lower-mass YSOs. Our estimates of the outflow parameters based only on the line-wing emission are likely lower limits, which makes the case of G31.41 even more extreme compared to typical outflows.
In conclusion, the outflow hypothesis seems to yield values for the outflow parameters that are too high. In addition, one should note that the typical parameters given in studies such as those quoted above refer to single-dish observations of pc-scale flows, an order of magnitude higher than that observed in G31.41. Although we cannot exclude that in G31.41 one is observing the earliest stages of the expansion, it is questionable that the parameters of a young, compact outflow are greater than those typical of much older, extended outflows.
Finally, we note that the dynamical time scale of the putative outflow is an order of magnitude shorter than the time needed to form typical hot core species – such as methyl cyanide – according to theoretical models (see e.g. Charnley et al. [@charn]). Our estimate of $t_{\rm dyn}$ is affected by large uncertainties, owing to the unknown inclination angle and the difficulty in tracing the whole extent of the lobes. However, to increase $t_{\rm dyn}$ by a factor $\ga$10 one has to assume that the outflow lies very close ($\la$6) to the line of sight and/or that we detect only the most compact part of lobes extending over a region 10 times larger than that imaged in our and CO maps. The latter explanation is ruled out by our SMA+30m combined maps that do not reveal any large-scale bipolar outflow. The former would imply a significant overlap between the blue- and red-shifted lobes in the plane of the sky, which is not seen in Fig. \[fhmc\]. Indeed, from the observed separation between the peaks of the red- and blue-shifted emission ($\sim$1), assuming that the intrinsic length of the lobes is comparable to the radius of the core measured in the plane of the sky ($\sim$1), one can obtain a rough estimate of the inclination angle of $\arcsin(0.5)=30\degr$. Correspondingly, $t_{\rm dyn}$ would increase by only a factor 1.7.
Given the number of problems encountered in the outflow scenario, we investigate also a second possibility, namely that the velocity gradient of the HMC is due to a rotating toroid.
### Rotating toroid {#storo}
In an attempt to shed light on the velocity field in the HMC and its molecular surroundings, we overlay in Fig. \[fpvmcnco\] the position-velocity plots along the velocity gradient in high- and low-density tracers. For the sake of completeness we show both the SMA and the combined 30m+SMA data in the and (2–1) transitions. Note that the cut was made along P.A.=68 and all plots were obtained after averaging the emission along the direction perpendicular to the cut. This has the twofold purpose of increasing the S/N of the plots and taking into account emission at all offsets along the (putative) rotation axis.
As already explained, the and emission close to the systemic velocity is highly opaque and does not convey any information on the HMC, but is instructive to study the high-velocity and/or low-density gas around it. Vice versa, the emission is an excellent HMC tracer, but is not detected at high velocities and/or beyond $\sim$2 from the center. Combining all these tracers is the only way to perform a detailed and complete analysis of the velocity field in this core. Indeed, Fig. \[fpvmcnco\] is very instructive. It shows that the velocity trend observed in is complementary to that seen in the CO isotopomers and fills the gap caused by the interferometer resolving out the extended, bulk CO emission.
A more thorough inspection of this figure reveals that, despite the similarity, the and plots are significantly different. While the and patterns match very well in the overlapping regions (i.e. between 89 and 94 and between 102 and 105 ), the emission is offset by $\sim$1 from the and patterns, to the SW in the blue-shifted part, and to the NE in the red-shifted part. An interesting difference between and , on the one hand, and , on the other, is that the high-velocity emission is found at relatively large offsets in , whereas in it appears to peak close to the HMC center.
Is it possible to find a coherent interpretation of all these facts in the rotating toroid scenario? A possible explanation is that of a self-gravitating structure with a heavy, compact stellar cluster at the center. In the outer regions (traced by ), the gravitational field is determined by the gas mass and the rotation curve flattens, whereas close to the center (where and are detected) the velocity field resembles Keplerian rotation, because the gas mass becomes comparable to the stellar mass. This scenario is analogous to that studied by Bertin & Lodato ([@belo]), who determined the rotation curve of a self-gravitating disk with a central star.
In order to check the viability of this interpretation, below we first estimate the mass of the HMC and then verify whether this suffices to support rotational equilibrium.
From the continuum flux (see Table \[tcont\]), we derive a mass of 1700 $M_\odot$ for the HMC, assuming a dust temperature of 100 K and a dust absorption coefficient at 1.3 mm of 0.005 cm$^2$ g$^{-1}$ (Kramer et al. [@kramer]). The temperature of 100 K is a lower limit, because the real temperature must be higher than or equal to the maximum brightness temperature of the lines measured in the synthesized beam ($\sim$95 K). The latter corresponds to the surface temperature of the HMC, because the ground state lines are optically thick and, as discussed by BEL05, the temperature inside the HMC is likely increasing. The mass obtained is affected by a large uncertainty mostly because of the poorly known dust properties: for example, BEL04 adopted an absorption coefficient of 0.02 cm$^2$ g$^{-1}$, which would imply a HMC mass of only 420 $M_\odot$.
In Fig. \[fpvfit\] we show the same position-velocity plot of the and lines as in Fig. \[fpvmcnco\], this time overlaying the pattern outlining the region inside which emission is expected for a Keplerian rotating and free-falling disk. The latter does not take into account the line-width nor the spectral and angular resolutions and has been obtained assuming that the gas velocity is the vector sum of a tangential component due to Keplerian rotation about a central mass, $M$, plus a radial component due to free-fall onto the same mass. Under these assumptions the velocity component along the line of sight can be expressed as $$V = \sqrt{G\,M}\frac{x}{R^\frac{3}{2}} + \sqrt{2G\,M}\frac{z}{R^\frac{3}{2}}~,$$ where $x$ and $z$ are the coordinates, respectively, along the disk plane and the line of sight, and $R=\sqrt{x^2+z^2}$ is the distance from the center of the disk. We also assume that R lies between the disk radius, $R_{\rm o}$, and a minimum inner radius, $R_{\rm i}$. The first term on the right hand side of the equation is the component due to Keplerian rotation, the second that due to free fall. The dashed pattern in Fig. \[fpvmcnco\] has been obtained by plotting the maximum and minimum velocities $V$ for $z$ varying across the disk, i.e. from $-\sqrt{R_{\rm o}^2-x^2}$ and $+\sqrt{R_{\rm o}^2-x^2}$, taking into account that the region $R<R_{\rm i}$ is forbidden. In our case, a satisfactory fit is obtained for $M=330~M_\odot$, $R_{\rm o}=3\arcsec$ (or 0.11 pc), and $R_{\rm i}=0\farcs15$ (or 0.0055 pc). Note that the signature of (pseudo-)Keplerian rotation is the “butterfly” shape of the plot, determined by the two “spurs” of emission at about $\pm$3 and $\pm$4 relative to the systemic velocity ($\sim$96.5 ), plus the presence of high-velocity emission at zero offset[^7]. It is worth stressing that the contribution of the (2–1) line is crucial to outline such a pattern, which cannot be recognized from the sole emission. This explains why BEL05 managed to fit their data assuming a flat or solid-body rotation curve, and Girart et al. ([@gira]), from a number of CH$_3$OH lines, found that the rotation velocity increases with radius.
One may wonder whether a HMC as massive as several 100 $M_\odot$ may be undergoing Keplerian rotation. As suggested by Cesaroni et al. ([@ppv]) and demonstrated by Beltrán et al. ([@bel11]), these massive, large rotating cores are to be considered transient toroidal structures feeding a cluster of YSOs rather than stable circumstellar accretion disks. In fact, the latter are stabilized by the central star(s), whose mass is greater than that of the disk, whereas the former are dynamically dominated by the gas mass and hence are short-lived. Although this scenario may be true in most cases, G31.41 might represent an exception. We speculate that this HMC could contain a large number of stars tightly packed in the central region. In this case the stars could have a stabilizing effect analogous to that of a single point-like object located at the HMC center, similar to the previously mentioned model by Bertin & Lodato ([@belo]).
With all this in mind, the pseudo-Keplerian pattern recognized in Fig. \[fpvfit\] suggests that the HMC mass should be comparable to the total mass of the embedded stars that are tightly packed at the center. Is this scenario plausible? Indeed, in the case of G10.62–0.38, a massive star forming region with a luminosity of $9.2\times10^5~L_\odot$, Sollins et al. ([@sollins]) suggest that within a radius of 0.03 pc, several O stars with a total mass of 175 have formed at the center of a flattened disk. In our case, the existence of (at least) two high-mass YSOs close to the HMC center and separated (in projection) by only 019 or 1480 AU has been proved by Cesaroni et al. ([@cesa10]), who detected the free-free continuum emission. In terms of mass, these two YSOs represent only the tip of the iceberg, because the total mass of the associated cluster is dominated by (undetected) low-mass stars. Assuming e.g. a Miller & Scalo ([@milsc]) mass function, one can estimate the cluster mass from that of the most massive star. For this we can use the value of 20–25 computed by Osorio et al. ([@osor]) from their model fit and obtain a total mass of the cluster of $\sim10^3$ . The same result is obtained by fixing the total luminosity of the cluster to the value of $2\times10^5~\Lsun$ quoted by Osorio et al. ([@osor]). In all likelihood $\sim10^3$ is an upper limit to the real stellar mass, because the cluster mass function may be highly incomplete in such a small region. However, this estimate indicates that the dynamical mass of $\sim$330 , obtained from the fit in Fig. \[fpvfit\], is probably dominated by the stellar mass, which would lend support to the application of Bertin & Lodato ([@belo]) model to the case of G31.41.
Finally, we note that the toroid interpretation is compatible with the hourglass-shaped morphology of the magnetic field (see Girart et al. [@gira]), whose symmetry axis is directed SE–NW, because the latter coincides with the rotation axis of the toroid. However, a caveat is in order. Girart et al. ([@gira]) find a significant correlation between rotation velocity and radius in the core, using different tracers. In particular, they conclude that in the region sampled by their observations (lying between $\sim$05 and 16), velocity increases with distance from the HMC center (see their Fig. 4). This result seems inconsistent with our hypothesis that the core is undergoing pseudo-Keplerian rotation, because in this case the rotation velocity should decrease with radius. Perhaps this discrepancy can be explained by the presence of the magnetic field. Higher angular resolution observations are needed to sample the velocity field inside the core and thus obtain a reliable, direct measurement of the rotation curve.
The “8-shaped” structure {#sotto}
-------------------------
The last aspect we will discuss in the context of the outflow/toroid controversy, is the double-peaked, “8-shaped” structure observed in the $K$=2 map at the systemic velocity (see Fig. \[fmapsd\]). The same morphology is seen in all components up to $K$=6, i.e. up to excitation energies of $\sim$300 K, whereas the $K$=8 map (energy of 513 K) presents a single peak at the HMC center (Fig. \[fmapsd\]). In Sect. \[score\] we interpreted these facts in terms of opacity and temperature gradients. Indeed, BEL05 found that the rotational temperature peaks toward the HMC center and from the ratio between our and (12–11) $K$=2 data we calculate an optical depth at the peak of the line in the range 8–77 across the HMC. The interplay between opacity and temperature can explain the existence of a dip at the HMC center, but does not justify the lack of circular symmetry in the low-$K$ maps. Here, we wish to find an explanation for the “8-shaped” feature and check whether this can better fit into the outflow or toroid model.
As a basis for our discussion, in Fig. \[fhmc\] we present an overlay of the blue- and red-shifted $K$=4 emission on the bulk emission in the same line. In this way, one is comparing the high- with the low-velocity emission.
In the outflow scenario, a naïve interpretation of this figure is that the high-velocity gas is leaking from the HMC through the axis of a “donut-like” structure seen edge-on, corresponding to the “8-shaped” feature in the map. In this case, the two peaks would coincide with the maxima of column density across the “donut”. Albeit plausible, this interpretation has a problem. The large opacity should prevent the detection of two distinct peaks, because the line brightness is independent of column density. Therefore, instead of the 8-shaped feature, one should see an elongated structure perpendicular to the bipolar outflow and peaking [*at the center*]{} of it (i.e. at the HMC center).
Explaining the presence of the 8-shaped structure appears to be a problem also in the toroid scenario. In fact, if the toroid is seen edge-on, the emission at the systemic velocity should peak in between the blue- and red-shifted emission. However, if the toroid is inclined with respect to the line of sight, the nearest and farthest sides of it are seen displaced from one another on the plane of the sky, symmetrically displaced with respect to the center. This is indeed what we see in Fig. \[fhmc\], where the two peaks lie along the (projection of the) rotation axis and are equally offset from the center. The situation depicted in this figure resembles that of the ring undergoing Keplerian rotation about the low-mass system GG Tau (see Fig. 6a of Guilloteau et al. [@ggtau]). Clearly, this comparison is inappropriate, because the scales of the two objects differ by more than an order of magnitude. Moreover, G31.41 is probably more similar to a “puffy pancake” rather than a “donut” or ring, because gas and dust are present at all radii, as demonstrated by the continuum and high-energy line emission peaking at the center of the HMC (see Figs. \[fmapsu\] and \[fmapsd\]). Note, however, that the emission from the lower energy lines may be arising mostly from the colder outer region and hence be confined in a ring. Finally, G31.41, unlike GG Tau, is embedded in a dense pc-scale envelope, which complicates the line and continuum radiative transfer. Despite all these differences, the comparison between the two objects is intriguing and may qualitatively explain the features observed in G31.41.
Further support for this interpretation is obtained by comparing the case of G31.41 to that of a source closer in mass than GG Tau. This is the Herbig Ae star MWC758, which is surrounded by a circumstellar disk, as demonstrated by the model fit of Isella et al. ([@isella]). A comparison between the channel maps presented by these authors (see their Fig. 2) with those in Fig. \[fmcnchm\] reveals a surprising similarity: in both cases the shape of the emission is circular and peaks at opposite positions with respect to the center in the blue- and red-shifted channels, but turns into an 8-shaped structure close to the systemic velocity. This fact may be the fingerprint of a rotating disk in G31.41, as well as in MWC758.
Summary and conclusions {#scon}
=======================
With the present study we have performed a deep analysis of the geometrical structure and kinematics of the HMC in the high-mass star-forming region G31.41. Thanks to new high angular resolution SMA images and complementary IRAM 30-m maps, we have investigated both the HMC, through the (12–11) transition, and the lower density surroundings, through and . We could thus search for a possible outflow powered by the star(s) embedded in the HMC. In particular, our main goal was to shed light on the nature of the NE–SW velocity gradient, previously detected in the HMC by various authors, and possibly distinguish between rotation and expansion.
Our new data confirm the presence of the velocity gradient basically in all HMC tracers observed, and indicate that a red–blue symmetry along the same axis is seen also in the CO isotpomers. We were unable to detect any bipolar outflow along the SE–NW direction, in contrast with the findings of Olmi et al. ([@olmi96b]). The latter was probably an artifact caused by very limited uv sampling and lack of zero-spacing information.
We find a hint of infall in the HMC, consistent with the results of Girart et al. ([@gira]). A comparison between the and CO maps suggests that these species are tracing the same phenomenon. Especially the (2–1) line emission appears to be the obvious prosecution of the (12–11) emission on a larger scale.
We have discussed whether the velocity gradient observed in G31.41 is a compact bipolar outflow or a rotating, geometrically thick, toroidal structure. The former hypothesis implies outflow parameters typical of high-mass stars with luminosities in excess of $10^5$ , whereas previous estimates by Osorio et al. ([@osor]) and Cesaroni et al. ([@cesa94; @cesa10]) indicate a significantly lower luminosity for G31.41. Moreover, such an outflow would be much more compact than typical bipolar outflows detected in high-mass star-forming regions and the dynamical time scale would be an order of magnitude shorter than that needed to form HMC spacies such as . It is also worth noting that a core undergoing expansion seems difficult to reconcile with the detection of infall.
The composite and position-velocity plot can be explained if the gas is undergoing both (pseudo-)Keplerian rotation and infall, suggesting that this massive object could be dynamically stabilized by a compact cluster of YSOs tightly packed inside a few 1000 AU from the center. In this scenario the emission could be tracing the outer regions, where the gravitational field is dominated by the gas and the rotation velocity tends to a constant value. Albeit intriguing, this interpretation remains speculative and could be proved only by producing synthetic and CO maps and position-velocity plots to be compared with the data. This numerical effort goes beyond the purposes of the present study.
We conclude that the case of G31.41 could be a scaled-up version of lower-mass YSOs, such as GG Tau and MWC758, where circumstellar disks have been detected and successfully modeled. We stress that whatever the interpretation, the observations of this object have produced an important finding for the study of high-mass star formation. We obtained iron-clad evidence for the existence of a velocity gradient across the HMC and demonstrated that this gradient is [*not*]{} due to multiple cores unresolved in the beam and moving at different velocities, but to the gas undergoing a smooth drift in (line-of-sight) velocity from the one end to the other of the core. If, as we believe, one is dealing with a rotating toroid, then this indicates that star formation in this HMC may proceed in a similar way to low-mass stars, namely through circumstellar, centrifugally supported disks “hidden” inside the densest interiors of the toroid itself. If, instead, the outflow interpretation were correct, such a sharp, neat velocity gradient would strongly suggest the existence of a very effective focusing mechanism for the outflow and the best candidate as a collimating agent would be a (yet undetected) circumstellar disk. Therefore, regardless of whether the G31.41 HMC is undergoing expansion or rotation, the role of disks in the mechanism of high-mass star formation seems to be strengthened by our findings. The advent of ALMA will be crucial to detect deeply embedded disks in HMCs.
We thank the SMA staff for their help during the observations. The GILDAS team is also acknowledged for the excellent software that we used to analyze our data.
Araya, E., Hofner, P., Kurtz, S., Olmi, L., & Linz, H. 2008, ApJ, 675, 420 Beltrán, M.T, Cesaroni, R., Neri, R., et al. 2004, ApJ, 601, L187 (BEL04) Beltrán, M.T, Cesaroni, R., Neri, R., et al. 2005, A&A, 435, 901 (BEL05) Beltrán, M.T, Cesaroni, R., Neri, R., & Codella, C. 2011, A&A, 525, A151 Benjamin, R.A., Churchwell, E., Babler, B.L., et al. 2003, PASP, 115, 953 Bertin, G., Lodato, G. 1999, A&A 350, 694 Cesaroni, R., Churchwell, E., Hofner, P., Walmsley, C.M., & Kurtz, S. 1994a, A&A, 288, 903 Cesaroni, R., Olmi, L., Walmsley, C.M., Churchwell, E., & Hofner, P. 1994b, ApJ, 435, L137 Cesaroni, R., Hofner, P., Walmsley, C.M., & Churchwell, E. 1998, A&A, 331, 709 Cesaroni, R., Galli, D., Lodato, G., Walmsley, C.M., & Zhang, Q. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil (Tucson: Univ. of Arizona Press), 197 Cesaroni, R., Hofner, P., Araya, E., & Kurtz, S. 2010, A&A, 509, 50 Charnley, S.B., Tielens, A.G.G.M., & Millar, T.J. 1992, ApJ, 399, L71 Gaume, R.A. & Mutel, R.L. 1987, ApJS, 65, 193 Gibb, A.G., Wyrowski, F., & Mundy, L.G. 2004, ApJ, 616, 301 Girart, J.M., Beltrán, M.T., Zhang, Q., Rao, R., & Estalella, R. 2009, Science, 324, 1408 Guilloteau, S., Dutrey, A., & Simon, M. 1999, A&A, 348, 570 Ho, P.T.P., Moran, J.M., & Lo, K.Y. 2004, ApJ, 616, L1 Isella, A., Natta, A., Wilner, D., Carpenter, J.M., & Testi, L. 2010, ApJ, 725, 1735 Keto, E.H. 2007, ApJ, 666, 976 Kramer, C., Richer, J., Mookerjea, B., Alves, J., Lada, C. 2003, A&A 399, 1073 Krumholz, M.R., Klein, R.I., McKee, C.F., Offner, S.S.R., & Cunningham, A.J. 2009, Science, 323, 754 Kuiper, R., Klahr, H., Beuther, H., & Henning, Th. 2010, ApJ, 722, 1556 López-Sepulcre, A., Codella, C., Cesaroni, R., Marcelino, N., & Walmsley, C.M. 2009, A&A, 499, 811 Miller, G.E. & Scalo, J.M. 1979, ApJ, 41, 513 Nakano, T. 1987, MNRAS, 224, 107 Olmi, L., Cesaroni, R., Neri, R., & Walmsley, C.M. 1996, A&A 315, 565 Osorio, M., Anglada, G., Lizano, S., & D’Alessio, P. 2009, ApJ 694, 29 Palla, F., Stahler, S.W. 1993, ApJ 418, 414 Sanders, D.B., Clemens, D.P., Scoville, N.Z., & Solomon, P.M. 1986, ApJS, 60, 1 Sault, R.J., Teuben, P.J., & Wright, M.C.H. 1995, in ASP Conf. Ser. 77, Astronomical Data Analysis Software and Systems IV, 433 Simon, M., Guilloteau, S., Dutrey, A. 2001, ApJ 545, 1034 Sollins, P.K., Zhang, Q., Keto, E., & Ho, P.T.P. 2005, ApJ 624, L49 Van Dishoeck, E.F., Blake, G.A., Draine, B.T., & Lunine, J.I. 1993, Protostars and Planets III, ed. E.H. Levy & J.I. Lunine, (Tucson: Univ. of Arizona Press), 163 Wilson, T.L. & Rood, R. 1994, ARA&A, 32, 191 Wu, Y., Wei, Y., & Zhao, M., 2004, A&A, 426, 503
[^1]: Based on observations carried out with the Submillimeter Array. The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.
[^2]: The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics, and is funded by the Smithsonian Institution and the Academia Sinica.
[^3]: The MIR cookbook can be found at https://www.cfa.harvard.edu/$\sim$cqi/mircook.html
[^4]: The GILDAS software is available at http://www.iram.fr/IRAMFR/GILDAS
[^5]: Parameters of the and transitions can be found, e.g., in the “splatalogue” database: http://www.splatalogue.net/
[^6]: For this purpose, program CLASS of the GILDAS package was used, after extracting the relevant spectra from the data cube and converting them to the appropriate format.
[^7]: The peak in the contour plot at 0 and $\sim$84 is likely due to an unidentified line of a core tracer, whereas the broad wing of emission extending up to $\sim$70 is genuine high-velocity (2–1) emission.
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abstract: 'We present a new class of solutions for the gas flows in elliptical galaxies containing massive central black holes (BH). Modified King model galaxies are assumed. Two source terms operate: mass loss from evolving stars, and a secularly declining heating by supernovae (SNIa). Relevant atomic physical processes are modeled in detail. Like the previous models investigated by Ciotti et al. (1991, CDPR), these new models first evolve through three consecutive evolutionary stages: wind, outflow, and inflow. At this point the presence of the BH alters dramatically the subsequent evolution, because the energy emitted by the BH can heat the surrounding gas to above virial temperatures, causing the formation of a hot expanding central bubble. Short and strong nuclear bursts of radiation ($\lbh$) are followed by longer periods during which the X-ray galaxy emission comes from the coronal gas ($\lx$). The range and approximate distribution spanned by $\lx$ are found to be in accordance with observations of X-ray early type galaxies. Moreover, although high accretion rates occur during bursting phases when the central BH has a luminosity characteristic of quasars, the total mass accreted is very small when compared to that predicted by stationary cooling-flow solutions and computed masses are in accord with putative BH nuclear masses. In the bursting phases the X-ray gas luminosity is low and the surface brightness profile is very low compared to pre-burst or to cooling flow models. We propose that these new models, while solving some long-standing problems of the cooling flow scenario, can provide a unified description of QSO-like objects and X-ray emitting elliptical galaxies, these being the same objects observed at two different evolutionary phases.'
author:
- 'Luca Ciotti and Jeremiah P. Ostriker'
title:
- 'Cooling flows and quasars:'
- 'Different aspects of the same phenomenon? Concepts'
---
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Introduction
============
As first revealed by [*Einstein*]{} observations, normal elliptical galaxies, both isolated or in groups and clusters, can be powerful X-ray sources with 0.5 - 4.5 KeV luminosities $\lx$ ranging from $\sim
10^{39}$ to $\sim 10^{42}$ erg s$^{-1}$. This emission is associated with hot gaseous halos within the galaxies, containing $\mgas=10^8 -
10^{11}\msol$ (see [@fab89]).
In order to explain this observational finding, a certain class of solutions designated cooling flow models have been proposed and extensively investigated (e.g., Fabian, Nulsen, & Canizares 1984; Sarazin & White 1987,1988; Vedder, Trester, & Canizares 1988). While these models have many attractive features, they are far from giving a totally satisfactory account of the X-ray properties of all elliptical galaxies, as most observed systems are much fainter in the X-rays than the models predict and have different radial profiles than expected. Moreover, the cooling flow models do not solve the question of [*where*]{} the cool gas is deposited: over a Hubble time an amount of material comparable to the mass of stars in the galactic core flows into the nucleus, but the expected distortions of the central optical surface brightness and velocity dispersion are not observed.
One possible solution to part of the previous set of problems was proposed by D’Ercole et al. (1989) and CDPR, who showed that the heating from SNIa could be effective in maintaining low luminosity galaxies in a wind phase over an Hubble time (and so preventing the gas from accumulating in the centre). But the most massive galaxies ultimately experience a central cooling catastrophe, leading to a situation similar to a cooling flow. Clearly a component of the explanation is missing and is possibly related to the fact (e.g., [@r84]) that many (perhaps most) early-type galaxies show a nuclear activity, and, according to the standard interpretation of the AGN phenomenon, a massive BH is at its origin. So it is natural to investigate the accretion of a galactic gas inflow onto galaxies within which lurk massive central BHs ($\mbh\sim 10^8\msol$).
Binney & Tabor (1995, BT) explored this problem with the aid of spherically symmetric numerical simulations, assuming an homogeneous release of energy in the inner kpc of their galaxy models during the inflow phases. Moreover, BT assumed that all the accretion luminosity was available for the gas heating, due to the interaction between a nuclear jet and the surrounding ISM.
In the present paper we explore, by numerical integration of the fully non-stationary equations of hydrodynamics, the modifications on the results of CDPR, assuming the presence of a massive BH the galaxy centre with detailed allowance for the effects on the flow of the radiation emitted by the central BH. As will be shown, the gas over the body of the galaxy is (as noted by BT) really optically thin, but nevertheless the effect of [*energy*]{} exchange between the nuclear [*radiation*]{} and the gas flow is dramatic. This effect was already known and extensively studied for accreting compact objects ([@omccwy76]; Cowie, Ostriker, & Stark 1978). In a successive paper ([@co97], Paper II), a quantitative analysis of various aspects of the scenario summarized in this Letter, together with an exaustive description of the input physics and its modelization, will be given.
Results
=======
All the results shown here refer to a model whose parameters are fixed following the line of CDPR. The stellar density profile is a King (1972) distribution, with total blue luminosity $\lb=5\times
10^{10}\lsol$, central velocity dispersion $\ssc=280$ km s$^{-1}$, and core radius $\rcs=350$ pc. The dark-matter halo is described by a quasi-isothermal density distribution, with $\mh/\mast=7.8$ and $\rch/\rcs =4.2$. The SNIa rate is the same as that in the King Reference Model of CDPR. The bolometric luminosity emitted by accretion onto the BH is $\lbh\equiv\eps c^2 \mdot$, where $c$ is the light velocity, and $\eps$ is the accretion efficiency, with $10^{-3}\lsim
\eps\lsim 10^{-1}$. The spectral distribution of $\lbh$ near the BH is assumed to be $\lbh(\nu)\propto
\lbh\nu^{-0.5}/(\nub^{0.7}+\nu^{0.7})$, where $h\nub$=1MeV. In the present model $\eps=0.1$.
The spherically symmetric hydrodynamical equations are integrated numerically using the Eulerian up-wind scheme with time splitting and artifical viscosity as used in CDPR. In the energy equation the contribution of Compton heating (and cooling) of the gas due to $\lbh$ and to the recycling of the radiation produced by the gas heated by the BH activity are included. We allow also for the effect of the photoionization in both cooling and heating of cold gas as well as momentum exchange between photons and electrons. At each radius the radiation field is integrated, considering the gas differential absorption on $\lbh (\nu)$ and using for the electrons the Klein-Nishina cross-section. In this way the absorbed fraction of $\lbh$ is computed self-consistently.
In Fig. 1b (solid line) the temporal evolution of the coronal X-ray luminosity $\lx$ of the gas in the 0.5–4.5 KeV band is shown over an Hubble time. The evolution up to the so-called [*cooling catastrophe*]{} ($t\simeq 9.4$ Gyr) is analogous to that described in CDPR, but after this time the Compton heating instability completely alters the flow evolution and its properties. At the cooling catastrophe negative infall velocities appear near the galaxy center, with $\mdot\sim 60\;\msol$yr$^{-1}$, and this accretion produces a strong, energetic feed-back producing a very high $\lbh$ (Fig. 1a). The gas in the central regions of the galaxy is strongly heated to temperatures comparable with the Compton temperature associated with $\lbh (\nu)$ ($\simeq 10^9$ K), and starts to expand, decreasing its density by more than two orders of magnitude, driving a shock wave outwards and producing a hot bubble of a few hundred parsecs in diameter. The net effect is, observationally, a large reduction of $\lx$ (Fig. 1b), and, hydrodynamically, the interruption of the galactic inflow and the consequent shut-off of $\lbh$. Then the radiative losses increase again, and, after a period of the order of the hot gas cooling time, the cycle repeats. In the model described here this time is of the order of $\sim 1$ Gyr. In the case of very high accretion the shock wave can reach the galaxy edge, and expel gas from the galaxy. At higher time resolution each burst shows a very complex structure, that will be discussed in detail in Paper II: the temporal blow-up of the first burst shown in Fig. 1 is plotted in Fig. 2, showing QSO-like luminosities. An important characteristic of all computed models – of which a single representative is here discussed – is that the fraction of $\lbh$ effectively absorbed by the gaseous halo is in the range $10^{-4}-10^{-2}$, but [*the gas flows are found to be invariably unstable due to Compton heating for all the explored efficiencies*]{}: in presence of a massive BH at the center of elliptical galaxies the possibility of a stationary cooling flow seems to be very remote.
In Fig. 3 we show the distribution of $\lx$ from 9 Gyr to 15 Gyr. The dashed histogram shows the model distribution of $\lx$ given in Fig. 1, and the solid line shows data for (non-boxy) early-type galaxies taken from Fig.1 of CDPR. Finally the dotted histogram is the distribution of $\lx$ for the same model, with the cooling flow assumption of $\eps=0$. We see that the model with $\eps =0.1$ has a distribution, over time, of $\lx$ surprisingly similar to that of observed galaxies, but the cooling flow model (as is well known) produces far too much radiation. In Fig. 4 the X-ray surface brightness profile ($\SX$) of the presented model is shown at two different epochs, before and during bursts (vertical arrows in Fig. 2). Also shown is the cooling flow profile for the same galaxy at $t=15$ Gyr. Note how $\SX$ is characterized by a well defined core before a burst, alleviating the problem of the too cuspy $\SX$ that afflicts cooling flow models (Canizares, Fabbiano, & Trinchieri 1987). Certainly interesting is the fate of the transient cold shell surrounding the hot bubble (especially in low-$\eps$ solutions) during the flaring activity, when the central gas surface brightness is very low. Due to Rayleigh-Taylor instability, the shell will break up, and perhaps cold fingers of gas should be observable inside the hot low density bubble, accreting on the central BH. So, the presence of the central heating source produces in a natural way a multiphase ISM on galactic scales, while the same phenomenon may be harder to obtain (Balbus 1991) in the cooling flow scenario.
Fig. 5 shows the spectra in the preburst (dotted) and during burst (solid) phases, compared to the cooling flow (dashed) spectrum. The emitted spectrum is never as soft in this set of models as it is in cooling flow models, and, during bursts, occasionally it will have a very hard tail.
Discussion and Conclusions
==========================
In this Letter we show how the presence of a massive central BH in early type galaxies is able to produce naturally both the observed X-ray underluminosity with respect to the pure cooling flow expectations, and the large observed scatter in $\lx$ at fixed $\lb$. As can be seen from Fig. 1, $\lx$ – except in the very short period of bursts – is always lower than that of the corresponding inflow model with $\eps=0$. Moreover, the statistical distribution of observed data compared with the amount of time spent at each $\lx$ by the model here discussed, is eloquent (Fig. 3). Finally, due to the strong feed-back on the gas flows of the radiation emitted by the accretion, the total mass accumulated by the BH over 15 Gyr is very low ($\sim 3\;10^8\msol$), to be compared with the $\sim 10^{10}\msol$ of the correspondent $\eps=0$-model. The same model in pure cooling-flow (without the initial SNIa driven wind-outflow phases) would have accumulated in its center $\sim 10^{11}\msol$ of gas.
From an observational point of view, it is interesting to note that during the accretion phases the galaxy luminosity is dominated by $\lbh$ (with highest values at $10^{46}-10^{47}$ erg s$^{-1}$), while during the quiescent BH phases the galaxy emission is due only to the diffuse hot gas $\lx$. The total energy emitted by the accretion when $\lbh >10^{42}$ is $\sim 7.5\,10^{61}$ erg, while during the same phases the total energy emitted by the coronal gas is $\sim
5.6\,10^{57}$ erg. The ratio between the total time spent by the galaxy when $\lbh >\lx$, and the total time spanned by the simulation is $\sim 10^{-2}$: very few galaxies should be caught in a AGN-like phase even though most contain central BHs.
Thus, the Compton heating instability could be an alternative possibility to that advocated by Fabian & Rees (1995) in order to explain why the nuclei of elliptical galaxies are not luminous sources of radiation as expected if they host a massive central BH. A clear prediction of this model is that some significant fraction of QSOs should be embedded in high temperature, low surface brightness X-ray halos.
In other works to be reported in Paper II we varied the efficiency in the range $10^{-3}\leq\eps\leq 10^{-1}$ and the galaxy luminosity in the range $10^{10}\leq\lb/\lsol\leq 10^{11}$, with results very similar to those shown in Figs. 2-5. We are well aware that in the real accretion phenomenon a disk geometry for the infalling gas seems to be inescapable. Then the accretion luminosity will be emitted preferentially along polar directions. It is clear that (at least) fully 2D hydrodynamical simulations are required for a better understanding of this problem, and to follow the development and the final fate of the gas instabilities. We are working in this direction, but expect that many of the quantitative features of the present work will be carried over to the more complicated calculations.
We would like to thank Giuseppe Bertin, James Binney, Annibale D’Ercole, Bruce Draine, Silvia Pellegrini, and Alvio Renzini for useful discussions and advice. This work was supported by NSF grant AST9424416 and in addition L.C. was supported by NSF grant AST9108103 and by the Italian Space Agency (ASI) with contracts ASI-94-RS-96 and ASI-95-RS-152.
Balbus, S. 1991, , 372, 25 Binney, J., & Tabor, G. 1995, , 276, 663 (BT) Canizares, C.R., Fabbiano, G., & Trinchieri, G. 1987, , 312, 503 Ciotti, L., D’Ercole, A., Pellegrini, S., & Renzini, A. 1995, , 376, 380 (CDPR) Ciotti, L., & Ostriker, J.P. 1997, in preparation (Paper II) Cowie, L.L., Ostriker, J.P., & Stark, A.A. 1978, , 226, 1041 D’Ercole, A., Renzini, A., Ciotti, L., & Pellegrini, S. 1989, , 341, L9 Fabbiano, G. 1989, , 27, 87 Fabian, A.C., Nulsen, P.E.J., & Canizares, C.R. 1984, Nature, 311, 733 Fabian, A.C., & Rees, M.J. 1995, , 277, L55 King, I. 1972, , 174, L123 Ostriker, J.P., McCray, R., Weaver, R., & Yahil, A. 1976, , 208, L61 Rees, M.J. 1984, , 22, 471 Sarazin, C.L. & White, R.E.III 1987, , 320, 32 Sarazin, C.L. & White, R.E.III 1988, , 331, 102 Vedder, P.W., Trester, J.J., & Canizares, C.R. 1988, , 332, 725
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abstract: 'Deep neural networks (DNNs) have emerged as a popular mathematical tool for function approximation due to their capability of modelling highly nonlinear functions. Their applications range from image classification and natural language processing to learning-based control. Despite their empirical successes, there is still a lack of theoretical understanding of the representative power of such deep architectures. In this work, we provide a theoretical analysis of the expressiveness of fully-connected, feedforward DNNs with 1-Lipschitz activation functions. In particular, we characterize the expressiveness of a DNN by its Lipchitz constant. By leveraging random matrix theory, we show that, given sufficiently large and randomly distributed weights, the expected upper and lower bounds of the Lipschitz constant of a DNN and hence their expressiveness increase exponentially with depth and polynomially with width, which gives rise to the benefit of the depth of DNN architectures for efficient function approximation. This observation is consistent with established results based on alternative expressiveness measures of DNNs. In contrast to most of the existing work, our analysis based on the Lipschitz properties of DNNs is applicable to a wider range of activation nonlinearities and potentially allows us to make sensible comparisons between the complexity of a DNN and the function to be approximated by the DNN. We consider this work to be a step towards understanding the expressive power of DNNs and towards designing appropriate deep architectures for practical applications such as system control.'
author:
- |
\
Dynamic Systems Lab, Institute for Aerospace Studies, University of Toronto, Canada\
Vector Institute for Artificial Intelligence, Canada
bibliography:
- 'reference.bib'
title: An Analysis of the Expressiveness of Deep Neural Network Architectures Based on Their Lipschitz Constants
---
Deep Neural Networks, Expressiveness of Deep Architectures, Lipschitz Constant, Learning-based Control
Introduction {#sec:introduction}
============
Given their capability to approximate highly nonlinear functions, deep neural networks (DNNs) have found increasing application in domains such as image classification [@krizhevsky2012imagenet; @googlenet], natural language processing [@hinton2012deep; @hannun2014deep], and learning-based control [@shi2019neural; @chen2019large; @zhou-cdc17]. As compared to their shallow counterparts, DNNs are often favoured in practice due to their compact representation of nonlinear functions [@montufar2017number]. Despite their practical successes, the theoretical understanding of the representative power of such deep architectures remains an active research topic addressed by both the machine learning and neuroscience community. In this work, we aim to contribute to the understanding of the expressiveness of DNNs by presenting a new perspective based on Lipschitz constant analysis that is interpretable for applications such as system control.
There are several recent works analyzing the expressive power of deep architectures. One notable work is [@NIPS2011_4350], where the authors show that, for a sum-product network, a deep network is exponentially more efficient than a shallow network in representing the same function. Following this work, several researchers then considered more practical DNNs with piecewise linear activation functions (e.g., rectified linear units (ReLU) and hard tanh) and showed that the expressiveness of a DNN measured by the number linear regions partitioned by the DNN grows exponentially with depth and polynomially with width [@pascanu2013number; @montufar2014number; @arora2016understanding; @serra2017bounding]. In parallel to the work on piecewise linear DNNs, [@raghu2017expressive] consider DNNs with independent and identically distributed (i.i.d.) Gaussian weight and bias parameters (i.e., random DNNs) and introduce a new measure of expressiveness based on the length of the output trajectory as the DNN traverses a one-dimensional trajectory in its input space. Similar to the other results, the authors show that the expressiveness of a DNN measured by the expected output trajectory length increases exponentially with the depth of the network.
While existing work has shown the exponential expressiveness of deep architectures, the measures of expressiveness are typically specific to the type of deep architectures being considered. For instance, for the sum-product networks considered in [@NIPS2011_4350], the measure of expressiveness is the number of monomials used to construct the polynomial function, and for DNNs with piecewise linear activation functions [@pascanu2013number; @montufar2014number; @arora2016understanding; @serra2017bounding]), the number of linear regions is used as the measure to characterize the complexity of the DNN. These specialized notions of expressivity prohibit sensible comparisons between the complexity of a DNN and the underlying function it approximates. While the expressiveness measure based on output trajectory length [@raghu2017expressive] is applicable to DNNs with more general activation functions, it is still not trivial to connect this measure to the properties of the function to be approximated by the DNN.
In this work, motivated by the theoretical analysis of DNNs in feedback control applications [@shi2019neural; @fazlyab2019efficient], we introduce an alternative perspective on the expressive power of DNNs based on their Lipschitz properties. Similar to [@raghu2017expressive], we consider a DNN with random weight parameters. By leveraging results from random matrix theory, we provide an analysis of the expressive power of DNNs based on their Lipschitz constant and establish connections with earlier results using alternative measures of DNN expressiveness. Our ultimate goal is to understand the implications of choosing particular neural network architectures for learning in feedback control applications.
Preliminaries
=============
We consider fully-connected DNNs, $f: \mathcal{X}\mapsto \mathcal{Y}$, that are defined as follows: $$\label{eqn:dnn_definition}
\begin{aligned}
\mathbf{h}_0(\mathbf{x}) = {\mathbf{x}},\:\:
\mathbf{h}_l({\mathbf{x}}) =\boldsymbol{\sigma}\left( \mathbf{W}_l\mathbf{h}_{l-1}({\mathbf{x}})+\mathbf{b}_l\right)\: \forall l = 1,...,L,\:\:
\mathbf{y}=\mathbf{W}_{L+1}\mathbf{h}_{L}({\mathbf{x}}) + \mathbf{b}_{L+1},
\end{aligned}$$ where ${\mathbf{x}}\in \mathcal{X}\subseteq {\mathbb{R}}^{n_0}$ is the input, $\mathbf{y}\in \mathcal{Y}\subseteq {\mathbb{R}}^{n_{L+1}}$ is the output, the subscripts $l = \{0,...,L+1\}$ denote the layer index with $l=0$ being the input layer, $l=1,...,L$ being the hidden layers, and $l=L+1$ being the output layer, $\mathbf{h}_l:\mathcal{X} \mapsto {\mathbb{R}}^{n_l}$ is the output from the $l$th layer with $\boldsymbol{\sigma}(\cdot)$ being the element-wise activation function and $n_l$ being the number of neurons in the $l$th layer, and $ \mathbf{W}_l\in {\mathbb{R}}^{n_l\times n_{l-1}}$ and $\mathbf{b}_l\in{\mathbb{R}}^{n_l}$ are the weight and bias parameters between layers $(l-1)$ and $l$. In our analysis, we focus on DNNs with 1-Lipschitz activation functions [@virmaux2018lipschitz], which include most commonly used activation functions such as ReLU, tanh, and sigmoid.
To facilitate our analysis, similar to [@raghu2017expressive], in this work, we consider DNNs with random weight matrices $\mathbf{W}_l$ whose elements are i.i.d. zero-mean Gaussian random variables $\mathcal{N}(0,\sigma_w^2)$, where $\sigma_w^2$ is the variance of the Gaussian distribution. Our goal is to analyze the expressiveness of such a DNN as we vary its architectural properties (i.e., width and depth).
Lipschitz Constant as a Measure of Expressiveness {#sec:lipschitz_constant}
=================================================
In this work, we characterize the expressiveness of a DNN by its Lipschitz constant. Intuitively, a larger Lipschitz constant implies that small changes in the DNN input can lead to large changes at the output, which provide greater flexibility to model nonlinear functions.
Formally, a function $f:\mathcal{X}\mapsto\mathcal{Y}$ is said to be Lipschitz continuous on $\mathcal{X}$ if $$\label{eqn:lipschitz_constant_definition}
(\exists \rho >0)\:(\forall {\mathbf{x}},{\mathbf{x}}'\in\mathcal{X}) \:\: ||f({\mathbf{x}})-f({\mathbf{x}}')||\le \rho||{\mathbf{x}}-{\mathbf{x}}'||,$$ and its Lipschitz constant on $\mathcal{X}$ is the smallest $\rho$ such that the inequality in holds. It is not hard to verify that common activation functions (e.g., ReLU, tanh, and sigmoid) are globally Lipschitz continuous. A DNN with such activation functions is a finite number of compositions of Lipschitz continuous functions and is thus Lipschitz continuous on its domain $\mathcal{X}$. Note that, in general, the Lipschitz continuity condition in is independent of the choice of the norm; in this work, we will consider Lipschitz continuity in the $l_2$-norm.
In the following subsections, we establish a connection between the expected Lipschitz constant of a DNN and its architecture (i.e., width and depth), and compare the result to existing results on the expressive power of DNNs in the literature. We summarize our main results in this manuscript and provide details of the derivations and proofs in the appendices.
Upper and Lower Bounds on the Lipschitz Constant of a DNN {#subsec:lipschitz_expressivness}
---------------------------------------------------------
As noted in [@fazlyab2019efficient; @virmaux2018lipschitz], the exact estimation of the Lipschitz constant of a DNN is NP-hard; however, for our purpose of understanding the expressiveness of DNNs, estimates of the upper and lower bounds on the Lipschitz constant of a DNNs based on their weight matrices are sufficient.
Recall that we consider a family of DNNs with 1-Lipschitz activation functions. By the Lipschitz continuity of composite functions, an upper bound on the Lipschitz constant of a DNN with 1-Lipschitz activation functions is the product of the spectral norms, or equivalently, of the maximum singular values of the weight matrices: $$\label{eqn:upper_bound_main}
\overline{\rho}(f({\mathbf{x}})) = \prod_{l=1}^{L+1} ||\mathbf{W}_{l}||_2,$$ where $ \overline{\rho}(f({\mathbf{x}}))$ denotes the upper bound on the Lipschitz constant of the DNN, $ ||\mathbf{W}_{l}||_2$ denotes the spectral norm or the maximum singular value of the weight matrix $\mathbf{W}_{l}$. As derived in [@combettes2019lipschitz], a lower bound $ \underline{\rho}(f({\mathbf{x}}))$ on the Lipschitz constant of a DNN is $$\label{eqn:lower_bound_main}
\underline{\rho}(f({\mathbf{x}})) = ||\mathbf{W}_{L+1}\mathbf{W}_{L}\cdots \mathbf{W}_{1}||_2,$$ which corresponds to the Lipschitz constant of a purely linear network (i.e., a network with activation nonlinearities removed).
Note that the upper and lower bounds on the Lipschitz constant of a DNN in and depend only on the maximum singular values of the weight matrices and their product. In the following analysis, we leverage random matrix theory to derive expressions of the bounds in and in terms of the width and depth of the DNN and the variance of the weight parameters $\sigma_w^2$.
Estimates of the Lipschitz Constant Bounds Based on Extreme Singular Value Theorem
----------------------------------------------------------------------------------
In this subsection, we establish a connection between the Lipschitz constant of a DNN and its architecture (i.e., width and depth) based on the extreme singular value theory for random matrices.
### Upper Bound
In this part, we show that, for a sufficiently large $\sigma_w$, the expected upper bound on the Lipschitz constant and hence the attainable expressiveness of a DNN increases exponentially with depth and polynomially with width. To start our discussion, we state the following result from random matrix theory on the extreme singular values of Gaussian random matrices:
Let $\mathbf{A}$ be an $(N \times n)$ matrix whose elements are independent standard normal random variables. Then, $\sqrt{N}-\sqrt{n}\le\mathbb{E}[\lambda_\text{min}(\mathbf{A})] \le \mathbb{E}[\lambda_\text{max}(\mathbf{A})]\le \sqrt{N}+\sqrt{n}$, where $\lambda_\text{min}$ and $\lambda_\text{max}$ denote the minimum and maximum singular values of $\mathbf{A}$, respectively, and $\mathbb{E}[\cdot]$ represents the expected value. \[thm:extreme\_singular\_value\]
Note that, for a Gaussian random matrix, the theorem above allows us to infer the extreme singular values of the matrix without explicitly knowing the values of its elements. By representing the weight parameters of a DNN as i.i.d. Gaussian random variables, we can leverage this result to estimate the upper bound of the Lipschitz constant . In particular, by applying Theorem \[thm:extreme\_singular\_value\], we prove the following theorem in App. \[app:upper\_bound\]:
Consider a DNN defined in , where the weight parameters are independent Gaussian random variables distributed as $\mathcal{N}(0,\sigma_w^2)$ with $\sigma_w^2$ denoting the variance of the Gaussian distribution, and where the activation functions are 1-Lipschitz. The expected Lipschitz constant of the DNN is upper bounded by $\prod_{l=1}^{L+1} \sigma_w\left(\sqrt{n_l}+ \sqrt{n_{l-1}}\right)$. \[theorem:upper\_bound\]
Theorem \[theorem:upper\_bound\] allows us to obtain an intuition about the expected attainable Lipschitz constant and thus the flexibility of a DNN as we vary its width $n_l$ for $l = 1,..., L$ and depth $L+1$. To compare to established results [@serra2017bounding; @raghu2017expressive], we set the width of the hidden layers to $n$ (i.e., $n_l = n$ for $l = 1,...,L$), then the expected Lipschitz constant of a DNN with Gaussian random weights is upper bounded by $O\left((2\sigma_w)^{L+1} n^{\frac{L+1}{2}}\right)$. For $\sigma_w \ge \frac{1}{2\sqrt{n}}$, this upper bound increases exponentially with depth and polynomially with width. This observation is consistent with the results on the expressiveness measured by the number of linear regions for piecewise linear networks [@serra2017bounding; @raghu2017expressive] and the expressiveness measured by the trajectory length for Gaussian random networks [@raghu2017expressive].
### Lower Bound {#subsubsec:lower_bound}
Similarly based on the extreme singular value theorem for random matrices, we present a conjecture on the lower bound of the Lipschitz constant . We include a justification of the conjecture in App. \[app:lower\_bound\] and empirically illustrate the result in Sec. \[sec:numerical\_examples\].
Consider a DNN defined in where the weight parameters are independent Gaussian random variables distributed as $\mathcal{N}(0,\sigma_w^2)$ and the activation functions are 1-Lipschitz. The Lipschitz constant of the DNN is approximately lower bounded by $\left(\sigma_w^{L+1} \prod_{l=1}^{L}\sqrt{n_l}\right)\left(\sqrt{n_{L+1}} +\sqrt{n_0}+O(\sqrt{n_0})\right)$. \[theorem:lower\_bound\]
Based on Conjecture \[theorem:lower\_bound\], if we consider a DNN with constant width $n$ (i.e., $n_l = n$ for $l = 1,...,L$), the Lipschitz constant of the DNN with independent Gaussian weight parameters is approximately lower bounded by $\Omega\left(\sigma_w^{L+1}n^{\frac{L}{2}}\right)$, which also increases exponentially in depth and polynomially in the width of the DNN given sufficiently large $\sigma_w$ (i.e., $\sigma_w\ge \frac{1}{\sqrt{n}}$). Interestingly, we note that, for the case where $n\gg 1$ and $L\gg 1$, this asymptotic lower bound based on the Lipschitz constant of the DNN coincides with the expressiveness lower bound based on the output trajectory length measure for DNNs with ReLU activation functions [@raghu2017expressive]. This connection is sensible since the expressiveness measure in [@raghu2017expressive] can be intuitively thought of as the extent to which the DNN stretches a trajectory in its input space, which is a property related to the Lipschitz constant of a DNN (see App. \[app:connection\] for further details).\
Note that, for both the upper and lower bound analysis, we require the magnitude of $\sigma_w$ to be sufficiently large. Intuitively, a small $\sigma_w$ means that the magnitude of the weights are small. In the extreme case, where all weights are zero, a deep architecture cannot be expressive in any notion of expressiveness (e.g., number of linear regions). We therefore require the spread of the weights $\sigma_w$ to be sufficiently large to exploit the expressivity of the deep layers. This lower bound is typically not restrictive; as an example, $1/\sqrt{n}$ is approximately 0.22 for $n = 20$.
### Differences Compared to Other Expressiveness Measures
In this work, we propose to use the Lipschitz constant of a DNN as a measure of its expressiveness. In contrast to existing expressiveness measures, a Lipschitz-based characterization has two benefits:
- *Less assumptions on the DNN:* As compared to previous work on piecewise linear DNNs [@pascanu2013number; @montufar2014number; @arora2016understanding; @serra2017bounding], by considering the Lipschitz constant as the expressiveness measure, we do not constrain ourself to DNNs with specific activation functions such as ReLUs or hard tanh. In our analysis, we only require the activation function to be 1-Lipschitz, which is satisfied by most commonly used activations that include but are not limited to ReLU, tanh, hard tanh, and sigmoid.
- *Towards understanding DNN expressiveness for practical applications:* In contrast to expressiveness measures such as the number of linear regions [@pascanu2013number; @montufar2014number; @arora2016understanding; @serra2017bounding] and trajectory length [@raghu2017expressive], the Lipschitz constant is a generic property for Lipschitz continuous nonlinear functions. For regression problems, the expressiveness characterization through the Lipschitz constant allows us to make sensible comparisons between a DNN and the function it approximates. For control applications, the Lipschitz constant also plays a critical role in stability analysis. The Lipschitz-based characterization of the expressiveness of a DNN has the potential to facilitate the design of deep architectures for safe and efficient learning in a closed-loop control setup.
Numerical Examples {#sec:numerical_examples}
==================
In this section, we provide numerical examples that illustrate the insights on the expressiveness of DNNs based on the results in Sec. \[sec:lipschitz\_constant\]. In particular, we show the connection between the architectural properties of a DNN and its expressiveness.
Bounds on the Lipschitz Constant of a DNN
-----------------------------------------
To visualize the results of Sec. , we randomly sample the weight parameters of DNNs from a zero-mean, unit variance Gaussian distribution and compare the upper and lower bounds on the Lipschitz constants of these DNNs as we increase its width and depth. To examine the quality of the estimated Lipschitz constant bounds from Sec. , we show a comparison of the estimated bounds computed based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] and the bounds computed directly based on and in Fig. \[fig:est\_vs\_act\]. From these plots, we see that there is a close correspondence between the Lipschitz constant bounds computed based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\], which assumes random matrices, and the bounds computed based on and based on the actual network weights. This result verifies that the bounds provided in Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] are good approximations of the bounds on the Lipschitz constant of a fixed DNN based on and . We note that here we compute the bounds in and directly based on the sampled weight parameters that are known for this simulation study; in general, to understand the implications of a DNN architecture based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\], we do not rely on knowing the weights explicitly.
![A comparison of the estimated lower and upper bounds of the Lipschitz constant of DNNs based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] *(dashed lines)*, and the lower and upper bounds computed based on and with the actual weight values *(solid lines)*.[]{data-label="fig:est_vs_act"}](./figures/upper_bound_vs_width "fig:"){width="42.00000%"} ![A comparison of the estimated lower and upper bounds of the Lipschitz constant of DNNs based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] *(dashed lines)*, and the lower and upper bounds computed based on and with the actual weight values *(solid lines)*.[]{data-label="fig:est_vs_act"}](./figures/lower_bound_vs_width_wo_legend "fig:"){width="42.00000%"}\
![A comparison of the estimated lower and upper bounds of the Lipschitz constant of DNNs based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] *(dashed lines)*, and the lower and upper bounds computed based on and with the actual weight values *(solid lines)*.[]{data-label="fig:est_vs_act"}](./figures/upper_bound_vs_depth "fig:"){width="42.00000%"} ![A comparison of the estimated lower and upper bounds of the Lipschitz constant of DNNs based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] *(dashed lines)*, and the lower and upper bounds computed based on and with the actual weight values *(solid lines)*.[]{data-label="fig:est_vs_act"}](./figures/lower_bound_vs_depth_wo_legend "fig:"){width="42.00000%"}
![Estimated lower and upper bounds on the Lipschitz constant of DNNs with varying widths and depths. The growth of the lower and upper bound is relatively faster in depth than in width. The dashed contour lines correspond to DNN architectures with equal numbers of neurons, and the solid contour lines correspond to levels of constant Lipschitz bounds.[]{data-label="fig:sim_est_bounds"}](./figures/upper_bound_depth_vs_width_contour "fig:"){width="45.00000%"} ![Estimated lower and upper bounds on the Lipschitz constant of DNNs with varying widths and depths. The growth of the lower and upper bound is relatively faster in depth than in width. The dashed contour lines correspond to DNN architectures with equal numbers of neurons, and the solid contour lines correspond to levels of constant Lipschitz bounds.[]{data-label="fig:sim_est_bounds"}](./figures/lower_bound_depth_vs_width_contour "fig:"){width="45.00000%"}
Figure \[fig:sim\_est\_bounds\] shows the upper and lower bounds of the Lipschitz constant based on Theorem \[theorem:upper\_bound\] and Conjecture \[theorem:lower\_bound\] for different DNN architectures. By inspecting horizontal slices and vertical slices of the plots in Fig. \[fig:sim\_est\_bounds\], which correspond to the top and bottom plots in Fig. \[fig:est\_vs\_act\], we see that the upper and lower bounds of the Lipschitz constant of a DNN increase exponentially with depth and polynomially with width. The dashed contour lines in the plots show DNN architectures with the same number of neurons. As we trace one of the contour lines from left to right, we see that increasing width and decreasing depth reduces the bounds of the Lipschitz constants, which indicates a decrease in the expressiveness of the deep architecture. Similar to the discussion in [@montufar2017number], based on our formulation, we also see that, given the same number of neurons, deeper networks are more compact representations of nonlinear functions.
Towards Learning Deep Models for Control
----------------------------------------
To illustrate the implication of the expressiveness of a DNN for control, we consider a simple system setup and examine the stability of the system when we use a DNN with different architectures in the loop. In particular, we consider a system that is represented by $$\dot{{\mathbf{x}}} = \mathbf{A}{\mathbf{x}}+ f({\mathbf{x}}),
\label{eqn:sim_simple_system}$$ where ${\mathbf{x}}$ is the state, $\mathbf{A}$ is Hurwitz, and $ f({\mathbf{x}})$ is a function parametrized by a DNN. By Lyapunov’s direct method, one can show that a condition that guarantees stability of the system is $$\rho(f({\mathbf{x}}))\le \lambda_\text{min}(\mathbf{Q})/\left(2 \lambda_\text{max} (\mathbf{P})\right),
\label{eqn:sim_lip_safe_upper_bound}$$ where $\rho(f({\mathbf{x}}))$ denotes the Lipschitz constant of the DNN, $\mathbf{Q}$ is a positive definite matrix, $\mathbf{P}$ is the corresponding solution to the Lyapunov equation $PA + A^TP = -Q$, and $\lambda_\text{min}$ and $\lambda_\text{max}$ are the minimum and maximum eigenvalues of a matrix, respectively.
For an illustration, we set $\mathbf{A} = \begin{scriptsize}\begin{bmatrix}
0 & 2700\\
-3600 & -5400
\end{bmatrix}\end{scriptsize}$. We compare five DNN architectures with different widths and depths but the same number of neurons. For each DNN architecture, we sample 50 DNNs with i.i.d. zero-mean, unit variance Gaussian weight parameters. We note that out of the five architectures, we know based on Theorem \[theorem:upper\_bound\] that the first case, a DNN with a hidden layer of 300 neurons, has an estimated upper bound on the Lipschitz constant less than the safe upper bound in , and system is stable. In contrast, as we can see from Fig. \[fig:sim\_est\_bounds\], when we decrease the width and increase the depth of a DNN, its Lipschitz constant increases and system is less likely to be stable. Table \[tab:sim\_stability\_summary\] shows empirical results for the relationship between the architectural properties of a DNN and the stability of the system. This means, in practice, one may want to carefully choose an appropriate DNN architecture, or, alternatively, regularize the weight parameters, to ensure stability of a learning-based control system. We consider our insights to be a step towards providing design guidelines for DNN architectures, for example, for closed-loop control applications.
Architecture (width $\times$ depth) $300 \times 1$ $100\times 3$ $50\times 6$ $20\times 15$ 10$\times$ 30
------------------------------------- ---------------- --------------- -------------- --------------- ---------------
Likelihood of stable system (%) 100 100 40 32 32
: Likelihood of stable system with different DNN architectures.
\[tab:sim\_stability\_summary\]
Discussion on the Assumption of Gaussian Random Weight Matrices {#sec:random_weights}
===============================================================
In this work, we considered DNNs with Gaussian random weight matrices to facilitate analysis of their expressiveness. In this section, we examine if this assumption is reasonable for practical applications. In particular, we examine, through some examples, the accuracy of estimating the maximum singular value of the weight matrices based on Theorem \[thm:extreme\_singular\_value\] when the assumption of Gaussian random matrices does not hold exactly.
---------------- ----------- ---------------- ----------- ----------------
True Norm Estimated Norm True Norm Estimated Norm
$\mathbf{W}_1$ 36 38.5 197 208
$\mathbf{W}_2$ 7.38 8.31 1.92 2.04
---------------- ----------- ---------------- ----------- ----------------
: True and estimated maximum singular values of weight matrices in trained networks.
\[tab:norm\_estimate\]
[r]{}[4.5cm]{}
To examine the properties of weight matrices in trained networks, we consider a regression problem. The true function to be approximated has two inputs and one output. Fig. \[fig:weight\_distributions\] shows the distributions of two weight matrices from two trained networks with different architectures, and Table \[tab:norm\_estimate\] summarizes their maximum singular values. By inspecting the distributions (Fig. \[fig:weight\_distributions\]), we see that the weights are not necessarily always Gaussian-distributed; however, the estimates of the maximum singular values of the matrices based on the assumption of random weights are very close to the true maximum singular values (Table \[tab:norm\_estimate\]). Based on Bai-Yin’s law for extreme singular values of random matrices with more general distributions [@rudelson2010non], we can infer that the expected maximum singular value based on Theorem \[thm:extreme\_singular\_value\] is an approximation of the true maximum singular value of a random matrix with an error of $O(\sigma_w \sqrt{n})$, where $\sigma_w$ is the standard deviation of the weight distribution and $n$ is the matrix column dimension. In future, we plan to explore the properties of the weight matrices of trained networks and examine their relation to random matrix theory.
Conclusion
==========
In this paper, we presented a new perspective on the expressiveness of DNNs based on their Lipschitz properties. Using random matrix theory, we showed that, given the spread of the weights is sufficiently large (i.e., $\sigma_w \ge \frac{1}{\sqrt{n_l}}$ for $l = 1,...,L$), the expressiveness of a DNN measured by its Lipschitz constant grows exponentially with depth and polynomially with width. This result is similar to the results based on other expressiveness measures discussed in the current literature. By considering the Lipschitz constant as a measure of DNN expressiveness, we can more sensibly understand the implication of being ‘deep’ in the context of function approximation for applications including safe learning-based control.
Proofs of Main Results in Sec. \[sec:lipschitz\_constant\]
==========================================================
Proof of Theorem \[theorem:upper\_bound\]: Upper Bound on Lipschitz Constant of a Gaussian Random DNN {#app:upper_bound}
-----------------------------------------------------------------------------------------------------
The following is a proof for Theorem \[theorem:upper\_bound\] presented in Sec. \[sec:lipschitz\_constant\]. In the following proof, based on the extreme singular value theorem for random matrices (Theorem \[thm:extreme\_singular\_value\]), we derive an expression for the upper bound on the Lipschitz constant of a DNN in terms of its width and depth.
Consider a random matrix $\mathbf{A}\in{\mathbb{R}}^{N\times n}$ whose elements are independent Gaussian random variables distributed as $\mathcal{N}(0,\sigma_w^2)$. As a result of Theorem \[thm:extreme\_singular\_value\] and the homogeneity of the matrix norm, the expected maximum singular value of $\mathbf{A}$ is upper bounded by $\mathbb{E}[\lambda_\text{max}(\mathbf{A})]\le \sigma_w(\sqrt{N}+\sqrt{n})$. By assumption, the elements of each weight matrix $\mathbf{W}_l$ are distributed as $\mathcal{N}(0,\sigma_w^2)$. The expected spectral norm, or equivalently the expected maximum singular value, of the weight matrices are upper bounded as follows: $$\label{eqn:max_sv_weights}
{\mathbb{E}}[||\mathbf{W}_l||_2] = {\mathbb{E}}[\lambda_\text{max}(\mathbf{W}_l)]\le \sigma_w(\sqrt{n_l}+\sqrt{n_{l-1}}).$$ Since the weight matrices are independent, by substituting into , we have the following expected upper bound on the Lipschitz constant of the DNN: $$\label{eqn:upper_bound_proof}
{\mathbb{E}}[\overline{\rho}(f(\mathbf{x}))] = \prod_{l=1}^{L+1} {\mathbb{E}}[||\mathbf{W}_{l}||_2]\le \prod_{l=1}^{L+1}\sigma_w \left(\sqrt{n_l}+ \sqrt{n_{l-1}}\right).$$ The expression in establishes a connection between the upper bound on the Lipschitz constant of a DNN and its architecture, which is represented by the dimensions of the weight matrices in this analysis. This result allows us to obtain insights on the expressiveness of a DNN without explicitly knowing the values of its weights.
Justification of Conjecture 3: Lower Bound on Lipschitz Constant of a Gaussian Random DNN {#app:lower_bound}
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To derive an estimate of the lower bound in , we first note that the product of random Gaussian matrices is in general not a Gaussian random matrix. In deriving the lower bound, we need to consider a more general class of matrices than in Theorem \[thm:extreme\_singular\_value\]:
Let $\mathbf{A}$ be an $N \times n$ matrix whose elements are independent random variables with zero mean, unit variance, and finite fourth moment. Suppose that the dimensions $N$ and $n$ grow to infinity with $N/n$ converging to a constant in $[0,1]$. Then, $\mathbb{E}[\lambda_\text{min}(\mathbf{A})] = \sqrt{N}-\sqrt{n} + O(\sqrt{n})$ and $ \mathbb{E}[\lambda_\text{max}(\mathbf{A})] = \sqrt{N}+\sqrt{n} + O(\sqrt{n})$ almost surely. \[thm:extreme\_singular\_value\_general\]
In contrast to Theorem \[thm:extreme\_singular\_value\], the above theorem is applicable to a wider class of random matrices with independent elements; however, this result is an asymptotic result in the limit of sufficiently large $N$ and $n$. For practical DNNs where the dimensions of the weight matrices are sufficiently large, this theorem allows us to derive an approximate lower bound for . We provide a justification of Conjecture \[theorem:lower\_bound\] presented in Sec. \[sec:lipschitz\_constant\] of our manuscript below:
We consider two random matrices $\mathbf{A}_1\in{\mathbb{R}}^{N\times n_1}$ and $\mathbf{A}_2\in{\mathbb{R}}^{n_2\times N}$ whose elements are independent zero-mean random variables with variances $\sigma_{a1}^2$ and $\sigma_{a2}^2$, respectively. The $i$th row and $j$th column element of the matrix product $\mathbf{A}_{21} = \mathbf{A}_2\mathbf{A}_1$ is $\sum_{k=1}^{N} a_{2,ik}a_{1,kj}$, where $a_{1,kj}$ denotes the $k$th row and $j$th column element of $\mathbf{A}_1$ and $a_{2,ik}$ denotes the $i$th row and $k$th column element of $\mathbf{A}_2$. Here, in our derivation, we make a conjecture that the elements of the product matrix of random matrices with elements being i.i.d. zero-mean random variables approximately preserve independence. Based on this conjecture, we derive an expression of the variance the elements of $\mathbf{A}_{21}$. Without loss of generality, we consider the $i$th row and $j$th column element of $\mathbf{A}_{21}$. Since, by assumption, the elements of $\mathbf{A}_1$ and $\mathbf{A}_2$ have zero mean and are i.i.d., the variance of the $i$th row and $j$th column element of $\mathbf{A}_{21}$ is $$\begin{aligned}
\sigma_{21}^2 &= \mathbb{V}\left[\sum_{k=1}^{N} a_{2,ik}a_{1,kj} \right]=\sum_{k=1}^{N}\: \mathbb{V}\left[a_{2,ik}a_{1,kj}\right]=N \sigma_{a1}^2 \sigma_{a2}^2,\end{aligned}$$ where $\mathbb{V}$ denotes the variance of a random variable, and $\mathbb{V}\left[a_{2,ik}a_{1,kj}\right]=\sigma_{a1}^2 \sigma_{a2}^2, \forall k= 1,2,...,N$ is the variance of the product of an element of $\mathbf{A}_1$ and an element of $\mathbf{A}_2$. The standard deviation of elements in the product of $\mathbf{A}_{21}$ can be written as $$\label{eqn:variance_of_product}
\sigma_{21} = \sqrt{N} \sigma_{a1}\sigma_{a2}.$$
By applying recursively, we can derive an estimate of the bound in , which is the spectral norm of the product of random matrices. In particular, a recursive relationship in the standard deviations of the product of random matrices can be written as $$\sigma_{w,1:l} = \sqrt{n_{l-1}}\sigma_{w,1:l-1}\sigma_{w},$$ where $\sigma_{w,1:l} $ denotes the standard deviation of the product of random matrices $\mathbf{W}_l\mathbf{W}_{l-1}\cdots \mathbf{W}_1$. For the product random matrix $\mathbf{W}_{L+1}\mathbf{W}_{L}\cdots \mathbf{W}_1$ in , we have $$\label{eqn:standard_deviation_prod}
\sigma_{1:L+1} = \sigma_w^{L+1}\prod_{l=1}^{L}\sqrt{n_l}.$$
As above, we make a conjecture that the elements of the product matrix constructed from the random weight matrices $\mathbf{W}_1, \mathbf{W}_2,..., \mathbf{W}_{L+1}$ are independent. Since the elements of the product matrix are the sums of products of independent zero-mean random variables by construction, the elements of the product matrix have zero mean. Moreover, since the elements of the weight matrices $\mathbf{W}_1, \mathbf{W}_2,..., \mathbf{W}_{L+1}$ are assumed to be Gaussian distributed, they have finite fourth moments. Further by the properties of the sum and product of random variables [@dufour2003properties], the elements of the product matrix constructed from the weight matrices $\mathbf{W}_1, \mathbf{W}_2,..., \mathbf{W}_{L+1}$ also have finite fourth moments. By Theorem \[thm:extreme\_singular\_value\_general\] and the homogeneity of matrix norms, a random matrix $\mathbf{M}$ whose elements are i.i.d. random variables with mean 0, variance $\sigma_w^2$, and finite fourth moment, the expected maximum singular value of $\mathbf{M}$ is given by $$\label{eqn:expected_max_sv_variance}
\mathbb{E}[\lambda_\text{max}(\mathbf{M})] = \sigma_m\left(\sqrt{N}+\sqrt{n} + O(\sqrt{n})\right).$$ Based on and , an estimate of the expected lower bound of the Lipschitz constant in is $$\label{eqn:lower_bound_proof}
\mathbb{E}[\underline{\rho}(f({\mathbf{x}}))] =\mathbb{E}[ ||\mathbf{W}_{L+1}\cdots \mathbf{W}_{1}||_2 ]= \left(\sigma_w^{L+1} \prod_{l=1}^{L}\sqrt{n_l}\right) \left(\sqrt{n_{L+1}} +\sqrt{n_0}+ O(\sqrt{n_0})\right).$$ Similar to the upper bound, this expected lower bound on the Lipschitz constant allows us to infer the Lipschitz constant of a DNN based on its architectural properties.
In Sec. \[sec:numerical\_examples\] of the manuscript, we empirically show that the expression in is a reasonable approximation of the lower bound of the Lipschitz constant of a DNN in . However, we note that, in our justification above, we make an assumption that the elements of the product matrix constructed from random matrices whose elements are i.i.d. zero-mean Gaussian random variables preserve independence. This is a conjecture that requires further investigation. We would like to further look into results on multiplications of random matrices to improve this result.
Connection to the Result Based on Output Trajectory Length {#app:connection}
==========================================================
In this appendix, we show a connection between our result and the result in [@raghu2017expressive]. Both our work and [@raghu2017expressive] consider DNNs with i.i.d. zero-mean Gaussian weight parameters. In our work, we use the Lipschitz constant as a measure of the expressiveness of a DNN, while in [@raghu2017expressive], the proposed expressiveness measure of a DNN is the expected length of an output trajectory as the DNN traverses a one-dimensional trajectory in its input space. Intuitively, as an input trajectory is passed through a DNN, it is deformed by the linear weight layers and the nonlinear activation layers; the output trajectory length measure in [@raghu2017expressive] is the extent to which the DNN ‘stretches’ a trajectory given in the input space.
By considering the expected output trajectory length as the expressiveness measure, [@raghu2017expressive] prove the following result:
Let $f({\mathbf{x}})$ be a DNN with ReLU activation functions and weights being i.i.d. Gaussian random variables $\mathcal{N}(0,\sigma_w^2)$, and let ${\mathbf{x}}(t)$ be a one-dimensional trajectory with ${\mathbf{x}}(t+\delta)$ having a non-trivial perpendicular component to ${\mathbf{x}}(t)$ for all $t,\delta$. Denote $\mathbf{h}_l({\mathbf{x}}(t)) = \mathbf{h}_l(t)$ as the image of the trajectory in the $l$th layer of the DNN. The expected output trajectory length of the DNN is lower bounded by $${\mathbb{E}}[\eta(\mathbf{h}_{L+1}(t))] \ge O\left( \frac{\sigma_w n}{\sqrt{n+1}}\right)^{L+1} \eta({\mathbf{x}}(t)),
\label{eqn:lower_bound_trajectory_length}$$ where $\eta({\mathbf{x}}(t))= \bigintsss_t \left\vert\left\vert\frac{d{\mathbf{x}}(t)}{dt}\right\vert\right\vert_2 dt$ is the trajectory length and $n$ is the width of the DNN.
![Expressiveness of a ReLU DNN measured by output trajectory length [@raghu2017expressive] versus the expressiveness by the proposed Lipschitz constant approach. The dots in grey correspond to the calculated expressiveness measures for DNNs with different widths and depths, and the red solid line is the identity line. The result is generated for DNNs with width and depth ranging between 30 and 100, respectively.[]{data-label="fig:trajlength_vs_lipschitz"}](./figures/trajlength_vs_lipschitz){width="8cm"}
Note that, if we consider the expected output trajectory length normalized by the input trajectory length (i.e., the ‘stretch’ of the trajectory), we can establish a connection with the lower bound in and the lower bound we derived based on Lipschitz constant expressiveness characterization in Sec. \[subsubsec:lower\_bound\]. In particular, in Sec. \[subsubsec:lower\_bound\], we showed that for a DNN with a constant width $n$ (i.e., $n_l = n$ for $l = 1,...,L$), the asymptotic lower bound on the Lipschitz constant of the DNN is $O\left( \sigma_w^{L+1} n^{\frac{L}{2}}\right)$. On the other hand, the normalized lower bound on the expected output trajectory in can be written as $O\left( \sigma_w^{L+1}\left(\frac{n}{\sqrt{n+1}}\right)^{L+1}\right)$. For $n\gg 1$ and $L\gg 1$, this asymptotic lower bound from coincides with the asymptotic lower bound we obtained based on the Lipschitz constant measure of expressiveness. Fig. \[fig:trajlength\_vs\_lipschitz\] illustrates this connection between our proposed expressiveness measure based on the Lipschitz constant of a DNN and the expressiveness measure based on the output trajectory length [@raghu2017expressive] for a set of ReLU DNNs with different widths and depths. From the plot, we see that, for DNNs with different architectures, the correlation between the asymptotic lower bounds based on these two measures of expressiveness (grey dots) approximately coincides with the identity line (red line).
The observed connection between the two measures of expressiveness of a DNN is sensible. If we consider the input trajectory to a DNN to be represented by a set of discrete points, the length of the output trajectory captures the extent of ‘stretch’ between pairs of points as they are passed through the DNN. Mathematically, the extent of ‘stretch’ or the distance between two points in a DNN’s output space in relation to the distance between the corresponding points in the input space is characterized by the Lipschitz property of the DNN.
|
---
abstract: |
At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit [@Neal1996; @Daniely; @Matthews2017GaussianProcess; @Lee2017; @Matthews2018GaussianProcess], thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function $f_\theta$ (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial.
We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function $f_\theta$ follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping.
Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.
author:
- |
Arthur Jacot\
École Polytechnique Fédérale de Lausanne\
`[email protected]` Franck Gabriel\
Imperial College London and École Polytechnique Fédérale de Lausanne\
`[email protected]` Clément Hongler\
École Polytechnique Fédérale de Lausanne\
`[email protected]`
bibliography:
- 'main\_NIPS.bib'
date: May 2018
title: |
Neural Tangent Kernel:\
Convergence and Generalization in Neural Networks
---
Introduction
============
Artificial neural networks (ANNs) have achieved impressive results in numerous areas of machine learning. While it has long been known that ANNs can approximate any function with sufficiently many hidden neurons [@Hornik1989; @Leshno], it is not known what the optimization of ANNs converges to. Indeed the loss surface of neural networks optimization problems is highly non-convex: it has a high number of saddle points which may slow down the convergence [@Dauphin2014]. A number of results [@Choromanska; @Pascanu2014; @Pennington2017] suggest that for wide enough networks, there are very few “bad” local minima, i.e. local minima with much higher cost than the global minimum. More recently, the investigation of the geometry of the loss landscape at initialization has been the subject of a precise study [@Karakida2018]. The analysis of the dynamics of training in the large-width limit for shallow networks has seen recent progress as well [@Mei2018]. To the best of the authors knowledge, the dynamics of deep networks has however remained an open problem until the present paper: see the contributions section below.
A particularly mysterious feature of ANNs is their good generalization properties in spite of their usual over-parametrization [@Sagun]. It seems paradoxical that a reasonably large neural network can fit random labels, while still obtaining good test accuracy when trained on real data [@Zhang]. It can be noted that in this case, kernel methods have the same properties [@Belkin].
In the infinite-width limit, ANNs have a Gaussian distribution described by a kernel [@Neal1996; @Daniely; @Matthews2017GaussianProcess; @Lee2017; @Matthews2018GaussianProcess]. These kernels are used in Bayesian inference or Support Vector Machines, yielding results comparable to ANNs trained with gradient descent [@Cho2009; @Lee2017]. We will see that in the same limit, the behavior of ANNs during training is described by a related kernel, which we call the neural tangent network (NTK).
Contribution
------------
We study the network function $f_\theta$ of an ANN, which maps an input vector to an output vector, where $\theta$ is the vector of the parameters of the ANN. In the limit as the widths of the hidden layers tend to infinity, the network function at initialization, $f_\theta$ converges to a Gaussian distribution [@Neal1996; @Daniely; @Matthews2017GaussianProcess; @Lee2017; @Matthews2018GaussianProcess].
In this paper, we investigate fully connected networks in this infinite-width limit, and describe the dynamics of the network function $f_\theta$ during training:
- During gradient descent, we show that the dynamics of $f_\theta$ follows that of the so-called *kernel gradient descent* in function space with respect to a limiting kernel, which only depends on the depth of the network, the choice of nonlinearity and the initialization variance.
- The convergence properties of ANNs during training can then be related to the positive-definiteness of the infinite-width limit NTK. In the case when the dataset is supported on a sphere, we prove this positive-definiteness using recent results on dual activation functions [@Daniely]. The values of the network function $f_\theta$ outside the training set is described by the NTK, which is crucial to understand how ANN generalize.
- For a least-squares regression loss, the network function $f_{\theta}$ follows a linear differential equation in the infinite-width limit, and the eigenfunctions of the Jacobian are the kernel principal components of the input data. This shows a direct connection to kernel methods and motivates the use of early stopping to reduce overfitting in the training of ANNs.
- Finally we investigate these theoretical results numerically for an artificial dataset (of points on the unit circle) and for the MNIST dataset. In particular we observe that the behavior of wide ANNs is close to the theoretical limit.
Neural networks {#sec:realization_function}
===============
In this article, we consider fully-connected ANNs with layers numbered from $0$ (input) to $L$ (output), each containing $n_{0},\ldots,n_{L}$ neurons, and with a Lipschitz, twice differentiable nonlinearity function $\sigma:\mathbb{R}\to\mathbb{R}$, with bounded second derivative [^1].
This paper focuses on the ANN *realization function* $F^{(L)} : \mathbb{R}^{P} \to \mathcal{F}$, mapping parameters $\theta$ to functions $f_\theta$ in a space $\mathcal{F}$. The dimension of the parameter space is $P = \sum_{\ell=0}^{L-1} (n_\ell + 1) n_{\ell+1}$: the parameters consist of the connection matrices $W^{(\ell)} \in \mathbb{R}^{n_\ell \times n_{\ell+1}}$ and bias vectors $b^{(\ell)} \in \mathbb{R}^{n_{\ell+1}}$ for $\ell=0, ..., L-1$. In our setup, the parameters are initialized as iid Gaussians $\mathcal{N}(0, 1)$.
For a fixed distribution $p^{in}$ on the input space $\mathbb{R}^{n_0}$, the function space $\mathcal{F}$ is defined as $\left\{ f : \mathbb{R}^{n_0} \to \mathbb{R}^{n_L} \right\} $. On this space, we consider the seminorm $|| \cdot ||_{p^{in}} $, defined in terms of the bilinear form $$\left<f, g\right>_{p^{in}} = \mathbb{E}_{x \sim p^{in}}\left[ f(x)^T g(x) \right].$$
In this paper, we assume that the input distribution $p^{in}$ is the empirical distribution on a finite dataset $x_1, ..., x_N$, i.e the sum of Dirac measures $ \frac{1}{N} \sum_{i=0}^N \delta_{x_i}$.
We define the network function by $f_\theta(x) := \tilde{\alpha}^{(L)}(x; \theta)$, where the functions $\tilde{\alpha}^{(\ell)}(\cdot; \theta) : \mathbb{R}^{n_0} \to\mathbb{R}^{n_\ell}$ (called *preactivations*) and $\alpha^{(\ell)}(\cdot; \theta):\mathbb{R}^{n_0} \to\mathbb{R}^{n_\ell}$ (called *activations*) are defined from the $0$-th to the $L$-th layer by: $$\begin{aligned}
\alpha^{(0)}(x; \theta) &= x \\
\tilde{\alpha}^{(\ell+1)}(x; \theta) &= \frac{1}{\sqrt{n_\ell}}W^{(\ell)} \alpha^{(\ell)}(x; \theta) + \beta b^{(\ell)} \\
\alpha^{(\ell)}(x; \theta) &= \sigma(\tilde{\alpha}^{(\ell)}(x; \theta)),\end{aligned}$$ where the nonlinearity $\sigma$ is applied entrywise. The scalar $\beta > 0$ is a parameter which allows us to tune the influence of the bias on the training.
\[rem:parametrization\] Our definition of the realization function $F^{(L)}$ slightly differs from the classical one. Usually, the factors $\frac{1}{\sqrt{n_\ell}}$ and the parameter $\beta$ are absent and the parameters are initialized using what is sometimes called LeCun initialization, taking $W^{(\ell)}_{ij} \sim \mathcal{N}(0, \frac{1}{n_\ell})$ and $b^{(\ell)}_{j} \sim \mathcal{N}(0, 1)$ (or sometimes $b^{(\ell)}_{j} = 0$) to compensate. While the set of representable functions $F^{(L)}(\mathbb{R}^P)$ is the same for both parametrizations (with or without the factors $\frac{1}{\sqrt{n_\ell}}$ and $\beta$), the derivatives of the realization function with respect to the connections $\partial_{W_{ij}^{(\ell)}} F^{(L)}$ and bias $\partial_{b_{j}^{(\ell)}} F^{(L)}$ are scaled by $\frac{1}{\sqrt{n_\ell}}$ and $\beta$ respectively in comparison to the classical parametrization.
The factors $\frac{1}{\sqrt{n_\ell}}$ are key to obtaining a consistent asymptotic behavior of neural networks as the widths of the hidden layers $n_1, ..., n_{L-1}$ grow to infinity. However a side-effect of these factors is that they reduce greatly the influence of the connection weights during training when $n_\ell$ is large: the factor $\beta$ is introduced to balance the influence of the bias and connection weights. In our numerical experiments, we take $\beta=0.1$ and use a learning rate of $1.0$, which is larger than usual, see Section \[sec:numerical-experiments\]. This gives a behaviour similar to that of a classical network of width $100$ with a learning rate of $0.01$.
Kernel gradient {#sec:kernel_gradient}
===============
The training of an ANN consists in optimizing $f_\theta$ in the function space $\mathcal{F}$ with respect to a functional cost $C : \mathcal{F} \to \mathbb{R}$, such as a regression or cross-entropy cost. Even for a convex functional cost $C$, the composite cost $C \circ F^{(L)} : \mathbb{R}^P \to \mathbb{R}$ is in general highly non-convex [@Choromanska]. We will show that during training, the network function $f_\theta$ follows a descent along the kernel gradient with respect to the Neural Tangent Kernel (NTK) which we introduce in Section \[sec:neural\_tangent\_kernel\]. This makes it possible to study the training of ANNs in the function space $\mathcal{F}$, on which the cost $C$ is convex.
A *multi-dimensional kernel* $K$ is a function $\mathbb{R}^{n_{0}}\times\mathbb{R}^{n_{0}}\to\mathbb{R}^{n_{L}\times n_{L}}$, which maps any pair $(x,x')$ to an $n_{L}\times n_{L}$-matrix such that $K(x, x') = K(x', x)^T$ (equivalently $K$ is a symmetric tensor in $\mathcal{F}\otimes\mathcal{F}$). Such a kernel defines a bilinear map on $\mathcal{F}$, taking the expectation over independent $x, x' \sim p^{in}$: $$\left<f, g \right>_K := \mathbb{E}_{x, x' \sim p^{in}} \left[f(x)^T K(x, x') g(x') \right].$$ The kernel $K$ is *positive definite with respect to $|| \cdot ||_{p^{in}} $* if $ || f ||_{p^{in}} > 0 \implies || f ||_K > 0$.
We denote by $ \mathcal{F}^{*} $ the dual of $ \mathcal{F} $ with respect to $ p^{in} $, i.e. the set of linear forms $ \mu: \mathcal {F} \to \mathbb{R} $ of the form $ \mu = \langle d, \cdot \rangle_{p^{in}} $ for some $ d \in \mathcal{F} $. Two elements of $\mathcal{F}$ define the same linear form if and only if they are equal on the data. The constructions in the paper do not depend on the element $d \in \mathcal{F}$ chosen in order to represent $ \mu$ as $\langle d, \cdot \rangle_{p^{in}}$. Using the fact that the partial application of the kernel $K_{i, \cdot}(x, \cdot)$ is a function in $\mathcal{F}$, we can define a map $\Phi_K : \mathcal{F}^* \to \mathcal{F}$ mapping a dual element $\mu = \left<d, \cdot\right>_{p^{in}}$ to the function $f_\mu = \Phi_K(\mu)$ with values: $$f_{\mu, i}(x) = \mu K_{i, \cdot}(x, \cdot) = \left< d, K_{i, \cdot}(x, \cdot) \right>_{p^{in}}.$$
For our setup, which is that of a finite dataset $ x_1, \ldots, x_n \in \mathbb{R}^{n_0} $, the cost functional $ C $ only depends on the values of $ f \in \mathcal{F} $ at the data points. As a result, the (functional) derivative of the cost $ C $ at a point $f_0\in\mathcal{F}$ can be viewed as an element of $ \mathcal{F}^{*} $, which we write $ \partial_f^{in} C |_{f_0} $. We denote by $ d |_{f_0} \in \mathcal{F} $, a corresponding dual element, such that $ \partial_f^{in} C|_{f_0} = \langle d |_{f_0}, \cdot \rangle_{p^{in}} $.
The *kernel gradient* $\nabla_K C|_{f_0} \in \mathcal{F}$ is defined as $\Phi_K \left( \partial_f^{in} C|_{f_0} \right) $. In contrast to $ \partial_f^{in} C $ which is only defined on the dataset, the kernel gradient generalizes to values $x$ outside the dataset thanks to the kernel $K$: $$\nabla_K C|_{f_0} (x) = \frac{1}{N} \sum_{j=1}^{N} K(x, x_j) d|_{f_0} (x_j) .$$
A time-dependent function $ f (t) $ follows the *kernel gradient descent with respect to $K$* if it satisfies the differential equation $$\partial_t f(t) = -\nabla_{K}C|_{f(t)}.$$ During kernel gradient descent, the cost $C(f(t))$ evolves as $$\partial_t C|_{f(t)} = -\left<d|_{f(t)}, \nabla_{K}C|_{f(t)} \right>_{p^{in}}=-\left\|d|_{f(t)} \right\|_{K}^{2}.$$ Convergence to a critical point of $C$ is hence guaranteed if the kernel $K$ is positive definite with respect to $|| \cdot ||_{p^{in}}$: the cost is then strictly decreasing except at points such that $||d|_{f(t)}||_{p^{in}} = 0$. If the cost is convex and bounded from below, the function $f(t)$ therefore converges to a global minimum as $t \to \infty$.
Random functions approximation {#sec:rfa}
------------------------------
As a starting point to understand the convergence of ANN gradient descent to kernel gradient descent in the infinite-width limit, we introduce a simple model, inspired by the approach of [@Rahimi2007].
A kernel $K$ can be approximated by a choice of $P$ random functions $f^{(p)}$ sampled independently from any distribution on $\mathcal{F}$ whose (non-centered) covariance is given by the kernel $K$: $$\mathbb{E}[f_{k}^{(p)}(x)f_{k'}^{(p)}(x')]=K_{kk'}(x,x').$$
These functions define a random linear parametrization $F^{lin}:\mathbb{R}^{P}\to\mathcal{F}$ $$\theta \mapsto f_{\theta}^{lin}=\frac{1}{\sqrt{P}}\sum_{p=1}^{P}\theta_{p}f^{(p)}.$$
The partial derivatives of the parametrization are given by $$\partial_{\theta_p} F^{lin}(\theta) = \frac{1}{\sqrt{P}} f^{(p)}.$$
Optimizing the cost $C\circ F^{lin}$ through gradient descent, the parameters follow the ODE: $$\partial_t \theta_{p}(t)=-\partial_{\theta_{p}}(C\circ F^{lin})(\theta(t))=-\frac{1}{\sqrt{P}}\partial_{f}^{in}C|_{f_{\theta(t)}^{lin}}\;f^{(p)}=-\frac{1}{\sqrt{P}} \left<d|_{f_{\theta(t)}^{lin}}, f^{(p)}\right>_{p^{in}}.$$ As a result the function $f_{\theta(t)}^{lin}$ evolves according to $$\partial_t f_{\theta(t)}^{lin}=\frac{1}{\sqrt{P}}\sum_{p=1}^{P} \partial_t \theta_{p}(t)f^{(p)}=-\frac{1}{P}\sum_{p=1}^{P}\left<d|_{f_{\theta(t)}^{lin}}, f^{(p)}\right>_{p^{in}} f^{(p)},$$ where the right-hand side is equal to the kernel gradient $-\nabla_{\tilde{K}} C$ with respect to the *tangent kernel* $$\tilde{K} =\sum_{p=1}^P \partial_{\theta_p} F^{lin}(\theta) \otimes \partial_{\theta_p} F^{lin}(\theta) = \frac{1}{P}\sum_{p=1}^P f^{(p)} \otimes f^{(p)}.$$ This is a random $n_L$-dimensional kernel with values $
\tilde{K}_{ii'}(x, x') = \frac{1}{P} \sum_{p=1}^P f^{(p)}_i(x) f^{(p)}_{i'}(x').
$
Performing gradient descent on the cost $ C \circ F^{lin}$ is therefore equivalent to performing kernel gradient descent with the tangent kernel $\tilde{K}$ in the function space. In the limit as $P \to \infty$, by the law of large numbers, the (random) tangent kernel $\tilde{K}$ tends to the fixed kernel $K$, which makes this method an approximation of kernel gradient descent with respect to the limiting kernel $K$.
Neural tangent kernel {#sec:neural_tangent_kernel}
=====================
For ANNs trained using gradient descent on the composition $C \circ F^{(L)}$, the situation is very similar to that studied in the Section \[sec:rfa\]. During training, the network function $f_\theta$ evolves along the (negative) kernel gradient $$\partial_t f_{\theta(t)} = -\nabla_{\Theta^{(L)}} C|_{f_{\theta(t)}}$$ with respect to the *neural tangent kernel* (NTK) $$\begin{aligned}
\Theta^{(L)}(\theta) &= \sum_{p=1}^P \partial_{\theta_p} F^{(L)}(\theta) \otimes \partial_{\theta_p} F^{(L)}(\theta).\end{aligned}$$ However, in contrast to $F^{lin}$, the realization function $F^{(L)}$ of ANNs is not linear. As a consequence, the derivatives $\partial_{\theta_p} F^{(L)}(\theta)$ and the neural tangent kernel depend on the parameters $\theta$. The NTK is therefore random at initialization and varies during training, which makes the analysis of the convergence of $f_\theta$ more delicate.
In the next subsections, we show that, in the infinite-width limit, the NTK becomes deterministic at initialization and stays constant during training. Since $f_\theta$ at initialization is Gaussian in the limit, the asymptotic behavior of $f_\theta$ during training can be explicited in the function space $ \mathcal{F} $.
Initialization
--------------
As observed in [@Neal1996; @Daniely; @Matthews2017GaussianProcess; @Lee2017; @Matthews2018GaussianProcess], the output functions $f_{\theta, i}$ for $i=1, ..., n_L$ tend to iid Gaussian processes in the infinite-width limit (a proof in our setup is given in the appendix):
\[prop:output\_limit\] For a network of depth $L$ at initialization, with a Lipschitz nonlinearity $\sigma$, and in the limit as $n_1, ..., n_{L-1} \to \infty$, the output functions $f_{\theta, k}$, for $k=1, ..., n_L$, tend (in law) to iid centered Gaussian processes of covariance $\Sigma^{(L)}$, where $\Sigma^{(L)}$ is defined recursively by: $$\begin{aligned}
\Sigma^{(1)}(x, x') &= \frac{1}{n_0} x^T x' + \beta^2 \\
\Sigma^{(L+1)}(x, x') &= \mathbb{E}_{f\sim\mathcal{N}\left(0,\Sigma^{\left(L\right)}\right)}[\sigma(f(x)) \sigma(f(x'))] + \beta^2,\end{aligned}$$ taking the expectation with respect to a centered Gaussian process $f$ of covariance $\Sigma^{(L)}$.
\[rem:no-problem-with-gauss-meas\] Strictly speaking, the existence of a suitable Gaussian measure with covariance $\Sigma^{(L)}$ is not needed: we only deal with the values of $f$ at $x, x'$ (the joint measure on $f(x), f(x')$ is simply a Gaussian vector in 2D). For the same reasons, in the proof of Proposition \[prop:output\_limit\] and Theorem \[thm:convergence\_NTK\_initialization\], we will freely speak of Gaussian processes without discussing their existence.
The first key result of our paper (proven in the appendix) is the following: in the same limit, the Neural Tangent Kernel (NTK) converges in probability to an explicit deterministic limit.
\[thm:convergence\_NTK\_initialization\] For a network of depth $L$ at initialization, with a Lipschitz nonlinearity $\sigma$, and in the limit as the layers width $n_1, ..., n_{L-1} \to \infty$, the NTK $\Theta^{(L)}$ converges in probability to a deterministic limiting kernel: $$\Theta^{(L)} \to \Theta^{(L)}_\infty \otimes Id_{n_L}.$$ The scalar kernel $\Theta^{(L)}_\infty : \mathbb{R}^{n_0} \times \mathbb{R}^{n_0} \to \mathbb{R}$ is defined recursively by $$\begin{aligned}
\Theta^{(1)}_\infty(x, x') &= \Sigma^{(1)}(x, x') \\
\Theta^{(L+1)}_\infty(x, x') &= \Theta^{(L)}_\infty(x, x') \dot{\Sigma}^{(L+1)}(x, x') + \Sigma^{(L+1)}(x, x'),\end{aligned}$$ where $$\dot{\Sigma}^{(L+1)}\left(x,x'\right)=
\mathbb{E}_{f\sim\mathcal{N}\left(0,\Sigma^{\left(L\right)}\right)}\left[\dot{\sigma}\left(f\left(x\right)\right)\dot{\sigma}\left(f\left(x'\right)\right)\right],$$ taking the expectation with respect to a centered Gaussian process $f$ of covariance $\Sigma^{(L)}$, and where $\dot{\sigma}$ denotes the derivative of $\sigma$.
By Rademacher’s theorem, $\dot{\sigma}$ is defined everywhere, except perhaps on a set of zero Lebesgue measure.
Note that the limiting $\Theta^{(L)}_\infty$ only depends on the choice of $\sigma$, the depth of the network and the variance of the parameters at initialization (which is equal to $1$ in our setting).
Training
--------
Our second key result is that the NTK stays asymptotically constant during training. This applies for a slightly more general definition of training: the parameters are updated according to a training direction $d_t \in \mathcal{F}$: $$\partial_t \theta_p(t) = \left< \partial_{\theta_p} F^{(L)}(\theta(t)), d_t \right>_{p^{in}}.$$ In the case of gradient descent, $d_t = -d|_{f_{\theta(t)}}$ (see Section \[sec:kernel\_gradient\]), but the direction may depend on another network, as is the case for e.g. Generative Adversarial Networks [@Goodfellow2014]. We only assume that the integral $\int_0^T \| d_t \|_{p^{in}} dt$ stays stochastically bounded as the width tends to infinity, which is verified for e.g. least-squares regression, see Section \[sec:least-squares\].
\[thm:conv-ntk-training\]
Assume that $ \sigma $ is a Lipschitz, twice differentiable nonlinearity function, with bounded second derivative. For any $T$ such that the integral $\int_0^T \| d_t \|_{p^{in}} dt$ stays stochastically bounded, as $n_1, ..., n_{L-1} \to \infty$, we have, uniformly for $t\in[0, T]$, $$\Theta^{(L)}(t) \to \Theta^{(L)}_\infty \otimes Id_{n_L}.$$ As a consequence, in this limit, the dynamics of $f_\theta$ is described by the differential equation $$\begin{aligned}
\partial_t f_{\theta(t)} = \Phi_{\Theta^{(L)}_\infty \otimes Id_{n_L}} \left( \left<d_{t}, \cdot \right>_{p^{in}} \right).\end{aligned}$$
As the proof of the theorem (in the appendix) shows, the variation during training of the individual activations in the hidden layers shrinks as their width grows. However their collective variation is significant, which allows the parameters of the lower layers to learn: in the formula of the limiting NTK $\Theta^{(L+1)}_\infty(x, x')$ in Theorem \[thm:convergence\_NTK\_initialization\], the second summand $\Sigma^{(L+1)}$ represents the learning due to the last layer, while the first summand represents the learning performed by the lower layers.
As discussed in Section \[sec:kernel\_gradient\], the convergence of kernel gradient descent to a critical point of the cost $C$ is guaranteed for positive definite kernels. The limiting NTK is positive definite if the span of the derivatives $\partial_{\theta_p} F^{(L)}$, $p=1, ..., P$ becomes dense in $\mathcal{F}$ w.r.t. the $p^{in}$-norm as the width grows to infinity. It seems natural to postulate that the span of the preactivations of the last layer (which themselves appear in $\partial_{\theta_p} F^{(L)}$, corresponding to the connection weights of the last layer) becomes dense in $\mathcal{F}$, for a large family of measures $ p^{in}$ and nonlinearities (see e.g. [@Hornik1989; @Leshno] for classical theorems about ANNs and approximation). In the case when the dataset is supported on a sphere, the positive-definiteness of the limiting NTK can be shown using Gaussian integration techniques and existing positive-definiteness criteria, as given by the following proposition, proven in Appendix \[Appendix-4\]:
\[prop:pos-def\] For a non-polynomial Lipschitz nonlinearity $ \sigma $, for any input dimension $ n_0 $, the restriction of the limiting NTK $ \Theta_\infty^{(L)} $ to the unit sphere $ \mathbb{S}^{n_0 - 1} = \{ x \in \mathbb{R}^{n_0} : x^T x =1 \} $ is positive-definite if $ L \geq 2 $.
Least-squares regression {#sec:least-squares}
========================
Given a goal function $f^*$ and input distribution $p^{in}$, the least-squares regression cost is $$C(f) = \frac{1}{2} ||f - f^*||^2_{p^{in}} = \frac{1}{2}\mathbb{E}_{x \sim p^{in}} \left[\|f(x) - f^*(x)\|^2 \right].$$ Theorems \[thm:convergence\_NTK\_initialization\] and \[thm:conv-ntk-training\] apply to an ANN trained on such a cost. Indeed the norm of the training direction $\|d(f)\|_{p^{in}} = \| f^* - f \|_{p^{in}}$ is strictly decreasing during training, bounding the integral. We are therefore interested in the behavior of a function $f_t$ during kernel gradient descent with a kernel $K$ (we are of course especially interested in the case $K = \Theta^{(L)}_\infty \otimes Id_{n_L}$): $$\begin{aligned}
\partial_t f_t = \Phi_K\left(\left<f^* - f, \cdot \right>_{p^{in}}\right).\end{aligned}$$ The solution of this differential equation can be expressed in terms of the map $\Pi : f \mapsto \Phi_K \left(\left<f, \cdot \right>_{p^{in}}\right)$: $$f_t = f^* + e^{-t \Pi}(f_0 - f^*)$$ where $e^{-t \Pi} = \sum_{k=0}^{\infty} \frac{(-t)^k}{k!} \Pi^k$ is the exponential of $-t \Pi$. If $\Pi$ can be diagonalized by eigenfunctions $f^{(i)}$ with eigenvalues $\lambda_i$, the exponential $e^{-t \Pi}$ has the same eigenfunctions with eigenvalues $e^{-t \lambda_i}$.
For a finite dataset $x_1, ..., x_N$ of size $N$, the map $\Pi$ takes the form $$\Pi(f)_k (x) = \frac{1}{N} \sum_{i=1}^N \sum_{k'=1}^{n_L} f_{k'}(x_i) K_{kk'}(x_i, x).$$ The map $\Pi$ has at most $Nn_L$ positive eigenfunctions, and they are the kernel principal components $f^{(1)}, ..., f^{(N n_L)}$ of the data with respect to to the kernel $K$ [@Scholkopf; @Shawe-Taylor]. The corresponding eigenvalues $\lambda_i$ is the variance captured by the component.
Decomposing the difference $(f^* - f_0) = \Delta^0_f + \Delta^1_f + ... + \Delta^{N n_L}_f$ along the eigenspaces of $\Pi$, the trajectory of the function $f_t$ reads $$f_{t} = f^* + \Delta^0_f + \sum_{i=1}^{N n_L} e^{-t \lambda_i} \Delta^i_f,$$ where $\Delta^0_f$ is in the kernel (null-space) of $\Pi$ and $\Delta^i_f \propto f^{(i)}$.
The above decomposition can be seen as a motivation for the use of early stopping. The convergence is indeed faster along the eigenspaces corresponding to larger eigenvalues $\lambda_i$. Early stopping hence focuses the convergence on the most relevant kernel principal components, while avoiding to fit the ones in eigenspaces with lower eigenvalues (such directions are typically the ‘noisier’ ones: for instance, in the case of the RBF kernel, lower eigenvalues correspond to high frequency functions).
Note that by the linearity of the map $e^{-t \Pi}$, if $f_0$ is initialized with a Gaussian distribution (as is the case for ANNs in the infinite-width limit), then $f_t$ is Gaussian for all times $t$. Assuming that the kernel is positive definite on the data (implying that the $Nn_L \times Nn_L$ Gram marix $\tilde{K}=\left(K_{kk'}(x_i, x_j) \right)_{ik, jk'}$ is invertible), as $t \to \infty$ limit, we get that $f_\infty = f^* + \Delta^0_f = f_0 - \sum_i \Delta^i_f$ takes the form $$f_{\infty, k}(x) = \kappa_{x, k}^T \tilde{K}^{-1} y^* + \left(f_0(x) - \kappa_{x, k}^T \tilde{K}^{-1} y_0\right),$$ with the $N n_l$-vectors $ \kappa_{x, k} $, $ y^* $ and $ y_0 $ given by $$\begin{aligned}
\kappa_{x, k} & = \left(K_{kk'}(x, x_i)\right)_{i, k'} \\
y^* & = \left(f^*_k(x_i)\right)_{i, k} \\
y_0 & = \left(f_{0, k}(x_i)\right)_{i, k}.\end{aligned}$$ The first term, the mean, has an important statistical interpretation: it is the maximum-a-posteriori (MAP) estimate given a Gaussian prior on functions $f_k\sim \mathcal{N}(0, \Theta^{(L)}_\infty)$ and the conditions $f_k(x_i)=f^*_k(x_i)$ . Equivalently, it is equal to the kernel ridge regression [@Shawe-Taylor] as the regularization goes to zero ($\lambda \to 0$). The second term is a centered Gaussian whose variance vanishes on the points of the dataset.
Numerical experiments {#sec:numerical-experiments}
=====================
In the following numerical experiments, fully connected ANNs of various widths are compared to the theoretical infinite-width limit. We choose the size of the hidden layers to all be equal to the same value $n := n_1 = ... = n_{L-1}$ and we take the ReLU nonlinearity $\sigma(x)=\max(0,x)$.
In the first two experiments, we consider the case $n_0=2$. Moreover, the input elements are taken on the unit circle. This can be motivated by the structure of high-dimensional data, where the centered data points often have roughly the same norm [^2].
In all experiments, we took $n_L = 1$ (note that by our results, a network with $n_L$ outputs behaves asymptotically like $n_L$ networks with scalar outputs trained independently). Finally, the value of the parameter $\beta$ is chosen as $0.1$, see Remark \[rem:parametrization\].
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Convergence of the NTK
----------------------
The first experiment illustrates the convergence of the NTK $\Theta^{(L)}$ of a network of depth $L=4$ for two different widths $n=500, 10000$. The function $\Theta^{(4)}(x_0, x)$ is plotted for a fixed $x_0=(1, 0)$ and $x=(cos(\gamma), sin(\gamma))$ on the unit circle in Figure \[fig:NTK\_convergence\]. To observe the distribution of the NTK, $10$ independent initializations are performed for both widths. The kernels are plotted at initialization $t=0$ and then after $200$ steps of gradient descent with learning rate $1.0$ (i.e. at $t=200$). We approximate the function $f^*(x) = x_1 x_2$ with a least-squares cost on random $\mathcal{N}(0, 1)$ inputs.
For the wider network, the NTK shows less variance and is smoother. It is interesting to note that the expectation of the NTK is very close for both networks widths. After $200$ steps of training, we observe that the NTK tends to “inflate”. As expected, this effect is much less apparent for the wider network ($n=10000$) where the NTK stays almost fixed, than for the smaller network ($n=500$).
Kernel regression
-----------------
For a regression cost, the infinite-width limit network function $f_{\theta(t)}$ has a Gaussian distribution for all times $t$ and in particular at convergence $t\to \infty $ (see Section \[sec:least-squares\]). We compared the theoretical Gaussian distribution at $t\to\infty$ to the distribution of the network function $f_{\theta(T)}$ of a finite-width network for a large time $T=1000$. For two different widths $n=50, 1000$ and for $10$ random initializations each, a network is trained on a least-squares cost on $4$ points of the unit circle for $1000$ steps with learning rate $1.0$ and then plotted in Figure \[fig:ANN\_regression\].
We also approximated the kernels $\Theta_\infty^{(4)}$ and $\Sigma^{(4)}$ using a large-width network ($n=10000$) and used them to calculate and plot the 10th, 50th and 90-th percentiles of the $t\to \infty$ limiting Gaussian distribution.
The distributions of the network functions are very similar for both widths: their mean and variance appear to be close to those of the limiting distribution $t \to \infty$. Even for relatively small widths ($n=50$), the NTK gives a good indication of the distribution of $f_{\theta(t)}$ as $t\to\infty$.
Convergence along a principal component
---------------------------------------
We now illustrate our result on the MNIST dataset of handwritten digits made up of grayscale images of dimension $28 \times 28$, yielding a dimension of $n_0 = 784$.
We computed the first 3 principal components of a batch of $N=512$ digits with respect to the NTK of a high-width network $n=10000$ (giving an approximation of the limiting kernel) using a power iteration method. The respective eigenvalues are $\lambda_1=0.0457$, $\lambda_2=0.00108$ and $\lambda_3=0.00078$. The kernel PCA is non-centered, the first component is therefore almost equal to the constant function, which explains the large gap between the first and second eigenvalues[^3]. The next two components are much more interesting as can be seen in Figure \[fig:MNIST\_kernel\_principal\_components\], where the batch is plotted with $x$ and $y$ coordinates corresponding to the 2nd and 3rd components.
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We have seen in Section \[sec:least-squares\] how the convergence of kernel gradient descent follows the kernel principal components. If the difference at initialization $f_0 - f^*$ is equal (or proportional) to one of the principal components $f^{(i)}$, then the function will converge along a straight line (in the function space) to $f^*$ at an exponential rate $e^{-\lambda_i t}$.
We tested whether ANNs of various widths $n=100, 1000, 10000$ behave in a similar manner. We set the goal of the regression cost to $f^* = f_{\theta(0)}+0.5 f^{(2)}$ and let the network converge. At each time step $t$, we decomposed the difference $f_{\theta(t)} - f^*$ into a component $g_t$ proportional to $f^{(2)}$ and another one $h_t$ orthogonal to $f^{(2)}$. In the infinite-width limit, the first component decays exponentially fast $||g_t||_{p^{in}} = 0.5 e^{-\lambda_2 t}$ while the second is null ($h_t=0$), as the function converges along a straight line.
As expected, we see in Figure \[fig:MNIST\_convergence\_outside\] that the wider the network, the less it deviates from the straight line (for each width $n$ we performed two independent trials). As the width grows, the trajectory along the 2nd principal component (shown in Figure \[fig:MNIST\_convergence\_inside\]) converges to the theoretical limit shown in blue.
A surprising observation is that smaller networks appear to converge faster than wider ones. This may be explained by the inflation of the NTK observed in our first experiment. Indeed, multiplying the NTK by a factor $a$ is equivalent to multiplying the learning rate by the same factor. However, note that since the NTK of large-width network is more stable during training, larger learning rates can in principle be taken. One must hence be careful when comparing the convergence speed in terms of the number of steps (rather than in terms of the time $t$): both the inflation effect and the learning rate must be taken into account.
Conclusion
==========
This paper introduces a new tool to study ANNs, the Neural Tangent Kernel (NTK), which describes the local dynamics of an ANN during gradient descent. This leads to a new connection between ANN training and kernel methods: in the infinite-width limit, an ANN can be described in the function space directly by the limit of the NTK, an explicit constant kernel $\Theta^{(L)}_\infty$, which only depends on its depth, nonlinearity and parameter initialization variance. More precisely, in this limit, ANN gradient descent is shown to be equivalent to a kernel gradient descent with respect to $\Theta^{(L)}_\infty$. The limit of the NTK is hence a powerful tool to understand the generalization properties of ANNs, and it allows one to study the influence of the depth and nonlinearity on the learning abilities of the network. The analysis of training using NTK allows one to relate convergence of ANN training with the positive-definiteness of the limiting NTK and leads to a characterization of the directions favored by early stopping methods.
Acknowledgements {#acknowledgements .unnumbered}
================
The authors thank K. Kytölä for many interesting discussions. The second author was supported by the ERC CG CRITICAL. The last author acknowledges support from the ERC SG Constamis, the NCCR SwissMAP, the Blavatnik Family Foundation and the Latsis Foundation.
Appendix {#sec:proofs}
========
This appendix is dedicated to proving the key results of this paper, namely Proposition \[prop:output\_limit\] and Theorems \[thm:convergence\_kernel\_initialization\] and \[thm:conv-ntk-training\], which describe the asymptotics of neural networks at initialization and during training.
We study the limit of the NTK as $n_1, ..., n_{L-1} \to \infty$ sequentially, i.e. we first take $n_1 \to \infty$, then $n_2 \to \infty $, etc. This leads to much simpler proofs, but our results could in principle be strengthened to the more general setting when $\min(n_1, ..., n_{L-1}) \to \infty$.
A natural choice of convergence to study the NTK is with respect to the operator norm on kernels: $$\lVert K \rVert_{op} = \max_{\lVert f \rVert_{p^{in}} \leq 1} \lVert f \rVert_K = \max_{\lVert f \rVert_{p^{in}} \leq 1}\sqrt {\mathbb{E}_{x, x'}[f(x)^T K(x, x')f(x')]},$$ where the expectation is taken over two independent $x, x' \sim p^{in}$. This norm depends on the input distribution $p^{in}$. In our setting, $p^{in}$ is taken to be the empirical measure of a finite dataset of distinct samples $x_1, ..., x_N$. As a result, the operator norm of $K$ is equal to the leading eigenvalue of the $N n_L \times N n_L$ Gram matrix $\left( K_{kk'}(x_i, x_j)\right)_{k, k' < n_L, i, j < N}$. In our setting, convergence in operator norm is hence equivalent to pointwise convergence of $ K $ on the dataset.
Asymptotics at Initialization
-----------------------------
It has already been observed [@Neal1996; @Lee2017] that the output functions $f_{\theta, i}$ for $i=1, ..., n_L$ tend to iid Gaussian processes in the infinite-width limit.
\[prop:output\_limit\] For a network of depth $L$ at initialization, with a Lipschitz nonlinearity $\sigma$, and in the limit as $n_1, ..., n_{L-1} \to \infty$ sequentially, the output functions $f_{\theta, k}$, for $k=1, ..., n_L$, tend (in law) to iid centered Gaussian processes of covariance $\Sigma^{(L)}$, where $\Sigma^{(L)}$ is defined recursively by: $$\begin{aligned}
\Sigma^{(1)}(x, x') &= \frac{1}{n_0} x^T x' + \beta^2 \\
\Sigma^{(L+1)}(x, x') &= \mathbb{E}_{f}[\sigma(f(x)) \sigma(f(x'))] + \beta^2,\end{aligned}$$ taking the expectation with respect to a centered Gaussian process $f$ of covariance $\Sigma^{(L)}$.
We prove the result by induction. When $L=1$, there are no hidden layers and $f_\theta$ is a random affine function of the form: $$f_\theta(x) = \frac{1}{\sqrt{n_0}} W^{(0)} x + \beta b^{(0)}.$$ All output functions $f_{\theta, k}$ are hence independent and have covariance $\Sigma^{(1)}$ as needed.
The key to the induction step is to consider an $(L+1)$-network as the following composition: an $L$-network $\mathbb{R}^{n_0} \to \mathbb{R}^{n_L}$ mapping the input to the pre-activations $\tilde{\alpha}^{(L)}_i$, followed by an elementwise application of the nonlinearity $\sigma$ and then a random affine map $\mathbb{R}^{n_L} \to \mathbb{R}^{n_{L+1}}$. The induction hypothesis gives that in the limit as sequentially $n_1, ..., n_{L-1} \to \infty$ the preactivations $\tilde{\alpha}^{(L)}_i$ tend to iid Gaussian processes with covariance $\Sigma^{(L)}$. The outputs $$f_{\theta, i} = \frac{1}{\sqrt{n_L}} W_i^{(L)} \alpha^{(L)} + \beta b_i^{(L)}$$ conditioned on the values of $\alpha^{(L)}$ are iid centered Gaussians with covariance $$\tilde{\Sigma}^{(L+1)}(x, x') = \frac{1}{n_L} \alpha^{(L)}(x;\theta)^T \alpha^{(L)}(x';\theta) + \beta^2.$$ By the law of large numbers, as $n_L \to \infty$, this covariance tends in probability to the expectation $$\tilde{\Sigma}^{(L+1)}(x, x') \to \Sigma^{(L+1)}(x, x') = \mathbb{E}_{f \sim \mathcal{N}(0, \Sigma^{(L)})}[\sigma(f(x)) \sigma(f(x'))] + \beta^2.$$ In particular the covariance is deterministic and hence independent of $\alpha^{(L)}$. As a consequence, the conditioned and unconditioned distributions of $f_{\theta, i}$ are equal in the limit: they are iid centered Gaussian of covariance $\Sigma^{(L+1)}$.
In the infinite-width limit, the neural tangent kernel, which is random at initialization, converges in probability to a deterministic limit.
\[thm:convergence\_kernel\_initialization\] For a network of depth $L$ at initialization, with a Lipschitz nonlinearity $\sigma$, and in the limit as the layers width $n_1, ..., n_{L-1} \to \infty$ sequentially, the NTK $\Theta^{(L)}$ converges in probability to a deterministic limiting kernel: $$\Theta^{(L)} \to \Theta^{(L)}_\infty \otimes Id_{n_L}.$$ The scalar kernel $\Theta^{(L)}_\infty : \mathbb{R}^{n_0} \times \mathbb{R}^{n_0} \to \mathbb{R}$ is defined recursively by $$\begin{aligned}
\Theta^{(1)}_\infty(x, x') &= \Sigma^{(1)}(x, x') \\
\Theta^{(L+1)}_\infty(x, x') &= \Theta^{(L)}_\infty(x, x') \dot{\Sigma}^{(L+1)}(x, x') + \Sigma^{(L+1)}(x, x'),\end{aligned}$$ where $$\dot{\Sigma}^{(L+1)}\left(x,x'\right)=
\mathbb{E}_{f\sim\mathcal{N}\left(0,\Sigma^{\left(L\right)}\right)}\left[\dot{\sigma}\left(f\left(x\right)\right)\dot{\sigma}\left(f\left(x'\right)\right)\right],$$ taking the expectation with respect to a centered Gaussian process $f$ of covariance $\Sigma^{(L)}$, and where $\dot{\sigma}$ denotes the derivative of $\sigma$.
The proof is again by induction. When $L=1$, there is no hidden layer and therefore no limit to be taken. The neural tangent kernel is a sum over the entries of $W^{(0)}$ and those of $b^{(0)}$: $$\begin{aligned}
\Theta_{kk'}(x, x') &= \frac{1}{n_0} \sum_{i=1}^{n_0} \sum_{j=1}^{n_1} x_i x'_i \delta_{jk}\delta_{jk'} + \beta^2 \sum_{j=1}^{n_1} \delta_{jk}\delta_{jk'} \\
&= \frac{1}{n_0} x^T x' \delta_{kk'} + \beta^2 \delta_{kk'} = \Sigma^{(1)}(x, x') \delta_{kk'}.\end{aligned}$$
Here again, the key to prove the induction step is the observation that a network of depth $L+1$ is an $L$-network mapping the inputs $x$ to the preactivations of the $L$-th layer $\tilde{\alpha}^{(L)}(x)$ followed by a nonlinearity and a random affine function. For a network of depth $ L + 1 $, let us therefore split the parameters into the parameters $\tilde{\theta}$ of the first $L$ layers and those of the last layer $(W^{(L)}, b^{(L)})$.
By Proposition \[prop:output\_limit\] and the induction hypothesis, as $n_1, ..., n_{L-1} \to \infty$ the pre-activations $\tilde{\alpha}^{(L)}_i$ are iid centered Gaussian with covariance $\Sigma^{(L)}$ and the neural tangent kernel $\Theta^{(L)}_{ii'}(x, x')$ of the smaller network converges to a deterministic limit: $$\left(\partial_{\tilde{\theta}} \tilde{\alpha}^{(L)}_{i}(x;\theta)\right)^T \partial_{\tilde{\theta}} \tilde{\alpha}^{(L)}_{i'}(x';\theta) \to \Theta^{(L)}_\infty(x, x') \delta_{ii'}.$$
We can split the neural tangent network into a sum over the parameters $\tilde{\theta}$ of the first $L$ layers and the remaining parameters $W^{(L)}$ and $b^{(L)}$.
For the first sum let us observe that by the chain rule: $$\partial_{\tilde{\theta}_p} f_{\theta, k}(x) = \frac{1}{\sqrt{n_L}} \sum_{i=1}^{n_L} \partial_{\tilde{\theta}_p} \tilde{\alpha}^{(L)}_{i}(x;\theta) \dot{\sigma}(\tilde{\alpha}^{(L)}_i(x;\theta)) W^{(L)}_{ik}.$$ By the induction hypothesis, the contribution of the parameters $\tilde{\theta}$ to the neural tangent kernel $\Theta^{(L+1)}_{kk'}(x, x')$ therefore converges as $n_1, ..., n_{L-1} \to \infty$: $$\begin{aligned}
&\frac{1}{n_L}\! \sum_{i, i'=1}^{n_L}\! \Theta^{(L)}_{ii'}(x, x') \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_i(x;\theta)\!\right) \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_{i'}(x';\theta)\!\right) W^{(L)}_{ik}W^{(L)}_{i'k'} \\
&\!\to \frac{1}{n_L}\!\sum_{i=1}^{n_L}\! \Theta^{(L)}_\infty(x, x') \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_i(x;\theta)\!\right) \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_i(x';\theta)\!\right) W^{(L)}_{ik}W^{(L)}_{ik'}\end{aligned}$$ By the law of large numbers, as $n_L \to \infty$, this tends to its expectation which is equal to $$\Theta^{(L)}_\infty(x, x') \dot{\Sigma}^{(L+1)}(x, x') \delta_{kk'}.$$ It is then easy to see that the second part of the neural tangent kernel, the sum over $W^{(L)}$ and $b^{(L)}$ converges to $\Sigma^{(L+1)} \delta_{kk'}$ as $n_1, ..., n_L \to \infty$.
Asymptotics during Training
---------------------------
Given a training direction $t \mapsto d_t \in \mathcal{F}$, a neural network is trained in the following manner: the parameters $\theta_p$ are initialized as iid $\mathcal{N}(0, 1)$ and follow the differential equation: $$\partial_t \theta_p(t) = \left< \partial_{\theta_p} F^{(L)}, d_t \right>_{p^{in}}.$$ In this context, in the infinite-width limit, the NTK stays constant during training:
\[thm:conv-ntk-training\] Assume that $ \sigma $ is a Lipschitz, twice differentiable nonlinearity function, with bounded second derivative. For any $T$ such that the integral $\int_0^T \| d_t \|_{p^{in}} dt$ stays stochastically bounded, as $n_1, ..., n_{L-1} \to \infty$ sequentially, we have, uniformly for $t\in[0, T]$, $$\Theta^{(L)}(t) \to \Theta^{(L)}_\infty \otimes Id_{n_L}.$$ As a consequence, in this limit, the dynamics of $f_\theta$ is described by the differential equation $$\begin{aligned}
\partial_t f_{\theta(t)} = \Phi_{\Theta^{(L)}_\infty \otimes Id_{n_L}} \left( \left<d_{t}, \cdot \right>_{p^{in}} \right).\end{aligned}$$
As in the previous theorem, the proof is by induction on the depth of the network. When $L=1$, the neural tangent kernel does not depend on the parameters, it is therefore constant during training.
For the induction step, we again split an $L+1$ network into a network of depth $L$ with parameters $\tilde{\theta}$ and top layer connection weights $W^{(L)}$ and bias $b^{(L)}$. The smaller network follows the training direction $$d'_{t} = \dot{\sigma}\left(\tilde{\alpha}^{(L)}(t)\right) \left(\frac{1}{\sqrt{n_L}}W^{(L)}(t)\right)^T d_t \label{eq:direct-small-network}$$ for $i=1, \ldots, n_L$, where the function $\tilde{\alpha}^{(L)}_i(t) $ is defined as $ \tilde{\alpha}^{(L)}_i(\cdot ; \theta(t))$. We now want to apply the induction hypothesis to the smaller network. For this, we need to show that $ \int_{0}^{T} \lVert d'_t \rVert_{p^{in}} \mathrm{d} t $ is stochastically bounded as $ n_1, \ldots, n_L \to \infty $. Since $ \sigma $ is a $c$-Lipschitz function, we have that $$\lVert d'_t \rVert_{p^{in}} \leq c \lVert \frac{1}{\sqrt{n_L}} W^{(L)} (t) \rVert_{op}
\lVert d_t \rVert_{p^{in}}.$$ To apply the induction hypothesis, we now need to bound $ \lVert \frac{1}{\sqrt{n_L}} W^{(L)} (t) \rVert_{op} $. For this, we use the following lemma, which is proven in Appendix \[Appendix-3\] below:
\[lem:control-w\] With the setting of Theorem \[thm:conv-ntk-training\], for a network of depth $ L + 1$, for any $ \ell =1, \ldots, L $, we have the convergence in probability: $$\lim_{n_L \to \infty} \cdots \lim_{n_1 \to \infty}
\sup_{t \in [0, T]} \lVert \frac{1}{\sqrt{n_\ell}} \left( W^{(\ell)}(t) - W^{(\ell)}(0) \right) \rVert_{op} = 0$$
From this lemma, to bound $ \lVert \frac{1}{\sqrt{n_L}} W^{(L)} (t) \rVert_{op} $, it is hence enough to bound $ \lVert \frac{1}{\sqrt{n_L}} W^{(L)} (0) \rVert_{op} $. From the law of large numbers, we obtain that the norm of each of the $ n_{L + 1} $ rows of $ W^{(L)} (0) $ is bounded, and hence that $ \lVert \frac{1}{\sqrt{n_L}} W^{(L)} (0) \rVert_{op} $ is bounded (keep in mind that $ n_{L + 1} $ is fixed, while $ n_1, \ldots, n_L $ grow). From the above considerations, we can apply the induction hypothesis to the smaller network, yielding, in the limit as $n_1, \ldots, n_L \to \infty$ (sequentially), that the dynamics is governed by the constant kernel $ \Theta^{(L)}_\infty $: $$\partial_t \tilde{\alpha}^{(L)}_{i}(t) = \frac{1}{\sqrt{n_L}} \Phi_{\Theta^{(L)}_\infty} \left( \left< \dot{\sigma}\left(\tilde{\alpha}^{(L)}_i(t)\right) \left(W^{(L)}_i(t)\right)^T d_t, \cdot \right>_{p^{in}} \right) .$$ At the same time, the parameters of the last layer evolve according to $$\begin{aligned}
\partial_t W^{(L)}_{ij}(t) &= \frac{1}{\sqrt{n_L}} \left< \alpha^{(L)}_i(t), d_{t, j} \right>_{p^{in}}.\end{aligned}$$
We want to give an upper bound on the variation of the weights columns $W_i^{(L)}(t)$ and of the activations $\tilde{\alpha}^{(L)}_{i}(t)$ during training in terms of $L^2$-norm and $p^{in}$-norm respectively. Applying the Cauchy-Schwarz inequality for each $ j $, summing and using $ \partial_t || \cdot || \leq || \partial_t \cdot || $), we have $$\begin{aligned}
\partial_t \left\lVert W^{(L)}_{i}(t) - W^{(L)}_{i}(0) \right\rVert_2 &\leq \frac{1}{\sqrt{n_L}} ||\alpha^{(L)}_i(t)||_{p^{in}} ||d_t||_{p^{in}}. \end{aligned}$$
Now, observing that the operator norm of $\Phi_{\Theta_\infty^{(L)}}$ is equal to $\vert \vert \Theta_\infty^{(L)}\vert \vert_{op}$, defined in the introduction of Appendix \[sec:proofs\], and using the Cauchy-Schwarz inequality, we get $$\begin{aligned}
\partial_t \left\lVert \tilde{\alpha}^{(L)}_{i}(t) - \tilde{\alpha}^{(L)}_{i}(0) \right\rVert_{p^{in}} &\leq \frac{1}{\sqrt{n_L}} \left\lVert \Theta^{(L)}_\infty \right\rVert_{op} \left\lVert \dot{\sigma}\left(\tilde{\alpha}^{(L)}_i(t)\right) \right\rVert_\infty \left\lVert W^{(L)}_i(t)\right\rVert_2 \left\lVert d_t \right\rVert_{p^{in}},\end{aligned}$$ where the sup norm $ \lVert \cdot \rVert_{\infty} $ is defined by $\left\lVert f \right\lVert_\infty = \sup_x | f(x) |.$
To bound both quantities simultaneously, study the derivative of the quantity $$A(t) = ||\alpha^{(L)}_i(0)||_{p^{in}} + c \left\lVert \tilde{\alpha}^{(L)}_{i}(t) - \tilde{\alpha}^{(L)}_{i}(0) \right\rVert_{p^{in}} + || W^{(L)}_i(0) ||_2 + \left\lVert W^{(L)}_{i}(t) - W^{(L)}_{i}(0) \right\rVert_2.$$ We have $$\begin{aligned}
\partial_t A(t) &\leq \frac{1}{\sqrt{n_L}} \left( c^2 \left\lVert \Theta^{(L)}_\infty \right\rVert_{op} \left\lVert W^{(L)}_i(t)\right\rVert_2 + ||\alpha^{(L)}_i(t)||_{p^{in}} \right) ||d_t||_{p^{in}} \\
&\leq \frac{\max\{c^2 \|\Theta^{(L)}_\infty \|_{op} , 1\} }{\sqrt{n_L}}\|d_t \|_{p^{in}} A(t),\end{aligned}$$ where, in the first inequality, we have used that $ | \dot{\sigma} | \leq c $ and, in the second inequality, that the sum $\lVert W^{(L)}_i(t) \rVert_2 + ||\alpha^{(L)}_i(t)||_{p^{in}}$ is bounded by $A(t)$. Applying Grönwall’s Lemma, we now get $$A(t) \leq A(0) \exp\left(\frac{\max\{c^2 \|\Theta^{(L)}_\infty \|_{op} , 1\} }{\sqrt{n_L}} \int_0^t \|d_s \|_{p^{in}} ds\right).$$ Note that $ \|\Theta^{(L)}_\infty \|_{op}$ is constant during training. Clearly the value inside of the exponential converges to zero in probability as $n_L \to \infty$ given that the integral $\int_0^t \|d_t \|_{p^{in}} ds$ stays stochastically bounded. The variations of the activations $\left\lVert \tilde{\alpha}^{(L)}_{i}(t) - \tilde{\alpha}^{(L)}_{i}(0) \right\rVert_{p^{in}}$ and weights $\left\lVert W^{(L)}_{i}(t) - W^{(L)}_{i}(0) \right\rVert_2$ are bounded by $c^{-1}(A(t) - A(0))$ and $A(t) - A(0)$ respectively, which converge to zero at rate $O\left(\frac{1}{\sqrt{n_L}}\right)$.
We can now use these bounds to control the variation of the NTK and to prove the theorem. To understand how the NTK evolves, we study the evolution of the derivatives with respect to the parameters. The derivatives with respect to the bias parameters of the top layer $\partial_{b^{(L)}_j}f_{\theta, j'}$ are always equal to $\delta_{jj'}$. The derivatives with respect to the connection weights of the top layer are given by $$\partial_{W^{(L)}_{ij}}f_{\theta, j'}(x) = \frac{1}{\sqrt{n_L}} \alpha^{(L)}_i(x ;\theta) \delta_{jj'}.$$ The pre-activations $\tilde{\alpha}^{(L)}_i$ evolve at a rate of $\frac{1}{\sqrt{n_L}}$ and so do the activations $\alpha^{(L)}_i$. The summands $\partial_{W^{(L)}_{ij}}f_{\theta, j'}(x) \otimes \partial_{W^{(L)}_{ij}}f_{\theta, j''}(x')$ of the NTK hence vary at rate of $n_L^{-3/2}$ which induces a variation of the NTK of rate $\frac{1}{\sqrt{n_L}}$.
Finally let us study the derivatives with respect to the parameters of the lower layers $$\partial_{\tilde{\theta}_k} f_{\theta, j}(x) = \frac{1}{\sqrt{n_L}} \sum_{i=1}^{n_L} \partial_{\tilde{\theta}_k} \tilde{\alpha}^{(L)}_{i}(x;\theta) \dot{\sigma}\left(\tilde{\alpha}^{(L)}_i(x;\theta)\right) W^{(L)}_{ij}.$$ Their contribution to the NTK $\Theta^{(L+1)}_{jj'}(x, x')$ is $$\begin{aligned}
&\frac{1}{n_L}\! \sum_{i, i'=1}^{n_L}\! \Theta^{(L)}_{ii'}(x, x') \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_i(x;\theta)\!\right) \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_{i'}(x';\theta)\!\right) W^{(L)}_{ij}W^{(L)}_{i'j'}.\end{aligned}$$ By the induction hypothesis, the NTK of the smaller network $\Theta^{(L)}$ tends to $\Theta^{(L)}_\infty \delta_{ii'}$ as $n_1, ..., n_{L-1} \to \infty$. The contribution therefore becomes $$\begin{aligned}
&\frac{1}{n_L}\! \sum_{i=1}^{n_L}\! \Theta^{(L)}_\infty(x, x') \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_i(x;\theta)\!\right) \dot{\sigma}\!\left(\!\tilde{\alpha}^{(L)}_{i}(x';\theta)\!\right) W^{(L)}_{ij}W^{(L)}_{ij'}.\end{aligned}$$ The connection weights $W^{(L)}_{ij}$ vary at rate $\frac{1}{\sqrt{n_L}}$, inducing a change of the same rate to the whole sum. We simply have to prove that the values $\dot{\sigma}(\tilde{\alpha}^{(L)}_i(x;\theta))$ also change at rate $\frac{1}{\sqrt{n_L}}$. Since the second derivative of $ \sigma $ is bounded, we have that $$\partial_t \left( \dot{\sigma}\left(\tilde{\alpha}^{(L)}_i(x;\theta(t))\right) \right) = O\left( \partial_t \tilde{\alpha}^{(L)}_i(x;\theta(t)) \right).$$ Since $ \partial_t \tilde{\alpha}^{(L)}_i(x;\theta(t)) $ goes to zero at a rate $\frac{1}{\sqrt{n_L}}$ by the bound on $ A(t) $ above, this concludes the proof.
It is somewhat counterintuitive that the variation of the activations of the hidden layers $\alpha^{(\ell)}_i$ during training goes to zero as the width becomes large[^4]. It is generally assumed that the purpose of the activations of the hidden layers is to learn “good” representations of the data during training. However note that even though the variation of each individual activation shrinks, the number of neurons grows, resulting in a significant collective effect. This explains why the training of the parameters of each layer $\ell$ has an influence on the network function $f_\theta$ even though it has asymptotically no influence on the individual activations of the layers $\ell'$ for $\ell<\ell'<L$.
A Priori Control during Training {#Appendix-3}
--------------------------------
The goal of this section is to prove Lemma \[lem:control-w\], which is a key ingredient in the proof of Theorem \[thm:conv-ntk-training\]. Let us first recall it:
\[lem:control-w\] With the setting of Theorem \[thm:conv-ntk-training\], for a network of depth $ L + 1$, for any $ \ell =1, \ldots, L $, we have the convergence in probability: $$\lim_{n_L \to \infty} \cdots \lim_{n_1 \to \infty}
\sup_{t \in [0, T]} \lVert \frac{1}{\sqrt{n_\ell}} \left( W^{(\ell)}(t) - W^{(\ell)}(0) \right) \rVert_{op} = 0$$
We prove the lemma for all $ \ell =1, \ldots, L $ simultaneously, by expressing the variation of the weights $ \frac{1}{\sqrt{n_\ell}} W^{(\ell)} $ and activations $ \frac{1}{\sqrt{n_\ell}} \tilde{\alpha}^{(\ell)} $ in terms of ‘back-propagated’ training directions $ d^{(1)}, \ldots, d^{(L)} $ associated with the lower layers and the NTKs of the corresponding subnetworks:
1. At all times, the evolution of the preactivations and weights is given by: $$\begin{aligned}
\partial_{t}\tilde{\alpha}^{(\ell)} & =\Phi_{\Theta^{(\ell)}} \left( <d_{t}^{(\ell)},\cdot>_{p^{in}} \right) \\
\partial_{t}W^{(\ell)} & =\frac{1}{\sqrt{n_{\ell}}}<\alpha^{(\ell)},d_{t}^{(\ell+1)}>_{p^{in}}, \end{aligned}$$ where the layer-wise training directions $ d^{(1)}, \ldots, d^{(L)} $ are defined recursively by $$\begin{aligned}
d_{t}^{\left(\ell\right)} & =\begin{cases}
d_{t} & \text{ if }\ell=L+1\\
\dot{\sigma}\left(\tilde{\alpha}^{\left(\ell\right)}\right)\left(\frac{1}{\sqrt{n_{\ell}}}W^{\left(\ell\right)}\right)^{T}d_{t}^{\left(\ell+1\right)} & \text{ if }\ell\leq L,
\end{cases}\end{aligned}$$ and where the sub-network NTKs $ \Theta^{(\ell)}$ satisfy $$\begin{aligned}
\Theta^{(1)} & =\left[\left[\frac{1}{\sqrt{n_{0}}}\alpha^{(0)}\right]^{T}\left[\frac{1}{\sqrt{n_{0}}}\alpha^{(0)}\right]\right]\otimes Id_{n_{\ell}}+\beta^{2}\otimes Id_{n_{\ell}}\\
\Theta^{(\ell+1)} & =\frac{1}{\sqrt{n_{\ell}}}W^{(\ell)}\dot{\sigma}(\tilde{\alpha}^{(\ell)})\Theta^{(\ell)}\dot{\sigma}(\tilde{\alpha}^{(\ell)})\frac{1}{\sqrt{n_{\ell}}}W^{(\ell)}\\
& +\left[\left[\frac{1}{\sqrt{n_{\ell}}}\alpha^{(\ell)}\right]^{T}\left[\frac{1}{\sqrt{n_{\ell}}}\alpha^{(\ell)}\right]\right]\otimes Id_{n_{\ell}}+\beta^{2}\otimes Id_{n_{\ell}}.\end{aligned}$$
2. Set $ w^{(k)} (t) := \left\Vert \frac{1}{\sqrt{n_{k}}}W^{(k)} (t)\right\Vert _{op} $ and $a^{(k)}\left(t\right):=\left\Vert \frac{1}{\sqrt{n_{k}}}\alpha^{\left(k\right)}\left(t\right)\right\Vert _{p^{in}}$. The identities of the previous step yield the following recursive bounds: $$\left\Vert d_{t}^{(\ell)}\right\Vert _{p^{in}}\le c w^{(\ell)}(t)\left\Vert d_{t}^{(\ell+1)}\right\Vert _{p^{in}},$$ where $c$ is the Lipschitz constant of $\sigma$. These bounds lead to $$\left\Vert d_{t}^{(\ell)}\right\Vert _{p^{in}}\leq c^{L+1-\ell}\prod_{k=\ell}^{L}w^{(k)}(t)\left\Vert d_{t}\right\Vert _{p^{in}}.$$ For the subnetworks NTKs we have the recursive bounds $$\begin{aligned}
\|\Theta^{(1)}\|_{op} & \le(a^{(0)}(t))^{2}+\beta^{2}.\\
\|\Theta^{(\ell+1)}\|_{op} & \le c^{2} (w^{(\ell)}(t) )^2
\| \Theta^{(\ell)} \|_{op}+(a^{(\ell)}(t))^{2}+\beta^{2},\end{aligned}$$ which lead to $$\|\Theta^{(\ell+1)}\|_{op}\leq\mathcal{P}\left(a^{(1)},\ldots,a^{(\ell)},w^{(1)},\ldots,w^{(\ell)}\right),$$ where $ \mathcal P $ is a polynomial which only depends on $ \ell, c, \beta $ and $ p^{in} $.
3. Set $$\begin{aligned}
\tilde{a}^{(k)}\left(t\right) & :=\left\Vert \frac{1}{\sqrt{n_{k}}}\left(\tilde{\alpha}^{\left(k\right)}\left(t\right)-\tilde{\alpha}^{\left(k\right)}\left(0\right)\right)\right\Vert _{p^{in}}\\
\tilde{w}^{(k)}\left(t\right) & :=\left\Vert \frac{1}{\sqrt{n_{k}}}\left(W^{\left(k\right)}\left(t\right)-W^{\left(k\right)}\left(0\right)\right)\right\Vert _{op}\end{aligned}$$ and define $$\begin{aligned}
A\left(t\right)=\sum_{k=1}^{L}a^{\left(k\right)}\left(0\right)+c\tilde{a}^{\left(k\right)}\left(t\right)+w^{\left(k\right)}\left(0\right)+\tilde{w}^{\left(k\right)}\left(t\right).\end{aligned}$$ Since $a^{\left(k\right)}\left(t\right)\leq a^{\left(k\right)}\left(0\right)+c\tilde{a}^{\left(k\right)}\left(t\right)$ and $w^{\left(k\right)}\left(t\right)\leq w^{\left(k\right)}\left(0\right)+\tilde{w}^{\left(k\right)}\left(t\right)$, controlling $A\left(t\right)$ will enable us to control the $a^{\left(k\right)}\left(t\right)$ and $w^{\left(k\right)}\left(t\right)$. Using the formula at the beginning of the first step, we obtain $$\begin{aligned}
\partial_{t}\tilde{a}^{\left(\ell\right)}\left(t\right) & \le\frac{1}{\sqrt{n_{\ell}}}\|\Theta^{(\ell)}(t)\|_{op}\|d_{t}^{(\ell)}\|_{p^{in}}\\
\partial_{t}\tilde{w}^{\left(\ell\right)}\left(t\right) & \le\frac{1}{\sqrt{n_{\ell}}}a^{\left(\ell\right)}\left(t\right)\|d_{t}^{(\ell+1)}\|_{p^{in}}.\end{aligned}$$ This allows one to bound the derivative of $A\left(t\right)$ as follows: $$\partial_{t}A\left(t\right)\le\sum_{\ell=1}^{L}\frac{c}{\sqrt{n_{\ell}}}\|\Theta^{(\ell)}(t)\|_{op}\|d_{t}^{(\ell)}\|_{p^{in}}+\frac{1}{\sqrt{n_{\ell}}}a^{\left(\ell\right)}\left(t\right)\|d_{t}^{(\ell+1)}\|_{p^{in}}.$$ Using the polynomial bounds on $\|\Theta^{(\ell)}(t)\|_{op}$ and $\|d_{t}^{(\ell+1)}\|_{p^{in}}$ in terms of the $a^{\left(k\right)}$ and $w^{\left(k\right)}$ for $k=1,\ldots\ell$ obtained in the previous step, we get that $$\text{\ensuremath{\partial_{t}A\left(t\right)\leq\frac{1}{\sqrt{\min\left\{ n_{1},\ldots,n_{L}\right\} }}\mathcal{Q}\left(w^{\left(1\right)}\left(t\right),\ldots,w^{\left(L\right)}\left(t\right),a^{\left(1\right)}\left(t\right),\ldots,a^{\left(L\right)}\left(t\right)\right)\|d_{t}\|_{p^{in}},}}$$ where the polynomial $Q$ only depends on $L,c,\beta$ and $p^{in}$ and has positive coefficients. As a result, we can use $a^{\left(k\right)}\left(t\right)\leq a^{\left(k\right)}\left(0\right)+c\tilde{a}^{\left(k\right)}\left(t\right)$ and $w^{\left(k\right)}\left(t\right)\leq w^{\left(k\right)}\left(0\right)+\tilde{w}^{\left(k\right)}\left(t\right)$ to get the polynomial bound $$\partial_{t}A\left(t\right)\leq\frac{1}{\sqrt{\min\left\{ n_{1},\ldots,n_{L}\right\} }}\tilde{\mathcal{Q}}\left(A\left(t\right)\right)\|d_{t}\|_{p^{in}}.$$
4. Let us now observe that $A\left(0\right)$ is stochastically bounded as we take the sequential limit $\lim_{n_{L}\to\infty}\cdots\lim_{n_{1}\to\infty}$ as in the statement of the lemma. In this limit, we indeed have that $w^{\left(\ell\right)}$and $a^{\left(\ell\right)}$ are convergent: we have $w^{\left(\ell\right)}\to0$, while $a^{\left(\ell\right)}$ converges by Proposition \[prop:output\_limit\].
The polynomial control we obtained on the derivative of $A\left(t\right)$ now allows one to use (a nonlinear form of, see e.g. [@dragomir]) Grönwall’s Lemma: we obtain that $A\left(t\right)$ stays uniformly bounded on $\left[0,\tau\right]$ for some $\tau=\tau\left(n_{1},\ldots,n_{L}\right)>0$, and that $\tau\to T$ as $\min\left(n_{1},\ldots,n_{L}\right)\to\infty$, owing to the $\frac{1}{\sqrt{\min\left\{ 1,\ldots,n_{L}\right\} }}$in front of the polynomial. Since $A\left(t\right)$ is bounded, the differential bound on $A\left(t\right)$ gives that the derivative $\partial_{t}A\left(t\right)$ converges uniformly to $0$ on $\left[0,\tau\right]$ for any $\tau<T$, and hence $A\left(t\right)\to A\left(0\right)$. This concludes the proof of the lemma.
Positive-Definiteness of $ \Theta_\infty^{(L)} $ {#Appendix-4}
------------------------------------------------
This subsection is devoted to the proof of Proposition \[prop:pos-def\], which we now recall:
\[prop:pos-def\]For a non-polynomial Lipschitz nonlinearity $\sigma$, for any input dimension $n_{0}$, the restriction of the limiting NTK $\Theta_{\infty}^{(L)}$ to the unit sphere $\mathbb{S}^{n_{0}-1}=\{x\in\mathbb{R}^{n_{0}}:x^{T}x=1\}$ is positive-definite if $L\geq2$.
A key ingredient for the proof of Proposition \[prop:pos-def\] is the following Lemma, which comes from [@Daniely].
\[lem:daniely\]Let $\hat{\mu}:\left[-1,1\right]\to\mathbb{R}$ denote the dual of a Lipschitz function $\mu:\mathbb{R}\to\mathbb{R}$, defined by $\hat{\mu}\left(\rho\right)=\mathbb{E}_{\left(X,Y\right)}\left[\mu\left(X\right)\mu\left(Y\right)\right]$ where $\left(X,Y\right)$ is a centered Gaussian vector of covariance $\Sigma$, with $$\Sigma=\begin{pmatrix}1 & \rho\\
\rho & 1
\end{pmatrix}.$$ If the expansion of $\mu$ in Hermite polynomials $\left(h_{i}\right)_{i\geq0}$ is given by $\mu=\sum_{i=0}^{\infty}a_{i}h_{i}$, we have $$\hat{\mu}\left(\rho\right)=\sum_{i=0}^{\infty}a_{i}^{2}\rho^{i}.$$
The other key ingredient for proving Proposition \[prop:pos-def\] is the following theorem, which is a slight reformulation of Theorem 1(b) in [@Gneiting], which itself is a generalization of a classical result of Schönberg:
\[thm:schoenberg\]For a function $f:\text{\ensuremath{\left[-1,1\right]}}\to\mathbb{R}$ with $f\left(\rho\right)=\sum_{n=0}^{\infty}b_{n}\rho^{n}$, the kernel $K_{f}^{\left(n_{0}\right)}:\mathbb{S}^{n_{0}-1}\times\mathbb{S}^{n_{0}-1}\to\mathbb{R}$ defined by $$K_{f}^{\left(n_{0}\right)}\left(x,x'\right)=f\left(x^{T}x'\right)$$ is positive-definite for any $n_{0}\geq1$ if and only if the coefficients $b_{n}$ are strictly positive for infinitely many even and infinitely many odd integers $n$.
With Lemma \[lem:daniely\] and Theorem \[thm:schoenberg\] above, we are now ready to prove Proposition \[prop:pos-def\].
We first decompose the limiting NTK $\Theta^{\left(L\right)}$ recursively, relate its positive-definiteness to that of the activation kernels, then show that the positive-definiteness of the activation kernels at level $2$ implies that of the higher levels, and finally show the positive-definiteness at level $2$ using Lemma \[lem:daniely\] and Theorem \[thm:schoenberg\]:
1. Observe that for any $L\geq1$, using the notation of Theorem \[thm:convergence\_NTK\_initialization\], we have $$\Theta^{\left(L+1\right)}=\dot{\Sigma}^{\left(L\right)}\Theta^{\left(L\right)}+\Sigma^{\left(L+1\right)}.$$ Note that the kernel $ \dot{\Sigma}^{\left(L\right)}\Theta^{\left(L\right)} $ is positive semi-definite, being the product of two positive semi-definite kernels. Hence, if we show that $\Sigma^{\left(L+1\right)}$ is positive-definite, this implies that $\Theta^{\left(L+1\right)}$ is positive-definite.
2. By definition, with the notation of Proposition \[prop:output\_limit\] we have $$\Sigma^{\left(L+1\right)}\left(x,x'\right)=\mathbb{E}_{f\sim\mathcal{N}\left(\text{0,}\Sigma^{\left(L\right)}\right)}\left[\sigma\left(f\left(x\right)\right)\sigma\left(f\left(x'\right)\right)\right]+\beta^{2}.$$ This gives, for any collection of coefficients $c_{1},\ldots,c_{d}\in\mathbb{R}$ and any pairwise distinct $x_{1},\ldots,x_{d}\in\mathbb{R}^{n_{0}}$, that $$\sum_{i,j=1}^{d}c_{i}c_{j}\Sigma^{\left(L+1\right)}\left(x_{i},x_{j}\right)=\mathbb{E}\left[\left(\sum_{i}c_{i}\sigma\left(f\left(x_{i}\right)\right)\right)^{2}\right]+\left(\beta\sum_{i}c_{i}\right)^{2}.$$ Hence the left-hand side only vanishes if $\sum c_{i}\sigma\left(f\left(x_{i}\right)\right)$ is almost surely zero. If $\Sigma^{\left(L\right)}$ is positive-definite, the Gaussian $\left(f\left(x_{i}\right)\right)_{i=1,\ldots d}$ is non-degenerate, so this only occurs when $c_{1}=\cdots=c_{d}=0$ since $\sigma$ is assumed to be non-constant. This shows that the positive-definiteness of $\Sigma^{\left(L+1\right)}$ is implied by that of $\Sigma^{\left(L\right)}$. By induction, if $\Sigma^{\left(2\right)}$ is positive-definite, we obtain that all $\Sigma^{\left(L\right)}$ with $L\geq2$ are positive-definite as well. By the first step this hence implies that $\Theta^{\left(L\right)}$ is positive-definite as well.
3. By the previous steps, to prove the proposition, it suffices to show the positive-definitess of $\Sigma^{\left(2\right)}$ on the unit sphere $\mathbb{S}^{n_{0}-1}$. We have $$\Sigma^{\left(2\right)}\left(x,x'\right)=\mathbb{E}_{\left(X,Y\right)\sim\mathcal{N}\left(0,\tilde{\Sigma}\right)}\left[\sigma\left(X\right)\sigma\left(Y\right)\right]+\beta^{2}$$ where $$\tilde{\Sigma}=\left(\begin{array}{cc}
\frac{1}{n_{0}}+\beta^{2} & \frac{1}{n_{0}}x^{T}x'+\beta^{2}\\
\frac{1}{n_{0}}x^{T}x+\beta^{2} & \frac{1}{n_{0}}+\beta^{2}
\end{array}\right).$$ A change of variables then yields $$\mathbb{E}_{\left(X,Y\right)\sim\mathcal{N}\left(0,\tilde{\Sigma}\right)}\left[\sigma\left(X\right)\sigma\left(Y\right)\right]+\beta^{2}=\hat{\mu}\left(\frac{n_{0}\beta^{2}+x^{T}x'}{n_{0}\beta^{2}+1}\right)+\beta^{2},\label{eq:from-sigma-to-mu-hat}$$ where $\hat{\mu}:\left[-1,1\right]\to\mathbb{R}$ is the dual in the sense of Lemma \[lem:daniely\] of the function $\mu:\mathbb{R}\to\mathbb{R}$ defined by $\mu\left(x\right)=\sigma\left(x\sqrt{\frac{1}{n_{0}}+\beta^{2}}\right)$.
4. Writing the expansion of $\mu$ in Hermite polynomials $\left(h_{i}\right)_{i\geq0}$ $$\mu=\sum_{i=0}^{\infty}a_{i}h_{i},$$ we obtain that $\hat{\mu}$ is given by the power series $$\hat{\mu}\left(\rho\right)=\sum_{i=0}^{\infty}a_{i}^{2}\rho^{i},$$ Since $\sigma$ is non-polynomial, so is $\mu$, and as a result, there is an infinite number of nonzero $a_{i}$’s in the above sum.
5. Using (\[eq:from-sigma-to-mu-hat\]) above, we obtain that $$\Sigma^{\left(2\right)}\left(x,x'\right)=\nu\left(x^{T}x'\right),$$ where $\nu:\mathbb{R}\to\mathbb{R}$ is defined by $$\nu\left(\rho\right)=\beta^{2}+\sum_{i=0}^{\infty}a_{i}\left(\frac{n_{0}\beta^{2}+\rho}{n_{0}\beta^{2}+1}\right)^{i},$$ where the $a_{i}$’s are the coefficients of the Hermite expansion of $\mu$. Now, observe that by the previous step, the power series expansion of $\nu$ contains both an infinite number of nonzero even terms and an infinite number of nonzero odd terms. This enables one to apply Theorem \[thm:schoenberg\] to obtain that $\Sigma^{\left(2\right)}$ is indeed positive-definite, thereby concluding the proof.
Using similar techniques to the one applied in the proof above, one can show a converse to Proposition \[prop:pos-def\]: if the nonlinearity $\sigma$ is a polynomial, the corresponding NTK $\Theta^{\left(2\right)}$ is not positive-definite $\mathbb{S}^{n_{0}-1}$ for certain input dimensions $n_{0}$.
[^1]: While these smoothness assumptions greatly simplify the proofs of our results, they do not seem to be strictly needed for the results to hold true.
[^2]: The classical example is for data following a Gaussian distribution $\mathcal{N}(0, Id_{n_0})$: as the dimension $n_0$ grows, all data points have approximately the same norm $\sqrt{n_0}$.
[^3]: It can be observed numerically, that if we choose $\beta=1.0$ instead of our recommended $0.1$, the gap between the first and the second principal component is about ten times bigger, which makes training more difficult.
[^4]: As a consequence, the pre-activations stay Gaussian during training as well, with the same covariance $\Sigma^{(\ell)}$.
|
---
abstract: 'We compute the arithmetic $\mathcal{L}$-invariants (of Greenberg–Benois) of twists of symmetric powers of $p$-adic Galois representations attached to Iwahori level Hilbert modular forms (under some technical conditions). Our method uses the automorphy of symmetric powers and the study of analytic Galois representations on $p$-adic families of automorphic forms over symplectic and unitary groups. Combining these families with some explicit plethysm in the representation theory of ${\operatorname{GL}}(2)$, we construct global Galois cohomology classes with coefficients in the symmetric powers and provide formulae for the $\mathcal{L}$-invariants in terms of logarithmic derivatives of Hecke eigenvalues.'
address:
- 'R. Harron: University of Wisconsin, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706'
- 'A. Jorza: University of Notre Dame, Department of Mathematics, 275 Hurley Hall, Notre Dame, IN 46556'
author:
- Robert Harron
- Andrei Jorza
bibliography:
- 'biblio.bib'
title: 'On symmetric power $\mathcal{L}$-invariants of Iwahori level Hilbert modular forms'
---
[^1]
Introduction {#introduction .unnumbered}
============
The $p$-adic interpolation of special values of $L$-functions has been critical to understanding their arithmetic, providing, for instance, the link between values of the Riemann zeta function and the class groups of cyclotomic fields. However, it can happen that the $p$-adic $L$-function of a motive $M$ vanishes at a point of interpolation despite the fact that the classical $L$-value is non-zero. To recuperate an interpolation property, one predicts that a *derivative* of the $p$-adic $L$-function is related to the $L$-value and one defines the (analytic) $\mathcal{L}$-invariant of $M$ as the ratio of these quantities. The name of the game then becomes to provide an *arithmetic* meaning for the $\mathcal{L}$-invariant and show that it is non-zero.
The phenomenon of $\mathcal{L}$-invariants first arose in the work of Ferrero–Greenberg [@FeG78] on $p$-adic Dirichlet $L$-functions. Later on, in their work [@mazur-tate-teitelbaum] on formulating a $p$-adic analogue of the Birch and Swinnerton-Dyer conjecture, Mazur–Tate–Teitelbaum encountered this behaviour in the $p$-adic $L$-function of an elliptic curve over ${\mathbb{Q}}$ with split multiplicative reduction at $p$. They conjectured a formula for the $\mathcal{L}$-invariant in terms of the $p$-adic Tate parameter of $E$ (and coined the term $\mathcal{L}$-invariant). This was proved in [@greenberg-stevens:p-adic-L] by Greenberg and Stevens by placing the elliptic curve in a Hida family, i.e. a $p$-adic family of modular forms of varying weight. Their proof proceeded in two steps: first, relate the Tate parameter to the derivative in the weight direction of the $U_p$-eigenvalue of the Hida family, and then use the functional equation of the two-variable $p$-adic $L$-function of the Hida family to relate this derivative to the analytic $\mathcal{L}$-invariant. We think of the Tate parameter formula as being a conjectural *arithmetic* $\mathcal{L}$-invariant and the Greenberg–Stevens method as first linking it to a derivative of a Hecke eigenvalue and then appealing to analytic properties of several-variable $p$-adic $L$-functions to connect with the analytic $\mathcal{L}$-invariant. In the case of a $p$-ordinary motive $M$, Greenberg used an in-depth study of the ordinary filtration to conjecture an arithmetic formula for the $\mathcal{L}$-invariant of $M$ in terms of its Galois cohomology ([@G94 Equation (23)]). In [@benois:L-invariant], Benois generalized this to the non-ordinary setting by passing to the category of $(\varphi,\Gamma)$-modules over the Robba ring and using triangulations. This article aims to establish the first step of the Greenberg–Stevens method for symmetric powers of Hilbert modular forms with respect to the Greenberg–Benois arithmetic $\mathcal{L}$-invariant; the second step being of a different nature (and the corresponding $p$-adic $L$-functions not known to exist!).
Before presenting the main results of this article, we introduce the trivial zero conjecture as formulated by Benois in [@benois:L-invariant]. Let $p$ be an odd prime and let $\rho:G_{{\mathbb{Q}}}\to {\operatorname{GL}}(n, \overline{{\mathbb{Q}}}_p)$ be a continuous Galois representation which is unramified at all but finitely many primes, and semistable at $p$. One has an $L$-function $L(\rho,s)=\prod_\ell L_\ell(\rho|_{G_{F_\ell}}, s)$ where $L_\ell(\rho|_{G_{{\mathbb{Q}}_\ell}},s) = \det(1-{\operatorname{Frob}}_\ell
\ell^{-s}|\rho^{I_{\ell}})^{-1}$ for $\ell\neq p$ (here ${\operatorname{Frob}}_\ell$ is the geometric Frobenius) while $L_p(\rho|_{G_{{\mathbb{Q}}_p}}, s) = \det(1-{\varphi}p^{-s}|{\operatorname{D}}_{{\operatorname{cris}}}(\rho|_{G_{{\mathbb{Q}}_p}}))^{-1}$. It is conjectured that $L(V,s)$ has meromorphic continuation to ${\mathbb{C}}$ and that there exist Gamma factors $\Gamma(\rho,s)$ and $\Gamma(\rho^*(1),s)$ such that $\Gamma(\rho,s)L(\rho,s)=\varepsilon(\rho,s)\Gamma(\rho^*(1),-s)L(\rho^*(1),-s)$ for an epsilon factor $\varepsilon(\rho,s)$ of the form $A\cdot B^s$. If $D\subset {\operatorname{D}}_{{\operatorname{st}}}(\rho|_{G_{{\mathbb{Q}}_p}})$ is a regular submodule (see §\[sect:regular submodules general\]), it is expected that there exists an analytic $p$-adic $L$-function $L_p(\rho, D, s)$ such that $$L_p(\rho,D,0)=\mathcal{E}(\rho,D)\frac{L(\rho,s)}{\Omega_\infty(\rho)}$$where $\mathcal{E}(\rho,D)$ is a product of Euler-like factors and $\Omega_\infty(\rho)$ is a transcendental period.
\[conjecture:zero\]If $\rho$ is critical, $L(\rho,0)\neq 0$, $L_p(\rho,D,s)$ has order of vanishing $e$ at $s=0$, and $D$ satisfies the conditions of [@benois:L-invariant §2.2.6] then $$\lim_{s\to 0}\frac{L_p(\rho,D,s)}{s^e}=(-1)^e\mathcal{L}(\rho,
D)\mathcal{E}^+(\rho,D)\frac{L(\rho,0)}{\Omega_\infty(\rho)}$$ where $\mathcal{E}^+(\rho,D)$ is defined in [@benois:L-invariant §2.3.2] and $\mathcal{L}(\rho,D)$ is the arithmetic $\mathcal{L}$-invariant defined in [@benois:L-invariant §2.2.2].
When $F$ is a number field and $\rho:G_F\to {\operatorname{GL}}(n,
\overline{{\mathbb{Q}}}_p)$ is a geometric Galois representation one may still formulate the above conjecture for ${\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho$ and a collection of regular submodules $D_v\subset {\operatorname{D}}_{{\operatorname{st}}}(\rho|_{G_{F_v}})$ for places $v\mid p$. In that case, it natural to follow Hida and define the arithmetic $\mathcal{L}$-invariant as $\mathcal{L}(\rho,\{D_v\})=\mathcal{L}({\operatorname{Ind}}_F^{{\mathbb{Q}}}, D)$ where $D=\bigoplus D_v$ is a regular submodule of ${\operatorname{D}}_{{\operatorname{st}}}(({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}})$.
We now describe the main results of this article. Let $F$ be a totally real field in which $p$ splits completely and let $\pi$ be a cohomological Hilbert modular form (cf. §\[sect:hmf def\]). Let $V_{2n}={\operatorname{Sym}}^{2n}\rho_{\pi,p}\otimes\det^{-n}\rho_{\pi,p}$ where $\rho_{\pi,p}$ is the $p$-adic representation of $G_F$ attached to $\pi$. In the following, when computing $\mathcal{L}$-invariants of $V_{2n}$, we will assume that the Bloch–Kato Selmer group $H^1_f(F, V_{2n})$ vanishes and technical condition C4 from §\[sect:benois-l-inv\]. We will also assume that $\pi_v$ is Iwahori spherical at $v\mid p$. Fixing a basis $e_1,e_2$ for ${\operatorname{D}}_{{\operatorname{st}}}(\rho_{\pi,p}|_{G_{F_v}})$, we fix a regular submodule $D_v\subset {\operatorname{D}}_{{\operatorname{st}}}(V_{2n}|_{G_{F_v}})$ as in §\[sect:regular submodules\]. We remark now that the most difficult case occurs when $V_{2n}$ is crystalline at $v$ and this article provides the first results in this case when $n>3$ (in the semistable cases, the result is either quite easy or, at least in the $p$-ordinary case, follows from the work of Hida [@hida:mazur-tate-teitelbaum]).
We begin with the case of $V_2$. Suppose that if $\pi_v$ for $v\mid p$ is an unramified principal series then the Satake parameters are distinct. Further suppose that the Hilbert eigenvariety $\mathcal{E}$ around $\pi$ and the refinement of $\pi_v$ giving the ordering $e_1,e_2$ is étale over the weight space around the point corresponding to $\pi$ (cf. §\[sect:sym2l\]). Finally, let $a_v$ be the analytic Hecke eigenvalue corresponding to the double coset $[{\operatorname{Iw}}{\operatorname{diag}}(p,1){\operatorname{Iw}}]$.
\[thmA\] Writing $a'_v$ for the derivative in the direction $(1,\ldots, 1; -1)$ in the weight space, we have $$\mathcal{L}(V_2, \{D_v\}) = \prod_{v\mid p}
\left(\frac{-2a_v'}{a_v}\right)$$
Such a formula for $p$-ordinary elliptic modular forms was obtained first by Hida [@hida:ad0] (under a condition on the Galois deformation ring) and then by the first author [@harron:thesis] (under the assumption $H^1_f({\mathbb{Q}}, V_2)$ used here). In [@mok:adjoint-L-invariant], Mok proved this result for finite slope elliptic modular forms.
For higher symmetric powers, we must assume $\pi$ is not CM, so that certain of its symmetric powers are actually cuspidal. When $F={\mathbb{Q}}$, the CM case has been dealt with in [@harron:CM] and [@harron-lei:CM].
For the case of $V_6$, we do as in [@harron:thesis; @harron:sym6-ordinary] and use the Ramakrishnan–Shahidi lift $\Pi$ of ${\operatorname{Sym}}^3\pi$ to ${\operatorname{GSp}}(4)$. Suppose that $\pi$ is not CM and that if $\pi_v$ for $v\mid p$ is an unramified principal series then the ratio of the Satake parameters is not in $\mu_{60}$ (this condition is necessary for the existence of global triangulations). Further suppose that the genus 2 Siegel–Hilbert eigenvariety $\mathcal{E}$ around $\Pi$ and the $p$-stabilization of $\Pi_v$ giving the ordering $e_1,e_2$ is étale over the weight space around the point corresponding to $\Pi$ (cf. §\[sect:sym6l\]). Finally, let $a_{v,1}$ and $a_{v,2}$ be the analytic Hecke eigenvalues corresponding to the double cosets $[{\operatorname{Iw}}{\operatorname{diag}}(1,p^{-1}, p^{-2}, p^{-1}){\operatorname{Iw}}]$ and $[{\operatorname{Iw}}{\operatorname{diag}}(1,1,p^{-1}, p^{-1}){\operatorname{Iw}}]$.
\[thmB\] If $\overrightarrow{u}=(u_1,u_2; u_0)$ is any direction in the weight space, i.e. $u_1\geq u_2\geq 0$, then $$\mathcal{L}(V_6, \{D_v\})=\prod_{v\mid p}
\left(\frac{-4\widetilde{\nabla}_{\overrightarrow{u}}a_{v,2}+3\widetilde{\nabla}_{\overrightarrow{u}}a_{v,1}}{u_1-2u_2}\right)$$ where we write $\widetilde{\nabla}_{\overrightarrow{u}}f =
(\nabla_{\overrightarrow{u}}f)/f$ for the logarithmic directional derivative of $f$ evaluated at the point above $\Pi$.
This generalizes the main result of [@harron:thesis; @harron:sym6-ordinary] which compute the arithmetic $\mathcal{L}$-invariant of $V_6$ in the case of $p$-ordinary elliptic modular forms.
The first computation of $\mathcal{L}$-invariants of $V_{2n}$ for general $n$ we present uses symplectic eigenvarieties and is, for now, conditional on the stabilization of the twisted trace formula (this is necessary for the construction of an analytic Galois representation). Suppose $\pi$ is not CM, and that for $v\mid p$ such that $\pi_v$ is unramified the ratio of the Satake parameters is not in $\mu_{\infty}$ (again, necessary for the existence of global triangulations). Suppose $\pi$ satisfies the hypotheses of Theorem \[t:automorphy of sym\] (2) and let $\Pi$ be a suitable twist of the cuspidal representation of ${\operatorname{GSp}}(2n, {\mathbb{A}}_F)$ from Theorem \[t:GL(2n+1) to Sp(2n)\]. Let $\mathcal{E}$ be Urban’s eigenvariety for ${\operatorname{GSp}}(2n)$ and let $a_{v,i}$ be the analytic Hecke eigenvalues from the proof of Lemma \[l:gsp eigenvariety\]. Suppose that the eigenvariety $\mathcal{E}$ is étale over the weight space at the $p$-stabilization of $\Pi$ corresponding to the ordering $e_1,e_2$.
\[thmC\] If $\overrightarrow{u}=(u_1,\ldots,u_n;u_0)$ is any direction in the weight space, then $$\mathcal{L}(V_{4n-2}, \{D_v\})=\prod_{v\mid
p}-\left(\frac{B_{n}\widetilde{\nabla}_{\overrightarrow{u}}a_{v,1}+B_{1}(\widetilde{\nabla}_{\overrightarrow{u}}a_{v,n-1}-2\widetilde{\nabla}_{\overrightarrow{u}}a_{v,n})+\sum_{i=2}^{n-1}B_{i}(\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i-1}-\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i})}{\sum_{i=1}^nu_iB_{n+1-i}}\right)$$ where we write $\displaystyle B_i=(-1)^i\binom{2n}{n+i}i$.
As mentioned earlier, the results of Hida [@hida:mazur-tate-teitelbaum] address the case where $V_{2n}$ is ordinary and semistable but not crystalline.
Our second computation of $\mathcal{L}$-invariants for $V_{2n}$ uses unitary groups, is also conditional on the stabilization of the twisted trace formula, and is more restrictive. It however has the advantage that work in progress of Eischen–Harris–Li–Skinner will provide several-variable $p$-adic $L$-functions for Hida families on unitary groups and thus the second step of the Greenberg–Stevens method may be closer at hand. As above suppose $\pi$ is not CM, and that for $v\mid p$ such that $\pi_v$ is unramified the ratio of the Satake parameters is not in $\mu_{\infty}$. Assume Conjecture \[t:GL(n) to U\_n\] and suppose $\pi$ satisfies the hypotheses of Theorem \[t:automorphy of sym\] (2) and Proposition \[p:sym hmf to unitary\] (the latter requires $\pi_v$ to be special at two finite places not above $p$). Let $E/F$ the CM extension and $\Pi$ the cuspidal representation of $U_{4n}({\mathbb{A}}_F)$ to which there is a transfer of a twist of ${\operatorname{Sym}}^{4n-1}\pi$ as in Proposition \[p:sym hmf to unitary\]. Let $\mathcal{E}$ be Chenevier’s eigenvariety and let $a_{v,i}$ be the analytic Hecke eigenvalues from the proof of Corollary \[c:unitary triangulation\]. Suppose that the eigenvariety $\mathcal{E}$ is étale over the weight space at the $p$-stabilization of $\Pi$ corresponding to the ordering $e_1,e_2$.
\[thmD\] If $\overrightarrow{u}=(u_1,\ldots,u_n;u_0)$ is any direction in the weight space, then $$\mathcal{L}(V_{8n-2}, \{D_v\})=\prod_{v\mid
p}\left(\frac{-\sum_{i=1}^{4n}(-1)^{i-1}\binom{4n-1}{i-1}\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i}}{\sum_{i=1}^{4n}(-1)^{i-1}\binom{4n-1}{i-1}u_i}\right)$$ and $$\mathcal{L}(V_{8n-6}, \{D_v\})=\prod_{v\mid
p}\left(\frac{-\sum_{i=1}^{4n}B_{i-1}\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i}}{\sum_{i=1}^{4n}u_iB_{i-1}}\right).$$ Here $B_i=B_{4n-1,4n-3,i}$ is the inverse Clebsch–Gordan coefficient of Proposition \[p:End to Sym\^n\], up to a scalar independent of $i$ given by $$B_i= (-1)^i\binom{4n-1}{i}((4n-1)^3-(4i+1)(4n-1)^2+(4i^2+2i)(4n-1)-2i^2).$$
The assumption that $\pi$ satisfy the hypotheses of Theorem \[t:automorphy of sym\] (2), i.e. that various ${\operatorname{Sym}}^{k}\pi$ be automorphic over $F$ is necessary for our computations. Ongoing work of Clozel and Thorne provides the automorphy of such symmetric powers when $k$ is small.
The paper is organized as follows. In Section \[sect:1\], we describe Benois’ definition of the arithmetic $\mathcal{L}$-invariant and triangulations in $p$-adic families. In Section \[sect:hmf\], we study Galois representations attached to Hilbert modular forms and functorial transfers to unitary and symplectic groups. Then, in Section \[sect:eigs\], we describe the unitary and symplectic eigenvarieties and global triangulations of certain analytic Galois representations. In section \[sect:l\], we prove Theorems \[thmA\] through \[thmD\]. Finally, the appendix discusses some plethysm for ${\operatorname{GL}}(2)$, relating the $B_{n,k,i}$ to inverse Clebsch–Gordon coefficients and proving an explicit formula for them.
Some basic notation {#sect:notation .unnumbered}
===================
Throughout this article, $p$ denotes a fixed odd prime. We will use $L$ to denote a finite extension of ${{\mathbb{Q}}_p}$. By a $p$-adic representation of a group $G$, we mean a continuous homomorphism $\rho:G\rightarrow{\operatorname{GL}}(V)$, where $V$ is a finite-dimensional vector space over $L$. Let $\mu_{p^\infty}$ denote the set of $p$-power roots of unity and let $\chi$ denote the $p$-adic cyclotomic character giving the action of whatever appropriate Galois group on it. Let $\Gamma$ denote the Galois group ${\operatorname{Gal}}({{\mathbb{Q}}_p}(\mu_{p^\infty})/{{\mathbb{Q}}_p})\cong{{\mathbb{Z}}_p}^\times$ and let $\gamma$ denote a fixed topological generator. If $F$ is a field, $G_F$ denotes its absolute Galois group and $H^\bullet(F,-)=H^\bullet(G_F,-)$ denotes it Galois cohomology.
(phi, Gamma)-modules and $\mathcal{L}$-invariants {#sect:1}
=================================================
This section contains a review of some pertinent content from [@benois:L-invariant] and [@liu:triangulation]. We refer the reader to these articles for further details.
(phi, Gamma)-modules over the Robba ring
----------------------------------------
For a real number $r$ with $0\leq r<1$ denote by $\mathcal{R}^r$ the set of power series $f(x)=\sum_{k\in {\mathbb{Z}}}a_kx^k$ holomorphic for $r\leq |x|_p<1$ with $a_k\in{{\mathbb{Q}}_p}$. Let $\mathcal{R}=\bigcup_{r<1}\mathcal{R}^r$ be the *Robba ring* (over ${{\mathbb{Q}}_p}$).
The Robba ring carries natural actions of a Frobenius, $\varphi$, and $\Gamma$ given as follows. If $f\in \mathcal{R}$, then $$(\varphi f)(x)=f((1+x)^p-1)$$ and for $\tau \in \Gamma$, $$(\tau f)(x)=f((1+x)^{\chi(\tau)}-1).$$
More generally, if $S$ is an affinoid algebra over ${{\mathbb{Q}}_p}$, define $\mathcal{R}_{S}=\mathcal{R}\widehat{\otimes}_{{{\mathbb{Q}}_p}} S$ (cf. [@liu:triangulation Proposition 2.1.5]). Extending the actions of $\varphi\otimes 1$ and $\tau\otimes1$ linearly, we get actions of $(\varphi,\Gamma)$ on $\mathcal{R}_S$; i.e. this should be thought of as the Robba ring over ${{\mathbb{Q}}_p}$ with coefficients in $S$.
A $(\varphi, \Gamma)$-module over $\mathcal{R}_L$ is a free $\mathcal{R}_L$-module $D_L$ of finite rank, equipped with a $\varphi$-semilinear Frobenius map $\varphi_{D_L}$ and a semilinear action of $\Gamma$ which commute with each other, such that the induced map $\varphi_{D_L}^*D_L=D_L\otimes_{\varphi}\mathcal{R}_L\to D_L$ is an isomorphism. More generally, a $(\varphi, \Gamma)$-module over $\mathcal{R}_S$ is a vector bundle $D_S$ (coherent, locally free sheaf) over $\mathcal{R}_S$ of finite rank, equipped with a semilinear Frobenius, $\varphi_{D_S}$, and a semilinear action of $\Gamma$, commuting with each other, such that $\varphi_{D_S}^\ast D_S\to
D_S$ is an isomorphism.
If $\delta:{{\mathbb{Q}}_p}^\times\to S^\times$ is a continuous character, define $\mathcal{R}_S(\delta)$ as the rank 1 $(\varphi,\Gamma)$-module $\mathcal{R}_S e_\delta$ with basis $e_\delta$ with $\varphi_{\mathcal{R}_S(\delta)}(x
e_\delta)=\varphi_{\mathcal{R}_S}(x) \delta(p)e_\delta$ and for $\tau \in \Gamma$, $\tau(x e_\delta) =
\tau(x)\delta(\chi(\tau))e_\delta$.
A $(\varphi,\Gamma)$-module $D_S$ is said to be *trianguline* if there exists an increasing (separated, exhaustive) filtration ${\operatorname{Fil}}_\bullet D_S$ such that the graded pieces are of the form $\mathcal{R}_S(\delta)\otimes_SM$ for some continuous $\delta:{{\mathbb{Q}}_p}^\times\to S^\times$ and locally free one-dimensional $M$ over $S$ with trivial $(\varphi,\Gamma)$ actions.
There is a functor $D_{{\operatorname{rig}},L}^\dagger$ associating to an $L$-linear continuous representation of $G_{{\mathbb{Q}}_p}$ a $(\varphi,\Gamma)$-module over $\mathcal{R}_{L}$ and more generally a functor $D_{{\operatorname{rig}},S}^\dagger$ associating to an $S$-linear continuous $G_{{\mathbb{Q}}_p}$-representation a $(\varphi, \Gamma)$-module over $\mathcal{R}_S$. The functor $D_{{\operatorname{rig}},L}^\dagger$ induces an isomorphism of categories between the category of $L$-linear continuous representations of $G_{{\mathbb{Q}}_p}$ and slope 0 $(\varphi,\Gamma)$-modules over $\mathcal{R}_L$.
There exist functors $\mathcal{D}_{{\operatorname{cris}}}$ (resp. $\mathcal{D}_{{\operatorname{st}}}$) attaching to a $(\varphi,\Gamma)$-module $D$ over $\mathcal{R}_L$ a filtered $\varphi$-module (resp. $(\varphi, N)$ over ${{\mathbb{Q}}_p}$ with coefficients in $L$ such that if $V$ is crystalline (resp. semistable) then $\mathcal{D}_{{\operatorname{cris}}}(D_{{\operatorname{rig}},L}^\dagger(V))\cong D_{{\operatorname{cris}}}(V)$ (resp. $\mathcal{D}_{{\operatorname{st}}}(D_{{\operatorname{rig}},L}^\dagger(V))\cong D_{{\operatorname{st}}}(V)$).
Suppose $V$ is a finite-dimensional $L$-linear continuous representation of $G_{{\mathbb{Q}}_p}$ and $D=D_{{\operatorname{rig}},L}^\dagger(V)$ is the associated $(\varphi,\Gamma)$-module over $\mathcal{R}_L$. It is a theorem of Ruochuan Liu ([@liu:herr]) that the Galois cohomology $H^\bullet({\mathbb{Q}}_p, V)$ can be computed as the cohomology $H^\bullet(D)$ of the Herr complex $0\to D\stackrel{f}{\to} D\oplus D\stackrel{g}{\to} D\to 0$ where the transition maps are $f(x) = (\varphi_D-1)x\oplus
(\gamma-1)x$ and $g(x,y) = (\gamma-1)x-(\varphi_D-1)y$. We denote by ${\operatorname{cl}}(x,y)$ the image of $x\oplus y\in D\oplus D$ in $H^1(D)$.
The Bloch–Kato local conditions $H_f^1({\mathbb{Q}}_p, V)=\ker \left(H^1({\mathbb{Q}}_p, V)\to H^1({\mathbb{Q}}_p, V\otimes_{{{\mathbb{Q}}_p}} B_{{\operatorname{cris}}})\right)$ and $H^1_{{\operatorname{st}}}({\mathbb{Q}}_p,V)$ (analogously defined) can also be computed directly using Herr’s complex. Indeed, if $\alpha=a\oplus b\in D\oplus D$ one gets an extension $D_\alpha=D\oplus \mathcal{R}_L e$, depending only on $cl(a,b)$, endowed with Frobenius and $\Gamma$-action defined by $(\varphi_D-1)(0\oplus e) = a\oplus 0$ and $(\gamma-1)(0\oplus e) = b\oplus 0$. Let $H_f^1(D)$ be the set of crystalline extensions $D_\alpha$, i.e. those satisfying $\dim_{{\mathbb{Q}}_p}\mathcal{D}_{{\operatorname{cris}}}(D_\alpha)=\dim_{{\mathbb{Q}}_p}\mathcal{D}_{{\operatorname{cris}}}(D)+1$ and $H_{{\operatorname{st}}}^1(D)$ be the set of semistable extensions (defined analogously). Then $$\begin{aligned}
H^1_f({\mathbb{Q}}_p,V)\cong H^1_f({{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\quad\textrm{and}\quad H^1_{{\operatorname{st}}}({\mathbb{Q}}_p,V)\cong H^1_{{\operatorname{st}}}({{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)).\end{aligned}$$
We end with the following computation of Benois ([@benois:L-invariant Proposition 1.5.9]). Let $\delta:{\mathbb{Q}}_p^\times\to L^\times$ be the character $\delta(x)=x^{-k}$ where $k\in {\mathbb{Z}}_{\geq 0}$. The rank 1 module $\mathcal{R}_L(\delta)$ is crystalline and so $\mathcal{D}_{{\operatorname{cris}}}(\mathcal{R}_L(\delta))=\mathcal{R}_L(\delta)^{\Gamma}\subset \mathcal{R}_L(\delta)$. This allows us to define the map $$i:\mathcal{D}_{{\operatorname{cris}}}(\mathcal{R}_L(\delta))\oplus
\mathcal{D}_{{\operatorname{cris}}}(\mathcal{R}_L(\delta))\to
H^1(\mathcal{R}_L(\delta))$$ by $i(x,y)={\operatorname{cl}}(-x, y\log\chi(\gamma))$. Then $i$ is an isomorphism and $H^1_f(\mathcal{R}_L(\delta))\cong i(\mathcal{D}_{{\operatorname{cris}}}(\mathcal{R}_L(\delta))\oplus 0)$. Moreover, defining $H^1_c(\mathcal{R}_L(\delta))=i(0\oplus \mathcal{D}_{{\operatorname{cris}}}(\mathcal{R}_L(\delta)))$, both $H^1_f(\mathcal{R}_L(\delta))$ and $H^1_c(\mathcal{R}_L(\delta))$ have rank 1.
Regular submodules and the Greenberg–Benois $\mathcal{L}$-invariant {#sect:benois-l-inv}
-------------------------------------------------------------------
\[sect:regular submodules general\]
(cf. [@benois:L-invariant §§2.1–2.2]) Let $\rho:G_{\mathbb{Q}}\rightarrow{\operatorname{GL}}(V)$ be a “geometric” $p$-adic representation of $G_{\mathbb{Q}}$, i.e. $\rho$ is unramified outside a finite set of places and it is potentially semistable at $p$. Let $S$ be a finite set of places containing the ramified ones as well as $p$ and $\infty$. We give an overview of Benois’ definition of the arithmetic $\mathcal{L}$-invariant of $V$. His definition requires five additional assumptions (C1–5) on $V$ described below. We also discuss $\mathcal{L}$-invariants of representations of $G_F$, where $F$ is a number field. We end by proving a lemma we later use in our computations of $\mathcal{L}$-invariants.
For $\ell\nmid p\infty$, define $H^1_f({\mathbb{Q}}_\ell,V)=\ker(H^1({\mathbb{Q}}_\ell,V)\rightarrow H^1(I_\ell,V))$, where $I_\ell$ denotes the inertia subgroup of $G_{{\mathbb{Q}}_\ell}$. When $\ell=p$, the Bloch–Kato local condition $H^1_f({\mathbb{Q}}_p,V)$ was defined in the previous section. Finally, let $H^1_f({\mathbb{R}},V)=H^1({\mathbb{R}},V)$. Let $G_{{\mathbb{Q}},S}={\operatorname{Gal}}({\mathbb{Q}}_S/{\mathbb{Q}})$, where ${\mathbb{Q}}_S$ is the maximal extension of ${\mathbb{Q}}$ unramified outside of $S$. Define the Bloch–Kato Selmer group of $V$ as $$H^1_f(V)=\ker\left(H^1(G_{{\mathbb{Q}},S}, V)\to \bigoplus_{v\in
S}H^1({\mathbb{Q}}_v,V)/H^1_f({\mathbb{Q}}_v, V)\right),$$ which does not depend on the choice of $S$.
As in [@benois:L-invariant §2.1.2] we will assume:
1. \[assumption:C1\]$H_f^1(V)=H^1_f(V^*(1))=0$;
2. \[assumption:C2\]$H^0(G_{{\mathbb{Q}},S},V)=H^0(G_{{\mathbb{Q}},S},V^*(1))=0$;
3. \[assumption:C3\]$V|_{G_{{\mathbb{Q}}_p}}$ is semistable and the semistable Frobenius $\varphi$ is semisimple at $1$ and $p^{-1}$;
4. \[assumption:C4\]${{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V|_{G_{{\mathbb{Q}}_p}})$ has no saturated subquotient isomorphic to some $U_{k,m}$ for $k\geq 1$ and $m\geq 0$ (cf. [@benois:L-invariant §2.1.2], where $U_{k,m}$ is the (unique) non-split crystalline $(\varphi,\Gamma)$-module extension of $\mathcal{R}(x^{-m})$ by $\mathcal{R}(|x|x^k)$).
A *regular submodule* $D$ of ${\operatorname{D}}_{{\operatorname{st}}}(V)$ is a $(\varphi,N)$-submodule such that $D\cong
{\operatorname{D}}_{{\operatorname{st}}}(V)/{\operatorname{Fil}}^0{\operatorname{D}}_{{\operatorname{st}}}(V)$ (as vector spaces) under the natural projection map. Given a regular submodule $D$, Benois constructs the filtration $$\begin{aligned}
D_{-1} &= (1-p^{-1}\varphi^{-1})D+N(D^{\varphi=1})\\
D_0&=D\\
D_1&=D+{\operatorname{D}}_{{\operatorname{st}}}(V)^{\varphi=1}\cap N^{-1}(D^{\varphi=p^{-1}})\end{aligned}$$
The filtration $D_\bullet$ on ${\operatorname{D}}_{{\operatorname{st}}}(V)$ gives a filtration $F_\bullet{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)$ by setting $$F_i{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)={{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)\cap (D_i\otimes_{{{\mathbb{Q}}_p}}
\mathcal{R}_L[1/t])$$ (here $t=\log(1+x)\in \mathcal{R}_L$).
Define the *exceptional subquotient* of $V$ to be $W=F_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)/F_{-1}{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)$, which is a $(\varphi,\Gamma)$ analogue of Greenberg’s $F^{00}/F^{11}$ (see [@G94 p. 157]). Benois shows there are unique decompositions $$\begin{aligned}
W&\cong W_0\oplus W_1\oplus M\\
{\operatorname{gr}}_0{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)&\cong W_0\oplus M_0\\
{\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)&\cong W_1\oplus M_1\end{aligned}$$ such that $W_0$ has rank $\dim H^0(W^*(1))$, $W_1$ has rank $\dim H^0(W)$. Moreover, $M_0$ and $M_1$ have equal rank and the sequence $0\to M_0\stackrel{f}{\to}M\stackrel{g}{\to}M_1\to 0$ is exact.
One has $$H^1(W)={\operatorname{coker}}(H^1(F_{-1}{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\rightarrow H^1(F_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))),$$ $$H^1_f(W)={\operatorname{coker}}(H^1_f(F_{-1}{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\rightarrow H^1_f(F_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))),$$ and $H^1(W)/H^1_f(W)$ has dimension $e_D={\operatorname{rk}}M_0+{\operatorname{rk}}W_0+{\operatorname{rk}}W_1$.
The (dual of the) Tate-Poitou exact sequence gives an exact sequence $$0\to H^1_f(V)\to H^1(G_{{\mathbb{Q}},S},V)\to \displaystyle \bigoplus_{v\in
S}\frac{H^1({\mathbb{Q}}_v,V)}{H^1_f({\mathbb{Q}}_v,V)}\to H^1_f(V^*(1))^\vee$$ where $V^*={\operatorname{Hom}}(V, {\mathbb{Q}}_p)$ and $A^\vee={\operatorname{Hom}}(A,
{\mathbb{Q}}/{\mathbb{Z}})$. Assumptions (C1) and (C2) above imply that $$H^1(G_{{\mathbb{Q}},S},V)\cong \displaystyle \bigoplus_{v\in
S}\frac{H^1({\mathbb{Q}}_v,V)}{H^1_f({\mathbb{Q}}_v,V)}$$ Note that $\displaystyle \bigoplus_{v\in
S}\frac{H^1({\mathbb{Q}}_v,V)}{H^1_f({\mathbb{Q}}_v,V)}$ contains the $e_D$-dimensional subspace $\displaystyle \frac{H^1(W)}{H^1_f(W)}\cong \frac{H^1(F_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_p))}{H^1_f({\mathbb{Q}}_p,V_p)}$. Define $H^1(D,V)\subset H^1(G_{{\mathbb{Q}},S},V)$ to be the set of classes whose image in $\displaystyle \frac{H^1({\mathbb{Q}}_p,V)}{H^1_f({\mathbb{Q}}_p,V)}$ lies in $\displaystyle \frac{H^1(W)}{H^1_f(W)}$.
From now on, assume
- $W_{0}=0$.
Since $\varphi$ acts as 1 on ${\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)$, [@benois:L-invariant Proposition 1.5.9] implies, assuming that the Hodge–Tate weights are nonnegative, that ${\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V)\cong \oplus \mathcal{R}_L(x^{-k_i})$ where the $k_i\geq 0$ are the Hodge–Tate weights. Thus one obtains a decomposition $H^1({{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\cong H^1_f({{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\oplus H^1_c({{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))$ by summing the decompositions for each $\mathcal{R}_L(x^{-k_i})$; as in the rank 1 case, $H^1_f({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\cong H^1_c({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\cong \mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))$. There exist linear maps $\rho_{D,?}:H^1(D,V)\to\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))$ for $?\in \{f,c\}$ making the following diagram commute: $$\xymatrix{
\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\ar[r]^\cong_{\iota_f}&H^1_f({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\\
H^1(D,V)\ar[r]\ar[u]^{\rho_{D,f}}\ar[d]_{\rho_{D,c}}&H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\ar[u]_{\pi_f}\ar[d]^{\pi_c}\\
\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\ar[r]^\cong_{\iota_c}&H^1_c({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))
}$$ Under the assumption that $W_0=0$, Benois shows that the linear map $\rho_{D,c}$ is invertible and defines the arithmetic $\mathcal{L}$-invariant as $$\mathcal{L}(V,D):={\det}\left(\rho_{D,f}\circ\rho_{D,c}^{-1}\big|\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V))\right).$$
In the case of a number field $F$ and a $p$-adic representation $\rho:G_F\to {\operatorname{GL}}(V)$ we follow Hida in defining an arithmetic $\mathcal{L}$-invariant. Suppose that $p$ splits completely in $F$, that $\rho$ is unramified almost everywhere and that at all $v\mid p$, $\rho|_{G_{F_v}}$ is semistable. Assume that $\rho$ satisfies conditions (C1–4) (with appropriate modifications). Then the representation ${\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho$ satisfies conditions (C1–4). Indeed, conditions (C3) and (C4) follow from the fact that $$({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}}\cong
\bigoplus_{v\mid p}{\operatorname{Ind}}_{F_v}^{{\mathbb{Q}}_p}(\rho|_{G_{F_v}})\cong
\bigoplus_{v\mid p}\rho|_{G_{F_v}}$$ since $p$ splits completely in $F$. Conditions (C1) and (C2) follow from Shapiro’s lemma.
For $v\mid p$, let $V_v:=V|_{G_{F_v}}$. Choose $D_v\subset
{\operatorname{D}}_{{\operatorname{st}}}(V_v)$ a regular submodule giving the modules $W_{0,v}$, $W_{1,v}$ and $M_v$. Then $D=\oplus_{v\mid p}D_v\subset
\oplus_{v\mid p}{\operatorname{D}}_{{\operatorname{st}}}(\rho_v)\cong
{\operatorname{D}}_{{\operatorname{st}}}(({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}})$ is a regular submodule and $W_0=\oplus_{v\mid p}W_{0,v}$, $W_1=\oplus_{v\mid p}W_{1,v}$ and $M=\oplus_{v\mid
p}M_v$. Assuming $W_{0,v}=0$ for every $v\mid p$ yields $W_0=0$ and we may define $$\mathcal{L}(\{D_v\}, \rho)=\mathcal{L}(D, {\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)$$ Note that ${\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}})\cong \bigoplus_{v\mid p}{\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_v)$.
\[l:rank 1 L invariant\] Suppose that ${\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_v)\cong \mathcal{R}$ for each $v\mid
p$. Let $H^1(\mathcal{R})\cong H^1_f(\mathcal{R})\oplus
H^1_c(\mathcal{R})$ with basis $x=(-1,0)$ and $y=(0,\log_p
\chi(\gamma))$. Suppose $c\in
H^1(D, {\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)$ is such that the image of $c$ in $H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_v))$ is $\xi_v=a_vx+b_vy$ with $b_v\neq
0$. Then $$\mathcal{L}(\{D_v\},\rho)=\prod_{v\mid p}\frac{a_v}{b_v}$$
Since $$H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}}))\cong
\bigoplus_{v\mid p}H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v))$$ and $$\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}}))\cong
\bigoplus_{v\mid p}\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v)),$$ the maps $\iota_f$, $\iota_c$, $\pi_f$, and $\pi_c$ are direct sums of the maps $$\begin{array}{c}
\iota_{f,v}:\mathcal{D}_{{\operatorname{cris}}}(\mathcal{R})\cong
H^1_{f}(\mathcal{R}),\quad
\iota_{c,v}:\mathcal{D}_{{\operatorname{cris}}}(\mathcal{R})\cong
H^1_{c}(\mathcal{R}),\\
\\
\pi_{f,v}:H^1(\mathcal{R})\to H^1_f(\mathcal{R})\quad\text{and}\quad
\pi_{c,v}:H^1(\mathcal{R})\to H^1_c(\mathcal{R}).
\end{array}$$ Let $\xi_v=a_vx+b_vy$ be the image of $c$ in $H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v))\cong H^1(\mathcal{R})$. We deduce that the maps $\rho_f$ (resp. $\rho_c$) are direct sums of the $\rho_{f,v}=\iota_{f,v}^{-1}\circ\pi_{f,v}$ (resp. $\rho_{c,v}=\iota_{c,v}^{-1}\circ\pi_{c,v}$). Then $a_v\in
\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v))$ is the image of $\xi_v$ under $\rho_f$ and $b_v\in
\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v))$ is the image of $\xi_v$ under $\rho_c$. We deduce that $$\begin{aligned}
\mathcal{L}(\{D_v\}, V)&=\det(\rho_f\circ\rho_c^{-1}|\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(({\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)|_{G_{{\mathbb{Q}}_p}})))\\
&=\prod_{v\mid p}\det(\rho_{f,v}\circ\rho_{c,v}^{-1}|\mathcal{D}_{{\operatorname{cris}}}({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v)))\\
&=\prod_{v\mid p}\frac{a_v}{b_v}\end{aligned}$$
Refined families of Galois representations {#sect:refined galois}
------------------------------------------
Lemma \[l:rank 1 L invariant\] provides a framework for computing arithmetic $\mathcal{L}$-invariants as long as one is able to produce cohomology classes $c\in H^1(D, {\operatorname{Ind}}_F^{{\mathbb{Q}}}\rho)$ and compute their projections to $H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(\rho_v))$. We will produce such cohomology classes using analytic Galois representations on eigenvarieties and we will compute explicitly the projections (in effect the $a_v$ and $b_v$ of Lemma \[l:rank 1 L invariant\]) using global triangulations of $(\varphi, \Gamma)$-modules.
We recall here the main result of [@liu:triangulation §5.2] on triangularization in refined families. Let $L$ be a finite extension of ${\mathbb{Q}}_p$. Suppose $X$ is a separated and reduced rigid analytic space over $L$ and $V_X$ is a locally free, coherent $\mathcal{O}_X$-module of rank $d$ with a continuous $\mathcal{O}_X$-linear action of the Galois group $G_{{\mathbb{Q}}_p}$. The family $V_X$ of Galois representations is said to be *refined* if there exist $\kappa_1,\ldots,\kappa_d\in \mathcal{O}(X)$, $F_1,\ldots,F_d\in \mathcal{O}(X)$, and a Zariski dense set of points $Z\subset X$ such that:
1. for $x\in X$, the Hodge–Tate weights of ${\operatorname{D}}_{{\operatorname{dR}}}(V_{x})$ are $\kappa_1(x),\ldots,\kappa_d(x)$,
2. for $z\in Z$, the representation $V_z$ is crystalline,
3. for $z\in Z$, $\kappa_1(z)<\ldots <\kappa_d(z)$,
4. the eigenvalues of $\varphi$ acting on ${\operatorname{D}}_{{\operatorname{cris}}}(V_z)$ are distinct and equal to $\{p^{\kappa_i(z)}F_i(z)\}$,
5. \[cond:5\]for $C\in{\mathbb{Z}}_{\geq0}$ the set $Z_C$, consisting of $z\in Z$ such that for all distinct subsets $I,J\subset \{1,\ldots,d\}$ of equal cardinality one has $|\sum_{i\in I}\kappa_i(z)-\sum_{j\in
J}\kappa_j(z)|>C$, accumulates at every $z\in Z$,
6. for each $1\leq i\leq d$, there exists a continuous character $\chi_i:\mathcal{O}_K^\times\to \mathcal{O}(X)^\times$ such that $\chi_{z,i}'(1)=\kappa_i$ and $\chi_{z,i}(u)=u^{\kappa_i(z)}$ for all $z\in Z$.
Given a refined family $V_X$, we define $\Delta_i:{{\mathbb{Q}}_p}^\times\to \mathcal{O}(X)^\times$ by $\Delta_i(p)=\prod_{j=1}^i F_j$ and for $u\in
{{\mathbb{Z}}_p}^\times$, $\Delta_i(u)=\prod_{j=1}^i\chi_j(u)$. Let $\delta_i=\Delta_i/\Delta_{i-1}$.
Let $V_X$ be a refined family. For all $z\in Z$, there is an induced *refinement* of $V_z$, i.e. a filtration $0=\mathcal{F}_0\subsetneq \mathcal{F}_1\subsetneq\ldots\subsetneq \mathcal{F}_d={\operatorname{D}}_{{\operatorname{cris}}}(V_z)$ of $\varphi$-submodules. It is determined by the condition that the eigenvalue of $\varphi_{{\operatorname{cris}}}$ on $\mathcal{F}_i/\mathcal{F}_{i-1}$ is $p^{\kappa_i(z)}F_i(z)$. We say that $z$ is *noncritical* if ${\operatorname{D}}_{{\operatorname{cris}}}(V_z)=\mathcal{F}_i+{\operatorname{Fil}}^{\kappa_{i+1}(z)}{\operatorname{D}}_{{\operatorname{cris}}}(V_z)$. We say that $z$ is *regular* if $\det\varphi$ on $\mathcal{F}_i$ has multiplicity one in ${\operatorname{D}}_{{\operatorname{cris}}}(\wedge^i V_z)$ for all $i$.
Then, [@liu:triangulation Theorem 5.2.10] gives:
\[t:triangulation\] If $z\in Z$ is regular and noncritical, then in an affinoid neighborhood $U$ of $z$, $V_{U}$ is trianguline with graded pieces isomorphic to $\mathcal{R}_{U}(\delta_1),\ldots,\mathcal{R}_{U}(\delta_d)$.
In fact, [@liu:triangulation Theorem 5.3.1] shows that $V_x$ is trianguline at all $x\in X$, but the triangulation is only made explicit over a proper birational transformation of $X$.
Symmetric powers of Hilbert modular forms {#sect:hmf}
=========================================
Hilbert modular forms and their Galois representations {#sect:hmf def}
------------------------------------------------------
Let $F/{\mathbb{Q}}$ be a totally real field of degree $d$ and let $I\subseteq{\operatorname{Hom}}_{{\mathbb{Q}}}(F, {\mathbb{C}})$ be a parametrization of the infinite places. We fix an embedding $\iota_\infty:\overline{{\mathbb{Q}}}{\hookrightarrow}{\mathbb{C}}$ thus identifying $I$ with a subset of ${\operatorname{Hom}}_{{\mathbb{Q}}}(F,{\overline{{\mathbb{Q}}}})$. We also fix $\iota_p:\overline{{\mathbb{Q}}}{\hookrightarrow}\overline{{\mathbb{Q}}}_p$ which identifies ${\operatorname{Hom}}_{{\mathbb{Q}}}(F,{\overline{{\mathbb{Q}}}}_p)$ with ${\operatorname{Hom}}_{{\mathbb{Q}}}(F,{\overline{{\mathbb{Q}}}})$. This determines a partition $I=\bigcup_{v\mid p}I_v$. Let $\varpi_v$ be a uniformizer for $F_v$ and $e_v$ be the ramification index of $F_v/{\mathbb{Q}}_p$.
Before defining Hilbert modular forms, we need some notation on representations of ${\operatorname{GL}}(2,{\mathbb{R}})$. Recall that the Weil group of ${\mathbb{R}}$ is $W_{{\mathbb{R}}}={\mathbb{C}}^\times\rtimes \{1,j\}$ where $j^2=-1$ and $jz=\overline{z}j$ for $z\in {\mathbb{C}}$. For an integer $n\geq 2$, let $\mathcal{D}_n$ be the essentially discrete series representation of ${\operatorname{GL}}(2,{\mathbb{R}})$ whose Langlands parameter is ${\varphi}_n:W_{{\mathbb{R}}}\to {\operatorname{GL}}(2, {\mathbb{C}})$ given by ${\varphi}_n(z) = \begin{pmatrix} (z/\overline{z})^{(n-1)/2}&\\ &(\overline{z}/z)^{(n-1)/2}\end{pmatrix}$ and ${\varphi}_n(j) = \begin{pmatrix} &1\\ (-1)^{n-1}&\end{pmatrix}$. The representation $\mathcal{D}_n$ is unitary and has central character ${\operatorname{sign}}^n$ and Blattner parameter $n$. More generally, if $t\in {\mathbb{C}}$, the representation $\mathcal{D}_n\otimes |\det|^t$ has associated Langlands parameter ${\varphi}_n\otimes |\cdot|^{2t}$. If $w\in {\mathbb{Z}}$, then $$H^1(\mathfrak{gl}_2,{\operatorname{SO}}(2),\mathcal{D}_n(-w/2)\otimes
V_{(w+n-2)/2, (w-n+2)/2}^\vee)\neq 0$$ where $V_{(a,b)} = {\operatorname{Sym}}^{a-b}{\mathbb{C}}^2\otimes \det^b$ is the representation of highest weight $(a,b)$.
By a cohomological Hilbert modular form of infinity type $(k_1, \ldots, k_d, w)$, we mean a cuspidal automorphic representation $\pi$ of ${\operatorname{GL}}(2, {\mathbb{A}}_F)$ such that
1. $k_i\equiv w\pmod{2}$ with $k_i\geq 2$ and
2. for every $i\in I$, $\pi_i\cong \mathcal{D}_{k_i}\otimes |\det|^{-w/2}$.
This is equivalent to the fact that $\pi_i$ has Langlands parameter $z\mapsto |z|^{-w}\begin{pmatrix}
(z/\overline{z})^{(k_i-1)/2}&\\ &
(z/\overline{z})^{-(k_i-1)/2}\end{pmatrix}$, $j\mapsto \begin{pmatrix} &1\\
(-1)^{k_i-1}&\end{pmatrix}$. Since $H^1(\mathfrak{gl}_2,{\operatorname{SO}}(2),\pi_i\otimes
V_{(-w+k_i-2)/2, (-w-k_i+2)/2}^\vee)\neq 0$, the representation $\pi$ can realized in the cohomology of the local system $\left(\bigotimes_i\left({\operatorname{Sym}}^{k_i-2}\otimes \det^{(-w-k_i+2)/2}\right)\right)^\vee$ over a suitable Hilbert modular variety (cf. [@raghuram-tanabe:hilbert §3.1.9]). When $F={\mathbb{Q}}$ and $w=2-k$ we recover the usual notion of an elliptic modular form of weight $k$.
If $p$ is a prime number, then (by Eichler, Shimura, Deligne, Wiles, Taylor, Blasius–Rogawski) there exists a continuous $p$-adic Galois representation $\rho_{\pi,p}:G_F\to {\operatorname{GL}}(2, \overline{{\mathbb{Q}}}_p)$ such that $L^S(\pi, s-1/2)=L^S(\rho_{\pi,p},s)$ for a finite set $S$ of places of $F$. Moreover, one has local-global compatibility: if $v\in S$, then ${\operatorname{WD}}(\rho_{\pi,p}|_{G_{F_v}})^{{\operatorname{Fr-ss}}}\cong{\operatorname{rec}}(\pi_v\otimes|\cdot|^{-1/2})$. When $v\nmid p$ this follows from the work of Carayol ([@carayol:hilbert-ell]) and when $v\mid p$ from the work of Saito ([@saito:hilbert-p]) and Skinner ([@skinner:hilbert]). Finally, the Hodge–Tate weights of $\rho_{\pi,p}|_{G_{F_v}}$ are $(w-k_v)/2$ and $(w+k_v-2)/2$. (For weight $k$ elliptic modular forms with $w=2-k$ this amounts to Hodge–Tate weights $1-k$ and $0$.)
We end this discussion with the following result on the irreducibility of symmetric powers.
\[l:hmf lie irreducible\] Suppose $\pi$ is not CM. Then $\rho_{\pi,p}$ and ${\operatorname{Sym}}^n\rho_{\pi,p}$ are Lie irreducible.
First, the remark at the end of [@skinner:hilbert] shows that $\rho_{\pi,p}$ is irreducible. We will apply [@patrikis:tate Proposition 1.0.14] to $\rho_{\pi,p}$ which states that an irreducible Galois representation of a compatible system is of the form ${\operatorname{Ind}}(\tau\otimes\sigma)$ where $\tau$ is Lie irreducible and $\sigma$ is Artin. Since $\pi$ is not CM, the irreducibility of $\rho_{\pi,p}$ implies that $\rho_{\pi,p}$ is either Lie irreducible or Artin. But $\pi$ is cohomological and $\rho_{\pi,p}|_{G_{F_v}}$ has Hodge–Tate weights $0$ and $1-k_v<0$ and so it cannot be Artin. Finally, the restriction of $\rho_{\pi,p}$ to any open subgroup will contain ${\operatorname{SL}}(2, \overline{{\mathbb{Q}}}_p)$. Since the symmetric power representation of ${\operatorname{SL}}(2)$ is irreducible, it follows that the restriction of ${\operatorname{Sym}}^n\rho_{\pi,p}$ to any open subgroup will be irreducible.
Regular submodules {#sect:regular submodules}
------------------
As previously discussed, $p$-adic $L$-functions are expected to be attached to Galois representations and a choice of regular submodule. In this section, we will describe the possible regular submodules in the case of twists of symmetric powers of Galois representations.
Suppose $F$ is a totally real field and $\pi$ is a cohomological Hilbert modular form of infinity type $(k_1,\ldots, k_d,w)$ over $F$. Let $V_{2n}={\operatorname{Sym}}^{2n}\rho_{\pi,p}\otimes\det^{-n}\rho_{\pi,p}$ and $V_{2n,v}=V_{2n}|_{G_{F_v}}$. We will classify the regular submodules of ${\operatorname{D}}_{{\operatorname{st}}}(V_{2n,v})$ in the case when $p$ splits completely in $F$ and $\pi$ is Iwahori spherical at places $v\mid p$.
Local-global compatibility describes the representation $\rho_{\pi,p}|_{G_{F_v}}$ completely whenever $v\nmid p$, but not so when $v\mid p$. We now make explicit the possibilities for the $p$-adic Galois representation at places $v\mid p$ and in the process we choose a suitable regular submodule of ${\operatorname{D}}_{{\operatorname{st}}}(V_{2n,v})$.
Since $p$ splits completely in $F$, $F_v\cong{\mathbb{Q}}_p$. The Galois representation $V=\rho_{\pi,p}|_{G_{F_v}}$ is de Rham with Hodge–Tate weights $(w-k_v)/2$ and $(w+k_{v}-2)/2$ where we denote by $v$ as well the unique infinite place in $I_v$. Since $\pi$ is Iwahori spherical there are two possibilities: either $\pi_v={\operatorname{St}}\otimes\mu$, where $\mu$ is an unramified character, or $\pi_v$ is the unramified principal series with characters $\mu_1$ and $\mu_2$.
If $\pi_v={\operatorname{St}}\otimes\mu$ then $V$ is semistable but not crystalline, $k_v$ is even, ${\operatorname{D}}_{{\operatorname{st}}}^*(V)={\mathbb{Q}}_p
e_1\oplus {\mathbb{Q}}_pe_2$, $\varphi=\begin{pmatrix} \lambda& \\
& p \lambda \end{pmatrix}$ for $\lambda=\mu({\operatorname{Frob}}_p)$ with $v_p(\lambda)=(k_v-2)/2$ and $N=\begin{pmatrix}0 &1\\
0&0 \end{pmatrix}$. The filtration jumps at the Hodge–Tate weights $(w-k_v)/2$ and $(w+k_v-2)/2$ and the proper filtered pieces are given by ${\mathbb{Q}}_p(e_2-\mathcal{L} e_1)$ for some $\mathcal{L} \in {\mathbb{Q}}_p$, which is the Fontaine–Mazur $\mathcal{L}$-invariant of $V$ (and the Benois $\mathcal{L}$-invariant of $V$). Writing $f_{i}=e_1^{n+i}e_2^{n-i}$, with $n\geq i\geq-n$, for the basis of ${\operatorname{D}}_{{\operatorname{st}}}(V_{2n,v})$, the Frobeniuns map $\varphi(f_i)=p^{-i}f_i$ is represented by a diagonal matrix while the monodromy is upper triangular with off-diagonal entries $2n,2n-1,\ldots,1$. The $(\varphi,N)$-stable submodules of ${\operatorname{D}}_{{\operatorname{st}}}(V_{2n,v})$ are the spans $\langle f_{n}, f_{n-1}, \ldots, f_i\rangle$. Note that $${\operatorname{Fil}}^0{\operatorname{D}}_{{\operatorname{dR}}}^*(V_{2n,v})=\left\langle(e_2-\mathcal{L} e_1)^{2n}, (e_2-\mathcal{L} e_1)^{2n-1}e_1,(e_2-\mathcal{L} e_1)^{2n-1}e_2,\ldots,(e_2-\mathcal{L} e_1)^ne_1e_2^{n-1},(e_2-\mathcal{L} e_1)^ne_2^n\right\rangle$$ and hence is $(n+1)$-dimensional. Thus, a regular submodule must be $n$-dimensional. The only $n$-dimensional $(\varphi,N)$-stable submodule is $D=\langle f_{n},\ldots,f_{1}\rangle$ and if $\mathcal{L} \neq 0$ (as is expected), then $D$ is regular. In this case, ${\operatorname{D}}_{{\operatorname{st}}}(V_{2n,v})^{\varphi=1}=\langle f_0\rangle$ and $D^{\varphi=p^{-1}}=\langle f_1\rangle$. Thus, $D_1=\langle f_n,\dots, f_0\rangle$ (since $N^{-1}(\langle f_1\rangle)=\langle f_0\rangle$) and $D_{-1}=\langle f_n,\dots,f_2\rangle$.
If $\pi_v$ is unramified then $V$ is crystalline. Let $L/{\mathbb{Q}}_p$ be the finite extension generated by the roots $\alpha=\mu_1(\varpi_v)$ and $\beta=\mu_2(\varpi_v)$ of $x^2-a_vx+p^{k_v-1}$ with $v_p(\alpha)\leq v_p(\beta)$. Then after base change to $L$, ${\operatorname{D}}_{{\operatorname{cris}}}(V)^*=Le_1 \oplus Le_2$. There are now two possibilities. Either the local representation $V$ splits as $\mu\oplus\mu^{-1}{\chi_{{\operatorname{cycl}}}}^{k-1}$, where $\mu$ is unramified, or $V$ is indecomposable. The former case is expected to occur only when $\pi$ is CM.
If $V$ splits then $\pi_v$ is ordinary ${\operatorname{D}}_{{\operatorname{cris}}}^*(V_{2n,v})=\bigoplus_{i=-n}^nLt^{i(k_v-1)}$ (basis $f_i = e_1^{n+i}e_2^{n-i}$), $\varphi$ has eigenvalues $\alpha^{2n}p^{n(k_v-1)}, \ldots, \alpha^{-2n}p^{-n(k_v-1)}$ where $\mu({\operatorname{Frob}}_p)=\alpha$. The de Rham tangent space is ${\operatorname{D}}_{{\operatorname{cris}}}^*(V_{2n})/{\operatorname{Fil}}^0{\operatorname{D}}_{{\operatorname{cris}}}^*(V_{2n})=L f_{1}\oplus
\cdots\oplus L f_{n}$ and so the only regular subspace is $D=L f_{1}\oplus \cdots\oplus
L f_{n}$. In this case the filtration on $D$ is given by $D_{-1}=D_0=D$ and $D_1=D\oplus Lf_0$.
If $V$ is not split, we will assume that $\alpha/\beta\not\in\mu_\infty$. Then, we choose $e_1$ and $e_2$ to be eigenvectors of $\varphi$ so that ${\operatorname{D}}_{{\operatorname{cris}}}^*(V)=L e_1\oplus Le_2$ with $\varphi=\begin{pmatrix} \alpha &\\ &\beta \end{pmatrix}$. Moreover, we can scale $e_1$ and $e_2$ so that the one-dimensional filtered piece is $\langle e_1+e_2\rangle$. We remark that $V$ is reducible if and only if it is ordinary. Again taking $f_i = e_1^{n+i}e_2^{n-i}$ as a basis, the Frobenius on ${\operatorname{D}}_{{\operatorname{cris}}}^*(V_{2n,v})$ is diagonal with $\varphi(f_i)=(\alpha/\beta)^i$. The de Rham tangent space is generated by homogeneous polynomials in $e_1$ and $e_2$ which are not divisible by $(e_1+e_2)^n$. Thus, any choice of $n$ basis vectors in $f_{n},\ldots, f_{-n}$ will generate a regular submodule. The assumption that $\alpha/\beta\not\in\mu_\infty$ implies that ${\varphi}(f_i)=f_i$ only for $i=0$. Since $\alpha$ and $\beta$ are Weil numbers of the same complex absolute value, the eigenvalue $p^{-1}$ does not show up. Therefore, no matter what choice of $D$ we take, we have $D_{-1}=D_0=D$. We choose the regular submodule $D=\langle f_n,f_{n-1},\ldots, f_1\rangle$. Since $f_0$ is not among the chosen basis vectors, $D_1 = D\oplus\langle f_0\rangle=\langle{f_n,f_{n-1},\dots,f_0}\rangle$.
In all cases, we have $D_0=D=\langle{f_n,f_{n-1},\dots,f_1}\rangle$ and $D\oplus\langle f_0\rangle=\langle{f_n,f_{n-1},\dots,f_0}\rangle$. Therefore, $${\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_{2n,v})\cong \mathcal{R}_L.$$
Automorphy of symmetric powers
------------------------------
Let $\pi$ be a cuspidal automorphic representation of ${\operatorname{GL}}(2,{\mathbb{A}}_F)$ over a totally real field $F$ as in §\[sect:hmf def\]. We say that ${\operatorname{Sym}}^n\pi$ is (cuspidal) automorphic on ${\operatorname{GL}}(n+1,{\mathbb{A}}_{F'})$ for a number field $F'/F$ if there exists a (cuspidal) automorphic representation $\Pi_n$ of ${\operatorname{GL}}(n+1,{\mathbb{A}}_{F'})$ such that $L(\Pi_n,{\operatorname{std}},s)=L({\operatorname{BC}}_{F'/F}(\pi),{\operatorname{Sym}}^n,s)$. A cuspidal representation $\sigma$ of ${\operatorname{GL}}(2, {\mathbb{A}}_{F})$ is said to be dihedral if it is isomorphic to its twist by a quadratic character, in which case there exists a CM extension $E/F$ and a character $\psi:{\mathbb{A}}_E^\times/E^\times\to {\mathbb{C}}^\times$ such that $\sigma\cong{\operatorname{AI}}_{E/F}\psi$; we say that $\sigma$ is tetrahedral (resp. octahedral) if ${\operatorname{Sym}}^2\sigma$ (resp. ${\operatorname{Sym}}^3\sigma$) is cuspidal and is isomorphic to its twist by a cubic (resp. quadratic) character; we say that $\sigma$ is solvable polyhedral if $\sigma$ is dihedral, tetrahedral or octahedral; we say that $\sigma$ is icosahedral if ${\operatorname{Sym}}^5\sigma\cong{\operatorname{Ad}}(\tau){\boxtimes}\sigma\otimes\omega_\sigma^2$ for some cuspidal representation $\tau$ of ${\operatorname{GL}}(2, {\mathbb{A}}_F)$ and if $\sigma$ is not solvable polyhedral. The cuspidal representation $\pi$, being associated with a regular Hilbert modular form, is either dihedral (if the Hilbert modular form is CM) or not polyhedral.
\[t:automorphy of sym\]Suppose $\pi$ is as in §\[sect:hmf def\]. Then
1. ${\operatorname{Sym}}^m\pi$ is automorphic for $m=2$ ([@gelbart-jacquet Theorem 9.3]), $m=3$ ([@kim-shahidi:sym3 Corollary 1.6] and [@kim-shahidi:cuspidality-sym Theorem 2.2.2]) and $m=4$ ([@kim:sym4 Theorem B] and [@kim-shahidi:cuspidality-sym 3.3.7]); it is cuspidal unless $\pi$ is CM.
2. Suppose $\pi$ is not CM. If ${\operatorname{Sym}}^5\pi$ is automorphic then it is cuspidal. If $m\geq 6$ suppose either that ${\operatorname{Sym}}^k\pi$ is automorphic for $k\leq 2m$ or that $\pi{\boxtimes}\tau$ is automorphic for any cuspidal representation $\tau$ of ${\operatorname{GL}}(r, {\mathbb{A}}_{F})$ where $r\leq \lfloor m/2+1\rfloor$. Then ${\operatorname{Sym}}^m\pi$ is cuspidal ([@ramakrishnan:remarks-sym Theorem A’]).
Results on the automorphy of ${\operatorname{Sym}}^n\pi$ for small values of $n$ have been obtained by Clozel and Thorne assuming conjectures on level raising and automorphy of tensor products. Potential automorphy results follow from automorphy lifting theorems.
Functorial transfers to unitary and symplectic groups
-----------------------------------------------------
To compute $\mathcal{L}$-invariants, we use $p$-adic families of Galois representations. However, since ${\operatorname{GL}}(n)$ for $n>2$ has no associated Shimura variety, to construct $p$-adic families of automorphic representations, we transfer to unitary or symplectic groups.
We begin with ${\operatorname{GSp}}(2n)$. Let $\omega_0$ be the $n\times n$ antidiagonal matrix with 1-s on the antidiagonal and let ${\operatorname{GSp}}(2n)$ be the reductive group of matrices $X\in {\operatorname{GL}}(2n)$ such that $X^TJX=\nu(X)J$ where $J=\begin{pmatrix} & \omega_0\\-\omega_0&\end{pmatrix}$ and $\nu(X)\in \mathbb{G}_m$ is the multiplier character. The diagonal maximal torus $T$ consists of matrices $t(x_1,\ldots,
x_g, z)$ with $(x_1,\ldots, x_g, zx_g^{-1},\ldots, zx_1^{-1})$ on the diagonal. The Weyl group $W$ of ${\operatorname{GSp}}(2n)$ is $S_n\rtimes
({\mathbb{Z}}/2 {\mathbb{Z}})^n$ and thus any $w\in W$ can be written as a pair $w=(\nu, \varepsilon)$ where $\nu\in S_n$ is a permutation and $\varepsilon:\{1,\ldots, n\}\to \{-1,1\}$ is a function. The Weyl group acts by conjugation on $T$ and $(\nu,
\varepsilon)$ takes $t(x_1, \ldots, x_n, z)$ to $t(x'_{\nu(1)},
\ldots, x'_{\nu(n)}, z)$ where $x'_i = x_i$ if $\varepsilon(i)=1$ and $x'_i = zx_i^{-1}$ if $\varepsilon(i)=-1$. We choose as a basis for $X^\bullet(T)$ the characters $e_i(t(x_i;z))=x_iz^{-1/2}$ and $e_0((x_i;z))=z^{1/2}$. Let $B$ be the Borel subgroup of upper triangular matrices, which corresponds to the choice of simple roots $e_i-e_{i+1}$ for $i<n$ and $2e_n$. Half the sum of positive roots is then $\rho = \sum_{i=1}^n(n+1-i)e_i$. The compact roots are $\pm(e_i-e_j)$ and so half the sum of compact roots is $\rho_c = \sum_{i=1}^n(n-2i+1)/2 e_i$. This gives the relationship between Harish-Chandra and Blattner parameters for ${\operatorname{GSp}}(4)$ as $\lambda_{\operatorname{HC}}+\sum_{i=1}^n ie_i = \lambda_{\operatorname{Blattner}}$.
\[t:GL(4) to GSp(4)\] Let $\pi$ be as in §\[sect:hmf def\] of infinity type $(k_1,\ldots,k_d,w)$. If $\pi$ is not CM there exists a cuspidal automorphic representation $\Pi$ of ${\operatorname{GSp}}(4,{\mathbb{A}}_{F})$ which is a strong lift of ${\operatorname{Sym}}^3\pi$ from ${\operatorname{GL}}(4,{\mathbb{A}}_F)$ such that for every infinite place $\tau$, $\Pi_\tau$ is the holomorphic discrete series with Harish-Chandra parameters $(2(k_\tau-1),k_\tau-1;-3w/2)$.
When $F={\mathbb{Q}}$ and the weight is even this is [@ramakrishnan-shahidi Theorem A$^\prime$]. Ramakrishnan remarks that the proof of this result should also work for totally real fields. We give a proof using Arthur’s results on the discrete spectrum of symplectic groups.
Let $\sigma$ be the cuspidal representation of ${\operatorname{GL}}(4,
{\mathbb{A}}_F)$ whose $L$-function coincides with that of ${\operatorname{Sym}}^3\pi$. Then $L(\wedge^2\sigma\otimes\det^3\rho_{\pi,p},s)=L(\wedge^2{\operatorname{Sym}}^3\rho_{\pi,p}\otimes\det^3\rho_{\pi,p},s)=\zeta(s)L({\operatorname{Sym}}^4\rho_{\pi,p}\otimes\det^2\rho_{\pi,p},s)$ has a pole at $s=1$ (for example because ${\operatorname{Sym}}^4\pi$ is cuspidal automorphic) so [@gan-takeda:local-langlands-GSp(4) Theorem 12.1] shows that there exists a globally generic (i.e., having a global Whittaker model) cuspidal representation $\Pi^g$ of ${\operatorname{GSp}}(4, {\mathbb{A}}_F)$ strongly equivalent to $\sigma$, and thus to ${\operatorname{Sym}}^3\pi$. For every infinite place $\tau$ the representation $\Pi^g_\tau$ will be a generic (nonholomorphic) discrete series representation. Let $\psi$ be the global $A$-parameter attached to $\Pi^g$; since $\Pi^g$ is globally generic with cuspidal lift to ${\operatorname{GL}}(4)$, $\psi$ will be simple generic and therefore the $A$-parameter $\psi$ is in fact an $L$-parameter and the component group of $\psi$ is trivial. Let $\Pi=(\otimes_{\tau\mid\infty}\Pi^h_\tau)\otimes(\otimes_{v\nmid\infty}\Pi^g_v)$ be the representation of ${\operatorname{GSp}}(4, {\mathbb{A}}_F)$ obtained using the holomorphic discrete series $\Pi^h_\tau$ in the same archimedean local $L$-packet as $\Pi^g_\tau$. Then Arthur’s description of the discrete automorphic spectrum for ${\operatorname{GSp}}(4)$ implies that $\Pi$ is an automorphic representation (for convenience, see [@mok:siegel-hilbert Theorem 2.2]). Since the representations $\Pi_\tau$ at infinite places $\tau$ are discrete series, they are also tempered and so [@wallach:constant Theorem 4.3] implies that $\Pi$ will also be cuspidal. By construction, $\Pi$ will be strongly equivalent to ${\operatorname{Sym}}^3\pi$.
It remains to compute the Harish-Chandra parameter of $\Pi_\tau$. Let ${\varphi}_{\pi_\tau}$ and ${\varphi}_{\Pi_\tau}$ be the Langlands parameters of $\pi_\tau$ respectively $\Pi_\tau$. Then ${\varphi}_{\Pi_\tau}={\operatorname{Sym}}^3{\varphi}_{\pi_\tau}$ and so $${\varphi}_{\Pi_\tau}(z)=|z|^{-3w}\begin{pmatrix}
(z/\overline{z})^{(3(k_\tau-1))/2}&&&\\ &
(z/\overline{z})^{(k_\tau-1)/2}&&\\ &&
(z/\overline{z})^{-(k_\tau-1)/2}&\\ &&&
(z/\overline{z})^{-(3(k_\tau-1))/2}\end{pmatrix}$$ The recipe from [@sorensen:gsp4 §2.1.2] shows that the $L$-packet defined by ${\varphi}_{\Pi_\tau}$ consists of the holomorphic and generic discrete series with Harish-Chandra parameters $(2(k_\tau-1),k_\tau-1;-3w/2)$.
For higher $n$ one does not yet have transfers from ${\operatorname{GL}}(n)$ to similitude symplectic groups, although one may first transfer from ${\operatorname{GL}}(2n+1)$ to ${\operatorname{Sp}}(2n)$ and then lift to ${\operatorname{GSp}}(2n)$.
\[t:GL(2n+1) to Sp(2n)\]Let $F$ be a totally real field and $\pi$ be a regular algebraic cuspidal automorphic self-dual representation of ${\operatorname{GL}}(2n+1, {\mathbb{A}}_F)$ with trivial central character. Then there exists a cusidal automorphic representation $\overline{\sigma}$ of ${\operatorname{Sp}}(2n, {\mathbb{A}}_F)$ which is a weak functorial transfer of $\pi$ such that $\overline{\sigma}$ is a holomorphic discrete series at infinite places. If, moreover, $\pi$ is the symmetric $2n$-th power of a cohomological Hilbert modular form then there exists a cuspidal representation $\sigma$ of ${\operatorname{GSp}}(2n,{\mathbb{A}}_F)$ which is a holomorphic discrete series at infinity and such that any irreducible component of the restriction $\sigma|_{{\operatorname{Sp}}(2n,{\mathbb{A}}_F)}$ is in the same global $L$-packet at $\overline{\sigma}$.
We will use [@ginzburg-rallis-soudry:descent Theorem 3.1] to produce an irreducible cuspidal globally generic functorial transfer $\tau$ of a cuspidal self-dual representation $\pi$ of ${\operatorname{GL}}(2n+1, {\mathbb{A}}_F)$ to ${\operatorname{Sp}}(2n, {\mathbb{A}}_F)$, and all citations in this paragraph are from [@ginzburg-rallis-soudry:descent]. In the notation of §2.3 the group $H$ is taken to be the metaplectic double cover $\widetilde{{\operatorname{Sp}}}(4n+2)$ of ${\operatorname{Sp}}(4n+2)$ (this is case 10 from (3.43)) in which case, in the notation of §2.4, $L(\pi,
\alpha^{(1)},s)=1$ and $L(\pi,\alpha^{(2)}, s)=L(\pi,{\operatorname{Sym}}^2,s)$ which has a pole at $s=1$ since $\pi\cong \pi^\vee$. Let $E_\pi$ be the irreducible representation of $H({\mathbb{A}}_F)$ of Theorem 2.1 generated by the residues of certain Eisenstein series attached to the representation $\pi$ thought of as a representation of the maximal parabolic of $H$. One finds $\tau$ as an irreducible summand of the automorphic representation generated by the Fourier–Jacobi coefficients of $E_\pi$ where, in the notation of §3.6, one takes $\gamma=1$. The fact that $\tau$ is cuspidal and globally generic and a strong transfer of $\pi$ then follows from Theorem 3.1.
Next, Arthur’s global classification of the discrete spectrum of ${\operatorname{Sp}}(2n)$ ([@arthur:endoscopic-classification Theorem 1.5.2]) implies that there exists a cuspidal automorphic representation $\tau'$ which is isomorphic to $\tau$ at all finite places and the representation $\tau'_\infty$ is a holomorphic discrete series in the same $L$-packet at $\tau_\infty$.
Finally, Proposition 12.2.2, Corollary 12.2.4 and Proposition 12.3.3 (the fact that $\pi$ is the symmetric $2n$-th power of a [*cohomological*]{} Hilbert modular form implies that hypothesis (2) of this proposition is satisfied) of [@patrikis:tate] imply the existence of a regular algebraic cuspidal automorphic representation $\sigma$ of ${\operatorname{GSp}}(2n, {\mathbb{A}}_F)$ such that if $v$ is either an infinite place or a finite place such that $\sigma_v$ is unramified then $\sigma_v|_{{\operatorname{Sp}}(2n, F_v)}$ contains $\tau'_v$. Moreover, the discrete series $\sigma_\infty$ is holomorphic or else its restriction to ${\operatorname{Sp}}(2n, {\mathbb{R}})$ would not contain the holomorphic discrete series $\tau'_\infty$.
One reason to seek a formula for symmetric power $\mathcal{L}$-invariants in terms of $p$-adic families on a certain reductive group is that it might yield a proof of a trivial zero conjecture for symmetric powers of Hilbert modular forms following the template of the Greenberg–Stevens proof of the Mazur–Tate–Teitelbaum conjecture. This method for proving such conjectures requires the existence of $p$-adic $L$-functions for these $p$-adic families. Whereas there has been little progress towards such $p$-adic families of $p$-adic $L$-functions on symplectic groups in general, the work of Eischen–Harris–Li–Skinner is expected to yield such $p$-adic $L$-functions in the case of unitary groups. We will therefore present a computation (however, under some restrictions) of the symmetric power $\mathcal{L}$-invariants using unitary groups.
For a CM extension $E$ of a totally real field $F$, let $U_n$ be the unitary group defined in [@bellaiche-chenevier:selmer Definition 6.2.2] which is definite at every finite place and if $n\neq 2{\text{ (mod }4)}$ is quasi-split at every nonsplit place of $F$. We denote by ${\operatorname{BC}}$ the local base change (see, for example, [@shin:clay09 §2.3]). Assume the following conjecture on strong base change that would follow from stabilization of the trace formula.
\[t:GL(n) to U\_n\] Let $\pi$ be a regular algebraic conjugate self-dual cuspidal automorphic representation of ${\operatorname{GL}}(n, {\mathbb{A}}_E)$ such that $\pi_v$ is the base change from $U$ of square integrable representations at ramified places of $E$ and is either unramified or the base change from $U$ of square integrable represenations at places of $F$ which are inert in $E$. Suppose, moreover, that for at least one inert prime the local representation is not unramified. Then there exists a (necessarily cuspidal) automorphic representation $\Pi$ of $U_n({\mathbb{A}}_F)$ such that $\Pi_w={\operatorname{BC}}(\pi_v)$ for all places $v$.
We will use this conjecture to transfer symmetric powers of Hilbert modular forms to unitary groups.
\[p:sym hmf to unitary\] Assume Conjecture \[t:GL(n) to U\_n\]. Let $\pi$ be an Iwahori spherical cohomological non-CM Hilbert modular form over a number field $F$ in which $p$ splits completely. Suppose that there exist places $w_1$ and $w_2$ not above $p$ with the property that $\pi_{w}$ is special for $w\in\{w_1,w_2\}$ and suppose that ${\operatorname{Sym}}^n\pi$ is cuspidal automorphic over $F$. Then there exists a CM extension $E/F$ in which $p$ splits completely and a Hecke character $\psi$ of $E$ such that $\psi\otimes{\operatorname{BC}}_{E/F}{\operatorname{Sym}}^n\pi$ is the base change of a cuspidal automorphic representation of $U_n({\mathbb{A}}_F)$.
There exists $\alpha\in \{\varpi_{v_1}, \varpi_{v_2}, \varpi_{v_1}\varpi_{v_2}\}$ having a square root in $\mathbb{F}_{p^{[F:{\mathbb{Q}}]}}$. Then $E=F(\sqrt{\alpha})$ is ramified over $F$ only at $w_1$ or $w_2$ (or both) and $p$ splits completely in $E$.
Suppose that $\Pi={\operatorname{BC}}_{E/F}{\operatorname{Sym}}^n\pi$ is cuspidal automorphic. Then $\Pi^{c\vee}\cong \Pi\otimes
{\operatorname{BC}}_{E/F}\omega_\pi^{-n}$. Choose a Hecke character $\psi:E^\times\backslash {\mathbb{A}}_E^\times\to
{\mathbb{C}}^\times$ such that $\psi|_{{\mathbb{A}}_F^\times} =
\omega_\pi^{-n}$. Then $(\psi\otimes\Pi)^{c\vee}\cong
\psi^{c\vee}\omega_\pi^{-n}\otimes\Pi\cong
\psi\otimes\Pi$. Since ${\operatorname{Sym}}^n\pi$ is Iwahori spherical it follows that $\Pi$ is Iwahori spherical. Moreover, if $v$ is a ramified place of $E/F$, $\pi_v$ is special by assumption and thus $\Pi_v$ is the base change from $U$ of a square integrable representation. Finally, at every finite place $\Pi_v$ is either unramified or special. Therefore, the hypotheses of Conjecture \[t:GL(n) to U\_n\] are satisfied and the conclusion follows.
It remains to show that $\Pi\cong{\operatorname{Sym}}^n{\operatorname{BC}}_{E/F}\pi$ is cuspidal automorphic. Writing $\pi_E={\operatorname{BC}}_{E/F}\pi$ we note that $\pi_E$ is cuspidal since $\pi$ is not CM. Therefore one may apply [@ramakrishnan:remarks-sym Theorem A’] to study the cuspidality of ${\operatorname{Sym}}^n\pi_E$. If $\pi_E$ is dihedral then $\pi_E\cong{\operatorname{Ind}}_{L}^E\eta$ for some quadratic extension $L/E$ contradicting the fact that $\rho_{\pi,p}$ is Lie irreducible.
If $\pi_E$ is tetrahedral then ${\operatorname{Sym}}^3\pi$ is cuspidal but ${\operatorname{BC}}_E{\operatorname{Sym}}^3\pi$ is not which implies that ${\operatorname{Sym}}^3\pi\cong{\operatorname{Ind}}_E^F\tau$ for an automorphic representation $\tau$ of ${\operatorname{GL}}(2, {\mathbb{A}}_F)$. It suffices to show that there exists a Galois representation attached to $\tau$ since then ${\operatorname{Sym}}^3\pi$ is not Lie irreducible contradicting Lemma \[l:hmf lie irreducible\]. Since a real place $v$ of $F$ splits completely in $E$, the $L$-parameter of $({\operatorname{Ind}}_E^F\tau)_v$ is the direct sum of the $L$-parameters of the local components of $\tau$ at the complex places over $v$. Therefore $\tau$ is regular algebraic, necessarily cuspidal, representation of ${\operatorname{GL}}(2, {\mathbb{A}}_E)$. Galois representations have been attached to such cuspidal representations by Harris–Lan–Taylor–Thorne, although one does not need such a general result; indeed, by twisting one may guarantee that the central character of $\tau$ is trivial in which case the Galois representation has been constructed by Mok ([@mok:siegel-hilbert]).
If $\pi_E$ is icosahedral, i.e., ${\operatorname{Sym}}^6\pi_E$ is not cuspidal but all lower symmetric powers are cuspidal then ${\operatorname{Sym}}^6\pi_E=\eta\boxplus\eta'$ where $\eta$ is a cuspidal representation of ${\operatorname{GL}}(3, {\mathbb{A}}_F)$ and $\eta'$ is a cuspidal representation of ${\operatorname{GL}}(4, {\mathbb{A}}_F)$ (cf. [@ramakrishnan:remarks-sym §4]). Lemmas 4.10 and 4.18 of [@ramakrishnan:remarks-sym] carry over to this setting and show that $\eta$ and $\eta'$ are regular, algebraic conjugate self-dual representations and thus that ${\operatorname{Sym}}^6\rho_{\pi,p}|_{G_E}$ is decomposable, which contradics Lemma \[l:hmf lie irreducible\].
Finally, if $\pi_E$ is octahedral then ${\operatorname{Sym}}^4\pi_E$ is not cuspidal. As in the case of ${\operatorname{Sym}}^6$ we may write ${\operatorname{Sym}}^4\pi_E=\eta\boxplus\eta'$ where $\eta$ is a regular algebraic cuspidal conjugate self-dual representation of ${\operatorname{GL}}(2, {\mathbb{A}}_E)$ and $\eta'$ is a regular algebraic cuspidal conjugate self-dual representation of ${\operatorname{GL}}(3, {\mathbb{A}}_E)$ yielding a contradiction as above.
$p$-adic families and Galois representations {#sect:eigs}
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As previously mentioned, we will compute explicitly the arithmetic $\mathcal{L}$-invariants attached to $V_{2n}$ using triangulations of $(\varphi,\Gamma)$-modules attached to analytic Galois representations on eigenvarieties. Which eigenvariety we choose will be dictated by the requirement that the formula from Lemma \[l:rank 1 L invariant\] needs to make sense. This translates into a lower bound for the rank of the reductive group whose eigenvariety we will use. In this section we make explicit the eigenvarieties under consideration and the analytic Hecke eigenvalues whose derivatives will control the arithmetic $\mathcal{L}$-invariant.
Urban’s eigenvarieties
----------------------
We recall Urban’s construction of eigenvarieties from [@urban:eigenvarieties Theorem 5.4.4], to which we refer for details. Let $G$ be a split reductive group over ${\mathbb{Q}}$ such that $G({\mathbb{R}})$ has discrete series representations. (This condition is satisfied by the restriction to ${\mathbb{Q}}$ of ${\operatorname{GL}}(2)$ and ${\operatorname{GSp}}(2n)$ over totally real fields and compact unitary groups attached to CM extensions.) We denote by $T$ a maximal (split) torus, by $B$ a Borel subgroup containing $T$ and by $B^-$ the opposite Borel, obtained as $B^-=w_BBw_B^{-1}$ for some $w_B\in W_{G,T}=N_G(T)/T$. For a character $\lambda\in X^\bullet(T)$ let $V_\lambda={\operatorname{Ind}}_{B^-}^G \lambda$ induced from the opposite Borel, the irreducible algebraic representation of $G$ of highest weight $\lambda$ with respect to the set of positive roots defined by the Borel $B$. Its dual is $V_\lambda^\vee\cong {\operatorname{Ind}}_B^G \left((-w_B)\lambda\right)$.
Let $\pi$ be a cuspidal automorphic representation of $G({\mathbb{A}}_{{\mathbb{Q}}})$ and assume that it has regular cohomological weight $\lambda\in X^\bullet(T)$, i.e. that $\pi$ can be realized in the cohomology of the local system $V_\lambda^\vee$; equivalently $\pi_\infty$ has central character equal to the central character of $\lambda$ and has a twist which is the discrete series representation of $G({\mathbb{R}})$ of Harish-Chandra parameter $\lambda$ (cf. [@urban:eigenvarieties §1.3.4]). For example, if $f$ is a classical modular form of weight $k\geq 2$ (as before $w=2-k$) then the cohomological weight of the associated automorphic representation is $(k-2, 0)\in X^\bullet(\mathbb{G}_m^2)$ as the Eichler–Shimura isomorphism implies that $f$ appears in the cohomology of $({\operatorname{Sym}}^{k-2})^\vee$.
Let ${\mathbb{A}}_{{\mathbb{Q}},f}^{(p)}$ be the finite adeles away from $p$ and let $K^p\subset G({\mathbb{A}}_{{\mathbb{Q}},f}^{(p)})$ be a compact open subgroup. Let $I_m'$ be a pro-$p$-Iwahori subgroup of $G({{\mathbb{Q}}_p})$ such that $\pi_f^{K^pI_m'}\neq 0$. We denote by $\mathcal{H}^p$ the Hecke algebra of $K^p$, $\mathcal{U}_p$ the Hecke algebra of the Iwahori subgroup $I_m=\{g\in G({{\mathbb{Z}}_p}):g{\text{ (mod }p)}\in B({{\mathbb{Z}}_p})\}$ and $\mathcal{H}=\mathcal{H}^p\otimes \mathcal{U}_p$. A $p$*-stabilization* $\nu$ of $\pi$ is an irreducible constituent of $\pi_f^{K^pI_m'}\otimes \varepsilon^{-1}$ as a $\mathcal{H}$-representation, where $\varepsilon$ is a character of $I_m/I_m'\cong T({\mathbb{Z}}/p^m {\mathbb{Z}})$ acting on $\pi_f^{K^pI_m'}$ (cf. [@urban:eigenvarieties p. 1689]). For $\nu$ finite slope of weight $\lambda$, Urban rescales by $|\lambda(t)|_p^{-1}$ the eigenvalue of the Hecke operator $U_t=I_mtI_m\in
\mathcal{U}_p$, where $t\in T({\mathbb{Q}}_p)$ is such that $|\alpha(t)|_p\leq1$ for all positive roots $\alpha$ (*finite slope* means the $U_t$ act invertibly; cf. [@urban:eigenvarieties p. 1689–1690]). This rescaled eigenvalue is denoted $\theta(U_t)$. The $p$-stabilization $\nu$ of $\pi$ is said to have *non-critical slope* if there is a Hecke operator $U_t$, with $|\alpha(t)|_p<1$ for all positive roots $\alpha$, such that for every simple root $\alpha$ $$v_p(\theta(U_t))<(\lambda(\alpha^\vee)+1)v_p(\alpha(t)).$$
\[t:urban\] Suppose $\nu$ has non-critical slope. Then there exists a rigid analytic “weight space” $\mathcal{W}$ of the weight $\lambda$ of $\pi$, a rigid analytic variety $\mathcal{E}$, a generically finite flat morphism $w:\mathcal{E}\to \mathcal{W}$ and a homomorphism $\theta:\mathcal{H}\to \mathcal{O}_{\mathcal{E}}$ such that:
1. there exists a dense set of points $\Sigma\subset \mathcal{E}$ and
2. for every point $z\in \Sigma$ there exists an irreducible cuspidal representation $\pi_z$ of $G({\mathbb{A}}_{{\mathbb{Q}}})$ of weight $w(z)$, with Iwahori spherical local representations at places over $p$, and a $p$-stabilization $\nu_z$ such that $\theta(T)$ is the normalized eigenvalue of $T\in \mathcal{H}$ on $\nu_{z}$ and
3. there exists $z_0\in \Sigma$ such that $\pi_{z_0}=\pi$ and $\nu_{z_0}=\nu$.
Suppose $F$ is a totally real field in which $p$ splits completely and $G/F$ is a split reductive group over $F$. Let $G'={\operatorname{Res}}_{F/{\mathbb{Q}}}G$ be the Weil restriction of scalars. Then a cuspidal automorphic representation $\pi$ of $G({\mathbb{A}}_F)$ can be thought of as a cuspidal representation of $G'({\mathbb{A}}_{{\mathbb{Q}}})$ and we may apply Urban’s construction above. Let $T$ be a maximal torus of $G$ and $T'={\operatorname{Res}}_{F/{\mathbb{Q}}}T$. Then $X^\bullet(T')\cong
X^\bullet(T)\cong\oplus_{G_{F/{\mathbb{Q}}}}X^\bullet(T_{/{\mathbb{Q}}_p})$ gives a one-to-one correspondence between the weights of automorphic representations of $G({\mathbb{A}}_F)$ and tuples $(w_1,\ldots)$ where $w_i\in X^\bullet(T_{/{\mathbb{Q}}_p})$. For each place $v\mid p$, given $t\in T({\mathbb{Q}}_p)$ one gets a Hecke operator $U_t$ acting on $\pi_v$ and the refinement $\nu_v$ with eigenvalue $\alpha_v$. Then the eigenvalue of $U_t$ on the refinement $\nu$ of $\pi$ thought of as a representation of $G'({\mathbb{A}}_{{\mathbb{Q}}})$ is $\prod_v \alpha_v$ and the slope of the renormalized eigenvalue is $$v_p(\theta(U_t))=\sum_v v_p(\lambda_v(t)) + \sum_v v_p(\alpha_v)=\sum_v v_p(\theta_v(U_t))$$ where $\theta_v(U_t)$ is the eigenvalue of $U_t$ on $\pi_v$ renormalized by the weight $\lambda_v$. If $\Phi$, $\Phi^+$ and $\Psi$ are the set of roots, positive roots, and simple roots of $G_{/{\mathbb{Q}}_p}$ with respect to a Borel, then the set of roots, positive roots, and simple roots of $G$ are $\prod_i \Phi$, $\prod_i \Phi^+$, and $\{0\oplus\cdots\oplus \alpha\oplus\cdots\oplus 0|\alpha\in \Psi\}$.
In the next sections, we will make explicit first what non-critical slope means in the case of the groups under consideration, and second, the relationship between the renormalized action of Hecke operators on $p$-stabilizations and their action on smooth representations of $p$-adic groups. The latter will allow us to study global triangulations of Galois representations in terms of analytic Hecke eigenvalues.
Eigenvarieties for Hilbert modular forms
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Suppose $\pi$ is a cohomological Hilbert modular form of infinity type $(k_1,\ldots, k_d, w)$ over a totally real field $F$ of degree $d$ in which $p$ splits completely. The representation $\pi$ has cohomological weight $\bigoplus_i ((-w+k_i-2)/2, (-w-k_i+2)/2)$. For each place $v\mid p$, let $\alpha_v$ be an eigenvalue of $U_p$ (corresponding to $t=\begin{pmatrix} 1&\\ &p^{-1}\end{pmatrix}$) acting on a refinement $\nu$ of $\pi$ in which case $\theta(U_p)=p^{(w+k_i-2)/2}\alpha_v$. Then $v_p(\theta(U_p))=\sum_i ((w+k_i-2)/2+v_p(\alpha_v))$. The simple roots of ${\operatorname{GL}}(2)_{/F}$ are of the form $(1,-1)_i$ where $i$ is an infinite place. As $((-w+k-2)/2,(-w-k+2)/2)\cdot(1,-1) = k-2$, the refinement $\nu$ has noncritical slope if and only if $$\sum_i ((w+k_i-2)/2+v_p(\alpha_v))<k_i-1$$ for every $i$, which is equivalent to $$\sum_i ((w+k_i-2)/2+v_p(\alpha_v))<\min(k_i)-1$$ We remark that for modular forms of weight $k$ (with $w=2-k$) this is the usual definition of noncritical slope.
\[l:hmf eigenvariety\] Let $\pi$ be a cohomological Hilbert modular form, $\nu$ a refinement of $\pi$ of noncritical slope as above, and $\mathcal{E}\to \mathcal{W}$ Urban’s eigenvariety around $(\pi,\nu)$. Shrinking $\mathcal{W}$ if necessary, there exists an analytic Galois representation $\rho_{\mathcal{E}}:G_F\to {\operatorname{GL}}(2,
\mathcal{O}_{\mathcal{E}})$ such that for $z\in \Sigma$, $z\circ \rho_{\mathcal{E}}=\rho_{\pi_z}$. If $\pi$ has Iwahori level at $v\mid p$ then $\rho_{\mathcal{E}}$ admits a refinement in the sense of §\[sect:refined galois\].
The existence of $\rho_{\mathcal{E}}$ is done by a standard argument, but we reproduce it here for convenience. If $v\nmid
p$ is a place such that $\pi_w$ is unramified and $z\in \Sigma$ then $z(\theta(T_v))$ is equal to the eigenvalue of $T_v$ on $\pi_z$. Local-global compatibility implies that ${\operatorname{Tr}}\rho_{\pi_z}({\operatorname{Frob}}_v)=z\circ\theta(T_v)$ and so $\theta(T_v)$ is an analytic function $T({\operatorname{Frob}}_v)$ specializing at points $z\in\Sigma$ to ${\operatorname{Tr}}\rho_{\pi_z}({\operatorname{Frob}}_v)$. As $\{{\operatorname{Frob}}_v|v\nmid Np\}$ is dense in $G_F$ we may define by continuity $T(g) = \lim T({\operatorname{Frob}}_v)$ where ${\operatorname{Frob}}_v\to g$. The function $T:G_F\to \mathcal{O}_{\mathcal{E}}$ is an analytic two-dimensional pseudorepresentation. Since $\pi$ is cuspidal, $\rho_\pi$ is irreducible (cf. [@skinner:hilbert p. 256]). This implies that shrinking $\mathcal{W}$, $T$ is the trace of an analytic Galois representation $\rho_{\mathcal{E}}$ (cf. [@jorza:thesis 4.2.6]).
If $\pi$ has Iwahori level at $v\mid p$ then $\pi_z$ will also have Iwahori level at $v\mid p$ for $z\in \Sigma$. Thus $\rho_{z,v}$ for $v\mid p$ is either unramified or Steinberg. If Steinberg then the unique refinement of $\pi_{z,v}$ has slope $k_v-2+w/2$. Shrinking the neighborhood $\mathcal{W}$ we may guarantee that the slope is constant. The classical points on $\mathcal{E}$ are dense and the ones satisfying $k_v-2+w/2$ constant lie in a subvariety and so there is a dense set $\Sigma$ of points on $\mathcal{E}$ corresponding to cohomological regular Hilbert modular forms.
Suppose $z\in \Sigma$ in which case $\rho_{\pi_z,p}|_{G_{F_v}}$ is crystalline for $v\mid p$. By local-global compatibility the eigenvalues of $\varphi_{{\operatorname{cris}}}$ acting on ${\operatorname{D}}_{{\operatorname{cris}}}(\rho_{\pi_z,p}|_{G_{F_v}}$ are $\alpha_vp^{-1/2}$ and $\beta_vp^{-1/2}$ ($F_v\cong {\mathbb{Q}}_p$ so $\varpi_v$ can be chosen to be $p$) and by choice of central character for $\pi$ we deduce that $\alpha_v \beta_v=p^{-w}$. Writing $a_v(z) = z(\theta(U_p))$ we get an analytic function on $\mathcal{E}$ such that for $z\in \Sigma$, the eigenvalues of $\varphi_{{\operatorname{cris}}}$ are $\alpha_v p^{-1/2} = a_v(z) p^{(-w-k_v+1)/2}$ and $\beta_v p^{-1/2}=p^{-w-1/2}\alpha_v^{-1} = a_v(z)^{-1} p^{(-w+k_v-3)/2}$. Let $\kappa_{v,1}(z) = (w-k_v)/2$ and $\kappa_{v,2}(z) = (w+k_v-2)/2$; let $F_1(z) = a_v(z)^{-1}p^{3/2}$ and $F_2(z) = a_v(z)p^{1/2}$. Then for $z\in \Sigma$, $\kappa_{1, v}(z)<\kappa_{2,v}(z)$ are the Hodge–Tate weights of the crystalline representation $\rho_{\pi_z,p}|_{G_{F_v}}$ and the eigenvalues of $\varphi_{{\operatorname{cris}}}$ are $F_1(z)p^{\kappa_1(z)}$ and $F_2(z)p^{\kappa_2(z)}$ which implies that $\rho_{\mathcal{E}}|_{G_{F_v}}$ admits a refinement as desired.
\[c:hmf triangulation\] Let $\pi$ be as in Lemma \[l:hmf eigenvariety\]. For $v\mid
p$, assume that $\pi_v$ is Iwahori spherical; if $\pi_v$ is an unramified principal series assume that $\alpha_v\neq\beta_v$. Then there exists a global triangulation $$\mathscr{D}_{{\operatorname{rig}}}^\dagger(\rho_{\mathcal{E},
p}|_{G_{F_v}})\sim \begin{pmatrix} \delta_1&*\\
&\delta_2\end{pmatrix}$$ with $\delta_1(p) = a_v(z)^{-1}p^{3/2}$, $\delta_1(u)=u^{(w-k_v)/2}$, $\delta_2(p)=a_v(z)p^{-1/2}$ and $\delta_2(u) = u^{(w+k_v-2)/2}$.
We only need to check that the refinement attached to the triangulation of $\rho_{\pi,p}|_{G_{F_v}}$ is noncritical and regular by Theorem \[t:triangulation\]. Noncriticality is immediate from the requirement that the Hodge–Tate weights be ordered increasingly (cf. [@liu:triangulation §5.2]). Regularity is equivalent to the fact that ${\varphi}_{{\operatorname{cris}}}$ has distinct eigenvalues, i.e. that $\alpha_v\neq \beta_v$.
Eigenvarieties for ${\operatorname{GSp}}(2n)$ {#sect:eigenvarieties for symplectic}
---------------------------------------------
Suppose $F/{\mathbb{Q}}$ is a number field in which $p$ splits completely, $g\geq 2$ an integer, and $\pi$ a cuspidal automorphic representation of ${\operatorname{GSp}}(2g, {\mathbb{A}}_F)$ such that for $v\mid\infty$ (writing $v\mid p$ for the associated place above $p$) the representation $\pi_v$ is the holomorphic discrete series representation with Langlands parameter, under the $2^g$-dimensional spin representation ${\operatorname{GSpin}}(2g+1, {\mathbb{C}}){\hookrightarrow}{\operatorname{GL}}(2^g,{\mathbb{C}})$, uniquely identified by $$z\mapsto |z|^{\mu_0}{\operatorname{diag}}((z/\overline{z})^{1/2\sum \varepsilon(i)(\mu_{v,i}+g+1-i)})$$ where $\varepsilon:\{1,2,\ldots,g\}\to \{-1,1\}$. If $V_\mu$ is the algebraic representation of ${\operatorname{GSp}}(2g)$ with highest weight $\mu_v=(\mu_{v,1},\ldots, \mu_{v,g}; \mu_0)$ then $H^\bullet(\mathfrak{gsp}_{2g}, {\operatorname{SU}}(g), \pi_v\otimes V_\mu^\vee)\neq 0$ and so $\pi_v$ has cohomological weight $\mu$ in the sense of Urban.
We denote by ${\operatorname{Iw}}$ the Iwahori subgroup of ${\operatorname{GSp}}(2g)$ of matrices which are upper triangular mod $p$. Let $\beta_0 = t(1,\ldots,1;p^{-1})$ and $\beta_j=t(1,\ldots,1,p^{-1},\ldots,p^{-1};p^{-2})$ where $p^{-1}$ appears $j$ times. The double coset $[{\operatorname{Iw}}\beta_i{\operatorname{Iw}}]$ acts on the finite dimensional vector space $\pi_v^{{\operatorname{Iw}}}$ for $v\mid p$.
\[l:hecke local gsp\]Let $\chi=\chi_1\otimes\cdots\otimes\chi_g\otimes\sigma$ be an unramified character of the diagonal torus of ${\operatorname{GSp}}(2g, {\mathbb{Q}}_p)$ and assume that the unitary induction $\mu={\operatorname{Ind}}_B^{{\operatorname{GSp}}(2g, {\mathbb{Q}}_p)}\chi$ is irreducible. Then $\mu^{{\operatorname{Iw}}}$ is $2^g g!$ dimensional and has a basis in which the operator $U_t=[{\operatorname{Iw}}t{\operatorname{Iw}}]$ for any $t=(p^{a_1}, \ldots, p^{a_g}; p^{a_0})$ is upper triangular. The basis is $\{e_w\}$ indexed by the Weyl group elements $w=(\nu,\varepsilon)\in W$ in the Bruhat ordering. The diagonal element of $U_t$ corresponding to $e_w$ is $$p^{g(g+1)/4a_0-\sum (n+1-j)a_j}\sigma(p)^{a_0}\prod_{j=1}^g \chi_j(p)^{a_{\nu(j)}\textrm{ or }a_0-a_{\nu(j)}}$$ where the exponent depends on whether $\varepsilon(\nu(j))=1$ or $-1$.
In particular, the Hecke operators $U_{p,i}=[{\operatorname{Iw}}\beta_{g-i} {\operatorname{Iw}}]$ are upper triangular whose diagonal elements are $$\left\{\displaystyle p^{c_{i,\nu,\varepsilon}}\sigma^{-2}(p)\prod_{\nu(j)>i}\chi_j^{-1}(p)\prod_{\nu(j)\leq i,\varepsilon(j)=-1}\chi_j^{-2}(p)\right\}\text{ for }
1\leq i\leq g-1$$ and $$\left\{\displaystyle
p^{c_{g,\nu,\varepsilon}}\sigma^{-1}(p)\prod_{\varepsilon(j)=-1}\chi_{j}^{-1}(p)\right\}\text{ for }i=g.$$ Here $$c_{i,\nu,\varepsilon}=\sum_{\nu(j)>i}(n+1-j)+2\sum_{\nu(j)\leq
i,\varepsilon(j)=-1}(n+1-j)-g(g+1)/2$$ and $$c_{g,\nu,\varepsilon}=\sum_{\varepsilon(j)=-1}(n+1-j)-g(g+1)/4.$$
The proof is inspired by [@genestier-tilouine Proposition 3.2.1]. Indeed, the Iwahori decomposition gives ${\operatorname{GSp}}(2g, {\mathbb{Q}}_p) = \cup_{w\in W} Bw{\operatorname{Iw}}$. Writing $e_w$ for $w\in W$ the function defined on the cell $Bw{\operatorname{Iw}}$ gives a basis for $\mu^{{\operatorname{Iw}}}$ and the fact that the Hecke action is upper triangular with respect to the basis $\{e_w\}$ in the Bruhat ordering is the first part of the proof of [@genestier-tilouine Proposition 3.2.1]. The computation of the diagonal elements only requires replacing the parahoric subgroup by the Iwahori subgroup and therefore the quotient of the Weyl group by the Weyl group of the parabolic with the whole Weyl group.
Let $\widetilde{\chi}=\chi\otimes\delta^{1/2}$ where $\delta$ is the modulus character of the upper triangular Borel subgroup, given by $\delta(t) =|\rho(t)|^2_p$. Write $^w
\widetilde{\chi}(t)=\delta^{1/2}(t)\delta^{1/2}(w^{-1}tw)\widetilde{\chi}(w^{-1}tw)$ and $^w\chi(t)=\chi(w^{-1}tw)$. Let $S$ be the Satake isomorphism from the Iwahori–Hecke algebra to the spherical Hecke algebra of the maximal torus $T$. Analogously to the second part of [@genestier-tilouine Proposition 3.2.1], the eigenvalue of $U_t$ on the one-dimensional subspace of $\mu^{{\operatorname{Iw}}}$ generated by $e_w$ is equal to $^w
\widetilde{\chi}(S(t))=\delta^{1/2}(t)\chi(w^{-1}tw)$. We now make explicit these eigenvalues. Recall that $w^{-1}tw=(p^{a'_{\nu(1)}}, \ldots, p^{a'_{\nu(g)}}; p^{a_0})$ where $a'_i = a_i$ if $\varepsilon(i)=1$ and $a'_i=a_0-a_i$ if $\varepsilon(i)=-1$. Moreover, $\delta^{1/2}(t) =
p^{g(g+1)/4a_0-\sum (n+1-j)a_j}$. Thus the $e_w$ eigenvalue of $U_t$ is $$p^{g(g+1)/4a_0-\sum (n+1-j)a_j}\sigma(p)^{a_0}\prod_{j=1}^g \chi_j(p)^{a_{\nu(j)}\textrm{ or }a_0-a_{\nu(j)}}$$ where the exponent depends on whether $\varepsilon(\nu(j))=1$ or $-1$.
We remark that in the final computation of $\mathcal{L}$-invariants the constants $c_{i,\nu,\varepsilon}$ do not matter.
\[l:siegel-hilbert noncritical slope\] Let $\pi$ be a cuspidal automorphic representation of ${\operatorname{GSp}}(2g,
{\mathbb{A}}_F)$ of cohomological weight $\lambda_v=\bigoplus
(\mu_{v,1},\ldots, \mu_{v,g};\mu_0)$ as above. Suppose $\nu$ is a $p$-stabilization of $\pi$ and $t\in
{\operatorname{Res}}_{F/{\mathbb{Q}}}T({\mathbb{Q}}_p)$ with associated Hecke operator $U_{v,t}=[{\operatorname{Iw}}t{\operatorname{Iw}}]$ acting on $\pi_v$. There exists an integer $m$ such that the refinement $\nu\otimes|\cdot|_{{\mathbb{A}}_F}^m$ of $\pi\otimes|\cdot|_{{\mathbb{A}}_F}^m$ has noncritical slope with respect to $U_t$.
Recall Urban’s convention that $\theta(U_{v,t}) = |\lambda_v(t)|_p^{-1}a_{v,t}$ where $a_{v,t}$ is the $U_{v,t}$ eigenvalue on the refinement $\nu$. The condition that $\nu$ have noncritical slope is that $$\sum_{v\mid
p}v_p(\theta(U_{v,t}))<(\lambda(\alpha^\vee)+1)v_p(\alpha(t))$$ for every simple root $\alpha$ of ${\operatorname{Res}}_{F/{\mathbb{Q}}}{\operatorname{GSp}}(2g)$. This is equivalent to $$\sum_{v\mid
p}\left(v_p(\lambda_v(t))+v_p(\alpha_{v,t})\right)<\min
\left((\mu_{v,i}-\mu_{v,i+1}+1)(a_i-a_{i+1}),
2(2\mu_{v,g}+1)a_g\right)$$ Suppose one replaces $\pi$ by $\pi_m=\pi\otimes|\cdot|^m$ for an integer $m$. Then the cohomological weight of $\pi_m$ is $\lambda_m'=\oplus_{v\mid p} (\mu_{v,i}; \mu_0+m)$ which implies that $v_p(\lambda_v'(t))=v_p(\lambda_v(t))+ma_0/2$. $v_p(\alpha_{v,t}') = v_p(\alpha_{v,t})+m$. Lemma \[l:hecke local gsp\] implies that $v_p(\alpha_{v,t}')=v_p(\alpha_{v,t})-ma_0$ because $(\chi_1\times\cdots\times\chi_g\rtimes\sigma)\otimes\eta\cong \chi_1\times\cdots\times\chi_g\rtimes\sigma\eta$. The result is now immediate as the right-hand side of the inequality does not change with $m$.
Before discussing analytic Galois representations over Siegel eigenvarieties we explain how to attach Galois representations to cohomological cuspidal automorphic representations of ${\operatorname{GSp}}(2n, {\mathbb{A}}_F)$ for a totally real $F$ using the endoscopic classification of cuspidal representations of symplectic groups due to Arthur, which is conditional on the stabilization of the twisted trace formula.
\[t:galois gsp\] Let $\pi$ be a cuspidal representation of ${\operatorname{GSp}}(2n, {\mathbb{A}}_F)$ of cohomological weight $\oplus (\mu_{v,1},\ldots, \mu_{v,n};\mu_0)$.
1. If $n=2$ there exists a spin Galois representation $\rho_{\pi,{\operatorname{spin}},p}:G_F\to {\operatorname{GSp}}(4, \overline{{\mathbb{Q}}}_p)$ such that if $v\nmid\infty p$ with $\pi_v$ is unramified then ${\operatorname{WD}}(\rho_{\pi,{\operatorname{spin}},p}|_{G_{F_v}})^{{\operatorname{Fr-ss}}}\cong {\operatorname{rec}}_{{\operatorname{GSp}}(4)}(\pi_v\otimes|\cdot|^{-3/2})$. If $v\mid p$ and $\pi_v$ is unramified then the crystalline representation $\rho_{\pi,{\operatorname{spin}},p}|_{G_{F_v}}$ has Hodge–Tate weights $(\mu_0-\mu_{v,1}-\mu_{v,2})/2+\{0,\mu_{v,2}+1, \mu_{v,1}+2, \mu_{v,1}+\mu_{v,2}+3\}$.
2. If $n\geq 2$ there exists a standard Galois representation $\rho_{\pi,{\operatorname{std}}, p}:G_F\to {\operatorname{GL}}(2n+1,{\mathbb{C}})$ such that if $v\nmid \infty p$ with $\pi_v$ the unramified principal series $\chi_1\times\cdots\times\chi_g\rtimes\sigma$ then $\rho_{\pi,{\operatorname{std}},p}|_{G_{F_v}}$ is unramified and local-global compatibility is satisfied. If $v\mid p$ and $\pi_v$ is the unramified principal series $\chi_1\times\cdots\times\chi_g\rtimes\sigma$ then $\rho_{\pi,{\operatorname{std}},p}|_{G_{F_v}}$ is crystalline with Hodge–Tate weights $0, \pm(\mu_{v,i}+n+1-i)$ and the eigenvalues of ${\varphi}_{{\operatorname{cris}}}$ are $\chi_1(p),\ldots,\chi_n(p),1,\chi_1^{-1}(p),\ldots,\chi_n^{-1}(p)$.
The first part follows from [@mok:siegel-hilbert Theorem 3.5].
Let $\pi_0$ be any irreducible constituent of $\pi|_{{\operatorname{Sp}}(2n,
{\mathbb{A}}_F)}$. If $v\mid\infty$ then $\pi_{0,v}$ will then be a discrete series representation with $L$-parameter uniquely defined by $$z\mapsto {\operatorname{diag}}((z/\overline{z})^{\mu_{v,i}+n+1-i}, 1, (z/\overline{z})^{-\mu_{v,i}-(n+1-i)})$$ cohomological weight $\oplus (\mu_{v,i})$.
Arthur’s endoscopic classification implies the existence of a transfer of $\pi_0$ from ${\operatorname{Sp}}(2n)$ to ${\operatorname{GL}}(2n+1)$ as follows (cf. [@scholze:torsion Corollary V.1.7]). Let $\eta:{}^L\!{\operatorname{Sp}}(2n)\to{}^L\!{\operatorname{GL}}(2n+1)$ be the standard inclusion of ${\operatorname{SO}}(2n+1,{\mathbb{C}}){\hookrightarrow}{\operatorname{GL}}(2n+1,{\mathbb{C}})$. There exists a partition $2n+1=\sum_{i=0}^r\ell_i k_i$ and cuspidal automorphic representations $\Pi_i$ of ${\operatorname{GL}}(k_i,{\mathbb{A}}_F)$ such that:
1. $\Pi_i^\vee\cong\Pi_i$ for all $i$;
2. writing ${\varphi}_{\pi_{0,v}}$ for the $L$-parameter of $\pi_{0,v}$ and ${\varphi}_{\Pi_{i,v}}$ for the $L$-parameter of $\Pi_{i,v}$, if $v$ is archimedean or $\pi_{0,v}$ is unramified then $$\eta\circ
{\varphi}_{\pi_{0,v}}=\oplus_{i=1}^r\oplus_{j=1}^{\ell_i}{\varphi}_{\Pi_{i,v}}|\cdot|_v^{j-(\ell_i+1)/2}$$
Next, if $\tau$ is a cuspidal automorphic representation of ${\operatorname{GL}}(k,{\mathbb{A}}_F)$ with cohomological weight $\oplus
(a_{v,1},\ldots, a_{v,k})$ and $\tau^\vee\cong\tau \otimes\chi$ for some Hecke character $\chi$ such that $\chi_v(-1)$ is independent of $v$ then by [@blght:calabi-yau-2 Theorem 1.1] there exists a continuous Galois representation $\rho_\tau:G_F\to {\operatorname{GL}}(k,\overline{{\mathbb{Q}}}_p)$ such that:
1. if $\tau_v$ is unramified and $v\nmid p$ then ${\operatorname{WD}}(\rho_\tau|_{G_{F_v}})^{{\operatorname{Fr-ss}}}\cong
{\operatorname{rec}}(\pi_v\otimes|\cdot|^{(1-k)/2})$;
2. if $\tau_v$ is unramified and $v\mid p$ then $\rho_\tau|_{G_{F_v}}$ is crystalline with Hodge–Tate weights $-a_{v,k+1-i}+k-i$ and ${\operatorname{WD}}(\rho_\tau|_{G_{F_v}})^{{\operatorname{Fr-ss}}}\cong
{\operatorname{rec}}(\pi_v\otimes|\cdot|^{(1-k)/2})$. (The Hodge–Tate weights are $(k-1)/2+$ the Harish-Chandra parameter.)
The discrepancy between the description above and [@blght:calabi-yau-2 Theorem 1.1] arises because we defined cohomological weight as the highest weight of the algebraic representation with the same central and infinitesimal characters as the discrete series, whereas in [@blght:calabi-yau-2 Theorem 1.1] one uses the dual.
We will apply this to (a suitable twist of) $\Pi_i$ and denote $\rho_{\Pi_i}$ the resulting Galois representation. We define $$\rho_{\pi,{\operatorname{std}},p}=\rho_{\pi_0}=\oplus_{i=1}^r\oplus_{j=1}^{\ell_i}\rho_{\Pi_i}\otimes|\cdot|^{(k_i-1)/2 + j - (\ell_i+1)/2}$$ First, from the description of Hodge–Tate weights above in the case of ${\operatorname{GL}}(k)$ it is immediate that the Hodge–Tate weights of the crystalline representation $\rho_{\pi_0,p}|_{G_{F_v}}$ for $v\mid p$ are the entries of the Harish-Chandra parameter of $\pi_{0,v}$ for the corresponding $v\mid\infty$. Each Galois representation $\rho_{\Pi_i}$ was twisted by $|\cdot|^{(k_i-1)/2}$ so that local-global compatibility holds without twisting. As a result, the eigenvalues of ${\varphi}_{{\operatorname{cris}}}$ on ${\operatorname{D}}_{{\operatorname{cris}}}(\rho_{\pi_0,p}|_{G_{F_v}})$ are $\chi_1(p),\ldots,\chi_n(p),1,\chi_1^{-1}(p),\ldots,\chi_n^{-1}(p)$, as desired. The statement at $v\nmid p$ with $\pi_{v}$ unramified is analogous.
\[l:gsp eigenvariety\] Let $\pi$ be a cohomological cuspidal automorphic representation of ${\operatorname{GSp}}(2g,{\mathbb{A}}_F)$, $\nu$ a $p$-stabilization of $\pi$ of noncritical slope and $\mathcal{E}\to \mathcal{W}$ Urban’s eigenvariety around $(\pi,\nu)$. If $\rho_{\pi,{\operatorname{std}},p}$ is irreducible, shrinking $\mathcal{W}$, there exists an analytic Galois representation $\rho_{\mathcal{E},{\operatorname{std}}}:G_F\to {\operatorname{GL}}(2g+1,
\mathcal{O}_{\mathcal{E}})$ such that for $z\in \Sigma$, $z\circ
\rho_{\mathcal{E},{\operatorname{std}}}=\rho_{\pi_z,{\operatorname{std}},p}$. If $\pi$ has Iwahori level at $v\mid p$ then $\rho_{\mathcal{E},{\operatorname{std}}}$ admits a refinement in the sense of §\[sect:refined galois\].
In the case $g=2$, if $\rho_{\pi,{\operatorname{spin}},p}$ is irreducible one obtains an analogous analytic spin Galois representation $\rho_{\mathcal{E},{\operatorname{spin}}}:G_F\to {\operatorname{GSp}}(4, \mathcal{O}_{\mathcal{E}})$ which admits a refinement if $\pi_v$ is Iwahori spherical at $v\mid p$.
We will follow the proof of Lemma \[l:hmf eigenvariety\]. First, the existence of $\rho_{\mathcal{E},{\operatorname{spin}}}$ and $\rho_{\mathcal{E},{\operatorname{std}}}$ follows analogously.
Next, we need to show that if $\pi_v$ is Iwahori spherical at $v\mid p$ then there is a dense set of points $\Sigma'\subset\Sigma\subset \mathcal{E}$ such that if $z\in \Sigma'$ then $\pi_{z,v}$ is unramified at $v\mid p$.
Let $z\in \Sigma$ and let $\pi_z$ be the associated cuspidal representation. Suppose $\pi_{z,v}$ is not unramified. Since it is Iwahori spherical it follows from [@tadic:symplectic Theorem 7.9] that for every $v\mid p$, $\pi_{z,v}={\operatorname{Ind}}\chi$ where $\chi=\chi_1\times\cdots\times \chi_g\rtimes\sigma$ such that one of the following is satisfied:
1. $\chi_i^2=1$ but $\chi_i\neq 1$ for at least 3 indices $i$,
2. $\chi_i=|\ \ |^{\pm 1}$ for at least one index $i$, or
3. $\chi_i\chi_j^{\pm 1}=|\ \ |$ for at least one pair $(i,j)$ and choice of exponent.
Denote by $\alpha_{v,i}$ the eigenvalue of $[{\operatorname{Iw}}\beta_{g-i}{\operatorname{Iw}}]$ acting on $\pi_{z,v}$. There exists a permutation $\nu$ and a function $\varepsilon:\{1,\ldots,g\}\to \{-1,1\}$ such that $\alpha_{v,i}$ are given by Lemma \[l:hecke local gsp\]. Solving, one obtains $$\chi_{\nu^{-1}(i)}(p)^{\varepsilon(i)}=p^{c_{i,\nu,\varepsilon}-c_{i-1,\nu,\varepsilon}}\frac{\alpha_{v,i-1}}{\alpha_{v,i}}$$ for $1<i<g$, $\chi_{\nu^{-1}(g)}(p)=p^{2c_{g,\nu,\varepsilon}-c_{g-1,\nu,\varepsilon}}\alpha_{v,g-1}/\alpha_{v,g}^2$ and $\chi_{\nu^{-1}(1)}(p)^{\varepsilon(1)}=p^{\mu_0-c_{1,\nu,\varepsilon}}\alpha_{v,1}$ where the last equality comes from the fact that $\det\pi_{z,v}(p)=\chi_1\cdots\chi_g\sigma^2(p)=p^{\mu_0}$.
But $\alpha_{v,i}=|\lambda_v(\beta_{g-i})|_p\cdot\theta(U_{v,i})$ where $\lambda_v(\beta_{g-i})=p^{\mu_{v,1}+\cdots+\mu_{v,i}-\mu_0}$ for $1\leq i<g$ and $\lambda_v(\beta_0)=p^{(\mu_{v,1}+\cdots+\mu_{v,g}-\mu_0)/2}$. We deduce that $$\chi_{\nu^{-1}(i)}(p)^{\varepsilon(i)}=p^{c_{i,\nu,\varepsilon}-c_{i-1,\nu,\varepsilon}+\mu_{v,i}}\frac{\theta(U_{v,i-1})}{\theta(U_{v,i})}$$ for $1<i<g$, $\chi_{\nu^{-1}(g)}(p)=p^{2c_{g,\nu,\varepsilon}-c_{g-1,\nu,\varepsilon}+\mu_{v,g}} \theta(U_{v,g-1})/\theta(U_{v,g})^2$ and $\chi_{\nu^{-1}(1)}(p)^{\varepsilon(1)}=p^{\mu_0-c_{1,\nu,\varepsilon}+\mu_{v,1}-\mu_0}\theta(U_{v,1})= p^{-c_{1,\nu,\varepsilon}+\mu_{v,1}}\theta(U_{v,1})$.
The functions $\theta(U_{v,i})$ are analytic on $\mathcal{E}$ and so $v_p(\theta(U_{v,i}))$ is locally constant. By shrinking $\mathcal{E}$ we may even assume they are constant. Since $\pi_{z,v}$ is Iwahori spherical but ramified it follows from the conditions listed above that $v_p(\chi_i(p))\in \{0,1\}$ or $v_p(\chi_i(p)\chi_j(p)^{\pm1})=1$. This, however, implies certain linear combinations of the weights $\mu_{v,i}$ and $\mu_0$ are constant, which is a contradiction as $\mathcal{E}$ maps to a full-dimensional open set in the weight space.
Suppose $z\in \Sigma'$ in which case $\rho_{\pi_z,{\operatorname{std}},p}|_{G_{F_v}}$ is crystalline for $v\mid
p$. Consider the analytic functions $a_{v,i}(z)=z(\theta(U_{v,i}))$ on $\mathcal{E}$. Let $\kappa_{n+1}(z) = 0$, $\kappa_{n+1\pm
i}(z)=\pm(\mu_{v,n+1-i}(z)+i)$ for $1\leq i\leq n$; Theorem \[t:galois gsp\] these are the Hodge–Tate weights, arranged increasingly, of $z\circ\rho_{\mathcal{E}}|_{G_{F_v}}$. By local-global compatibility the eigenvalues of ${\varphi}_{{\operatorname{cris}}}$ acting on ${\operatorname{D}}_{{\operatorname{cris}}}(\rho_{\pi_z,p}|_{G_{F_v}})$ are $\chi_i(p)^{\pm 1}$ and $1$. We will use the formulae above to construct the analytic functions $F_k$.
Let $F_{n+1}(z)=1$. For $1<i<n$ let $$F_{n+1\pm (n+1-i)}(z)=\left(p^{c_{i,\nu,\varepsilon}-c_{i-1,\nu,\varepsilon}-i}\frac{a_{v,i-1}(z)}{a_{v,i}(z)}\right)^{\pm
1}.$$ Let $F_{n+1\pm (n+1-n)}(z) = \left(p^{2c_{g,\nu,\varepsilon}-c_{g-1,\nu,\varepsilon}-n} a_{v,g-1}/a_{v,g}^2\right)^{\pm 1}$ and $F_{n+1\pm (n+1-1)}(z)=\left(p^{-c_{1,\nu,\varepsilon}-1}a_{v,1}\right)^{\pm 1}$. Thus $p^{\kappa_{n+1\pm i}}F_{n+1\pm i}=\chi_{\nu^{-1}(i)}(p)^{\pm \varepsilon(i)}$ for $1<i\leq n$ and $p^{\kappa_{n+1\pm 1}}F_{n+1\pm 1}=\chi_{\nu^{-1}(1)}(p)^{\pm 1}$. By Theorem \[t:galois gsp\] these are the eigenvalues of ${\varphi}_{{\operatorname{cris}}}$ and so $\rho_{\mathcal{E},{\operatorname{std}},p}$ admits a refinement.
Finally, we need to construct a refinement for $\rho_{\mathcal{E},{\operatorname{spin}},p}$ in the genus $n=2$ case. The eigenvalues of ${\varphi}_{{\operatorname{cris}}}$ in this case acting on $\chi_1\times\chi_2\rtimes\sigma$ are $p^{-3/2}\times\{\sigma(p), \sigma(p)\chi_1(p), \sigma(p)\chi_2(p), \sigma(p)\chi_1(p)\chi_2(p)\}$. Let $\kappa_1 = (\mu_0-\mu_{v,1}-\mu_{v,2})/2$, $\kappa_2 = (\mu_0-\mu_{v,1}+\mu_{v,2})/2+1$, $\kappa_3 = (\mu_0+\mu_{v,1}-\mu_{v,2})/2+2$ and $\kappa_4=(\mu_0+\mu_{v,1}+\mu_{v,2})/2+3$. For simplicity of notation we will assume that $\nu=1$ and $\varepsilon=1$, the other cases being analogous. (Later we will choose this refinement anyway.) Then $\sigma(p)=p^{c_2+(\mu_0-\mu_{v,1}-\mu_{v,2})/2}a_{v,2}^{-1}$, $\chi_2(p)=p^{\mu_{v,1}-c_1}a_{v,1}$ and $\chi_1(p) = p^{c_1-2c_2+\mu_{v,2}}a_{v,2}^{2}a_{v,1}^{-1}$. Write $F_1=p^{c_2-3/2}a_{v,2}^{-1}$, $F_2=p^{c_1-c_2-1-3/2}(a_{v,2}/a_{v,1})$, $F_3=p^{c_2-c_1-2-3/2}(a_{v,1}/a_{v,2})$ and $F_4=p^{-c_2-3-3/2}a_{v,2}$ which are analytic and satisfy $p^{\kappa_1}F_1=p^{-3/2}\sigma(p)$, $p^{\kappa_2}F_2=p^{-3/2}\sigma(p)\chi_1(p)$, $p^{\kappa_3}F_3=p^{-3/2}\sigma(p)\chi_2(p)$ and $p^{\kappa_4}F_4=p^{-3/2}\sigma(p)\chi_1(p)\chi_2(p)$, which are the eigenvalues of ${\varphi}_{{\operatorname{cris}}}$. Thus $\rho_{\mathcal{E},{\operatorname{spin}},p}$ has a refinement.
\[c:gsp4 triangulation\] Let $\pi$ be a Hilbert modular form of infinity type $(k_1,\ldots, k_d;w)$. Suppose $\pi$ is not CM and let $\Pi$ be the cuspidal representation of ${\operatorname{GSp}}(4,{\mathbb{A}}_F)$ from Theorem \[t:GL(4) to GSp(4)\]. Let $\nu$ be a $p$-stabilization of $\Pi$ and let $m\in {\mathbb{Z}}$ such that $\Pi\otimes|\cdot|^m$ has noncritical slope. For $v\mid
p$, assume that $\pi_v$ is Iwahori spherical; if $\pi_v$ is an unramified principal series assume that $\alpha_v/\beta_v\notin\mu_{60}$. Then $\rho_{\mathcal{E},{\operatorname{spin}},p}$ has a global triangulation whose graded pieces $\mathcal{R}(\delta_i)$ are such that $\delta_i(u)=u^{\kappa_i}$ and $\delta_i(p)=F_i$ from the proof of Lemma \[l:gsp eigenvariety\].
Since the Hodge–Tate weights in the triangulation are ordered increasingly the only thing left to check is that the associated refinement is regular, i.e., that $\det\varphi$ on the filtered piece $\mathcal{F}_i$ has multiplicity one in ${\operatorname{D}}_{{\operatorname{cris}}}(\wedge^i \rho_{\Pi})$. This is equivalent to showing that for each $i\in \{1,2,3,4\}$, each product of $i$ terms in $\{\alpha_v^3,\alpha_v^2 \beta_v, \alpha_v \beta_v^2, \beta_v^3\}$ occurs once, which can be checked if $\alpha_v/\beta_v\notin\mu_{60}$.
\[c:gsp triangulation\] Let $\pi$ be a Hilbert modular form of infinity type $(k_1,\ldots, k_d;w)$. Suppose $\pi$ is not CM and let $\Pi$ be the cuspidal representation of ${\operatorname{GSp}}(2n,{\mathbb{A}}_F)$ from Theorem \[t:GL(2n+1) to Sp(2n)\]. Let $\nu$ be a $p$-stabilization of $\Pi$ and let $m\in {\mathbb{Z}}$ such that $\Pi\otimes|\cdot|^m$ has noncritical slope. For $v\mid
p$, assume that $\pi_v$ is Iwahori spherical; if $\pi_v$ is an unramified principal series assume that $\alpha_v/\beta_v\notin\mu_{\infty}$. Then $\rho_{\mathcal{E},{\operatorname{std}},p}$ has a global triangulation whose graded pieces $\mathcal{R}(\delta_i)$ are such that $\delta_i(u)=u^{\kappa_i}$ and $\delta_i(p)=F_i$ from the proof of Lemma \[l:gsp eigenvariety\].
As in the previous corollary we only need to check that for each $1\leq i\leq 2n+1$ each product of $i$ terms in $\{(\alpha_v/\beta_v)^k|-n\leq k\leq n\}$ occurs only once. Again, this can be checked when $\alpha_v/\beta_v\notin\mu_\infty$.
Eigenvarieties for unitary groups {#sect:eigenvarieties for unitary}
---------------------------------
One could reproduce the results of §\[sect:eigenvarieties for symplectic\] in the context of unitary groups. Indeed, the endoscopic classification for unitary groups was completed by Mok and compact unitary groups of course have discrete series so all the results translate into this context, again under the assumption of stabilization of the twisted trace formula. The main reason for redoing the computations using unitary groups is work in progress of Eischen–Harris–Li–Skinner and Eischen–Wan which will produce $p$-adic $L$-functions for unitary groups.
Let $F/{\mathbb{Q}}$ be a totally real field in which $p$ splits completely and $E/F$ a CM extension in which $p$ splits completely as well. Suppose $U$ is a definite unitary group over $F$, in $n$ variables, attached to $E/F$. Suppose $\pi$ is an irreducible (necessarily cuspidal) automorphic representation $\pi$ of $U({\mathbb{A}}_F)$ of cohomological weight $\bigoplus_{v\mid\infty} (\mu_{v,1},\ldots, \mu_{v,n})$. Then the restriction to $W_{{\mathbb{C}}}$ of the $L$-parameter of $\pi_v$ is given by $$z\mapsto {\operatorname{diag}}((z/\overline{z})^{\mu_{v,i}+(n+1)/2-i})$$ (cf. [@bellaiche-chenevier:selmer §6.7]).
If for some $v\mid p$ the representation $\pi_v$ has Iwahori level then the Hecke operators $U_{v,i}=[{\operatorname{Iw}}e_i^\vee(p){\operatorname{Iw}}]$ act on $\pi_v$ where $e_i^\vee$ is dual to the character $e_i$ isolating the $i$-th entry on $T$. For consistency of notation with the previous section we remark that Urban’s $\theta(U_{v,i})$ is denoted by $\delta^{1/2}\psi_{\pi,\mathcal{R}}$ in [@bellaiche-chenevier:selmer §7.2.2]. If $\pi_v=\chi_1\times\cdots\times\chi_n$ is an unramified principal series then $\pi_v^{{\operatorname{Iw}}}$ is $n$ dimensional and the Hecke operators $U_{v,i}$ can be simultaneously written in upper triangular form with diagonal entries $\chi_i(p)p^{-(n-1)/2}$.
\[t:galois unitary\] Suppose $U$ and $\pi$ are as above, with $\pi$ of cohomological weight $\bigoplus_{v\mid\infty} (\mu_{v,1},\ldots,
\mu_{v,n})$. Then there exists a continuous Galois representation $\rho_{\pi,p}:G_E\to {\operatorname{GL}}(n,
\overline{{\mathbb{Q}}}_p)$ such that:
1. If $v\nmid p\infty$ and $\pi_v$ is unramified then $\rho_{\pi,p}|_{G_{E_w}}$ is unramified for $w\mid p$ and ${\operatorname{WD}}(\rho_{\pi,p}|_{G_{E_w}})^{{\operatorname{Fr-ss}}}\cong{\operatorname{rec}}({\operatorname{BC}}_{E_w/F_v}(\pi_v)\otimes|\cdot|^{-(n-1)/2})$.
2. If $v\mid p$, since it splits in $E$ we may write $v=w \overline{w}$ where $w$ is the finite place of $E$ corresponding to $\iota_p:\overline{{\mathbb{Q}}}\to \overline{{\mathbb{Q}}}_p$. If $\pi_v=\chi_1\times\cdots\times\chi_n$ is unramified then $\rho_{\pi,p}|_{G_{E_w}}$ is crystalline with Hodge–Tate weights $-\mu_{v,i}+i$ and ${\varphi}_{{\operatorname{cris}}}$ has eigenvalues $\chi_i(p)p^{-(n-1)/2}$.
The proof is analogous to that of Theorem \[t:galois gsp\] as the transfer from $U$ to ${\operatorname{GL}}$ is the content of [@mok:functoriality1] (cf. [@scholze:torsion Corollary V.1.7]). The statement about Hodge-Tate weights follows by appealing to [@blght:calabi-yau-2 Theorem 1.2] rather than [@blght:calabi-yau-2 Theorem 1.1].
The literature contains base change results for both isometry unitary and similitude unitary groups to various degrees of generality. We remark that one may deduce base change for isometry unitary groups from the analogous results for similitude results using algebraic liftings of automorphic representations ([@patrikis:tate Proposition 12.3.3]).
The main theorem of [@chenevier:unitary-eigenvarieties] implies that if $4\mid n$, which we will asume, then the conclusion of Theorem \[t:urban\] holds for $\pi$ and a $p$-stabilization $\nu$. Moreover, if $\rho_{\pi,p}$ is irreducible then Theorem \[t:galois unitary\] implies the existence of an analytic Galois representation $\rho_{\mathcal{E},p}:G_E\to {\operatorname{GL}}(n, \mathcal{O}_{\mathcal{E}})$ interpolating, as before, the Galois representations attached to the classical regular points on $\mathcal{E}$.
\[c:unitary triangulation\] Let $F$ be a totally real field in which $p$ splits completely. Let $\pi$ be a Hilbert modular form over $F$, of infinity type $(k_1,\ldots, k_d;w)$, suppose there exist finite places $w_1, w_2$ not above $p$ with the property that $\pi_w$ is special for $w\in \{w_1,w_2\}$, and suppose that $\pi_v$ is Iwahori spherical for $v\mid p$ and that if $\pi_v$ is unramified with Satake parameters $\alpha_v$ and $\beta_v$ then $\alpha_v/\beta_v\notin\mu_\infty$. Suppose $\pi$ is not CM. Let $E$ be a CM extension of $F$, $\psi$ a Hecke character of $E$, and $\Pi$ a cuspidal automorphic representation of $U({\mathbb{A}}_F)$ such that $\Pi=\psi\otimes{\operatorname{BC}}_{E/F}{\operatorname{Sym}}^n\pi$ as in Proposition \[p:sym hmf to unitary\]. Let $\mathcal{E}$ and $\rho_{\mathcal{E},p}$ as above. Then $\rho_{\mathcal{E},p}|_{G_{E_w}}$ for $w\mid v\mid p$ a finite place of $E$ admits a triangulation with graded pieces $\mathcal{R}(\delta_i)$ such that $\delta_i(u)=u^{\kappa_i}$ for $u\in {\mathbb{Z}}_p^\times$ and $\delta_i(p)=F_i$ where $\kappa_i=-\mu_{v,i}+i$ and $F_i=p^{(n-1)/2-i}a_{v,i}$ where $a_{v,i}=\theta(U_{v,i})$ is analytic over $\mathcal{E}$.
Theorem \[t:galois unitary\] implies that at regular classical points which are unramified at $v\mid p$ the analytic functions $\kappa_i$ give the Hodge–Tate weights of $\rho_{\mathcal{E},p}|_{G_{E_w}}$. Thus it suffices to check that $p^{\kappa_i}F_i$ gives the eigenvalues of ${\varphi}_{{\operatorname{cris}}}$ at such regular unramified crystalline points. This follows from the fact that the eigenvalue of $U_{v,i}$ on $\pi_v$ equals $a_{v,i}$ times $|\lambda(e_i^\vee(p))|_p^{-1}\delta^{-1/2}(e_i^\vee(p))$. Finally, the condition $\alpha_v/\beta_v\notin\mu_\infty$ implies the existence of the global triangulation as in the proof of Corollary \[c:gsp triangulation\].
Computing the $\mathcal{L}$-invariants {#sect:l}
======================================
Let $F$ be a totally real field in which the prime $p$ splits completely and let $\pi$ be a non-CM cohomological Hilbert modular form of infinity type $(k_1,\ldots,k_d;w)$. Let $V_{2n}=\rho_{\pi,p}\otimes\det^{-n}\rho_{\pi,p}$. Suppose that for $v\mid p$, $\pi_v$ is Iwahori spherical, which is equivalent to the requirement that $\rho_{\pi,p}|_{G_{F_v}}$ be semistable. For each such $v$ let $D_v\subset
{\operatorname{D}}_{{\operatorname{st}}}(V_{2n,v})$ be the regular submodule chosen in §\[sect:regular submodules\].
Under the hypotheses (C1–4), we will compute $\mathcal{L}(V_{2n}, D)$ in terms of logarithmic derivatives of analytic Hecke eigenvalues over eigenvarieties. We will assume the existence of a rigid analytic space $\mathcal{E}\to
\mathcal{W}$ which is étale at a weight $w_0$ over which one has a point $z_0\in \mathcal{E}$ such that $z_0\circ\rho_{\mathcal{E},p}\cong
\psi\otimes{\operatorname{Sym}}^m\rho_{\pi,p}$ for some Hecke character $\psi$ and some $m\geq n$. We will moreover assume that $z_0$ corresponds to the $p$-stabilization of $\Pi_v$ coming from the $p$-stabilization of $\pi_v$ that gave rise to the regular submodule $D_v$. Assume there exists a global analytic triangulation of $\mathscr{D}_{{\operatorname{rig}}}^\dagger(\rho_{\mathcal{E},p}|_{G_{F_v}})$ with graded pieces $\mathcal{R}(\delta_i)$.
\[l:derivative galois\] Let $\overrightarrow{u}$ be a direction in $\mathcal{W}$ and let $\nabla_{\overrightarrow{u}}\rho_{\mathcal{E},p}$ be the tangent space to $\rho_{\mathcal{E},p}$ in the $\overrightarrow{u}$-direction, which makes sense under the assumption that $\mathcal{E}\to \mathcal{W}$ is étale at $z_0$. Then $c_{\overrightarrow{u}}=(z_0\circ\rho_{\mathcal{E},p})^{-1}\nabla_{\overrightarrow{u}}\rho_{\mathcal{E},p}$ is a cohomology class in $H^1(F, {\operatorname{End}}(z_0\circ\rho_{\mathcal{E},p}))$ and the natural projection $c_{\overrightarrow{u},n}\in H^1(F, V_{2n})$ lies in fact in the Selmer group $H^1(\{D_v\}, V_{2n})$.
Note that ${\operatorname{End}}(z_0\circ\rho_{\mathcal{E},p})\cong{\operatorname{End}}(\psi\otimes{\operatorname{Sym}}^m\rho_{\pi,p})\cong\oplus_{i=0}^m
V_{2i}$ and the natural projection on cohomology arises from the natural projection of this representation to $V_{2n}$.
One needs to check two things. The first, that the cohomology classes are unramified at $v\notin S\cup\{w\mid p\}$ follows along the same lines as [@hida:mazur-tate-teitelbaum Lemma 1.3]. The second is that the image of the cohomology class in $H^1(F_v,V_{2n})/H^1_f(F_v, V_{2n})$ lands in $H^1(F_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_{2n,v}))/H^1_f(F_v, V_{2n})$. But Proposition \[p:End to Sym\^n\] implies that the natural projection $c_{\overrightarrow{u},n}$ (in the notation of this lemma) lies entirely in the span of $e_1^ie_2^{n-i}$ for $2i\leq n$. By the choice of regular submodular $D_v$, this implies that ${\operatorname{res}}_vc_{\overrightarrow{u},n}\in H^1(F_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_{2n,v}))$, as desired.
\[p:l-invariant formula\] In the notation of the previous lemma, suppose $\mathscr{D}_{{\operatorname{rig}}}^\dagger(\rho_{\mathcal{E},p}|_{G_{F_v}})$ has an analytic triangulation with graded pieces $\delta_1,\ldots, \delta_{m+1}$ where $\delta_i(u)=u^{\kappa_i}$ and $\delta_i(p)=p^{\kappa_i}F_i$. Then $$\mathcal{L}(V_{2n}, \{D_v\})=\prod_{v\mid p}\left(-\frac{\sum_i B_{m,n,i-1}(\nabla_{\overrightarrow{u}}F_i)/F_i}{\sum_i B_{m,n,i-1}\kappa_i}\right)$$ as long as this formula makes sense. Here, the coefficients $B_{m,n,i}$ are given in Remark \[r:Bnki\] and are basically alternating binomial coefficients multiplied by inverse Clebsch–Gordon coefficients.
Lemma \[l:rank 1 L invariant\] implies that $\mathcal{L}(V_{2n}, \{D_v\})=\prod_{v\mid p} (a_v/b_v)$ where the projection $c_{\overrightarrow{u}, v}$ to $H^1({\operatorname{gr}}_1{{\operatorname{D}}_{{\operatorname{rig}}}^\dagger}(V_{2n,v}))\cong H^1(\mathcal{R})$ is written as $a_v(-1,0)+b_v(0,\log_p\chi(\gamma))$. Explicitly, $$\frac{a_v}{b_v}=\frac{c_{\overrightarrow{u},
v}(p)}{c_{\overrightarrow{u}, v}(u)/-\log_p(u)}$$ for $u\in {\mathbb{Z}}_p^\times$. But Proposition \[p:End to Sym\^n\] implies that $$c_{\overrightarrow{u}, v}=\sum
B_{m,n,i}(\nabla_{\overrightarrow{u}}\delta_i)/\delta_i$$ The result now follows from the fact that $\delta_i(p)=F_i$, $\delta_i(u)=u^{\kappa_i}$ and $(\nabla_{\overrightarrow{u}}u^{\kappa})/u^{\kappa} = \nabla_{\overrightarrow{u}}\kappa \log_p(u)$.
In the remaining sections we apply Proposition \[p:l-invariant formula\] to obtain explicit formulae for $\mathcal{L}$-invariants for relevant symmetric powers in terms of logarithmic derivatives of analytic Hecke eigenvalues.
We will assume that $F$ is a totally real field in which $p$ splits completely and $\pi$ is a cohomological Hilbert modular form with infinity type $(k_1,\ldots, k_d;w)$. At $v\mid p$ we assume that $\pi_v$ is Iwahori spherical. In the computation of the $\mathcal{L}$-invariant of $V_{2n}$ we will assume that $H^1_f(F, V_{2n})=0$. Throughout we will consider the refinement of $\pi$ corresponding to the ordering $e_1,e_2$ of the basis of ${\operatorname{D}}_{{\operatorname{st}}}(\rho_{\pi,p}|_{G_{F_v}})$, which gives a suitable refinement of any automorphic form equivalent to ${\operatorname{Sym}}^m \pi$ using the ordering $e_1^m, e_1^{m-1}e_2, \ldots, e_2^m$. We will assume that $V_{2n,v}$ satisfies condition (C4).
Symmetric squares {#sect:sym2l}
-----------------
Suppose for $v\mid p$ such that $\pi_v$ is unramified that the two Satake parameters are distinct. Let $\mathcal{E}$ be the eigenvariety from Lemma \[l:hmf eigenvariety\]. Suppose that $\mathcal{E}$ is étale over the weight space at the chosen refinement of $\pi_v$.
\[t:l invariant formula hmf\]Writing $a'_v$ for the derivative in the direction $(1,\ldots, 1; -1)$ in the weight space we compute $$\mathcal{L}(V_2, \{D_v\}) = \prod_{v\mid p}
\left(\frac{-2a_v'}{a_v}\right)$$
Recall from Corollary \[c:hmf triangulation\] that $\kappa_1=(w-k_v)/2$, $\kappa_2=(w+k_v-2)/2$, $F_1=a_v^{-1}p^{3/2}$ and $F_2=a_vp^{1/2}$. The result now follows directly from Proposition \[p:l-invariant formula\] and the fact that $B_{1,1,0}=1$ and $B_{1,1,1}=-1$.
Symmetric sixth powers {#sect:sym6l}
----------------------
Assume that $\pi$ is not CM. Suppose for $v\mid p$ such that $\pi_v$ is unramified that $\alpha_v/\beta_v\notin\mu_{60}$. Let $\Pi$ be a suitably twisted Ramakrishnan–Shahidi lift of ${\operatorname{Sym}}^3\pi$ such that the chosen refinement has noncritical slope (cf. Theorem \[t:GL(4) to GSp(4)\] and Lemma \[l:siegel-hilbert noncritical slope\]). Let $\mathcal{E}$ be Urban’s eigenvariety for ${\operatorname{GSp}}(4)$ and let $a_{v,1}$ and $a_{v,2}$ be the analytic Hecke eigenvalues from the proof of Lemma \[l:gsp eigenvariety\]. Suppose that the eigenvariety $\mathcal{E}$ is étale over the weight space at the chosen refinement of $\Pi$.
\[t:l invariant formula gsp4\] If $\overrightarrow{u}=(u_1,u_2; u_0)$ is any direction in the weight space, i.e. $u_1\geq u_2\geq 0$, such that the denominator below is non-zero, then $$\mathcal{L}(V_6, \{D_v\})=\prod_{v\mid p}
\left(\frac{-4\widetilde{\nabla}_{\overrightarrow{u}}a_{v,2}+3\widetilde{\nabla}_{\overrightarrow{u}}a_{v,1}}{u_1-2u_2}\right)$$ where we write $\widetilde{\nabla}_{\overrightarrow{u}}f =
(\nabla_{\overrightarrow{u}}f)/f$.
Recall from Corollary \[c:gsp4 triangulation\] that $\kappa_1=(\mu_0-\mu_{v,1}-\mu_{v,2})/2$, $\kappa_2=(\mu_0-\mu_{v,1}+\mu_{v,2})/2+1$, $\kappa_3=(\mu_0+\mu_{v,1}-\mu_{v,2})/2+2$, $\kappa_4=(\mu_0+\mu_{v,1}+\mu_{v,2})/2+3$ giving $\nabla_{\overrightarrow{u}}\kappa_1=(u_0-u_1-u_2)/2$, $\nabla_{\overrightarrow{u}}\kappa_2=(u_0-u_1+u_2)/2$, $\nabla_{\overrightarrow{u}}\kappa_3=(u_0+u_1-u_2)/2$, $\nabla_{\overrightarrow{u}}\kappa_4=(u_0+u_1+u_2)/2$. Similarly, $\widetilde{\nabla}_{\overrightarrow{u}}F_1=-\widetilde{\nabla}_{\overrightarrow{u}}a_{v,2}$, $\widetilde{\nabla}_{\overrightarrow{u}}F_2=\widetilde{\nabla}_{\overrightarrow{u}}a_{v,2}-\widetilde{\nabla}_{\overrightarrow{u}}a_{v,1}$, $\widetilde{\nabla}_{\overrightarrow{u}}F_3=\widetilde{\nabla}_{\overrightarrow{u}}a_{v,1}-\widetilde{\nabla}_{\overrightarrow{u}}a_{v,2}$, $\widetilde{\nabla}_{\overrightarrow{u}}F_4=\widetilde{\nabla}_{\overrightarrow{u}}a_{v,2}$. Using that $(B_{3,3,i})_i\sim (1,-3,3,-1)$ we deduce the formula.
Symmetric powers via symplectic groups {#sect:symgspl}
--------------------------------------
We remark that the results of this paragraph are conditional on the stabilization of the twisted trace formula (cf. Theorem \[t:galois gsp\]).
Assume that $\pi$ is not CM. Suppose for $v\mid p$ such that $\pi_v$ is unramified that $\alpha_v/\beta_v\notin\mu_{\infty}$. Suppose $\pi$ satisfies the hypotheses of Theorem \[t:automorphy of sym\] (2) and let $\Pi$ be a suitable (as before, from Lemma \[l:siegel-hilbert noncritical slope\]) twist of the cuspidal representation of ${\operatorname{GSp}}(2n, {\mathbb{A}}_F)$ from Theorem \[t:GL(2n+1) to Sp(2n)\]. Let $\mathcal{E}$ be Urban’s eigenvariety for ${\operatorname{GSp}}(2n)$ and let $a_{v,i}$ be the analytic Hecke eigenvalues from the proof of Lemma \[l:gsp eigenvariety\]. Suppose that the eigenvariety $\mathcal{E}$ is étale over the weight space at the chosen refinement of $\Pi$.
\[t:l invariant formula gsp\] If $\overrightarrow{u}=(u_1,\ldots,u_n;u_0)$ is any direction in the weight space, such that the denominator below is non-zero, then $$\mathcal{L}(V_{4n-2}, \{D_v\})=\prod_{v\mid
p}-\left(\frac{B_{n}\widetilde{\nabla}_{\overrightarrow{u}}a_{v,1}+B_{1}(\widetilde{\nabla}_{\overrightarrow{u}}a_{v,n-1}-2\widetilde{\nabla}_{\overrightarrow{u}}a_{v,n})+\sum_{i=2}^{n-1}B_{i}(\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i-1}-\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i})}{\sum_{i=1}^nu_iB_{n+1-i}}\right)$$ where we write $B_i=(-1)^i\binom{2n}{n+i}i$.
Given that Remark \[r:Bnki\] gives explicit values for the $B_{n,k,i}$, we know there will be directions where the denominator is non-zero.
Combine the formulae for the triangulation of the standard Galois representation from the proof of Lemma \[l:gsp eigenvariety\] with Proposition \[p:l-invariant formula\]. Finally, compute $$\begin{aligned}
B_i&=B_{2n,2n-1,n+i}-B_{2n,2n-1,n-i}\\
&=(-1)^{n+i}\binom{2n}{n+i}(2n-2(n+i))-(-1)^{n-i}\binom{2n}{n-i}(2n-2(n-i))\\
&=(-1)^{n+1}2 \cdot(-1)^i\binom{2n}{n+i}i\end{aligned}$$ and the result follows because all the $B_i$ can be scaled by the same factor.
Symmetric powers via unitary groups {#sect:symunitaryl}
-----------------------------------
We remark that the results of this paragraph as well are conditional on the stabilization of the twisted trace formula.
Assume that $\pi$ is not CM. Suppose for $v\mid p$ such that $\pi_v$ is unramified that $\alpha_v/\beta_v\notin\mu_{\infty}$. Suppose $\pi$ satisfies the hypotheses of Theorem \[t:automorphy of sym\] (2) and Proposition \[p:sym hmf to unitary\]. Let $E/F$ the CM extension and $\Pi$ the cuspidal representation of $U_{4n}({\mathbb{A}}_F)$ which a transfer of a twist of ${\operatorname{Sym}}^{4n-1}\pi$ as in Proposition \[p:sym hmf to unitary\]. Let $\mathcal{E}$ be Chenevier’s eigenvariety and let $a_{v,i}$ be the analytic Hecke eigenvalues from the proof of Corollary \[c:unitary triangulation\]. Suppose that the eigenvariety $\mathcal{E}$ is étale over the weight space at the chosen refinement of $\Pi$.
\[t:l invariant formula unitary\] If $\overrightarrow{u}=(u_1,\ldots,u_n;u_0)$ is any direction in the weight space, then $$\mathcal{L}(V_{8n-2}, \{D_v\})=\prod_{v\mid
p}\left(\frac{-\sum_{i=1}^{4n}(-1)^{i-1}\binom{4n-1}{i-1}\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i}}{\sum_{i=1}^{4n}(-1)^{i-1}\binom{4n-1}{i-1}u_i}\right)$$ and $$\mathcal{L}(V_{8n-6}, \{D_v\})=\prod_{v\mid
p}\left(\frac{-\sum_{i=1}^{4n}B_{i-1}\widetilde{\nabla}_{\overrightarrow{u}}a_{v,i}}{\sum_{i=1}^{4n}u_iB_{i-1}}\right)$$ Here $B_i=B_{4n-1,4n-3,i}$ is the inverse Clebsch–Gordan coefficient of Proposition \[p:End to Sym\^n\], up to a scalar independent of $i$ given by $$B_i= (-1)^i\binom{4n-1}{i}((4n-1)^3-(4i+1)(4n-1)^2+(4i^2+2i)(4n-1)-2i^2).$$
Note that we cannot simply appeal to Proposition \[p:l-invariant formula\] as the analytic Galois representation on the unitary eigenvariety is a representation of $G_E$ and not $G_F$. However, note that since $E/F$ is finite and $V_{2m}$ is a characteristic 0 vector space, the inflation-restriction sequence gives $H^1(G_{F,S},V_{2m})\cong H^1(G_{E,S_E},V_{2m})^{G_{E/F}}$. We have constructed a cohomology class $c_{\overrightarrow{u}}\in
H^1(G_{E,S_E},V_{m})$ and by construction $c_{\overrightarrow{u}}$ is invariant under the nontrivial element of ${\operatorname{Gal}}(E/F)$ (complex conjugation acts trivially on $V_{2m}$). Thus it descends to a cohomology class in $H^1(G_{F,S}, V_{2m})$. Since $p$ splits completely in both $E$ and $F$, this local component of the descended class is the same as that of the original class. This implies that the conclusion of Proposition \[p:l-invariant formula\] stays the same and the formulae follow from Corollary \[c:unitary triangulation\] as before.
Appendix: Plethysm for ${\operatorname{GL}}(2)$ {#sect:plethysm}
===============================================
Let $V$ denote the standard two-dimensional representation of ${\operatorname{GL}}(2)$ (or ${\operatorname{SL}}(2)$) (via matrix multiplication). In this section, we study the decomposition $${\operatorname{End}}{\operatorname{Sym}}^n V\cong{\operatorname{Sym}}^nV\otimes({\operatorname{Sym}}^nV)^\vee\cong\bigoplus_{k=0}^n{\operatorname{Sym}}^{2k}V\otimes{\det}^{-k}$$ as representations of ${\operatorname{GL}}(2)$, or alternatively, of ${\operatorname{End}}{\operatorname{Sym}}^n V\cong\bigoplus_{k=0}^n{\operatorname{Sym}}^{2k}V$ as representations of ${\operatorname{SL}}(2)$.
In general, if $\mathfrak{g}$ is a Lie algebra acting on a finite dimensional vector space $W$, then $\mathfrak{g}$ acts on $W^\vee$ by $(X f)(w) = f(-Xw)$. Moreover, $\mathfrak{g}$ acts on endomorphisms $f:W\to W$ by $(Xf)(w) = X(f(w))+f(-Xw)$ and on $W\otimes W^\vee$ by $X(u\otimes w^\vee) = (X u)\otimes w^\vee+u\otimes (X w^\vee)$. We deduce that there exists a $\mathfrak{g}$-equivariant isomorphism $W\otimes W^\vee\cong {\operatorname{End}}(W)$ sending $u\otimes w^\vee$ to the endomorphism $x\mapsto w^\vee(x) u$.
Let $L=\begin{pmatrix} 0&0\\1&0\end{pmatrix}$ and $R=\begin{pmatrix} 0&1\\ 0&0\end{pmatrix}$ be lowering and raising matrices in ${\mathfrak{sl}}(2)$. We choose the basis $(e_1,e_2)$ of $V$ such that $L e_1=e_2, Le_2=0$ and $Re_1=0, Re_2=e_1$. For $m\geq 0$, we denote $V_m = {\operatorname{Sym}}^mV$ the representation of ${\mathfrak{sl}}(2)$ of highest weight $m$. Define $g_{m,0},\ldots, g_{m,m}$ as the basis of $V_m$, thought of as a subset of $V^{\otimes m}$, by $$g_{m,i}=\binom{m}{i}^{-1}\sum_{\underline{j}}e_{j_1}\otimes\cdots\otimes e_{j_m}:=e_1^{m-1}e_2^i$$ where the sum is over $m$-tuples $\underline{j}=(j_1,\dots,j_m)\in\{1,2\}^m$ with $j_k=1$ for exactly $m-i$ values of $k$. (When $i\notin \{0,\ldots,m\}$ we simply define $g_i^{(m)}$ to be 0.) The operators $L$ and $R$ act on ${\operatorname{Sym}}^mV$ via $$\begin{aligned}
Lg_{m,i} &= (m-i)g_{m,i+1}\\
Rg_{m,i} &= ig_{m,i-1}.\end{aligned}$$
Let $m,n,p\in \mathbb{Z}$ such that $V_p$ appears as a subrepresentation of $V_m\otimes V_n$, i.e. $p\in \{m+n, m+n-2,
\ldots, |m-n|\}$. Denote by $\Xi_{m,n,p}:V_m\otimes V_n\to V_p$ the nontrivial $\mathfrak{sl}(2)$-equivariant projection from [@carter-flath-saito:6j Lemma 2.7.4] and let $$\psi_{m,n,p}=\left((\sqrt{-1})^{(m+n-p)/2}\frac{m!n!}{((m+n-p)/2)!}\right)\Xi_{m,n,p}$$ which will again be $\mathfrak{sl}(2)$-equivariant. Denote by $C_{m,n,p}^{u,v,w}$ be the “inverse Clebsch–Gordan” coefficients such that $$\psi_{m,n,p}(g_{m,u}\otimes g_{n,v})=\sum_{w=0}^p C_{m,n,p}^{u,v,w} g_{p,w}$$
\[l:cg\] The coefficients $C_{m,n,p}^{u,v,w}$ satisfy the recurrence relations $$\begin{aligned}
(p-w)C_{m,n,p}^{u,v,w}&=(m-u)C_{m,n,p}^{u+1,v,w+1}+(n-v)C_{m,n,p}^{u,v+1,w+1}\\
wC_{m,n,p}^{u,v,w}&=uC_{m,n,p}^{u-1,v,w-1}+vC_{m,n,p}^{u,v-1,w-1}\end{aligned}$$ and the initial values in the case when $u+v=(m+n-p)/2$ are given by $$C_{m,n,p}^{u,v,0}=(-1)^{u}(m-u)!(n-v)!$$ and $C_{m,n,p}^{u,v,w}=0$ for $w>0$.
Since the map $\psi_{m,n,p}$ is $\mathfrak{sl}(2)$-equivariant we get $$\begin{aligned}
L\psi_{m,n,p}(g_{m,u}\otimes
g_{n,v})&=\psi_{m,n,p}((Lg_{m,u})\otimes g_{n,v}+g_{m,u}\otimes
(Lg_{n,v}))\\
&=(m-u)\psi_{m,n,p}(g_{m,u+1}\otimes g_{n,v})+(n-v)\psi_{m,n,p}(g_{m,u}\otimes
g_{n,v+1})\\
&=(m-u)\sum_w C_{m,n,p}^{u+1,v,w}g_{p,w}+(n-v)\sum_w C_{m,n,p}^{u,v+1,w}g_{p,w}.\end{aligned}$$ At the same time this is $$\begin{aligned}
L\psi_{m,n,p}(g_{m,u}\otimes
g_{n,v})&=\sum_w C_{m,n,p}^{u,v,w} Lg_{p,w}\\
&=\sum_w C_{m,n,p}^{u,v,w} (p-w)g_{p,w+1},\end{aligned}$$ which gives, after identifying the coefficients of $g_{p,w+1}$, the recurrence $$(p-w)C_{m,n,p}^{u,v,w}=(m-u)C_{m,n,p}^{u+1,v,w+1}+(n-v)C_{m,n,p}^{u,v+1,w+1}$$ The second recurrence formula is obtained analogously applying the operator $R$ to the definition of the coefficients $C_{m,n,p}^{u,v,w}$.
In [@carter-flath-saito:6j Lemma 2.7.4] $g_{m,u}$ is denoted $e_{m/2, m/2-u}$ and the content of the lemma is that $$\Xi_{m,n,p}(g_{m,u}\otimes g_{n,v}) =
\left((\sqrt{-1})^{(m+n-p)/2}(-1)^{u}\frac{((m+n-p)/2)!(m-u)!(n-v)!}{m!n!}\right)g_{p,0}$$ when $u+v=(m+n-p)/2$. The result follows from the definition of $\psi_{m,n,p}$.
Lemma \[l:cg\] gives an explicit map $V_n\otimes V_n\to
V_{2k}$ for $k\leq n$. To make explicit the map ${\operatorname{End}}(V_n)\to
V_{2k}$, we start with $V_n\cong V_n^\vee$, which is noncanonical as an isomorphism of vector spaces, but can be chosen uniquely (up to scalars) as follows to make the isomorphism $\mathfrak{g}$-equivariant.
Let $(e_1^\vee,e_2^\vee)$ be the basis of $V$ dual to $(e_1,e_2)$, in which case the dual basis to $g_{n,i}$ is $$g_{n,i}^\vee=\sum_{\underline{j}}e_{j_1}^\vee\otimes\cdots\otimes e_{j_n}^\vee,$$ where again the sum is over $\underline{j}$ with $j_k=1$ for exactly $n-i$ values of $k$. The map ${\varphi}$ sending $e_1\mapsto -e_2^\vee$ and $e_2\mapsto e_1^\vee$ is an $\mathfrak{sl}(2)$-equivariant isomorphism $V\cong V^\vee$ and this leads to the $\mathfrak{sl}(2)$-equivariant isomorphism ${\varphi}_n:V_n\to V_n^\vee$ sending $g_{n,i}\mapsto (-1)^{n-i}\binom{n}{i}^{-1}g_{n,n-i}^\vee$. This implies $$\begin{aligned}
Lg_i^\vee &=-(n+1-i)g_{i-1}^\vee\\
Rg_i^\vee &=-(i+1)g_{i+1}^\vee.\end{aligned}$$
We remark that, as an ${\mathfrak{sl}}(2)$-representation, $V_m$ has weights $\{m,m-2,\ldots,-m\}$ and $L$ maps the weight $w$ eigenspace to the weight $w-2$ eigenspace, while $R$ goes in the other direction. Moreover, the vector $g_{n,i}\otimes g_{n,j}^\vee\in
V_n\otimes V_n^\vee$ has weight $2(j-i)$ and this implies that $$v_{2k}=\sum_{i=0}^{n-k} \binom{k+i}{i}g_{n,i}\otimes
g_{n,k+i}^\vee$$ has (highest) weight $2k$. Computationally, this vector suffices to make explicit the projection ${\operatorname{End}}(V_n)\to V_{2k}$. Indeed, $V_{2k}$ has basis $\{(2k-i)!L^iv_{2k}|i=0,\ldots, 2k\}$ and this basis is proportional to the $(g_{2k,0},\ldots,g_{2k,2k})$. Thus the projection ${\operatorname{End}}(V_n)\cong V_n\otimes V_n^\vee\to V_{2k}$ can be computed by finding the projection of $g_{n,i}\otimes
g_{n,j}^\vee$ to $V_{2k}$ in terms of the basis $\{(2k-i)!L^iv_{2k}|i=0,\ldots, 2k\}$, which amounts to a matrix inversion.
However, we will obtain a closed expression for the projection map using the inverse Clebsch–Gordan coefficients from Lemma \[l:cg\]. The endomorphism $g_{n,i}\otimes g_{n,j}^\vee\in
V_n\otimes V_n^\vee\cong{\operatorname{End}}(V_n)$ projects to $V_{2k}$ and the composite map is $$\begin{aligned}
\psi_{n,n,2k}\circ (1\otimes{\varphi}_n^{-1})(g_{n,i}\otimes
g_{n,j}^\vee)&=(-1)^j\binom{n}{j}\psi_{n,n,2k}(g_{n,i}\otimes
g_{n,n-j})\\
&=(-1)^j\binom{n}{j}\sum_{w=0}^{2k}C_{n,n,2k}^{i,n-j,w}g_{2k,w}\end{aligned}$$
We arrive at the main result of this section:
\[p:End to Sym\^n\]Suppose the representation $V$ has basis $(e_1,e_2)$ and $V_n={\operatorname{Sym}}^nV$ has basis $(g_{n,0},\dots,g_{n,n})$. Suppose $T\in{\operatorname{End}}({\operatorname{Sym}}^nV)$ has an upper triangular matrix with $(a_0, \ldots, a_{n})$ on the diagonal with respect to this basis. Then the projection of $T$ to $V_{2k}$ is $$\begin{pmatrix}*&\cdots &*&\displaystyle \sum_{i=0}^{n}B_{n,k,i}a_i&0&\cdots&0 \end{pmatrix}$$ with respect to the basis $(g_{2k,i})$ of $V_{2k}$ where $$\begin{aligned}
B_{n,k,i}=\sum_{a+b=k}(-1)^{a}\binom{n}{i}\binom{i}{a}\binom{n-i}{b}(n-i+a)!(i+b)!\end{aligned}$$ Here, the explicit coordinate is the middle one, i.e. the coefficient of $g_{2k,k}$, and we use the usual convention that $\binom{x}{y}=0$ if $y<0$ or $y>x$.
That $T$ is upper triangular implies that it is a linear combination of the form $\displaystyle \sum_{i\leq j}
\alpha_{i,j} g_{n,i}\otimes g_{n,j}^\vee$ (here $a_i=\alpha_{i,i}$). But $g_{n,i}\otimes g_{n,j}^\vee$ has weight $2(j-i)\geq 0$ and so $T\subset \bigoplus_{w\geq 0} ({\operatorname{End}}V_n)_w$. Therefore its projection to $V_{2k}$ belongs to $\bigoplus_{w\geq 0}(V_{2k})_w$, which is spanned by $g_{2k,u}$ for $u\leq k$. This implies that the coefficients of $g_{2k,u}$ for $u>k$ are $0$. Note that the projection ${\operatorname{End}}V_n\to (V_{2k})_0$ to the weight 0 eigenspace factors through $({\operatorname{End}}V_n)_0\to
(V_{2k})_0$ and so the projection of $T$ to $(V_{2k})_0$ only depends on the image $\sum a_i g_i\otimes g_i^\vee$ of $T$ in $({\operatorname{End}}{\operatorname{Sym}}^nV)_0$.
The coefficient $B_{n,k,i}$ is then the coefficient of $g_{2k,k}$ in the projection to $V_{2k}$ of $g_{n,i}\otimes
g_{n,i}^\vee$. By the discussion above this is $$B_{n,k,i}=(-1)^i\binom{n}{i}C_{n,n,2k}^{i,n-i,k}$$ Lemma \[l:cg\] implies inductively that for $m\leq k$: $$C_{n,n,2k}^{i,n-i,k}=\sum_{a+b=m}\binom{i}{a}\binom{n-i}{b}\binom{k}{m}^{-1}C_{n,n,2k}^{i-a,n-i-b,k-m}$$ and so $$C_{n,n,2k}^{i,n-i,k}=\sum_{a+b=k}(-1)^{i-a}\binom{i}{a}\binom{n-i}{b}(n-i+a)!(i+b)!$$ and the formula follows.
\[r:Bnki\] We end with a computation of the special values of $B_{n,k,i}$ which are involved in our formulae for $\mathcal{L}$-invariants. In the main formula for $B_{n,k,i}$ in Proposition \[p:End to Sym\^n\], the indices $a$ and $b$ satisfy $a+b=k$, $a\leq i$ and $b\leq n-i$. When $k=n$ the only posibility is $(a,b)=(i,n-i)$, when $k=n-1$ the two possibilities are $a\in \{i-1,i\}$, and when $k=n-2$ the three possibilities are $a\in \{i-2,i-1,i\}$. Thus $$\begin{aligned}
B_{n,n,i}&=(-1)^i(n!)^2\binom{n}{i},\\
B_{n,n-1,i}&=\binom{n}{i}\left((-1)^{i-1}\binom{i}{i-1}\binom{n-i}{n-i}(n-1)!n!+(-1)^i\binom{i}{i}\binom{n-i}{n-1-i}n!(n-1)!\right)\\
&=(-1)^in!(n-1)!\binom{n}{i}(n-2i),\\
B_{n,n-2,i}&=(-1)^i\binom{n}{i}\left(\binom{i}{2}(n-2)!n!-\binom{i}{1}\binom{n-i}{1}((n-1)!)^2+\binom{n-i}{2}n!(n-2)!\right)\\
&=(-1)^i\binom{n}{i}(n-2)!(n-1)!(n^3-(4i+1)n^2+(4i^2+2i)n-2i^2)\end{aligned}$$
#### [**Acknowledgements**]{}
We are grateful to Joël Bellaïche, Frank Calegari, Mladen Dimitrov, Matthew Emerton, Piper Harron, Ruochuan Liu, Dinakar Ramakrishnan, Sug Woo Shin, Claus Sorensen, and Jacques Tilouine. We also thank Paul Terwilliger for pointing us to the reference [@carter-flath-saito:6j] that lead us to the closed-form expression for the $B_{n,k,i}$.
[^1]: The first author is partially supported by NSA Young Investigator Grant \#H98230-13-1-0223 and NSF RTG Grant “Number Theory and Algebraic Geometry at the University of Wisconsin”.
|
UNIVERSITY OF CALIFORNIA\
SANTA CRUZ
**STRING HOMOLOGY AND LIE ALGEBRA STRUCTURES**
A dissertation submitted in partial satisfaction of the\
requirements for the degree of\
DOCTOR OF PHILOSOPHY\
in\
MATHEMATICS\
by\
**Felicia Y. Tabing**
June 2015
The Dissertation of Felicia Y. Tabing\
is approved: \
------------------------------------------------------------------------
\
Professor Hirotaka Tamanoi \
------------------------------------------------------------------------
\
Professor Geoffrey Mason \
------------------------------------------------------------------------
\
Professor Richard Montgomery
------------------------------------------------------------------------
\
Tyrus Miller\
Vice Provost and Dean of Graduate Studies
Copyright $\textcopyright$ by\
Felicia Y. Tabing\
2015
**Abstract**\
String Homology and Lie Algebra Structures
by\
**Felicia Y. Tabing**
Chas and Sullivan introduced string homology in [@CS], which is the equivariant homology of the loop space with the $S^1$ action on loops by rotation. Craig Westerland computed the string homology for spheres with coefficients in ${\mathbb Z}/2{\mathbb Z}$ [@We] and in Somnath Basu’s dissertation [@Ba], he computes the string homology and string bracket for spheres over rational coefficients, and he finds that the bracket is trivial. In this paper, we compute string homology and the string bracket for spheres with integer coefficients, treating the odd- and even-dimensional cases separately. We use the Gysin sequence and Leray-Serre spectral sequence to aid in our computations. We find that over the integers, the string Lie algebra bracket structure is more interesting, and not always zero as in [@Ba]. The string bracket turns out to be non-zero on torsion coming from string homology.\
We also make some computations of the Goldman Lie algebra structure, and more generally, the string Lie algebra structure of closed, orientable surfaces.
To Michael Kusuda.
**Acknowledgements**
\
I would like to express my deepest gratitude to my advisor Hirotaka Tamanoi for his support of me over the past few years. He has been incredibly patient with me, and I am very grateful for him pushing me to learn how to work on my own. I enjoyed the time spent in his office, learning about algebraic topology. I also very much valued hearing his views on life, which I recall when I am having a hard time.\
I wish to thank the rest of my thesis committee, Geoff Mason and Richard Montgomery. Richard, for being available to chat about mathematics, and introducing me to Bill Goldman.\
I am most grateful to Debra Lewis for the help and guidance she has given me. Her support of me through all my years as a graduate student, and even as an undergraduate, has been invaluable. She was always available to talk when I was feeling anxious and needed moral support, and provided me with encouragement.\
Teaching has been one the most enjoyable part of graduate school. I would like to acknowledge Frank Ba[ü]{}erle, who was a teaching mentor to me. I hope one day that I can be as much of a compassionate and patient teacher as he is.\
I would like to thank the following fellow graduate students, current and former, for their friendship, support, mathematical and non-mathematical conversations: Alex Beloi, Victor Bermudez, Jonathan Chi, Michael Campbell, Jamison Barsotti, Sean Gasiorek, Rob Carman, Gabriel Martins, Mitchell Owen, Vinod Sastry, Shawn Tsosie, and Wei Yuan. Liz Pannell deserves a special mention for her friendship and starting the Noetherian Ring with me. I am grateful to Danquynh Nguyen, for being so generous and sharing her delicious food with me, as I would often come to campus without food, and very hungry. Jean Verrette, whom I am grateful to have been paired with as roommates at the Algebraic Topology Summer School at MSRI.\
I am grateful for the support of my parents, Sylvia and German Tabing, for not discouraging me from mathematics, even though when I was a kid I said I wanted to be surgeon. I want to acknowledge Linda and Harry Kusuda, for their support and believing in me. I also want to thank Guy Gov and Annie Nguyen, with whom I can forget about my academic worries and have fun.\
Lastly, I am greatly indebted to Michael Kusuda for his never-ending love an support. He was my greatest supporter, made sure I was well fed, and took care of my every need. With his support, I was allowed to concentrate on learning mathematics, and I am extremely grateful to him.
Introduction
============
The term *String Topology* came from the paper of the same name by Moira Chas and Dennis Sullivan in 1999. This paper discussed the various algebraic structures that arose from the homology of the free loop space. This paper came out of trying to generalize the Lie algebra structure that William M. Goldman described by the intersection and concatenation of loops on surfaces [@Go].\
Chapter 1 is an introduction to the Goldman Lie algebra, and we explore its structure. In particular we consider the structure of the Lie algebra for the closed torus, including computations showing it is finitely generated.\
Chapter 2 introduces string topology background needed for the rest of this paper, and the various algebra structures of loop homology and string homology.\
Chapter 3 contains the computations of the integral string homology and string bracket structure for spheres, where some torsion phenomena appear. In our computations, we use the Leray-Serre spectral sequence, and the Gysin exact sequence.\
Chapter 4 explores the string homology and bracket structure of surfaces.
The Goldman Lie Algebra
=======================
The Goldman Lie algebra was introduced by William M. Goldman in 1986 [@Go].\
Throughout, let $\Sigma_{g,n}$ denote an oriented, genus $g$ surface with $n \geq 0$ boundary components. Denote $\hat{\pi}(\Sigma_{g,n})$ to be the set of free homotopy classes of loops on $\Sigma_{g,n}$, where the surface is not mentioned in the notation of $\hat{\pi}$ when it is clear from the context that we are talking about some fixed surface. Recall the following. The set of free homotopy classes of loops on a surface $\Sigma_{g,n}$ is in one-to-one correspondence with conjugacy classes of $\pi_1(\Sigma_{g,n})$. We can represent homotopy classes of loops by cyclically reduced words with letters the generators of the fundamental group. Fix a surface $\Sigma_{g,n}$ and an orientation of $\Sigma_{g,n}$. Let $\alpha, \beta \in \hat{\pi}$. The **Goldman bracket** of $\alpha$ and $\beta$ is defined to be $$\begin{aligned}
[\alpha, \beta]=\sum_{p \in \alpha \cap \beta} \epsilon (p) \alpha *_p \beta\end{aligned}$$ where $\alpha$ and $\beta$ intersect in transverse double points $p$, and $\epsilon (p)$ is the sign of the intersection, or $\epsilon (p)=1$ if the ordered vectors in the tangent space to $\Sigma_{g,n}$ tangent to loop $\alpha$ and $\beta$ match the orientation of the surface, and $\epsilon (p)=-1$ otherwise.
![Loops $aab$ and $b$ on the torus with one boundary component.\[fig:sigma11\]](sigma11.jpg)
We compute $[aab,b]$ on the surface $\Sigma_{1,1}$. The loops represented by words $aab$ and $b$ are shown in Figure \[fig:sigma11\], with intersection points $p_1$ and $p_2$. At the intersection point $p_1$, we smooth the intersection by creating a new loop, $aabb$, by following the red loop $aab$ in the direction of its orientation at $p_1$, and when returning to $p_1$, we now follow the blue loop $b$ in the direction of its orientation. When we return back to $p_1$, we close the loop. At the intersection $p_2$, we do the same, and create the loop $abab$. We get that $[aab,b]=\pm(aabb+abab)$ where the sign depends on the chosen orientation of $\Sigma_{1,1}$. (Goldman) The Goldman bracket is well defined, skew-symmetric, and satisfies the Jacobi identity [@Go]\
We can extend the bracket linearly to ${\mathbb Z}[\hat{\pi}]$ (or ${\mathbb Q}[\hat{\pi}]$), the free module over ${\mathbb Z}$ (or ${\mathbb Q}$) with basis $\hat{\pi}$, to get a bilinear map $$\begin{aligned}
[ -,-]: {\mathbb Z}\hat{\pi}\times {\mathbb Z}\hat{\pi} \rightarrow {\mathbb Z}\hat{\pi}\end{aligned}$$. Thus, ${\mathbb Z}\hat{\pi}$ is a Lie algebra with bracket $[-,-]$, which we call the *Goldman Lie Algebra*, denoted by $\mathfrak{G}$ throughout the rest of this chapter. When it is unclear what the surface we are referring to, we use $\mathfrak{G}_{\Sigma_{g,n}}$
Goldman Lie Algebra Structure
-----------------------------
What is the Lie Algebra structure of the Goldman Lie algebra? So far, the center of the Goldman Lie algebra is known, but much of the structure is still a mystery. (Etingof) The center of $\mathfrak{G}_{\Sigma_{g,0}}$ is spanned by the contractible loop [@Et]. (Kabiraj) The center of $\mathfrak{G}_{\Sigma_{g,n}}$ is generated by peripheral loops [@Ka]\
A question posed by Chas [@Ch] is whether or not $\mathfrak{G}$ is finitely generated. In Goldman’s paper [@Go], he also introduces what is called the homological Goldman Lie algebra. This Lie algebra is defined on intersection form on the first homology group of a surface. It is known that this Lie algebra is indeed finitely generated [@KKT], but of course, the homological Goldman Lie algebra is simpler.\
The closed torus is a special case. $\mathfrak{G}_{\Sigma_{1,0}}$ is finitely generated. Recall that we can represent free homotopy classes of loops on $\Sigma_{1,0}$ by cyclically reduced words in two letters, $a$ and $b$, and we can represent all homotopy classes of loops on the torus by the word $a^lb^k$ for $k,l\in {\mathbb Z}$. \[prop:torusstructure\] The Goldman bracket structure of $\mathfrak{G}_{\Sigma_{0,1}}$ is given by $$\begin{aligned}
[a^ib^j,a^kb^l]=(il-jk)a^{i+k}b^{j+l}\end{aligned}$$ \[thm:torus\]$\mathfrak{G}_{\Sigma_{1,0}}$ is finitely generated when considered as a Lie algebra over ${\mathbb Q}$.. We denote a contractible loop by $1$. We claim that $\mathfrak{G}_{\Sigma_{1,0}}$ is generated by $\{a,b,a^{-1},b^{-1}\}$. This will take many steps. We will first show that we can generate certain homotopy classes of loops. Below, we assume $n\neq 0$.
1. $a^nb^1=[a,a^{n-1}b]$, which we get inductively,
2. $a^n=[b^{-1},-\frac{1}{n}a^nb]$
3. $ab^n=[b,-ab^{n-1}]$
4. $b^n=[a^{-1},-\frac{1}{n}ab^n]$
5. $a^nb^n=[a^n,\frac{1}{n^2}b^n]$
6. $a^{-n}b=[a^{-1},-a^{-n+1}b]$
7. $a^{-n}=[b^{-1},\frac{1}{n}a^{-n}b]$
8. $a^{-1}b^n-[a^{-1},-\frac{1}{n}b^n]$
9. $a^{-n}b^n=[a^{-n},-\frac{1}{n^2}b^n]$
10. $ab^{-n}=[b^{-1},ab^{n+1}]$ which we get inductively,
11. $b^{-1}=[a^{-1},\frac{1}{n}ab^{-n}]$
12. $a^{-n}b^{-n}=[a^{-n},\frac{1}{n^2}b^{-n}]$
13. $a^nb^{-n}=[a^n,-\frac{1}{n^2}b^{-n}]$.
14. From 13. and 9. for $n=1$, we get $a^0b^0=1=[ab^{-1},\frac{1}{2}a^{-1}b]$.
We still have a few more cases to show, namely how to generate the homotopy class of the loop $a^ib^j$ in the following cases.
1. Suppose $i,j>0$.
1. Suppose $i<j$, then $j=i+r$ for some $r\in {\mathbb Z}-\{0\}$.\
Then $\frac{1}{ar}[a^ib^i,b^j]=a^ib^j$.
2. Suppose $i>j$, then $i=j+r$ for $r\in {\mathbb Z}-\{0\}$.\
Then $-\frac{1}{br}[a^jb^j,a^r]=a^ib^j$.
2. Suppose $i<0<j$.
1. Suppose $|i|<|j|$, then $j=-i+r$ for $r\in {\mathbb Z}-\{0\}$.\
Then $\frac{1}{ar}[a^ib^{-i},b^r]=a^ib^j$.
2. Suppose $|a|>|b|$, then $i=-j+r$ for $r\in {\mathbb Z}-\{0\}$.\
Then $-\frac{1}{br}[a^{-j}b^{j},a^r]=a^ib^j$.
3. The case $i,j<0$, and $i\neq j$ is similar to Case 1.
4. The case $b<0<a$ is similar to Case 2.
Thus, everything in $\mathfrak{G}_{\Sigma_{1,0}}$ can be generated as a Lie algebra over ${\mathbb Q}$. We can refine the generators of $\mathfrak{G}_{\Sigma_{0,1}}$ to a smaller basis, namely $\{a, a^{-1}b^{-1}+b+1,b\}$ We show that we generate the basis elements mentioned in the proof of Theorem \[thm:torus\]. $$\begin{aligned}
[a^{-1}b^{-1}+b+1,b]&=-a^{-1},\\
[a^{-1}b^{-1}+b+1,a]&=b^{-1}-ab,\\
[a,b]&=ab.\end{aligned}$$ $\mathfrak{G}_{\Sigma_{1,0}}$ as a Lie algebra over ${\mathbb Q}$ is not nilpotent, nor solvable, since $[\mathfrak{G}_{\Sigma_{1,0}} , \mathfrak{G}_{\Sigma_{1,0}}] =\mathfrak{G}_{\Sigma_{1,0}}$. $\mathfrak{G}_{\Sigma_{1,0}}$ is not finitely generated as a Lie algebra over $\mathbb{Z}$. We will show that the set $\{(n-1)a^n\}_{n>2, n\in {\mathbb Z}}$ cannot be generated. Suppose to the contrary that we can generate $(n-1)a^n$, so there exists $(i_s,j_s),(k_s,l_s) \in {\mathbb Z}^2$ such that $$\begin{aligned}
\sum_{s=1}^t\pm [a^{i_s}b^{j_s},a^{k_s},b^{l_s}]=(n-1)a^n.\end{aligned}$$ Then, as in Proposition \[prop:torusstructure\], $$\begin{aligned}
\sum_{s=1}^t\pm [a^{i_s}b^{j_s},a^{k_s},b^{l_s}]=\sum_{s=1}^t\pm (i_sl_s-j_sk_s)a^{i_s+k_s}b^{j_s+l_s}.\end{aligned}$$ We need that $i_s+k_s=n$ and $j_s+l_s=0$, so $i_sl_s-j_sk_s=-i_sj_s-j_sn+j_si_s=-j_sn$. So $n\mid i_sl_s-j_sk_s$, and $n\mid \sum_{s=1}^t\pm (i_sl_s-j_sk_s)$, so $n\mid (n-1)$, which is a contradiction. We conjecture that $\mathfrak{G}_{\Sigma_{g,n}}$ for $g\geq 1$ and $n>1$ is not finitely generated. For the particular case for a punctured torus, the peripheral loop is given by a commutator word. We noticed in using Chas’ program for computing the bracket seems to not generate a commutator word, nor products of commutators. This needs more work, but this would mean we have a set $\{(aba^{-1}b^{-1})^n\}_{n \in {\mathbb Z}}$ of infinitely many homotopy classes of loops that each cannot be generated by any other homotopy classes of loops. \[prop:derived\] The derived Lie algebra for $\mathfrak{G}_{\Sigma_{1,0}}$ is given by $$\begin{aligned}
[\mathfrak{G}_{\Sigma_{1,0}},\mathfrak{G}_{\Sigma_{1,0}}]=\langle d(a^ib^j),na^n,nb^n \rangle\end{aligned}$$ for $d=gcd(i,j)$ and $n\in {\mathbb Z}-\{0\}$. We first show $[\mathfrak{G}_{\Sigma_{1,0}},\mathfrak{G}_{\Sigma_{1,0}}]\subset \langle d(a^ib^j),na^n,nb^n \rangle$
1. Suppose $d=gcd(i,j)$, $i,j \neq 0$ and $ma^ib^j \in [\mathfrak{G}_{\Sigma_{1,0}},\mathfrak{G}_{\Sigma_{1,0}}]$ for some $m\in {\mathbb Z}$. Write $xi+yj=d$ for some $x,y\in {\mathbb Z}$ and $ma^ib^j=[a^kb^l,a^p,b^q]$ for $k,l,p,q \in {\mathbb Z}$. But $$\begin{aligned}
\label{align:gcdd}
[a^kb^l,a^p,b^q]=(kq-lp)a^{k+p}b^{l+q}\end{aligned}$$ so we have that $k+p=i$, $l+q=j$, and $kq-lp=kj-li=d(k(\frac{j}{d})-l(\frac{i}{d}))=m$. Thus $d\mid m$.
2. Suppose that $n \neq 0$ and that $[a^ib^j,a^kb^l]=ma^n$ for $i,j,k,l,m\in {\mathbb Z}$. Then $i+k=n$, $j+l=0$, so $$\begin{aligned}
\label{align:n0}
[a^ib^j,a^kb^l]=-jna^n\end{aligned}$$ so $n\mid m$.
3. Showing that for $n \neq 0$ and $n\mid m$ for $mb^n \in [\mathfrak{G}_{\Sigma_{1,0}},\mathfrak{G}_{\Sigma_{1,0}}]$ is similar to Case 2.\
To show the other containment, we can consider the equality \[align:gcdd\] with $k=y$ and $l=-x$ for Case 1, we can consider the equality \[align:n0\] with $j=-1$, and we can do something similar for Case 3.
The lower central series for $\mathfrak{G}_{\Sigma_{1,0}}$ stabilizes, i.e. $$\begin{aligned}
[\mathfrak{G}_{\Sigma_{1,0}},G_i]=\langle d(a^ib^j),na^n,nb^n \rangle\end{aligned}$$ where $d=gcd(i,j)$, $n\in {\mathbb Z}-\{0\}$, for all $i\geq 0$, and $G_i=[\mathfrak{G}_{\Sigma_{1,0}},G_{i-1}]$ defined inductively, where $G_0=\mathfrak{G}_{\Sigma_{1,0}}$. For $i=1$, this is just Proposition \[prop:derived\]. For $i=2$, we need to show that $$\begin{aligned}
[ \mathfrak{G}_{\Sigma_{1,0}}, [\mathfrak{G}_{\Sigma_{1,0}}, \mathfrak{G}_{\Sigma_{1,0}} ]]=\langle d(a^ib^j),na^n,nb^n \rangle .\end{aligned}$$ The “$\subset $” containment is clear. First, consider $a^ib^{-1}\in \mathfrak{G}_{\Sigma_{1,0}}$ and $a^{n-i}b \in \langle d(a^ib^j),na^n,nb^n\rangle$ (since $gcd(n-i,1)=1$). We have that $$\begin{aligned}
[a^ib^{-1},a^{n-i}b]=na^n.\end{aligned}$$ In a similar way, we can show that $nb^n \in [ \mathfrak{G}_{\Sigma_{1,0}}, [\mathfrak{G}_{\Sigma_{1,0}}, \mathfrak{G}_{\Sigma_{1,0}} ]]$.\
Now consider $d=gcd(i.j)$, so we can write $d=xi+yj$. Consider $a^{i+y}b^{j-x} \in \mathfrak{G}_{\Sigma_{1,0}}$ and $ a^{-y}b^x \in [\mathfrak{G}_{\Sigma_{1,0}} , \mathfrak{G}_{\Sigma_{1,0}} ]$ (since $1=\frac{i}{d}x+\frac{j}{d}y$ implies $gcd(x,y)=1$. We have $$\begin{aligned}
[a^{i+y}b^{j-x} , a^{-y}b^x]=da^ib^j.\end{aligned}$$ Thus, it follows that the lower central series stabilizes. $\mathfrak{G}_{\Sigma_{1,0}}$ as a Lie algebra over ${\mathbb Z}$ is not nilpotent.
String Topology Preliminaries
=============================
Here we describe the basic algebraic structures appearing in the homology and equivariant homology of the free loop space, as described by Chas and Sullivan in *String Topology*. Throughout this paper, let $M$ be a manifold of dimension $d$, $\Omega M$ the based loop space of $M$, and the space of all continuous, piecewise smooth loops on $M$, $LM=Map(S^1, M)$, the free loop space of $M$. Note that $LM$ can be considered to be an infinite-dimensional manifold, and it is topologised with the compact-open topology. We will consider homology and cohomology with integer coefficients, unless otherwise stated. We denote the usual homology of the free loop space of $M$ as $H_*(LM)$ and equivariant homology will be denoted by $H_*^{S^1}(LM)$.
Loop Homology Algebra Preliminaries
-----------------------------------
We first describe the Chas-Sullivan loop product, which Chas and Sullivan defined on the chain level of $LM$, the space of all continuous, piecewise smooth loops on $M$.\
The loop product is a combination of the intersection product and the product given by the concatenation of loops. It is defined transversally at the chain level.\
Consider an $i$-chain of loops in $LM$. We can think of a simplex in this chain as a map $\sigma : \Delta_i \rightarrow LM$ or as a map $\sigma : \Delta_i \otimes S^1 \rightarrow M$. So we can think of an $i$ chain of loops as a map from a simplex with loops above it into $M$. Intuitively, if we have an $i$-chain and a $j$-chain of loops where the marked points intersect transversally, then we get a new $i+j-d$-chain of loops consisting of the intersecting marked points, and at each marked point, the new loop is formed by going around the $i$-chain loops then around the $j$-chain of loops. This description at the chain level can pass to homology to form the *Chas-Sullivan loop product*.\
Here we give a more precise description of the product given in *String Topology and Cyclic Homology* [@CHV]. Let $Map(8,M)=Map(S^1\vee S^1)$ be the space of continuous, piecewise smooth maps from the figure eight, or the wedge sum of two circles to $M$. This is topologised with the compact-open topology and can be considered as an infinite-dimensional manifold, but we need piecewise smooth in order for it to be some sort of manifold. It can also be viewed as a subspace of $LM \times LM$ where the loops agree at $0$.\
Consider the following diagram. The left square is a pullback diagram.:
$$\begin{CD}
LM \times LM @< \rho_{\text{in}}<< Map(8,M) @> \rho_{\text{out}} >> LM\\
@VV{ev \times ev}V @VV{ev}V\\
M \times M @<\Delta << M
\end{CD}$$
\
where $\rho_{\text{in}}$ is the restriction of the figure eight to the two different loops, and $\rho_{\text{out}}$ is where the figure eight loop is considered as one loop. The maps $ev$ are the evaluation of the loops at $0$, and $\Delta$ is the diagonal map. $ev$ is a locally trivial fibration so if $\eta_{\Delta}$ is a tubular neighborhood of the diagonal embedding, then a tubular neighborhood of the $\eta_{\rho_{in}}=(ev\times ev)^{-1}(\eta_{\Delta})$ is homeomorphic to $ev^*(TM)=ev^*(\Delta(M))$. We can have a tubular neighborhood since $\rho_{in}$ is a codimension $d$ embedding. Since $ev^*(\eta_{\Delta})$ is the pullback of $\eta_{\Delta}$, which has fiber dimension $d$ since it is the normal bundle, the pullback $ev^*(\eta_{\Delta})$ has fibers isomorphic to fibers of $\eta_{\Delta}$, so $ev^*(\eta_{\Delta})$ also has fiber dimension $d$. This means that the normal bundle of $map(8,M)$ in $LM\times LM$ has codimension $d$. Since in the above diagram, the left square is a pullback diagram of fiber bundles, we have that $\rho_{in}$ is a codimension $d$ embedding.\
The induced maps on homology go in the wrong direction, in order to remedy this, we need to turn the map $\rho_{\text{in}}$ around somehow. We do this by using the Pontrjagin-Thom collapse map:\
$$LM \times LM \rightarrow LM \times LM / LM \times LM -ev^*(TM) \cong Thom(Map(8,M))$$\
Define the umkehr map $(\rho_{in})_!$ containing the induced map on homology above as follows:\
$$\begin{aligned}
(\rho_{in})_!: H_*(LM)\otimes H_*(LM)\cong H_*(LM \times LM)\rightarrow & H_*(Thom(Map(8,M)))\\ &\cong H_{*-d}(Map(8,M))\end{aligned}$$ The last isomorphism is given by the Thom Isomorphism by taking the cap product with $u\in H^d(Thom(Map(8,M)))$, the Thom class given by the orientation. \[def:loopproduct\] The following composition gives the *Chas-Sullivan loop product* (or just *loop product*): $$\begin{aligned}
-\bullet - =(\rho_{out})_* \circ (\rho_{in})_!: H_*(LM \times LM) \rightarrow H_{*-d}(Map(8,M)) \rightarrow H_{*-d}(LM)\end{aligned}$$ This product can be extended to homology. It is convenient to regrade the loop homology as follows: $$\begin{aligned}
\mathbb{H}_*(LM):=H_{*+d}(LM)\end{aligned}$$\
we can rewrite the product: $$-\bullet -= \mathbb{H}_*(LM) \otimes\mathbb{H}_*(LM) \rightarrow \mathbb{H}_{*}(LM)$$ We may drop the $LM$ from the notation and denote loop homology by $\mathbb{H}_*$ when it is clear which manifold we are referring to. (Chas-Sullivan) $(\mathbb{H}_*(LM),\bullet )$ is an associative, graded, commutative algebra. There is a *Batalin-Vilkovisky operator* denoted by $\Delta$, which comes from the natural action given by rotation of loops, $$\begin{aligned}
\rho:S^1\times LM \rightarrow LM\end{aligned}$$ given by $\rho(t,\gamma)(s)=\gamma(s+t)$. This action defines a degree one operator on loop homology: $\Delta:\mathbb{H}_*(LM)\rightarrow \mathbb{H}_{*+1}(LM)$ given by $\delta(\alpha)=\rho_*([S^1]\otimes \alpha)$ for $\alpha \in H_k(LM)$. (Chas-Sullivan) \[thm:loopbracket\] $(\mathbb{H}_*(LM), \Delta )$ is a Batalin-Vilkovisky algebra,
1. $(\mathbb{H}_*(LM), \bullet)$ is a graded, commutative, associative algebra
2. $\Delta \circ \Delta =0$
3. $(-1)^{|\alpha |} \Delta (\alpha \bullet \beta ) - (-1)^{ |\alpha |} \Delta (\alpha) \bullet \beta - \alpha \bullet \Delta (\beta)$ is a derivation in each variable.
We can also define a Lie bracket with $\bullet$ and $\Delta$ as in part 3 of Theorem \[thm:loopbracket\]. The *loop bracket* is defined as $$\begin{aligned}
\{\alpha , \beta \}:= (-1)^{|\alpha |} \Delta (\alpha \bullet \beta ) - (-1)^{ |\alpha |} \Delta (\alpha) \bullet \beta - \alpha \bullet \Delta (\beta)\end{aligned}$$ which is the deviation of $\Delta$ from being a derivation of $\bullet$. (Chas-Sullivan) $(\mathbb{H}_*(LM),\bullet , \{-,-\})$ has the structure of a Gerstenhaber algebra,
1. $(\mathbb{H}_*(LM), \bullet)$ is a graded, commutative, associative algebra
2. $\{-,-\}$ is a degree $1$ Lie bracket,
1. ${\alpha , \beta }=(-1)^{ (|\alpha | +1)(|\beta |+1)+1} \{\beta ,\alpha \}$,
2. $\{ \alpha , \{ \beta , \gamma \} \} =\{ \{ \alpha, \beta \} , \gamma \} + (-1)^{ (|\alpha | +1)(|\beta |+1)} \{ \beta , \{ \alpha , \gamma \} \}$,
3. $\{ \alpha , \beta \bullet \gamma \} = \{\alpha , \beta \} \bullet \gamma + (-1)^{ (|\alpha | -1)|\beta |} \beta \bullet \{ \alpha , \gamma \}$.
String Homology Algebra Preliminaries
-------------------------------------
Now we consider algebraic structures on the equivariant homology of the free loop space with respect to the action of rotation of loops, $H_*^{S^1}(LM)$. Consider the fibration $$\begin{aligned}
S^2 \rightarrow LM \times ES^1 \rightarrow LM \times_{S^1} ES^1.\end{aligned}$$ This induces a long exact sequence on homology, the Gysin sequence from which we will use to describe a Lie bracket on $H_*^{S^1}(LM)$. $$\begin{aligned}
\cdots \rightarrow \mathbb{H}_{*-d}(LM) \xrightarrow{e} H_*^{S^1}(LM) \xrightarrow{\cap} H_{*-2}^{S^1}(LM) \xrightarrow{M} \mathbb{H}_{*-d-1}(LM) \rightarrow \cdots\end{aligned}$$ where $e$ and $M$ are informally called the “erasing map” and “marking map,” respectively. The map $e$ forgets the marked points on the loops, and the map $M$ puts markings back on the loops in all possible places. We have that $M$ is a homomorphism of graded Lie algebras, it preserves the brackets, going from the string bracket to the loop bracket. The map $e$ is the induced fibration map. For the rest of this paper, it will be clear from the context which space we are referring to, so we often drop the $LM$ from the homology notation. Note that $e\circ M=0$ by exactness, and $\Delta = M \circ e$. \[defn:stringbracket\] For two classes $\alpha , \beta \in H_*^{S^1}(LM)$, we can define the *string bracket* by $$\begin{aligned}
[\alpha , \beta ] = (-1)^{| \alpha | -d }e(M(\alpha) \bullet M(\beta) ) \end{aligned}$$ where $\bullet$ was the loop product mentioned in Definition \[def:loopproduct\]. (Chas-Sullivan) $(H_*^{S^1}(LM),[-,-])$ is a graded Lie algebra, with Lie bracket of degree $2-d$.\
More precisely, our bracket is a map: $$\begin{aligned}
[-,-]: H_i^{S^1}(LM)\times H_j^{S^1}(LM) \rightarrow H_{i+j+2-d}^{S^1}(LM).\end{aligned}$$ In the following chapters, we compute the $H_*^{S^1}(LS^n)$ for all $n\in {\mathbb N}$ and we compute the structure of the string bracket.
Computations of String Homology and the String Bracket
======================================================
In this chapter, we compute explicitly the integral string homology and the string bracket for spheres. Somnath Basu made some computations of rational string homology for spheres in his Ph.D. thesis [@Ba] using rational homotopy theory and minimal models. Craig Westerland also made computations of string homology over ${\mathbb Z}_2$ for spheres in *String Homology of Spheres and Projective Spaces* [@We] using a spectral sequence. We separate the computations for the even- and odd-dimensional spheres. First, we compute particular examples, $S^1$, $S^3$, and $S^2$, to get a better hold on the computation, then generalize to the higher-dimensional spheres. We use primarily the Gysin exact sequence, and the Leray-Serre spectral sequence to aid in computations of string homology. We find that there is a lot of interesting torsion in integral string homology, and the bracket structure is not always zero.
String Homology and String Bracket of $S^1$
-------------------------------------------
We compute the string homology of $S^1$ using the Gysin sequence for the circle bundle $$\begin{aligned}
\nonumber S^1 \rightarrow LS^1 \times ES^1 \rightarrow LS^1 \times_{S^1} ES^1\end{aligned}$$\
Basu computed this in his thesis, but here we use elementary techniques.\
Recall that the non-equivariant homology of $LS^1$ is given as follows [@CJY], [@He], $$\nonumber \mathbb{H}_*(LS^1)=\Lambda_{{\mathbb Z}}[a]\otimes {\mathbb Z}[x,x^{-1}], \; |a |=-1, |x|=0.$$ where $\mathbb{H}_*(LS^1)=H_{*+1}(LS^1)$ and $a$ corresponds to the dual of $[S^1]$ under the geometric grading [@Se], [@CJY].\
The BV-operator ($\Delta=M\circ e$) acts on generators of $\mathbb{H}_*(LS^1)$ as follows, [@Me]: $$\begin{aligned}
&\Delta (a \otimes x^i)& =i(1\otimes x^{i})\nonumber \\
&\Delta (1\otimes x^i) & =0.\nonumber \end{aligned}$$ Consider the Gysin sequence for the above circle bundle:\
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& H\^[S\^1]{}\_1(LS\^1) & H\^[S\^1]{}\_[-1]{}(LS\^1)0 & \_[-1]{}(LS\^1)\_[n[Z]{}]{} [Z]{}(ax\^n)\
& H\^[S\^1]{}\_0(LS\^1) & 0 &\
\
]{};
(m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge node\[above\] [$c$]{} (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) ;
\
The end of the Gysin sequence gives us that ${H^{S^1}_0(LS^1)\cong \bigoplus\limits_{n\in {\mathbb Z}} {\mathbb Z}(e(a\otimes x^n))}$. Using the information from the BV-operator, ${M\circ e(a \otimes x^n)=\Delta (a \otimes x^n)}=n(1 \otimes x^n)$. Since $e$ is surjective and ${ker(e)=im(M)= \bigoplus\limits_{n\in {\mathbb Z}} n{\mathbb Z}(1 \otimes x^n)}$, we have that ${H^{S^1}_1(LS^1) \cong \mathbb{H}_{0}(LS^1) / ker(e) \cong \bigoplus\limits_{n\in {\mathbb Z}} {\mathbb Z}/n{\mathbb Z}(1 \otimes x^n) \oplus {\mathbb Z}(1\otimes 1)} $. From the beginning of the Gysin sequence, we have ${im(c)=ker(M)={\mathbb Z}(a\otimes 1)}$, and since $c$ is injective, ${H^{S^1}_2(LS^1) \cong {\mathbb Z}(a \otimes 1)}$. Summarizing, we get the following remark. $$\begin{aligned}
& H^{S^1}_0(LS^1) &\cong \bigoplus_{n\in {\mathbb Z}} {\mathbb Z}(e(a\otimes x^n))\nonumber \\
& H^{S^1}_1(LS^1) &\cong H^{S^1}_{2i+1}(LS^1) \cong \bigoplus_{n\in {\mathbb Z}-\{0\}} {\mathbb Z}/n{\mathbb Z}(1 \otimes x^n)\oplus {\mathbb Z}(1 \otimes 1), \hspace*{.5cm} i\geq 0\nonumber \\
& H^{S^1}_2(LS^1) &\cong H^{S^1}_{2i}(LS^1) \cong {\mathbb Z}(a \otimes 1), \hspace*{.5cm} i\geq 1\nonumber \end{aligned}$$\
The string bracket, ${[-,-]: H_i^{S^1}(LS^1) \otimes H_j^{S^1}(LS^1) \rightarrow H_{i+j+1}^{S^1}(LS^1)}$ is a degree $+1$ map, and it is only nontrivial on generators of degree zero since the marking map is trivial for generators of degree greater than zero. For ${a \otimes x^n}$, ${a \otimes x^m}$ in ${H^{S^1}_0(LS^1)}$, $$\begin{aligned}
[e(a \otimes x^n),e( a \otimes x^m)] & =(-1)^{-1}e(M(e(a \otimes x^n))\bullet M(e(a \otimes x^m))) \\
&=-e(n(1 \otimes x^n) \bullet m(1 \otimes x^m))\\
& =-nm(e(1 \otimes x^{n+m}))\\
&=-nm(1 \otimes x^{n+m})\end{aligned}$$ So ${[a \otimes x^n, a \otimes x^m]=0}$ if $n+m \neq 0$ and $n+m$ divides $nm$. If $n+m=0$ then ${[a \otimes x^n, a \otimes x^m]=nm(1\otimes 1)}$. We can conclude that the bracket is only nontrivial for the torsion elements.
String Homology and String Bracket of $S^3$
-------------------------------------------
We compute the equivariant homology of $S^3$ using the Gysin sequence for the circle bundle $$\begin{aligned}
\label{eqn:threecirclebundle}
S^1 \rightarrow LS^3 \times ES^1 \rightarrow LS^3 \times_{S^1} ES^1\end{aligned}$$ and the Serre homology spectral sequence for $$LS^3 \rightarrow LS^3 \times_{S^1} ES^1 \rightarrow {\mathbb C}P^\infty\nonumber$$ First we compute the equivariant cohomology of $LS^3$ and then translate it to equivariant homology. We also compute the erasing ($e$) and marking ($M$) maps, as in Chas and Sullivan’s paper, to compute the String Bracket.\
### First Few Equivariant Homology Groups of $LS^3$
By equivariant homology, we mean the homology of the Borel construction from the natural action of $S^1$ on $LS^3$ by rotation, denoted by $H_*^{S^1}(LS^3)=H_*(LS^3 \times_{S^1} ES^1)$. We calculate the first few equivariant homology groups of $LS^3$ to aid in our computation of the equivariant cohomology of $LS^3$.\
Recall that the non-equivariant homology of $LS^3$ is given as follows: $$\mathbb{H}_*(LS^3)=\Lambda_{{\mathbb Z}}[\alpha]\otimes {\mathbb Z}[y], \; |\alpha |=-3, |y|=2 \nonumber$$ where $\mathbb{H}_*(LS^3)=H_{*+3}(LS^3)$ and $\alpha$ corresponds to the dual of $[S^3]$ under the usual grading [@Se], [@CJY].\
To compute the equivariant homology of $LS^3$, we consider the Gysin sequence for the following fibration: $$S^1\rightarrow LS^3 \times ES^1 \rightarrow LS^3 \times_{S^1} ES^1.\nonumber$$ and the BV-operator ($\Delta=M\circ e$), which acts on generators of $\mathbb{H}_*(LS^3)$ as follows [@T], [@Me]: $$\begin{aligned}
\Delta (\alpha \otimes y^i) & =i(1\otimes y^{i-1})\nonumber \\
\Delta (1\otimes y^i) & =0.\nonumber \end{aligned}$$ The Gysin exact sequence:\
\(m) \[ matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_4(LS\^3) & H\^[S\^1]{}\_2(LS\^3) & \_[0]{}(LS\^3)[Z]{}(11)\
& H\^[S\^1]{}\_3(LS\^3) & H\^[S\^1]{}\_1(LS\^3) & \_[-1]{}(LS\^3)[Z]{}(y)\
& H\^[S\^1]{}\_2(LS\^3) & H\^[S\^1]{}\_0(LS\^3) & \_[-2]{}(LS\^3)0\
& H\^[S\^1]{}\_1(LS\^3) & 0 & \_[-3]{}(LS\^3)[Z]{}(1)\
& H\^[S\^1]{}\_0(LS\^3) &0 & 0\
]{};
(m-1-2) edge (m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-5-2) (m-5-2) edge (m-5-3) (m-5-3) edge\[orange\] node\[above\] [$M$]{}(m-5-4);
\
The short exact sequence in the last two rows shows that $H^{S^1}_0(LS^3)\cong {\mathbb Z}(\alpha \otimes 1)$. The short exact sequence in the third and fourth row, $$0\rightarrow H_1^{S^1}(LS^3) \rightarrow 0\nonumber$$ shows that $H^{S^1}_1(LS^3)\cong 0$. Thus, we obtain a short exact sequence from the second and third row, $$0\rightarrow {\mathbb Z}(\alpha \otimes y) \rightarrow H^{S^1}_2(LS^3) \rightarrow {\mathbb Z}(\alpha \otimes 1) \rightarrow 0. \nonumber$$ Since the last non-zero term in the sequence is free, the sequence splits, giving $H^{S^1}_2(LS^3)\cong {\mathbb Z}(\alpha \otimes y) \oplus {\mathbb Z}(\alpha \otimes 1)$. To calculate $H^{S^1}_3(LS^3)$, we use the BV operator. The injective map $e$ in the exact sequence (1.2) means that $e(\alpha \otimes y)= \alpha \otimes y$. Since $\Delta(\alpha \otimes y)=M \circ e (\alpha \otimes y)=1 \otimes 1$, the map $M$ in the first row of the Gysin sequence above is surjective, so the connecting map $e$ from the first to the second row of the Gysin sequence has kernel ${\mathbb Z}(1 \otimes 1) $. Thus we have a short exact sequence, $$0 \rightarrow H^{S^1}_3(LS^3) \rightarrow 0\nonumber$$ So $H^{S^1}_3(LS^3) \cong 0$.\
We may be able to continue computing the rest of the equivariant homology groups of $LS^3$ in this way, but we eventually reach extension issues. In summary, we have the following remark: $$\begin{aligned}
& H^{S^1}_0(LS^3) &={\mathbb Z}(\alpha \otimes 1)\nonumber \\
& H^{S^1}_1(LS^3) &=0\nonumber \\
& H^{S^1}_2(LS^3) &={\mathbb Z}(\alpha \otimes y) \oplus {\mathbb Z}(\alpha \otimes 1)\nonumber \\
& H^{S^1}_3(LS^3) &=0.\nonumber \end{aligned}$$
### Equivariant Cohomology of $LS^3$
Consider the fibration $$LS^3 \rightarrow LS^3\times_{S^1}ES^1 \rightarrow {\mathbb C}P^{\infty}\nonumber$$ and the cohomology Leray-Serre spectral sequence associated with it.\
We use the fact that we know the ordinary cohomology of $LS^3$ and ${\mathbb C}P^{\infty}$, since the $E_{\infty}$ page converges to $H^*_{S^1}(LS^3)$, the equivariant cohomology of $S^3$. [@CJY] $$H^*(LS^3)\cong H^*(\Omega S^3)\otimes H^*(S^3)\cong \Gamma[y]\otimes \Lambda [a] \hspace*{.5cm}|a|=3, |y|=2, y_i=\frac{y^i}{i!} \nonumber$$ $$H^*({\mathbb C}P^{\infty})\cong {\mathbb Z}[x], \;|x|=2\nonumber$$ Below is the $E_2$ page of the spectral sequence. All of the nonzero entries are ${\mathbb Z}$ generated by the entry. The arrows are the $d_2$ maps.\
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10&& y\_5 &0&…&&&&&&&&&&\
9 && ay\_3 &0&&&&&&&&&&&&&&&&&&&\
8 && y\_4 &0&y\_4x&0&y\_4x\^2&0&y\_4x\^3&0&y\_4x\^4&0&y\_4x\^5&&\
7 && ay\_2 &0&ay\_2x&0&ay\_2x\^2&0&ay\_2x\^3&0&ay\_2x\^4&0&ay\_2x\^5&&\
6 && y\_3 &0&y\_3x&0&y\_3x\^2&0&y\_3x\^3&0&y\_3x\^4&0&y\_3x\^5&&\
5 && ay\_1&0&ay\_1x&0&ay\_1x\^2&0&ay\_1x\^3&0&ay\_1x\^4&0&ay\_1x\^5&&\
4 && y\_2 &0 & y\_2x& 0& y\_2x\^2 & 0 & y\_2x\^3 & 0&y\_2x\^4&0&y\_2x\^5&&\
3 && a & 0 & ax & 0 & ax\^2 & 0 & ax\^3 & 0 & ax\^4 & 0 & ax\^5 &&\
2 && y\_1 & 0 & y\_1x & 0 & y\_1x\^2 & 0 & y\_1x\^3 & 0 & y\_1x\^4 & 0 & y\_1x\^5 &&\
1 && 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &&\
0 && 1 & 0 & x & 0 & x\^2 & 0 & x\^3 & 0 & x\^4 & 0 & x\^5 &&\
&& & & & & &&&&&&&&\
E\_2 & & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 &
\
; (m-13-2.north) – (m-1-2.north); (m-12-1.east) – (m-12-13.east); (m-9-3) – node\[above\] [$0$]{} (m-10-5); (m-8-3) – node\[above\] [$\cong$]{}(m-9-5); (m-7-3) – node\[above\] [$0$]{}(m-8-5); (m-6-3) – node\[above\] [$\times 2$]{}(m-7-5); (m-5-3) – node\[above\] [$0$]{}(m-6-5); (m-4-3) – node\[above\] [$\times 3$]{}(m-5-5); (m-3-3) – node\[above\] [$0$]{}(m-4-5); (m-2-3) – node\[above\] [$\times 4$]{}(m-3-5); (m-11-5) – node\[above\] [$0$]{}(m-12-7); (m-5-5) – node\[above\] [$0$]{}(m-6-7); (m-6-7) – node\[above\] [$\times 2$]{}(m-7-9); (m-7-9) – node\[above\] [$0$]{}(m-8-11); (m-6-5) – node\[above\] [$\times 2$]{}(m-7-7); (m-7-7) – node\[above\] [$0$]{}(m-8-9); (m-7-5) – node\[above\] [$0$]{}(m-8-7); (m-8-7) – node\[above\] [$\cong$]{}(m-9-9); (m-8-5) – node\[above\] [$\cong$]{}(m-9-7);
We can figure out the first few equivariant cohomology groups easily. It can immediately be seen that $H_{S^1}^0(LS^3)\cong {\mathbb Z}(1\otimes 1)$ and $H_{S^1}^1(LS^3)\cong 0$. For $H_{S^1}^2(LS^3)$, the differential maps $d_2$ going to and from the generators along the diagonal, $y_1$ and $x$, are zero, so these generators survive to the $E_{\infty}$ page. In the filtration of $H_{S^1}^2(LS^3)$ corresponding to this spectral sequence, we obtain $0 \subset {\mathbb Z}y_1 \subset H_{S^1}^2(LS^3)$ where $H_{S^1}^2(LS^3)/{\mathbb Z}x \cong {\mathbb Z}y_1$. Thus $H_{S^1}^2(LS^3) \cong {\mathbb Z}y_1 \oplus {\mathbb Z}x$.\
The derivation property of the differentials in the Serre spectral sequence makes the computation of the $d_2$ differentials easier. We only need to know the image of $x$, $y_1$, and $a$ through $d_2$ to know the image of the other generators in the $E^2$ grid. We see immediately that $d_2 (x)=0$ and $d_2 (y_1) = 0$. From the multiplicative property of the sequence, we can conclude that $d_2(x^i) =0$ and $d_2(y_i) =0$ for all $i\geq 1$. We computed $H^{S^1}_3(LS^3)\cong 0 $ above, and using the Universal Coefficient Theorem, we find that $H_{S^1}^3(LS^3)\cong 0 $ also. This means that on the $E_{\infty}$ page of the spectral sequence, there should only be zeros along the third diagonal. This gives that $d_2: {\mathbb Z}a \rightarrow {\mathbb Z}y_1x$ should be an isomorphism. Since $a$ and $y_1x$ are the generators of these isomorphic groups $d_2(a)=\pm y_1x$. Let us assume $d_2(a)=y_1x$. Also, $$d_2(ay_i)=d(a)y_i=y_1xy_i=y_1x\frac{y_1^i}{i!}=(i+1)y_{i+1}x\nonumber$$ To summarize: $$\begin{aligned}
& d_2(x^i) &=0\nonumber \\
& d_(y_i) &=0\nonumber \\
& d_2(a)&=y_1x\nonumber \\
& d_2(ay_i) &=(i+1)y_{i+1}x.\nonumber \end{aligned}$$\
These calculations correspond to the red arrows on the $E_2$ page above. The spectral sequence collapses at the $E_3$ page since there can never be nonzero differentials after the $E_2$ page because there is nothing for these differentials to hit, so $E_3=E_{\infty}$.\
Let’s take a look at the $E_{\infty}$ page:\
\(m) \[matrix of math nodes, nodes in empty cells,nodes=[minimum width=4ex, minimum height=4ex,outer sep=0pt]{}, column sep=1ex,row sep=1ex\]
9 &&& &&&&&&&&&&&&\
8 && y\_4 &0&[Z]{}y\_4x/4[Z]{}&0&[Z]{}y\_4x\^2/4[Z]{}&0&[Z]{}y\_4x\^3/4[Z]{}&0&[Z]{}y\_4x\^4/4[Z]{}&…&&\
7 && 0&0&0&0&0&0&0&0&0&…&&\
6 && y\_3 &0&[Z]{}y\_3x/ 3[Z]{}&0&[Z]{}y\_3x\^2/ 3[Z]{}&0&[Z]{}y\_3x\^3 3[Z]{}&0&[Z]{}y\_3x\^4/3[Z]{}&…&&\
5 && 0&0&0&0&0&0&0&0&0&…&&\
4 && y\_2 &0 & [Z]{}y\_2x/2[Z]{}& 0& [Z]{}y\_2x\^2/ 2[Z]{}& 0 & [Z]{}y\_2x\^3/ 2[Z]{}& 0& [Z]{}y\_2x\^4/ 2[Z]{}&…&&\
3 && 0 & 0 & 0 & 0 & 0 & 0 & 0& 0 & 0 & …&&\
2 && y\_1 & 0 &0& 0 & 0& 0 & 0 & 0 & 0 & …&&\
1 && 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & …&&\
0 && 1 & 0 & x & 0 & x\^2 & 0 & x\^3 & 0 & x\^4 & …&&\
&& & & & & &&&&&&&&\
E\_ & & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & &
\
; (m-12-2.north) – (m-1-2.north); (m-11-1.east) – (m-11-12.east);
After some work, and with the assumption that $H_{S^1}^i(LS^3)$ is just the direct sum of the diagonal on the $E_\infty$ page shown above, we can make the following remark. $$\begin{aligned}
H_{S^1}^{2i+1}(LS^3) &=0\nonumber \\
H_{S^1}^{2i}(LS^3) &={\mathbb Z}y_i \oplus {\mathbb Z}x^i \oplus \sum_{j=2}^{i-1} {\mathbb Z}y_j x^{i-j} /j{\mathbb Z}, \hspace*{.5cm} i>0\nonumber \\
H^0_{S^1}(LS^3) &={\mathbb Z}1\nonumber .
\end{aligned}$$
### Equivariant Homology of $LS^3$
Using the results above and the Universal Coefficient Theorem, we get the following:
$$\begin{aligned}
H_{2i}^{S^1}(LS^3)&={\mathbb Z}(\alpha \otimes y^i) \oplus {\mathbb Z}x_i ,\hspace*{.5cm} i \geq 0\nonumber \\
H_{2i+1}^{S^1}(LS^3)&=\sum_{j=2}^{i}{\mathbb Z}(\alpha \otimes y^i)x/j{\mathbb Z}, \hspace*{.5cm} i \geq 2\nonumber \\
H_1^{S^1}(LS^3)&=H_3^{S^1}(LS^3)=0.\nonumber \end{aligned}$$ Note that $\alpha \otimes y^i$ is dual to $y_i$ and $1\otimes y^i$ is dual to $ay_i$. This matches the findings of Basu [@Ba] and Westerland [@We] using the Universal Coefficient Theorem.
### The Spectral Sequence Associated with the Gysin Sequence
To determine the erasing and marking maps, we will translate the Gysin sequence into a spectral sequence and see how they arise in the computation of the spectral sequence. We are using the fact that $H^*(S^1)=\Lambda \omega$ where $|\omega|=1$. Note that on the $E_{\infty}$ page, $a=\omega y_1$.\
\(m) \[matrix of math nodes, nodes in empty cells,nodes=[minimum width=4ex, minimum height=4ex,outer sep=0pt]{}, column sep=.5ex,row sep=1ex\]
1 && [Z]{}& 0 & [Z]{}y\_1 [Z]{}x & 0 & [Z]{}y\_2 [Z]{}x\^2 & 0 & [Z]{}y\_3 [Z]{}x\^3 [Z]{}y\_2 x/2[Z]{}&\
0 && [Z]{}1 & 0 & [Z]{}y\_1 [Z]{}x & 0 & [Z]{}y\_2 [Z]{}x\^2 & 0 & [Z]{}y\_3 [Z]{}x\^3 [Z]{}y\_2 x/2[Z]{}&\
&& & & & & &&&&&&&\
E\_[2]{} & & 0 & 1 & 2 & 3 & 4 & 5 & 6 &
\
; (m-4-2.north) – (m-1-2.north); (m-3-1.east) – (m-3-9.east); (m-1-3) – node\[above\] [$d_2$]{} (m-2-5); (m-1-5) – node\[above\] [$d_2$]{} (m-2-7); (m-1-7) – node\[above\] [$d_2$]{} (m-2-9);;
\(m) \[matrix of math nodes, nodes in empty cells,nodes=[minimum width=4ex, minimum height=4ex,outer sep=0pt]{}, column sep=1ex,row sep=1ex\]
1 && 0 & 0 & [Z]{}a & 0 & [Z]{}ay\_1 & 0 & [Z]{}ay\_2 & 0 & [Z]{}ay\_3 &&\
0 &&[Z]{}1 & 0 & [Z]{}y\_1 & 0 & [Z]{}y\_2 & 0 & [Z]{}y\_3 & 0 & [Z]{}y\_4 &&\
&& & & & & &&&&&&&\
E\_ & & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &
\
; (m-4-2.north) – (m-1-2.north); (m-3-1.east) – (m-3-13.east);
First, we will compute the differential maps for the fibration (1.1) in the spectral sequence. Note that the $E_{\infty}=\cdots =E_4=E_3$ because all of the differential maps $d_i$ for $i\geq 3$ are $0$.\
Since $H^1(LS^3)=0$, the $E_{\infty}$ page has zeros along the $1$ diagonal. This means for the map $d_2: {\mathbb Z}\omega \rightarrow {\mathbb Z}y_1 \oplus {\mathbb Z}x$, $ker(d_2)=0$ so the map is injective. Since the entry $E_{\infty}^{2,0}={\mathbb Z}y_1$, the image of $d_2$ must be ${\mathbb Z}x$, so we can say that $d_2(\omega)=x$ (up to a sign). We have that $d_2(y_1)=0$ and $d_2(x)=0$ also. Using the multiplicative structure of the spectral sequence, we obtain the following remark.\
$$\begin{aligned}
d_2(\omega y_i) & =xy_i\nonumber \\
d_2(\omega x^i)&=x^{i+1}\nonumber \\
d_2(\omega y_i x^j) & = y_i x^{j+1}\nonumber \end{aligned}$$
### The Erasing Map $e$
In the Gysin sequence above for the circle fibration \[eqn:threecirclebundle\], the erasing map $e:$ can be viewed as the map induced by $\epsilon$, the projection map in the fibration \[eqn:threecirclebundle\], so $e=\epsilon_*$. Since we are interested in seeing how the erasing map acts on specific generators, we will instead look at the dual map $\epsilon^*$.\
The map $\epsilon^*:H_{S^1}^i(LS^3) \rightarrow H^i(LS^3)$ can be derived from the spectral sequence of the above fibration \[eqn:threecirclebundle\]. The map $\epsilon^*$ is the composition of the surjection map $E_2^{i,0}=H_{S^1}^i(LS^3) \rightarrow E_{\infty}^{i,0}=E_2^{i,0}/im(d_2)$ and the inclusion map $E_{\infty}^{i,0}\rightarrow H^i(LS^3)$ [@Mc]. This means that an image of a generator in the bottom row of the $E^2$ page of the spectral sequence by $\epsilon^*$ is nonzero if it survives to the $E^{\infty}$ page, and a generator’s image is zero if it does not survive. The following remark is immediate. $$\begin{aligned}
\epsilon ^*(x^i)&=0 \nonumber \\
\epsilon ^*(y_i)&=y_i \nonumber \\
\epsilon ^*(y_jx^i)&=0\nonumber\\\end{aligned}$$\
To dualize $\epsilon^*$ to obtain $e$, we need the Kronecker pairing as in the computation of $M$. $$\begin{aligned}
e=\epsilon _* :H_*(LS^3) & \rightarrow H_*^{S^1}(LS^3)\nonumber \\
\alpha \otimes y^i & \mapsto \alpha \otimes y^i\nonumber \\
1\otimes y^i & \mapsto (\alpha \otimes y^{i+1})x \hspace*{.25cm}\nonumber \\
1 \otimes 1 & \mapsto 0. \nonumber\end{aligned}$$ Since $|\alpha \otimes y^i |=2i$, $e(\alpha \otimes y^i )=kx_i+l(\alpha \otimes y^i) $ for $k,l\in {\mathbb Z}$. We have, $$\begin{aligned}
<\epsilon ^*(y_i),\alpha \otimes y^i >&=<y_i,\alpha \otimes y^i >=1
\\& =<y_i, \epsilon_*(\alpha \otimes y^i )> \\
&=<y_i,kx_i+l(\alpha \otimes y^i )>\\
&=k<y_i,x_i>+l<y_i,\alpha \otimes y^i >=l \end{aligned}$$ and $$\begin{aligned}
<\epsilon ^*(x^i),\alpha \otimes y^i >&=<0,\alpha \otimes y^i >=0\\
&=<x^i, \epsilon_*(\alpha \otimes y^i )>\\
&=<y^i,kx_i+\alpha \otimes y^i >\\
&=k<x^i,x_i>+l<x^i,\alpha \otimes y^i >=k. \nonumber\end{aligned}$$ Therefore $e(\alpha \otimes y^i )=\epsilon_*(\alpha \otimes y^i )=\alpha \otimes y^i $.\
Since $H_3^{S^1}(LS^3)=0$ and $|1\otimes 1|=3$, we must have $e(1\otimes 1)=0$. Since $|1 \otimes y^i |=3+2i$, these generators are of odd degree so they cannot be paired with generators in cohomology since $H^{2i+1}_{S^1}(LS^3) =0$ for $i \geq 0$, so we need to use another technique to find the image of $1 \otimes y^i$. For this we will go back to the Gysin sequence for the fibration (1.1).\
For $i=1$ we look at the following piece of the Gysin sequence.\
\(m) \[ matrix of math nodes, row sep=2em, column sep=2em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_4(LS\^3))[Z]{}(y\^2) [Z]{}x\_2 & \_[2]{}(LS\^3)[Z]{}(1y) &\
& H\^[S\^1]{}\_5(LS\^3) [Z]{}/2 [Z]{}((y\^2)x) & 0\
]{};
(m-1-2) edge\[orange\] node\[above\] [$M$]{}(m-1-3) (m-1-3) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge (m-2-3) ;
\
Since $e$ is surjective and $ker(e)=im(M)\cong 2{\mathbb Z}(1\otimes y)$ by Lemma 6.2, we have that $im(e)\cong {\mathbb Z}/ 2{\mathbb Z}(1\otimes y)$. Thus $e(1\otimes y)=\pm(\alpha \otimes y^2)x$.\
In general, we have the following.\
\(m) \[ matrix of math nodes, row sep=2em, column sep=.5em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_[2i+2]{}(LS\^3)[Z]{}(y\^[i+1]{}) [Z]{}x\_[i+1]{} & \_[2i]{}(LS\^3)[Z]{}(1y\^i) &\
& H\^[S\^1]{}\_[2i+3]{}(LS\^3)\_[j=2]{}\^[i+1]{}[Z]{}\_j(y\^j)x\_[i-j +2]{} & &\
& H\^[S\^1]{}\_[2i+1]{}(LS\^3) \_[j=2]{}\^[i]{}[Z]{}\_j(y\^j)x\_[i-j +1]{} & 0\
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(m-1-2) edge\[orange\] node\[above\] [$M$]{}(m-1-3) (m-1-3) edge\[out=354,in=174,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2)
(m-2-2) edge\[orange, out=355,in=175\] node\[descr,yshift=0.3ex\] [$c$]{} (m-3-2) (m-3-2)edge\[black\] node\[above\] [$M$]{}(m-3-3) ;
\
We have that $ker(e)=im(M)=(i+1){\mathbb Z}(1\otimes y^i)$, so $im(e)\cong ({\mathbb Z}_{i+1} {\mathbb Z})(1\otimes y^i)$. Since the map $c$ is given by the cap product with $x\in H^2_{S^1}(LS^3)$, $c((\alpha \otimes y^j)x_{i-j+2})=(\alpha \otimes y^j)x_{i-j+1}$. So $ker(c)=im(e) \cong {\mathbb Z}_j (\alpha \otimes y^{i+1})x$. We can conclude that $e(1 \otimes y^i)=(\alpha \otimes y^{i+1})x$.
### The Marking Map $M$
We consider the dual of the marking map, $M^*:H^i(LS^3) \rightarrow H^{i-1}_{S^1}(LS^3)$. This can be derived from the spectral sequence of the circle fibration \[eqn:threecirclebundle\]. $M_*$ is the composition of the surjective map $H^i(LS^3)\longrightarrow E^{i-1,1}_{\infty}\cong H^i(LS^3)/E_{\infty}^{i,0}$ and the injective map $E^{i-1,1}_{\infty}\cong ker(d_2) \longrightarrow E^{i-1,1}_{2}$ [@Mc]. $$\begin{aligned}
M^*(y_i)&=0\nonumber \\
M^*(ay_i) &= (i+1)y_{i+1}\nonumber \end{aligned}$$
Since the kernel of the differential $d_2: (E_2^{-1,1}\cong 0) \longrightarrow (E_2^{1,0}\cong 0)$ is $0$, for $1\in H^0(LS^3)$, $M^*(1)=0$. To find the image of $y_i$, we consider the composition $M^*:(H^{2i}(LS^3)\cong {\mathbb Z}y_i) \longrightarrow (E_{\infty}^{2i-1,1}\cong H^{2i}(LS^3)/E_{\infty}^{2i,0} \cong {\mathbb Z}y_i/{\mathbb Z}y_i\cong 0)$, so $M^*(y_i)=0$. The image of $ay_i$ can be determined by identifying $ay_i$ with $(i+1)\omega y_{i+1}$. Then, $$\begin{aligned}
M^*: H^{2i+1} &\rightarrow E_{\infty}^{2i+2,1} \cong ker(d_2) \longrightarrow E_2^{2i+2,1} \nonumber \\
ay_i &\mapsto (i+1)\omega y_{i+1} \xmapsto{\phantom{ E_2^{2i+2,1} }} (i+1)\omega y_{i+1}=(i+1) y_{i+1}\nonumber \end{aligned}$$ so $M^*(ay_i)=(i+1)y_{i+1}$.
To dualize $M_*$ to obtain $M$, we need the Kronecker pairing [@Br].
The Kronecker pairing is a map $$\begin{aligned}
<-,->:H^i(X)\otimes H_i(X) \rightarrow {\mathbb Z}\nonumber \end{aligned}$$ such that for $\alpha=[f] \in H^i(X)$ and $\gamma=[c]\in H_i(X)$ then $$\begin{aligned}
<\alpha , \gamma >=f(c).\nonumber \end{aligned}$$ Alternatively, for $\beta :H^i(X) \rightarrow Hom(H_i(X))$, the map from the universal coefficient theorem, $$<\alpha , \gamma >=f(c) \in {\mathbb Z}. \nonumber$$\
The Kronecker pairing satisfies the following property, which will be used to dualize the map $M ^*$: $$<f^*(\alpha),\gamma >=<\alpha , f_*(\gamma )>\nonumber$$ $$\begin{aligned}
M=M_*: H_*^{S^1}(LS^3)& \longrightarrow H_{*+1}(LS^3)\nonumber \\
\alpha \otimes 1 & \xmapsto{\phantom{ H_*^{S^1}(LS^3)}} 0\nonumber \\
\alpha \otimes y^i & \xmapsto{\phantom{ H_*^{S^1}(LS^3)}} i(1\otimes y^{i-1})\nonumber \\
x_i & \xmapsto{\phantom{ H_*^{S^1}(LS^3)}} 0\nonumber \\
(\alpha \otimes y^j)x_i & \xmapsto{\phantom{ H_*^{S^1}(LS^3)}} 0\nonumber \end{aligned}$$ Since $H_1(LS^3)=0$, it is immediate that $M(\alpha \otimes 1)=0$. To find the image of $\alpha \otimes y^i\in H_{2i}^{S^1}(LS^3)$, since $M$ is a map of degree $+1$, the only possible generator of $H_*(LS^3)$ of degree $2i+1$ is $ 1\otimes y^{i-1}$. Let $M_*(\alpha \otimes y^i)=k (1\otimes y^{i-1})$ for $k\in {\mathbb Z}$. Then $$\begin{aligned}
<M^*( ay_{i-1}), y^i>&=<ay_{i-1},M_*(\alpha \otimes y^i)>\\
&=<ay_{i-1},k(1 \otimes y^{i-1})>=k\\
& =<iy_i,\alpha \otimes y^i>=i.\end{aligned}$$ This implies $k=i$, so $M(\alpha \otimes y^i)=i(1\otimes y^{i-1})$.\
We must have $M(x_i)=0$ since $|x_i|=2i$ and the only generator on cohomology that it can be paired with is $ay^{i-1}$, which is not dual to $x_i$. Similarly, $(\alpha \otimes y^j)x_i$ gets sent to zero by $M$ since it is torsion, mapping into a free group.
### The String Bracket $[-,-]$
Recall the string bracket from Definition \[defn:stringbracket\] $$\begin{aligned}
[a,b]=(-1)^{(|a|-3})e(M(a)\bullet M(b))\end{aligned}$$ of degree $-1$ for $LS^3$.\
The only possible non-zero bracket is from the pair $\alpha \otimes y^i, \alpha \otimes y^j$, as M maps all other generators of $H_*^{S^1}(LS^3)$ to $0$, thus the string bracket of these generators are also $0$. We see that when $i\geq 1$ and $j \geq 1$, $$\begin{aligned}
\nonumber [\alpha \otimes y^i,\alpha \otimes y^j]&=(-1)^{2i-3}e(M(\alpha \otimes y^i)\bullet M( \alpha \otimes y^j))\\
&=-e(i(1\otimes y^{i-1}) \cdot j(1 \otimes y^{j-1})) \\
&=-e((ij)(1\otimes y^{i+j-2}))\\
&=-ij(\alpha \otimes y^{i+j-1})x. \end{aligned}$$ So the bracket is equal to zero if both $i=1$ and $j=1$ or if $(i+j-1)\mid ij$ and non-zero in all other cases.\
As it turns out, the only non-zero brackets are torsion elements, which corresponds to the findings of [@Ba], which are that the brackets are all trivial when considering rational string homology of $S^3$.
String Homology and the String Bracket of Odd Spheres
-----------------------------------------------------
### String Homology for Odd Spheres
We try to compute the string homology for odd spheres using only the Gysin sequence for the following fibration: $$\begin{aligned}
\label{eqn:oddcirclebundle}
S^1 \rightarrow LS^n \times ES^1 \rightarrow LS^n \times_{S^1} ES^1 \end{aligned}$$ for $n$ odd.\
Recall that the loop homology is given as follows by [@CJY]: $$\mathbb{H}_*(LS^n)=\Lambda[a]\otimes {\mathbb Z}[u] \nonumber$$ where $a$ corresponds to the dual of $[S^n]$, so $|a|=-n$ and $|u|=n-1$ after re-grading.\
The BV-operator acts on the generators as follows, [@Me]: $$\begin{aligned}
\Delta(a\otimes u^i)= & i(1 \otimes u^{i-1})\\
\Delta (1 \otimes u^i)= & 0.\end{aligned}$$ Throughout this section, we consider $n$ to be odd. Consider the bottom of the Gysin sequence. Let $H^{S^1}_i$ denote $H^{S^1}_i(LS^n)$ and $\mathbb{H}_i$ denote $\mathbb{H}_i(LS^n)$\
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& H\^[S\^1]{}\_0[Z]{}&0 &\
]{};
(m-1-2) edge node\[above\] [$\cong$]{}(m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge node\[above\] [$\cong$]{} (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-5-2) (m-5-2) edge (m-5-3) (m-5-3) edge\[orange\] node\[above\] [$M$]{}(m-5-4) (m-5-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-6-2) (m-6-2) edge node\[above\] [$\cong$]{} (m-6-3) (m-6-3) edge\[orange\] node\[above\] [$M$]{}(m-6-4) (m-8-2) edge node\[above\] [$\cong$]{} (m-8-3) (m-8-3) edge\[orange\] node\[above\] [$M$]{}(m-8-4) (m-8-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-9-2) (m-9-2) edge node\[above\] [$\cong$]{} (m-9-3) (m-9-3) edge\[orange\] node\[above\] [$M$]{}(m-9-4) (m-9-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-10-2) (m-10-2) edge node\[above\] [$\cong$]{} (m-10-3) (m-10-3) edge\[orange\] node\[above\] [$M$]{}(m-10-4) (m-10-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-11-2) (m-11-2) edge node\[above\] [$\cong$]{}(m-11-3) (m-11-3) edge\[orange\] node\[above\] [$M$]{}(m-11-4) (m-11-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e,\cong$]{}(m-12-2) (m-12-2) edge (m-12-3);
\
The maps $H^{S^1}_i\longrightarrow H^{S^1}_{i-2}$ are given by the cap product with the class generator $x\in H^2(\mathbb{C}P^{\infty})$. Since $H^{S^1}_2(LS^n)\cong {\mathbb Z}$, we denote the generator by $\gamma$, which is dual to $x$. We use the notation $\gamma_i=\frac{\gamma^i}{i!}$, dual to $x^i$. Since the maps given by the cap product are isomorphisms between where the loop homology is zero, we have that $$\begin{aligned}
H^{S^1}_{2i+1}(LS^n)=& 0, \hspace*{.5cm} 0\leq i \leq \frac{n-3}{2}\\
H^{S^1}_{2i}(LS^n)=& {\mathbb Z}(\gamma_i), \hspace*{.5cm} 1 \leq i \leq \frac{n-3}{2}.\end{aligned}$$ Note that for even degrees, the generator $\gamma_i$ increases subscript as isomorphisms in the sequence are given by cap product with $x$, dual to the cup product with $x$.\
To determine $H^{S^1}_{n-1}(LS^n)$, note that we have a short exact sequence, $$0\longrightarrow {\mathbb Z}(a \otimes u) \longrightarrow H^{S^1}_{n-1} \longrightarrow {\mathbb Z}(\gamma_{\frac{n-3}{2}}) \longrightarrow 0\nonumber$$ that splits since the last term is free. Thus $H^{S^1}_{n-1}(LS^n)\cong {\mathbb Z}e(a \otimes u) \oplus (\gamma_{\frac{n-1}{2}})$. We use the notation of $e(-)$ to denote that the generator comes from the erasing map. Using the BV-operator to determing the marking map $M:H^{S^1}_{n-1} \rightarrow \mathbb{H}_0$, we have that $M(a \otimes u)=1\otimes 1$, so the erasing map $e:\mathbb{H}_0\rightarrow H^{S^1}_n$ is zero, thus $H_n^{S^1}(LS^n)\cong 0$. \[lem:marking\] $M(\gamma_{\frac{n-1}{2}})=0$, or more generally, the marking map sends generators coming from $H_*(\mathbb{C}P^{\infty})$ to zero. In the circle bundle (\[eqn:oddcirclebundle\]) the marking map is an umkehr map coming from the projection map. Notice that $\mathbb{C}P^{\infty}=BS^1=\{pt\}\times ES^1 \subset LS^n \times_{S^1} ES^1$. Since $\pi^{-1}(\{pt\}\times_{S^1} ES^1)=\{pt\} \times ES^1$, which is contractible, then $M$ maps generators from $\mathbb{C}P^{\infty}$ into a contractible space, thus $M(\gamma_{i})=0$ for any $i$, where $\gamma_i$ denotes a generator coming from the homology of $\mathbb{C}P^{\infty}$.\
With the knowledge that $M(\gamma_{\frac{n-1}{2}})=0$, the cap product map $H^{S^1}_{n+1}\rightarrow H^{S^1}_{n-1}$ is injective with image isomorphic to ${\mathbb Z}(\gamma_{\frac{n-1}{2}})$ so $H^{S^1}_{n+1}\cong {\mathbb Z}(\gamma_{\frac{n+1}{2}})$.\
Now consider the next piece of the Gysin sequence where loop homology is non-zero.\
\(m) \[ matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex \] [ H\^[S\^1]{}\_[2n+1]{} & H\^[S\^1]{}\_[2n-1]{} & \_[n]{}0\
H\^[S\^1]{}\_[2n]{} & H\^[S\^1]{}\_[2n-2]{} [Z]{}[Z]{}& \_[n-1]{}[Z]{}(1u)\
H\^[S\^1]{}\_[2n-1]{} & H\^[S\^1]{}\_[2n-3]{}0 & \_[n-2]{}[Z]{}(au\^2)\
H\^[S\^1]{}\_[2n-2]{} & H\^[S\^1]{}\_[2n-4]{}[Z]{}(\_[n-2]{}) & \_[n-3]{}0\
H\^[S\^1]{}\_[2n-3]{}0 & H\^[S\^1]{}\_[2n-5]{}0 & 0\
&&\
H\^[S\^1]{}\_[n+3]{} [Z]{}(\_[n-2]{}) & H\^[S\^1]{}\_[n+1]{}[Z]{}(\_) & \_[2]{}0\
H\^[S\^1]{}\_[n+2]{} & H\^[S\^1]{}\_n & \_[1]{}0\
]{};
(m-1-1) edge node\[above\] [$\cong$]{}(m-1-2) (m-1-2) edge\[orange\] node\[above\] [$M$]{}(m-1-3) (m-1-3) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-1) (m-2-1) edge (m-2-2) (m-2-2) edge\[orange\] node\[above\] [$M$]{}(m-2-3) (m-2-3) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-1) (m-3-1) edge (m-3-2) (m-3-2) edge\[orange\] node\[above\] [$M$]{} (m-3-3) (m-3-3) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-1) (m-4-1) edge (m-4-2) (m-4-2) edge\[orange\] node\[above\] [$M$]{} (m-4-3) (m-4-3) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-5-1) (m-5-1) edge (m-5-2) (m-5-2) edge\[orange\] node\[above\] [$M$]{}(m-5-3)
(m-7-1) edge node\[above\] [$\cong$]{} (m-7-2) (m-7-2) edge\[orange\] node\[above\] [$M$]{}(m-7-3) (m-7-3) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-8-1)
(m-8-1) edge node\[above\] [$\cong$]{} (m-8-2) (m-8-2) edge\[orange\] node\[above\] [$M$]{}(m-8-3)
;
\
In the third and fourth row above, we have a short exact sequence with $H^{S^1}_{2n-2}$ in the center, which splits, so $H^{S^1}_{2n-2}\cong {\mathbb Z}(e(a\otimes u^2)) \oplus {\mathbb Z}(\gamma_{n-1})$. Mapping $H^{S^1}_{2n-2}$ through $M$, we have $M(e(a\otimes u^2))=2(1\otimes u)$ given by the BV-operator. Thus $H^{S^1}_{2n-1}\cong {\mathbb Z}/2{\mathbb Z}(e(1 \otimes u))$. Since the cap product map $H^{S^1}_{2n}\rightarrow H^{S^1}_{2n-2}$ is injective with image ${\mathbb Z}(\gamma_{n-1})$, $H^{S^1}_{2n} \cong {\mathbb Z}(\gamma_{n})$. Summarizing, we have $$\begin{aligned}
H^{S^1}_{2i+1}(LS^n)\cong & 0, \hspace*{.5cm} & \frac{n-1}{2}\leq i \leq n-2 \\
H^{S^1}_{2i}(LS^n)\cong & {\mathbb Z}(\gamma_i), \hspace*{.5cm} & \frac{n+1}{2} \leq i \leq n-2 \\
H^{S^1}_{n-2}(LS^n)\cong & {\mathbb Z}e(a\otimes u^2) \oplus {\mathbb Z}(\gamma_{n-1}) \\
H^{S^1}_{n-1}(LS^n)\cong & {\mathbb Z}/2{\mathbb Z}(e(1 \otimes u)) \\
H^{S^1}_{2n}(LS^n)\cong & {\mathbb Z}(\gamma_{n}) \\
H^{S^1}_{2i+1} \cong & {\mathbb Z}_2 (1 \otimes u)\gamma_{i-n-1} \hspace*{.5cm}& n-1 \leq i \leq \frac{3n-5}{2} \\
H^{S^1}_{2i} \cong & {\mathbb Z}\gamma_i \hspace*{.5cm} & \frac{n}{2} \leq i \leq \frac{3n-5}{2} \end{aligned}$$ Now assume the following holds for all $k\in {\mathbb N}$:
$$\label{eqn:oddinduction}
H^{S^1}_{2i} \cong \left\{
\begin{array}{rl}
{\mathbb Z}\gamma_i & \text{if } (n-1)\nmid 2i,\\
{\mathbb Z}\gamma_i \oplus {\mathbb Z}(e(a\otimes u^i)) & \text{if } (n-1)\vert 2i.
\end{array} \right.$$
for $1\leq i \leq \frac{(k+1)(n-1)-2}{2}$ and $$H_{2i+1}^{S^1}\cong t_k \nonumber$$ for $\frac{k(n-1)}{2}\leq i \leq \frac{(k+1)(n-1)-2}{2}$, where $t_k$ is a torsion group of order $k!$. (We would like to be able to say that $H_{2i+1}^{S^1}\cong {\mathbb Z}_k(e(1\otimes u^{k-1}))\gamma_{i-\frac{k}{2}(n-1)}\oplus \cdots \oplus {\mathbb Z}_3 e(1\otimes u^2)\gamma_{i-\frac{3}{2}(n-1)} \oplus {\mathbb Z}_2 e(1\otimes u)\gamma_{i-n+1}$, but there are extension issues that are difficult to resolve, so we cannot say which torsion group $H_{2i+1}^{S^1}$ should be.)\
Consider the $k+1$-th non-zero piece of the Gysin sequence:\
\[descr/.style=[fill=white,inner sep=1.5pt]{}\] (m) \[ matrix of math nodes, row sep=2em, column sep=.5em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_[(k+1)(n-1)+4]{} & H\^[S\^1]{}\_[(k+1)(n-1)+2]{} & 0\
& H\^[S\^1]{}\_[(k+1)(n-1)+3]{} & H\^[S\^1]{}\_[(k+1)(n-1)+1]{}& 0\
& H\^[S\^1]{}\_[(k+1)(n-1)+2]{} & H\^[S\^1]{}\_[(k+1)(n-1)]{} & [Z]{}(1u\^k)\
& H\^[S\^1]{}\_[(k+1)(n-1)+1]{} & H\^[S\^1]{}\_[(k+1)(n-1)-1]{}\_[j=2]{}\^[k]{}[Z]{}\_j &[Z]{}(a u\^[k+1]{})\
& H\^[S\^1]{}\_[(k+1)(n-1)]{} & H\^[S\^1]{}\_[(k+1)(n-1)-2]{}[Z]{}(\_) & 0\
]{};
(m-1-2) edge node\[above\] [$\cong$]{}(m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-5-2) (m-5-2) edge (m-5-3) (m-5-3) edge\[orange\] node\[above\] [$M$]{}(m-5-4)
;
\
Thus, we can extract a short exact sequence from the last two lines of the Gysin sequence above, giving us $H^{S^1}_{(k+1)(n-1)}\cong {\mathbb Z}(e(a\otimes u^{k+1}))\oplus {\mathbb Z}(\gamma_{\frac{(k+1)(n-1)}{2}})$. It can be seen that $H^{S^1}_{(k+1)(n-1)+2}\cong {\mathbb Z}(\gamma_{\frac{(k+1)(n-1)+2}{2}})$ and that\
$torsion(H^{S^1}_{(k+1)(n-1)+1})\cong t_{k+1}$. Since $H^{S^1}_{(k+1)(n-1)+1}$ is all torsion since it is sandwiched between a short exact sequence of torsion groups, we have $H^{S^1}_{(k+1)(n-1)+1}\cong t_{k+1}$. Since loop homology $\mathbb{H}_{i}(LS^n)$ is zero for $(k+1)(n-1)+2-n \leq i \leq (k+2)(n-1)-1-n$), we obtain the analogous statements of (\[eqn:oddinduction\]) for $k+1$. In summary, we get the following theorem. For $n$ odd, $$H^{S^1}_{2i}(LS^n) \cong \left\{
\begin{array}{rl}
{\mathbb Z}\gamma_i & \text{if } (n-1)\nmid 2i,\\
{\mathbb Z}\gamma_i \oplus {\mathbb Z}(e(a\otimes u^i)) & \text{if } (n-1)\vert 2i. \nonumber
\end{array} \right.$$ for $1\leq i \leq \frac{(k+1)(n-1)-2}{2}$ and $$H_{2i+1}^{S^1}(LS^n)\cong t_k \nonumber$$ for $\frac{k(n-1)}{2}\leq i \leq \frac{(k+1)(n-1)-2}{2}$, where $t_k$ is a torsion group of order $k!$. All other $j$ that does not fall into the above categories, we have that $H_i^{S^1}(LS^n) \cong H_{i-2}^{S^1}(LS^n)$
### The String Bracket for Odd Spheres
The string bracket is a degree $2-n$ map. The only possible non-zero bracket is of the generators $e(a\otimes u^i)$, since the marking map $M$ sends all other generators to zero. $$\begin{aligned}
[e(a\otimes u^i), e(a\otimes u^j)]=ije(1\otimes u^{i+j-2})\end{aligned}$$ where $e(1\otimes u^{i+j-2})$ is a generator of ${\mathbb Z}_{i+j-1}$, so the bracket is only zero when $i+j-1$ divides $ij$ $$\begin{aligned}
[e(a\otimes u^i), e(a\otimes u^j)]=&e(M(e(a\otimes u^i))\bullet M(e(a\otimes u^j)) \nonumber \\
=&e(i(1\otimes u^{i-1})\bullet j(1\otimes u^{j-1})) \nonumber \\
=&e(ij(1\otimes u^{i+j-2})). \nonumber \end{aligned}$$
String Homology and the String Bracket of Even Spheres
------------------------------------------------------
### Computations for $S^4$
Here we only use information from the Gysin sequence. As before, we know that $$\begin{aligned}
\mathbb{H}_*(LS^4,{\mathbb Z})=\frac{\Lambda (b) \otimes {\mathbb Z}[a,v]}{(a^2,ab,2av)}\end{aligned}$$ where $|a|=-4$,$|b|=-1$, and $|v|=6$, so all of the generators look like $av^k$, $bv^k$, $v^k$, where $|av^k|=-4+6k$, $|bv^k|=-1+6k$, $|v^k|=6^k$ [@CJY].\
We also know how the BV-operator acts, $$\begin{aligned}
\Delta (v^k)&=0\\
\Delta (av^k) &=0\\
\Delta (bv^k) &=(2k+1)v^k\end{aligned}$$ from [@Me]. Let’s consider the bottom of the Gysin sequence:\
\(m) \[ matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_7(LS\^4) & H\^[S\^1]{}\_5(LS\^4) & \_[2]{}(LS\^4)[Z]{}\_2 (av)\
& H\^[S\^1]{}\_6(LS\^4) & H\^[S\^1]{}\_4(LS\^4) & \_[1]{}(LS\^4)0\
& H\^[S\^1]{}\_5(LS\^4) & H\^[S\^1]{}\_3(LS\^4) & \_[0]{}(LS\^4)[Z]{}(1)\
& H\^[S\^1]{}\_4(LS\^4) & H\^[S\^1]{}\_2(LS\^4) & \_[-1]{}(LS\^4)[Z]{}(b)\
& H\^[S\^1]{}\_3(LS\^4) & H\^[S\^1]{}\_1(LS\^4) & \_[-2]{}(LS\^4)0\
& H\^[S\^1]{}\_2(LS\^4) & H\^[S\^1]{}\_0(LS\^4) & \_[-3]{}(LS\^4)0\
& H\^[S\^1]{}\_1(LS\^4) & 0 & \_[-4]{}(LS\^4)[Z]{}(a)\
& H\^[S\^1]{}\_0(LS\^4) &0 & 0.\
]{};
(m-1-2) edge (m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-5-2) (m-5-2) edge (m-5-3) (m-5-3) edge\[orange\] node\[above\] [$M$]{} (m-5-4) (m-5-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-6-2) (m-6-2) edge (m-6-3) (m-6-3) edge\[orange\] node\[above\] [$M$]{} (m-6-4) (m-6-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-7-2) (m-7-2) edge (m-7-3) (m-7-3) edge\[orange\] node\[above\] [$M$]{} (m-7-4) (m-7-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-8-2) (m-8-2) edge (m-8-3) (m-8-3) edge\[orange\] node\[above\] [$M$]{}(m-8-4);
\
The Gysin sequence, along with the BV-operator, allow us to determine that $$\begin{aligned}
H_0^{S^1}(LS^4)& \cong {\mathbb Z}(e(a))\\
H_1^{S^1}(LS^4)& \cong 0\\
H_2^{S^1}(LS^4)& \cong {\mathbb Z}(\gamma)\\
H_3^{S^1}(LS^4)& \cong {\mathbb Z}(e(b))\\
H_4^{S^1}(LS^4)& \cong {\mathbb Z}(\gamma_2)\\
H_5^{S^1}(LS^4)& \cong 0\\
H_6^{S^1}(LS^4)& \cong {\mathbb Z}(\gamma_3)\oplus {\mathbb Z}_2(e(av))\\
H_7^{S^1}(LS^4)& \cong 0.\\\end{aligned}$$ Here, $\gamma$ is the generator from $H_2({\mathbb C}P^{\infty})$. Continuing up the Gysin sequence inductively, the $k$th piece of the sequence is as follows.\
\(m) \[ matrix of math nodes, row sep=2em, column sep=1em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_[6k+7]{}(LS\^4) & H\^[S\^1]{}\_[6k+5]{}(LS\^4) & \_[6k+2]{}(LS\^4)[Z]{}\_2 (av\^[k+1]{})\
& H\^[S\^1]{}\_[6k+6]{}(LS\^4) & H\^[S\^1]{}\_[6k+4]{}(LS\^4) & \_[6k+1]{}(LS\^4)0\
& H\^[S\^1]{}\_[6k+5]{}(LS\^4) & H\^[S\^1]{}\_[6k+3]{}(LS\^4) & \_[6k]{}(LS\^4)[Z]{}(v\^k)\
& H\^[S\^1]{}\_[6k+4]{}(LS\^4) & H\^[S\^1]{}\_[6k+2]{}(LS\^4) & \_[6k-1]{}(LS\^4)[Z]{}(bv\^k)\
& H\^[S\^1]{}\_[6k+3]{}(LS\^4) & H\^[S\^1]{}\_[6k+1]{}(LS\^4) & \_[6k-2]{}(LS\^4)0\
& H\^[S\^1]{}\_[6k+2]{}(LS\^4) & H\^[S\^1]{}\_[6k]{}(LS\^4) & \_[6k-3]{}(LS\^4)0\
]{};
(m-1-2) edge (m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-5-2) (m-5-2) edge (m-5-3) (m-5-3) edge\[orange\] node\[above\] [$M$]{} (m-5-4) (m-5-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-6-2) (m-6-2) edge node\[above\] [$\cong$]{}(m-6-3) (m-6-3) edge\[orange\] node\[above\] [$M$]{} (m-6-4);
\
Using the Poincaré polynomial for $H_*^{S^1}(LS^4,{\mathbb Z}_2)$ from [@We], $$\begin{aligned}
\left(\frac{1}{1-t^6}\right)\left( t^3+\frac{1+t^7}{1-t^2}\right)\end{aligned}$$ which we can rewrite in a more useful way as follows $$\begin{aligned}
\sum_{k=0}^{\infty}(k+1)(t^{6k}+t^{6k+2}+t^{6k+3}+t^{6k+4})+\sum_{k=1}^{\infty}k(t^{6k+1}+t^{6k+5}).\end{aligned}$$ From this, we see that $H_{6k+2}^{S^1}(LS^4,{\mathbb Z}_2)\cong \bigoplus\limits_{i=1}^{k} {\mathbb Z}_2$ Using the Universal Coefficient Theorem, $$\begin{aligned}
0\rightarrow H_{6k+2}^{S^1}(LS^4,{\mathbb Z}) \otimes {\mathbb Z}_2 \rightarrow \bigoplus_{i=1}^{k} {\mathbb Z}_2 \rightarrow Tor(H_{6k+1}^{S^1}(LS^4,{\mathbb Z}),{\mathbb Z}_2)\rightarrow 0\end{aligned}$$ Since $H_{6k+1}^{S^1}(LS^4),{\mathbb Z})=0$, We have that $H_{6k+2}^{S^1}(LS^4,{\mathbb Z})\otimes {\mathbb Z}_2 \cong \bigoplus\limits_{i=1}^{k} {\mathbb Z}_2$. We knot that $H_{6k+2}^{S^1}(LS^4,{\mathbb Z})$ has a summand $\bigoplus_i {\mathbb Z}_{2^{l}}$ where $\sum l=k$, then we have $$\begin{aligned}
H_{6k+2}^{S^1}(LS^4,{\mathbb Z})\otimes {\mathbb Z}_2 \cong\bigoplus_i {\mathbb Z}_{2^{l}} \cong \bigoplus_{i=1}^{k} {\mathbb Z}_2.\end{aligned}$$ So we must have that each $l=1$. Therefore, we have the following proposition. $$\begin{aligned}
H^{S^1}_{6k+7} & \cong 0\\
H^{S^1}_{6k+6} & \cong {\mathbb Z}(\gamma_{3k+3}) \oplus {\mathbb Z}_2({e(av^{k+1})})\bigoplus_{i=1}^{k+1} [{\mathbb Z}_2(e(av^{i})\gamma)] \oplus C_k\\
H^{S^1}_{6k+5} & \cong 0\\
H^{S^1}_{6k+4} & \cong {\mathbb Z}(\gamma_{3k+2}) \bigoplus_{i=1}^{k+1} [{\mathbb Z}_2(e(av^{i})\gamma)] \oplus C_k \\
H^{S^1}_{6k+3} & \cong {\mathbb Z}e(bv^k)\\
H^{S^1}_{6k+2} & \cong H^{S^{1}}_{6k} \cong {\mathbb Z}(\gamma_{3k}) \oplus C_{k-1}. \\\end{aligned}$$ $C_k$ is some torsion group of order $\prod_{i=1}^{k}(2i+1)$. The torsion comes from the fact that $M(e(bv^k))=\Delta(bv^k)=(2k+1)v^k$, which is where all of the odd torsion groups come from.
### The String Bracket for $S^4$
The string bracket for $S^4$ is of degree $-2$. We know that the marking map $M$ maps generators from $H_*({\mathbb C}P^{\infty})$ to zero (Lemma \[lem:marking\]), and it takes all the torsion to zero since for those cases, $M$ maps into zero or into a free group. Thus, the only possible case for the bracket to be nonzero is for the generators $e(bv^k)$. We have that $$\begin{aligned}
[e(bv^k),e(bv^l)]&=(-1)e(\Delta(bv^k)\bullet \Delta (bv^l))\\
&=-e((2k+1)v^k \bullet (2l+1)v^l)\\
&=-(2k+1)(2l+1)e(v^{k+l})\\
&=-4kle(v^{k+l}).\end{aligned}$$ We know that $e(v^{k+l})$ has order $2(k+l)+1$,so the bracket is zero when $k \neq 0$, $l\neq 0$, and $(2k+2l+1)|(4kl+2k+2l+1)$, or when $4kl|(2k+2l+1)$, but the latter number is odd, so this can never happen. Thus, the bracket is always nontrivial in this case.\
When $k=0$ or $l=0$, the bracket is zero.
### Computations for $S^2$
We just state results for the string homology computations for $S^2$, as the computations for $S^4$ are more illustrative. From [@CJY] , we have $$\begin{aligned}
\mathbb{H}_*(LS^2)=\frac{\Lambda(b)\otimes {\mathbb Z}[a,v]}{a^2, ab, 2av}\end{aligned}$$ where $|a|=-2$, $|b|=-1$, and $|v|=2$. The BV-operator acts as follows, $$\begin{aligned}
\Delta(v^k)&=0\\
\Delta(av^k)&=0\\
\Delta(bv^k)&=(2k+1)v^k+av^{k+1}.\end{aligned}$$ $$\begin{aligned}
H_0^{S^1}(LS^2) & \cong {\mathbb Z}e(a)\\
H_2^{S^1}(LS^2) & \cong {\mathbb Z}e(a)\gamma \oplus {\mathbb Z}_2 e(av)\\
H_4^{S^1}(LS^2) & \cong {\mathbb Z}\oplus {\mathbb Z}_2 \oplus {\mathbb Z}_6 \\
H_{2i+1}^{S^1}(LS^2) & \cong {\mathbb Z}e(bv^i)\\
H_{2j}^{S^1}(LS^2) & \cong {\mathbb Z}\oplus C_k\end{aligned}$$ where $i \geq 0$, $j \geq 3$, and $C_k$ is a torsion group of order $\prod_{k=1}^{j-1}(4j-2-4k)$. The string bracket is only non-zero in odd degrees. $$\begin{aligned}
[e(bv^i),e(bv^j)]=-4ije(v^{i+j})\end{aligned}$$ which is not always zero since $e(v^{i+j})$ is torsion, and all other brackets are zero. $$\begin{aligned}
[e(bv^i),e(bv^j)]& = (-1)^{2i+1-2}e(\Delta(bv^i)\bullet \Delta(bv^j))\\
&=-e(((2i+1)v^i+av^{i+1})\bullet((2j+1)v^j+av^{j+1}))\\
&=-e((2i+1)(2j+1)v^{i+j}+(2i+2j+2)av^{i+j+1})\\
&=-(2i+1)(2j+1)e(v^{i+j})\\
&=-4ije(v^{i+j}).\end{aligned}$$
### String Homology for Even Spheres
We have from [@CJY] that, for $n$ even. $$\begin{aligned}
\mathbb{H}_*(LS^{n},{\mathbb Z}) \cong \frac{\Lambda(b) \otimes {\mathbb Z}[a,v]}{(a^2,ab,2av)}\end{aligned}$$ where $|a|=-n$, $|b|=-1$ and $|v|=2n-2$. By [@Me] we have $$\begin{aligned}
\Delta(v^k) &=0\\
\Delta(av^k) &=0\\
\Delta(bv^k) &= (2k+1)v^k.\end{aligned}$$ To keep track of things, $|av^k|=k(2n-2)-n$, $|bv^k|=k(2n-2)-1$, $|v^k|=k(2n-2)$. Let us consider the bottom of the Gysin sequence:\
\(m) \[ matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_[3n-1]{} & H\^[S\^1]{}\_[3n-2]{} & \_[2n-2]{}[Z]{}(v\^2)\
& H\^[S\^1]{}\_[3n-2]{} & H\^[S\^1]{}\_[3n-4]{}& \_[2n-3]{}[Z]{}(bv)\
&&&\
& H\^[S\^1]{}\_[2n-1]{} & H\^[S\^1]{}\_[2n-3]{}0 & \_[n-2]{}[Z]{}\_2 (av)\
&&&\
& H\^[S\^1]{}\_[n+1]{} & H\^[S\^1]{}\_[n-1]{}[Z]{}& \_[0]{}[Z]{}(v)\
& H\^[S\^1]{}\_[n]{} & H\^[S\^1]{}\_[n-2]{} & \_[-1]{}[Z]{}(b)\
&&&\
& H\^[S\^1]{}\_2 & H\^[S\^1]{}\_0 & \_[-n+1]{}0\
& H\^[S\^1]{}\_1 & 0 & \_[-n]{}[Z]{}(a)\
& H\^[S\^1]{}\_0 &0 &\
]{};
(m-1-2) edge node\[above\] [$\cong$]{}(m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge node\[above\] [$\cong$]{} (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4)
(m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4)
(m-6-2) edge (m-6-3) (m-6-3) edge\[orange\] node\[above\] [$M$]{}(m-6-4) (m-6-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-7-2) (m-7-2) edge node\[above\] [$\cong$]{} (m-7-3) (m-7-3) edge\[orange\] node\[above\] [$M$]{}(m-7-4) (m-9-2) (m-9-2) edge node\[above\] [$\cong$]{} (m-9-3) (m-9-3) edge\[orange\] node\[above\] [$M$]{}(m-9-4) (m-9-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{}(m-10-2) (m-10-2) edge node\[above\] [$\cong$]{}(m-10-3) (m-10-3) edge\[orange\] node\[above\] [$M$]{}(m-10-4) (m-10-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e,\cong$]{}(m-11-2) (m-11-2) edge (m-11-3);
\
from this sequence and knowledge of the BV-operator, we get $$\begin{aligned}
H_0^{S^1}(LS^n)&\cong {\mathbb Z}a\\
H_1^{S^1}(LS^n)&\cong 0\\
H_2^{S^1}(LS^n)&\cong {\mathbb Z}\gamma \\
H_3^{S^1}(LS^n)&\cong 0 \\
H_4^{S^1}(LS^n)&\cong {\mathbb Z}\gamma_2\\
\vdots &\\
H_{n-1}^{S^1}(LS^n)&\cong {\mathbb Z}e(b)\\
H_{n}^{S^1}(LS^n)&\cong {\mathbb Z}e(v) \oplus {\mathbb Z}\gamma_{\frac{n}{2}} \\
\vdots &\\
H_{2n-2}^{S^1}(LS^n)&\cong {\mathbb Z}_2 e(av) \oplus {\mathbb Z}\oplus {\mathbb Z}\gamma_{\frac{2n-2}{2}} \\
H_{2n-1}^{S^1}(LS^n)&\cong 0 \\
H_{2n}^{S^1}(LS^n)&\cong {\mathbb Z}_2 \oplus {\mathbb Z}\oplus {\mathbb Z}\\
\vdots & \\
H_{3n-3}^{S^1}(LS^n)&\cong {\mathbb Z}e(bv)\\
H_{3n-2}^{S^1}(LS^n)&\cong {\mathbb Z}_2 \oplus {\mathbb Z}_3 e(v^3) \oplus {\mathbb Z}\oplus {\mathbb Z}\\
\vdots & \end{aligned}$$ $$\begin{aligned}
H_{4n-4}^{S^1}(LS^n)&\cong {\mathbb Z}_2 \oplus {\mathbb Z}_2 \oplus {\mathbb Z}_3 \oplus {\mathbb Z}\oplus {\mathbb Z}\\
H_{4n-3}^{S^1}(LS^n)&\cong 0\\
\vdots \\
H_{5n-5}^{S^1}(LS^n)&\cong {\mathbb Z}e(bv^2)\\
H_{5n-4}^{S^1}(LS^n)&\cong {\mathbb Z}_2 \oplus {\mathbb Z}_2 \oplus {\mathbb Z}_2 \oplus {\mathbb Z}_3 \oplus {\mathbb Z}\oplus {\mathbb Z}\\
\vdots &\end{aligned}$$ where all of the odd degree homology are isomorphic, and all even degree homology are isomorphic, or $H_i^{S^1}\cong H_{i-2}^{S^1}$m in the gaps denoted by the vertical dots. The $k$-th piece of the sequence is as follows:
\(m) \[ matrix of math nodes, row sep=2em, column sep=2.5em, text height=1.5ex, text depth=0.25ex \] [ & H\^[S\^1]{}\_[k(2n-2)+n+1]{} & H\^[S\^1]{}\_[k(2n-2)+n-1]{} & \_[k(2n-2)]{}[Z]{}(v\^k)\
& H\^[S\^1]{}\_[k(2n-2)+n]{} & H\^[S\^1]{}\_[k(2n-2)-2+n]{}& \_[k(2n-2)-1]{}[Z]{}(bv\^k)\
&&&\
& H\^[S\^1]{}\_[k(2n-2)+1]{} & H\^[S\^1]{}\_[k(2n-2)-1]{}0 & \_[k(2n-2)-n]{}[Z]{}\_2 (av\^k)\
]{};
(m-1-2) edge node\[above\] [$\cong$]{}(m-1-3) (m-1-3) edge\[orange\] node\[above\] [$M$]{}(m-1-4) (m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge node\[above\] [$\cong$]{} (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4)
(m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4)
;
\
inductively, we have that $$\begin{aligned}
H_{k(2n-2)-2}^{S^1} \cong {\mathbb Z}_2^{k-1}\oplus C_k \oplus {\mathbb Z}\oplus {\mathbb Z}\end{aligned}$$ where $C_k$ is a torsion group of order $\prod_{i=1}^{k-1}(2i+1)$. The bottom of the above Gysin sequence gives the short exact sequence $$\begin{aligned}
0 \rightarrow {\mathbb Z}_2 av^k \rightarrow H^{S^1}_{k(2n-2)} \rightarrow H_{k(2n-2)-2}^{S^1} \rightarrow 0\end{aligned}$$ which gives $$\begin{aligned}
H^{S^1}_{k(2n-2)}\cong {\mathbb Z}_2 e(av^k) \oplus {\mathbb Z}_2^{k-1}\oplus \left(\text{torsion group of order } \sum_{i=1}^{k-1}(2i+1)\right) \oplus {\mathbb Z}\oplus {\mathbb Z}.\end{aligned}$$ Note that the even torsion can be resolved using the results by Westerland in [@We] (as in the above example for $S^4$). From the top of the above Gysin sequence, we get the following. $$\begin{aligned}
H_{k(2n-2)+1}^{S^1}(LS^n)& \cong 0\\
\vdots & \\
H_{k(2n-2)+n-1}^{S^1}(LS^n) & \cong {\mathbb Z}e(bv^k)\\
H_{k(2n-2)+n}^{S^1}(LS^n) & \cong {\mathbb Z}_2^{k} \oplus C_k \oplus {\mathbb Z}\oplus {\mathbb Z}.\end{aligned}$$ Summarizing, we get the following theorem. Suppose $n$ is even. $$\begin{aligned}
H_{k(2n-2)-2}^{S^1}(LS^n)&\cong {\mathbb Z}_2^{k-1}\oplus C_k \oplus {\mathbb Z}\oplus {\mathbb Z}\\
H_{k(2n-2)+1}^{S^1}(LS^n)& \cong 0\\
\vdots & \\
H_{k(2n-2)+n-1}^{S^1}(LS^n) & \cong {\mathbb Z}e(bv^k)\\
H_{k(2n-2)+n}^{S^1}(LS^n) & \cong {\mathbb Z}_2^{k} \oplus C_k \oplus {\mathbb Z}\oplus {\mathbb Z}.\end{aligned}$$ where all other unstated homology, we have $H_{i}^{S^1}(LS^n)\cong H_{i-1}^{S^1}(LS^n)$. .
### The String Bracket for Even Spheres
The string bracket is always zero except on the generators $e(bv^j)$, $$\begin{aligned}
[e(bv^k),e(bv^l)]=-(4kl+2k+2l+1)e(v^{k+l})\end{aligned}$$ but $e(v^{k+l})$ has order $2(k+l)+1$ so it is not always zero. $$\begin{aligned}
[e(bv^k),e(bv^l)]&=(-1)^{k(2n-2)-1-n}e(M(e(bv^k))\bullet M(e(bv^l))\\
&=-(4kl+2k+2l+1)e(v^{k+l})\end{aligned}$$
String Homology and String Bracket Computations for Surfaces
============================================================
In this chapter we compute the string homology and string bracket for surfaces.
The Torus
---------
We can compute the loop homology of the torus $T$ quite easily since we already know the loop homology of $S^1$. We compute the loop homology and BV-operator using the following,
$$\begin{aligned}
\mathbb{H}_*(LT)\cong \mathbb{H_*(LS^1)}\otimes \mathbb{H_*(LS^1)}\end{aligned}$$
and that we can compute the BV-operator as follows: $$\begin{aligned}
\Delta_T(a\otimes b)=\Delta(a)\otimes b + (-1)^{|a|+1}a \otimes \Delta(b).\end{aligned}$$ We obtain the following: $$\begin{aligned}
\mathbb{H}_{-2}(LT)& \cong \bigoplus_{(n,m)\in {\mathbb Z}^2}{\mathbb Z}(1_{nm})\\
\mathbb{H}_{-1}(LT)& \cong \bigoplus_{(n,m)\in {\mathbb Z}^2}{\mathbb Z}x_{nm} \bigoplus_{(n,m)\in {\mathbb Z}^2} {\mathbb Z}y_{nm}\\
\mathbb{H}_{0}(LT)& \cong \bigoplus_{(n,m)\in {\mathbb Z}^2} {\mathbb Z}z_{nm}\end{aligned}$$ the loop product: $$\begin{aligned}
x_{nm}\bullet y_{kl}& =1_{n+k,m+l}\\
x_{nm}\bullet z_{kl}&=x_{n+k,m+l}\\
y_{nm}\bullet x_{kl}& = y_{n+k,m+l}\\
z_{nm}\bullet z_{kl} & = z_{n+k,m+l}\end{aligned}$$
and the BV-operator:
$$\begin{aligned}
\Delta(1_{nm})& =nx_{nm}+my_{nm}\\
\Delta(x_{nm})& = nz_{nm}\\
\Delta(y_{nm}) & = -mz_{nm}\\
\Delta(z_{nm}) & =0.\end{aligned}$$
To calculate string homology of the torus, we use the Gysin sequence:
\(m) \[ matrix of math nodes, row sep=2em, column sep=.5em, text height=1.5ex, text depth=0.25ex \] [ & & &0\
& H\^[S\^1]{}\_3(LT)&H\^[S\^1]{}\_1(LT) & \_[0]{}(LT) \_[(n,m)[Z]{}\^2]{} [Z]{}z\_[nm]{}\
& H\^[S\^1]{}\_2(LT) & H\^[S\^1]{}\_0(LT) & \_[-1]{}(LT) \_[(n,m)[Z]{}\^2]{}[Z]{}x\_[nm]{} \_[(n,m)[Z]{}\^2]{} [Z]{}y\_[nm]{}\
& H\^[S\^1]{}\_1(LT) & 0 & \_[-2]{}(LT)\_[(n,m)[Z]{}\^1]{}[Z]{}(1\_[nm]{}\
& H\^[S\^1]{}\_0(LT) & 0 &\
\
]{};
(m-1-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-2-2) (m-2-2) edge node\[above\] [$c$]{} (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge (m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=355,in=175,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-5-2) (m-5-2) edge (m-5-3) ;
\
we get the following:
$$\begin{aligned}
H_0^{S^1}(LT) \cong & \bigoplus_{(n,m)\in {\mathbb Z}^2}{\mathbb Z}e(1_{nm})\\
H_1^{S^1}(LT) \cong & \bigoplus_{\substack{(n,m)\in {\mathbb Z}^2 \\ gcd(n,m)=d \\ d\neq n,m}} {\mathbb Z}_d (q_{n_1}e(x_{nm})+ q_{m_1}e(y_{nm})) \bigoplus _{\substack{n|m\\ nd=m}}{\mathbb Z}_n (e(x_nm)+de(y_{nm})\\
& \bigoplus_{\substack{m|n \\ md=n}} {\mathbb Z}_m (de(x_{nm})+e(y_{nm})) \bigoplus_{(n,m)\in \Delta} {\mathbb Z}_n (e(x_{nn})+e(y_nn))\\
& \bigoplus_{gcd(n,m)=1}{\mathbb Z}(q_ne(x_{nm})+q_me(y_{nm})) \bigoplus_{gcd(n,m)=d}{\mathbb Z}(q_ne(x_{nm})+q_me(y_{nm}))\\
& \bigoplus_{n|m}{\mathbb Z}_e(y_{nm}) \bigoplus_{m|n}e(x_{nm})\bigoplus_{(n,n)\in \Delta}{\mathbb Z}e(y_{nn})\oplus {\mathbb Z}e(x_{00})\oplus e(y_{00})\end{aligned}$$
where for $H_1^{S^1}(LT)$, $(n,m)\in {\mathbb Z}-{(0,0)}$ for all $n,m$, and $\Delta$ is the diagonal in ${\mathbb Z}\times {\mathbb Z}$, and each $q_i$ are polynomials in the quotients that show up in the division algorithm for $n,m$. Note that the generator in the third to the last summand could also have been chosen to be ${\mathbb Z}e(x_nn)$. To calculate $H^{S^1}_2(LT)$, we look at the second and third line in the Gysin sequence above. Using that $\mathbb{H}_0(LT)$ is free, and $\Delta$ maps torsion to zero, the marking map is nonzero only on the free elements of $H_1^{S^1}(LT)$, we can extract the following short exact sequence:\
$$\begin{aligned}
0\rightarrow \frac{\mathbb{H}_0(LT)}{im(M)=im(\Delta)}\rightarrow H^{S^1}_2(LT) \rightarrow {\mathbb Z}1_{00} \rightarrow 0\end{aligned}$$ where the last part of the sequence comes from the kernel of $M$ being generated by $1_{00}$. We get that $$\begin{aligned}
\frac{\mathbb{H}_0(LT)}{im(M)}=\bigoplus_{(n,n)\in \Delta -(0,0)} {\mathbb Z}_n z_{nn} \bigoplus_{m|n} {\mathbb Z}_n z_{nm} \bigoplus_{n|m} {\mathbb Z}_m z_{nm} \bigoplus_{\substack{gcd(m,n)=d \\ d\neq n,m}} {\mathbb Z}_{q_n n -mq_m} z_{nm}.
\end{aligned}$$ Thus, we have $$\begin{aligned}
H_2^{S^1}(LT) \cong \frac{\mathbb{H}_0(LT)}{im(M)} \oplus {\mathbb Z}1_{00}\gamma\end{aligned}$$ From the very top of the Gysin sequence pictured above, we get that $H_3^{S^1}(LT) \cong ker(M)$, thus we have $$\begin{aligned}
H_3^{S^1}(LT) \cong tor(H_1^{S^1}(LT)) \oplus {\mathbb Z}x_{00}\gamma \oplus {\mathbb Z}y_{00}\gamma .\end{aligned}$$ Since loop homology higher than two is zero, we obtain the following isomorphisms: $$\begin{aligned}
H_{2k}^{S^1}(LT) & \cong H_2^{S^1}(LT)\\
H_{2k+1}^{S^1}(LT)& \cong H_3^{S^1}(LT)\end{aligned}$$ for all $k\geq 2$.
### Torus Bracket Computations {#torus-bracket-computations .unnumbered}
For $[-,-]: H_0^{S^1}(LT) \otimes H_0^{S^1}(LT) \rightarrow H_0^{S^1}(LT)$, $$\begin{aligned}
[e(1_{nm}),e(1_{kl})]=(nl-mk)e(1_{(n+k),(m+l)}).\end{aligned}$$ which corresponds to the Goldman bracket for the torus in Proposition \[prop:torusstructure\].\
For $[-,-]: H_0^{S^1}(LT) \otimes H_1^{S^1}(LT) \rightarrow H_1^{S^1}(LT)$, $$\begin{aligned}
[e(1_nm),e(q_1x_{nm}+q_2y_{kl})]=(q_1k-q_2l)e(nx_{n+k,m+l}+my_{n+k,m+l}).\end{aligned}$$ For $[-,-]: H_1^{S^1}(LT) \otimes H_1^{S^1}(LT) \rightarrow H_2^{S^1}(LT)$, $$\begin{aligned}
[e(q_1x_{nm}+q_2y_{nm}),e(q_3x_{kl}+q_4y_{kl})]=(q_1n-q_2m)(q_3k-q_4l)e(z_{n+k,m+l})\end{aligned}$$ which is torsion, and not always zero. All other brackets are zero.
### String Lie Algebra Structure on the Torus {#string-lie-algebra-structure-on-the-torus .unnumbered}
The center of the String Lie algebra can be directly computed from the above bracket results. Let $\mathfrak{g}(T)$ denote the String Lie algebra for the closed torus, the center of the Lie algebra is as follows: $$\begin{aligned}
Z(\mathfrak{g}(T))& \cong H_*^(S^1)(L_0(T) \bigoplus_{k\in {\mathbb Z}- \{ 0 \} } {\mathbb Z}e(y_{k0})\bigoplus_{l\in {\mathbb Z}- \{0\}} {\mathbb Z}e(x_{0l}) \\ & \bigoplus_{(m,n)\in {\mathbb Z}^2 -\{(0,0)\}} {\mathbb Z}e(z_{mn}) \oplus \text{ tor}(H_1^{S^1}(LT)) \oplus H_2^{S^1}(LT)\end{aligned}$$ where $L_0(T)$ denotes the connected component of the loop space $LT$ containing the contractible loops.
Genus $g>1$
-----------
Using that $L(BG)=\amalg_{[\gamma]} BC(\gamma)$, for a closed, orientable surface $\Sigma_g$ of genus $g>1$, we have $$\begin{aligned}
L\Sigma_g=LB(\pi_1(\Sigma_g,*))=\coprod_{[\gamma]\in \hat{\pi}} BC(\gamma)\end{aligned}$$ where $[\gamma ]$ is a conjugacy class in $\pi_1(\Sigma_g)$, or an element of $\hat{\pi}$. By Kupers [@Ku], $C(\gamma)\cong {\mathbb Z}$ for $\gamma \neq e$, where $e$ is the identity in $\pi_1(\Sigma_g)$. This gives us that $BC(\gamma)=S^1$. For the centralizer of $e$, $C(e)=\pi_1(\Sigma_g)$ since everything commutes with $e$, so $BC(e)=\Sigma_g$. Therefore, we have $$\begin{aligned}
L\Sigma_g=\coprod_{[\gamma]\neq e \in \hat{\pi}}S^1 \cup \Sigma_g.\end{aligned}$$ We can follow [@Ku], and also knowing the loop homology of $S^1$ and $\Sigma_g$, we get the following (Kupers, [@Ku], Theorem 2.2) $$\begin{aligned}
\mathbb{H}_{-2}(L\Sigma _g) &= \bigoplus_{\gamma \in \hat{\pi}} {\mathbb Z}[\gamma ]\\
\mathbb{H}_{-1}(L\Sigma _g) &=H_1(\Sigma _g)\bigoplus_{\gamma \in \hat{\pi}-\{e\}} {\mathbb Z}\beta_{\gamma}\\
\mathbb{H}_0(L\Sigma _g) &={\mathbb Z}1\\
\mathbb{H}_i(L \Sigma_g) &=0 \hspace*{.5cm}\text{for}\hspace*{.2cm} i>0.\end{aligned}$$ Let $H_1(\Sigma_g)=\bigoplus\limits_{i=1}^g{\mathbb Z}a_i \bigoplus\limits_{j=1}^g{\mathbb Z}b_i$. Let $\kappa_{\gamma}$ be the generator of $C[\gamma ]$, then $\gamma = \kappa_{\gamma}^{l_{\gamma}}$, where $l_{\gamma}\in {\mathbb Z}$. The loop product can be computed as follows [@Ku], $$\begin{aligned}
1 \bullet 1 &=1\\
\beta_{\gamma} \bullet 1 &= \beta_{\gamma}\\
a_i \bullet 1 &= a_i\\
b_j \bullet 1 &= b_i\\
a_i \bullet a_j &=0\\
b_i \bullet b_j &=0\\
a_i \bullet b_j &= \delta_{ij}\\
[\gamma] \bullet 1 &= [\gamma]\\
\beta_{[\gamma_1]}\bullet \beta_{[\gamma_2]} &= \frac{[\beta_{[\gamma_1]}, \beta_{[\gamma_2]}]}{l_{\gamma_1}\cdot l_{\gamma_2}}\end{aligned}$$ where (4.3)-(4.6) are just the homology intersection product on $H_1(\Sigma_g)$ and the bracket in (4.9) is the Goldman bracket. We also can compute the BV-operator as in [@Ku], where the only non-trivial result is $\Delta ([\gamma])=l_{\gamma}\beta_{\gamma}$.\
As before, we can use the Gysin sequence to compute string homology. Consider the bottom of the Gysin sequence,\
\(m) \[ matrix of math nodes, row sep=2em, column sep=1em, text height=1.5ex, text depth=0.25ex \] [ & & &0\
& H\^[S\^1]{}\_3(L\_g)&H\^[S\^1]{}\_1(L\_g) & \_[0]{}(L\_g) [Z]{}1\
& H\^[S\^1]{}\_2(L\_g) & H\^[S\^1]{}\_0(L\_g) & \_[-1]{}(L\_g)\_[-{e}]{} [Z]{}\_ H\_1(L\_g)\
& H\^[S\^1]{}\_1(L\_g) & 0 & \_[-2]{}(L\_g)\_ [Z]{}\[\]\
& H\^[S\^1]{}\_0(L\_g) & 0 &\
\
]{};
(m-1-4) edge\[out=345,in=160,cyan\] node\[descr,yshift=0.1ex\] [$e$]{} (m-2-2) (m-2-2) edge node\[above\] [$c$]{} (m-2-3) (m-2-3) edge\[orange\] node\[above\] [$M$]{}(m-2-4) (m-2-4) edge\[out=345,in=160,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-3-2) (m-3-2) edge node\[above\] [$c$]{}(m-3-3) (m-3-3) edge\[orange\] node\[above\] [$M$]{} (m-3-4) (m-3-4) edge\[out=345,in=165,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-4-2) (m-4-2) edge (m-4-3) (m-4-3) edge\[orange\] node\[above\] [$M$]{} (m-4-4) (m-4-4) edge\[out=345,in=160,cyan\] node\[descr,yshift=0.3ex\] [$e$]{} (m-5-2) (m-5-2) edge (m-5-3) ;
\
we get that $H_0^{S^1}(L\Sigma_g)\cong \bigoplus\limits_{\gamma \in \hat{\pi}} {\mathbb Z}e([\gamma])$. Since $M \circ e ([\gamma])=\Delta ([\gamma])=l_{\gamma}\beta_{[\gamma]}$ for $[\gamma]\neq e$, $\Delta ([e])=0$ and $\Delta (a_i)=\Delta(b_j)=0$, we get that $H_1^{S^1}(L\Sigma_g) \cong \mathbb{H}_{-1}(L \Sigma_g)/ \bigoplus\limits_{\gamma \in \hat{\pi}-\{e\}}({\mathbb Z}l_{\gamma}\beta_{\gamma})$. Since $M: H_1^{S^1}(L \Sigma_g) \rightarrow {\mathbb Z}1$ is the zero map, $H_2^{S^1}(L \Sigma_g)$ sits in the following short exact sequence $$\begin{aligned}
0 \rightarrow {\mathbb Z}_1 \rightarrow H_2^{S^1}(L\Sigma_g) \rightarrow im(c) \rightarrow 0.\end{aligned}$$ Since $im(c)=ker(M)={\mathbb Z}e([e])$, then the above exact sequence splits and $H_2^{S^1}(L\Sigma_g)\cong {\mathbb Z}e(1) \oplus {\mathbb Z}s$ (EXPLAIN $s$). Therefore, we have the following: $$\begin{aligned}
H_0^{S^1}(L\Sigma_g)& \cong \bigoplus\limits_{\gamma \in \hat{\pi}} {\mathbb Z}e([\gamma])\\
H_1^{S^1}(L\Sigma_g) & \cong \bigoplus\limits_{i=1}^g{\mathbb Z}e(a_i) \bigoplus\limits_{j=1}^g{\mathbb Z}e(b_i) \bigoplus\limits_{\gamma \in \hat{\pi}-\{e\}}{\mathbb Z}_{l_{\gamma}}e(\beta_{\gamma})\\
H_2^{S^1}(L\Sigma_g) & \cong {\mathbb Z}e(1) \oplus {\mathbb Z}s\\
H_{2i+1}^{S^1}(L \Sigma_g) & \cong H_1^{S^1}(L \Sigma_g), \hspace*{.2cm} i\geq 1\\
H_{2i}^{S^1}(L \Sigma_g) & \cong H_2^{S^1}(L \Sigma_g), \hspace*{.2cm} i \geq 1.\end{aligned}$$ From this, we can compute the string bracket for $\Sigma_g$. The only non-trivial bracket is the Goldman bracket. $$\begin{aligned}
\label{algn:surfacegoldman}
[e([\gamma_1]),e([\gamma_2])]=[\gamma_1,\gamma_2] \end{aligned}$$ The second bracket in \[algn:surfacegoldman\] is the Goldman bracket. So the only non-trivial string bracket of string homology of $\Sigma_g$ is the Goldman bracket as in Chapter 1.
[99]{} S. Basu, *Transversal String Topology and Invariants of Manifolds*, (2011) G.E. Bredon, *Topology and Geometry*, Graduate Texts in Mathematics; **139** (1993). M. Chas, The Goldman bracket and the intersection of curves on surfaces, *Contemporary Mathematics*, **Vol. 639**, 2015 M. Chas and D. Sullivan, String Topology, *Ann. Math.*,(1999) R. Cohen, J.D.S. Jones, J. Yan, The Loop Homology Algebra of Spheres and Projective Spaces, *Progress in Mathematics*, **Vol. 215**, 77-92 (2002) R. Cohen, K. Hess, A. A. Voronov, *String Topology and Cyclic Homology*, Advanced courses in mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, (2006) P. Etingof, Casimirs of the Goldman Lie algebra of a closed surface, *Int. Math Res. Notices*, **Vol. 2006**, (2006) W.M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations, *Invent. math*, **85**, 263-302 (1986) R. Hepworth, *String Topology for Lie Groups* A. Kabiraj, Center of the Goldman Algebra, arXiv:1412.2331 \[math.GT\], (2014) N. Kawazumi, Y. Kuno, K. Toda, Generators of the Homological Goldman Lie Algebra, *Osaka K. Math*, **51**, 665-671, (2014) A.P.M. Kupers, *An elementary proof of the string topology structure of compact oriented surfaces*, arXiv:1110.1158 \[math.AT\], (2011) J. McCleary, A User’s Guide to Spectral Sequences, *Cambridge University Press*, (2001) L. Menichi, String topology for spheres, *Comment. Math. Helv.* **84**, no 1, 135-157, (2009) N. Seeliger, *Loop homology of spheres and complex projective spaces*, arXiv:1104.5219 \[math.AT\], (2011) H. Tamanoi, Batalin-Vilkovisky Lie algebra structure on the loop homology of complex Stiefel manifolds, *International Mathematics Research Notices*, **Vol. 2006**, Article ID 97193, 1-23 (2006) C. Westerland, String homology of spheres and projective spaces, *Algebraic and Geometric Topology*, **7**, 309-325, (2007)
|
---
abstract: 'A graph theoretic perspective is taken for a range of phenomena in continuum physics in order to develop representations for analysis of large scale, high-fidelity solutions to these problems. Of interest are phenomena described by partial differential equations, with solutions being obtained by computation. The motivation is to gain insight that may otherwise be difficult to attain because of the high dimensionality of computed solutions. We consider graph theoretic representations that are made possible by low-dimensional states defined on the systems. These states are typically functionals of the high-dimensional solutions, and therefore retain important aspects of the high-fidelity information present in the original, computed solutions. Our approach is rooted in regarding each state as a vertex on a graph and identifying edges via processes that are induced either by numerical solution strategies, or by the physics. Correspondences are drawn between the sampling of stationary states, or the time evolution of dynamic phenomena, and the analytic machinery of graph theory. A collection of computations is examined in this framework and new insights to them are presented through analysis of the corresponding graphs.'
author:
- |
R. Banerjee, K. Sagiyama, G.H. Teichert\
Department of Mechanical Engineering\
\
K. Garikipati[^1]\
Departments of Mechanical Engineering & Mathematics\
Michigan Insititute for Computational Discovery & Engineering\
\
University of Michigan
bibliography:
- 'refs.bib'
title: 'A graph theoretic framework for representation, exploration and analysis on computed states of physical systems'
---
Introduction
============
In this communication, we explore the casting of large-scale computations of continuum physics in the framework of graph theory. The motivation is to work with low-dimensional representations that encode the fidelity of very high-dimensional, computed solutions, and to develop effective methods to explore and extract further information that could have relevance to decision-making on natural or engineered systems. In this first presentation of ideas, the treatment is deterministic. No probabilistic considerations are invoked.
Of specific interest here are initial and boundary value problems (IBVPs) that span the range from stationary or steady-state systems through first- and second-order dynamics. Each computation of the IBVPs is at high spatial and/or temporal resolution, making for a high-dimensional and/or long time series numerical solution. While dimensionality reduction techniques such as proper orthogonal decomposition [@Sirovich1987; @Berkooz1993; @Rathnam2003] or tensor decomposition [@Tucker1966; @Hitchcock1927] methods and their variants have been widely applied to high-dimensional problems, and compelling progress continues to be made on them, our approach is different. With an ultimate view to decision-making on these systems, we consider functionals defined on the high-dimensional numerical solutions, and that are induced by physics on the system-wide scale. Examples of these functionals include the lift, drag and thrust in computational fluid dynamics, the average strain, load at yield, failure strain or dissipated energy in computational solid mechanics, and phase volumes, total and interfacial free energies in computational materials physics. We consider as the state of the physical system, a low-dimensional Euclidean vector (typically of dimension $\sim \mathcal{O}(10^1)$) whose components are such functionals. The crux of our approach is to treat the states as vertices on a graph, which can then be completed by identifying edges between vertex pairs. We seek additionally to introduce edges that are induced by either (a) the numerical solution technique, or (b) a transition guided by some physical property. In the first case the existence of an edge is determined by convergence of a solution step between states. More alternatives present themselves for edge definition via the physics. We present examples that help make these notions more precise. The edges enable graph traversal by time as well as by enumeration of states guided by the numerics or physics in the low-dimensional space.
Graph theory has a well-appreciated relevance to physical systems, due in some measure to the fact that it underlies network analysis [@Newman2010]. Applications including traffic flow, electric power grids and neural circuits are well-known. However, its direct use in representing and analyzing physical phenomena that are described by partial differential equations appears relatively under-explored except for the following work: Graph vertices and edges have been used to represent vortices and their interactions, respectively, for the analysis of turbulent flows in computational fluid dynamics [@Nair2015; @Taira2016; @Scarsoglio2016]. Recently, graphs have been used to represent problem components such as governing equations, constitutive relations and initial/boundary conditions in the numerical framework of IBVPs [@WangSun2018a]. Graph vertices and edges also have been used to represent variables and relations between them, respectively, in a game theoretic approach to discovering constitutive response functions for material failure [@WangSun2018b].
The following sections consider computations of IBVPs for stationary and steady-state systems (Section \[sec:stationary\]), non-dissipative dynamics (Section \[sec:non-diss-dynamics\]) and dissipative dynamics (Section \[sec:diss-dynamics\]), and connect them to specific types of graphs. The standard machinery of graph theoretic definitions and results is invoked for this purpose. Specific examples are then drawn from computed solutions in a rather extensive Section \[sec:computations\]. Inferences and insights, not directly available from the high-dimensional numerical solutions, are drawn from analysis of the graphs using well-established concepts from graph theory [@West2000; @Newman2010], as well as algorithms that are motivated by the numerical solution or physics specific to each system. Closing remarks are made in Section \[sec:closingremarks\].
Stationary and steady-state systems represented as graphs {#sec:stationary}
=========================================================
Consider a domain $\Omega \subset \mathbb{R}^3$, and the following stationary or steady-state BVP:
$$\begin{aligned}
{3}
\nabla\cdot\boldsymbol{\sigma} + \boldsymbol{f} &= \boldsymbol{0}, \quad &&\text{in} \quad &&\Omega\label{eq:stationary-a}\\
\boldsymbol{u} &= \overline{\boldsymbol{u}}(\boldsymbol{p}), \quad &&\text{on} \quad &&\partial\Omega_u\\
\boldsymbol{\sigma}\boldsymbol{n} &= \boldsymbol{\sigma_n}(\boldsymbol{p}), \quad &&\text{on} \quad &&\partial\Omega_\sigma.\label{eq:stationary-c}\end{aligned}$$
Here, $\boldsymbol{\sigma}(\boldsymbol{u},\boldsymbol{\alpha};\boldsymbol{p})$ is a (vector or tensor) flux, which is, in general, a functional of its arguments $\boldsymbol{u}$ (the vector of primal variables) and $\boldsymbol{\alpha}$ (the vector of internal state variables), and is parameterized by $\boldsymbol{p}$. We assume that the continuous problem defined by Equations – is discretized and solved by a numerical method such as finite elements, finite volumes or finite differences, or by Fourier methods among many other possibilities. We refer to this as the numerical problem, and it could be high dimensional in terms of the degrees of freedom (or unknowns).
States of the system $\mathscr{S}_i \in \mathbb{R}^k$, $i = 1,\dots N$ are low-dimensional vectors with $k$ being much smaller than the dimensionality of the numerical problem. Each $\mathscr{S}_i$ is, in general, a functional of the numerical solution. For stationary or steady state phenomena, different states, $i = 1,\dots N$ are obtained for parameter sets $\boldsymbol{p}_i$, boundary conditions $\overline{\boldsymbol{u}}_i, \boldsymbol{\sigma}_{n_i}$, and boundary decompositions $\partial\Omega = \partial\Omega_{u_i}\cup\partial\Omega_{\sigma_i}$. Also, denote by $\mathscr{T}_{ij}$ a transition from $\mathscr{S}_j$ to $\mathscr{S}_i$ that represents either (a) the nonlinear solution scheme, which, with initial guess $\mathscr{S}_j$, arrives at $\mathscr{S}_i$,[^2] or (b) a change in a physical quantity between $\mathscr{S}_j$ and $\mathscr{S}_i$, which we will refer to as a *transition quantity*. To fix ideas, one may consider, as examples of transition quantities, energies and volume fractions of chemical species, among many others. In what follows, we will continue to refer to transition quantities in the abstract, but will make this more specific with examples of graphs in Section \[sec:computations\]. With this elementary terminology in hand, we can place the description of the states of stationary/steady-state problems and the transitions between them in the graph theoretic setting. For this purpose, we use the standard terminology of graph theory laid out, for instance, by West [@West2000].
Vertices, edges and paths; graph properties {#sec:stationary-graphs}
-------------------------------------------
A graph $G(V,E)$ can be constructed such that its set of vertices $V = \{\mathscr{S}_i\}_{i = 1,\dots N}$ and its set of edges $E = \{\mathscr{T}_{ij}\}_{i,j= 1,\dots M}$, for $M \le N$. A number of properties of $G$ can be recognized. Some of them directly reflect definitions of graph theoretic elements. Others are manifestations of the theorems of graph theory and their corollaries. Terminology will be used interchangeably: a vertex for a computed state, an edge for a transition.
![The graph $G(V,E)$, where $V = \{\mathscr{S}_i,\mathscr{S}_j,\mathscr{S}_k,\mathscr{S}_l\}$. Only the edges representing solution schemes/transitions $\{\mathscr{T}_{ij},\mathscr{T}_{ji}\}$ and $\{\mathscr{T}_{lk},\mathscr{T}_{kl}\}$ have been labelled. The graph $G$ is undirected and is also a clique, representing computations on linear, stationary systems, as well as non-dissipative, dynamical systems under unconditionally stable time integration schemes.[]{data-label="fig:statgraph"}](figures/graphs-stationary.pdf)
1. Graph $G$ is connected if any computed state $\mathscr{S}_i$ must be reached from some initial guess $\mathscr{S}_j$ via a transition $\mathscr{T}_{ij}$ (nonlinear solution step or change in transition quantity). Alternately, certain choices of solution scheme or transition quantity could leave a state $\mathscr{S}_k$ as an unconnected vertex, and the graph $G$ will be unconnected.
2. The reversibility of linear, non-dissipative systems implies that an edge between $\mathscr{S}_j$ and $\mathscr{S}_i$ can be traversed in either direction: $\mathscr{T}_{ij}$ or $\mathscr{T}_{ji}$. In this sense, linear non-dissipative BVPs are represented by undirected graphs.
3. The graph $G$ is a clique if it represents a linear, non-dissipative BVP with $E$ defined by numerical solutions between states. In this case, any computed state $\mathscr{S}_i$ can be reached from a starting guess $\mathscr{S}_j$ via the edge $\mathscr{T}_{ij}$. The degree of every vertex is $N-1$: $d_G(\mathscr{S}_i) = N-1,\;\forall i$. The graph $G$ is $(N-1)$-regular.
4. A vertex $\mathscr{S}_j$ may not be connected to $\mathscr{S}_i$ if the BVP defining the graph is nonlinear and the corresponding states are too distant from each other, with respect to a suitable metric. In such cases, the nonlinear solution scheme may not converge and there may be no edge between $\mathscr{S}_j$ and $\mathscr{S}_i$. The graph $G$ is not a clique in this case. A vertex has degree $d_G(\mathscr{S}_i) \le N-1$. For stiff nonlinear BVPs, the strict inequality is expected to hold. If $E$ is defined by a transition quantity between states, there may not exist a transition to $\mathscr{S}_i$ from $\mathscr{S}_j$ and $G$ is not a clique: $d_G(\mathscr{S}_i) < N-1$.
5. The above property motivates the definition of an edge weight: $$\text{If}\;\exists \mathscr{T}_{ij},\;\text{then}\quad w_s(\mathscr{T}_{ij}) = \Vert \mathscr{S}_j - \mathscr{S}_i\Vert_V
\label{eq:weight-s}$$ where $\Vert\bullet\Vert_V$ is a suitable norm on $V$.
6. A vertex $\mathscr{S}_i \approx \mathscr{S}$ can be added to $V$ by solving an inverse problem. That is, $\boldsymbol{p}_i$, $\overline{\boldsymbol{u}}(\boldsymbol{p}_i)$, $\boldsymbol{\sigma}_n(\boldsymbol{p}_i)$ can be optimized and $\partial\Omega_{u_i}$ and $\partial\Omega_{\sigma_i}$ chosen so that $\Vert\mathscr{S}_i - \mathscr{S}\Vert < \varepsilon$ for a numerical tolerance $0<\varepsilon \ll 1$.
7. Typical computations will yield paths on subgraphs of $G$; that is, each state $\mathscr{S}_i$ is only visited once, and the transition (solution step or change in transition quantity) $\mathscr{T}_{ij}$ is only traversed once. Eulerian tours result if a state $\mathscr{S}_i$ may be revisited, but the transition $\mathscr{T}_{ij}$ is only traversed once, so that subsequent visits to $\mathscr{S}_i$ are from other states $\mathscr{S}_k \neq \mathscr{S}_j$ with transitions $\mathscr{T}_{ik} \neq \mathscr{T}_{ij}$. Cycles follow naturally.
8. A branch is created at vertex $\mathscr{S}_i$ if this state can only be reached along edge (transition) $\mathscr{T}_{ij}$, but has two or more solution edges (transitions) leading out from it: $\mathscr{T}_{i_1 i},\mathscr{T}_{i_2 i},\dots$. The states $\mathscr{S}_{i_1},\mathscr{S}_{i_2},\dots$ differ in parameters $\boldsymbol{p}_{i_1},\boldsymbol{p}_{i_2},\dots$, boundary conditions $\{\overline{\boldsymbol{u}}_{i_1}, \boldsymbol{\sigma}_{n_{i_1}}\}, \{\overline{\boldsymbol{u}}_{i_2}, \boldsymbol{\sigma}_{n_{i_2}}\},\dots$, or boundary decompositions $\partial\Omega = \partial\Omega_{u_{i_1}}\cup\partial\Omega_{\sigma_{i_1}} = \partial\Omega_{u_{i_2}}\cup\partial\Omega_{\sigma_{i_2}},\dots$.
Because computed solutions of linear BVPs can attain any state from any other, the corresponding clique graphs are somewhat less interesting. Figure \[fig:statgraph\] illustrates a clique that could be generated by such a system. The graphs become more interesting if some edges are not present, such as due to the choice of transition quantity. The absence of edges (transitions) in systems representing nonlinear BVPs leads to graphs that are not cliques (Property 4). In the absence of a transition between states $\mathscr{S}_j$ and $\mathscr{S}_i$, however, there will typically be a multi-edge path $\mathscr{P}_{ij} = \mathscr{T}_{ik}\cup\mathscr{T}_{kl}\dots\cup\mathscr{T}_{mn}\cup\mathscr{T}_{nj}$. The graph in this case holds information about states that cannot be attained directly from each other, but can be reached via intermediate states. Such examples are presented and analyzed in Sections \[sec:nonconvexelasticity\]-\[sec:DNS-MLgraphs\]
Non-dissipative dynamical systems {#sec:non-diss-dynamics}
=================================
We consider second-order dynamics in the form of the following IBVP:
$$\begin{aligned}
{3}
\rho \frac{\partial^2\boldsymbol{u}}{\partial t^2} + \nabla\cdot\boldsymbol{\sigma} + \boldsymbol{f} &= \boldsymbol{0}, \quad &&\text{in} \quad &&\Omega\label{eq:sec-order-dyn-a}\\
\boldsymbol{u} &= \overline{\boldsymbol{u}}(\boldsymbol{p}), \quad &&\text{on} \quad &&\partial\Omega_u\label{eq:sec-order-dyn-b}\\
\boldsymbol{\sigma}\boldsymbol{n} &= \boldsymbol{\sigma_n}(\boldsymbol{p}), \quad &&\text{on} \quad &&\partial\Omega_\sigma.\label{eq:sec-order-dyn-c}\\
\boldsymbol{u} &=\boldsymbol{u}_0, \quad &&\text{at} \quad && t = 0\label{eq:sec-order-dyn-d}\\
\dot{\boldsymbol{u}} &=\boldsymbol{v}_0, \quad &&\text{at} \quad && t = 0\label{eq:sec-order-dyn-e}.\end{aligned}$$
For (non)linear elastodynamics (the prominent example of this type of IBVP), $\boldsymbol{u} \in \mathbb{R}^{n_\text{dim}}$ is the displacement field, its second-order time derivative is the acceleration, with $\rho$ being the mass density. In this case, initial conditions on the primal variable, which is the displacement, and its velocity appear in Equations and . For this class of IBVPs, the natural definition of states is for them to be parameterized by time, $t$. The state $\mathscr{S}_i$ is *first* attained at $t = t_i$.
Properties of graphs representing non-dissipative dynamical systems {#sec:non-diss-dynamics-properties}
-------------------------------------------------------------------
With IBVPs now in consideration, the notion of initial guess state is replaced by an initial state from which the system evolves dynamically to another state. The edge $\mathscr{T}_{ij}$ represents a single time step between initial state $\mathscr{S}_j$ and final state $\mathscr{S}_i$. Over the time step, a nonlinear (in general) solution step occurs, as well as changes in any transition quantities. The two possible distinct approaches to defining edges thus collapse to the time step. These ideas will be revisited with graph examples in Section \[sec:computations\]. The following observations can be made regarding the properties introduced for stationary systems:
1. Property 1 holds with the replacement of initial guesses with initial states.
2. The time reversal of non-dissipative dynamical systems, such as second-order elastodynamics, implies that edges can be traversed in either direction between states $\mathscr{S}_j$ and $\mathscr{S}_i$. The graph $G$ is undirected; Property 2 holds.
3. Assuming unconditionally stable, high-order time integration schemes, all states $\mathscr{S}_i$ admitted by a linear IBVP can be reached by the solution scheme, starting from any other state, $\mathscr{S}_j$. The high-order accuracy is needed to ensure that state $\mathscr{S}_i$ is attained up to the desired tolerance. Property 3 holds; linear IBVPs are represented by cliques. A graph with $N$ vertices is $(N-1)$-regular.
4. The assumption of an unconditionally stable, high-order integration scheme implies that, even for nonlinear IBVPs (nonlinear elastodynamics, for example), a state $\mathscr{S}_i$ can always be reached in a single time step from initial condition $\mathscr{S}_j$. Property 4 is modified to state that the graph $G$ is a clique even for nonlinear IBVPs.
5. The definition of an edge weight in Property 5 continues to hold. However, also refer to Property 9 below.
6. Property 6 on addition of a vertex holds. The numerical stability conferred on the Jacobian of the forward problem by an unconditionally stable scheme also is reflected in the solution of the inverse problem, if solved by adjoint methods.
7. Property 7 holds; of special interest here are cycles that correspond to periodic solutions, or orbits, of the IBVP.
8. Property 8 holds unchanged.
The natural parameterization introduced by time leads to additional features in the graphs of non-dissipative dynamical systems.
- Another definition of an edge weight can be introduced as a function of the time between first visits to the given states. Thus, $$w_t(\mathscr{T}_{ij}) = f(\vert t_{i} - t_{j}\vert)
\label{eq:weight-t}$$ where it bears emphasis that $t_i$ is the first time that $\mathscr{S}_i$ is attained. Clearly, the existence of periodic orbits implies cycles on the graph, and that $\mathscr{S}_i$ also may be visited at times $\{t^1_i, t^2_i,\dots,t^n_i\}$. This definition of weight collapses to the time step $\Delta t_{ij}$ between first visits to the states $\mathscr{S}_j$ and $\mathscr{S}_i$. Other weight definitions are possible as we will see in Section \[sec:DNS-MLgraphs\].
Figure \[fig:statgraph\] continues to represent the important properties of graphs of non-dissipative dynamical systems. Variations on parameter sets $\boldsymbol{p}$, boundary conditions, boundary decompositions $\partial\Omega = \partial\Omega_{u}\cup\partial\Omega_{\sigma}$, and initial conditions lead to distinct trajectories in the dynamics, each of which can be represented by its own graph.
Dissipative dynamical systems {#sec:diss-dynamics}
=============================
We consider first-order dynamics in the form of the following IBVP:
$$\begin{aligned}
{3}
\rho \frac{\partial u}{\partial t} + \nabla\cdot\boldsymbol{\sigma} + f &= \boldsymbol{0}, \quad &&\text{in} \quad &&\Omega\label{eq:first-order-dyn-a}\\
u &= \overline{u}(\boldsymbol{p}), \quad &&\text{on} \quad &&\partial\Omega_u\label{eq:first-order-dyn-b}\\
\boldsymbol{\sigma}\boldsymbol{n} &= \boldsymbol{\sigma_n}(\boldsymbol{p}), \quad &&\text{on} \quad &&\partial\Omega_\sigma.\label{eq:first-order-dyn-c}\\
u &=u_0, \quad &&\text{at} \quad && t = 0\label{eq:first-order-dyn-d}.\end{aligned}$$
Note that now, $u \in \mathbb{R}$. For first-order dynamics such as heat conduction and mass transport (of the Fickian, chemical potential-driven, conservative or non-conservative phase field type), $\rho$ is either the heat capacity or equals one, respectively. The flux is of the form $\boldsymbol{\sigma} = -M\nabla u$ for Fourier heat conduction and Fickian diffusion. For conservative phase field models, such as the Cahn-Hilliard equations [@CahnHilliard1958], $\boldsymbol{\sigma} = -M(\nabla\mu(u) -\kappa \nabla\nabla^2 u)$, where $\mu(u)$ is the chemical potential obtained as the derivative of a non-convex function with multiple (at least two) minima, and $\kappa$ represents an interfacial energy. Non-conservative phase field models (Allen-Cahn, [@AllenCahn1979]) have $\boldsymbol{\sigma} = -\kappa M\nabla u$, and $f(u)$ is obtained as $M$ multiplied by the derivative of a non-convex function with multiple (at least two) minima. Such examples are presented and explored in Sections \[sec:CHdynamicsgraphs\] and \[sec:DNS-MLgraphs\]. Again, different states, $\mathscr{S}_i$, are defined by the time at which they were attained, $t = t_i$. An important aspect of the graphs in this case is that states cannot be revisited, as we point out below. Variations in parameter sets $\boldsymbol{p}_i$, boundary conditions, boundary decompositions $\partial\Omega = \partial\Omega_{u_i}\cup\partial\Omega_{\sigma_i}$, and initial conditions correspond to distinct trajectories, each represented by its own graph, or sub-graph as illustrated in Figure \[fig:dissipgraph\].
Dissipative dynamical systems also can be of second order, if the PDE in Equation (\[eq:sec-order-dyn-a\]) is extended to include a first order term: $$\rho \frac{\partial^2\boldsymbol{u}}{\partial t^2} + \gamma \frac{\partial \boldsymbol{u}}{\partial t} + \nabla\cdot\boldsymbol{\sigma} + \boldsymbol{f} = \boldsymbol{0}, \quad \text{in} \quad \Omega\label{eq:sec-order-dissip-dyn}$$ Dissipation can be introduced to stationary and steady-state systems such as Equation (\[eq:stationary-a\]) via an internal variable that is itself governed by first-order dynamics: $$\frac{\partial \boldsymbol{\alpha}}{\partial t} = \boldsymbol{g}(\boldsymbol{u},\boldsymbol{\alpha}), \quad \text{in} \quad \Omega\label{eq:dissip-stationary}$$ Rate formulations of continuum plasticity fit the description in Equation , where $\boldsymbol{\alpha}$ would be the equivalent plastic strain.
![(a) The graph $G(V,E)$ representing a dissipative dynamical system, where $V = \{\mathscr{S}_{i_1},\mathscr{S}_{j_1},\dots,\mathscr{S}_{i_2},\mathscr{S}_{j_2},\dots,\mathscr{S}_{i_3},\mathscr{S}_{j_3},\dots\}$. The dissipative nature of the dynamical systems represented in this case implies that the existence of edge $\mathscr{T}_{ij}$ means the non-existence of $\mathscr{T}_{ji}$, rendering $G$ a directed graph. If the initial states of sub-graphs $G_1 = (V_1,E_1), G_2 = (V_2,E_2), \dots$ are chosen arbitrarily, then $G$ is, in general an unconnected graph of disjoint trees. See Property 15. (b) It is possible for the graph $G(V,E)$ to contain a vertex that is reached from multiple vertices when representing, for example, an equilibirium state that is attained from multiple initial states.[]{data-label="fig:dissipgraph"}](figures/graphs-dissipative.pdf)
![(a) The graph $G(V,E)$ representing a dissipative dynamical system, where $V = \{\mathscr{S}_{i_1},\mathscr{S}_{j_1},\dots,\mathscr{S}_{i_2},\mathscr{S}_{j_2},\dots,\mathscr{S}_{i_3},\mathscr{S}_{j_3},\dots\}$. The dissipative nature of the dynamical systems represented in this case implies that the existence of edge $\mathscr{T}_{ij}$ means the non-existence of $\mathscr{T}_{ji}$, rendering $G$ a directed graph. If the initial states of sub-graphs $G_1 = (V_1,E_1), G_2 = (V_2,E_2), \dots$ are chosen arbitrarily, then $G$ is, in general an unconnected graph of disjoint trees. See Property 15. (b) It is possible for the graph $G(V,E)$ to contain a vertex that is reached from multiple vertices when representing, for example, an equilibirium state that is attained from multiple initial states.[]{data-label="fig:dissipgraph"}](figures/graphs-dissipative-B.pdf)
Properties of graphs representing dissipative dynamical systems {#sec:diss-dynamics-properties}
---------------------------------------------------------------
We maintain the convention that $\mathscr{T}_{ij}$ represents a single time step of the computation between states $\mathscr{S}_j$ and $\mathscr{S}_i$. The following observations can be made regarding the properties introduced previously:
1. Property 1 holds with a vertex $\mathscr{S}_j$ being the initial state from which $\mathscr{S}_i$ is attained along edge $\mathscr{T}_{ij}$.
2. The dissipative nature of the systems now being considered means a loss of time reversal symmetry. Consequently, the edge between $\mathscr{S}_j$ and $\mathscr{S}_i$ cannot be traversed in both directions. The graph is directed: if $\exists \mathscr{T}_{ij}, \Longrightarrow \nexists \mathscr{T}_{ji}$. This is a central aspect of graphs representing dissipative dynamical systems, and is illustrated by arrows on directed edges in Figure \[fig:dissipgraph\].
3. Even assuming unconditionally stable, high-order time integration schemes, a state $\mathscr{S}_j$ admitted by a linear IBVP is connected to only certain other states $\{\mathscr{S}_i,\mathscr{S}_k,\dots\}$. The loss of time time reversal symmetry and the broader property of dissipation prevent $G$ from being a clique. This theme is further echoed in Properties 7 and 10-15.
4. Property 4 on the existence of an edge between a pair of states draws from the dissipative nature of the dynamical system, which confers added stability to integration schemes. States $\mathscr{S}_j$ and $\mathscr{S}_i$ may be distant from each other in a suitable norm, but the existence/non-existence of the edge $\mathscr{T}_{ij}$ also depends on Properties 10-15, induced by the dissipative dynamics, rather than solely on $w_s(\mathscr{T}_{ij})$.
5. Properties 5 and 9 regarding edge weights of non-dissipative dynamical systems hold for dissipative dynamical systems.
6. Property 6 on addition of a vertex holds. The added stability from dissipation will be reflected in the solution of inverse problems by adjoint methods to determine states added as vertices to the graph.
7. Property 7 suffers a major restriction: Dissipation means that no walk exists that visits the same state $\mathscr{S}_i$ more than once. It follows that the solution step $\mathscr{T}_{ij}$ can only be traversed once. Cycles do not occur. Paths are the only admitted walks.
8. Property 8 holds unchanged.
9. Property 9 on time-defined weights holds.
The following properties are particular to graphs representing dissipative dynamical systems:
- There does not exist a path starting at vertex $\mathscr{S}_j$ and ending at $\mathscr{S}_i$ if the times satisfy $t_i < t_j$.
- If vertex $\mathscr{S}_i$ cannot be reached within a single edge by starting from $\mathscr{S}_j$, then either $t_i < t_j$, or $\Vert\mathscr{S}_j - \mathscr{S}_i\Vert_V > \varepsilon$ for some $\varepsilon > 0$, implying that $\mathscr{S}_i$ is “distant” from $\mathscr{S}_j$.
- For the directed graph, the quantities in $\mathscr{S}_j \in \mathbb{R}^k$ can be defined to include at least one whose rates along any directed edge, $\mathscr{T}_{ij}$ are either non-negative or non-positive. These could be observed quantities drawn from $\boldsymbol{u}$ or internal variables drawn from $\boldsymbol{\alpha}$, already governed by Equation (\[eq:dissip-stationary\]). $$\dot{f}(\boldsymbol{u}) \ge 0,\quad\text{or}\; \dot{f}(\boldsymbol{u}) \le 0, \qquad \dot{\alpha}^A \ge 0,\quad\text{or}\; \dot{\alpha}^A \le 0,
\label{eq:entropy-quant}$$ where $A$ denotes the components of $\boldsymbol{\alpha}$. We will use the term *entropy quantities* for $f(\boldsymbol{u})$ and $\alpha^A$ of the type in Equation by analogy with the non-negativity of the entropy rate.
- Note that the states that define the vertices of $G$ must contain entropy quantities if the graph is to properly represent a dissipative dynamical process.
- Dissipation implies that all walks are paths, and therefore no cycles are allowed. It follows, then, that these graphs are trees [@West2000].
- A tree, $G_1(V_1,E_1)$, has as its root an initial state $\mathscr{S}_{i_1}$, and remains disjoint from the tree, $G_2(V_2,E_2)$, with root $\mathscr{S}_{i_2}$, unless the vertices on $G_1$ and $G_2$ are chosen to ensure that the entropy quantities are identical for some states $\mathscr{S}_{j_1} \in V_1$ and $\mathscr{S}_{j_2} \in V_2$. The graph $G = G_1 \cup G_2 \cup \dots$ is not, in general, fully-connected, as illustrated in Figure \[fig:dissipgraph\]a. However, Figure \[fig:dissipgraph\]b illustrates a situation in which the sub-trees $G_1$ and $G_2$ have been connected, by choosing vertices $\mathscr{S}_{k_1}$ and $\mathscr{S}_{k_2}$, and the transitions $\mathscr{T}_{{l_1}{k_1}}$ and $\mathscr{T}_{{l_1}{k_2}}$ so that a branch exists at vertex $\mathscr{S}_{l_1}$.
Graph theoretic representations of computed solutions {#sec:computations}
=====================================================
We consider examples that span the systems outlined abstractly in Sections \[sec:stationary\]–\[sec:diss-dynamics\]. In each case, we present an outline of the partial differential equations and computational physics frameworks followed by discussions of the graphs induced by the computed numerical solutions.
Graphs constructed on solutions to non-dissipative elastodynamics and linear elasticity {#sec:linelasticity}
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![A clique representing either a non-dissipative dynamical system of elastodynamics computed with an unconditionally stable, second-order accurate solution scheme, or a stationary problem of linear, quasistatic elasticity. The deformation has been scaled $40\times$ for clarity.[]{data-label="fig:stat-non-dissip-comput-graph"}](figures/graphs-stationary-computations1.pdf)
The graph in Figure \[fig:stat-non-dissip-comput-graph\] is a clique representing a non-dissipative elastodynamics problem. Each vertex shows a computed state solved with an unconditionally stable, second-order scheme, which ensures that each state is attainable from any other. The superposed field represents the Euclidean norm of the displacement, $\boldsymbol{u}$. The state $\mathscr{S}_i$ represents the initial condition, and the cycle $\mathscr{S}_i \rightarrow\mathscr{S}_j \rightarrow\mathscr{S}_k\rightarrow\mathscr{S}_l\rightarrow\mathscr{S}_i$ is a complete period; represented by a cycle on the graph. The edges, $\mathscr{T}_{ji}, \dots \mathscr{T}_{il}$, all have the same time-based weights $w_t$ defined in Equation . However, their solution norm-based weights $w_s$, defined via Equation , will differ. This graph is isomorphic to one in which the vertices represent stationary states of deformation of a quasi-statically strained, linearly elastic solid. The computed states, of course, would be different from those in the figure. In such a stationary problem of linearized elasticity, any state can be attained from any other state, thus maintaining the cliqueness of the graph. The edges now represent solution steps between states, and admit only the solution norm-based weights, $w_s$. The graph is undirected, reflecting the time-reversal symmetry of elastodynamics, Equation , and the reversibility of the stationary problem of elasticity, Equation . Both these properties are inherited by the respective numerical schemes: an unconditionally stable, second-order accurate method for the elastodynamics problem, and a linear solve for linearized elasticity. In both these systems of equations the constitutive relation $\boldsymbol{\sigma} = \mathrm{sym}[\partial \hat{\psi}/\partial\nabla^\text{s}\boldsymbol{u}]$ holds for a strain energy density function $\hat{\psi}(\nabla^\text{s}\boldsymbol{u})$, where $\nabla^\text{s}$ is the symmetric gradient.
Graphs on stationary states of gradient-regularized, non-convex elasticity at finite strain {#sec:nonconvexelasticity}
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We next consider graphs induced by the states that arise as free energy minima [@Rudrarajuetal2016; @SagiyamaGarikipati2017a; @SagiyamaGarikipati2017b] in a gradient-regularized model of non-convex elasticity at finite strain [@Toupin1962; @Barsch1984]. In summarizing the problem we begin with the free energy density function, now written as $\bar{\psi}(F_{iJ},F_{iJ,K})$, where, with $u_i$ denoting the displacement vector, and $X_I$ denoting the reference position in coordinate notation, the deformation gradient tensor $F_{iJ}$ and its gradient $F_{iJ,K}$ are
$$F_{iJ} = \delta_{iJ} + \frac{\partial u_i}{\partial X_J}; \qquad F_{iJ,K} = \frac{\partial F_{iJ}}{\partial X_K}
\label{eq:defgrad}$$
As illustrated in Figure \[fig:freeenergy1\] the solid undergoes a transition from a high temperature cubic crystal structure, in which the free energy density $\bar{\psi}$ is fully convex, to a low temperature tetragonal crystal structure, in which a non-convex component appears in $\bar{\psi}$. We focus on describing the response of the solid in this regime, where the non-convex component, $\widetilde{\psi}$ must be accounted for in an appropriate constitutive description by writing it as a function of suitably parameterized strain quantities. For details of the parameterization the interested reader is directed to the the work of Barsch & Krumhansl [@Barsch1984].
![Illustration of the free energy density component, $\widetilde{\psi}$, as a surface undergoing a convex to non-convex transition in a suitably parameterized, reduced strain space. The parameterization does not appear explicitly, but is represented by the transformation from a cubic crystal structure to three tetragonal structures with the same symmetry group.[]{data-label="fig:freeenergy1"}](figures/cubic-tetragonal.pdf)
The non-convexity in $ \widetilde{\psi}$ gives rise to finely fluctuating solution fields. They model martensitic microstructures, which can be solved for if the free energy density is coercified by a strain gradient contribution: $$\bar{\psi}(F_{iJ}, F_{iJ,K}) = \widetilde{\psi}(F_{iJ}) + \lambda F_{iJ,K}F_{iJ,K}.
\label{eq:freeenergy}$$ where $\lambda$ is related to the energy of interfaces.
The stress and higher-order stress, respectively, are then written as:
$$\begin{aligned}
P_{iJ} &= \frac{\partial \bar{\psi}}{\partial F_{iJ}}\label{eq:stressP} \\
B_{iJK} &= \frac{\partial \bar{\psi}}{\partial F_{iJ,K}}\label{eq:stressB}\end{aligned}$$
The governing system of partial differential equations is:
$$\begin{aligned}
{2}
P_{iJ,J} - B_{iJK,JK} &= 0 &&\mathrm{in} ~\Omega\label{eq:strongformgradelasticity-a}\\
u_{i} &= \bar{u}_i &&\mathrm{on} ~\partial\Omega_{i}^u\label{eq:strongformgradelasticity-b}\\
P_{iJ}N_J - DB_{iJK}N_KN_J - 2D_J(B_{iJK})N_K & &&\phantom{.} \nonumber\\
- B_{iJK}D_JN_K + (b^L_LN_JN_K-b_{JK})B_{iJK} & = T_{i} &&\mathrm{on} ~\partial\Omega_{i}^T\label{eq:strongformgradelasticity-c}\\
Du_i &= 0 &&\mathrm{on} ~\partial\Omega_{i}^m\label{eq:strongformgradelasticity-d}\\
B_{iJK}N_JN_K &= 0 &&\mathrm{on} ~\partial\Omega_{i}^M\label{eq:strongformgradelasticity-e}\end{aligned}$$
where, $\partial\Omega= \partial\Omega_{i}^u \cup \partial\Omega_{i}^T$ and $\partial\Omega= \partial\Omega_{i}^m \cup \partial\Omega_{i}^M$ represent distinct decompositions of the smooth boundary. Here, $b_{IJ}=-D_{I}N_J=-D_{J}N_I$ are components of the second fundamental form of the smooth boundary. The governing partial differential equation in is in conservation form of , which can be seen by identifying $\sigma_{iJ} = P_{iJ} - B_{iJK,K}$. It also is nonlinear and fourth-order as is apparent upon substituting Equations , and for $B_{iJK,JK}$ in Equation . The Dirichlet boundary condition in has the same form as for conventional elasticity. Equation is the extension of , while and are additional requirements, all arising from the fourth-order differential nature of this problem.[^3]
With the mathematical formulation of the problem outlined above, we have computed a number of martensitic microstructures, some of which appear in Figure \[fig:microstructures\]. The three tetragonal crystal structures in Figure \[fig:freeenergy1\] occur in the martensitic microstructures of Figure \[fig:microstructures\], with each tetragonal variant being represented by the same color in the two figures. A more detailed exposition of this problem from the mathematical and computational points of view has been laid out elsewhere [@Rudrarajuetal2014; @Rudrarajuetal2016; @Sagiyamaetal2016; @SagiyamaGarikipati2017b]. Using isogeometric analytic methods, described in the preceding references, for the ease of representing finite-dimensional functions of high-order continuity, we have obtained numerical solutions to boundary value problems posed on the system of equations (\[eq:defgrad\]-\[eq:strongformgradelasticity-e\]). With periodic boundary conditions, we have obtained a range of microstructures that appear in Figure \[fig:microstructures\].
![Example microstructures obtained with the nonconvex model of elasticity around a mean deformation gradient $\frac{1}{\text{meas}(\Omega)}\int_\Omega \boldsymbol{F}\mathrm{d}V = \boldsymbol{1}$.[]{data-label="fig:microstructures"}](figures/microstructures.pdf)
Because of the differing orientations of the tetragonal variants, the elastic response fluctuates rapidly over each microstructure shown in Figure \[fig:microstructures\]. In solids of interest for materials physics applications (e.g., batteries, electronics and structural alloys) the sub-domains with a uniform variant are on the scale of microns or less, and it is often of interest to model their effective properties; for instance, the effective stress-strain response. With the aim of carrying out such homogenization numerically (a study that will be described in detail elsewhere), we have computed $2770$ elastic states for the microstructure appearing in the upper left of Figure \[fig:microstructures\]. These strains were imposed with Dirichlet boundary conditions and that define a bijective mapping to a set of uniform deformation gradients[^4] denoted by $\widehat{\boldsymbol{F}}$, from which the Green-Lagrange strain is obtained as $\widehat{\boldsymbol{E}} = \frac{1}{2}(\widehat{\boldsymbol{F}}^\mathrm{T}\widehat{\boldsymbol{F}} - \boldsymbol{1})$. Because of its symmetry, each tensor-valued $\widehat{\boldsymbol{E}}$ is a point in $\mathbb{R}^6$.
![An example of a strain state $\mathscr{S}_i \equiv\widehat{\boldsymbol{E}}_i$ imposed on the microstructure appearing in the upper left of Figure \[fig:microstructures\].[]{data-label="fig:micro_deform"}](figures/micro_deform.pdf)
### Graphs induced by strain states {#sec:graphsinducedstrains}
The $N = 2770$ strain states, which we use to parameterize the set of boundary conditions, are labelled by $\widehat{\boldsymbol{E}}_i \in \mathbb{R}^6,\; i = 1,\dots N$. Identifying each state with a graph vertex, we formally write $\mathscr{S}_i \equiv \widehat{\boldsymbol{E}}_i$. These states are chosen from a larger set of states (vertices), $V = \{ \mathscr{S}_i\}_{i=1,\dots M} \equiv \{\widehat{\boldsymbol{E}}_i\}_{i=1,\dots M}$, where $M > N$, of states whose tensor components are generated by a Sobol’ sequence, chosen for its space-filling property, in $\mathbb{R}^6$. Starting with an arbitrary strain state, $\mathscr{S}_j \equiv \widehat{\boldsymbol{E}}_j$ from the set $V$, we apply the boundary conditions that define a uniform deformation gradient $\widehat{\boldsymbol{F}}_j$ (unique up to rotations) relative to an undeformed reference state of the crystal occupying region $\Omega \subset \mathbb{R}^3$. The corresponding stationary elastic state of the chosen microstructure is thus computed and satisfies $\frac{1}{\text{meas}(\Omega)}\int_\Omega\boldsymbol{F}\mathrm{d}V = \widehat{\boldsymbol{F}}_j$. We introduce another set $\widetilde{V}$, which will contain those states whose elastic response has been solved for, and initialize it with the state $\mathscr{S}_j \equiv \widehat{\boldsymbol{E}}_j$. So, $\widetilde{V} = \{\mathscr{S}_j \}$. Next, we search for another state, $\mathscr{S}_k \equiv \widehat{\boldsymbol{E}}_k$, subjected to Dirichlet boundary conditions corresponding to the uniform deformation gradient $\widehat{\boldsymbol{F}}_k$ (unique up to rotations). For many choices of $\widehat{\boldsymbol{E}}_k$, the stiffness induced by the microstructure will render the preceding boundary value problem too stiff for convergence of the nonlinear solver. However, if $\widehat{\boldsymbol{E}}_k$ is within an $\varepsilon$-ball of $\widehat{\boldsymbol{E}}_j$ in the Frobenius norm $\Vert\bullet\Vert_\text{F}$, the relative deformation gradient $\widehat{\boldsymbol{F}}_k\widehat{\boldsymbol{F}}_j^{-1}$ applied to the elastic state $\mathscr{S}_j \equiv \widehat{\boldsymbol{E}}_j$ allows the computation of the elastic state labelled by $\mathscr{S}_k \equiv \widehat{\boldsymbol{E}}_k$. This step induces the transition in states $\mathscr{S}_j\rightarrow \mathscr{S}_k$, i.e., the edge $\mathscr{T}_{kj}$. We expand $\widetilde{V}$ to $\widetilde{V} = \{\mathscr{S}_j,\mathscr{S}_k \}$, and initialize $\widetilde{E} = \{\mathscr{T}_{kj}\}$. In subsequent steps, we search for solutions to the elastic states corresponding to strains drawn from $V$, with initial (reference) states drawn from $\widetilde{V}$. If the initial state is $\mathscr{S}_l \equiv \widehat{\boldsymbol{E}}_l$, and the target state is $\mathscr{S}_m \equiv \widehat{\boldsymbol{E}}_m$, the relative deformation gradient is $\widehat{\boldsymbol{F}}_m\widehat{\boldsymbol{F}}_l^{-1}$ and the transition between states is $\mathscr{S}_l\rightarrow \mathscr{S}_m$, allowing an expansion to $\widetilde{V} = \{\mathscr{S}_j,\mathscr{S}_k,\dots,\mathscr{S}_l,\mathscr{S}_m\}$, and the set of edges to $\widetilde{E} = \{\mathscr{T}_{kj}, \dots,\mathscr{T}_{ml} \}$. The induced graph is $\widetilde{G} = (\widetilde{V},\widetilde{E})$, with $\widetilde{V} \subset V, \widetilde{E} \subset E$.
If $\varepsilon$ is small enough we find that the nonlinear solver converges for the transitions $\mathscr{S}_l\rightarrow \mathscr{S}_m$ and $\mathscr{S}_m\rightarrow \mathscr{S}_l$, making the edges $\mathscr{T}_{ml}$ undirected. For a given $\varepsilon$, the nonlinear solution may, however, fail to converge for some pairs $\{\mathscr{S}_l,\mathscr{S}_m\}$. If the elastic state $\mathscr{S}_m$ was sought, then it is not added to $\widetilde{V}$, and the edge $\mathscr{T}_{ml}$ is not added to $\widetilde{E}$.[^5] The generation of the induced graph is laid out in Algorithm \[algo:strainstates\].
Graph generation induced by strain states.
\[algo:strainstates\]
We have applied Algorithm \[algo:strainstates\] to generate graphs as large as $\vert\widetilde{V}\vert = 2770$. A circular layout of one of the graphs induced by strain states of the microstructure in the upper left corner of Figure \[fig:microstructures\] during its generation by Algorithm \[algo:strainstates\] appears in Figure \[fig:microstructureStrains\]. For clarity and ease of viewing, we have dispensed with the $\mathscr{S},\mathscr{T}$ labelling, and only have indicated vertices (states) by their numbers. Because of the density of edges in larger graphs, we have restricted ourselves, for purposes of illustration, to analyzing a graph of size $\vert\widetilde{V}\vert = 128$ in the remainder of Section \[sec:nonconvexelasticity\].
![Generation of the graph of strain states by Algorithm \[algo:strainstates\], showing a central, densely connected component in circular layout and a few unconnected vertices whose edges remain to be determined at the stage shown. This graph was generated by exploring the strain states of the microstructure in the upper left corner of Figure \[fig:microstructures\].[]{data-label="fig:microstructureStrains"}](figures/gradelast_sample2.pdf)
### Graph layouts; eigenvector centrality and degree centrality reveal the importance of individual strain states to graph traversal {#sec:elastgraphlayouts}
The high degrees of most vertices makes the circular layout in Figure \[fig:microstructureStrains\] unsuitable for visualization. For this reason we also have presented the graph in the Kamada-Kawai layout [@Kamada1989], which is based on finding local minima of an edge length-dependent energy defined on the two dimensional planar graph. This energy is written as [@Koren2005] $$U = \sum\limits_{\mathscr{T}_{ij}\in \widetilde{E}}w(\mathscr{T}_{ij})d^2(\mathscr{T}_{ij}) = \sum\limits_{p=1}^2 \boldsymbol{x}_p^\text{T}\boldsymbol{L} \boldsymbol{x}_p
\label{eq:graphenergy}$$ where $d(\mathscr{T}_{ij})$ is the length of edge $\mathscr{T}_{ij}$ in the planar layout of the graph, $\boldsymbol{x}_p \in \mathbb{R}^N$ with $N = \vert\widetilde{V}\vert$, is the vector of the $p^\text{th}$ coordinate of the vertices ($p = 1,2$), and $\boldsymbol{L}$ is the Laplacian matrix of $\widetilde{G}$ [@Newman2010]. This matrix is constructed from the adjacency matrix, $\boldsymbol{A}$, and the degree matrix, $\boldsymbol{D}$ as $\boldsymbol{L} = \boldsymbol{D} - \boldsymbol{A}$, where $\boldsymbol{A}$ and $\boldsymbol{D}$ are defined as $$A_{ij} = \begin{cases}
w(\mathscr{T}_{ij}) &\text{if}\; \exists\, \mathscr{T}_{ij}\\
0 &\text{otherwise}
\end{cases},\quad
D_{ij} = \begin{cases}\sum\limits_{k=1}^N A_{ik}&\text{if}\; i = j\\
0 &\text{otherwise}
\end{cases}$$ For unweighted matrices, the same definitions work with $w(\mathscr{T}_{ij}) = 1$ if $\exists\;\mathscr{T}_{ij}$. Note that $\boldsymbol{A}$ is symmetric for undirected graphs, such as we have for this problem, and $\boldsymbol{D}$ is diagonal. The minimization problem gives $$(\boldsymbol{x}^\text{min}_1,\boldsymbol{x}^\text{min}_2) = \text{arg}\;\min\limits_{(\boldsymbol{x}_1,\boldsymbol{x}_2)} \sum\limits_{p=1}^2 \boldsymbol{x}_p^\text{T}\boldsymbol{L} \boldsymbol{x}_p
\label{eq:graphenergymin}$$
![The graph of strain states in Kamada-Kawai layout. The vertices are shaded by their eigenvector centrality, while the area of each vertex symbol is proportional to the Frobenius norm of the strain $\Vert\widehat{\boldsymbol{E}}_i\Vert_\text{F}$.[]{data-label="fig:ms_kamadakawai"}](figures/ms_kamadakawai.pdf)
The Kamada-Kawai layout in Figure \[fig:ms\_kamadakawai\] is favored for the visual appeal achieved by energy minimization as defined by Equation . Each vertex symbol has its area scaled by $\Vert\widehat{\boldsymbol{E}}\Vert_\text{F}$. We also computed the eigenvector centrality of $\widetilde{G}$, to elucidate the extent to which a vertex and its associated strain state are connected to (have reversible strain paths to) other states, or to other well-connected states, or both [@Newman2010]. This is a measure of the importance of a strain state (vertex) to navigation of paths through the entire set $\widetilde{V}$ (alternately, the graph) generated by Algorithm \[algo:strainstates\]. The eigenvector centrality is represented by the shading of vertices.
The ability to visualize the distribution of the related, but simpler, degree centrality (or simply the degree of a vertex), over $\widetilde{G}$ reveals the strain states that are easy or difficult to attain. The Kamada-Kawai layout places the low degree centrality vertices at the periphery, and removed from the inner vertices that have high degree centrality, because their smaller numbers of edges allow a lower energy penalty in Equation even while the edges themselves are longer. This layout thus better delineates the lower degree vertices. Also of interest is the spectral layout [@Koren2005] in Figure \[fig:ms\_spectral\], which is effective in accentuating the vertices with lower eigenvector centrality. This format shows the eigenvector components of each vertex for the second and third smallest eigenvalues of $\boldsymbol{L}$ on the Cartesian axes.[^6] Low eigenvector centrality vertices fall further out along these axes, helping to distinguish them from high eigenvector centrality vertices. Clearly, strain state $\widehat{\boldsymbol{E}}_{114}$ is related to the fewest states by this measure, followed by $\widehat{\boldsymbol{E}}_{21},\widehat{\boldsymbol{E}}_{3},\widehat{\boldsymbol{E}}_{30}$ and $\widehat{\boldsymbol{E}}_{25}$ in that order. For brevity we define $\widetilde{V}_\text{LEV} = \{\mathscr{S}_{114},\mathscr{S}_{21},\mathscr{S}_{3},\mathscr{S}_{30},\mathscr{S}_{25} \}$. The corresponding strain states for $\widetilde{\boldsymbol{E}}_{114}$ and $\widetilde{\boldsymbol{E}}_{21}$ are $$\begin{aligned}
{2}
\widehat{\boldsymbol{E}}_{114} &=
\begin{bmatrix}
-0.015 & 0.017 & 0.002\\
0.017 & -0.032 & 0.023\\
0.002 & 0.023 & 0.003
\end{bmatrix},\quad
\widehat{\boldsymbol{E}}_{21} &=
\begin{bmatrix}
-0.035 & 0.025 & -0.004\\
0.025 & -0.010 & 0.013\\
-0.004 & 0.013 & 0.025
\end{bmatrix}\end{aligned}$$
![The graph of strain states in a spectral layout. The vertices are shaded by their eigenvector centralities, while the area of each vertex symbol is proportional to the Frobeius norm of the strain $\Vert\widehat{\boldsymbol{E}}_i\Vert_\text{F}$. The lower the eigenvector centrality of a vertex is, the longer are the edges connecting it to its neighboring vertices.[]{data-label="fig:ms_spectral"}](figures/ms_spectral.pdf)
### Measure of states and eigenvector centrality of vertices {#sec:elastcentrality}
Since, in Algorithm \[algo:strainstates\], the (non-)existence of edges is determined by (non-)convergence of the nonlinear solver, we investigated the strain states $\widehat{\boldsymbol{E}}_i$ of the three vertices $\mathscr{S}_i$ that lie furthest from the origin in the spectral layout. As a measure of the relative “extremity” of these states, we computed the Frobenius norms $\Vert\widehat{\boldsymbol{E}}_i\Vert_\text{F}$, which we represent by linearly scaling the area of the vertices in the graph layouts. This measure has relevance because states with larger norms $\Vert\widehat{\boldsymbol{E}}\Vert_\text{F}$ lie further from the origin in $\mathbb{R}^6$, and also from low strain states. The strain state with maximum Frobenius norm (the most distant state from the origin in this norm in $\mathbb{R}^6$) is $$\widehat{\boldsymbol{E}}_{126} =
\begin{bmatrix}
-0.038 & -0.013 & -0.095\\
-0.013 & -0.004 & -0.047\\
-0.095 & -0.047 & 0.025
\end{bmatrix}$$ and the maximum norm of the difference in states is $\Vert\widehat{\boldsymbol{E}}_{126} - \widehat{\boldsymbol{E}}_{86} \Vert_\text{F} = 0.1855$. From Figure \[fig:ms\_kamadakawai\], it emerges that while $\mathscr{S}_{126}$ has low eigenvector centrality, it is not one of the five lowest: $\mathscr{S}_{126} \notin \widetilde{V}_\text{LEV}$. Thus, while $\widehat{\boldsymbol{E}}_{126}$ is a distant state by two measures in $\mathbb{R}^6$, it is far from being the least visited. We conclude that the magnitude of strain or of strain difference is not an indicator of states that have fewer reversible strain steps to other states. Instead, the interaction of the strain state with the underlying microstructure may play a dominant role in determining its accessibility.
### Cliques indicate sets of mutually accessible strain states {#sec:elastcliquecycle}
The observations that there is a distribution of eigenvector centrality and degree centrality motivates investigation of other measures of connectedness between the strain states. The existence of cliques and cycles between states becomes relevant in this context. Using the Kamada-Kawai layout for its visual clarity, Figure \[fig:ms\_clique\] highlights the vertices belonging to the largest clique in red. Recall that this set of strain states are all reversibly attainable from each other. Of course, vertices belonging to a clique, say $\widetilde{G}_\alpha$, display a smaller maximum of the Frobenius norm of the difference between strain states, when compared with the maximum computed over the entire graph $\widetilde{G}$, $$\max\limits_{\mathscr{S}_i,\mathscr{S}_j\in\widetilde{G}_\alpha} \Vert \widehat{\boldsymbol{E}}_i - \widehat{\boldsymbol{E}}_j \Vert_\text{F} \le \max\limits_{\mathscr{S}_k,\mathscr{S}_l\in\widetilde{G}} \Vert \widehat{\boldsymbol{E}}_k - \widehat{\boldsymbol{E}}_l \Vert_\text{F}$$ where the sub-graph over which the maximum on the left hand-side of the inequality is computed is any clique $\widetilde{G}_\alpha \subset \widetilde{G}$.
### Cycles {#sec:elastcycles}
The number of cycles can be of interest, especially in sparsely connected graphs, where they are likely to be repeatedly traversed in the course of many walks. For the graph of strain states, cycles identify sequences of deformation that return to the starting state without reversing path, and suggest a measure of elastic cyclability of the microstructure. The graph under consideration harbors a large clique seen in Figure \[fig:ms\_clique\], which as we now demonstrate, contains a very large number of cycles. The number of cycles in an undirected graph scales exponentially with $\vert\widetilde{V}\vert$: Consider the clique in Figure \[fig:ms\_clique\] as a subgraph $\widetilde{G}_\beta\subset\widetilde{G}$. It has $\vert\widetilde{V}_\beta\vert = 34$ and $\vert\widetilde{E}_\beta \vert= \binom{\vert\widetilde{V}_\beta\vert}{2} = 561$. Since $\widetilde{G}_\beta$ is undirected, it is the union of $\sum_{p = 1}^{561}2^p$ directed graphs. Using the Rocha-Thatte algorithm [@Rocha2015] the number of cycles in each of these directed graphs can be found in $p\sim 1,\dots, \binom{34}{2}$ iterations (the size of the longest path in each corresponding graph). Thus, the number of iterations needed to find all the cycles in $\widetilde{G}_\beta$ scales as $\sim \sum_{p = 1}^{561}p\cdot 2^p$. The graph $\widetilde{G} \supset \widetilde{G}_\beta$ itself will have many more cycles, which we have not attempted to enumerate.
![The largest clique in the graph of strain states is shown in red.[]{data-label="fig:ms_clique"}](figures/ms_clique.pdf)
### Shortest paths {#sec:elastshortestpath}
![The shortest paths between the vertices in the pairs $\{\mathscr{S}_{86},\mathscr{S}_{126} \}$ (largest separation in $\widetilde{G}$), $\{\mathscr{S}_{115},\mathscr{S}_{118} \}$ (largest separation in the largest clique $\widetilde{G}_\beta \subset \widetilde{G}$) and $\{\mathscr{S}_{3},\mathscr{S}_{114} \}$ (largest separation in the Kamada-Kawai layout of $\widetilde{G}$).[]{data-label="fig:ms_shortestpath"}](figures/path_all.pdf)
The final aspect of the graph we have studied is the shortest path between vertex pairs whose strain states are maximally separated over $\widetilde{G}$. Let $\{\mathscr{S}_i, \mathscr{S}_j\}$ be the unordered pair satisfying $$\{\mathscr{S}_i,\mathscr{S}_j \} = \text{arg}\,\max\limits_{\mathscr{S}_k,\mathscr{S}_l\in\widetilde{V}} \Vert \widehat{\boldsymbol{E}}_k - \widehat{\boldsymbol{E}}_l \Vert_\text{F}
\label{eq:maxstraindiff}$$ We find the shortest path between $\mathscr{S}_i, \mathscr{S}_j$. That is, if $\widetilde{V}_1 = \{\mathscr{S}_i,\mathscr{S}_k,\dots \mathscr{S}_l,\mathscr{S}_j \}$, $\widetilde{V}_2 = \{\mathscr{S}_i,\mathscr{S}_m,\dots \mathscr{S}_n,\mathscr{S}_j \}$, …, $\widetilde{V}_O = \{\mathscr{S}_i,\mathscr{S}_o,\dots \mathscr{S}_p,\mathscr{S}_j \}$ are distinct sets whose vertices are arranged in a sequence to define paths between $\mathscr{S}_i$ and $\mathscr{S}_j$, with corresponding edge sets $\widetilde{E}_1,\widetilde{E}_2,\dots \widetilde{E}_O$, we seek $$\widetilde{E}_S = \text{arg} \min\limits_{\widetilde{E}_1,\dots \widetilde{E}_O} \vert \widetilde{E}_K\vert$$ Thus, $\widetilde{E}_S$ gives us the most efficient path through the most distantly separated strain states of those sampled to generate data for numerical homogenization of the microstructure. For the strain states graph, $\widetilde{G}$, the pair corresponding to Equation is $\{\mathscr{S}_{86}, \mathscr{S}_{126}\}$. Figure \[fig:ms\_shortestpath\] shows the shortest path between the vertices corresponding to this pair, as well as for $\{ \mathscr{S}_{115},\mathscr{S}_{118}\}$, which is the corresponding pair for the clique $\widetilde{G}_\beta$. For comparison, we also have shown the shortest path between $\{\mathscr{S}_3, \mathscr{S}_{114}\}$, which is the most separated pair in the Kamada-Kawai layout.
### Components {#sec:elastcomponents}
We note that the rapidly fluctuating strains $\boldsymbol{E}$ over each of the martensitic microstructures in Figure \[fig:microstructures\] are associated with a finely corrugated energy surface in the high-dimensional space in which we have obtained these numerical solutions.[^7] As a consequence, while, physically, there exist trajectories in strain space $\boldsymbol{E}(\boldsymbol{\xi}) \in \mathbb{R}^6$, with a non-monotonic dependence on a vector parameter $\boldsymbol{\xi}$, that transform one microstructure of Figure \[fig:microstructures\] into another, it is very challenging to numerically trace such transitions. Even if trajectories were identified, numerical solutions along them could prove stiff to the point of non-convergence. For this reason, the graph associated with each microstructure in Figure \[fig:microstructures\], say $\widetilde{G}_{\text{ms}_1}, \widetilde{G}_{\text{ms}_2}, \dots$, forms a connected component of a larger graph $G = \widetilde{G}_{\text{ms}_1}\cup \widetilde{G}_{\text{ms}_2}\cup\dots$, which is itself not fully connected. The problem of numerically finding those edges that would make $G$ fully connected is of mainly mathematical, rather than practical interest because of the difficulty of following such trajectories.
\[fig:stat-nonlin-graph\]
Graphs constructed on time series solutions of a dissipative dynamical system {#sec:CHdynamicsgraphs}
------------------------------------------------------------------------------
Thermodynamic dissipation is near-ubiquitous in dynamic physical processes. Here, we consider a first-order, dissipative, dynamical problem related to phase transformations in materials physics and biophysics, and seek to study it in the context of the properties observed in the abstract in Section \[sec:diss-dynamics\].
### First-order dynamics of a two-species, phase-separating system {#sec:CHsystem}
We base our treatment on the Cahn-Hilliard equation written for two species whose compositions $u_1$ and $u_2$ satisfy $-1 \le u_1,u_2 \le 1$. The Cahn-Hilliard equation is a conservative phase field method that models phase separation as the consequence of an instability in a nearly uniform composition field. The description is based on a free energy density function of the form $$\psi(u_1,u_2,\nabla u_1,\nabla u_2) = \widetilde{\psi}(u_1,u_2) + \frac{1}{2}\kappa_1\Vert \nabla u_1\Vert^2 + \frac{1}{2}\kappa_2\Vert \nabla u_2\Vert^2,
\label{eq:CHfreeenergy}$$ where $\widetilde{\psi}(u_1,u_2)$ is non-convex with respect to $u_1$ and $u_2$, and the gradient terms with coefficients $\kappa_1$ and $\kappa_2$ represent interfacial energies. Of interest here is a function of the form $$\widetilde{\psi}(u_1,u_2) = \frac{3 d}{2 s^4} (u_1^2+u_2^2)^2 + \frac{d}{s^3} u_2 (u_2^2-3u_1^2) - \frac{3 d}{2 s^2} (u_1^2+u_2^2), \quad d,s > 0
\label{eq:homogenergy3well2field}$$ This “homogeneous” component of the free energy density is illustrated in Figure \[fig:freeenergy2\]. The three local minima represent distinct phases that are explained below.
![The non-convex free energy density function, $\widetilde{\psi}(u_1,u_2)$ that gives rise to separation into three phases corresponding to the three wells.[]{data-label="fig:freeenergy2"}](figures/surf_plot.png)
Chemical potentials are defined via variational derivatives of the total free energy density:
$$\begin{aligned}
{2}
\mu_1 &= \frac{\delta \psi}{\delta u_1} &&= \frac{\partial\widetilde{\psi}}{\partial u_1} - \kappa_1\nabla^2 u_1
\label{eq:chempot1}\\
\mu_2 &= \frac{\delta \psi}{\delta u_2} &&= \frac{\partial\widetilde{\psi}}{\partial u_2} - \kappa_2\nabla^2 u_2
\label{eq:chempot2}\end{aligned}$$
The governing equations are first-order in time and of conservation form
$$\begin{aligned}
{3}
\frac{\partial u_i}{\partial t} &= M_i\nabla^2\mu_i &&\text{in} &&\Omega\times[0,T]\\
&= M_i(\nabla^2\frac{\partial\widetilde{\psi}}{\partial u_i} - \kappa_i \nabla^4 u_i) &&\text{in} &&\Omega\times[0,T]\label{eq:CHgoveq}\\
\nabla\left(\frac{\partial\widetilde{\psi}}{\partial u_i} - \kappa_i\nabla^2 u_i\right)\cdot\boldsymbol{n} &=0 &&\text{on} &&\partial\Omega\times[0,T]\label{eq:CHzeroflux}\\
\nabla u_i\cdot\boldsymbol{n} &= 0 &&\text{on} &&\partial\Omega\times[0,T]\label{eq:CHhodirichlet}\end{aligned}$$
for $i = 1,2$. The surface normal is $\boldsymbol{n}$. The coefficients defining $\widetilde{\psi}$ in Equations and , and the mobilities, $M_1, M_2$ in the governing equations appear in Table \[tbl:ch3well2field\].
Parameter $d$ $s$ $\kappa_1$ $\kappa_2$ $M_1$ $M_2$
----------- ------- ------- ------------- ------------ ------- -------
Value $0.4$ $0.7$ $1$ or $10$ $1$ $0.1$ $0.1$
: Parameters for the two-species phase-separation problem.
\[tbl:ch3well2field\]
### Graphs constructed on dynamic states indexed by time {#sec:CHgraphs}
Figure \[fig:dissip2-comp\] shows graphs, $G_1$ on the left and $G_2$ on the right, which represent distinct IBVPs modelled by Equations (\[eq:CHfreeenergy\]–\[eq:CHhodirichlet\]). A state of the system can be defined as the vector $\mathscr{S}_i = (\theta_{1_i}, \theta_{2_i}, \theta_{3_i}, \Pi_i, \Gamma_i, \Gamma_{12_i})$, corresponding to time $t = t_i$. Here, $\theta_{1_i},\theta_{2_i},\theta_{3_i}$ are the volumes of the phases corresponding to the minima at $(0.35,0.61), (-0.35,0.61)$ and $(0,-0.7)$, respectively, in the $u_1-u_2$ plane, $\Pi_i = \int_\Omega \psi_i\mathrm{d}V$ is the total free energy of the state, $\Gamma_i$ is the total interfacial energy of the state and $\Gamma_{12_i}$ is the interfacial energy between the $\theta_{1}$ and $\theta_2$ phases in that state. The edges, $\mathscr{T}_{ij}$ are defined by the nonlinear, time-stepping solution scheme as proposed in Section \[sec:diss-dynamics-properties\]. In Figure \[fig:dissip2-comp\] the phases with volumes $\theta_{1},\theta_2$ and $\theta_3$ appear in green, yellow and red, respectively. The initial conditions in states $\mathscr{S}_0$ are randomized for each graph and appear as they were computed. The dissipation in this problem is manifested in a decreasing free energy density, $\Pi$ as the system progresses through time-parameterized states as illustrated in Figure \[fig:dissip2-comp\]. Since $\dot{\Pi}\le 0$, $\Pi$ itself is an entropy quantity, and renders the graphs directed following the discussion of properties of such systems in Section \[sec:diss-dynamics-properties\]. Since the free energy, $\Pi_i$, of a state, $\mathscr{S}_i$, depends on the path taken to reach the state, it follows that a path, starting at $\mathscr{S}_i$, but not following exactly the sub-path $\mathscr{S}_i\rightarrow\mathscr{S}_j\rightarrow\mathscr{S}_k\rightarrow\mathscr{S}_l$ already present in the graph, cannot arrive at $\mathscr{S}_l$. At least the value of $\Pi$ will differ. The five states shown in Figure \[fig:dissip2-comp\] for each graph at $t = t_0,t_1,t_2,t_3,t_4$ are only for illustrative purposes. Our computations yielded many more states: $\vert V_1\vert= 1125$ and $\vert V_2\vert = 1252$.
![Graphs, $G_1$ (left) and $G_2$ (right) representing computations on two IBVPs of the Cahn-Hilliard equation with distinct, randomized initial conditions and differing gradient (interface) energy parameters. Separation is seen into three phases, but corresponding states in each graph are different due to the distinct initial conditions and gradient energy parameters.[]{data-label="fig:dissip2-comp"}](figures/graphs-dissip2-comp.pdf)
Because the graphs $G_1$ and $G_2$ were generated from the system of partial differential equations (\[eq:CHfreeenergy\]–\[eq:CHhodirichlet\]), which impose a sequence of states by their dissipative character, these directed graphs are linear, unbranched trees. Their unweighted adjacency matrix entries are $$A_{ij} =\begin{cases}
1 & \text{if}\; j = i-1\\
0 & \text{otherwise}
\end{cases}$$ yielding $\boldsymbol{A}_1$ and $\boldsymbol{A}_2$ that are lower sub-diagonal, unlike the symmetric adjacency matrix for the strain states graph (see Section \[sec:elastgraphlayouts\]). Because of their simplicity, there is not much insight to be gained by analyzing these graphs. However, having constructed the graphs by extracting the relatively low-dimensional state vectors $\mathscr{S}_i \in \mathbb{R}^6$ from the original finite element computations on $\sim 10^6$ degrees of freedom (appearing in Figure \[fig:dissip2-comp\]), we are afforded other approaches to study the system.
### Dynamics of low-dimensional states {#sec:CHevolstates}
The differences that can be seen between corresponding states, say $\mathscr{S}_2$, on $G_1$ and $G_2$ in Figure \[fig:dissip2-comp\] arise from different values of $\kappa_1$ used in Equation to generate the respective graphs. Recall that the coefficients $\kappa_1$ and $\kappa_2$ impose penalties on gradients of $u_1$ and $u_2$, respectively, thereby introducing interfacial energy densities in Equation . By increasing $\kappa_1$ by an order of magnitude (from $1$ to $10$), interfaces between the phases with volumes $\theta_1$ and $\theta_2$ (green and yellow) are made more sharply unfavorable in the computations that generate $G_2$. This leads to the shorter total interface length between these phases, which is apparent on comparing the states $\mathscr{S}_2,\mathscr{S}_3$ and $\mathscr{S}_4$ across graphs $G_1$ and $G_2$ in Figure \[fig:dissip2-comp\]. It is instructive to study the divergence of the system dynamics in these two cases via the state quantities $\mathscr{S}_i = (\theta_{1_i}, \theta_{2_i}, \theta_{3_i}, \Pi_i, \Gamma_i, \Gamma_{12_i})$ represented in the graph vertices.
Figure \[fig:theta\] shows the volumes $\theta_1,\theta_2$ and $\theta_3$ evolving with time.[^8] At the final time of $t = 2162$ for the IBVP represented by $G_1$, these quantities are $\theta_1 = 1070, \theta_2 = 1068$ and $\theta_3 = 1062$ for a mean $\theta_\text{m} = 1066.67$. At the same time for the longer running computation represented by $G_2$, the corresponding values are $\theta_1 = 1055, \theta_2 = 1069$ and $\theta_3 = 1077$ with a mean $\theta_\text{m} = 1067$. The deviation of the curve of $\theta_1$ from $\theta_2$ and $\theta_3$ for $G_2$ appears to be driven by the specific instance of randomized initial conditions, especially given that the higher $\kappa_1$ for the corresponding IBVP introduces a mechanism that does not favor $\theta_1$ or $\theta_2$ relative to each other. We conclude, therefore, that the phase volumes are not the ideal quantities for conveying insight to the macroscopic evolution of the system.
![Evolution of phases $\theta_1,\theta_2,\theta_3$ for graphs $G_1$ (left) and $G_2$ (right).[]{data-label="fig:theta"}](figures/equal_theta-time.png "fig:") ![Evolution of phases $\theta_1,\theta_2,\theta_3$ for graphs $G_1$ (left) and $G_2$ (right).[]{data-label="fig:theta"}](figures/unequal_theta-time.png "fig:")
The free energies in Figure \[fig:CHenergies\] offer more insight. We first note that the total free energy, $\Pi$, is decreasing for all except very early times.[^9] The distinction between $G_1$ and $G_2$ is brought out by the positive definite interfacial free energies, especially $\Gamma_{12}$, which corresponds to the $\theta_1-\theta_2$ phase interface: At $t = 2162$ this quantity for $G_1$ is $\Gamma_{12} = 452$, and for $G_2$, it is $\Gamma_{12} = 77$. Related differences are seen in the total interfacial free energy, $\Gamma$, which includes a contribution from $\Gamma_{12}$, The physical relevance is clear: the higher penalty imposed on $\theta_1-\theta_2$ interfaces by $\kappa_1 = 10$ decreases their total length in the computations represented in $G_2$ (see the green-yellow interfaces in Figure \[fig:dissip2-comp\]) leading to a significantly lower interfacial energy.
![Evolution of total free energy, $\Pi$, total interfacial energy, $\Gamma$ and $\theta_1-\theta_2$ interfacial energy $\Gamma_{12}$ for graphs $G_1$ (left) and $G_2$ (right).[]{data-label="fig:CHenergies"}](figures/equal_energy-time.png "fig:") ![Evolution of total free energy, $\Pi$, total interfacial energy, $\Gamma$ and $\theta_1-\theta_2$ interfacial energy $\Gamma_{12}$ for graphs $G_1$ (left) and $G_2$ (right).[]{data-label="fig:CHenergies"}](figures/unequal_energy-time.png "fig:")
### Dissipative dynamics as an organizing principle for graph generation {#sec:CHorgprinciples}
It is noteworthy that, while the physics of the first-order dissipative dynamical system in Equations (\[eq:CHfreeenergy\]–\[eq:CHhodirichlet\]) may be considered more complex than the stationary system in Equations (\[eq:defgrad\]-\[eq:freeenergy\]) by some measures, the complexity of the resulting graphs is reversed. The non-branching, time-series dynamics and thermodynamic dissipation jointly impose an organizing principle, so that the graphs representing the IBVPs of the Cahn-Hilliard equation are linear, directed trees. An organizing principle, however, needs to be imposed on the states of stationary systems. By identifying the nonlinear solution scheme between strain states as an organizing principle that induces edges, the corresponding graphs are imbued with rich structure as elucidated in Section \[sec:nonconvexelasticity\]. The graphs induced by first-order dynamics could have greater complexity if, instead of a single sequence of states dictated by the temporal evolution of the dynamics, different solution paths were explored over time as branches from some chosen states. The graphs $G_1$ and $G_2$, however, would remain trees; cliques and cycles would still be absent.
Graphs constructed on the states of a dissipative dynamical system without time series data {#sec:DNS-MLgraphs}
-------------------------------------------------------------------------------------------
We continue to explore the role of thermodynamic dissipation as an organizing principle for graph generation. However, we consider states that have not been indexed by time, unlike the case in Section \[sec:CHdynamicsgraphs\]. As we demonstrate, the second law of thermodynamics leads to a transition quantity that induces edges between states to construct graphs. When considered in light of the properties identified in Section \[sec:diss-dynamics\], the corresponding graphs are richer than those in Section \[sec:CHdynamicsgraphs\].
### First-order dynamics of a phase transforming binary alloy system
Phase field methods such as the Cahn-Hilliard equation, whose states generated the graphs of Section \[sec:CHdynamicsgraphs\], are characterized by their imposition of first-order dynamics to traverse a free energy landscape. The dynamics of phase field methods are subject to the second law’s requirement of non-increasing free energy. Another variant of phase field methods is the Allen-Cahn equation [@AllenCahn1979], which differs from the Cahn-Hilliard equation in having non-conservative dynamics, and representing phase transformations by nucleation and growth of precipitates. It too has widely been applied to model precipitate formation during phase transformations in binary alloy systems. The basis of this treatment also lies in a free energy density, now defined as a function of composition, $c$, an order parameter, $\eta$, its gradient $\nabla\eta$ and the elastic deformation gradient, $\boldsymbol{F}^\text{e}$, [@Kim1999; @jietal2014] $$\psi(c,\eta,\nabla\eta,\boldsymbol{F}^\text{e}) = \psi_\text{c}(c,\eta) + \psi_\text{grad}(\nabla\eta) + \psi_\text{e}(\boldsymbol{F}^\text{e}(\eta,\boldsymbol{F}),\eta)
\label{eq:Pi}$$ where $\psi_\text{c}(c,\eta)$ is the local chemical free energy density, $\psi_\text{grad}(c,\eta)$ is the gradient energy term, and $\psi_\text{e}(\boldsymbol{F}^\text{e}(\eta,\boldsymbol{F}),\eta)$ is the elastic strain energy density. The deformation gradient, $\boldsymbol{F}$, reappears, and its elastic component is $\boldsymbol{F}^\text{e}$, which is defined below. We use $$\begin{aligned}
\psi_\text{c}(c,\eta) &= \psi_\text{c}^\alpha(c^\alpha)\left(1-h(\eta)\right)
+\psi_\text{c}^{\beta^\prime}(c^{\beta^\prime})h(\eta)
+\omega \widetilde{\psi}(\eta)\label{eq:chemfreeenergy}\\
h(\eta) &= 3\eta^2 - 2\eta^3\\
\widetilde{\psi}(\eta) &= \eta^2-2\eta^3+\eta^4 \label{eqn:Landau}\end{aligned}$$ Here, $\psi_\text{c}$ is written for a Mg-Y alloy in terms of contributions from the matrix phase, $\alpha$ (Mg) and precipitate phase, $\beta^\prime$ (Mg-Y). The regularized Heaviside function $h(\eta)$ interpolates smoothly between the phases, using $h(0) = 0$, $h(1) = 1$, and $h'(0) = h'(1) = 0$. As seen from Equation the $\alpha$ phase corresponds to $\eta = 0$, and the $\beta^\prime$ phase to $\eta = 1$. The Landau free energy density, $\widetilde{\psi}$, drives the structural change in the alloy, and has wells for the $\alpha$ and $\beta^\prime$ phases, respectively, at $\eta = 0$ and $\eta = 1$. The functions $\psi_\text{c}^\alpha(c^\alpha)$ and $\psi_\text{c}^{\beta^\prime}(c^{\beta^\prime})$ are written as quadratic approximations: $$\begin{aligned}
\psi_\text{c}^\alpha(c^\alpha) &\approx A^\alpha(c^\alpha - c^\alpha_0)^2 + B^\alpha \label{eqn:f_alpha}\\
\psi_\text{c}^{\beta^\prime}(c^{\beta^\prime}) &\approx A^{\beta^\prime}(c^{\beta^\prime} - c^{\beta^\prime}_0)^2 + B^{\beta^\prime} \label{eqn:f_beta}\end{aligned}$$ where the parameters appear in Table \[tab:chem2\]. The quantities $c^\alpha$ and $c^{\beta^\prime}$ are obtained from constraint conditions on chemical potentials and the alloy composition, and have the forms: $$\begin{aligned}
c^\alpha &= \frac{A^{\beta^\prime}\left[c - c^{\beta^\prime}_0h(\eta)\right] + A^\alpha c^\alpha_0h(\eta)}{A^\alpha h(\eta) + A^{\beta^\prime}(1 - h(\eta))} \label{eqn:calpha}\\
c^{\beta^\prime} &= \frac{A^\alpha\left[c - c^\alpha_0(1-h(\eta))\right] + A^{\beta^\prime} c^{\beta^\prime}_0(1-h(\eta))}{A^\alpha h(\eta) + A^{\beta^\prime}(1 - h(\eta))} \label{eqn:cbeta}\end{aligned}$$ The resulting local free energy density, $\psi_\text{c}$, appears in Figure \[fig:localFreeEnergy\].
![Plot of the local chemical free energy density, $\psi_\text{c}$ , in the Mg-Y alloy for transformations between $\alpha$ (Mg) and $\beta^\prime$ (Mg-Y) phases. The wells at $\eta = 0$ and $\eta = 1$ correspond to the $\alpha$ and $\beta^\prime$ phases.[]{data-label="fig:localFreeEnergy"}](figures/localFreeEnergy.pdf){width="55.00000%"}
---------------------- --------- -----
$A^\alpha$ 6.2999 GPa
$B^\alpha$ -1.6062 GPa
$c^\alpha_0$ 0.2635
$A^{\beta^\prime}$ 704.23 GPa
$B^{\beta^\prime}$ -1.5725 GPa
$c^{\beta^\prime}_0$ 0.1273
---------------------- --------- -----
: Parameters in the quadratic chemical free energy density descriptions and .
\[tab:chem2\]
The gradient energy term is defined via a second order tensor $\boldsymbol{\kappa}$ $$\begin{aligned}
\psi_\text{grad}(\nabla\eta) &= \frac{1}{2}\nabla\eta\cdot\boldsymbol{\kappa}\nabla\eta
\label{eqn:grad_energy}\end{aligned}$$ and is anisotropic if $\boldsymbol{\kappa}$ is an anisotropic tensor. The components of $\boldsymbol{\kappa}$ are related to the barrier height $\omega$, the interface thickness, and the anisotropic interfacial energies based on the equilibrium solution for the one-dimensional problem and neglecting elasticity. Details of these approximations appear in the work of Kim et al. [@Kim1999] and Teichert & Garikipati [@Teichert2018], while the interfacial energies themselves were computed and reported by Liu et al. [@Liu2013]. Following Teichert & Garikipati [@Teichert2018] we use $$\begin{aligned}
\boldsymbol{\kappa} =
\begin{bmatrix}
0.1413 & 0 & 0\\
0 & 0.002993 & 0\\
0 & 0 & 0.1197
\end{bmatrix},\;\omega = 0.115896\end{aligned}$$
The strain energy density function is an anisotropic St. Venant-Kirchhoff model. The elasticity constants are modeled as dependent on the order parameter to represent the difference in elasticity between the two phases. The total strain energy of the precipitate-matrix system is driven by a strain mismatch between the crystal structures of the matrix, $\alpha$-, and precipitate, $\beta^\prime$-phases. The stress-free transformation tensor of the $\beta^\prime$ precipitate, $\boldsymbol{F}_{\beta^\prime}$ (see Table \[tab:eigenstrain\]), and the order parameter determine the misfit eigenstrain, represented by $\boldsymbol{F}^\lambda$.
$c_{\mathrm{Y}} = 0.125$
------------------------- --------------------------
$F_{\beta^\prime_{11}}$ 1.0307
$F_{\beta^\prime_{22}}$ 1.0196
$F_{\beta^\prime_{33}}$ 0.9998
: Components of the deformation gradient tensor, representing the eigenstrain in the Mg-Y $\beta^\prime$ precipitate [@Natarajan2017].
\[tab:eigenstrain\]
The multiplicative decomposition of the total deformation gradient into elastic and the misfit components is also defined:
$$\begin{aligned}
\psi_\text{e}(\boldsymbol{F}^\text{e}(\eta,\boldsymbol{F}),\eta) &= \frac{1}{2}\boldsymbol{E}^\text{e}:(\mathbb{C}^\alpha(1-h(\eta) + \mathbb{C}^{\beta^\prime}h(\eta)):\boldsymbol{E}^\text{e}\label{eqn:SVK}\\
\boldsymbol{E}^\text{e} &= \frac{1}{2}\left({\boldsymbol{F}^\text{e}}^\text{T}\boldsymbol{F}^\text{e} - \boldsymbol{1}\right)\\
\boldsymbol{F}^\text{e}(\eta,\boldsymbol{F}) &= \boldsymbol{F}{\boldsymbol{F}^\lambda}^{-1}(\eta)\label{eqn:FeFlam}\\
\boldsymbol{F}^\lambda(\eta) &= \boldsymbol{1}(1-h(\eta)) + \boldsymbol{F}_{\beta^\prime}h(\eta)\end{aligned}$$
where $\boldsymbol{1}$ is the second-order isotropic tensor. The components of the matrix phase elasticity tensor $\mathbb{C}^\alpha$ were calculated by Ji and co-workers [@jietal2014], and correspond well with experimental data. The components of the precipitate phase elasticity tensor, $\mathbb{C}^{\beta^\prime}$ were obtained by density functional theory (see Table \[tab:elasticity\]).
Mg ($\alpha$) $\beta^\prime$
--------------------- --------------- ----------------
$\mathbb{C}_{1111}$ 62.6 78.8
$\mathbb{C}_{2222}$ 62.6 62.9
$\mathbb{C}_{3333}$ 64.9 65.6
$\mathbb{C}_{1122}$ 26.0 24.6
$\mathbb{C}_{2233}$ 20.9 19.9
$\mathbb{C}_{3311}$ 20.9 23.1
$\mathbb{C}_{1212}$ 18.3 11.9
$\mathbb{C}_{2323}$ 13.3 11.6
$\mathbb{C}_{3131}$ 13.3 8.46
: Elasticity constants used for the Mg matrix [@jietal2014] and the $\beta^\prime$ precipitate (calculated by Anirudh Natarajan, unpublished data) (GPa).
\[tab:elasticity\]
The phase field dynamics are modeled by the diffusion and Allen-Cahn equations, of the following forms [@AllenCahn1979; @Kim1999], posed on the reference configuration, $\Omega$:
$$\begin{aligned}
{3}
\frac{\partial c}{\partial t} &= -\nabla\cdot\boldsymbol{J} &&\text{in} &&\Omega\times [0,T]
\label{eq:fick}\\
\frac{\partial \eta}{\partial t} &= -L\mu_\eta &&\text{in} &&\Omega\times [0,T]
\label{eq:AC}\\
\boldsymbol{J}\cdot\boldsymbol{n} &= 0 &&\text{on} &&\partial\Omega\times [0,T]\\
\nabla\eta\boldsymbol{\kappa}\cdot\nabla\eta &= 0 &&\text{on} &&\partial\Omega\times [0,T]\label{eq:ACbc}\end{aligned}$$
where the flux is defined by $\boldsymbol{J} := -M\nabla\mu_c$, $M$ is the mobility, and $L$ is a kinetic coefficient. The chemical potentials $\mu_c = \delta \Pi/\delta c$ and $\mu_\eta = \delta \Pi/\delta \eta$ are found using standard variational methods, giving the following expressions when assuming $\nabla\eta\cdot\boldsymbol{\kappa}\boldsymbol{n} = 0$ on $\partial \Omega$ (resulting from requiring equilibrium with respect to $\eta$ at the boundary, $\partial\Omega$): $$\begin{aligned}
\mu_c &= \frac{\partial \psi_\text{c}^\alpha}{\partial c}\left(1-h(\eta)\right)+\frac{\partial \psi_\text{c}^{\beta^\prime}}{\partial c}h(\eta)\\
\mu_\eta &= \left[\psi_\text{c}^{\beta^\prime} - \psi_\text{c}^\alpha - \mu_c(c^{\beta^\prime} - c^\alpha)\right]\frac{\partial h}{\partial \eta} - \nabla\cdot\boldsymbol{\kappa}\nabla\eta + \omega\frac{\partial \widetilde{\psi}}{\partial \eta} + \frac{\partial \psi_\text{e}}{\partial \eta}\end{aligned}$$ where $$\begin{aligned}
\frac{\partial \psi}{\partial \eta} &= \left(\frac{1}{2}\boldsymbol{E}:(\mathbb{C}^{\beta^\prime} - \mathbb{C}^\alpha):\boldsymbol{E} -
\boldsymbol{P}:\left(\boldsymbol{F}^\text{e}(\boldsymbol{F}^{\beta^\prime} -\boldsymbol{1})\boldsymbol{F}^{\lambda^{-1}}\right)\right)\frac{\partial h}{\partial \eta}\end{aligned}$$ The phase field equations (\[eq:fick\]-\[eq:ACbc\]) are coupled with nonlinear elasticity posed in terms of the first Piola-Kirchhoff stress, $\boldsymbol{P} = \partial\psi_\text{e}/\partial \boldsymbol{F}^\text{e}$ on the reference configuration $\Omega$:
$$\begin{aligned}
\mathrm{Div}\left(\boldsymbol{P}{\boldsymbol{F}^\lambda}^{-\mathsf{T}}\right) &= \boldsymbol{0} \text{ in } \Omega
\label{eq:elaststrongformgov}\\
\left(\boldsymbol{P}{\boldsymbol{F}^\lambda}^{-\mathsf{T}}\right)\boldsymbol{N} &= \boldsymbol{0} \text{ on } \partial\Omega_{0_T}: \;\text{Neumann boundary conditions}\\
\boldsymbol{u}\cdot\boldsymbol{N}&=0 \text{ on }\partial\Omega_{0_u}: \; \text{Dirichlet boundary conditions}
\label{eq:elaststrongformbc}\end{aligned}$$
### Graph generation by states of the binary alloy system without indexing by time
Traditionally, studies of precipitate formation, some examples of which have been referred to above [@Kim1999; @Liu2013; @jietal2014; @Teichert2018], have solved the phase field equations (\[eq:fick\]-\[eq:ACbc\]) with or without elastic effects (\[eq:elaststrongformgov\]–\[eq:elaststrongformbc\]). Motivated by the search for equilibrium precipitate shapes resulting from phase transformations, we have recently exploited the principle of energy minimization as an alternative to phase field dynamics in a binary alloy system. We define the states of the system as $\mathscr{S}_i = (a_i,b_i,c_i,t_{1_i},\dots,t_{8_i},c_{\text{p}_i},\Pi_i)$. Here, $a_i,b_i$ and $c_i$ define the dimensions of a rectangular prism that bounds the precipitate, $t_{1_i},\dots,t_{8_i}$ are parameters that define the control points in a spline basis for the precipitate shape, $c_{\text{p}_i}$ is the alloy concentration and as in Sections \[sec:nonconvexelasticity\] and \[sec:CHdynamicsgraphs\] $\Pi_i = \int_{\Omega}\psi_i\mathrm{d}V$ is the total free energy of the state. Using direct numerical simulation, we obtain $\sim \mathcal{O}(10^5)$ states of the system for a single precipitate. Our approach is to combine machine learning representations, sensitivity analysis and surrogate optimization to find a local minimum of $\Pi$, which corresponds to an equilibrium configuration for the precipitate-matrix system. Figure \[fig:DNS-MLppt\] shows the energy-minimizing precipitate geometry at four stages from a sequence that converges toward a local minimum using this approach. Details of the methods, convergence to minima, comparisons with experiment and computational cost tradeoffs have been presented elsewhere [@Teichert2018].
![The energy-minimizing precipitate geometry at each stage from a sequence that converges toward a local minimum using an approach that draws on machine learning, sensitivity analysis and surrogate optimization.[]{data-label="fig:DNS-MLppt"}](figures/DNSML_iter0000){width="70.00000%"}
![The energy-minimizing precipitate geometry at each stage from a sequence that converges toward a local minimum using an approach that draws on machine learning, sensitivity analysis and surrogate optimization.[]{data-label="fig:DNS-MLppt"}](figures/DNSML_iter0001){width="70.00000%"}
![The energy-minimizing precipitate geometry at each stage from a sequence that converges toward a local minimum using an approach that draws on machine learning, sensitivity analysis and surrogate optimization.[]{data-label="fig:DNS-MLppt"}](figures/DNSML_iter0002){width="70.00000%"}
![The energy-minimizing precipitate geometry at each stage from a sequence that converges toward a local minimum using an approach that draws on machine learning, sensitivity analysis and surrogate optimization.[]{data-label="fig:DNS-MLppt"}](figures/DNSML_iter0003){width="70.00000%"}
The governing equations and induce a dissipative character to the IBVP. For this reason, graphs developed from states computed by these equations supplemented by the elasticity equations (\[eq:elaststrongformgov\]–\[eq:elaststrongformbc\]) would be isomorphic to those in Section \[sec:CHgraphs\] and Figure \[fig:dissip2-comp\]. In particular, a tree with no branches would result for each IBVP solved. In contrast, the pre-computed states introduced above, $\mathscr{S}_i = (a_i,b_i,c_i,t_{1_i},\dots,t_{8_i},c_{\text{p}_i},\Pi_i)$, have not been connected by edges defined by a nonlinear time-stepping solution scheme, as was the case in Section \[sec:CHgraphs\]. This presents other approaches for edge definition by a transition quantity and exploration of the physics via graphs. It is another instantiation of exploration and analysis by graph principles already demonstrated in Section \[sec:nonconvexelasticity\].
### Graph completion and exploration principles induced by first-order dissipative dynamics
**Graph chemical potentials and edge definition via a transition quantity**: As a first step, we linearly transform the components $(a_i,\dots,c_{\text{p}_i})$ to each lie in $[0,1]$ and denote this sub-vector as $\boldsymbol{\Xi}_i$. The state can then be written as $\mathscr{S}_i = (\boldsymbol{\Xi}_i,\Pi_i)$. We observe that $\mathscr{S} \in \mathbb{R}^{13}$ is an effective, lower dimensional representation of the state than the spatio-temporal field $\boldsymbol{\zeta} := (c,\eta,\boldsymbol{u})$ in a direct numerical simulation that has $\sim \mathcal{O}(10^6)$ degrees of freedom. Denoting the changes in states between $\mathscr{S}_j$ and $\mathscr{S}_i$ by $\Delta \boldsymbol{\Xi}_{ij} = \boldsymbol{\Xi}_j - \boldsymbol{\Xi}_i$, and $\Delta\Pi_{ij} = \Pi_j - \Pi_i$, the term $$\mu_{\Xi_{ij}} := \Delta \Pi_{ij}/\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert
\label{eq:graphchempot1}$$ becomes a generalized chemical potential defined on the graph.
Returning to the task of defining edges between the vertices (states), we note that the principle of maximum dissipation of free energy between states (alternately, a steepest gradient principle) provides one criterion: Given a state $(\boldsymbol{\Xi}_j,\Pi_j)$, we introduce a directed edge to a subsequent state $(\boldsymbol{\Xi}_i,\Pi_i)$ provided the graph chemical potential, $\mu_{\Xi_{ij}} \le 0$ and $\mathscr{S}_i$ minimizes $\mu_{\Xi_{ij}}$ over pairs $\{\mathscr{S}_j,\mathscr{S}_k\}$. Formally stated, $$\text{Given}\;\mathscr{S}_j,\,\exists \mathscr{T}_{ij}\,\text{iff}\, \mu_{\Xi_{ij}} \le 0, \;\text{and}\; \mu_{\Xi_{ij}} = \min\limits_{k} \mu_{\Xi_{kj}}
\label{eq:graphchempot2}$$
Accordingly, we place a directed edge, $\mathscr{T}_{ij}$ from $\mathscr{S}_j$ to $\mathscr{S}_i$. The graph chemical potential in Equation thus is a transition quantity as discussed in Section \[sec:diss-dynamics-properties\] and defines edges via Equation with maximum dissipation as an organizing principle. The resulting tree graph is shown in Figure \[fig:ppt\_dist1\] in a circular layout. In this graph, edge weights are defined by Equation as the Euclidean distance between states $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$ and are represented by edge thickness. The vertex areas are proportional to the logarithm of the free energy, $\log\Pi_i$.
![A directed tree graph connecting precipitate states $\mathscr{S}_i = \{\boldsymbol{\Xi}_i,\Pi_i\}$, with edge thickness representing weights defined by Euclidean distance $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, and vertex area proportional to $\log\Pi_i$.[]{data-label="fig:ppt_dist1"}](figures/ppt_distance_weights.pdf){width=".8\textwidth"}
**Most and least favored energy minimization paths on the graph**: With the graph chemical potential as a transition quantity and maximum dissipation as an organizing principle for definition of edges, the minimum energy state can be identified. In Figure \[fig:ppt\_mu\] it is the vertex corresponding to $\mathscr{S}_0$, the state representing the equilibrium precipitate shape. The remaining vertices are numbered in ascending order of energies, $\Pi_i$. Starting at any vertex corresponding to $\mathscr{S}_i \neq \mathscr{S}_0$, a path is immediately traceable whose each edge represents the maximally dissipative transition among all admissible ones. All paths end in $\mathscr{S}_0$, emphasizing this state’s minimum energy property. Paths to $\mathscr{S}_0$ have different numbers of edges–a property that emerges from criterion . Note that this layout was presaged by the graph union $G = G_1\cup G_2$ in Figure \[fig:dissipgraph\]b.
The five steepest paths between states $\mathscr{S}_k$ and $\mathscr{S}_0$, defined by the magnitude of the (negative) total chemical potential, $\mu_{\Xi_{0k}}$, have been highlighted by red colored vertices. These five paths are identified by their leaf nodes $\{\mathscr{S}_{2182},\mathscr{S}_{2184},\mathscr{S}_{2185},\mathscr{S}_{2186},\mathscr{S}_{2187}\}$ arranged in order of decreasing (increasingly negative) $\mu_{\Xi_{0k}}$. The five most gradual paths by this same criterion are in blue, with leaf nodes $\{\mathscr{S}_{26},\mathscr{S}_{41},\mathscr{S}_{44},\mathscr{S}_{56},\mathscr{S}_{64}\}$ arranged in order of decreasing $\mu_{\Xi_{0k}}$. It also is of interest to note the change in precipitate geometry along the steepest and the most gradual paths. Not surprisingly, there are notable changes between the geometric states along the steepest path, and barely discernible changes along the most gradual path. The edge thickness continues to represent the weights defined by the inter-state Euclidean distances, $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, and the vertex area is proportional to $\log\Pi_i$. Exploration of the graph on the basis of $\mu_\Xi$ thus reveals all admissible, as well as most and least favored paths for energy minimization to the equilibrium precipitate shape.
![The directed tree graph highlighting the five steepest (red) and five most gradual (blue) energy paths, defined by the magnitude of the (negative) total chemical potential, $\mu_{\Xi_{0k}}$. On the right, note the changes in geometry of the precipitate between states lying along the steepest path, in comparison to the barely discernible changes along the most gradual path. Edge thickness represents weights defined by Euclidean distance $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, and vertex area is proportional to $\log\Pi_i$.[]{data-label="fig:ppt_mu"}](figures/ppt_smallLarge_totalmu.pdf){width="\textwidth"}
![The directed tree graph highlighting the five steepest (red) and five most gradual (blue) energy paths, defined by the magnitude of the (negative) total chemical potential, $\mu_{\Xi_{0k}}$. On the right, note the changes in geometry of the precipitate between states lying along the steepest path, in comparison to the barely discernible changes along the most gradual path. Edge thickness represents weights defined by Euclidean distance $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, and vertex area is proportional to $\log\Pi_i$.[]{data-label="fig:ppt_mu"}](figures/ppt_steepest.pdf){width="\textwidth"}
**A graph time for edge transition**: We also introduce a time-like scalar $\tau_i$ for each state $\mathscr{S}_i$, and restrict it to vary such that $\text{sgn}(\Delta\tau_{ij}) = -\text{sgn}(\Delta\Pi_{ij})$. We proceed to arrive at estimates for $\Delta\tau_{ij}$, guided by the graph chemical potential, $\mu_\Xi$. As with $\mu_\Xi$ this is a notion of “graph time”. Guided by Equation , we first observe that we can define an energy gradient-driven analog to the Allen-Cahn equation on the graph with $\Lambda \ge 0$ now denoting a “kinetic” coefficient on the graph:
$$\begin{aligned}
\frac{\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert }{\Delta \tau_{ij}} &= -\Lambda\mu_{\Xi_{ij}}\nonumber\\
\implies \Delta \tau_{ij} &= -\frac{\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert}{\Lambda\mu_{\Xi_{ij}}}\nonumber\end{aligned}$$
Recalling $\mu_{\Xi_{ij}} = \Delta \Pi_{ij}/\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, where $\Delta \Pi_{ij} \le 0$ we arrive at $$\Delta \tau_{ij} \sim \frac{\Vert\Delta\boldsymbol{\Xi}_{ij}\Vert^2}{\vert\Delta\Pi_{ij}\vert}
\label{eq:statetime}$$ requiring, of course, that $\Delta\Pi_{ij} \neq 0$. Equation suggests that the time to traverse an edge is related to the squared “distance” between states in $\mathbb{R}^{12}$ scaled by the magnitude of the corresponding energy decrease. This is in agreement with the physics of first-order kinetic processes according to which a rate rises with greater energy decreases, or the transition time decreases. Conversely, an increased “distance” between states in $\mathbb{R}^{12}$ decreases the rate, increasing the transition time. With this measure of graph time, we can explore the time required to traverse a leaf-to-root path. Figure \[fig:ppt\_time\] shows the five such paths with greatest time of traversal. It is notable that these are distinct from the five most gradual energy paths in Figure \[fig:ppt\_mu\]: The cumulative effect of transitions across each edge determines this traversal time, rather than the average steepness of the paths. Also shown are the changes in precipitate geometry along the path with the maximum time of traversal. Following Equation , large transition times result from larger changes in geometry, $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, over the first two edges, followed by small energy changes, $\Delta\Pi_{ij}$, over the last two edges. Also note that the weights have now been defined by graph time of transition via Equation , and are represented by edge thickness in Figure \[fig:ppt\_time\].
![The five leaf-to-root paths with the greatest traversal times are plotted with states in red. Here, the edge thickness represents weights defined by graph time of transition of an edge: $\Delta\tau_{ij}$. The path with maximum traversal time is shown with precipitate geometries of the states. Vertex area continues to be scaled by $\log\Pi_i$.[]{data-label="fig:ppt_time"}](figures/ppt_longest_time_num.pdf){width="\textwidth"}
![The five leaf-to-root paths with the greatest traversal times are plotted with states in red. Here, the edge thickness represents weights defined by graph time of transition of an edge: $\Delta\tau_{ij}$. The path with maximum traversal time is shown with precipitate geometries of the states. Vertex area continues to be scaled by $\log\Pi_i$.[]{data-label="fig:ppt_time"}](figures/ppt_slowest.pdf){width="\textwidth"}
**PageRank as a measure of the relative importance of states for energy minimization paths**: The final aspect that we study is centrality of the vertices representing states. The motivation is quite clear: As the graph illustrates in Figures \[fig:ppt\_mu\] and \[fig:ppt\_time\], the vertex representing state $\mathscr{S}_0$ has the highest in-degree, while this measure decreases with increasing neighbor separation from it. Figure \[fig:ppt\_pagerank\] uses the PageRank of the corresponding vertices [@Newman2010], to show the relative importance of states in enabling the dynamic process of energy minimization. Since this measure scales the in-degree by the out-degree, it indicates the states that pool many others while allowing paths to fewer downstream states, thus “focusing” the flow of the dynamic process of energy minimization. The weights are defined by the Euclidean distance between states and are represented by edge thickness. The areas of the vertices are proportional to $\log\Pi_i$.
![The directed tree graph with vertices shaded according to the log of the PageRank, where red is the highest PageRank and blue is the lowest. Edge thickness represents weights defined by distance $\Vert\Delta \boldsymbol{\Xi}_{ij}\Vert$, and vertex area is proportional to $\log\Pi_i$.[]{data-label="fig:ppt_pagerank"}](figures/ppt_logPageRank_num.pdf){width=".8\textwidth"}
Closing remarks {#sec:closingremarks}
===============
We make the case that graph theory offers a framework for representation, exploration and analysis of large scale computed solutions. The fundamental insight required is that high-dimensional field solutions typically admit functional representations of low-dimensional states, which are the vertices of a graph. Transitions between states, which could either be a (nonlinear) solution step, or be defined by the change in a physical property, are the edges of the graph. With this foundation, isomorphisms can be identified between the computational and physical framework of states and transitions on one hand, and graph vertices and edges on the other. In this isomorphism, many properties of the numerical solution procedure and of the dynamical physical system are in correspondence with properties of graphs. This includes standard notions of weights, directedness of edges, connectedness of graphs, components, cliques and cycles. Other correspondences arise when considering specific dynamical systems as graphs.
The framework-setting observations that have been summarized above are rendered concretely in considering four systems. In this regard, the nature of the graphs constructed on non-dissipative elastodynamics and linear elasticity are simple. The approach becomes more profound for the graphs constructed on states of gradient-regularized, non-convex elasticity, and graphs on first-order dissipative dynamical systems with and without time-series data. Here, the reach of the graph theoretic approach is apparent in its introduction of a framework to organize these solutions, and then to explore and analyze them. Some observations are useful in this setting:
- Comparing the three more consequential cases that were recalled above, we note that perhaps the greatest utility of the graph theoretic approach occurs where the data set of solutions needs an organizing principle to be imposed on it. This was the case with the nonlinear solution step inducing edges for the graph of strain states, and the maximum dissipation principle doing the same for the graph on states of the binary alloy system. These graphs are rich in having many branches, and interesting paths, cliques and cycles. Notably, the graph theoretic framework provides insight to the physics of these systems in terms of accessibility of strain states, paths between them, energy minimizing paths, and notions of graph chemical potentials, graph time and several others. In contrast, the graph on the three-phase transforming system indexed by time naturally has a linear, tree graph without branches. In this case, the strict condition of the second law of thermodynamics imposes directedness and barriers to arbitrary introduction of edges (transitions) between vertices (states).
- It is particularly insightful to note how two physical systems, both of which would be solved by imposing first-order dynamics in the traditional computational physics approaches, lead to very different graphs if the dynamics are eliminated in one case. Here, the graph constructed on states of the binary alloy system using maximum dissipation to induce edges between states reveals a rich array of possible paths to energy minimization, which yield to exploration and analysis. In contrast the time series information in the graph on the three phase transforming system has a single path and less room for further exploration.
- It is important to observe that the notion of low-dimensional states underlying the framework here is quite different from dimensionality reduction as it works in proper orthogonal decomposition, tensor decomposition, multi-fidelity modelling and related methods. High-fidelity, and therefore high-dimensional, *field solutions* are assumed here (although the approach works for low-dimensional field solutions, also). The low-dimensional *functionals* that form states are extracted from the high-fidelity, high-dimensional field solutions. These functionals and states are natural, “aggregate” physical quantities relevant to the dynamical physical system. In addition to the average strain, phase volume fractions, various global energy measures, precipitate shape parameters and average compositions employed here, we recall a few others: lift, drag and thrust in fluid dynamics, load at yield and failure strain in solid mechanics, charge state and voltage in electro-chemistry, and the magnetic moment in electro-magnetism.
It will be instructive to extend these graph theoretic ideas, and explore examples more deeply than the preliminary consideration of stationary or dynamical, dissipative or non-dissipative systems in this first communication. Such a broader framework would bring the graph theoretic perspective of computational physics closer to decision science for natural and engineered systems.
Acknowledgements {#acknowledgements .unnumbered}
================
We gratefully acknowledge the support of Toyota Research Institute, Award \#849910, “Computational framework for data-driven, predictive, multi-scale and multi-physics modeling of battery materials"; NSF DMREF grant: DMR1436154, “DMREF: Integrated Computational Framework for Designing Dynamically Controlled Alloy-Oxide Heterostructures"; Sandia National Laboratories via its LDRD mechanism: 746300, “Material Variability Research”; and the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Award \#DE-SC0008637 that funds the PRedictive Integrated Structural Materials Science (PRISMS) Center at University of Michigan. Computing resources were provided in part by the NSF via grant 1531752 MRI: Acquisition of Conflux, A Novel Platform for Data-Driven Computational Physics (Tech. Monitor: Ed Walker). This work also used the Extreme Science and Engineering Discovery Environment (XSEDE) Comet at the San Diego Supercomputer Center and Stampede2 at the Texas Advance Computing Center through allocations TG-MSS160003 and TG-DMR180072.
[^1]: Corresponding author, [[email protected]]{}
[^2]: For elasticity posed as a stationary problem, $\mathscr{S}_j$ and $\mathscr{S}_i$ also admit the added physical interpretations of reference and current, or deformed, states (configurations), respectively. However, for other problems, such as steady heat conduction only the numerical interpretations of initial guess ($\mathscr{S}_j$) and converged solution ($\mathscr{S}_i$) are available.
[^3]: The Neumann boundary condition, is notably more complex than its conventional counterpart, which would have only the first term on the left hand-side. Equation is the higher-order Dirichlet boundary condition applied to the normal gradient of the displacement field, and Equation is the higher-order Neumann boundary condition on the higher-order stress, $\boldsymbol{B}$. Adopting the physical interpretation of $\boldsymbol{B}$ as a couple stress [@Toupin1962], the homogeneous form of this boundary condition, if extended to the atomic scale, states that there is no boundary mechanism to impose a generalized moment across atomic bonds.
[^4]: Of course, because of the microstructure of martensitic variants, and the anisotropy induced by it, the actual deformation gradient, $\boldsymbol{F}$ is non-uniform, but satisfies $\frac{1}{\text{meas}(\Omega)}\int_\Omega \boldsymbol{F}\text{d}V = \widehat{\boldsymbol{F}}$.
[^5]: We found that if the nonlinear solver fails to converge for transition $\mathscr{S}_l\rightarrow \mathscr{S}_m$, it also fails for $\mathscr{S}_m\rightarrow \mathscr{S}_l$.Then, both the edges $\mathscr{T}_{ml}$ and $\mathscr{T}_{lm}$ do not exist.
[^6]: The minimum eigenvalue of $\boldsymbol{L}$ is zero.
[^7]: This finely corrugated energy surface arises from the non-convex component, $\widetilde{\psi}$ of the free energy density in Equation .
[^8]: The dynamics have been shown with respect to time rather than states in order to allow a physically meaningful comparison between the graphs. Because different time steps were used in the computations, states indexed by the same number in $G_1$ and $G_2$ do not necessarily correspond to the same time in the dynamics, unlike the schematic illustration in Figure \[fig:dissip2-comp\].
[^9]: The free energy $\Pi$ does show increases at very early times $t \le 2.9$ for $G_1$ and $t \le 20$ for $G_2$. This is a numerical artifact that is not germane to the graph theoretic methods being proposed in this communication. The backward Euler time integration scheme used for this nonlinear IBVP does not guarantee a non-increasing free energy. The very rapid dynamics of spinodal decomposition, which occurs at early times in this problem requires much smaller time steps to ensure non-increasing free energy than were taken for this computation.
|
---
author:
- Jacob Szeftel
- Nicolas Sandeau
- Michel Abou Ghantous
title: The Josephson Effect Revisited
---
$\textbf{1}$-Introduction
=========================
The Josephson effect was initially observed[@and; @sha] in the kind of circuit sketched in Fig.\[jos1\] and has kept arousing an unabated interest, in particular because of its relevance to quantum computation[@dou; @dev; @ydev] and more fundamental issues[@koi; @koiz]. For simplicity, both superconducting leads $A,B$ are assumed here to be made out of the same material. They are separated by a thin ($<10^{-9}m$) insulating film, enabling electrons to tunnel through it. If $A,B$ were made of a normal metal, a constant current $I=\frac{U_s}{R+R_t}$ would flow through the circuit. Nevertheless, this simple setup has attracted considerable attention because of Josephson’s predictions[@jos]:
1. there should be $\left\langle I\right\rangle\neq 0$ for $\left\langle U\right\rangle=0$ ($\left\langle I\right\rangle,\left\langle U\right\rangle)$ refer to time $t$ averaged values of $I(t),U(t)$);
2. $I(t),U(t)$ should oscillate at frequency $\omega=\frac{2e\left\langle U\right\rangle}{\hbar}$ with $e$ being the electron charge.
However the characteristic $I(U)$, reproduced in Fig.\[row\], indicates rather $\left\langle I\right\rangle(\left\langle U\right\rangle=0)=0$ with $\frac{d\left\langle I\right\rangle}{d\left\langle U\right\rangle}(\left\langle U\right\rangle=0)> 0$. Likewise, since the origin $\left\langle I\right\rangle=\left\langle U\right\rangle=0$ is not indicated in Fig.\[sha1\] and the accuracy is poor, claim $1$ cannot be validated on the basis of the experimental data. Besides, a periodic signal was indeed observed[@sha], but in the $RF$ range, i.e. $\omega<100MHz$, rather than in the microwave one, i.e. $\omega>1GHz$, as inferred from Josephson’s formula, given the measured $\left\langle U\right\rangle$ values, which does not buttress the validity of $2$ either. Consequently, the huge trove of experimental data, documenting the electrodynamical behaviour of the Josephson junction, have been interpreted so far by resorting[@mcc] to an *empirical* formula, relating $I(t),U(t)$ to Ginzburg and Landau’s phase $\Phi_{GL}(t)$, introduced[@gin] to describe the behaviour of the persistent current on a *phenomenological* basis. Unfortunately, $\Phi_{GL}$ is *not* observable and the $\Phi_{GL}$ based formula has long been recognised[@mcc] to account poorly for the observed characteristic $I(U)$. Consequently, the observations, made on the Josephson junction, namely the above mentioned $RF$ signal, the microwave assisted effect[@sha] and the negative resistance behaviour[@sha; @mcc], are still ill-understood, so that a *physical* explanation of the Josephson effect is needed. Therefore, this purpose will be tentatively achieved below by studying the time-periodic tunneling motion[@schi] of bound electron pairs[@sz4; @sz5] through the insulating barrier.
The outline is as follows : the expression of the tunneling current, conveyed by the independent electrons, is recalled in section $2$, whereas that of the bound electron current is worked out in section $3$; this enables us to solve, in section $4$, the electrodynamical equation of motion of the circuit, depicted in Fig.\[jos1\]; sections $5,6$ deal respectively with the microwave mediated Josephson effect and the negative resistance induced signal. The results are summarised in the conclusion.
$\textbf{2}$-Random Tunneling
=============================
As in our previous work[@sz5; @sz1; @sz2; @sz3; @sz4], the present analysis will proceed within the framework of the two-fluid model, for which the conduction electrons comprise superconducting and independent electrons, in respective concentration $c_s,c_n$. The superconducting and independent electrons are organized, respectively, as a many bound electron[@sz5] (MBE), BCS-like[@bcs] state, characterised by its chemical potential $\mu$, and a Fermi gas[@ash] of Fermi energy $E_F$. Assuming $U=U_A-U_B,eU>0$, the current, conveyed by the independent electrons, will flow from $A$ toward $B$ and there is $eU=E_F^A-E_F^B$, with $E_F^A,E_F^B$ being the Fermi energy in electrodes $A,B$, respectively. Hence, since the experiments are carried out at low temperature, the corresponding current density $j_n$ is inferred from the properties of the Fermi gas[@ash] to read
$$\label{jn}
j_n=\frac{e^2\rho(E_F^A)v_FT}{2}U\Rightarrow R_t\propto \frac{1}{\rho(E_F^A)}\quad,$$
with $\rho(E_F),v_F,T$ standing for the one-electron density of states at the Fermi level, the Fermi velocity and the one-electron transmission coefficient through the insulating barrier ($\Rightarrow 0<T<1$). Two remarks are in order, regarding Eq.(\[jn\])
- the independent electrons contribute thence the current $I_n(t)=U(t)/R_t$ to the total current $I(t)$. However, despite $I_n$ obeying Ohm’s law, the tunneling electrons suffer no energy loss inside the insulating barrier;
- because $c_n$ is expected to grow[@sz5] at the expense of $c_s$ with growing $\left| I\right|$, this implies that $\rho(E_F)$ and $R_t$ will, respectively, increase and decrease with increasing $\left| I\right|$.
$\textbf{3}$-Coherent Tunneling
===============================
Unlike the random diffusion of independent electrons across the insulating barrier, the tunneling motion of bound electrons takes place as a time-periodic oscillation to be analysed below. Their energy per unit volume $\mathcal{E}$ depends[@sz4] on $c_s$ only and is related to their chemical potential $\mu$ by $\mu=\frac{\partial\mathcal{E}}{\partial c_s}$. Before any electron crosses the barrier, the total energy of the whole bound electron system, including the leads $A,B$, reads $$\label{wh1}
\mathcal{E}_i=2\mathcal{E}(c_e)+ec_eU\quad,$$ with $c_e$ referring to the bound electron concentration at thermal equilibrium. Let $n>>1$ of bound electrons cross the barrier from $A$ toward $B$. The total energy becomes $$\label{wh2}
\mathcal{E}_f=\mathcal{E}(c_e+\frac{n}{V})+\mathcal{E}(c_e-\frac{n}{V})+e(c_e-\frac{n}{V})U\quad,$$ with $V$ being the volume, taken to be equal for both leads $A,B$. Energy conservation requires $\mathcal{E}_i=\mathcal{E}_f$, which leads finally to $$\label{wh3}
n=\frac{eV}{\frac{\partial\mu}{\partial c_s}(c_e)}U\quad.$$
The twofold degenerate wave-functions $\varphi_i,\varphi_f$, associated with the eigenvalue $\mathcal{E}_i=\mathcal{E}_f$, read $$\label{eig1}
\begin{array}{l}
\varphi_i=\varphi_A(c_e)\otimes\varphi_B(c_e)\\
\varphi_f=\varphi_A(c_e-\frac{n}{V})\otimes\varphi_B(c_e+\frac{n}{V})
\end{array}\quad ,$$ with $\varphi(c_s)$ being the MBE, $c_s$ dependent eigenfunction[@bcs; @sz5]. The coherent tunneling motion of $n$ electrons across the barrier is thence described by the wave-function $\psi(t)$, solution of the Schrödinger equation $$\label{sch1}
\begin{array}{c}
i\frac{\partial\psi}{\partial t}=H\psi\\
H=\omega_t\sigma_x\quad,\quad
\omega_t=\left\langle\varphi_i\left|V_b\right|\varphi_f\right\rangle
\end{array}\quad .$$ The Hamiltonian $H$ and the potential barrier $V_b$, hindering the electron motion through the Josephson junction and including the applied voltage $U$, are expressed in frequency unit, $\frac{V\mathcal{E}_i}{\hbar}$ is taken as the origin of energy, whereas $\psi$ and the Pauli matrix[@abr] $\sigma_x$ have been projected onto the basis $\{\varphi_i,\varphi_f\}$. The tunneling frequency $\omega_t$ is taken to lie in the RF range, i.e. $\omega_t<100MHz$, as reported by Shapiro[@sha]. Finally Eq.(\[sch1\]) is solved[@abr] to yield $$\label{psi1}
\psi(t)= \cos\left(\frac{\omega_t t}{2}\right)\varphi_i-i\sin\left(\frac{\omega_t t}{2}\right)\varphi_f\quad ,$$ whence the charge $Q_s,-Q_s$, piling up in $A,B$ respectively, is inferred, thanks to Eq.\[wh3\], to read $$Q_s(t)=-ne\left|\left\langle\psi(t)|\varphi_f\right\rangle\right|^2=C_eU\sin^2\left(\frac{\omega_t t}{2}\right)\quad,$$ with the effective capacitance $C_e=-\frac{e^2V}{\frac{\partial\mu}{\partial c_s}(c_e)}$. Since $\frac{\partial\mu}{\partial c_s}<0$ has been shown to be a prerequisite for the existence of persistent currents at thermal equilibrium[@sz5; @sz4], it implies that $C_e>0$. In addition, given the estimate[@sz5] of $\frac{\partial\mu}{\partial c_s}$, it may take a very large value up to $C_e\approx 1F$. At last, by contrast with $I_n$ being incoherent, the bound electrons contribute an *oscillating* current $I_s(t)={\dot Q}_s=\frac{dQ_s}{dt}$ to $I(t)$.
$\textbf{4}$-Electrodynamical Behaviour
=======================================
The total current $I(t)$ comprises $3$ contributions, namely $I_n=\frac{U}{R_t},I_s={\dot Q}_s$ and a component $C\dot U$, loading the Josephson capacitor, so that the electrodynamical equation of motion reads $$U_s=U+RI\quad,\quad I=\frac{U}{R_t}+{\dot Q}_s+C\dot U\quad,$$ which is finally recast into $$\label{eqm}
\dot U=\frac{U_s-U\left(1+\frac{R}{R_t}+\frac{RC_e\omega_t}{2}\sin\left(\omega_t t\right)\right)}{R\left(C+C_e\sin^2\left(\frac{\omega_t t}{2}\right)\right)}\quad.$$ It is worth noticing that, due to $\left|\frac{C_e}{C}\right|>>1$, the denominator in the right-hand side of Eq.(\[eqm\]) would vanish for $C_e<0$, at some $t$ value, so that Eq.(\[eqm\]) cannot be solved unless $C_e>0\Rightarrow \frac{\partial\mu}{\partial c_s}<0$, which confirms a *previous*[@sz5; @sz4] conclusion, derived independently.
$\left|\frac{\partial\mu}{\partial c_s}\right|$ is expected[@sz5] to increase with increasing $\left|I\right|$ and to vanish for $\left|I\right|>I_M$, the maximum value of the bound electron current, because the sample goes thereby normal. Consequently for practical purposes, Eq.(\[eqm\]) has been solved by assuming $R_t\left(\left|I\right|\leq I_M\right)=R_0g\left(\left|\frac{I}{I_M}\right|\right)+R_n$, $R_t\left(\left|I\right|>I_M\right)=R_n$ with $\frac{R_0}{R_n}>>1$, $C_e\left(\left|I\right|\leq I_M\right)=C_0g\left(\left|\frac{I}{I_M}\right|\right)$, $C_e\left(\left|I\right|>I_M\right)=0$ with $\frac{C_0}{C}>>1$ and $C$ being the capacitance of the Josephson junction and finally $g(x)=1-x^2$. Regardless of the initial condition $U(0)$, the solution $U(t)$ of Eq.(\[eqm\]) becomes time-periodic, i.e. $U\left(t\right)=U\left(t+\frac{2\pi}{\omega_t}\right),\forall t$, after a short transient regime.
Eq.(\[eqm\]) has been solved with the assignments $C=1pF,C_0=1mF,R=10\Omega,R_n=100\Omega,R_0=10K\Omega$, and the corresponding $U(t)$ have been plotted in Fig.\[per\]. The large slope $\left|\frac{dU}{dt}(0)\right|>>1$ stems from $\frac{C_0}{C}>>1$. Since no experimental data of $U(t),R_0,R_n,C_e,C$ have been reported in the literature to the best of our knowledge, no comparison between observed and calculated results can be done. Nevertheless, the large $u_M>>1$ values, seen in Fig.\[per\] ($u_M$ has been found to increase very steeply with $U_s$ decreasing toward $0$), have been observed[@sha].
The characteristics $I(U)$, plotted in Fig.\[char\], have been reckoned as $$\left\langle f\right\rangle=\frac{\omega}{2\pi}\int_0^{\frac{2\pi}{\omega}}f(u)du\quad,$$ with $f=U,I$. In all cases, there is $\left\langle I\right\rangle(0)=0$ with $\frac{d\left\langle I\right\rangle}{d\left\langle U\right\rangle}(0)> 0$ in agreement with the experimental data in Fig.\[row\]. However the slope $\frac{d\left\langle I\right\rangle}{d\left\langle U\right\rangle}(0)$, calculated for $\omega_t=100MHz$, is much larger than the one at $\omega_t=1MHz$. Accordingly, the characteristics, reproduced in Figs.\[row\],\[sha1\], differ markedly by their slope at the origin, which might thence hint at very different tunneling frequencies. At last, there are no observed $\left\langle I\right\rangle$ data in Fig.\[sha1\] over a broad $\left\langle U\right\rangle$ range, starting from $\left\langle U\right\rangle\approx 0$ up to a value big enough for the sample to go into the normal state, characterised by constant $I=I_n>I_M$. This feature might result[@sha] from $U_s\propto\sin(\omega_p t)$ with $\omega_p=60Hz$. Thus since the tunneling frequency $\omega_t$ is expected to decrease exponentially[@schi] with increasing $n$ and thence $U$, this entails that the signal could indeed no longer be observed for $\omega_t<\omega_p$.
$\textbf{5}$-Microwave Mediated Tunneling
=========================================
By irradiating the Josephson junction, depicted in Fig.\[jos1\], with an electromagnetic microwave, Shapiro observed[@sha] the step-like characteristic $I(U)$, recalled in Fig.\[step\]. The discontinuities of $\frac{d\left\langle I\right\rangle}{d\left\langle U\right\rangle}$, showing up at $\left\langle U\right\rangle=\frac{m\hbar\omega}{2e}$ with $m>0$ being an integer, brought forward a cogent proof that the MBE state comprises an *even* number of electrons. In order to explain this experimental result, let us begin with studying the microwave induced tunneling of *one* bound electron pair across the $U_m=\frac{m\hbar\omega}{2e}$ biased barrier. The corresponding Hilbert space, describing the system *before* and *after* crossing, is subtended by the basis $\left\{\varphi_i=\varphi_A(c_e)\otimes\varphi_B(c_e),\varphi_1=\varphi_A(c_e+\frac{2}{V})\otimes\varphi_B(c_e-\frac{2}{V})\right\}$ of respective energies $V\mathcal{E}_i,V\mathcal{E}_i+m\hbar\omega$. The tunneling motion of one electron pair is then described by $\psi_0(t)$, solution of the Schrödinger equation $$\label{sch2}
\begin{array}{c}
i\frac{\partial\psi_0}{\partial t}=H_0(t)\psi\\
H_0=m\omega\sigma_z+2\left(\omega_t+\omega_r\sin\left(\omega t\right)\right)\sigma_x
\end{array}\quad .$$ The Hamiltonian $H_0$ is expressed in frequency unit, $\frac{V\mathcal{E}_i}{\hbar}+\frac{m\omega}{2}$ is taken as the origin of energy, $\omega_r$ stands for the dipolar, off-diagonal matrix element[@boy] (the microwave power is $\propto\omega^2_r$), and $\sigma_z,\sigma_x$ are Pauli matrices[@abr], projected onto $\left\{\varphi_i,\varphi_1\right\}$. It is worth pointing out that Eq.(\[sch2\]) could be readily solved like Eq.(\[sch1\]), if $H_0$ were $t$ independent. Accordingly, in order to get rid of the $t$ dependence of $H_0$, we shall take advantage of a procedure devised for nonlinear optics[@sz6; @sz7].
To that end, $H_0$ is first recast into $$\label{pq}
H_0=P_0+f(t)\sigma_x\quad,$$ for which $P_0=m\omega\sigma_z+2\omega_t\sigma_x$ is a Hermitian, $2\times 2$, $t$ independent matrix, such that $\left(P_0\right)_{1,1}+\left(P_0\right)_{2,2}=0$, $\left(P_0\right)_{2,2}-\left(P_0\right)_{1,1}=m\omega$, and $f(t)=\omega_r\sin\left(\omega t\right)$ is a real function of period $=\frac{2\pi}{\omega}$, having the dimension of a frequency, such that $\left\langle f\right\rangle=\int_0^{\frac{2\pi}{\omega}}f(t)dt=0$. Then $H_0$ is projected onto $\left\{\psi_-,\psi_+\right\}$, the eigenbasis of $P_0$ $$\label{tht}
G=TH_0T^{-1}=\epsilon\sigma_z+d(t)\sigma_z+g(t)\sigma_x\quad.$$ $T$ is the unitary transfer matrix from $\left\{\varphi_i,\varphi_1\right\}$ to $\left\{\psi_-,\psi_+\right\}$ and $\sigma_z,\sigma_x$ have been projected onto $\left\{\psi_-,\psi_+\right\}$. The corresponding eigenvalues are $\mp\frac{\epsilon}{2}$ with $\epsilon=\sqrt{(m\omega)^2+\omega_t^2}\approx m\omega$ because of $\omega_t<<\omega$, while the real functions $d(t),g(t)$ have the same properties as $f(t)$ in Eq.(\[pq\]). Let us now introduce[@sz6; @sz7] the unitary transformation $R_1(t)$, operating in the Hilbert space, subtended by $\left\{\psi_-,\psi_+\right\}$ $$\label{uni}
R_1(t)=e^{i\Phi(t)}\left|\psi_-\right\rangle\left\langle\psi_-\right|+e^{-i\Phi(t)}\left|\psi_+\right\rangle\left\langle\psi_+\right|\quad ,$$ with the dimensionless $\Phi(t)=\frac{\omega t}{2}-\int_0^t d(u)du$. We then look for $\psi_1=R_1^{-1}\psi_0$, solution of the Schrödinger equation $$\label{sch3}
\begin{array}{c}
i\frac{\partial\psi_1}{\partial t}=H_1\psi_1\quad,H_1=R_1^{-1}GR_1-iR_1^{-1}\dot R_1\quad\\
H_1=P_1+\Re(z_1(t))\sigma_x+\Im(z_1(t))\sigma_y\\
P_1=\epsilon\sigma_z+2\omega_1\sigma_x
\end{array}\quad,$$ for which the Hermitian $2\times 2$ matrix $P_1$ has the same properties as $P_0$ in Eq.(\[pq\]), except for $\left(P_1\right)_{2,2}-\left(P_1\right)_{1,1}\approx(m-1)\omega,\left(P_1\right)_{2,1}=\omega_1=\omega_r/2$ instead of $\left(P_0\right)_{2,2}-\left(P_1\right)_{1,1}=m\omega,\left(P_0\right)_{2,1}=\omega_t$, the Pauli matrices $\sigma_z,\sigma_x,\sigma_y$ have been projected onto $\left\{\psi_-,\psi_+\right\}$, and $\Re(z_1(t)),\Im(z_1(t))$ which are the real and imaginary parts of the complex function $z_1(t)$, have the same properties as $f(t)$ in Eq.(\[pq\]). Consequently, iterating this procedure $m$ of times yields finally $$\label{sch4}
\begin{array}{c}
i\frac{\partial\psi_m}{\partial t}=H_m\psi_m\\
H_m=P_m+\Re(z_m(t))\sigma_x+\Im(z_m(t))\sigma_y\\
P_m=\eta\sigma_z+2\left(\Re(\omega_m)\sigma_x+\Im(\omega_m)\sigma_y\right)
\end{array}\quad,$$ for which the Pauli matrices $\sigma_z,\sigma_x,\sigma_y$ have been projected onto the eigenbasis of $P_m$, $\left\{\psi_-,\psi_+\right\}$, and $\eta\approx0$, $|\omega_m|<<\omega_r$. The Fourier series $\Re(z_m(t)),\Im(z_m(t))$ of fundamental frequency $\omega$ play no role, because the resonance condition[@abr] $\left|\left(P_m\right)_{1,1}-\left(P_m\right)_{2,2}\right|=\omega$ is not fulfilled due to $\left|\left(P_m\right)_{1,1}-\left(P_m\right)_{2,2}\right|=|\eta|<<\omega$, so that Eq.(\[sch4\]) is finally solved, similarly to Eq.(\[sch1\]), to give $$\psi_m= \cos\left(\frac{|\omega_m| t}{2}\right)\psi_--i\sin\left(\frac{|\omega_m| t}{2}\right)\psi_+\quad .$$ The solution of Eq.(\[sch2\]) is thereby inferred to read $$\psi_0(t)= \left(\prod_{i=1,m}R_i(t)\right)\psi_m(t)\quad .$$ $U_m$ can be fitted to get $\eta=0$. Thus, for the sake of illustration, calculated $|\omega_m|$ and $\delta_m=1-\frac{2eU_m}{m\hbar\omega}$ are indicated in table \[tab\]. As expected, $|\omega_m|$ decreases steeply with increasing $m$ but, remarkably enough, $|\omega_{2m+1}|$ decreases more slowly than $|\omega_{2m}|$, all the more so since $\omega_t$ is weaker. This property ensues[@abr; @boy] from $\omega_{2m}=0,\forall m$ for $\omega_t=0$.
$\omega_t=$ $100MHz$ $\omega_t=$ $1MHz$
------ ------------------------------- ------------------- ------------------------------- -------------------
$m$ $\frac{|\omega_m|}{\omega_r}$ $\delta_m$ $\frac{|\omega_m|}{\omega_r}$ $\delta_m$
$1$ $0.5$ $2\times 10^{-4}$ $0.5$ $2\times 10^{-8}$
$2$ $5\times 10^{-5}$ $8\times 10^{-5}$ $5\times 10^{-7}$ $3\times 10^{-5}$
$3$ $3\times 10^{-6}$ $3\times 10^{-5}$ $3\times 10^{-6}$ $8\times 10^{-6}$
$4$ $10^{-10}$ $2\times 10^{-5}$ $6\times 10^{-13}$ $4\times 10^{-6}$
$5$ $5\times 10^{-12}$ $10^{-5}$ $5\times 10^{-12}$ $3\times 10^{-6}$
$6$ $ 5\times 10^{-13}$ $7\times 10^{-6}$ $5\times 10^{-13}$ $2\times 10^{-6}$
$7$ $5\times 10^{-13}$ $5\times 10^{-6}$ $5\times 10^{-13}$ $10^{-6}$
$8$ $7\times 10^{-13}$ $4\times 10^{-6}$ $7\times 10^{-13}$ $9\times 10^{-7}$
$9$ $10^{-12}$ $3\times 10^{-6}$ $10^{-12}$ $7\times 10^{-7}$
$10$ $3\times 10^{-12}$ $3\times 10^{-6}$ $3\times 10^{-12}$ $6\times 10^{-7}$
$11$ $4\times 10^{-12}$ $2\times 10^{-6}$ $4\times 10^{-12}$ $5\times 10^{-7}$
: calculated $|\omega_m|,\delta_m$ values with $\omega=10GHz$, $\omega_r=100MHz$ and $\omega_t=100MHz,1MHz$.[]{data-label="tab"}
Let us neglect $\frac{2eU_m}{V\mathcal{E}_i}<10^{-20}$, so that the energy of $\psi_0$ is taken to be constant and equal to $V\mathcal{E}_i$. The coherent tunneling of $n>>2$ of bound electrons will thence be described in the basis $\left\{\psi_0,\varphi_f\right\}$ by Eq.(\[psi1\]), except for $\left\langle U\right\rangle-U_m$, $\left\langle I_m\right\rangle$ showing up instead of $\left\langle U\right\rangle$, $\left\langle I\right\rangle$, respectively, which entails that $\left\langle I_m\right\rangle\left(\left\langle U\right\rangle-U_m\right)=\left\langle I\right\rangle\left(\left\langle U\right\rangle\right)$, as illustrated by Fig.\[char\]. Likewise, the contributions $\left\langle I_{m=1,2,3...}\right\rangle$ will add up eventually to give the step-like characteristic $I(U)$, recalled in Fig.\[step\]. At last, Shapiro noticed[@sha] that some contributions $\left\langle I_{m}\right\rangle$ were missing in Fig.\[step\]. As explained above in section $4$, this might result from the corresponding $|\omega_m|<\omega_p$ and thence would confirm $\omega_t<<\omega$.
$\textbf{6}$-Negative Resistance
================================
Signals $U(t),I(t)\propto\sin(\omega t)$, with the RF frequency $\omega$ defined by the resonance condition $LC\omega^2=1$, have been observed[@mcc] in the kind of setup, sketched in Fig.\[jos2\]. Due to $\omega\neq\omega_t$, the bound electron tunneling plays no role and the oscillation rather stems from $R_t(I)$ decreasing[@sz5] down to $R_n$ with $|I|$ increasing up to $I_M$, as indicated in section 4. Accordingly, since the voltage drop across the coil is equal to $L\dot I$ and $U,I$ are related together by $I=\frac{U}{R_t}+C\dot U$, the electrodynamical equation of motion reads $$\label{neg}
\ddot U=\omega^2(U_s-U)-\frac{\dot U}{R_tC}\quad .$$ Linearising Eq.(\[neg\]) around the fixed point $U_0=U_s\Rightarrow I_0=\frac{U_s}{R_t(I_0)}$ yields the differential equation $$\label{lin}
\ddot U=-\omega^2U-\frac{\dot U}{R_eC}\quad ,$$ with the effective resistance $R_e$, defined by $R_e=R_t(I_0)+I_0\frac{dR_t}{dI}(I_0)$. Due to $\frac{dR_t}{dI}<0$, the fixed point may be unstable in case of negative resistance $R_e<0$, which will give rise to an oscillating solution of Eq.(\[neg\]), $U(t)\propto\sin(\omega t)$. As a matter of fact, integrating Eq.(\[neg\]) leads to the sine-wave, depicted in Fig.\[sin\]. Note that, unlike $U(t)$ in Fig.\[per\], every harmonic $\propto\sin(m\omega t)$ with $m>1$ is efficiently smothered by the resonating $L,C$ circuit due to $LC(m\omega)^2\ne 1$ for $m>1$. At last, we have checked that Eq.(\[neg\]) has no sine-wave solution for $\frac{R_0}{R_n}<50$ or $U_s>R_nI_M$, because those inequalities entail that $R_e>0$.
$\textbf{7}$-Conclusion
=======================
All experimental results[@and; @sha], illustrating the Josephson effect, have been accounted for on the basis of bound electrons tunneling periodically across the insulating barrier. Likewise, the very existence of the Josephson effect has been shown to be conditioned by $\frac{\partial\mu}{\partial c_s}<0$, which had previously been recognized as a prerequisite for persistent currents[@sz4] and thermal equilibrium[@sz5] too. The negative resistance feature[@mcc] has been ascribed to the tunneling resistance of independent electrons decreasing with increasing current, flowing through the superconducting electrodes, which confirms the validity of an analysis of the superconducting-normal transition[@sz5]. This work makes no use of Ginzburg and Landau’s phase which conversely plays a paramount role in the mainstream view[@dou; @dev; @koi; @koiz].
Remarkably enough, the signature of the Josephson effect, namely the periodic current due to bound electrons, has no counterpart in the microscopic realm. For instance, the electrons, involved in a covalent bond, cannot tunnel between the two bound atoms because of their thermal decay toward the bonding groundstate. As for the Josephson effect, the bonding eigenfunction and its associated energy would read $\varphi_b=\frac{\varphi_i+\varphi_f}{\sqrt{2}}$ and $V\mathcal{E}_i-\frac{\hbar\omega_t}{2}$, respectively, but the relaxation from the tunneling state $\psi(t)$ in Eq.(\[sch1\]) toward $\varphi_b$ might occur only inside the insulating barrier, which is impossible because the valence band, being fully occupied, can thence accomodate no additional electron.
[0]{} . . . . . . . . . . . . . . . . . . . . . .
|
---
author:
- Christian Ortiz Pauyac
- Mairbek Chshiev
- Aurelien Manchon
- 'Sergey A. Nikolaev'
title: |
Supplementary Materials:\
Spin Hall and spin swapping torques in diffusive ferromagnets
---
List of model and transport parameters {#list-of-model-and-transport-parameters .unnumbered}
======================================
---------------------------------------------------------------------------------------------- -------------------------------------------------------------------------
$\begin{aligned} k_{F} \end{aligned}$ $\qquad$ Fermi wavevector
$\begin{aligned} m \end{aligned}$ $\qquad$ Electron’s effective mass
$\begin{aligned} J \end{aligned}$ $\qquad$ Exchange constant in the $s$-$d$ model
$\begin{aligned} v_{i} \end{aligned}$ $\qquad$ Impurity potential
$\begin{aligned} n_{i} \end{aligned}$ $\qquad$ Impurity concentration
$\begin{aligned} \xi_{SO} \end{aligned}$ $\qquad$ Dimensionless spin-orbit coupling constant
$\begin{aligned} \varepsilon_{F}=\frac{\hbar^{2}k_{F}^{2}}{2m} \end{aligned}$ $\qquad$ Fermi energy
$\begin{aligned} v_{F}=\frac{\hbar k_{F}}{m} \end{aligned}$ $\qquad$ Fermi velocity
$\begin{aligned} \beta=\frac{J}{2\varepsilon_{F}} \end{aligned}$ $\qquad$ Spin polarization factor
$\begin{aligned} D_{0}=\frac{mk_{F}}{2\pi^{2}\hbar^{2}} \end{aligned}$ $\qquad$ Spin independent density of states per spin at the Fermi level
$\begin{aligned} \frac{1}{\tau_{0}}=\frac{2\pi v_{i}^{2}n_{i}D_{0}}{\hbar} \end{aligned}$ $\qquad$ Spin independent relaxation time
$\begin{aligned} \frac{1}{\tau_{L}}=\frac{2J}{\hbar} \end{aligned}$ $\qquad$ Larmor precession time
$\begin{aligned} \frac{1}{\tau_{\phi}}=\frac{4J^{2}\tau_{0}}{\hbar^{2}} \end{aligned}$ $\qquad$ Spin dephasing relaxation time
$\begin{aligned} \frac{1}{\tau_{sf}}=\frac{8}{9}\frac{\xi_{SO}^{2}}{\tau_{0}} \end{aligned}$ $\qquad$ Spin-flip relaxation time
$\begin{aligned} l_{F}=\tau_{0}v_{F} \end{aligned}$ $\qquad$ Mean-free path
$\begin{aligned} D=\frac{\tau_{0}v_{F}^{2}}{3} \end{aligned}$ $\qquad$ Diffusion coefficient
$\begin{aligned} \alpha_{sw}=\frac{2}{3}\xi_{SO} \end{aligned}$ $\qquad$ Dimensionless spin swapping constant
$\begin{aligned} \alpha_{sj}=\frac{\xi_{SO}}{l_{F}k_{F}} \end{aligned}$ $\qquad$ Dimensionless side-jump constant
$\begin{aligned} \alpha_{sk}=\frac{v_{i}mk_{F}}{3\pi\hbar^{2}}\xi_{SO} \end{aligned}$ $\qquad$ Dimensionless skew scattering constant
---------------------------------------------------------------------------------------------- -------------------------------------------------------------------------
General formalism
=================
We start with a free-electron Hamiltonian $\hat{\mathcal{H}}_{0}$ and its Fourier transform $\hat{\mathcal{H}}_{\boldsymbol{k}}$: $$\hat{\mathcal{H}}_{0}=-\frac{\hbar^{2}}{2m}\nabla^{2}\hat{\sigma}_{0}+J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\qquad\xrightarrow{\,\,\mathcal{F}\,\,}\qquad\hat{\mathcal{H}}_{\boldsymbol{k}}=\frac{\hbar^{2}_{}\boldsymbol{k}^{2}_{}}{2m}\hat{\sigma}_{0}+J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}.$$ Here, the first term stands for the kinetic energy, where $m$ and $\boldsymbol{k}$ are the electron’s effective mass and wave vector, respectively; the second term refers to the exchange interaction in the so-called $s$-$d$ model, where $\boldsymbol{m}=(\cos{\phi}\sin{\theta},\sin{\phi}\sin{\theta},\cos{\theta})$ is the magnetization unit vector parametrized in spherical coordinates, $J$ is the exchange coupling parameter, $\hat{\boldsymbol{\sigma}}$ is the Pauli matrix vector, and $\hat{\sigma}_{0}$ is the identity matrix. The unperturbed Green’s function is defined by $\hat{\mathcal{H}}_{\boldsymbol{k}}$: $$\hat{G}_{0,\boldsymbol{k}E}^{R(A)}=\left[E-\hat{\mathcal{H}}_{\boldsymbol{k}}\pm i\eta\right]^{-1}=\sum\limits_{s=\pm}\frac{|s\rangle\langle s|}{E-E_{\boldsymbol{k}s}\pm i\eta}=\frac{1}{2}\sum\limits_{s=\pm}\frac{\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k}s}\pm i\eta},
\label{eq:gzero}$$ where $s$ refers to the spin index, $|s\rangle$ is the corresponding eigenstate: $$|s\rangle=\left( \begin{array}{c}
se^{-i\phi}\sqrt{\frac{1+s\cos{\theta}}{2}}\\
\sqrt{\frac{1-s\cos{\theta}}{2}}\end{array} \right),$$ $E_{\boldsymbol{k}s}=E_{\boldsymbol{k}}+sJ$, $E_{\boldsymbol{k}}=\hbar^{2}\boldsymbol{k}^{2}/2m$, and $\eta$ is a positive infinitesimal.
Next, we consider the impurity Hamiltonian with spin-orbit coupling: $$\label{eq:imppot}
\hat{\mathcal{H}}_{\mathrm{imp}}=\sum\limits_{\boldsymbol{R}_{i}}V(\boldsymbol{r}-\boldsymbol{R}_{i})\hat{\sigma}_{0}+\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{\boldsymbol{R}_{i}}\left(\nabla V(\boldsymbol{r}-\boldsymbol{R}_{i})\times\hat{\boldsymbol{p}}\right)\cdot\hat{\boldsymbol{\sigma}},$$ where $\hat{\boldsymbol{p}}=-i\hbar\partial_{\boldsymbol{r}}$ is the momentum operator, $V(\boldsymbol{r}-\boldsymbol{R}_{i})=v_{i}\delta(\boldsymbol{r}-\boldsymbol{R}_{i})$ is the on-site potential at the impurity site $\boldsymbol{R}_{i}$, $\xi_{SO}$ is the spin-orbit coupling parameter (defined as a dimensionless quantity), and $k_{F}$ is the Fermi wavevector. Here, we neglect the localization effects and electron-electron correlations, and assume a short-range impurity potential. In the reciprocal space, it can be written as:[@rammer1] $$\hat{\mathcal{H}}_{\boldsymbol{k}\boldsymbol{k'}}=\Omega\langle\boldsymbol{k}|\hat{\mathcal{H}}_{\mathrm{imp}}|\boldsymbol{k'}\rangle,$$ where the momentum eigenstates are defined as $\langle\boldsymbol{r}|\boldsymbol{k}\rangle=\Omega^{-1/2}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}$, and $\Omega$ is the volume of the system. Then, by using the following identities: $$\int\limits_{\Omega}d\boldsymbol{r}\,f(\boldsymbol{r})\delta(\boldsymbol{r}-\boldsymbol{r}_{i})=f(\boldsymbol{r}_{i}),\qquad\int\limits_{\Omega}d\boldsymbol{r}\,f(\boldsymbol{r})\nabla\delta(\boldsymbol{r}-\boldsymbol{r}_{i})=-\int\limits_{\Omega}d\boldsymbol{r}\,\nabla f(\boldsymbol{r})\delta(\boldsymbol{r}-\boldsymbol{r}_{i}),$$ we obtain: $$\begin{aligned}
\hat{\mathcal{H}}_{\boldsymbol{k}\boldsymbol{k'}}&=\sum\limits_{\boldsymbol{R}_{i}}\int\limits_{\Omega}d\boldsymbol{r}\,\left[v_{i}\delta(\boldsymbol{r}-\boldsymbol{R}_{i})\hat{\sigma}_{0}e^{-i(\boldsymbol{k}-\boldsymbol{k'})\cdot\boldsymbol{r}}-iv_{i}\frac{\xi_{SO}}{k_{F}^{2}}e^{-i\boldsymbol{k}\cdot\boldsymbol{r}}\Big(\nabla\delta(\boldsymbol{r}-\boldsymbol{R}_{i})\times\partial_{\boldsymbol{r}}\Big)\cdot\hat{\boldsymbol{\sigma}}e^{i\boldsymbol{k'}\cdot\boldsymbol{r}} \right]\\
&=V(\boldsymbol{k}-\boldsymbol{k'})\left[\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{k}\times\boldsymbol{k'})\right],
\end{aligned}
\label{eq:potsoi}$$ where $V(\boldsymbol{k}-\boldsymbol{k'})$ is the Fourier transform of the impurity on-site potential: $$V(\boldsymbol{k}-\boldsymbol{k'})=v_{i}\sum_{\boldsymbol{R}_{i}}e^{-i(\boldsymbol{k}-\boldsymbol{k'})\cdot\boldsymbol{R}_{i}}.$$
We proceed to write a kinetic equation by means of the Keldysh formalism: $$\underline{\hat{G}}^{-1}=\hat{G}_{0}^{-1}-\underline{\Sigma},\qquad
\underline{\hat{G}}=\left( \begin{array}{cc}
\hat{G}^{R} &\hat{G}^{K}\\
0&\hat{G}^{A}\end{array} \right),\qquad
\underline{\hat{\Sigma}}=\left( \begin{array}{cc}
\hat{\Sigma}^{R} &\hat{\Sigma}^{K}\\
0&\hat{\Sigma}^{A}\end{array} \right),$$ where $\underline{\hat{G}}$ and $\underline{\hat{\Sigma}}$ are the Green’s function and self-energy in the Keldysh space; the indexes $R$, $A$ and $K$ stand for the retarded, advanced and Keldysh components, respectively, and $\hat{G}_{0}^{-1}=i\hbar\partial_{t}-\hat{\mathcal{H}}_{0}$. In the semiclassical approximation, a set of diffusive equations for the non-equilibrium charge and spin densities can be derived through the distribution function $\hat{g}_{\boldsymbol{k}}\equiv \hat{g}_{\boldsymbol{k}}(\boldsymbol{R},T)$ defined as the Wigner representation of the Keldysh Green’s function $\hat{G}^{K}$:[@rammer2] $$\begin{aligned}
\hat{G}^{K}(\boldsymbol{r}_{1},t_{1};\boldsymbol{r}_{2},t_{2})&\,\,\xrightarrow{\mathcal{W}}\hat{G}^{K}(\boldsymbol{R}+\frac{\boldsymbol{r}}{2},T+\frac{t}{2};\boldsymbol{R}-\frac{\boldsymbol{r}}{2},T-\frac{t}{2})\equiv\hat{G}^{K}(\boldsymbol{r},t;\boldsymbol{R},T)\\
&\,\,\xrightarrow{\mathcal{F}}\hat{G}^{K}(\boldsymbol{r},t;\boldsymbol{R},T)=\int\frac{dE}{2\pi}\frac{d\boldsymbol{k}}{(2\pi)^{3}}\,\,\hat{g}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)e^{-\frac{i}{\hbar}Et}e^{i\boldsymbol{r}\cdot\boldsymbol{k}}\\
&\,\,\xrightarrow{\,\,}\hat{g}_{\boldsymbol{k}}=i\int\frac{dE}{2\pi}\,\hat{g}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T),
\end{aligned}$$ where the relative $\boldsymbol{r}=\boldsymbol{r}_{1}-\boldsymbol{r}_{2}$, $t=t_{1}-t_{2}$ and center-of-mass $\boldsymbol{R}=(\boldsymbol{r}_{1}+\boldsymbol{r}_{2})/2$, $T=(t_{1}+t_{2})/2$ coordinates are introduced. In the dilute limit, we can employ the Kadanoff-Baym anzats: $$\underline{\hat{G}}_{\boldsymbol{k}E}(\boldsymbol{R},T)=\left( \begin{array}{cc}
\hat{G}^{R}_{\boldsymbol{k}E} &\hat{g}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)\\
0&\hat{G}^{A}_{\boldsymbol{k}E}\end{array} \right)$$ and $$\hat{g}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)=\hat{G}^{R}_{\boldsymbol{k}E}\,\hat{g}_{\boldsymbol{k}}(\boldsymbol{R},T)-\hat{g}_{\boldsymbol{k}}(\boldsymbol{R},T)\hat{G}^{A}_{\boldsymbol{k}E}.
\label{eq:anzats}$$ The Keldysh Green’s function $\hat{G}^{K}$ satisfies the Kadanoff-Baym equation:[@rammer2] $$[\hat{G}^{R}]^{-1}\ast\hat{G}^{K}-\hat{G}^{K}\ast[\hat{G}^{A}]^{-1}=\hat{\Sigma}^{K}\ast\hat{G}^{A}-\hat{G}^{R}\ast\hat{\Sigma}^{K},$$ Having applied the Wigner transformation, we use the so-called gradient approximation, where the convolution $\mathcal{A}\ast\mathcal{B}$ of two functions is expressed as: $$\begin{aligned}
\left(\mathcal{A}\ast\mathcal{B}\right)_{\boldsymbol{k}E}(\boldsymbol{R},T)&\simeq\mathcal{A}\mathcal{B}-\frac{i\hbar}{2}\left(\partial_{T}\mathcal{A}\partial_{E}\mathcal{B}-\partial_{E}\mathcal{A}\partial_{T}\mathcal{B}\right)\\
&-\frac{i}{2}\left(\nabla_{\boldsymbol{k}}\mathcal{A}\cdot\nabla_{\boldsymbol{R}}\mathcal{B}-\nabla_{\boldsymbol{R}}\mathcal{A}\cdot\nabla_{\boldsymbol{k}}\mathcal{B}\right).
\end{aligned}$$ Taking into account that $\hat{G}^{R(A)}$ and $\hat{\Sigma}^{R(A)}$ do not depend on the center-of-mass coordinates, we obtain: $$\begin{aligned}
i\hbar\partial_{T}\hat{g}^{K}+[\hat{g}^{K},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]&+\frac{i}{2}\left\{\nabla_{\boldsymbol{k}}\hat{\mathcal{H}}_{\boldsymbol{k}},\nabla_{\boldsymbol{R}}\hat{g}^{K}\right\}=\hat{\Sigma}^{K}\hat{G}^{A}-\hat{G}^{R}\hat{\Sigma}^{K}+\hat{\Sigma}^{R}\hat{g}^{K}-\hat{g}^{K}\hat{\Sigma}^{A}\\
&-\frac{i\hbar}{2}\left(\partial_{T}\hat{\Sigma}^{K}\partial_{E}\hat{G}^{A}+\partial_{E}\hat{G}^{R}\partial_{T}\hat{\Sigma}^{K} \right)
+\frac{i\hbar}{2}\left(\partial_{E}\hat{\Sigma}^{R}\partial_{T}\hat{g}^{K}+\partial_{T}\hat{g}^{K}\partial_{E}\hat{\Sigma}^{A} \right)\\
&-\frac{i}{2}\left(\nabla_{\boldsymbol{k}}\hat{\Sigma}^{R}\cdot\nabla_{\boldsymbol{R}}\hat{g}^{K}+\nabla_{\boldsymbol{R}}\hat{g}^{K}\cdot\nabla_{\boldsymbol{k}}\hat{\Sigma}^{A}\right)
+\frac{i}{2}\left(\nabla_{\boldsymbol{R}}\hat{\Sigma}^{K}\cdot\nabla_{\boldsymbol{k}}\hat{G}^{A}+\nabla_{\boldsymbol{k}}\hat{G}^{R}\cdot\nabla_{\boldsymbol{R}}\hat{\Sigma}^{K}\right),
\label{eq:full}
\end{aligned}$$ where $[\cdot\,,\cdot]$ and $\{\cdot\,,\cdot\}$ stand for a commutator and anticommutator, respectively. In steady state, we have: $$\begin{aligned}
\,[\hat{g}^{K},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]&+\frac{i}{2}\left\{\nabla_{\boldsymbol{k}}\hat{\mathcal{H}}_{\boldsymbol{k}},\nabla_{\boldsymbol{R}}\hat{g}^{K}\right\}=\hat{\Sigma}^{K}\hat{G}^{A}-\hat{G}^{R}\hat{\Sigma}^{K}+\hat{\Sigma}^{R}\hat{g}^{K}-\hat{g}^{K}\hat{\Sigma}^{A}\\
&-\frac{i}{2}\left(\nabla_{\boldsymbol{k}}\hat{\Sigma}^{R}\cdot\nabla_{\boldsymbol{R}}\hat{g}^{K}+\nabla_{\boldsymbol{R}}\hat{g}^{K}\cdot\nabla_{\boldsymbol{k}}\hat{\Sigma}^{A}\right)
+\frac{i}{2}\left(\nabla_{\boldsymbol{R}}\hat{\Sigma}^{K}\cdot\nabla_{\boldsymbol{k}}\hat{G}^{A}+\nabla_{\boldsymbol{k}}\hat{G}^{R}\cdot\nabla_{\boldsymbol{R}}\hat{\Sigma}^{K}\right).
\label{eq:kelfull}
\end{aligned}$$ Finally, in the dilute limit, we can assume that the self-energy is almost constant and neglect its derivatives on the right-hand side: $$[\hat{g}^{K},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]+\frac{i}{2}\left\{\nabla_{\boldsymbol{k}}\hat{\mathcal{H}}_{\boldsymbol{k}},\nabla_{\boldsymbol{R}}\hat{g}^{K}\right\}=\hat{\Sigma}^{K}\hat{G}^{A}-\hat{G}^{R}\hat{\Sigma}^{K}+\hat{\Sigma}^{R}\hat{g}^{K}-\hat{g}^{K}\hat{\Sigma}^{A},$$ or $$[\hat{g}^{K},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]+i\frac{\hbar^{2}}{m}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\,\hat{g}^{K}=\hat{\Sigma}^{K}\hat{G}^{A}-\hat{G}^{R}\hat{\Sigma}^{K}+\hat{\Sigma}^{R}\hat{g}^{K}-\hat{g}^{K}\hat{\Sigma}^{A}.
\label{eq:keldysh}$$
Self-energy
===========
Self-consistent Born approximation
----------------------------------
Let us consider first and second orders of the diagrammatic expansion for the scattering off the static impurity potential (Fig. 1$a$). The Green’s function $\underline{\hat{G}}$ and the corresponding self-energy $\underline{\hat{\Sigma}}$ are defined in the wave vector representation as follows: $$\begin{aligned}
\underline{\hat{G}}(\boldsymbol{k},t_{1};\boldsymbol{k'},t_{2})&=\underline{\hat{G}}_{0}(\boldsymbol{k},t_{1};\boldsymbol{k'},t_{2})+\sum\limits_{\{\boldsymbol{k}_{i}\}} \underline{\hat{G}}_{0}(\boldsymbol{k},t_{1};\boldsymbol{k}_{1},t_{2}) \langle\boldsymbol{k}_{1}|\hat{\mathcal{H}}_{\mathrm{imp}}|\boldsymbol{k}_{2}\rangle\underline{\hat{G}}_{0}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k'},t_{2})\\
&+\sum\limits_{\{\boldsymbol{k}_{i}\}} \underline{\hat{G}}_{0}(\boldsymbol{k},t_{1};\boldsymbol{k}_{1},t_{2}) \langle\boldsymbol{k}_{1}|\hat{\mathcal{H}}_{\mathrm{imp}}|\boldsymbol{k}_{2}\rangle\underline{\hat{G}}_{0}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})\langle\boldsymbol{k}_{3}|\hat{\mathcal{H}}_{\mathrm{imp}}|\boldsymbol{k}_{4}\rangle\underline{\hat{G}}_{0}(\boldsymbol{k}_{4},t_{1};\boldsymbol{k'},t_{2})+...\\
&=\underline{\hat{G}}_{0}(\boldsymbol{k},t_{1};\boldsymbol{k'},t_{2})+\sum\limits_{\{\boldsymbol{k}_{i}\}} \underline{\hat{G}}_{0}(\boldsymbol{k},t_{1};\boldsymbol{k}_{1},t_{2})\underline{\hat{\Sigma}}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{2},t_{2})\underline{\hat{G}}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k'},t_{2}),
\end{aligned}
\label{eq:selfscat}$$ where $\underline{\hat{G}}_{0}$ is a free propagator. To consider a particle moving in a random potential we take the average over different spatial configurations of the ensemble of $N$ impurities. Upon impurity-averaging the first order term in Eq. (\[eq:selfscat\]) is a constant and can be renormalized away. For the second order term, we take into account all two-line irreducible diagrams corresponding to the double scattering off the same impurity and neglect the so-called crossing diagrams, where impurity lines cross and give a small contribution to the region of interest, $E\simeq E_{F}$ and $k\simeq k_{F}$.[@rammer1] This is nothing else but the self-consistent Born approximation (Fig. 1$b$). Using Eq. (\[eq:potsoi\]) for the impurity potential, the self-energy is given by three terms: $$\begin{aligned}
\underline{\hat{\Sigma}}^{1a}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{4},t_{2})=\frac{1}{\Omega^{2}}\sum\limits_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}}\underline{\hat{G}}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})\right\rangle,
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{1b}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{4},t_{2})&=\frac{i}{\Omega^{2}}\frac{\xi_{SO}}{k_{F}^{2}}\sum\limits_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}}\left[[(\boldsymbol{k}_{1}\times\boldsymbol{k}_{2})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})\right.\\
&\left.+\,\underline{\hat{G}}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})[(\boldsymbol{k}_{3}\times\boldsymbol{k}_{4})\cdot\hat{\boldsymbol{\sigma}}]\right]\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})\right\rangle,
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{1c}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{4},t_{2})&=-\frac{1}{\Omega^{2}}\frac{\xi_{SO}^{2}}{k^{4}_{F}}\sum\limits_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}}[(\boldsymbol{k}_{1}\times\boldsymbol{k}_{2})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})[(\boldsymbol{k}_{3}\times\boldsymbol{k}_{4})\cdot\hat{\boldsymbol{\sigma}}]\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})\right\rangle,\\
\end{aligned}$$ where impurity averaging leads to: $$\begin{aligned}
\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})\right\rangle&= v_{i}^{2} \left\langle\sum_{\boldsymbol{R}_{i}}e^{-i(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})\cdot\boldsymbol{R}_{i}}e^{-i(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})\cdot\boldsymbol{R}_{i}}\right\rangle=v_{i}^{2}N\delta_{\boldsymbol{k}_{1}+\boldsymbol{k}_{3},\boldsymbol{k}_{2}+\boldsymbol{k}_{4}}.
\end{aligned}$$ To proceed with the Wigner transformation, we change the variables: $$\boldsymbol{k}_{1}=\boldsymbol{k}+\frac{\boldsymbol{q}}{2}\qquad\boldsymbol{k}_{2}=\boldsymbol{k'}+\frac{\boldsymbol{q'}}{2}\qquad\boldsymbol{k}_{3}=\boldsymbol{k'}-\frac{\boldsymbol{q'}}{2} \qquad\boldsymbol{k}_{4}=\boldsymbol{k}-\frac{\boldsymbol{q}}{2}$$ and $$T=\frac{t_{1}+t_{2}}{2}\qquad t=t_{1}-t_{2},$$ that gives: $$\begin{aligned}
\underline{\hat{\Sigma}}^{1a}(\boldsymbol{k},t;\boldsymbol{q},T)=\frac{1}{\Omega^{2}}\sum\limits_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}}\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\delta_{\boldsymbol{q},\boldsymbol{q'}},
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{1b}(\boldsymbol{k},t;\boldsymbol{q},T)&=\frac{i}{\Omega^{2}}\frac{\xi_{SO}}{k_{F}^{2}}\sum\limits_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}}\left[\big[((\boldsymbol{k}+\frac{\boldsymbol{q}}{2})\times(\boldsymbol{k'}+\frac{\boldsymbol{q'}}{2}))\cdot\hat{\boldsymbol{\sigma}}\big]\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\right.\\
&\left.+\,\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\big[((\boldsymbol{k'}-\frac{\boldsymbol{q'}}{2})\times(\boldsymbol{k}-\frac{\boldsymbol{q'}}{2})))\cdot\hat{\boldsymbol{\sigma}}\big]\right]\delta_{\boldsymbol{q},\boldsymbol{q'}},
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{1c}(\boldsymbol{k},t;\boldsymbol{q},T)&=-\frac{1}{\Omega^{2}}\frac{\xi_{SO}^{2}}{k^{4}_{F}}\sum\limits_{\boldsymbol{k}_{2}\boldsymbol{k}_{3}}\big[((\boldsymbol{k}+\frac{\boldsymbol{q}}{2})\times(\boldsymbol{k'}+\frac{\boldsymbol{q'}}{2}))\cdot\hat{\boldsymbol{\sigma}}\big]\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\big[((\boldsymbol{k'}-\frac{\boldsymbol{q'}}{2})\times(\boldsymbol{k}-\frac{\boldsymbol{q}}{2}))\cdot\hat{\boldsymbol{\sigma}}\big]\delta_{\boldsymbol{q},\boldsymbol{q'}}.
\end{aligned}$$ The Kronecker function reflects that translation invariance is recovered, and we have in the continuum limit: $$\begin{aligned}
\underline{\hat{\Sigma}}^{1a}(\boldsymbol{k},t;\boldsymbol{q},T)=v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q},T),
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{1b}(\boldsymbol{k},t;\boldsymbol{q},T)&=\underline{\hat{\Sigma}}^{sw}(\boldsymbol{k},t;\boldsymbol{q},T)+\underline{\hat{\Sigma}}^{sj}(\boldsymbol{k},t;\boldsymbol{q},T)\\
&= iv_{i}^{2}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left[(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}},\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q},T)\right]\\
&+iv_{i}^{2}n_{i}\frac{\xi_{SO}}{2k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left\{[\boldsymbol{q}\times(\boldsymbol{k}'-\boldsymbol{k})]\cdot\hat{\boldsymbol{\sigma}},\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q},T)\right\},
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{1c}(\boldsymbol{k},t;\boldsymbol{q},T)&=-v_{i}^{2}n_{i}\frac{\xi^{2}_{SO}}{k_{F}^{4}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}[(\boldsymbol{k}\times\boldsymbol{k'}+\frac{1}{2}\boldsymbol{q}\times\boldsymbol{k'}+\frac{1}{2}\boldsymbol{k}\times\boldsymbol{q})\cdot\hat{\boldsymbol{\sigma}}]\\
&\cdot\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q},T)[(\boldsymbol{k'}\times\boldsymbol{k}-\frac{1}{2}\boldsymbol{k'}\times\boldsymbol{q}-\frac{1}{2}\boldsymbol{q}\times\boldsymbol{k})\cdot\hat{\boldsymbol{\sigma}}].
\end{aligned}$$ Having Fourier transformed with respect to $\boldsymbol{q}$, that gives the Wigner coordinate $\boldsymbol{R}$: $$\begin{aligned}
\hat{G}(\boldsymbol{r}_{1};\boldsymbol{r}_{4})&=\int\frac{d\boldsymbol{k}_{1}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k}_{4}}{(2\pi)^{3}}\hat{G}(\boldsymbol{k}_{1};\boldsymbol{k}_{4})e^{i(\boldsymbol{k}_{1}\boldsymbol{r}_{1}-\boldsymbol{k}_{4}\boldsymbol{r}_{4})}\\
&=\int\frac{d\boldsymbol{k}}{(2\pi)^{3}}\int\frac{d\boldsymbol{q}}{(2\pi)^{3}}\hat{G}(\boldsymbol{k}+\frac{1}{2}\boldsymbol{q};\boldsymbol{k}-\frac{1}{2}\boldsymbol{q})e^{i(\boldsymbol{k}+\frac{1}{2}\boldsymbol{q})\cdot(\boldsymbol{R}+\frac{1}{2}\boldsymbol{r})}e^{-i(\boldsymbol{k}-\frac{1}{2}\boldsymbol{q})\cdot(\boldsymbol{R}-\frac{1}{2}\boldsymbol{r})}\\
&=\int\frac{d\boldsymbol{k}}{(2\pi)^{3}}\int\frac{d\boldsymbol{q}}{(2\pi)^{3}}\hat{G}(\boldsymbol{k};\boldsymbol{q})e^{i\boldsymbol{q}\cdot\boldsymbol{R}}e^{i\boldsymbol{k}\cdot\boldsymbol{r}}=\hat{G}(\boldsymbol{r};\boldsymbol{R}),
\end{aligned}$$ we get the final form for the self-energy in the mixed representation: $$\begin{aligned}
\underline{\hat{\Sigma}}^{1a}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R},T),\\
\underline{\hat{\Sigma}}^{1b}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=\underline{\hat{\Sigma}}^{sw}_{\boldsymbol{k}E}(\boldsymbol{R},T)+\underline{\hat{\Sigma}}^{sj}_{\boldsymbol{k}E}(\boldsymbol{R},T)\\
&= iv_{i}^{2}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left[(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}},\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\right]\\
&+v_{i}^{2}n_{i}\frac{\xi_{SO}}{2k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\nabla_{\boldsymbol{R}}\left\{(\boldsymbol{k}'-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R};T)\right\},\\
\underline{\hat{\Sigma}}^{1c}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=-v_{i}^{2}n_{i}\frac{\xi^{2}_{SO}}{k_{F}^{4}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}[(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R},T)[(\boldsymbol{k'}\times\boldsymbol{k})\cdot\hat{\boldsymbol{\sigma}}].
\end{aligned}$$ At the level of the self-consistent Born approximation, the self-energy is given by the following contributions. The first term $\underline{\hat{\Sigma}}^{1a}$ stands for the standard elastic scattering off the on-site impurity potential. To first order of $\xi_{SO}$, there are two terms, $\underline{\hat{\Sigma}}^{sw}$ and $\underline{\hat{\Sigma}}^{sj}$, related to the side-jump and spin swapping contributions, respectively. Finally, the second order of $\xi_{SO}$ yields the Elliot-Yafet spin relaxation mechanism (all gradient terms $\sim\xi_{SO}^{2}\nabla_{\boldsymbol{R}}$ are neglected on account of their smallness).
Let us rewrite these contributions for the retarded, advanced and Keldysh Green’s functions (taking into account that $\hat{G}^{R(A)}$ does not depend on the center-of-mass coordinates): $$\underline{\hat{\Sigma}}_{\boldsymbol{k}E}(\boldsymbol{R},T)=\underline{\hat{\Sigma}}^{1a}_{\boldsymbol{k}E}(\boldsymbol{R},T)+\underline{\hat{\Sigma}}^{1b}_{\boldsymbol{k}E}(\boldsymbol{R},T)+\underline{\hat{\Sigma}}^{1c}_{\boldsymbol{k}E}(\boldsymbol{R},T),$$ which is equivalent to: $$\hat{\Sigma}^{R}_{\boldsymbol{k}E}=v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\right]\hat{G}^{R}_{\boldsymbol{k'}E}\left[\hat{\sigma}_{0}-i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\right],
\label{eq:sret}$$ $$\hat{\Sigma}^{A}_{\boldsymbol{k}E}=v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\right]\hat{G}^{A}_{\boldsymbol{k'}E}\left[\hat{\sigma}_{0}-i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\right],
\label{eq:sadv}$$ and $$\begin{aligned}
\hat{\Sigma}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\right]\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\left[\hat{\sigma}_{0}-i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\right]\\
&+v_{i}^{2}n_{i}\frac{\xi_{SO}}{2k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\nabla_{\boldsymbol{R}}\left\{(\boldsymbol{k}'-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R};T)\right\}.
\label{eq:skel}
\end{aligned}$$
Skew-scattering
---------------
To take into account skew-scattering, one has to go beyond the Born approximation. Starting from third order diagrams (Fig. 1$c$), we obtain the following expressions for the self-energy to first order in $\xi_{SO}$: $$\begin{aligned}
\underline{\hat{\Sigma}}^{2a}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{6},t_{2})=\frac{i}{\Omega^{3}}\frac{\xi_{SO}}{k_{F}^{2}}\sum\limits_{\boldsymbol{k}_{2},\boldsymbol{k}_{3},\boldsymbol{k}_{4},\boldsymbol{k}_{5}}&[(\boldsymbol{k}_{1}\times\boldsymbol{k}_{2})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}_{0}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})\underline{\hat{G}}_{0}(\boldsymbol{k}_{4},t_{1};\boldsymbol{k}_{5},t_{2})\\
&\cdot\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})V(\boldsymbol{k}_{5}-\boldsymbol{k}_{6})\right\rangle,
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{2b}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{6},t_{2})=\frac{i}{\Omega^{3}}\frac{\xi_{SO}}{k_{F}^{2}}\sum\limits_{\boldsymbol{k}_{2},\boldsymbol{k}_{3},\boldsymbol{k}_{4},\boldsymbol{k}_{5}}&\underline{\hat{G}}_{0}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})[(\boldsymbol{k}_{3}\times\boldsymbol{k}_{4})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}_{0}(\boldsymbol{k}_{4},t_{1};\boldsymbol{k}_{5},t_{2})\\
&\cdot\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})V(\boldsymbol{k}_{5}-\boldsymbol{k}_{6})\right\rangle,
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{2c}(\boldsymbol{k}_{1},t_{1};\boldsymbol{k}_{6},t_{2})=\frac{i}{\Omega^{3}}\frac{\xi_{SO}}{k_{F}^{2}}\sum\limits_{\boldsymbol{k}_{2},\boldsymbol{k}_{3},\boldsymbol{k}_{4},\boldsymbol{k}_{5}}&\underline{\hat{G}}_{0}(\boldsymbol{k}_{2},t_{1};\boldsymbol{k}_{3},t_{2})\underline{\hat{G}}_{0}(\boldsymbol{k}_{4},t_{1};\boldsymbol{k}_{5},t_{2})[(\boldsymbol{k}_{5}\times\boldsymbol{k}_{6})\cdot\hat{\boldsymbol{\sigma}}]\\
&\cdot\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})V(\boldsymbol{k}_{5}-\boldsymbol{k}_{6})\right\rangle,
\end{aligned}$$ where impurity averaging in the triple scattering off the same impurity potential is implied: $$\begin{aligned}
\left\langle V(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})V(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})V(\boldsymbol{k}_{5}-\boldsymbol{k}_{6})\right\rangle&= v_{i}^{3}\left\langle \sum_{\boldsymbol{R}_{i}}e^{-i(\boldsymbol{k}_{1}-\boldsymbol{k}_{2})\cdot\boldsymbol{R}_{i}}e^{-i(\boldsymbol{k}_{3}-\boldsymbol{k}_{4})\cdot\boldsymbol{R}_{i}}e^{-i(\boldsymbol{k}_{5}-\boldsymbol{k}_{6})\cdot\boldsymbol{R}_{i}}\right\rangle\\
&=v_{i}^{3}N\delta_{\boldsymbol{k}_{1}+\boldsymbol{k}_{3}+\boldsymbol{k}_{5},\boldsymbol{k}_{2}+\boldsymbol{k}_{4}+\boldsymbol{k}_{6}}.
\end{aligned}$$ Here, we do not consider triple scatterings off the on-site impurity potential without spin-orbit coupling, which gives a negligible correction to the elastic relaxation time ($\sim\frac{\beta v_{i}}{\tau_{0}}$). Changing the variables: $$\begin{aligned}
\boldsymbol{k}_{1}&=\boldsymbol{k}+\frac{\boldsymbol{q}}{2}\qquad \boldsymbol{k}_{2}=\boldsymbol{k'}+\frac{\boldsymbol{q'}}{2}\qquad \boldsymbol{k}_{3}=\boldsymbol{k'}-\frac{\boldsymbol{q'}}{2},\\
\boldsymbol{k}_{4}&=\boldsymbol{k''}+\frac{\boldsymbol{q''}}{2}\qquad \boldsymbol{k}_{5}=\boldsymbol{k''}-\frac{\boldsymbol{q''}}{2}\qquad \boldsymbol{k}_{6}=\boldsymbol{k}-\frac{\boldsymbol{q}}{2}
\end{aligned}$$ and $$T=\frac{t_{1}+t_{2}}{2}\qquad t=t_{1}-t_{2}$$ leads in the continuum limit to: $$\begin{aligned}
\underline{\hat{\Sigma}}^{2a}(\boldsymbol{k},t;\boldsymbol{q},T)&=iv_{i}^{3}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\int\frac{d\boldsymbol{q'}}{(2\pi)^{3}}\\
&\left[\left(\boldsymbol{k}+\frac{\boldsymbol{q}}{2}\right)\times\left(\boldsymbol{k'}+\frac{\boldsymbol{q'}}{2}\right)\cdot\hat{\boldsymbol{\sigma}}\right]\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\underline{\hat{G}}(\boldsymbol{k''},t;\boldsymbol{q}-\boldsymbol{q'},T),
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{2b}(\boldsymbol{k},t;\boldsymbol{q},T)&=iv_{i}^{3}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\int\frac{d\boldsymbol{q'}}{(2\pi)^{3}}\\
&\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\left[\left(\boldsymbol{k'}-\frac{\boldsymbol{q'}}{2}\right)\times\left(\boldsymbol{k''}+\frac{\boldsymbol{q}}{2}-\frac{\boldsymbol{q'}}{2}\right)\cdot\hat{\boldsymbol{\sigma}}\right]\underline{\hat{G}}(\boldsymbol{k''},t;\boldsymbol{q}-\boldsymbol{q'},T),
\end{aligned}$$ $$\begin{aligned}
\underline{\hat{\Sigma}}^{2c}(\boldsymbol{k},t;\boldsymbol{q},T)&=iv_{i}^{3}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\int\frac{d\boldsymbol{q'}}{(2\pi)^{3}}\\
&\underline{\hat{G}}(\boldsymbol{k'},t;\boldsymbol{q'},T)\underline{\hat{G}}(\boldsymbol{k''},t;\boldsymbol{q}-\boldsymbol{q'},T)\left[\left(\boldsymbol{k''}-\frac{\boldsymbol{q}}{2}+\frac{\boldsymbol{q'}}{2}\right)\times\left(\boldsymbol{k}-\frac{\boldsymbol{q}}{2}\right)\cdot\hat{\boldsymbol{\sigma}}\right].
\end{aligned}$$ Let us assume that any inhomogeneity in a system is smooth, so we can neglect all gradient terms $\sim\boldsymbol{q}$. Having Fourier transformed with respect to $\boldsymbol{q}$ and $t$, we obtain: $$\underline{\hat{\Sigma'}}_{\boldsymbol{k}E}(\boldsymbol{R},T)=\underline{\hat{\Sigma}}^{2a}_{\boldsymbol{k}E}(\boldsymbol{R},T)+\underline{\hat{\Sigma}}^{2b}_{\boldsymbol{k}E}(\boldsymbol{R},T)+\underline{\hat{\Sigma}}^{2c}_{\boldsymbol{k}E}(\boldsymbol{R},T),$$ where $$\underline{\hat{\Sigma}}^{2a}_{\boldsymbol{k}E}(\boldsymbol{R},T)=iv_{i}^{3}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}[(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\underline{\hat{G}}_{\boldsymbol{k''}E}(\boldsymbol{R},T),$$ $$\underline{\hat{\Sigma}}^{2b}_{\boldsymbol{k}E}(\boldsymbol{R},T)=iv_{i}^{3}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R},T)[(\boldsymbol{k'}\times\boldsymbol{k''})\cdot\hat{\boldsymbol{\sigma}}]\underline{\hat{G}}_{\boldsymbol{k''}E}(\boldsymbol{R},T),$$ $$\underline{\hat{\Sigma}}^{2c}_{\boldsymbol{k}E}(\boldsymbol{R},T)=iv_{i}^{3}n_{i}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\underline{\hat{G}}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\underline{\hat{G}}_{\boldsymbol{k''}E}(\boldsymbol{R},T)[(\boldsymbol{k''}\times\boldsymbol{k})\cdot\hat{\boldsymbol{\sigma}}].$$
To first order in $\xi_{SO}$, we can express the retarded and advanced Green’s functions by using the Sokhotski formula: $$\begin{aligned}
\hat{G}^{R(A)}_{\boldsymbol{k}E}&=\left(\hat{E}-\hat{\mathcal{H}}_{\boldsymbol{k}}-\hat{\Sigma}^{R(A)}_{\boldsymbol{k}E}\right)^{-1}\approx\frac{1}{2}\sum\limits_{s=\pm}\frac{\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k}s}\pm i\frac{\hbar}{2\tau_{s}}}\\
&=\frac{1}{2}\sum\limits_{s=\pm}\left(\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\right)\left[\mp i\pi\delta(E-E_{\boldsymbol{k}s})\right],
\end{aligned}$$ where $\tau_{s}$ is the spin dependent relaxation time. As a result, the retarded and advanced components of the skew-scattering self-energy vanish, and we deal with its Keldysh part, which survives for $\hat{\Sigma}^{2a}$ and $\hat{\Sigma}^{2c}$ only: $$\begin{aligned}
\hat{\Sigma'}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=i\frac{\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\left(\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}^{K}_{\boldsymbol{k''}E}(\boldsymbol{R},T)+\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\hat{G}^{A}_{\boldsymbol{k''}E} \right)\\
&+i\frac{\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\left(\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}^{K}_{\boldsymbol{k''}E}(\boldsymbol{R},T)+\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\hat{G}^{A}_{\boldsymbol{k''}E} \right)(\boldsymbol{k''}\times\boldsymbol{k})\cdot\hat{\boldsymbol{\sigma}}
\end{aligned}$$ or $$\begin{aligned}
\hat{\Sigma'}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=i\frac{\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\hat{G}^{A}_{\boldsymbol{k''}E}\\
&+i\frac{\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}^{K}_{\boldsymbol{k''}E}(\boldsymbol{R},T)(\boldsymbol{k''}\times\boldsymbol{k})\cdot\hat{\boldsymbol{\sigma}}.
\end{aligned}$$ This expression can be further simplified, as we integrate over $\boldsymbol{k}$: $$\begin{aligned}
\mp\frac{i\pi}{2}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\sum\limits_{s=\pm}\left(\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\right)\delta(E-E_{\boldsymbol{k}s})=\mp\frac{i\pi}{2}(D^{\uparrow}+D^{\downarrow})\left(\hat{\sigma}_{0}+\delta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\right),
\end{aligned}$$ where $D^{\uparrow(\downarrow)}$ is the spin-dependent density of states and $\delta=(D^{\uparrow}-D^{\downarrow})/(D^{\uparrow}+D^{\downarrow})$, so the final form of $\hat{\Sigma'}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)$ is given by: $$\begin{aligned}
\hat{\Sigma'}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=-\frac{\pi}{2}\frac{\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}(D^{\uparrow}+D^{\downarrow})\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\left[\hat{\sigma}_{0}+\delta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\\
&\,\,\,\,\,-\frac{\pi}{2}\frac{\xi_{SO}}{k_{F}^{2}}v_{i}^{3}n_{i}(D^{\uparrow}+D^{\downarrow})\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left[\hat{\sigma}_{0}+\delta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}.
\end{aligned}$$ In the weak exchange coupling limit $(J\ll\varepsilon_{F})$, we can express $D^{\uparrow(\downarrow)}\approx D_{0}(1\mp\beta)$, where $\beta=\frac{J}{2\varepsilon_{F}}$ is the spin polarization factor and $D_{0}=\frac{mk_{F}}{2\pi^{2}\hbar^{2}}$ is the spin independent density of states per spin at the Fermi level. Then, we obtain: $$\begin{aligned}
\hat{\Sigma'}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=-\pi v_{i}^{3}n_{i}D_{0}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\\
&\,\,\,\,\,-\pi v_{i}^{3}n_{i}D_{0}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}
\end{aligned}$$ or $$\begin{aligned}
\hat{\Sigma'}^{K}_{\boldsymbol{k}E}(\boldsymbol{R},T)&=-\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}}\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\\
&\,\,\,\,\,-\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\hat{g}^{K}_{\boldsymbol{k'}E}(\boldsymbol{R},T)(\boldsymbol{k}\times\boldsymbol{k'})\cdot\hat{\boldsymbol{\sigma}},
\end{aligned}
\label{eq:sskew}$$ where $\frac{1}{\tau_{0}}=2\pi v_{i}^{2}n_{i}D_{0}/\hbar$ is the spin independent relaxation time.
Relaxation time
===============
The imaginary part of the retarded and advanced self-energies is related to the momentum relaxation time, which is given by the elastic scattering off the on-site impurity potential and Elliot-Yafet mechanism: $$\hat{\Sigma}^{R(A)}_{\boldsymbol{k}E}=\mp i\frac{\hbar}{2\hat{\tau}_{\boldsymbol{k}}}.$$ Taking into account Eqs. (\[eq:sret\]) and (\[eq:gzero\]), we get: $$\frac{1}{\hat{\tau}_{\boldsymbol{k}}}=\frac{\pi v_{i}^{2}n_{i}}{\hbar}\sum_{s=\pm}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right]\left(\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right)\left[\hat{\sigma}_{0}-i\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right]\delta(E-E_{\boldsymbol{k'}s}),$$ where $\boldsymbol{n}=\boldsymbol{k}\times\boldsymbol{k'}$. In terms of spherical coordinates, $d\boldsymbol{k}=k^{2}dk\,\sin{\theta}d\theta\,d\phi$ with $k\in[0,k_{F}]$, $\theta\in[0,\pi]$ and $\phi\in[0,2\pi]$, it is easy to show: $$\int k_{i}d\boldsymbol{k}=0,$$ where $k_{i}$ is the $i$th cartesian coordinate of $\boldsymbol{k}$, and all terms linear in $\boldsymbol{n}$ vanish. Thus, we get: $$\frac{1}{\hat{\tau}_{\boldsymbol{k}}}=\frac{\pi v_{i}^{2}n_{i}}{\hbar}\sum_{s=\pm}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}+\frac{\xi_{SO}^{2}}{k_{F}^{4}}(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}})(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}})+s\,\frac{\xi_{SO}^{2}}{k_{F}^{4}}(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}})(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}) \right]\delta(E-E_{\boldsymbol{k'}s}),$$ or having used $(\boldsymbol{a}\cdot\hat{\boldsymbol{\sigma}})(\boldsymbol{b}\cdot\hat{\boldsymbol{\sigma}})=(\boldsymbol{a}\cdot\boldsymbol{b})\hat{\sigma}_{0}+i(\boldsymbol{a}\times\boldsymbol{b})\cdot\hat{\boldsymbol{\sigma}}$: $$\begin{aligned}
\frac{1}{\hat{\tau}_{\boldsymbol{k}}}&=\frac{\pi v_{i}^{2}n_{i}}{\hbar}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[(1+\frac{\xi_{SO}^{2}}{k_{F}^{4}}\boldsymbol{n}^{2})\hat{\sigma}_{0}(\delta(E-E_{\boldsymbol{k'}+})+\delta(E-E_{\boldsymbol{k'}-}))\right.\\
&\left.+\left(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}+\frac{\xi_{SO}^{2}}{k_{F}^{4}}\hat{\boldsymbol{\sigma}}\cdot(2\boldsymbol{n}(\boldsymbol{n}\cdot\boldsymbol{m})-\boldsymbol{m}\boldsymbol{n}^{2})\right)(\delta(E-E_{\boldsymbol{k'}+})-\delta(E-E_{\boldsymbol{k'}-}))\right].
\label{eq:tau2}
\end{aligned}$$ Let us also consider the following expressions: $$\begin{aligned}
\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}k'_{i}k'_{j}\,(\delta(E-E_{\boldsymbol{k'}+})\pm\delta(E-E_{\boldsymbol{k'}-}))=0\qquad\mathrm{for}\,\,\, i\ne j,\\
\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}{k'}_{i}^{2}\,(\delta(E-E_{\boldsymbol{k'}+})\pm\delta(E-E_{\boldsymbol{k'}-}))=\frac{1}{6\pi^{2}}\int d{k'}{k'}^{4}\,(\delta(E-E_{\boldsymbol{k'}+})\pm\delta(E-E_{\boldsymbol{k'}-})),\\
\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\boldsymbol{n}^{2}\,(\delta(E-E_{\boldsymbol{k'}+})\pm\delta(E-E_{\boldsymbol{k'}-}))=\frac{1}{3\pi^{2}}\,k^{2}\int d{k'}{k'}^{4}\,(\delta(E-E_{\boldsymbol{k'}+})\pm\delta(E-E_{\boldsymbol{k'}-})).
\end{aligned}$$ We can rewrite Eq. (\[eq:tau2\]) as: $$\begin{aligned}
\frac{1}{\hat{\tau}_{\boldsymbol{k}}}&=\frac{v_{i}^{2}n_{i}}{2\pi\hbar}\int dk'\,\left[({k'}^{2}+\frac{2}{3}\frac{\xi_{SO}^{2}}{k_{F}^{4}}k^{2}{k'}^{4})\hat{\sigma}_{0}(\delta(E-E_{\boldsymbol{k'}+})+\delta(E-E_{\boldsymbol{k'}-}))\right.\\
&\left.+\left(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}{k'}^{2}-\frac{2}{3}\frac{\xi_{SO}^{2}}{k_{F}^{4}}{k'}^{4}(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{k})(\boldsymbol{k}\cdot\boldsymbol{m})\right)(\delta(E-E_{\boldsymbol{k'}+})-\delta(E-E_{\boldsymbol{k'}-}))\right].
\end{aligned}$$ Next, we can employ the following relation for the delta-function: $$\delta(E-E_{\boldsymbol{k}s})=\frac{\delta(k-k_{s})}{\frac{\hbar^{2}k_{s}}{m}},$$ where $k_{\pm}=\sqrt{2m(\varepsilon_{F}\mp J)}/\hbar$, and $\varepsilon_{F}=\frac{\hbar^{2}k_{F}^{2}}{2m}$ is the Fermi energy. Then, integrating over $k'$ gives: $$\begin{aligned}
\frac{1}{\hat{\tau}_{\boldsymbol{k}}}=\frac{v_{i}^{2}n_{i}}{2\pi\hbar}\Big(\frac{m}{\hbar^{2}}(k_{+}+k_{-})\hat{\sigma}_{0}&+\frac{2m}{3\hbar^{2}}\frac{\xi_{SO}^{2}}{k_{F}^{4}}k^{2}(k^{3}_{+}+k^{3}_{-})\hat{\sigma}_{0} +\frac{m}{\hbar^{2}}(k_{+}-k_{-}) \hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\\
&-\frac{2m}{3\hbar^{2}}\frac{\xi_{SO}^{2}}{k_{F}^{4}}(k^{3}_{+}-k^{3}_{-})(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{k})(\boldsymbol{k}\cdot\boldsymbol{m}) \Big).
\end{aligned}$$ Finally, in the weak exchange coupling limit $(J\ll\varepsilon_{F})$, we can perform a Taylor expansion, $k_{\pm}\approx k_{F}(1\mp\beta)$. Neglecting higher order terms $\sim\beta\xi_{SO}^{2}$, we obtain: $$\frac{1}{\hat{\tau}_{\boldsymbol{k}}}=\frac{1}{\tau_{0}}\left(\hat{\sigma_{0}}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}+\frac{2}{3}\frac{\xi_{SO}^{2}}{k_{F}^{2}}k^{2}\hat{\sigma}_{0} \right),$$ where $\frac{1}{\tau_{0}}=2\pi v^{2}_{i}n_{i}D_{0}/\hbar$ is the spin-independent relaxation time due to scattering off the impurity potential.
Averaged velocity operator
==========================
For educational purposes, let us derive the averaged velocity operator in diffusive ferromagnets with extrinsic spin-orbit coupling.[@velo] Within the Lippmann-Schwinger equation, the scattered state $\parallel\!\boldsymbol{k},s\rangle$ can be written in the first order of $\hat{\mathcal{H}}_{\mathrm{imp}}$: $$\begin{aligned}
\parallel\!\boldsymbol{k},s\rangle&=|\boldsymbol{k},s\rangle+\sum_{\boldsymbol{k'}}\hat{G}^{R}_{0,\boldsymbol{k'}}\langle\boldsymbol{k'}|\hat{\mathcal{H}}_{\mathrm{imp}}|\boldsymbol{k}\rangle|\boldsymbol{k'},s\rangle\\
&=|\boldsymbol{k},s\rangle-\frac{1}{\Omega}\frac{i\pi}{2}\sum_{\boldsymbol{k'}s'}(\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})(\hat{\sigma
_{0}}-i\frac{\xi_{SO}}{k_{F}^{2}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{k}\times\boldsymbol{k'}))V(\boldsymbol{k'}-\boldsymbol{k})\delta(E-E_{\boldsymbol{k'}s'})|\boldsymbol{k'},s\rangle,
\end{aligned}$$ and $$\begin{aligned}
\langle\boldsymbol{k},s\!\parallel&=\langle\boldsymbol{k},s|+\sum_{\boldsymbol{k'}}\langle\boldsymbol{k'},s|\langle\boldsymbol{k}|\hat{\mathcal{H}}_{\mathrm{imp}}|\boldsymbol{k'}\rangle\hat{G}^{R}_{0,\boldsymbol{k'}}\\
&=\langle\boldsymbol{k},s|+\frac{1}{\Omega}\frac{i\pi}{2}\sum_{\boldsymbol{k'}s'}\langle\boldsymbol{k'},s| (\hat{\sigma
_{0}}+i\frac{\xi_{SO}}{k_{F}^{2}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{k}\times\boldsymbol{k'}))(\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})V(\boldsymbol{k}-\boldsymbol{k'})\delta(E-E_{\boldsymbol{k'}s'}).
\end{aligned}$$ The corresponding matrix elements of the velocity operator can be found as: $$\begin{aligned}
\boldsymbol{v}_{\boldsymbol{k}\boldsymbol{k'}}^{ss'}=-\frac{i}{\hbar}\langle\boldsymbol{k},s\!\parallel[\hat{\boldsymbol{r}},\hat{\mathcal{H}}]\parallel\!\boldsymbol{k'},s'\rangle=-\frac{i}{\hbar}\langle\boldsymbol{k},s\!\parallel[\hat{\boldsymbol{r}},\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{imp}}]\parallel\!\boldsymbol{k'},s'\rangle,
\end{aligned}$$ where $$-\frac{i}{\hbar}[\hat{\boldsymbol{r}},\hat{\mathcal{H}}]=-\frac{i}{\hbar}[\hat{\boldsymbol{r}},\hat{\mathcal{H}}_{0}+\hat{\mathcal{H}}_{\mathrm{imp}}]=-\frac{i\hbar}{m}\nabla\hat{\sigma}_{0}+\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{\boldsymbol{R}_{i}}\hat{\boldsymbol{\sigma}}\times\nabla V(\boldsymbol{r}-\boldsymbol{R}_{i}).$$ Let us express these terms separately neglecting higher order terms $\sim\xi_{SO}^{2}$: $$\begin{aligned}
&-\frac{i}{\hbar}\langle\boldsymbol{k},s\!\parallel[\hat{\boldsymbol{r}},\hat{\mathcal{H}}_{0}]\parallel\!\boldsymbol{k'},s'\rangle=-\frac{i\hbar}{m}\langle\boldsymbol{k},s\!\parallel\nabla\parallel\!\boldsymbol{k'},s'\rangle\\
&=-\frac{i\hbar}{m}\langle\boldsymbol{k}|\nabla|\boldsymbol{k'}\rangle\delta_{ss'} - \frac{i\pi}{2}\sum_{s''}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}V(\boldsymbol{k''}-\boldsymbol{k'})\delta(E-E_{\boldsymbol{k''}s''})\langle\boldsymbol{k}|\nabla|\boldsymbol{k''}\rangle\langle s|(\hat{\sigma}_{0}+s''\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})(\hat{\sigma
_{0}}-i\frac{\xi_{SO}}{k_{F}^{2}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{k'}\times\boldsymbol{k''}))|s'\rangle\\
&\qquad\qquad\qquad\qquad\,\,\,\,\,\, + \frac{i\pi}{2}\sum_{s''}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}V(\boldsymbol{k}-\boldsymbol{k''})\delta(E-E_{\boldsymbol{k''}s''})\langle s|(\hat{\sigma
_{0}}+i\frac{\xi_{SO}}{k_{F}^{2}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{k}\times\boldsymbol{k''}))(\hat{\sigma}_{0}+s''\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})|s'\rangle\langle\boldsymbol{k''}|\nabla|\boldsymbol{k'}\rangle,
\end{aligned}$$ and $$\begin{aligned}
&-\frac{i}{\hbar}\langle\boldsymbol{k},s\!\parallel[\hat{\boldsymbol{r}},\hat{\mathcal{H}}_{\mathrm{imp}}]\parallel\!\boldsymbol{k'},s'\rangle=\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{\boldsymbol{R}_{i}}\langle\boldsymbol{k},s\!\parallel\hat{
\boldsymbol{\sigma}}\times\nabla V(\boldsymbol{r}-\boldsymbol{R}_{i})\parallel\!\boldsymbol{k'},s'\rangle\\
&=\frac{i}{\Omega}\frac{\xi_{SO}}{\hbar k_{F}^{2}}\langle s|\hat{\boldsymbol{\sigma}}|s'\rangle\times(\boldsymbol{k}-\boldsymbol{k'})V(\boldsymbol{k}-\boldsymbol{k'})\\
& +\frac{1}{\Omega} \frac{\pi}{2}\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{s''}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}V(\boldsymbol{k''}-\boldsymbol{k'})V(\boldsymbol{k}-\boldsymbol{k''})\delta(E-E_{\boldsymbol{k''}s''})\langle s|\hat{
\boldsymbol{\sigma}}\times(\boldsymbol{k}-\boldsymbol{k''})(\hat{\sigma}_{0}+s''\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})|s'\rangle\\
& -\frac{1}{\Omega} \frac{\pi}{2}\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{s''}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}V(\boldsymbol{k}-\boldsymbol{k''})V(\boldsymbol{k''}-\boldsymbol{k'})\delta(E-E_{\boldsymbol{k''}s''})\langle s|(\hat{\sigma}_{0}+s''\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})\hat{
\boldsymbol{\sigma}}\times(\boldsymbol{k''}-\boldsymbol{k'})| s'\rangle.
\end{aligned}$$ Upon impurity averaging we obtain: $$\begin{aligned}
\boldsymbol{v}_{\boldsymbol{k}}^{ss'}&=\frac{\hbar}{m}\boldsymbol{k}\,\delta_{ss'}+v_{i}^{2}n_{i}\frac{\pi}{2}\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{s''}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\delta(E-E_{\boldsymbol{k''}s''})\langle s|\left\{\hat{
\boldsymbol{\sigma}}\times(\boldsymbol{k}-\boldsymbol{k''}),(\hat{\sigma}_{0}+s''\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})\right\}|s'\rangle\\
&=\frac{\hbar}{m}\boldsymbol{k}\,\delta_{ss'}+v_{i}^{2}n_{i}\pi\frac{\xi_{SO}}{\hbar k_{F}^{2}}\sum_{s''}\int\frac{d\boldsymbol{k''}}{(2\pi)^{3}}\delta(E-E_{\boldsymbol{k''}s''})\langle s|\hat{\boldsymbol{\sigma}}\times\boldsymbol{k}+s''\boldsymbol{m}\times\boldsymbol{k}|s'\rangle,
\end{aligned}$$ or in the limit $J\ll\varepsilon_{F}$: $$\hat{\boldsymbol{v}}_{\boldsymbol{k}}=\frac{\hbar}{m}\boldsymbol{k}\,\hat{\sigma}_{0}+\frac{1}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\hat{\boldsymbol{\sigma}}\times\boldsymbol{k}-\frac{\beta}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{m}\times\boldsymbol{k}\,\hat{\sigma}_{0}.
\label{eq:av}$$
Quantum transport equations
===========================
Having integrated Eq. (\[eq:keldysh\]) over energy, we arrive at the kinetic equation written for the distribution function $\hat{g}_{\boldsymbol{k}}$: $$-i[\hat{g}_{\boldsymbol{k}},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]+\frac{\hbar^{2}}{m}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\,\hat{g}_{\boldsymbol{k}}=coll,
\label{eq:keld2}$$ where the collision integral is defined as: $$coll=\mathcal{J}_{\boldsymbol{k}}+\mathcal{I}_{\boldsymbol{k}},$$ and $$\mathcal{J}_{\boldsymbol{k}}=\int\frac{dE}{2\pi}\left(\hat{\Sigma}^{K}_{\boldsymbol{k}E}\hat{G}^{A}_{\boldsymbol{k}E}-\hat{G}^{R}_{\boldsymbol{k}E}\hat{\Sigma}^{K}_{\boldsymbol{k}E} \right),$$ $$\mathcal{I}_{\boldsymbol{k}}=\int\frac{dE}{2\pi}\left(\hat{\Sigma}^{R}_{\boldsymbol{k}E}\hat{g}^{K}_{\boldsymbol{k}E}-\hat{g}^{K}_{\boldsymbol{k}E}\hat{\Sigma}^{A}_{\boldsymbol{k}E} \right).$$ Let us proceed with its detailed derivation. Taking into account the Kadanoff-Baym anzats (\[eq:anzats\]) for $\hat{g}^{K}_{\boldsymbol{k}E}$, the integration over energy (up to a given Fermi level $\varepsilon_{F}$) can be performed by using the residue theorem: $$\begin{aligned}
\int\frac{dE}{2\pi}\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}&=\int\frac{dE}{8\pi}\sum\limits_{s,s'}\frac{\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k}s}+i\frac{\hbar}{2\tau_{s}}}\hat{g}_{\boldsymbol{k'}}\frac{\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k'}s'}-i\frac{\hbar}{2\tau_{s'}}}\\
&=-\frac{i}{4}\sum_{s,s'}\frac{(\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})\hat{g}_{\boldsymbol{k'}}(\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})}{\varepsilon_{F}-E_{\boldsymbol{k'}s'}-i\frac{\hbar}{2}\left(\frac{1}{\tau_{s'}}+\frac{1}{\tau_{s}}\right)},
\end{aligned}$$ and $$\begin{aligned}
\int\frac{dE}{2\pi}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k}E}&=\int\frac{dE}{8\pi}\sum\limits_{s,s'}\frac{\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k'}s'}+i\frac{\hbar}{2\tau_{s'}}}\hat{g}_{\boldsymbol{k'}}\frac{\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k}s}-i\frac{\hbar}{2\tau_{s}}}\\
&=\frac{i}{4}\sum_{s,s'}\frac{(\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})\hat{g}_{\boldsymbol{k'}}(\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})}{\varepsilon_{F}-E_{\boldsymbol{k'}s'}+i\frac{\hbar}{2}\left(\frac{1}{\tau_{s}}+\frac{1}{\tau_{s'}}\right)},
\end{aligned}$$ while $$\int\frac{dE}{2\pi}\hat{G}^{R(A)}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{R(A)}_{\boldsymbol{k'}E}=0,$$ where the retarded and advanced Green’s functions are defined as: $$\hat{G}_{\boldsymbol{k}E}^{R(A)}=\frac{1}{2}\sum\limits_{s=\pm}\frac{\hat{\sigma}_{0}+s\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}}{E-E_{\boldsymbol{k}s}\pm i\frac{\hbar}{2\tau_{s}}}.$$ Assuming the scattering term in the denominator to be small and transport properties to be described solely by the electrons close to the Fermi level, we can rewrite these expressions with the Sokhotski formula: $$\begin{aligned}
\int\frac{dE}{2\pi}\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}&=\frac{\pi}{2}\sum_{s'}\hat{g}_{\boldsymbol{k'}}(\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})\delta(\varepsilon_{F}-E_{\boldsymbol{k'}s'}),
\end{aligned}$$ and $$\begin{aligned}
\int\frac{dE}{2\pi}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k}E}=\frac{\pi}{2}\sum_{s'}(\hat{\sigma}_{0}+s'\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m})\hat{g}_{\boldsymbol{k'}}\delta(\varepsilon_{F}-E_{\boldsymbol{k'}s'}).
\end{aligned}$$
Starting from the Born approximation (\[eq:skel\]), we have: $$\begin{aligned}
\hat{\Sigma}^{K}_{\boldsymbol{k}E}\hat{G}^{A}_{\boldsymbol{k}E}&=v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k}E}+i\frac{\xi_{SO}}{k^{2}_{F}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k}E} \right.\\
&\left.-i\frac{\xi_{SO}}{k_{F}^{2}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{A}_{\boldsymbol{k}E}+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{A}_{\boldsymbol{k}E}\right]\\
&+\frac{v_{i}^{2}n_{i}}{2}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left\{(\boldsymbol{k'}-\boldsymbol{k}\}\times\hat{\boldsymbol{\sigma}},\hat{G}^{R}_{\boldsymbol{k'}E}\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\right\}\hat{G}^{A}_{\boldsymbol{k}E},
\end{aligned}$$ and $$\begin{aligned}
\hat{G}^{R}_{\boldsymbol{k}E}\hat{\Sigma}^{K}_{\boldsymbol{k}E}&=-v_{i}^{2}n_{i}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\left[\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}+i\frac{\xi_{SO}}{k_{F}^{2}}\hat{G}^{R}_{\boldsymbol{k}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E} \right.\\
&\left.-i\frac{\xi_{SO}}{k_{F}^{2}}\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\hat{G}^{R}_{\boldsymbol{k}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right]\\
&-\frac{v_{i}^{2}n_{i}}{2}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\hat{G}^{R}_{\boldsymbol{k}E}\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\right\}.
\end{aligned}$$ By using the following relations: $$\begin{aligned}
\hat{\sigma}_{a}\hat{\sigma}_{b}&=i\varepsilon_{abc}\,\hat{\sigma}_{c}+\delta_{ab}\hat{\sigma}_{0},\\
(\boldsymbol{a}\cdot\hat{\boldsymbol{\sigma}})(\boldsymbol{b}\cdot\hat{\boldsymbol{\sigma}})&=(\boldsymbol{a}\cdot\boldsymbol{b})\hat{\sigma}_{0}+i(\boldsymbol{a}\times\boldsymbol{b})\cdot\hat{\boldsymbol{\sigma}},
\end{aligned}$$ these terms give: $$\begin{aligned}
\int\frac{dE}{2\pi}\left[\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\right]&=\pi\hat{g}_{\boldsymbol{k'}}\delta_{T}+\pi\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J},
\end{aligned}$$ $$\begin{aligned}
&\int\frac{dE}{2\pi} i\frac{\xi_{SO}}{k_{F}^{2}}\left[[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}]\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}]\right]=\\
&=i\pi\frac{\xi_{SO}}{k_{F}^{2}}[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}]\delta_{T}+i\pi\frac{\xi_{SO}}{k_{F}^{2}}\left(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}-\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \right)\delta_{J}\\
&-\pi\frac{\xi_{SO}}{k_{F}^{2}}\{(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}\}\delta_{J},
\end{aligned}$$ $$\begin{aligned}
&\int\frac{dE}{2\pi} \frac{\xi^{2}_{SO}}{k_{F}^{4}}\left[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right]=\\
&=\pi\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{T}
+\pi\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&+i\pi\frac{\xi^{2}_{SO}}{k_{F}^{4}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{J}
-i\pi\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\delta_{J},
\end{aligned}$$ $$\begin{aligned}
&\int\frac{dE}{2\pi} \left[\left\{(\boldsymbol{k'}-\boldsymbol{k}\}\times\hat{\boldsymbol{\sigma}},\hat{G}^{R}_{\boldsymbol{k'}E}\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\right\}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\right\} \right]=\\
&=\pi\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\right\}\delta_{T}
+\pi\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\{\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}},\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\}\right\}\delta_{J},
\end{aligned}$$ where the following notations are used: $$\begin{aligned}
\delta_{T}&=\delta(\varepsilon_{F}-E_{\boldsymbol{k'}+})+\delta(\varepsilon_{F}-E_{\boldsymbol{k'}-}),\\
\delta_{J}&=\frac{1}{2}\left[\delta(\varepsilon_{F}-E_{\boldsymbol{k'}+})-\delta(\varepsilon_{F}-E_{\boldsymbol{k'}-})\right].
\end{aligned}$$ For the skew-scattering self-energy Eq. (\[eq:sskew\]), we have: $$\begin{aligned}
\hat{\Sigma'}^{K}_{\boldsymbol{k}E}\hat{G}^{A}_{\boldsymbol{k}E}&=-\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\hat{G}^{A}_{\boldsymbol{k}E}\\
&-\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{A}_{\boldsymbol{k}E},
\end{aligned}$$ and $$\begin{aligned}
\hat{G}^{R}_{\boldsymbol{k}E}\hat{\Sigma'}^{K}_{\boldsymbol{k}E}&=\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\hat{G}^{R}_{\boldsymbol{k}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right]\\
&+\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\hat{G}^{R}_{\boldsymbol{k}E}\left[\hat{\sigma}_{0}-\beta\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\right]\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},
\end{aligned}$$ that gives: $$\begin{aligned}
&\int\frac{dE}{2\pi}\,\left[\left\{\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\right\}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\left\{\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\right\}\right]=\\
&=\pi\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\}\delta_{T}+\pi\left\{\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\right\}\delta_{J},
\end{aligned}$$ $$\begin{aligned}
&\int\frac{dE}{2\pi}\,\left[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{A}_{\boldsymbol{k}E}\right.\\
&\qquad\qquad\left.+\hat{G}^{R}_{\boldsymbol{k}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}+\hat{G}^{R}_{\boldsymbol{k}E}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{g}_{\boldsymbol{k'}}\hat{G}^{A}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right]=\\
&=\pi\left(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}+\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right) \delta_{T}+\pi\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \}\delta_{J}+\pi\boldsymbol{m}^{2}\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&\qquad\qquad+i\pi\left((\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}-\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \hat{g}_{\boldsymbol{k'}} (\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}} \right)\delta_{J}.
\end{aligned}$$
Finally, we obtain: $$\begin{aligned}
\mathcal{J}_{\boldsymbol{k}}&=\pi a_{1}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[\hat{g}_{\boldsymbol{k'}}\delta_{T}+\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J}\\
&\qquad\qquad+i\frac{\xi_{SO}}{k_{F}^{2}}[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}]\delta_{T}+i\frac{\xi_{SO}}{k_{F}^{2}}\left(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}-\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \right)\delta_{J}-\frac{\xi_{SO}}{k_{F}^{2}}\{(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}\}\delta_{J}\\
&\qquad\qquad+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{T}
+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&\qquad\qquad+i\frac{\xi^{2}_{SO}}{k_{F}^{4}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{J}
-i\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\delta_{J}\\
&\qquad\qquad+\frac{1}{2}\frac{\xi_{SO}}{k_{F}^{2}}\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\right\}\delta_{T}
+\frac{1}{2}\frac{\xi_{SO}}{k_{F}^{2}}\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\{\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}},\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\}\right\}\delta_{J}\\
&-\pi a_{2}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\}\delta_{T}+\left\{\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\right\}\delta_{J}\\
&\qquad\qquad-\beta\left(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}+\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right) \delta_{T}-\beta\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \}\delta_{J}-\beta\boldsymbol{m}^{2}\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&\qquad\qquad-i\beta\left((\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}-\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \hat{g}_{\boldsymbol{k'}} (\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}} \right)\delta_{J}\Big],
\end{aligned}$$ where $a_{1}=n_{i}v_{i}^{2}$ and $a_{2}=\frac{\hbar}{2}\frac{v_{i}}{\tau_{0}}\frac{\xi_{SO}}{k_{F}^{2}}$.
In a similar manner, by using Eqs. (\[eq:sret\]) and (\[eq:sadv\]) we proceed with the second part of the collision integral $\mathcal{I}_{\boldsymbol{k}}$: $$\begin{aligned}
\mathcal{I}_{\boldsymbol{k}}&=-a_{1}\int\frac{dE}{2\pi}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\,\Big[\big(\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\big)\hat{G}^{R}_{\boldsymbol{k'}E}(\hat{\sigma}_{0}-i\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}})\hat{g}_{\boldsymbol{k}}\hat{G}^{A}_{\boldsymbol{k}E}\\
&\qquad\qquad\qquad\qquad\qquad+\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k}}\big(\hat{\sigma}_{0}+i\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\big)\hat{G}^{A}_{\boldsymbol{k'}E}(\hat{\sigma}_{0}-i\frac{\xi_{SO}}{k_{F}^{2}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}})\Big],
\end{aligned}$$ where $$\begin{aligned}
\int\frac{dE}{2\pi}\left[\hat{G}^{R}_{\boldsymbol{k'}E}\hat{g}_{\boldsymbol{k}}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k}}\hat{G}^{A}_{\boldsymbol{k'}E}\right]&=\pi\hat{g}_{\boldsymbol{k}}\delta_{T}+\pi\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J},
\end{aligned}$$ $$\begin{aligned}
&\int\frac{dE}{2\pi} i\frac{\xi_{SO}}{k_{F}^{2}}\left[[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{G}^{R}_{\boldsymbol{k'}E}]\hat{g}_{\boldsymbol{k}}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k}}[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{G}^{A}_{\boldsymbol{k'}E}]\right]=\\
&=-\pi\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\}\delta_{J},
\end{aligned}$$ $$\begin{aligned}
&\int\frac{dE}{2\pi} \frac{\xi^{2}_{SO}}{k_{F}^{4}}\left[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{R}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k}}\hat{G}^{A}_{\boldsymbol{k}E}+\hat{G}^{R}_{\boldsymbol{k}E}\hat{g}_{\boldsymbol{k}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{G}^{A}_{\boldsymbol{k'}E}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\right]=\\
&=\pi \frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}^{2}\hat{g}_{\boldsymbol{k}}\delta_{T}+\pi \frac{\xi^{2}_{SO}}{k_{F}^{4}}(2(\boldsymbol{n}\cdot\boldsymbol{m})\boldsymbol{n}-\boldsymbol{n}^{2}\boldsymbol{m})\cdot\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\}\delta_{J},
\end{aligned}$$ so we obtain: $$\begin{aligned}
\mathcal{I}_{\boldsymbol{k}}&=-\pi a_{1}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[\hat{g}_{\boldsymbol{k}}\delta_{T}+\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J}-\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\}\delta_{J}\\
&\qquad\qquad+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}^{2}\hat{g}_{\boldsymbol{k}}\delta_{T}+\frac{\xi^{2}_{SO}}{k_{F}^{4}}(2(\boldsymbol{n}\cdot\boldsymbol{m})\boldsymbol{n}-\boldsymbol{n}^{2}\boldsymbol{m})\cdot\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\}\delta_{J}\Big].
\end{aligned}$$ Finally, the collision integral is written as: $$\begin{aligned}
coll&=\pi a_{1}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[(\hat{g}_{\boldsymbol{k'}}-\hat{g}_{\boldsymbol{k}})\delta_{T}+\{\hat{g}_{\boldsymbol{k'}}-\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J}\\
&\qquad\qquad+i\frac{\xi_{SO}}{k_{F}^{2}}[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}]\delta_{T}+i\frac{\xi_{SO}}{k_{F}^{2}}\left(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}-\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \right)\delta_{J}-\frac{\xi_{SO}}{k_{F}^{2}}\{(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}},\hat{g}_{\boldsymbol{k'}}\}\delta_{J}\\
&\qquad\qquad+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{T}
+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&\qquad\qquad+i\frac{\xi^{2}_{SO}}{k_{F}^{4}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{J}
-i\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\delta_{J}\\
&\qquad\qquad-\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}^{2}\hat{g}_{\boldsymbol{k}}\delta_{T}-\frac{\xi^{2}_{SO}}{k_{F}^{4}}(2(\boldsymbol{n}\cdot\boldsymbol{m})\boldsymbol{n}-\boldsymbol{n}^{2}\boldsymbol{m})\cdot\{\hat{g}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\}\delta_{J}\\
&\qquad\qquad+\frac{1}{2}\frac{\xi_{SO}}{k_{F}^{2}}\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}}\right\}\delta_{T}
+\frac{1}{2}\frac{\xi_{SO}}{k_{F}^{2}}\left\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\{\nabla_{\boldsymbol{R}}\hat{g}_{\boldsymbol{k'}},\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\}\right\}\delta_{J}\Big]\\
&-\pi a_{2}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\}\delta_{T}+\left\{\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\right\}\delta_{J}\\
&\qquad\qquad-\beta\left(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{g}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}+\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right) \delta_{T}-\beta\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{g}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \}\delta_{J}-\beta\boldsymbol{m}^{2}\{\hat{g}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&\qquad\qquad-i\beta\left((\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{g}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}-\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \hat{g}_{\boldsymbol{k'}} (\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}} \right)\delta_{J}\Big].
\end{aligned}
\label{eq:collision1}$$ By neglecting higher order terms $\beta\delta_{J}\sim\beta^{2}$ in skew-scattering and introducing a more familiar distribution function $\hat{g}_{\boldsymbol{k}}=\hat{\sigma}_{0}-2\hat{h}_{\boldsymbol{k}}$, we get: $$\begin{aligned}
coll&=-2\pi a_{1}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[(\hat{h}_{\boldsymbol{k'}}-\hat{h}_{\boldsymbol{k}})\delta_{T}+\{\hat{h}_{\boldsymbol{k'}}-\hat{h}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J}\\
&\qquad\qquad+i\frac{\xi_{SO}}{k_{F}^{2}}[\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\hat{h}_{\boldsymbol{k'}}]\delta_{T}+i\frac{\xi_{SO}}{k_{F}^{2}}(\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}-\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}})\delta_{J}-\frac{\xi_{SO}}{k_{F}^{2}}\{(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}},\hat{h}_{\boldsymbol{k'}}\}\delta_{J}\\
&\qquad\qquad+\frac{1}{2}\frac{\xi_{SO}}{k_{F}^{2}}\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\nabla_{\boldsymbol{R}}\hat{h}_{\boldsymbol{k'}}\}\delta_{T}
+\frac{1}{2}\frac{\xi_{SO}}{k_{F}^{2}}\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\{\nabla_{\boldsymbol{R}}\hat{h}_{\boldsymbol{k'}},\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\}\}\delta_{J}\\
&\qquad\qquad+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{T}
+\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\boldsymbol{m}\{\hat{h}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}} \}\delta_{J}\\
&\qquad\qquad+i\frac{\xi^{2}_{SO}}{k_{F}^{4}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\delta_{J}
-i\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}(\boldsymbol{n}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}\delta_{J}\\
&\qquad\qquad-\frac{\xi^{2}_{SO}}{k_{F}^{4}}\boldsymbol{n}^{2}\hat{h}_{\boldsymbol{k}}\delta_{T}-\frac{\xi^{2}_{SO}}{k_{F}^{4}}(2(\boldsymbol{n}\cdot\boldsymbol{m})\boldsymbol{n}-\boldsymbol{n}^{2}\boldsymbol{m})\cdot\{\hat{h}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\}\delta_{J}\Big]\\
&+2\pi a_{2}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\Big[\{\hat{h}_{\boldsymbol{k'}},\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\}\delta_{T}+\{\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}},\{\hat{h}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\}\delta_{J}-\beta(\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\hat{h}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}+\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} ) \delta_{T}\Big],
\end{aligned}
\label{eq:collision2}$$ while the Keldysh equation (\[eq:keld2\]) is rewritten as: $$-2\left(-i[\hat{h}_{\boldsymbol{k}},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]+\frac{\hbar^{2}}{m}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\,\hat{h}_{\boldsymbol{k}} \right)=coll.
\label{eq:keld3}$$
Ferromagnetic solution without spin-orbit coupling
==================================================
Let us consider Eq. (\[eq:keld3\]) without extrinsic spin-orbit coupling: $$\begin{aligned}
-i[\hat{h}_{\boldsymbol{k}}, J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]+\frac{\hbar^2}{m}(\boldsymbol{k} \cdot \nabla_{\boldsymbol{R}}) \hat{h}_{\boldsymbol{k}}= \pi a_{1}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\big[(\hat{h}_{\boldsymbol{k'}}-\hat{h}_{\boldsymbol{k}})\delta_{T}+\{\hat{h}_{\boldsymbol{k'}}-\hat{h}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J}\big].
\end{aligned}$$ By introducing $\Omega=i\hbar/\tau_0$, ${\hat}U = \hat{\boldsymbol{\sigma}} \cdot \boldsymbol{m}$, and: $$\begin{aligned}
\label{eq:k_fer}
\hat{K}=-\frac{\hbar^2}{m}(\boldsymbol{k}\cdot \nabla_{\boldsymbol{R}}) \hat{h}_{\boldsymbol{k}}+\pi a_{1}\int\frac{d\boldsymbol{k'}}{(2\pi)^{3}}\big[\hat{h}_{\boldsymbol{k'}}\delta_{T}+\{\hat{h}_{\boldsymbol{k'}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}\}\delta_{J}\big],
\end{aligned}$$ in the weak exchange coupling limit ($J\ll\varepsilon_{F}$) we have: $$\begin{aligned}
\hat{h}_{\boldsymbol{k}} = \frac{i}{\Omega} \hat{K} + \frac{\beta}{2}\{\hat{h}_{\boldsymbol{k}}, \hat{U}\} + \frac{J}{\Omega}[\hat{U}, \hat{h}_{\boldsymbol{k}}].
\end{aligned}$$ This equation is solved iteratively: $$\begin{aligned}
\hat{h}_{\boldsymbol{k}} &= \frac{i}{\Omega}\left(1+2\frac{J^{2}}{\Omega^{2}}+8\frac{J^{4}}{\Omega^{4}}+32\frac{J^{6}}{\Omega^{6}}+... \right) \hat{K} + \frac{i}{\Omega}\frac{\beta^{2}}{2}\left(1+\beta^{2}+\beta^{4}+... \right)\hat{K} \\
&+ \frac{i}{\Omega}\frac{\beta}{2}\left(1+\beta^{2}+\beta^{4}+... \right)\{\hat{U},\hat{K}\}+\frac{i}{\Omega}\frac{J}{\Omega}\left(1+4\frac{J^{2}}{\Omega^{2}}+16\frac{J^{4}}{\Omega^{4}}+... \right)[\hat{U},\hat{K}]\\
&+ \frac{i}{\Omega}\frac{\beta^{2}}{2}\left(1+\beta^{2}+\beta^{4}+... \right)\hat{U}\hat{K}\hat{U}-2\frac{i}{\Omega}\frac{J^{2}}{\Omega^{2}}\left(1+4\frac{J^{2}}{\Omega^{2}}+16\frac{J^{4}}{\Omega^{4}}+... \right)\hat{U}\hat{K}\hat{U},
\end{aligned}$$ or by using $1+x+x^{2}+x^{3}...=\frac{1}{1-x}$ for $x\ll 1$: $$\begin{aligned}
\hat{h}_{\boldsymbol{k}} &=\frac{i}{\Omega}\left(\frac{\Omega^{2}-2J^{2}}{\Omega^{2}-4J^{2}}+\frac{\beta^{2}}{2(1-\beta^{2})} \right)\hat{K}+\frac{i}{\Omega}\frac{\beta}{2}\frac{1}{1-\beta^{2}}\{\hat{U},\hat{K}\}\\
&+\frac{iJ}{\Omega^{2}-4J^{2}}[\hat{U},\hat{K}]+\frac{i}{\Omega}\left(\frac{\beta^{2}}{2(1-\beta^{2})}-\frac{2J^{2}}{\Omega^{2}-4J^{2}} \right)\hat{U}\hat{K}\hat{U}.
\end{aligned}$$ Since $J^{2}/\Omega^{2}\ll1$ and $\beta^{2}\ll1$, this solution is well justified. By substituting $\Omega$, $\hat{K}$ and $\hat{U}$ and removing the delta-functions, we have: $$\begin{aligned}
\frac{\hbar}{\tau_{0}}\hat{h}_{\boldsymbol{k}}=& -\frac{\hbar^{2}}{m}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\hat{h}_{\boldsymbol{k}}+\frac{\tau_{0}^{2}}{m}\frac{2J^{2}}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\left(\hat{h}_{\boldsymbol{k}}-\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \hat{h}_{\boldsymbol{k}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right) \\
&-\frac{\beta^{2}}{2(1-\beta^{2})}\frac{\hbar^{2}}{m}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\left(\hat{h}_{\boldsymbol{k}}+\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \hat{h}_{\boldsymbol{k}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \right)-\frac{\beta}{2(1-\beta^{2})}\frac{\hbar^{2}}{m}(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\{\hat{h}_{\boldsymbol{k}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \}\\
&+\frac{\hbar}{\tau_{0}}\int\frac{d\check{\boldsymbol{k'}}}{4\pi}\,\hat{h}_{\boldsymbol{k'}} +\frac{\tau_{0}}{\hbar}\frac{2J^{2}}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}}\left(\int \frac{d\check{\boldsymbol{k'}}}{4\pi}\,\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \hat{h}_{\boldsymbol{k'}}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}-\int \frac{d\check{\boldsymbol{k'}}}{4\pi}\,\hat{h}_{\boldsymbol{k'}}\right)\\
&-\frac{iJ}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}} \int\frac{d\check{\boldsymbol{k'}}}{4\pi}\,[\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m},\hat{h}_{\boldsymbol{k'}}] +\frac{\tau_{0}\hbar}{m}\frac{iJ}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}} (\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})[\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m},\hat{h}_{\boldsymbol{k}}],
\label{eq:fer1}
\end{aligned}$$ where $\check{\boldsymbol{k}}=\boldsymbol{k}/|\boldsymbol{k}|$. In the diffusive limit $v_{F}\tau\ll L$, where $L$ is the system size, we can partition the distribution function $\hat{h}_{\boldsymbol{k}}$ into the isotropic charge $\mu_{c}$ and spin $\boldsymbol{\mu}$ and anisotropic $\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k}}$ components, $\hat{h}_{\boldsymbol{k}}=\mu_{c}\hat{\sigma}_{0}+\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}}+\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k}}$. This form is nothing else but the generalized $p$-wave approximation for the distribution function. Upon integrating Eq. (\[eq:fer1\]) multiplied by $\check{\boldsymbol{k}}$ over $d\check{\boldsymbol{k}}/4\pi$ and neglecting higher order terms $\sim \beta^{2}$, we obtain the following expression for $\hat{\boldsymbol{j}}$: $$\begin{aligned}
\label{eq:j_fer}
\frac{\hbar}{\tau_{0}}\hat{\boldsymbol{j}}=&-\frac{\hbar^{2}}{m}k\nabla\left(\mu_{c}\hat{\sigma}_{0}+\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}}+\beta\mu_{c}\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}+\beta\boldsymbol{\mu}\cdot{\boldsymbol{m}}\hat{\sigma}_{0}\right)\\
&-\frac{\tau_{0}\hbar}{m}\frac{2J}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}}k\nabla\,\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{m}\times\boldsymbol{\mu})-\frac{\tau_{0}^{2}}{m}\frac{4J^{2}}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}}k\nabla\,\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu})).
\end{aligned}$$
The charge and spin currents (its $j$th component in the spin space) can be defined as: $$\tilde{\boldsymbol{j}}^{C}=\frac{1}{4}\int\frac{d\check{\boldsymbol{k}}}{4\pi}\mathrm{Tr}\,\{\hat{\boldsymbol{v}}_{\boldsymbol{k}},\hat{h}_{\boldsymbol{k}}\}=\frac{v_{F}}{6}\mathrm{Tr}\,\hat{\boldsymbol{j}},$$ and $$\tilde{\boldsymbol{J}}^{S}_{j}=\frac{1}{4}\int\frac{d\check{\boldsymbol{k}}}{4\pi}\mathrm{Tr}\,\big[\hat{\sigma}_{j}\{\hat{\boldsymbol{v}}_{\boldsymbol{k}},\hat{h}_{\boldsymbol{k}}\}\big]=\frac{v_{F}}{6}\mathrm{Tr}\,\big[\hat{\sigma}_{j}\,\hat{\boldsymbol{j}}\big],$$ where the velocity operator is defined as $\hat{\boldsymbol{v}}_{\boldsymbol{k}}=\frac{\hbar}{m}\boldsymbol{k}\hat{\sigma}_{0}$, and $v_{F}=\frac{\hbar}{m} k_{F}$ is the Fermi velocity. Thus, neglecting higher order terms $\sim J^{3}$ gives: $$\tilde{\boldsymbol{j}}^{C} = - D\nabla (\mu_c + \beta \boldsymbol{\mu} \cdot \boldsymbol{m}),
\label{eq:cc_fer}$$ and $$\frac{\tilde{\boldsymbol{J}}^{S}_{j}}{D}= -\nabla( \mu_{j}+\beta\mu_c m_{j})-\frac{\tau_{0}}{\tau_{L}}\nabla(\boldsymbol{m}\times\boldsymbol{\mu})_{j}-\frac{\tau_{0}}{\tau_{\phi}}\nabla(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu}))_{j},
\label{eq:sc_fer}$$ where $D=\tau_{0}v_{F}^{2}/3$ is the diffusion coefficient, $1/\tau_{L}=2J/\hbar$ is the Larmor precession time, and $1/\tau_{\phi}=4J^{2}\tau_{0}/\hbar^{2}$ is the spin dephasing time.
The corresponding equations for the charge and spin densities are obtained by integrating Eq. (\[eq:fer1\]) over $\check{\boldsymbol{k}}$ and neglecting terms $\sim J\nabla^{2}$ and $\sim\beta\nabla^{2}$: $$-\frac{\hbar^{2}}{m}\frac{k}{3}\nabla \cdot \hat{\boldsymbol{j}}+\frac{\tau_{0}}{\hbar}\frac{4J^{2}}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu}))+\frac{2J}{1+\frac{4J^{2}\tau_{0}^{2}}{\hbar^{2}}}\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{m}\times\boldsymbol{\mu})=0.$$ Taking $\mathrm{Tr}\,[...]$ and $\mathrm{Tr}\,[\hat{\boldsymbol{\sigma}}...]$ and neglecting terms $\sim J^{3}$ leads to: $$0=D\nabla^{2}(\mu_{c}+\beta\boldsymbol{\mu}\cdot\boldsymbol{m})=-\nabla\cdot\tilde{\boldsymbol{j}}^{C}$$ and $$0=-\nabla\cdot\tilde{\boldsymbol{J}}^{S}+\frac{1}{\tau_{L}}(\boldsymbol{m}\times\boldsymbol{\mu})+\frac{1}{\tau_{\phi}}(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu}))$$ for the charge and spin components, respectively. Finally, by recovering time-dependence from Eq. (\[eq:kelfull\]) we obtain: $$\partial_{T}\mu_{c}=-\nabla\cdot\tilde{\boldsymbol{j}}^{C}
\label{eq:ca_fer}$$ and $$\partial_{T}\boldsymbol{\mu}=-\nabla\cdot\tilde{\boldsymbol{J}}^{S}+\frac{1}{\tau_{L}}(\boldsymbol{m}\times\boldsymbol{\mu})+\frac{1}{\tau_{\phi}}(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu})).
\label{eq:sa_fer}$$ Thus, Eqs. (\[eq:cc\_fer\]), (\[eq:sc\_fer\]), (\[eq:ca\_fer\]) and (\[eq:sa\_fer\]) define a set of the drift-diffusion equations for ferromagnets in the absence of extrinsic spin-orbit coupling.
Ferromagnetic solution with spin-orbit coupling
===============================================
To derive drift-diffusion equations including extrinsic spin-orbit coupling, we employ the same $p$-wave approximation for $\hat{h}_{\boldsymbol{k}}$. Then, we have: $$\begin{gathered}
-i[\hat{h}_{\boldsymbol{k}},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}]=2J\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{\mu}\times\boldsymbol{m})-i[\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k}},J\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m}],\\
(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\hat{h}_{\boldsymbol{k}}=(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\mu_{c}\hat{\sigma}_{0}+(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}}+(\boldsymbol{k}\cdot\nabla_{\boldsymbol{R}})\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k}}\end{gathered}$$ for the left-hand side of the Keldysh equation (\[eq:keld3\]), and: $$\begin{gathered}
\begin{aligned}
\{\nabla_{\boldsymbol{R}}\hat{h}_{\boldsymbol{k'}},(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}}\}&=2\left(\nabla_{\boldsymbol{R}}\times(\boldsymbol{k'}-\boldsymbol{k}) \right)\cdot\hat{\boldsymbol{\sigma}}\mu_{c}-2(\boldsymbol{k'}-\boldsymbol{k})\cdot\left(\nabla_{\boldsymbol{R}}\times\boldsymbol{\mu}\right)\hat{\sigma}_{0}\\
&+\{\nabla_{\boldsymbol{R}}\,\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k'}},(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}}\},
\end{aligned}\\
\begin{aligned}
\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\{\nabla_{\boldsymbol{R}}\hat{h}_{\boldsymbol{k'}},\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\}\}&=4(\boldsymbol{k'}-\boldsymbol{k})\cdot(\boldsymbol{m}\times\nabla_{\boldsymbol{R}} \mu_{c})\hat{\sigma}_{0}+4\hat{\boldsymbol{\sigma}}\cdot(\nabla_{\boldsymbol{R}}\times(\boldsymbol{k'}-\boldsymbol{k}))\boldsymbol{\mu}\cdot\boldsymbol{m}\\
&+\{(\boldsymbol{k'}-\boldsymbol{k})\times\hat{\boldsymbol{\sigma}},\{\nabla_{\boldsymbol{R}}\,\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k'}},\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\}\}
\end{aligned}\\
\begin{aligned}
\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{h}_{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}-\boldsymbol{n}^{2}\hat{h}_{\boldsymbol{k}}=2(\boldsymbol{n}\times(\boldsymbol{n}\times\boldsymbol{\mu}))\cdot\hat{\boldsymbol{\sigma}}+\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k'}}\boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}-\boldsymbol{n}^{2}\hat{\boldsymbol{j}}\cdot\check{\boldsymbol{k}}
\end{aligned}\end{gathered}$$ for the collision integral (\[eq:collision2\]). Upon integrating Eq. (\[eq:keld3\]) over $d\check{\boldsymbol{k}}/4\pi$ and neglecting terms $\sim \xi_{SO}^{2}\beta$, we obtain in the limit $J\ll\varepsilon_{F}$: $$\begin{aligned}
\label{eq:gen1}
2J\hat{\boldsymbol{\sigma}}\cdot(\boldsymbol{\mu}\times\boldsymbol{m})+\frac{1}{3}\frac{\hbar^{2}k}{m}\,\nabla_{\boldsymbol{R}}\cdot\hat{\boldsymbol{j}}=&-\frac{8}{9}\frac{\hbar}{\tau_{0}}\frac{{k}^{2}}{k_{F}^{2}}\xi_{SO}^{2}\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}}\\
&+\frac{1}{6}\frac{\hbar}{\tau_{0}}\frac{\xi_{SO}}{k_{F}}\big(\nabla_{\boldsymbol{R}}\cdot(\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}})-\nabla_{\boldsymbol{R}}\cdot(\hat{\boldsymbol{\sigma}}\times\hat{\boldsymbol{j}}) \big)\\
&+\frac{1}{6}\frac{\hbar}{\tau_{0}}\frac{\xi_{SO}}{k_{F}}\beta\big[\nabla_{\boldsymbol{R}}\cdot(\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}})+\nabla_{\boldsymbol{R}}\cdot(\hat{\boldsymbol{\sigma}}\times\hat{\boldsymbol{j}}),\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\big]\\
&+\frac{2}{3}\frac{\hbar}{\tau_{0}}\frac{\xi_{SO}}{k_{F}}\beta\,\nabla_{\boldsymbol{R}}\cdot(\boldsymbol{m}\times\hat{\boldsymbol{j}}).
\end{aligned}$$ One more equation is derived by averaging Eq. (\[eq:keld3\]) over $\check{\boldsymbol{k}}$ multiplied by $\check{\boldsymbol{k}}$ and neglecting terms $\sim \xi_{SO}^{2}\beta$: $$\begin{aligned}
\label{eq:gen2}
-iJ\big[\,\hat{\boldsymbol{j}},\,\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \big]+\frac{\hbar^{2}k}{m}\nabla_{\boldsymbol{R}}(\mu_{c}\hat{\sigma}_{0}+\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}})=&-\frac{\hbar}{\tau_{0}}\Big(1+\frac{2}{3}\frac{k^{2}}{k_{F}^{2}}\xi_{SO}^{2}\Big)\hat{\boldsymbol{j}}+\frac{1}{2}\frac{\hbar}{\tau_{0}}\beta\big\{\hat{\boldsymbol{j}},\hat{\boldsymbol{\sigma}}\cdot\boldsymbol{m} \big\}\\
&-\frac{i}{3}\frac{\hbar}{\tau_{0}}\frac{k}{k_{F}}\xi_{SO}\big(\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}}+\hat{\boldsymbol{\sigma}}\times\hat{\boldsymbol{j}} \big)\\
&+\frac{i}{3}\frac{\hbar}{\tau_{0}}\frac{k}{k_{F}}\xi_{SO}\beta\big(\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\,\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}}+\hat{\boldsymbol{\sigma}}\times\hat{\boldsymbol{j}}\,\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\big)\\
&+\frac{1}{3}\frac{\hbar}{\tau_{0}}\frac{k}{k_{F}}\xi_{SO}\beta\big(\hat{\boldsymbol{\sigma}}\cdot\hat{\boldsymbol{j}}+\hat{\boldsymbol{j}}\cdot\hat{\boldsymbol{\sigma}})\boldsymbol{m}
-\frac{1}{3}\frac{\hbar}{\tau_{0}}\frac{k}{k_{F}}\xi_{SO}\beta\big\{\hat{\boldsymbol{\sigma}},\hat{\boldsymbol{j}}\cdot\boldsymbol{m} \big\}\\
&+\frac{\hbar}{\tau_{0}}\frac{k}{k_{F}}\frac{\xi_{SO}}{k_{F}}\big(\nabla_{\boldsymbol{R}}\times\hat{\boldsymbol{\sigma}}\,(\mu_{c}-\beta \boldsymbol{\mu}\cdot\boldsymbol{m})+\nabla_{\boldsymbol{R}}\times(\boldsymbol{\mu}-\beta\mu_{c}\boldsymbol{m})\,\hat{\sigma}_{0} \big)\\
&+\frac{1}{6}\frac{m}{\pi\hbar}\frac{v_{i}}{\tau_{0}}\xi_{SO}k\big(\hat{\boldsymbol{\sigma}}\times\hat{\boldsymbol{j}}-\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}}\big)\\
&+\frac{1}{3}\frac{m}{\pi\hbar}\frac{v_{i}}{\tau_{0}}\xi_{SO}\beta k\big(\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\,\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}}-\hat{\boldsymbol{\sigma}}\times\hat{\boldsymbol{j}}\,\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\big)\\
&+\frac{1}{6}\frac{m}{\pi\hbar}\frac{v_{i}}{\tau_{0}}\xi_{SO}\beta k\big(\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\,\hat{\boldsymbol{\sigma}}\times \hat{\boldsymbol{j}}-\hat{\boldsymbol{j}}\times\hat{\boldsymbol{\sigma}}\,\boldsymbol{m}\cdot\hat{\boldsymbol{\sigma}}\big)\\
&-\frac{2}{3}\frac{m}{\pi\hbar}\frac{v_{i}}{\tau_{0}}\xi_{SO}\beta k\,\boldsymbol{m}\times \hat{\boldsymbol{j}}.
\end{aligned}$$ The equations above define a set of the generalized drift-diffusion equations, which can now be solved approximately while keeping leading orders in $\xi_{SO}$ and $\beta$. Then, starting from a ferromagnetic solution given by Eq. (\[eq:j\_fer\]) the anisotropic component of the density matrix is obtained by solving Eq. (\[eq:gen2\]): $$\begin{aligned}
\hat{\boldsymbol{j}}&=-\tau_{0}v_{F}\nabla\hat{\mu}_{0}+\left(\frac{\xi_{SO}}{k_{F}}+\frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar}\xi_{SO} \right)\nabla\times\boldsymbol{\mu}\hat{\sigma}_{0}-\frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar}\xi_{SO}\beta (\nabla\times\boldsymbol{m})\mu_{c}\hat{\sigma}_{0}\\
&-\left(\frac{2}{3}\xi_{SO}\beta\tau_{0}v_{F}-\frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar}\xi_{SO}\frac{\tau_{0}}{\tau_{L}} \right)\nabla\times(\boldsymbol{m}\times\boldsymbol{\mu})\hat{\sigma}_{0}+\left(\frac{\xi_{SO}}{k_{F}}+ \frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar}\xi_{SO}\right)\nabla\times\hat{\boldsymbol{\sigma}}\mu_{c}\\
&-\frac{\xi_{SO}}{k_{F}}\beta\nabla\times(\boldsymbol{m}\times(\hat{\boldsymbol{\sigma}}\times\boldsymbol{\mu}))-\frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar}\xi_{SO}\beta\nabla\times((\boldsymbol{m}\times\hat{\boldsymbol{\sigma}})\times\boldsymbol{\mu})-\frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar}\xi_{SO}\beta(\nabla\times\hat{\boldsymbol{\sigma}})\boldsymbol{\mu}\cdot\boldsymbol{m}\\
&+\left(\frac{\xi_{SO}}{k_{F}}+\frac{\tau_{0}v_{i}k_{F}^{2}}{3\pi\hbar} \right)\frac{\tau_{0}}{\tau_{L}}\nabla\times(\hat{\boldsymbol{\sigma}}\times\boldsymbol{m})\mu_{c} - \frac{2}{3}\tau_{0}v_{F}\xi_{SO}\nabla\times(\hat{\boldsymbol{\sigma}}\times(\boldsymbol{\mu}-\beta\mu_{c}\boldsymbol{m}))\\
&-\frac{2}{3}\tau_{0}v_{F}\xi_{SO}\frac{\tau_{0}}{\tau_{L}}\nabla\times\left((\hat{\boldsymbol{\sigma}}\times\boldsymbol{m})\times\boldsymbol{\mu}+\hat{\boldsymbol{\sigma}}\times(\boldsymbol{m}\times\boldsymbol{\mu}) \right),
\end{aligned}$$ where $$\hat{\mu}_{0}=(\mu_{c}+\beta\boldsymbol{\mu}\cdot\boldsymbol{m})\hat{\sigma}_{0}+(\boldsymbol{\mu}+\beta\mu_{c}\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}+\frac{\tau_{0}}{\tau_{L}}(\boldsymbol{m}\times\boldsymbol{\mu})\cdot\hat{\boldsymbol{\sigma}}+\frac{\tau_{0}}{\tau_{\phi}}(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu}))\cdot\hat{\boldsymbol{\sigma}}.$$ Here, the first term of $\hat{\boldsymbol{j}}$ comes from the ferromagnetic solution by moving the right-hand side of Eq. (\[eq:gen2\]) into Eq. (\[eq:k\_fer\]). Plugging this solution in Eq. (\[eq:gen1\]) leads to: $$\begin{aligned}
\label{eq:j_full}
\frac{2J}{\hbar}(\boldsymbol{\mu}\times\boldsymbol{m})\cdot\hat{\boldsymbol{\sigma}}&+\frac{8}{9}\frac{\xi_{SO}^{2}}{\tau_{0}}\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}}=\\
&=D\nabla\cdot\Big[ \nabla\hat{\mu}_{0}+\alpha_{sj}\big[\hat{\boldsymbol{\sigma}}\times\nabla(2\mu_{c}-\beta\boldsymbol{\mu}\cdot\boldsymbol{m})-\nabla\times(2\boldsymbol{\mu}-\beta\mu_{c}\boldsymbol{m})\big]\\
&+\alpha_{sk}\big[\hat{\boldsymbol{\sigma}}\times\nabla(\mu_{c}-\beta\boldsymbol{\mu}\cdot\boldsymbol{m})-\nabla\times(\boldsymbol{\mu}-\beta\mu_{c}\boldsymbol{m})\big]+\alpha_{sw}\nabla\times(\hat{\boldsymbol{\sigma}}\times\boldsymbol{\mu})\\
&-\nabla\times\big[(\alpha_{sj}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sk}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sw}\beta)(\hat{\boldsymbol{\sigma}}\times\boldsymbol{m})\mu_{c} + (\alpha_{sj}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sk}\frac{\tau_{0}}{\tau_{L}}-\alpha_{sw}\beta)(\boldsymbol{m}\times\boldsymbol{\mu}) \big]\\
&+\nabla\times\big[(\alpha_{sj}\beta+\alpha_{sk}\beta+\alpha_{sw}\frac{\tau_{0}}{\tau_{L}})\hat{\boldsymbol{\sigma}}\times(\boldsymbol{m}\times\boldsymbol{\mu}) - (\alpha_{sj}\beta+\alpha_{sk}\beta-\alpha_{sw}\frac{\tau_{0}}{\tau_{L}})(\hat{\boldsymbol{\sigma}}\times\boldsymbol{m})\times\boldsymbol{\mu} \big]\Big],
\end{aligned}$$ where $\alpha_{sw}=\frac{2\xi_{SO}}{3}$, $\alpha_{sj}=\frac{\xi_{SO}}{l_{F}k_{F}}$ and $\alpha_{sk}=\frac{v_{i}mk_{F}}{3\pi\hbar^{2}}\xi_{SO}$ are the spin swapping, side-jump and skew-scattering coefficients, respectively, and $l_{F}=\tau_{0}v_{F}$ is the mean-free path. As seen, Eq. (\[eq:j\_full\]) can be regarded as a generalized continuity equation for the density matrix $\mu_{c}\hat{\sigma}_{0}+\boldsymbol{\mu}\cdot\hat{\boldsymbol{\sigma}}$, and its right-hand side is nothing else but the divergence of the full current $\boldsymbol{j}^{C}\hat{\sigma}_{0}+\boldsymbol{J}^{S}\cdot\hat{\boldsymbol{\sigma}}$, where the dot product is over spin components. Thus, the corresponding expressions for the charge and spin currents (its $j$th spin component) can be readily written as: $$\begin{aligned}
\label{eq:finalcharge}
\boldsymbol{j}^{C}/D&=\tilde{\boldsymbol{j}}^{C}/D+\alpha_{sj}\nabla\times(2\boldsymbol{\mu}-\beta\mu_{c}\boldsymbol{m})+\alpha_{sk}\nabla\times(\boldsymbol{\mu}-\beta\mu_{c}\boldsymbol{m})+(\alpha_{sj}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sk}\frac{\tau_{0}}{\tau_{L}}-\alpha_{sw}\beta)\nabla\times(\boldsymbol{m}\times\boldsymbol{\mu})
\end{aligned}$$ and $$\begin{aligned}
\label{eq:finalspin}
\boldsymbol{J}_{j}^{S}/D&=\tilde{\boldsymbol{J}}_{j}^{S}/D+\alpha_{sj}\nabla\times\boldsymbol{e}_{j}(2\mu_{c}-\beta\boldsymbol{\mu}\cdot\boldsymbol{m})+\alpha_{sk}\nabla\times\boldsymbol{e}_{j}(\mu_{c}-\beta\boldsymbol{\mu}\cdot\boldsymbol{m})-\alpha_{sw}\nabla\times(\boldsymbol{e}_{j}\times\boldsymbol{\mu})\\
&+\nabla\times (\alpha_{sj}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sk}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sw}\beta)(\boldsymbol{e}_{j}\times\boldsymbol{m})\mu_{c} - \nabla\times(\alpha_{sj}\beta+\alpha_{sk}\beta+\alpha_{sw}\frac{\tau_{0}}{\tau_{L}})(\boldsymbol{e}_{j}\times(\boldsymbol{m}\times\boldsymbol{\mu})) \\
& + \nabla\times(\alpha_{sj}\beta+\alpha_{sk}\beta-\alpha_{sw}\frac{\tau_{0}}{\tau_{L}})((\boldsymbol{e}_{j}\times\boldsymbol{m})\times\boldsymbol{\mu}),
\end{aligned}$$ or $$\begin{aligned}
J_{ij}^{S}/D&=\tilde{J}_{ij}^{S}/D-\alpha_{sj}\epsilon_{ijk}\nabla_{k}(2\mu_{c}-\beta\mu_{n}m_{n})-\alpha_{sk}\epsilon_{ijk}\nabla_{k}(\mu_{c}-\beta\mu_{n}m_{n})-\alpha_{sw}(\delta_{ij}\nabla_{k}\mu_{k}-\nabla_{j}\mu_{i})\\
&+(\alpha_{sj}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sk}\frac{\tau_{0}}{\tau_{L}}+\alpha_{sw}\beta)(\delta_{ij}\nabla_{k}m_{k}-m_{i}\nabla_{j})\mu_{c} \\
& - (\alpha_{sj}\beta+\alpha_{sk}\beta+\alpha_{sw}\frac{\tau_{0}}{\tau_{L}})\epsilon_{ikn}\nabla_{k}(m_{n}\mu_{j}-\mu_{n}m_{j}) \\
& +(\alpha_{sj}\beta+\alpha_{sk}\beta-\alpha_{sw}\frac{\tau_{0}}{\tau_{L}})(\epsilon_{ikn}\nabla_{k}m_{n}\mu_{j}+\epsilon_{ijk}\nabla_{k}m_{n}\mu_{n}),
\end{aligned}$$ where $\epsilon_{ijk}$ is the Levi-Civita symbol, and summation over repeated indexes is implied. Here, the first and second subscripts correspond to the spatial and spin components, respectively. Finally, by recovering time dependence in Eq. (\[eq:j\_full\]) we obtain the remaining equations for the charge and spin densities: $$\label{eq:finalchden}
\partial_{T}\mu_{c}=D\nabla^{2}(\mu_{c}+\beta\boldsymbol{\mu}\cdot\boldsymbol{m})=-\nabla\cdot\boldsymbol{j}^{C}$$ and $$\label{eq:finalspden}
\partial_{T}\boldsymbol{\mu}=-\nabla\cdot\boldsymbol{J}^{S}+\frac{1}{\tau_{L}}(\boldsymbol{m}\times\boldsymbol{\mu})+\frac{1}{\tau_{\phi}}(\boldsymbol{m}\times(\boldsymbol{m}\times\boldsymbol{\mu}))-\frac{1}{\tau_{sf}}\boldsymbol{\mu},$$ where $1/\tau_{sf}=8\xi_{SO}^{2}/9\tau_{0}$ is the spin-flip relaxation time. The set of Eqs. (\[eq:finalcharge\]), (\[eq:finalspin\]), (\[eq:finalchden\]) and (\[eq:finalspden\]) is the central result of this work.
Spin swapping symmetry
======================
In this section we verify the symmetry of the spin swapping term and compare our results with the previous ones derived for normal metals.
In their original work Dyakonov and Lifshits give the following definition of the spin current $q_{ij}$ due to scattering off the spin-orbit coupling potential:[@perel] $$\label{lifshits}
q_{ij}=q_{ij}^{(0)}-\alpha_{sh}\epsilon_{ijk}q^{(0)}_{k}+\alpha_{sw}(q_{ji}^{(0)}-\delta_{ij}q_{kk}^{(0)}),$$ where $q^{(0)}_{k}$ and $q_{ij}^{(0)}$ stand for the primary charge and spin currents in the absence of spin-orbit coupling, respectively, and $\alpha_{sh}$ represents the overall spin Hall effect. Thus, it is argued that the spin swapping effect always appears in the form given above.
Let us consider our solution in the case of normal metals ($\beta=0$ and $J=0$): $$\begin{aligned}
\boldsymbol{J}_{j}^{S}&=-D\nabla\mu_{j}+D\alpha_{sh}\nabla\times\boldsymbol{e}_{j}\mu_{c}-D\alpha_{sw}\nabla\times(\boldsymbol{e}_{j}\times\boldsymbol{\mu})
\end{aligned}$$ or $$J_{ij}^{S}=-D\nabla_{i}\mu_{j}-D\alpha_{sh}\epsilon_{ijk}\nabla_{k}\mu_{c}+D\alpha_{sw}(\nabla_{j}\mu_{i}-\delta_{ij}\nabla_{k}\mu_{k}).$$ Taking into account that $q_{ij}^{(0)}\approx-D\nabla_{i}\mu_{j}$ and $q_{k}^{(0)}\approx-D\nabla_{k}\mu_{c}$, it is seen that Eq. (\[eq:finalspin\]) displays the correct symmetry up to a sign coming from the definition of the spin-orbit coupling potential, Eq. (\[eq:imppot\]). This form is also in agreement with some previously published results.[@brataas; @raimondi]
Finally, it is worth comparing our equations with those that fail to include spin swapping in the form given by Eq. (\[lifshits\]). For example, in Ref. \[\] the spin swapping term appeared with the following symmetry: $$\begin{aligned}
e^{2}\boldsymbol{J}_{j}^{S}/\sigma_{N}&=-\nabla\mu_{j}/2+\alpha_{sj}\boldsymbol{e}_{j}\times\nabla\mu_{c}-\alpha_{sw}\boldsymbol{e}_{j}\times(\nabla\times\boldsymbol{\mu})/2\\
&=-\nabla\mu_{j}/2+\alpha_{sj}\boldsymbol{e}_{j}\times\nabla\mu_{c}-\alpha_{sw}(\nabla\mu_{j}-\nabla_{j}\boldsymbol{\mu})/2,
\end{aligned}$$ where $\sigma_{N}$ is the bulk conductivity. It is clear that the symmetry of spin swapping is wrong: e.g. $q_{xx}$ should contain a term $\sim\alpha_{sw}(-q^{(0)}_{yy}-q^{(0)}_{zz})$, which is absent in the expression above. There is the same symmetry problem in Eq. (2) of Ref. \[\], where the spin swapping term reads as $-\alpha_{sw}\hat{\boldsymbol{\sigma}}\times\nabla\times\boldsymbol{\mu}$ (or $-\alpha_{sw}\boldsymbol{e}_{j}\times\nabla\times\boldsymbol{\mu}$ for the spin current component).
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|
---
abstract: |
For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity and the limit of value functions of $\lambda$-discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian Theorem—that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and super- optimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and super- solutions, which lead to the Tauberian Theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.
[**Keywords:**]{} [Dynamic programming principle, Abel mean, Cesaro mean, differential games, zero-sum games]{}
[**MSC2010**]{} 91A25, 49L20, 49N70, 91A23, 40E05
author:
- |
Dmitry Khlopin\
[*[email protected]*]{}
title: 'Tauberian theorem for value functions[^1]'
---
Introduction
============
Hardy proved (see, for example, [@Hardy1949 Sect. 6.8]) that, for a bounded continuous function $g$, the limit of long run averages and the limit of discounted averages (Cesaro mean and Abel mean, respectively) $$\frac{1}{T} \int_{0}^T g(t)\,dt,\qquad
\lambda \int_{0}^\infty e^{-\lambda t}g(t)\,dt$$ coincide if there exists at least one of these limits. This result and its generalizations have many applications (see, for example, [@Bingham; @Feller; @Korevaar]).
Let us consider the analogs of this Tauberian theorem for asymptotics of optimal values in game-theoretic problem statements. What if we optimize the Abel mean and/or Cesaro mean and then consider the limit of the optimal values corresponding to them? Such a limit value (as the discount tends to zero) was first considered in [@Kh_Blackwell] for a stochastic formulation. As proved in [@BK], for a stochastic two-person game with a finite number of states and actions, optimal long-time averages and optimal discounted averages share the common limit. For more details on the limit value for Abel mean and/or Cesaro mean in other stochastic formulations, see [@JN; @SorinBig; @LV; @Renault2011; @Sorin; @Vigeral; @Ziliotto2016].
In the deterministic case, the question of existence of limit values arose in the control theory, time and again; one may at the very least note [@chl1991; @colklie; @Gaitsgori1986]. In the ergodic case (more generally, in the nonexpansive-like case) such limits exist and, moreover, they are usually independent of the initial state, which was demonstrated in [@ArisawaLions; @Barles2000; @wKAM1998]; although the results were released roughly at the same time, the methods of obtaining them were thoroughly different. For the latest results on existence of limits of such values (for the nonexpansive-like case), see also [@BCQ2013 Sect. 3.4],[@CQ2015; @QG2013; @LQR; @QuinRenault].
A Tauberian theorem (the equality of limit values) was proved for discrete time systems in [@Lehrer]. The same result was obtained under an additional assumption that one of those limits is a constant function for control problems [@arisconst] and for differential games [@AlvBarditrue]. Note that, even in simple cases of control problems, the limits may not be constant functions [@grune; @QuinRenault]. In the general (non-ergodic) case, a Tauberian theorem for very general dynamic systems was first proved in paper [@barton]. Then, in [@Khlopin3], a Tauberian theorem was proposed for differential games (under Isaacs’s condition). Later, in [@Ziliotto], a very general approach to proving Tauberian theorems was proposed for games with two players with opposite goals in discrete setting. In particular, it implies the same result for recursive games [@LV]. Note that, in addition to uniform and exponential payoff families, the Tauberian theorems can be formulated for arbitrary probability densities. The corresponding results are known for discrete time systems [@1993; @Renault2013] and for optimal control and games [@CT15; @LQR; @Ziliotto2016].
The cornerstone of papers [@Khlopin3; @barton] is the construction of near-optimal strategies for one of the averages by pieces of near-optimal strategies for another average. It requires some assumptions on players’ strategies, in particular, the Dynamic Programming Principle; in addition, for games, there must be the existence of saddle point (see [@Khlopin3; @CT15], and unpublished work [@KhlopinArXiv]). Paper [@Ziliotto] exhibits a more subtle approach. In stochastic games, the value is a fixed point of the Shapley operator for the corresponding game. Parameterized (by the discount or the finite horizon) families of the corresponding Shapley operators were embedded into certain Lipschitz continuous families of nonexpansive operators, and the corresponding Tauberian theorem was proved for the fixed points of the latter operators.
The main aim of this paper is to obtain the Tauberian theorem as a direct consequence of the Dynamic Programming Principle without any technical assumptions on strategies, payoff functions, or anything else. To this end, we introduce a mapping (called a game value map) that assigns to every payoff the corresponding value function. In the general case, this map can be constructed only by payoff functions corresponding to the Abel mean and Cesaro mean as payoffs, no strategies required. Considering a properly chosen chain of payoffs, using the monotonicity of the game value map and sub- and super- optimality principles, we obtain one-sided inequalities on asymptotics leading to all Tauberian theorems. Since no additional assumptions are imposed on the players’ strategies (compare with [@Khlopin3; @KhlopinArXiv]), neither topological nor measurable structures are used (compare with [@Ziliotto2016]) to prove the Tauberian theorem itself, not even the existence of saddle point (as in [@Khlopin3; @CT15; @Ziliotto]) is required, although a reduction to the typical formalization will apparently require some of these.
We also apply this general theorem to zero-sum dynamic games in continuous and discrete settings and to differential games without a saddle point.
The structure of the paper is as follows. We start by formulating the Tauberian theorem for dynamic games in continuous setting (Theorem \[normal1\]) in Sect. \[abstractgame\]. Then, we consider the general statement: we define the concept of a game value map and formulate the Tauberian theorem for this map (Theorem \[normal2\]) and the one-side inequalities on asymptotics (Propositions \[av\_bw\]–\[bw\_av\_\]). Sect. \[proof\] contains the proofs of Theorems \[normal1\] and \[normal2\]. Sections \[apply\] and \[diff\] are devoted to the Tauberian theorem for games in discrete setting (Theorem \[normal3\]) and for differential games (Theorem \[maintheoremdiff\]), respectively. Also, the Tauberian theorem for a very simple stochastic game model (Corollary \[normal11\]) is shown in Sect. \[apply\]. The unwieldy and cumbersome proofs of propositions are confined to Appendix.
A dynamic zero-sum game {#abstractgame}
=======================
[**Dynamic system.**]{} Set ${{\mathbb{R}}}_+{\stackrel{\triangle}{=}}{{\mathbb{R}}}_{\geq 0}.$ Assume the following items are given:
- a nonempty set $\Omega$, the state space;
- a nonempty set ${{\mathbb{K}}}$ of maps from ${{\mathbb{R}}}_+$ to $\Omega$;
- a running cost $g:\Omega\mapsto [0,1]$; for each process $z\in {{\mathbb{K}}}$, the map $t\mapsto g(z(t))$ is assumed to be Borel measurable.
[**On payoffs.**]{} Let us now define a time average ${v}_T(z)$ and a discount average ${w}_\lambda(z)$ for each process $z\in {{\mathbb{K}}}$ by the following rules: $$\begin{aligned}
{v}_T(z){\stackrel{\triangle}{=}}\frac{1}{T}\int_{0}^T g(z(t))\,dt,\quad
{w}_\lambda(z){\stackrel{\triangle}{=}}\lambda\int_{0}^\infty e^{-\lambda t} g(z(t))\,dt\qquad \forall T,\lambda>0,z\in{{\mathbb{K}}}.
\end{aligned}$$ Note that the definitions are valid, and the means lie within $[0,1].$
[**On lower games.**]{} For all $\omega\in\Omega$, let there be given non-empty sets ${{\mathcal{L}}}(\omega)$ and ${{\mathcal{M}}}(\omega).$ Let, for all $\omega\in\Omega$, each pair $(l,m)\in{{\mathcal{L}}}(\omega)\times{{\mathcal{M}}}(\omega)$ of players’ rules generate a unique process $z[\omega,l,m]\in {{\mathbb{K}}}$ such that $z[\omega,l,m](0)=\omega$.
The lower game is conducted in the following way: for a given $\omega\in\Omega,$ the first player shows $l\in {{\mathcal{L}}}(\omega)$, and then, the second player chooses $m\in {{\mathcal{M}}}(\omega)$. The value function of this game is $$\begin{aligned}
{{\mathbb{V}}}[c](\omega){\stackrel{\triangle}{=}}\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)}c(z[\omega,l,m])\qquad \forall\omega\in\Omega.
\label{295}\end{aligned}$$ For instance, for every $T,\lambda>0$, the payoffs ${v}_T,{w}_\lambda$ generate the following value functions: $$\begin{aligned}
{\mathcal{V}}_T(\omega)&{\stackrel{\triangle}{=}}&{{\mathbb{V}}}[{v}_T](\omega)=\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)}\frac{1}{T}\int_{0}^T
g(z[\omega,l,m](t))\,dt\qquad \forall\omega\in\Omega,\\
{\mathcal{W}}_\lambda(\omega)&{\stackrel{\triangle}{=}}&{{\mathbb{V}}}[{w}_\lambda](\omega)=\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)}\int_{0}^\infty \lambda
e^{-\lambda t} g(z[\omega,l,m](t))\,dt\qquad
\forall\omega\in\Omega.\end{aligned}$$
Let us say that the payoff family ${v}_T (T>0)$ enjoys [*the Dynamic Programming Principle*]{} iff, for all $T>0$, the value function ${\mathcal{V}}_{T}$ coincides with the value functions for the following payoffs: $${{\mathbb{K}}}\ni z\mapsto \frac{1}{T}\int_{0}^h g(z(t))\,dt+\frac{T-h}{T}{\mathcal{V}}_{T-h}(z(h))\quad \forall h\in(0,T).$$
Let us say that the payoff family ${w}_\lambda (\lambda>0)$ enjoys [*the Dynamic Programming Principle*]{} iff, for all $\lambda>0$, the value function ${\mathcal{W}}_\lambda$ coincides with the value functions for the following payoffs: $${{\mathbb{K}}}\ni z\mapsto
\lambda\int_{0}^h e^{-\lambda t}g(z(t))\,dt+e^{-\lambda h}{\mathcal{W}}_{\lambda}(z(h)) \quad\forall h>0.$$
\[normal1\] Assume that the payoff families ${v}_T (T>0)$ and ${w}_\lambda (\lambda>0)$ enjoy the Dynamic Programming Principle.
Then, the following two statements are equivalent:
$(\imath)$
: The family of functions ${\mathcal{V}}_T$ $(T>0)$ converges uniformly on $\Omega$ as $T\uparrow\infty.$
$(\imath\imath)$
: The family of functions ${\mathcal{W}}_\lambda$ $(\lambda>0)$ converges uniformly on $\Omega$ as $\lambda\downarrow0$.
Moreover, when at least one of these statements holds, we have $$\lim_{T\uparrow\infty}{\mathcal{V}}_T(\omega)=
\ \lim_{\lambda\downarrow 0}{\mathcal{W}}_\lambda(\omega)\quad \forall\omega\in\Omega.$$
For the proof of this theorem, refer to Sect. \[proof\].
[**On abstract control systems.**]{}We can obtain the Tauberian theorem for an abstract control system. Following [@barton], assume the sets $\Omega$, ${{\mathbb{K}}}$ to be given; for all $\omega\in\Omega$, let ${{\mathcal{L}}}(\omega)$ be the set of all feasible processes $z\in{{\mathbb{K}}}$ that begin at $\omega$. Let ${{\mathcal{M}}}(\omega)$ be a singleton for all $\omega\in\Omega$.
Now, Theorem \[normal1\] implies
\[u1\] Assume that the payoff families ${v}_T (T>0)$ and ${w}_\lambda (\lambda>0)$ enjoy the Dynamic Programming Principle.
Then, the following two statements are equivalent:
$(\imath)$
: The maps $\Omega\ni\omega\mapsto
\sup_{z\in{{\mathcal{L}}}(\omega)} {v}_T(z)$ converge uniformly on $\Omega$ as $T\uparrow\infty.$
$(\imath\imath)$
: The maps $\Omega\ni\omega\mapsto\sup_{z\in{{\mathcal{L}}}(\omega)} {w}_\lambda(z)$ converge uniformly on $\Omega$ as $\lambda\downarrow0$.
Moreover, when at least one of these statements holds, we have $$\lim_{T\uparrow\infty}\sup_{z\in{{\mathcal{L}}}(\omega)} {v}_T(z)=
\lim_{\lambda\downarrow 0}\sup_{z\in{{\mathcal{L}}}(\omega)} {w}_\lambda(z)\quad \forall\omega\in\Omega.$$
In [@barton], it is stated that the Tauberian theorem holds for an abstract control system if ${{\mathbb{K}}}$ is closed with respect to concatenation. This condition can be refined, see [@khlopin2016].
Theorem \[normal1\] and Corollary \[u1\] look similar to Tauberian theorems for dynamic zero-sum game and control system with continuous setting, respectively, in the most general statement. Nevertheless, let us sketch the examples when this similarity is misleading. In certain game problems with information discrimination, the second player has to choose $m$ from ${{\mathcal{M}}}(\omega,l)$ instead of from ${{\mathcal{M}}}(\omega)$ [@KhCh2000]. Value functions with $\inf_{l}\sup_{m}\inf_{\tau}$ instead of $\sup_{l}\inf_{m}$ are applied in Hamilton-Jacobi-Isaacs variational inequalities [@subb (17.7)] and pursuit-evasion-defense problems [@Fisac (12)]. Finally, for instance, for the control problem, a maximization of the expectation of the payoff with respect to some probability distribution (see, for instance, [@oksendal (11.3.2)], [@lacker (2.5)]) is not covered by Corollary \[u1\]. For this reason, in the next section, we introduce a mapping (called a game value map) that assigns to each payoff the corresponding value function, and then we formulate the Tauberian theorem for game value map.
General statement. {#side}
==================
[**On game value map.**]{}Let the sets $\Omega$ and ${{\mathbb{K}}}$, running cost $g$, and payoffs ${v}_T,{w}_\lambda$ be as before. Denote by $\mathfrak{U}$ the set of all bounded maps from $\Omega$ to ${{\mathbb{R}}}$; denote by $\mathfrak{C}$ a non-empty set of maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$. Thereinafter, the set $\mathfrak{C}$ incorporates all conceivable payoffs, and the set $\mathfrak{U}$ contains all value functions for all games with payoffs $c\in\mathfrak{C}.$
Let $\mathfrak{C}$ satisfy the following condition: $$Ac+B\in\mathfrak{C}\ \mathrm{ for\ all }\ A\geq 0,B\in{{\mathbb{R}}}\ \mathrm{ if } \ c\in\mathfrak{C}.
\label{conditions}$$ In this section, we also assume that ${v}_T,{w}_\lambda\in\mathfrak{C}$ for all positive $\lambda,T.$
A map $V$ from $\mathfrak{C}$ to $\mathfrak{U}$ is called [*a game value map*]{} if the following conditions hold: $$\begin{aligned}
\label{conditions1}
&\ &V[Ac+B]=A\,V[c]+B\ \textrm{ for all } c\in\mathfrak{C}, A\geq 0, B\in{{\mathbb{R}}},\\
&\ &V[c_1](\omega)\leq V[c_2](\omega)\ \textrm{ for all } \omega\in\Omega\ \textrm{ if } c_1(z)\leq c_2(z) \ \textrm{for all }
z\in{{\mathbb{K}}}.
\label{conditions2}\end{aligned}$$
[**On Dynamic Programming Principle.**]{} For all positive $\lambda,T,h>0$ and every function $U_*:\Omega\to{{\mathbb{R}}}$, define payoffs $\zeta^{U_*}_{h,T}:{{\mathbb{K}}}\to{{\mathbb{R}}}$, $\xi^{U_*}_{h,\lambda}:{{\mathbb{K}}}\to{{\mathbb{R}}}$ as follows: $$\begin{aligned}
\zeta^{U_*}_{h,T}(z)&{\stackrel{\triangle}{=}}&
\frac{1}{T+h}\int_{0}^h g(z(t))\,dt+\frac{T}{T+h}{U_*}(z(h))\qquad\forall z\in{{\mathbb{K}}};\\
\xi^{U_*}_{h,\lambda}(z)&{\stackrel{\triangle}{=}}&
\lambda\int_{0}^h e^{-\lambda t}g(z(t))\,dt+e^{-\lambda h}{U_*}(z(h))\qquad\forall z\in{{\mathbb{K}}}.\end{aligned}$$
For a game value map $V$, let us say that a family of $U_T\in\mathfrak{U} (T>0)$ is [*a subsolution*]{} ([*a supersolution*]{}) for the family of payoffs ${v}_T (T>0)$ if, for every $\varepsilon>0$, there exists natural $\bar{T}$ such that, for all natural $h,T>\bar{T}$, the payoff $\zeta^{U_T}_{h,T}$ lies in $\mathfrak{C}$ and enjoys $$U_{T+h}\leq V[\zeta^{U_T}_{h,T}]+{\varepsilon}\ \
\Big(U_{T+h}\geq V[\zeta^{U_T}_{h,T}]-{\varepsilon}\Big).$$
For a game value map $V$, let us say that a family of $U_\lambda\in\mathfrak{U} (\lambda>0)$ is [*a subsolution*]{} ([*a supersolution*]{}) for the family of payoffs ${w}_\lambda (\lambda>0)$ if, for every $\varepsilon>0$, there exists natural $\bar{T}$ such that, for all natural $h>\bar{T}$ and positive $\lambda<1/\bar{T}$, the payoff $\xi^{U_\lambda}_{h,\lambda}\in\mathfrak{C}$ lies in $\mathfrak{C}$ and enjoys $$U_{\lambda}\leq V[\xi^{U_\lambda}_{h,\lambda}]+{\varepsilon}\ \
\Big(U_{\lambda}\geq V[\xi^{U_\lambda}_{h,\lambda}]-{\varepsilon}\Big).$$
For similar definitions, refer to the suboptimality principle [@Bardi Definition III.2.31] (also referred to as ‘stability with respect to second player’ [@ks]) and [@Bardi Sect. VI.4] for discrete setting.
For a game value map $V$, let us say that the family of payoffs ${v}_T(T>0)$ (resp., ${w}_\lambda(\lambda>0)$) enjoys [*the weak Dynamic Programming Principle*]{} iff their value functions ($V[{v}_T]$ and, resp., $V[{w}_\lambda]$) are, at the same time, a subsolution and a supersolution for this payoff family.
In particular, the family of payoffs ${v}_T(T>0)$ (resp., ${w}_\lambda(\lambda>0)$) enjoys [the weak Dynamic Programming Principle]{} if $$V[{v}_{T+h}]=V[\zeta^{V[{v}_T]}_{h,T}],\quad\Big(V[{w}_{\lambda}]=V[\xi^{V[{w}_\lambda]}_{h,\lambda}]\Big)\qquad
\forall h,T\in{{\mathbb{N}}}, \lambda>0.$$
\[normal2\] Let there be given a game value map $V:\mathfrak{C}\to\mathfrak{U}.$ Let ${v}_T,{w}_\lambda\in\mathfrak{C}$ for all $\lambda,T>0.$ Assume that payoffs ${v}_T (T>0)$ and payoffs ${w}_\lambda (\lambda>0)$ enjoy the weak Dynamic Programming Principle.
Then, the following two statements are equivalent:
$(\imath)$
: The family of functions $V[{v}_T]$ $(T>0)$ converges uniformly on $\Omega$ as $T\uparrow\infty.$
$(\imath\imath)$
: The family of functions $V[{w}_\lambda]$ $(\lambda>0)$ converges uniformly on $\Omega$ as $\lambda\downarrow0$.
Moreover, when at least one of these statements holds, we have $$\lim_{T\uparrow\infty}V[{v}_T](\omega)=\lim_{\lambda\downarrow 0}V[{w}_\lambda](\omega)\quad \forall\omega\in\Omega.$$
This theorem generalizes Theorem \[normal1\]. For its proof, refer to Sect. \[proof\].
[**One-sided Tauberian theorems for bounds from above.**]{} In Appendix \[A\], we prove the following proposition:
\[av\_bw\] For a game value map $V$, let a family of functions $U_T\in\mathfrak{U} (T>0)$ and the family of functions $V[{w}_\lambda](\lambda> 0)$ be a subsolution for payoffs ${v}_T$ and a supersolution for payoffs ${w}_\lambda $, respectively.
Further, let $U_T (T>0)$ satisfy $$\label{slowlyT}
\limsup_{T\uparrow\infty}\sup_{p\in[1,p_0]}\sup_{\omega\in\Omega}\,
\big(\,U_{T}(\omega)-U_{Tp}(\omega)\big)\leq 0.\quad\forall p_0>1.$$
Then, for every $\varepsilon>0$, there exists a natural $N$ such that $$V[{w}_{\lambda}](\omega)\geq U_{1/\lambda}(\omega)-\varepsilon\qquad\forall \omega\in\Omega,\lambda\in(0,1/N).$$
In Appendix \[B\], we prove a similar proposition, where the payoff families are swapped:
\[bw\_av\] For a game value map $V$, let a family of functions $U_\lambda\in\mathfrak{U} (\lambda>0)$ and the family of functions $V[{v}_T](T> 0)$ be a subsolution for payoffs ${w}_\lambda$ and a supersolution for payoffs ${v}_T$, respectively.
Further, let $U_\lambda (\lambda>0)$ satisfy $$\label{slowlyl}
\limsup_{\lambda\downarrow 0}\sup_{p\in[1,p_0]}
\sup_{\omega\in\Omega}\,\big(\,U_{\lambda}(\omega)-U_{p\lambda}(\omega)\big)\leq 0\qquad \forall p_0>1.$$
Then, for every $\varepsilon>0$, there exists a natural $N$ such that $$V[{v}_T](\omega)\geq U_{1/T}(\omega)-\varepsilon\qquad \forall \omega\in\Omega,T>N.$$
The inequalities similar to [$(\ref{slowlyT})$]{} will be found in other Tauberian Theorems, (see [@Bingham Definition 4.1.4],[@Hardy1949 Sect.6.2],[@MN1981 Theorem 4.1]). Also, we may take $p=p_0$ instead of $\sup_{p\in[1,p_0]}$ in [$(\ref{slowlyT})$]{} and [$(\ref{slowlyl})$]{} for $U_T{\stackrel{\triangle}{=}}V[{v}_T]$ and $U_\lambda{\stackrel{\triangle}{=}}V[{w}_\lambda]$ using [$(\ref{2012})$]{} and [$(\ref{2022})$]{} (see Appendix \[AA\]), respectively. On the other hand, conditions [$(\ref{slowlyT})$]{} and [$(\ref{slowlyl})$]{} are tight enough for $U_T{\stackrel{\triangle}{=}}V[{v}_T]$ and $U_\lambda{\stackrel{\triangle}{=}}V[{w}_\lambda]$, respectively. In particular, if in Proposition \[bw\_av\] one replaces [$(\ref{slowlyl})$]{} with the following condition $$\lim_{p_0\downarrow 1}\limsup_{\lambda\downarrow 0}\sup_{p\in[1,p_0]}
\sup_{\omega\in\Omega}\,\big|\,V[{w}_{\lambda}](\omega)-V[{w}_{p\lambda}](\omega)\big|=0,$$ which holds for all game value maps from [$(\ref{2022})$]{}, then Proposition \[bw\_av\] would fail for a certain game value map (see, for example, [@barton Sect. 4] for control problems). [**One-sided Tauberian theorems for bounds from below.**]{}Applying these propositions to $\mathfrak{C}^*{\stackrel{\triangle}{=}}\{-c\,|\,c\in\mathfrak{C}\}$, $V^*[c]\equiv -V[-c]$, $g^*\equiv 1-g$ instead of $\mathfrak{C},V,g$, we obtain:
\[av\_bw\_\] For a game value map $V$, let a family of functions $U_T\in\mathfrak{U} (T>0)$ and the family of functions $V[{w}_\lambda](\lambda> 0)$ be a supersolution for the payoffs ${v}_T$ and a subsolution for the payoffs ${w}_\lambda $, respectively.
Further, let $U_T (T>0)$ satisfy $$\label{slowlyT_}
\limsup_{T\uparrow\infty}\sup_{p\in[1,p_0]}\sup_{\omega\in\Omega}\,
\big(\,U_{Tp}(\omega)-U_{T}(\omega)\big)\leq 0.\quad\forall p_0>1.$$
Then, for every $\varepsilon>0$, there exists a natural $N$ such that $$V[{w}_{\lambda}](\omega)\leq U_{1/\lambda}(\omega)+\varepsilon\qquad\forall \omega\in\Omega,\lambda\in(0,1/N).$$
\[bw\_av\_\] For a game value map $V$, let a family of functions $U_\lambda\in\mathfrak{U} (\lambda>0)$ and the family of functions $V[{v}_T](T> 0)$ be a supersolution for the payoffs ${w}_\lambda$ and a subsolution for the payoffs ${v}_T $, respectively.
Further, let $U_\lambda (\lambda>0)$ satisfy $$\label{slowlyl_}
\limsup_{\lambda\downarrow 0}\sup_{p\in[1,p_0]}
\sup_{\omega\in\Omega}\,\big(\,U_{p\lambda}(\omega)-U_{\lambda}(\omega)\big)\leq 0\qquad \forall p_0>1.$$
Then, for every $\varepsilon>0$, there exists a natural $N$ such that $$V[{v}_T](\omega)\leq U_{1/T}(\omega)+\varepsilon\qquad \forall \omega\in\Omega,T>N.$$
For simplicity we could define $\mathfrak{C}$ as the set of all bounded maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$. It would be sufficient for proofs of all theorems of this article. We do not do it due to the following reasons. First, in stochastic frameworks, each payoff $c$ has to be measurable with respect to some measurable space, see, for instance, Corollary \[normal11\] in Sect. \[apply\]. Secondly, all proofs use merely the boundedness of ${v}_T,{w}_\lambda$, and the additional requirement does not appear to help to obtain the bounds.
Also note that in the definitions of subsolution and supersolution, we consider only natural $h$ and $T$. However, this strengthening is useless for continuous setting, whereas in discrete-time setting it allows a direct usage of the corresponding Dynamic Programming Principle (see Section \[apply\]).
Proofs of main results {#proof}
======================
[**Proof of Theorem \[normal2\].**]{}Since the payoffs ${v}_T (T>0)$ and the payoffs ${w}_\lambda (\lambda>0)$ enjoy the weak Dynamic Programming Principle, we see that $V[{v}_T] (T>0)$ and $V[{w}_\lambda](\lambda>0)$ are simultaneously super- and subsolutions with respect to $V$ for the payoffs ${v}_T$ and for the payoffs ${w}_\lambda$ respectively.
If at least one of the considered limits exists and is uniform in $\Omega,$ then, either the limit of $V[{v}_T]$ as $T\uparrow\infty,$ or the limit of $V[{w}_\lambda]$ as $\lambda\downarrow 0$ exists and is uniform in $\omega\in\Omega.$
Assume that it is the limit of $V[{v}_T]$ as $T\uparrow\infty.$ It follows that [$(\ref{slowlyT})$]{} and [$(\ref{slowlyT_})$]{} hold for $U_T=V[v_T]$. From Propositions \[av\_bw\] and \[av\_bw\_\], we infer that, for all $\varepsilon>0$, $|V[{v}_T](\omega)-V[{w}_{1/T}](\omega)|<\varepsilon$ holds for all $\omega\in\Omega$ if $T$ is sufficient large. Thus, the limit of $V[{w}_\lambda]$ as $\lambda\downarrow 0$ exists, is uniform, and coincides with the limit of $V[{v}_T]$ as $T\uparrow\infty.$
The case of the limit for $V[{w}_\lambda]$ is considered analogously. It is only necessary to apply conditions [$(\ref{slowlyl})$]{} and [$(\ref{slowlyl_})$]{} and Propositions \[bw\_av\] and \[bw\_av\_\].
[**Proof of Theorem \[normal1\].**]{}Let the set of all bounded maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$ be the set $\mathfrak{C}$. This set satisfies [$(\ref{conditions})$]{}, and ${w}_\lambda, {v}_T$, $\zeta^{{\mathcal{V}}_T}_{T,h},\xi^{{\mathcal{W}}_\lambda}_{\lambda,h}\in\mathfrak{C}$ for all positive $\lambda,T$ and natural $h$. Note that the map ${{\mathbb{V}}}$ (see [$(\ref{295})$]{}) takes each payoff $c:{{\mathbb{K}}}\to{{\mathbb{R}}}$ to a function ${{\mathbb{V}}}[c]:\Omega\to{{\mathbb{R}}}$. Since ${{\mathbb{V}}}$ satisfies conditions [$(\ref{conditions1})$]{}–[$(\ref{conditions2})$]{}, this map is a game value map.
To finish the proof, we can now apply Theorem \[normal2\].
Tauberian theorem in discrete time setting {#apply}
==========================================
[**On dynamics.**]{} Let there be given sets $\Omega$, ${{\mathbb{K}}}$ and a running cost $g$, as before. Assume that we would like to consider any process as a function from $\{0,1,2,\dots\}$ to $\Omega$. Since, for such a function, its definition can be completed in the form $$\begin{aligned}
z(k+t){\stackrel{\triangle}{=}}z(k)\quad \forall k\in\{0,1,2,\dots\}, t\in(0,1), \label{579}\end{aligned}$$ we can propose that this function is from ${{\mathbb{R}}}_+$ to $\Omega$ and lies in ${{\mathbb{K}}}$. Thus, in this section, we can assume that all $z\in{{\mathbb{K}}}$ satisfy [$(\ref{579})$]{}.
[**On payoffs.**]{}Recall that $\mathfrak{U}$ is the set of all bounded maps from $\Omega$ to ${{\mathbb{R}}}$. Let us consider a non-empty set $\mathfrak{C}$ of maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$ and a game value map $V:\mathfrak{C}\to \mathfrak{U}$ satisfying conditions [$(\ref{conditions})$]{} and [$(\ref{conditions1})$]{},[$(\ref{conditions2})$]{}, respectively.
For all $\mu\in(0,1), n\in{{\mathbb{N}}}$, define payoffs $\bar{v}_n:{{\mathbb{K}}}\to{{\mathbb{R}}}, \bar{w}_\mu:{{\mathbb{K}}}\to{{\mathbb{R}}}$ as follows: $$\begin{aligned}
\bar{{v}}_n(z)=\frac{1}{n}\sum_{t=0}^{n-1} g(z(t))\in [0,1],\quad
\bar{{w}}_\mu(z)=\mu\sum_{t=0}^\infty (1-\mu)^{t} g(z(t))\in[0,1]\qquad \forall z\in{{{\mathbb{K}}}}.\end{aligned}$$ Also, assume that $\bar{v}_n, \bar{w}_\mu\in\mathfrak{C}$ for all $\mu\in(0,1), n\in{{\mathbb{N}}}$.
[**On Dynamic Programming Principle.**]{} Let us say that the family of payoffs $\bar{{v}}_n ({n\in{{\mathbb{N}}}})$ enjoys [*the Dynamic Programming Principle*]{} iff, for all $n,h\in{{\mathbb{N}}}$, the payoff $$\label{617z}
\bar{{{\mathbb{K}}}}\ni z\mapsto \frac{1}{n+h}\sum_{t=0}^{h-1} {g}(z(t))+\frac{n}{n+h}V[\bar{{v}}_{n}](z(h))$$ lies in $\mathfrak{C}$ and the value of $V$ for this payoff coincides with $V[\bar{{v}}_{n+h}]$
Let us say that the family of payoffs $\bar{{w}}_\mu (\mu>0)$ enjoys [*the Dynamic Programming Principle*]{} iff, for all $\mu\in(0,1)$, the function $V[\bar{{w}}_\mu]$ coincides with values of $V$ for payoffs $$\label{617x}
{{\mathbb{K}}}\ni z\mapsto
\mu\sum_{t=0}^{h-1} (1-\mu)^{t} {g}(z(t))+(1-\mu)^{h}V[\bar{{w}}_\mu](z(h)) \quad\forall h\in{{\mathbb{N}}},$$ and each of these payoffs lies in $\mathfrak{C}$.
\[normal3\] Let all processes $z\in{{\mathbb{K}}}$ satisfy [$(\ref{579})$]{}. Let, for a game value map $V:\mathfrak{C}\to\mathfrak{U}$, the payoff families $\bar{{v}}_n ({n\in{{\mathbb{N}}}})$ and $\bar{{w}}_\mu (\mu>0)$ enjoy the Dynamic Programming Principle.
Then, the following two statements are equivalent:
$(\imath)$
: The sequence of functions $V[\bar{{v}}_n]$ $(n\in{{\mathbb{N}}})$ converges uniformly on $\Omega$ as $n\uparrow\infty.$
$(\imath\imath)$
: The family of functions $V[\bar{{w}}_\mu]$ $(0<\mu<1) $converges uniformly on $\Omega$ as $\mu\downarrow0$.
Moreover, when at least one of these statements holds, we have $$\lim_{n\uparrow\infty}V[\bar{{v}}_n](\omega)=
\lim_{\mu\downarrow 0}V[\bar{{w}}_\mu](\omega)\quad \forall\omega\in\Omega.$$
[**Proof of Theorem \[normal3\].**]{} We will use estimate [$(\ref{2012})$]{} and Lemma \[765\] proved in Appendix \[AA\].
Denote by $\mathfrak{C}_{b}$ the set of all bounded maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$. We can assume $\mathfrak{C}\subset\mathfrak{C}_{b}$; otherwise, we would always use $\mathfrak{C}\cap\mathfrak{C}_{b}$ instead of $\mathfrak{C}.$ Now, by Lemma \[765\], we can set ${V}[c]\in\mathfrak{U}$ for all $c\in\mathfrak{C}_{b}\setminus\mathfrak{C}$ such that conditions [$(\ref{conditions1})$]{}–[$(\ref{conditions2})$]{} keep to hold. Thus, we obtain the game value map $V:\mathfrak{C}_{b}\to\mathfrak{U}.$
Consider a function $\mu^*:{{\mathbb{R}}}_{>0}\to (0,1)$ defined as follows: $\mu^*(\lambda)=1-e^{-\lambda}$ for all $\lambda>0$. Then, $\int_{t}^{t+1}\lambda e^{-\lambda r}\,dr=e^{-\lambda t}\mu^*(\lambda)=
(1-\mu^*(\lambda))^{t}\mu^*(\lambda)$ for all $t\geq 0,\lambda>0$. Now, [$(\ref{579})$]{} implies $\bar{{w}}_{\mu^*(\lambda)}\equiv {w}_{\lambda}$ for all $\lambda>0$. Note that $\mu^*(0+)=0+.$ Then, the limit of $V[\bar{{w}}_\mu]$ as $\mu\downarrow 0$ exists and is uniform in $\Omega$ iff the limit of $V[{{w}}_\lambda]$ as $\lambda\downarrow 0$ exists and is uniform in $\Omega$. Moreover, in this case, these limits coincide.
Also, it is easy to see that $\bar{{v}}_n\equiv {v}_n$ for all $n\in{{\mathbb{N}}}.$ Since the payoffs ${v}_{T}(T>0)$ are bounded, we have ${v}_{T}\in\mathfrak{C}_b$ for all $T>0.$ For each $T>0$, we can choose $n\in{{\mathbb{N}}}$ such that $T\in(n-1,n]$, hence, we obtain $$|V[\bar{{v}}_{n}](\omega)-V[{{v}}_{T}](\omega)|{\stackrel{(\ref{2012})}{\leq}} \frac{2(n-T)}{T}\leq\frac{2}{T}
\qquad\forall \omega\in\Omega.$$ Thus, the limit of $V[\bar{{v}}_T]$ as $T\uparrow \infty$ exists and is uniform in $\Omega$ iff the limit of $V[\bar{{v}}_n]$ as $n\to\infty$ exists and is uniform in $\Omega$. Moreover, in this case, these limits coincide.
At last, thanks to [$(\ref{579})$]{} and $\bar{{w}}_{\mu^*(\lambda)}\equiv {w}_{\lambda}$, $\bar{{v}}_n\equiv {v}_n$ for all $n\in{{\mathbb{N}}},\lambda>0,$ we have that, for all natural $T=n,h\in{{\mathbb{N}}}$ and positive $\lambda,$ payoffs [$(\ref{617z})$]{} and [$(\ref{617x})$]{} coincide with $\zeta^{V[{v}_T]}_{T,h}$ and $\xi^{V[{w}_\lambda]}_{\lambda,h}$, respectively. Therefore, the payoff families ${w}_\lambda (\lambda>0)$ and ${v}_T (T>0)$ enjoy the weak Dynamic Programming Principle.
We have verified all conditions of Theorem \[normal2\]. Moreover, the corresponding limits in Theorem \[normal2\] and in Theorem \[normal3\] exist and are uniform in $\omega\in\Omega$ only simultaneously. Thanks to Theorem \[normal2\], Theorem \[normal3\] is proved.
Let us showcase the application of this approach in a stochastic framework.
Consider a $\sigma$-algebra ${{\mathcal{A}}}$ on ${{\mathbb{K}}}$. Assume that $\mathfrak{C}_{{{\mathcal{A}}}}$ is the set of all ${{\mathcal{A}}}$-measurable bounded maps of ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$. It is easy to see that $\mathfrak{C}_{{\mathcal{A}}}$ satisfies the condition [$(\ref{conditions})$]{}.
Similarly to Section \[abstractgame\], for all $\omega\in\Omega$, let there be given non-empty sets ${{\mathcal{L}}}(\omega)$ and ${{\mathcal{M}}}(\omega)$. Let, for all $\omega\in\Omega$, each pair $(l,m)\in{{\mathcal{L}}}(\omega)\times{{\mathcal{M}}}(\omega)$ of players’ rules induce a probability distribution ${{\mathbb{P}}}^\omega_{lm}$ (over $({{\mathbb{K}}},{{\mathcal{A}}})$) along with its mathematical expectation ${{\mathbb{E}}}^\omega_{lm}$. Similarly to [$(\ref{295})$]{}, define the map $W:\mathfrak{C}_{{{\mathcal{A}}}}\to \mathfrak{U}$ by the following rule: for each $c\in\mathfrak{C}$, $$\begin{aligned}
W[c](\omega){\stackrel{\triangle}{=}}\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)} {{\mathbb{E}}}^\omega_{lm} c\qquad \forall\omega\in\Omega.
\label{295_}\end{aligned}$$ Evidently, $W:\mathfrak{C}\to \mathfrak{U}$ satisfies conditions [$(\ref{conditions1})$]{},[$(\ref{conditions2})$]{}. Thus, $W$ is a game value map.
Applying Theorem \[normal3\] for this game value map $W,$ we obtain
\[normal11\] Let all processes $z\in{{\mathbb{K}}}$ satisfy [$(\ref{579})$]{}. Also, assume that for all $\mu\in(0,1)$, $n\in{{\mathbb{N}}}$ the payoffs $\bar{v}_n, \bar{w}_\mu$ are ${{\mathcal{A}}}$-measurable.
Let, for the game value map $W:\mathfrak{C}\to\mathfrak{U}$ (see [$(\ref{295_})$]{}), the payoff families $\bar{{v}}_n ({n\in{{\mathbb{N}}}})$ and $\bar{{w}}_\mu (0<\mu<1)$ enjoy the Dynamic Programming Principle.
Then, the following two statements are equivalent:
$(\imath)$
: The maps $\displaystyle\Omega\ni\omega\mapsto\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)} {{\mathbb{E}}}^\omega_{lm} \bar{{v}}_n$ converge uniformly on $\Omega$ as $n\uparrow\infty.$
$(\imath\imath)$
: The maps $\displaystyle\Omega\ni\omega\mapsto\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)} {{\mathbb{E}}}^\omega_{lm} \bar{{w}}_\mu$ converge uniformly on $\Omega$ as $\mu\downarrow0$.
Moreover, when at least one of these statements holds, we have $$\lim_{n\uparrow\infty}
\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)} {{\mathbb{E}}}^\omega_{lm} \bar{{v}}_n=
\lim_{\mu\downarrow 0}
\sup_{l\in {{\mathcal{L}}}(\omega)}\inf_{m\in {{\mathcal{M}}}(\omega)} {{\mathbb{E}}}^\omega_{lm} \bar{{w}}_\mu\quad \forall\omega\in\Omega.$$
Differential games without saddle point. {#diff}
========================================
[**Dynamic equation.**]{} Consider a nonlinear system in ${{\mathbb{R}}}^m$ controlled by two players, $$\label{sys}
\dot{x}=f(x,a,b),\ x(0)\in {{\mathbb{R}}}^m, a(t)\in {{\mathbb{A}}},\ b(t)\in {{\mathbb{B}}};$$ here, ${{\mathbb{A}}}$ and ${{\mathbb{B}}}$ are non-empty compact subsets of finite-dimensional Euclidean spaces.
In this section, we assume that
1. the functions $f:{{\mathbb{R}}}^m \times {{\mathbb{A}}} \times {{\mathbb{B}}}\to{{\mathbb{R}}}^m$, $g:{{\mathbb{R}}}^m \times {{\mathbb{A}}} \times {{\mathbb{B}}}\to [0,1]$ are continuous;
2. these functions are Lipschitz continuous in the state variable, namely, for a constant $L>0$, $$\big|\big|f(x,a,b)-f(y,a,b)\big|\big|+\big|g(x,a,b)-g(y,a,b)\big|\leq L\big|\big|x-y\big|\big|
\ \forall x,y\in{{\mathbb{R}}}^m,a\in {{\mathbb{A}}},b\in {{\mathbb{B}}}.$$
Denote by $B({{\mathbb{R}}}_+,{{\mathbb{A}}})$ and by $B({{\mathbb{R}}}_+,{{\mathbb{B}}})$ the sets of all Borel measurable functions ${{\mathbb{R}}}_+\ni t\mapsto a(t)\in
{{\mathbb{A}}}$ and ${{\mathbb{R}}}_+\ni t\mapsto b(t)\in {{\mathbb{B}}}$, respectively. Now, for each pair $(a,b)\in B({{\mathbb{R}}}_+,{{\mathbb{A}}})\times B({{\mathbb{R}}}_+,{{\mathbb{B}}})$, for every initial condition $x(0)=x_*$, system [$(\ref{sys})$]{} generates the unique solution $x(\cdot)=y(\cdot;x_*,a,b)$ defined for the whole ${{\mathbb{R}}}_+$. Denote by $Y(x_*)$ the set of all such solutions with $x(0)=x_*$.
Consider a set ${{\mathbb{X}}}\subset{{\mathbb{R}}}^m$ that is strongly invariant with respect to system [$(\ref{sys})$]{}, i.e., let $x(t)\in{{\mathbb{X}}}$ for all $t\in{{\mathbb{R}}}_+$, $x_*\in{{\mathbb{X}}},x\in Y(x_*)$. Set ${{\mathbb{Y}}}{\stackrel{\triangle}{=}}\cup_{x_*\in{{\mathbb{X}}}} Y(x_*).$
[**On strategies of players.**]{} Like before, let the goal of the first player be to maximize the payoff function and let the task of the second one be to minimize it. Our payoff functions are as follows: for all $\lambda,T>0,(x,a,b)\in{{\mathbb{Y}}}\times B({{\mathbb{R}}}_+,{{\mathbb{A}}})\times B({{\mathbb{R}}}_+,{{\mathbb{B}}}),$ $$\begin{aligned}
{v}_T(x,a,b){\stackrel{\triangle}{=}}\frac{1}{T}\int_{0}^T g(x(t),a(t),b(t))\,dt,\\
{w}_\lambda(x,a,b){\stackrel{\triangle}{=}}\lambda\int_{0}^\infty e^{-\lambda t} g(x(t),a(t),b(t))\,dt.
\end{aligned}$$
In the general case, without Isaacs’s condition, the lower game value and the upper game value depend on choosing the formalization of strategies of players [@ks Ch. XVI], [@subb Subsect. 14], [@Bardi]. For simplicity, we now assume that the first player announces a nonanticipating strategy (see [@EK; @Rox; @RN]) and another, knowing it, selects an admissible measurable control.
A map $\alpha : B({{\mathbb{R}}}_+,{{\mathbb{B}}})\mapsto B({{\mathbb{R}}}_+,{{\mathbb{A}}})$ is called [*a nonanticipating strategy*]{} for the first player if, for all $t>0$ and $b,b'\in B({{\mathbb{R}}}_+,{{\mathbb{B}}})$, $b|_{[0,t]} = b'|_{[0,t]}$ implies that $\alpha[b]|_{[0,t]} = \alpha[b']|_{[0,t]}$. We denote by ${{\mathcal{A}}}$ the set of all nonanticipating strategies for the first player.
Now, for all $T,\lambda>0,$ we can define lower game values, $$\begin{aligned}
{\mathcal{V}}_T(x_*)&{\stackrel{\triangle}{=}}&\sup_{\alpha\in{{\mathcal{A}}}}\inf_{b\in B({{\mathbb{R}}}_+,{{\mathbb{B}}})}
{v}_T(y(\cdot;x_*,\alpha(b),b),\alpha(b),b)\qquad\forall x_*\in{{\mathbb{X}}},\\
{\mathcal{W}}_\lambda(x_*)&{\stackrel{\triangle}{=}}&\sup_{\alpha\in{{\mathcal{A}}}}\inf_{b\in B({{\mathbb{R}}}_+,{{\mathbb{B}}})}
{w}_\lambda(y(\cdot;x_*,\alpha(b),b),\alpha(b),b)\qquad\forall x_*\in{{\mathbb{X}}}.
\end{aligned}$$
[**Constructing a game value map.**]{} Let us set $$\Omega{\stackrel{\triangle}{=}}{{\mathbb{X}}}\times{{\mathbb{A}}}\times{{\mathbb{B}}},\quad {{\mathbb{K}}}{\stackrel{\triangle}{=}}{{\mathbb{Y}}}\times B({{\mathbb{R}}}_+,{{\mathbb{A}}})\times B({{\mathbb{R}}}_+,{{\mathbb{B}}}).$$ Let ${\mathfrak{C}}$ be the set of all bounded maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$; this set satisfies [$(\ref{conditions})$]{}, and ${w}_\lambda, {v}_T$, $\zeta^{{\mathcal{V}}_T}_{T,h},\xi^{{\mathcal{W}}_\lambda}_{\lambda,h}\in\mathfrak{C}$ for each positive $\lambda,T$ and natural $h$. For every map $c\in{\mathfrak{C}}$, we can consider the following value function: $$\begin{aligned}
V[c](x_*,a_*,b_*){\stackrel{\triangle}{=}}\sup_{\alpha\in{{\mathcal{A}}}}\inf_{b\in B({{\mathbb{R}}}_+,{{\mathbb{B}}})}
c(y(\cdot;x_*,\alpha(b),b),\alpha(b),b)\qquad\forall \omega=(x_*,a_*,b_*)\in\Omega.\end{aligned}$$ It is easy prove that $V:\mathfrak{C}\to\mathfrak{U}$ satisfies [$(\ref{conditions1})$]{},[$(\ref{conditions2})$]{}. Thus, we obtain the game value map $V.$
The dynamic programming principle with respect to nonanticipating strategies for Bolza functionals (particularly, for the payoffs $v_T$) is well-known, see [@evans; @subb]. Such a principle for the payoff function ${w}_\lambda$ follows from [@BardiGame Theorem VIII.1.9]. All conditions of Theorem \[normal2\] verify. Moreover, $V[{w}_\lambda]\equiv{\mathcal{W}}_\lambda$, $V[{v}_T]\equiv{\mathcal{V}}_T$ are independent of $a_*,b_*$. Now, thanks to Theorem \[normal2\], we obtain
\[maintheoremdiff\] Let $f,g$ be as before, and let a non-empty set ${{\mathbb{X}}}\subset{{\mathbb{R}}}^m$ be strongly invariant with respect to [$(\ref{sys})$]{}.
Then, for a function $U_*:{{\mathbb{X}}}\to [0,1]$, $U_*$ is a limit of ${\mathcal{V}}_T$ as $T\uparrow \infty$ that is uniform in ${{\mathbb{X}}}$ iff $U_*$ is a limit of ${\mathcal{W}}_\lambda$ as $\lambda\downarrow 0$ that is uniform in ${{\mathbb{X}}}.$
Under Isaacs’s condition, this theorem was proved in [@Khlopin3].
Acknowledgements {#acknowledgements .unnumbered}
================
I would like to express my gratitude to Ya.V. Salii for the translation. I am grateful to an anonymous referee for helpful comments. This study was supported by the Russian Foundation for the Program of Basic Researches of the Ural Branch of the Russian Academy of Sciences (Project 15-16-1-8 “Control Synthesis under Incomplete Data, the Reachability Problems and Dynamic Optimization”) and by by the Russian Foundation for Basic Research, project no. 16-01-00505.
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Auxiliary statements. {#AA}
=====================
Assume a non-empty set $\mathfrak{C}$ satisfies [$(\ref{conditions})$]{} and a game value map $V:\mathfrak{C}\to\mathfrak{U}$ enjoys [$(\ref{conditions1})$]{}, [$(\ref{conditions2})$]{}.
[**Two estimates.**]{} Consider $T>0,r>1$, and $z\in{{\mathbb{K}}}.$ Assume ${v}_T,{v}_{rT}\in\mathfrak{C}$. Now, $$\begin{aligned}
|{v}_T(z)-{v}_{rT}(z)|&=&
\Big|\frac{1}{T}\int_0^{T} g(z(t))\,dt-\frac{1}{rT}\int_0^{rT} g(z(t))\,dt\Big|\\
&\leq& \Big|\frac{1}{T}-\frac{1}{rT}\Big|\int_0^T g(z(t))\,dt+\frac{1}{rT} \int_T^{rT}g(z(t))\,dt\\
&\leq&
1-\frac{1}{r}+\frac{r-1}{r}=\frac{2r-2}{r}<2(r-1).\end{aligned}$$ So, ${v}_{rT}(z)-2(r-1)\leq {v}_T(z)\leq{v}_{rT}(z)+2(r-1)$ for all $z\in{{\mathbb{K}}}$. Thanks to [$(\ref{conditions1})$]{},[$(\ref{conditions2})$]{}, we obtain $$\begin{aligned}
\label{2012}
\Big|V[{v}_T](\omega)-V[{v}_{rT}](\omega)\Big|\leq 2(r-1)\qquad \forall\omega\in\Omega.\end{aligned}$$
Consider $\lambda>0,r>1$, and $z\in{{\mathbb{K}}}.$ Assume ${w}_\lambda,{w}_{r\lambda}\in\mathfrak{C}$. Now, $$\begin{aligned}
|{w}_\lambda(z)-{w}_{r\lambda}(z)|&=&
\Big|\lambda\int_0^{\infty} e^{-\lambda t} g(z(t))\,dt-
r\lambda\int_0^{\infty}e^{-r\lambda t} g(z(t))\,dt\Big|\\
&\leq&
\int_0^{\infty} |\lambda e^{-\lambda t}-r\lambda e^{-r\lambda t}|g(z(t))\,dt\\
&\leq&
\int_0^{\infty} |\lambda e^{-\lambda t}-r\lambda e^{-r\lambda t}|\,dt\\
&=&
2\int_0^{\infty} \max\{\lambda e^{-\lambda t},r\lambda e^{-r\lambda t}\}\,dt-\int_0^{\infty} \lambda e^{-\lambda
t}\,dt-\int_0^{\infty} r\lambda e^{-r\lambda t}\,dt\\
&\leq&
2\int_0^{\infty} r\lambda e^{-\lambda t}\,dt-2=2(r-1).\end{aligned}$$ Thus, $$\begin{aligned}
\label{2022}
\Big|V[{w}_\lambda](\omega)-V[{w}_{r\lambda}](\omega)\Big|\leq 2(r-1)
\qquad \forall\omega\in\Omega.\end{aligned}$$
[**On extension of $V$.**]{} Denote by $\mathfrak{C}_{b}$ the set of all bounded maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$. Assume $\mathfrak{C}\subset\mathfrak{C}_b$.
Note that [$(\ref{conditions})$]{} holds for $\mathfrak{C}_{b}$. Define a map ${{\mathbb{V}}}_b:\mathfrak{C}_{b}\to\mathfrak{U}$ as follows: $${{\mathbb{V}}}_b[c'](\omega){\stackrel{\triangle}{=}}\sup\{V[c](\omega)\,|\,c\in\mathfrak{C}, c\leq c'\}
\qquad \forall \omega\in{\Omega},c'\in\mathfrak{C}_b.$$ Since, for all $R\in{{\mathbb{R}}},c\in\mathfrak{C}$, the inequality $-R\leq c\leq R$ implies $ -R=V[-R]\leq V[c]\leq V[R]=R$ by [$(\ref{conditions1})$]{},[$(\ref{conditions2})$]{}, the function ${{\mathbb{V}}}_b[c']$ is well-defined and bounded for every $c'\in\mathfrak{C}_{b}$. It is easy to prove that [$(\ref{conditions1})$]{} and [$(\ref{conditions2})$]{} hold for ${{\mathbb{V}}}_b.$ In addition, $${{\mathbb{V}}}_b[c''](\omega){\stackrel{\triangle}{=}}\sup\{V[c](\omega)\,|\,c\in\mathfrak{C}, c\leq c''\}{\stackrel{(\ref{conditions2})}{=}}V[c''](\omega)
\qquad \forall \omega\in{\Omega},c''\in\mathfrak{C}.$$ So, ${{\mathbb{V}}}_b|_{\mathfrak{C}}\equiv V.$ Thus, we have proved
\[765\] Let a non-empty set $\mathfrak{C}\subset\mathfrak{C}_b$ satisfy [$(\ref{conditions})$]{} and a map $V:\mathfrak{C}\to\mathfrak{U}$ enjoy [$(\ref{conditions1})$]{} and [$(\ref{conditions2})$]{}. Then, the map ${{\mathbb{V}}}_b:\mathfrak{C}_{b}\to\mathfrak{U}$ enjoys [$(\ref{conditions1})$]{} and [$(\ref{conditions2})$]{} and is an extension of $V$.
The proof of Proposition \[av\_bw\]. {#A}
=====================================
[**On simplicity of notation.**]{} For all payoff functions $c$, let us also use the following notation: $${\left[c\right]_{\omega}}\equiv V[c](\omega)\qquad\forall\omega\in\Omega.$$ For instance, for a function $U_*:\Omega\to{{\mathbb{R}}}$ and $\omega\in\Omega$, the symbol ${\left[U_*(z(1))\right]_{\omega}}$ means $V[c](\omega)$ for the payoff function ${{\mathbb{K}}}\ni z\mapsto c(z)=U_*(z(1))\in{{\mathbb{R}}}$. The symbol ${\left[U_*(z_1(1))\right]_{\omega}}$ also means the same for the same payoff. Moreover, the symbols $${\Big[\int_{0}^h a(t)g(z(t))\,dt+ U_*(z(h))\Big]_{\omega}},\
{\left[{\left[U_*(z_1(1))\right]_{z(1)}}\right]_{\omega}},\
{\Big[\int_{0}^h b(t)g(z_1(t))\,dt+ {\left[c\right]_{z_1(h)}}\Big]_{z(h')}}$$ denote the value of the game value map for the payoff function ${{\mathbb{K}}}\ni z\mapsto \int_{0}^h a(t)g(z(t))\,dt+ U_*(z(h))\in{{\mathbb{R}}}$ at $\omega$, for the payoff function ${{\mathbb{K}}}\ni z\mapsto c_1(z)={\left[U_*(z_1(1))\right]_{z(1)}}\in{{\mathbb{R}}}$ at $\omega$, for the payoff function $${{\mathbb{K}}}\ni z_1\mapsto \int_{0}^h b(t)g(z_1(t))\,dt+V[c](z_1(h))\in{{\mathbb{R}}}$$ at $z(h')$, respectively. Now, the expression ${\left[{\left[U_*(z_5(1))\right]_{z_{2}(1)}}\right]_{\omega}}$ is equal to ${\left[{\left[U_*(z_1(1))\right]_{z(1)}}\right]_{\omega}}$, similarly to e.g. the equivalence of $\int_A f(x)\int_B g(y)dydx$ and $\int_Af(r)\int_B g(s)dsdr.$
[**Auxiliary estimates.**]{} Recall that $\mathfrak{C}_{b}$ is the set of all bounded maps from ${{\mathbb{K}}}$ to ${{\mathbb{R}}}$. Without loss of generality, we assume that $\mathfrak{C}\subset\mathfrak{C}_{b}$; otherwise, we could always use $\mathfrak{C}\cap\mathfrak{C}_{b}$ instead of $\mathfrak{C}.$ Further, we can assume that $\mathfrak{C}=\mathfrak{C}_{b}$; otherwise, applying Lemma \[765\], we could always define $V$ on $\mathfrak{C}_{b}\setminus \mathfrak{C}$ such that conditions [$(\ref{conditions1})$]{}–[$(\ref{conditions2})$]{} keep to hold. Thus, we obtain the value $V[c]$ for all bounded payoffs $c:{{\mathbb{K}}}\to{{\mathbb{R}}}$.
Set $$\kappa(T,p_0){\stackrel{\triangle}{=}}\sup_{p\in[1,p_0]}\sup_{\omega\in\Omega}\,
\big(\,U_{Tp^{-1}}(\omega)-U_{T}(\omega)\big)\quad\forall T>0,p_0>1.$$
Fix some positive ${\varepsilon}<1/4$. Applying $2{\varepsilon}\leq {\varepsilon}^{1/2}$, we can choose some natural $k\geq 2$ such that ${\varepsilon}^{1/2}\leq k{\varepsilon}\leq (k+1){\varepsilon}\leq 2{{\varepsilon}}^{1/2}$. Set $p{\stackrel{\triangle}{=}}1+\frac{1}{k}$; we have $$\frac{{\varepsilon}}{p-1}\leq\frac{p{\varepsilon}}{p-1}=(k+1){\varepsilon}\leq 2{{\varepsilon}}^{1/2},\quad 2(p-1)=\frac{2}{k}\leq 2{{\varepsilon}}^{1/2}.$$
By the definitions of the subsolution and the supersolution, for each ${\varepsilon}>0$, there exists a positive $\bar{T}$ such that, for all $\lambda\in(0,1/\bar{T}),\omega\in\Omega$, and natural $h>\bar{T}$,$T>\bar{T}+h$, $$\begin{aligned}
U_{T}(\omega)&\leq&V\big[\zeta^{U_{T-h}}_{h,T}\big](\omega)+{\varepsilon}/2=
{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+\frac{T-h}{T} U_{T-h}(z(h))+{\varepsilon}/2\right]_{\omega}},\label{14421}\\
V[{w}_{\lambda}](\omega)&\geq&V\big[\xi^{V[{w}_{\lambda}]}_{h,\lambda}\Big](\omega)-{\varepsilon}/3={\left[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+e^{-\lambda h}
V[{w}_{\lambda}](z(h))-{\varepsilon}/3\right]_{\omega}}. \label{1442}\end{aligned}$$ By [$(\ref{slowlyT})$]{}, there exists $\hat{T}>\max\{\bar{T},\frac{1}{\ln p}\}$ such that $\kappa(T,p')\leq\kappa(T,p)\leq{\varepsilon}/2$ holds for all $T>\hat{T}$, $p'\in(1,p]$.
[**The special choice of $\lambda$.**]{}Fix every $\lambda<\frac{1}{(k+1)\hat{T}}$ such that $\frac{\ln p}{\lambda}$ is a natural number; set $$\begin{aligned}
\label{1444}
h{\stackrel{\triangle}{=}}\frac{\ln p}{\lambda},\ T{\stackrel{\triangle}{=}}\frac{p\ln p}{\lambda(p-1)}=(k+1)h,\ q{\stackrel{\triangle}{=}}p^{-1}=e^{-\lambda h}
=\frac{T-h}{T},\ \end{aligned}$$ Observe that $h$, $T=(k+1)h,$ and $T-h=kh$ are natural. It is easy to verify that $p>1$ implies that $\ln p<p-1< p\ln p$. Then, $$\begin{aligned}
\label{1445}
\frac{p}{\lambda}>T=\frac{p\ln p}{\lambda(p-1)}=\frac{h}{1-q}>\frac{1}{\lambda}>(k+1)\hat{T}.\end{aligned}$$ In particular, from the inequalities $T=h(k+1)>(k+1)\hat{T}$ and $T-h\geq h>\hat{T}$, it follows that $T,h,\lambda$ satisfy [$(\ref{14421})$]{},[$(\ref{1442})$]{}. Also, $T\lambda=\frac{p\ln p}{p-1}\in(1,p)$ guarantees that $$\begin{aligned}
\label{1446}
U_{1/\lambda}(\omega)\leq U_{T}(\omega)+\kappa(T,T\lambda)\leq U_{T}(\omega)+{\varepsilon}/2
\quad\forall \omega\in\Omega.\end{aligned}$$
[**Forward-tracking.**]{}For all $\omega\in\Omega,$ from [$(\ref{14421})$]{} it follows that $$\begin{aligned}
{U_{T}(\omega)} &{\stackrel{(\ref{14421})}{\leq}}&
{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+\frac{T-h}{T} U_{T-h}(z(h))+{\varepsilon}/2\right]_{\omega}}\nonumber\\
&\leq&{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+\frac{T-h}{T} U_{T}(z(h))+{\varepsilon}/2+\kappa(T,p)\right]_{\omega}}\nonumber\\
&\leq&{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+q U_{T}(z(h))\right]_{\omega}}+{\varepsilon}.\label{144210}\end{aligned}$$ The inequality will still hold if we replace the symbol $z$ inside the square brackets with e.g. $z_1$. We have $$\begin{aligned}
{U_{T}(\omega)} &{\stackrel{(\ref{144210})}{\leq}}&{\left[\frac{1}{T}\int_{0}^{h}g(z_1(t))\,dt+q U_{T}(z_1(h))\right]_{\omega}}+{\varepsilon}.\end{aligned}$$ In particular, for all $z\in{{\mathbb{K}}}$, we get $$\begin{aligned}
{U_{T}(z(h))} &{\stackrel{(\ref{144210})}{\leq}}&{\left[\frac{1}{T}\int_{0}^{h}g(z_1(t))\,dt+q U_{T}(z_1(h))\right]_{z(h)}}+{\varepsilon}.\end{aligned}$$ Substituting the corresponding part into [$(\ref{144210})$]{}, we obtain $$\begin{aligned}
{U_{T}(\omega)}
&\leq&{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+q U_{T}(z(h))\right]_{\omega}}+{\varepsilon}\\
&{\stackrel{(\ref{conditions2})}{\leq}}&{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\left[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt+q^{2} U_{T}(z_1(h))\right]_{z(h)}}+q{\varepsilon}\right]_{\omega}}+{\varepsilon}\\
&{\stackrel{(\ref{conditions1})}{=}}&{\left[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\left[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt+q^{2} U_{T}(z_1(h))\right]_{z(h)}}\right]_{\omega}}+{\varepsilon}(1+q).\\\end{aligned}$$ Repeating, we obtain $$\begin{aligned}
{U_{T}(\omega)}
&\leq&{\bigg[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\Big[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt}\\
&\ &{+{\Big[\frac{1}{T}\int_{0}^{h}q^2 g(z_2(t))\,dt+q^{3}
U_{T}(z_2(h))\Big]_{z_1(h)}}+q^2{\varepsilon}\Big]_{z(h)}}}{\bigg]_{\omega}}+{\varepsilon}(1+q)\\
&=&{\bigg[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\Big[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt}\\
&\ &{+{\Big[\frac{1}{T}\int_{0}^{h}q^2 g(z_2(t))\,dt+q^{3} U_{T}(z_2(h))\Big]_{z_1(h)}}\Big]_{z(h)}}}{\bigg]_{\omega}}+{\varepsilon}(1+q+q^2).\end{aligned}$$ Proceeding in a similar way, for all $n\in{{\mathbb{N}}}$, $\omega\in\Omega$, we obtain $$\begin{aligned}
{U_{T}(\omega)}
&\leq&{\Bigg[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\bigg[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt+ {\Big[\frac{1}{T}\int_{0}^{h}q^2 g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+{\Big[\frac{1}{T}\int_{0}^{h}q^n g(z_n(t))\,dt+q^{n+1}
U_{T}(z_n(h))\Big]_{z_{n-1}(h)}}\dots\Big]_{z_1(h)}}\bigg]_{z(h)}}\Bigg]_{\omega}}\\
&\ &+{\varepsilon}(1+q+q^2+\dots+q^n).\end{aligned}$$
[**Backtracking.**]{} Since $U_{T}$ is bounded, we can choose natural $n$ such that $U_T q^{n+1}\leq {\varepsilon}^{1/2}$. Then, $$\begin{aligned}
{U_{T}(\omega)}
&\leq&{\Bigg[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\bigg[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt+ {\Big[\frac{1}{T}\int_{0}^{h}q^2 g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+{\Big[\frac{1}{T}\int_{0}^{h}q^n
g(z_n(t))\,dt+{\varepsilon}^{1/2}\Big]_{z_{n-1}(h)}}\dots\Big]_{z_1(h)}}\bigg]_{z(h)}}\Bigg]_{\omega}}+\frac{{\varepsilon}}{1-q}\\
&\leq&{\Bigg[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\bigg[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt+ {\Big[\frac{1}{T}\int_{0}^{h}q^2 g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+{\Big[\frac{1}{T}\int_{0}^{h}q^n
g(z_n(t))\,dt\Big]_{z_{n-1}(h)}}\dots\Big]_{z_1(h)}}\bigg]_{z(h)}}\Bigg]_{\omega}}+{\varepsilon}^{1/2}+\frac{{\varepsilon}}{1-q}.\end{aligned}$$ By the choice of $p$, we have $\frac{{\varepsilon}}{1-q}=\frac{p{\varepsilon}}{p-1}\leq 2{{\varepsilon}}^{1/2}.$ In addition, $T\lambda\geq 1$ by [$(\ref{1445})$]{}. Now, $g\geq0$ leads to $$\begin{aligned}
U_{T}(\omega) &\leq&{\Bigg[\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+
{\bigg[\frac{1}{T}\int_{0}^{h}q g(z_1(t))\,dt+ {\Big[\frac{1}{T}\int_{0}^{h}q^2 g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+{\Big[\frac{1}{T}\int_{0}^{h}q^n
g(z_n(t))\,dt\Big]_{z_{n-1}(h)}}\dots\Big]_{z_1(h)}}\bigg]_{z(h)}}\Bigg]_{\omega}}+{\varepsilon}^{1/2}+2{{\varepsilon}}^{1/2}\\
&\leq&{\Bigg[\int_{0}^{h}\lambda g(z(t))\,dt+
q{\bigg[\int_{0}^{h}\lambda g(z_1(t))\,dt+ q{\Big[\int_{0}^{h} \lambda g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+q{\Big[\int_{0}^{h}\lambda
g(z_n(t))\,dt\Big]_{z_{n-1}(h)}}\dots\Big]_{z_1(h)}}\bigg]_{z(h)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}.\end{aligned}$$ Recall that $\lambda=\frac{\ln p}{h}$ by [$(\ref{1444})$]{}. Also, thanks to [$(\ref{1444})$]{}, we have $pe^{-\lambda t}\geq pe^{-\lambda h}=pq=1$ for all $t\in[0,h]$. It follows from $g\geq 0$ and $V[{w}_\lambda]\geq 0$ that $$\begin{aligned}
U_{T}(\omega) &\leq&{\Bigg[\int_{0}^{h}p\lambda e^{-\lambda t}g(z(t))\,dt+
q{\bigg[\int_{0}^{h}p\lambda e^{-\lambda t}g(z_1(t))\,dt}}\\
&\ &{{+\dots+q{\Big[\int_{0}^{h}p\lambda e^{-\lambda t}
g(z_n(t))\,dt+V[{w}_\lambda](z_{n}(h))\Big]_{z_{n-1}(h)}}\dots\bigg]_{z(h)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}\\
&{\stackrel{(\ref{conditions1})}{=}}&p{\Bigg[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+
q{\bigg[\int_{0}^{h}\lambda e^{-\lambda t}g(z_1(t))\,dt}}\\
&\ &{{+\dots+q{\Big[\int_{0}^{h}\lambda e^{-\lambda t}
g(z_n(t))\,dt+qV[{w}_\lambda](z_{n}(h))\Big]_{z_{n-1}(h)}}\dots\bigg]_{z(h)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}.\end{aligned}$$ Since $V[{w}_\lambda]$ is a supersolution (see [$(\ref{1442})$]{}), in view of $q=e^{-\lambda h}$, we obtain $$q{\Big[\int_{0}^{h}\lambda e^{-\lambda t} g(z_n(t))\,dt+e^{-\lambda h}V[{w}_\lambda](z_{n}(h))\Big]_{z_{n-1}(h)}}\leq e^{-\lambda
h}V[{w}_\lambda](z_{n-1}(h))+q{\varepsilon}/3.$$ Thus, $$\begin{aligned}
U_{T}(\omega)
&\leq&p{\Bigg[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+
q{\bigg[\int_{0}^{h}\lambda e^{-\lambda t}g(z_1(t))\,dt}}
+\dots+q{\Big[\int_{0}^{h}\lambda e^{-\lambda t} g(z_{n-1}(t))\,dt}\\&\ &+{{{e^{-\lambda
h}V[{w}_\lambda](z_{n-1}(h))\Big]_{z_{n-2}(h)}}\dots\bigg]_{z(h)}}\Bigg]_{\omega}}+pq^{n}{\varepsilon}/3+3{{\varepsilon}}^{1/2}.\end{aligned}$$ Proceeding in a similar way, we have $$\begin{aligned}
U_{T}(\omega) &\leq&
p{\left[\int_{0}^{h}\lambda e^{-\lambda t} g(z(t))\,dt+e^{-\lambda h}V[{w}_\lambda](z(h))\right]_{\omega}}
+p(q+q^2+\dots+q^{n}){\varepsilon}/2+3{{\varepsilon}}^{1/2}\\
&\leq&p V[{w}_\lambda](\omega)+p(1+q+q^2+\dots+q^{n}){\varepsilon}/3+3{{\varepsilon}}^{1/2}\\
&\leq&
p V[{w}_\lambda](\omega)+2p{{\varepsilon}}^{1/2}/3+3{{\varepsilon}}^{1/2}.\end{aligned}$$ Using $p\leq 1+\frac{1}{2}$ and $(p-1)V[{w}_\lambda]\leq p-1\leq {{\varepsilon}}^{1/2}$, we obtain $$\begin{aligned}
\nonumber
U_{1/\lambda}{\stackrel{(\ref{1446})}{\leq}}U_{T}+{\varepsilon}/2&\leq&
p V[{w}_\lambda]+5{{\varepsilon}}^{1/2}+{\varepsilon}/2\\
&\leq& V[{w}_\lambda]+(p-1)+5{{\varepsilon}}^{1/2}+{\varepsilon}/2\leq
V[{w}_\lambda]+6{{\varepsilon}}^{1/2}+{\varepsilon}/2\label{2250}\end{aligned}$$ for all positive $\lambda<\frac{1}{(k+1)\hat{T}}$ such that $\frac{\ln p}{\lambda}$ is a natural number.
[**The general case.**]{} For positive $\lambda'<\frac{1}{(k+1)\hat{T}}$, we can choose positive $r>1$ such that $\frac{r\ln p}{\lambda'}$ is a natural number and $0\leq \frac{r\ln p}{\lambda'}-\frac{\ln p}{\lambda'}\leq 1$. Recall that, by the choice of $\hat{T}$, we have $\hat{T}
\ln p\geq 1.$ By the choice of $\hat{T}$ and $\lambda',$ we obtain $$1\leq r\leq 1+\frac{\lambda'}{\ln p}\leq 1+\frac{1}{k+1}<p.$$ First, thanks to [$(\ref{2022})$]{}, we get $$V[{w}_{\lambda'/r}]{\stackrel{(\ref{2022})}{\leq}} V[{w}_{\lambda'}]+2(r-1)\leq V[{w}_{\lambda'}]+2(p-1)\leq
V[{w}_{\lambda'}]+2{{\varepsilon}}^{1/2}.$$ Secondly, $\lambda'/r<\lambda'<\frac{1}{(k+1)\hat{T}}$ guarantees [$(\ref{2250})$]{} for $\lambda=\lambda'/r$ because $\frac{\ln p}{\lambda}$ is natural. At last, by the definition of $\kappa$ and by the choice of $\hat{T}$ and $\lambda',$ we get $$U_{{1}/{\lambda'}}-U_{{r}/{\lambda'}}\leq \kappa({r}/{\lambda'},r)\leq
\kappa(r/\lambda',p)\leq {\varepsilon}/2.$$
Thus, we obtain $$U_{{1}/{\lambda'}}\leq U_{{r}/{\lambda'}}+{\varepsilon}/2{\stackrel{(\ref{2250})}{\leq}}
V[{w}_{\lambda'/r}]+6{{\varepsilon}}^{1/2}+{\varepsilon}\leq V[{w}_{\lambda'}]+8{{\varepsilon}}^{1/2}+{\varepsilon}\leq V[{w}_{\lambda'}]+9{{\varepsilon}}^{1/2}$$ for all sufficiently small positive $\lambda'$. By arbitrariness of positive ${\varepsilon}$, the proof is complete.
The proof of Proposition \[bw\_av\]. {#B}
====================================
We will continue the notation of the previous section for values of the game value map $V$. For instance, the symbols ${\Big[\int_{0}^h a(t)g(z(t))\,dt+ U_*(z(h))\Big]_{\omega}}$, ${\Big[\int_{0}^h b(t)g(z_1(t))\,dt+ {\left[c\right]_{z_1(h)}}\Big]_{z(h')}}$ denote the values of the game value map for a payoff ${{\mathbb{K}}}\ni z\mapsto \int_{0}^h a(t)g(z(t))\,dt+ U_*(z(h))\in{{\mathbb{R}}}$ at $\omega$ and for a payoff ${{\mathbb{K}}}\ni z_1\mapsto \int_{0}^h b(t)g(z_1(t))\,dt+V[c](z_1(h))\in{{\mathbb{R}}}$ at $z(h')$, respectively.
Also, as in the proof of Proposition \[av\_bw\], we can assume that $\mathfrak{C}$ coincides with the set of all bounded functions $c:{{\mathbb{K}}}\to{{\mathbb{R}}}$ and the game value $V[c]$ is correct for all bounded payoffs $c:{{\mathbb{K}}}\to{{\mathbb{R}}}$ and satisfies [$(\ref{conditions1})$]{}–[$(\ref{conditions2})$]{}.
[**Auxiliary estimates.**]{}For each positive ${\varepsilon}<1/4$, we can choose natural $k\geq 2$ such that ${\varepsilon}^{1/2}\leq k{\varepsilon}\leq (k+1){\varepsilon}\leq
2{{\varepsilon}}^{1/2}.$ Set $p{\stackrel{\triangle}{=}}1+\frac{1}{k}.$ Choose natural $n$ such that $2p^{-n}<{\varepsilon}^{1/2}.$ Now, $$\begin{aligned}
\frac{{\varepsilon}}{p-1}\leq\frac{p{\varepsilon}}{p-1}=(k+1){\varepsilon}\leq 2{{\varepsilon}}^{1/2},\quad 2(p-1)=\frac{2}{k}\leq 2{\varepsilon}^{1/2},\quad
2p^{-n}<{\varepsilon}^{1/2}.\end{aligned}$$
Set $$\kappa(\lambda,p_0){\stackrel{\triangle}{=}}\sup_{p'\in[1,p_0]}
\sup_{\omega\in\Omega}\,\big(\,U_{\lambda}(\omega)-U_{p'\lambda}(\omega)\big)\qquad \forall p_0>1,\lambda>0.$$ By [$(\ref{slowlyl})$]{}, there exists positive $\bar{T}$ such that $\kappa(\lambda,p)\leq{\varepsilon}/2$ holds for all positive $\lambda<1/\bar{T}$.
By the definitions of the subsolution and the supersolution, there exists positive $\hat{T}>\bar{T}$ such that, for all $\omega\in\Omega$, $\lambda\in(0,1/\hat{T})$ and natural $h>\hat{T},T>h+\hat{T}$, one has $$\begin{aligned}
U_{\lambda}(\omega)&\leq& {\left[
\lambda\int_{0}^{h}e^{-\lambda t}g(z(t))\,dt+e^{-\lambda h}U_{\lambda}(z(h))+{\varepsilon}/2\right]_{\omega}},\label{150131}\\
V[{v}_{T}](\omega)&\geq&{\left[
\frac{1}{T}\int_{0}^{h}g(z(t))\,dt+\frac{T-h}{T}V[{v}_{T-h}](z(h))-{\varepsilon}/3\right]_{\omega}}.\label{1501}\end{aligned}$$ Note that, for all $\lambda\in (0,1/\hat{T})$, $U_{\lambda}$ is bounded; then, $e^{-\lambda h_0}U_{\lambda}<{\varepsilon}/2$ for all sufficiently large natural $h_0$. Now, [$(\ref{150131})$]{},[$(\ref{conditions2})$]{}, and $g\leq 1$ imply that $U_{\lambda}(\omega)\leq \lambda\int_{0}^{h}e^{-\lambda t}\,dt+{\varepsilon}\leq 2$ for all $\lambda\in (0,1/\hat{T})$.
[**The special choice of $T$.**]{} Fix every natural $T>kp^{n+1}\hat{T}>k{p^{n+1}}\bar{T}$ such that $T(k+1)^{-n-1}$ is also natural. Then, $T(1-p^{-1})p^{-n}=Tk^n(k+1)^{-n-1}$ is natural as well. Set $$h{\stackrel{\triangle}{=}}T(1-p^{-1})=\frac{T}{k+1},\quad q{\stackrel{\triangle}{=}}p^{-1},\quad
\lambda{\stackrel{\triangle}{=}}\frac{p\ln p}{T(p-1)}=\frac{\ln p}{h}.$$ In view of $p\ln p>p-1>\ln p$, we also have $$\begin{aligned}
\frac{p}{\lambda}>\frac{p\ln p}{\lambda(p-1)}=T>\frac{1}{\lambda}>\frac{\ln
p}{\lambda(p-1)}=\frac{T}{p}=\frac{h(k+1)}{p}=\frac{h}{p-1}=kh>kp^{n}\hat{T},\nonumber\\
p/\lambda>T>1/\lambda>T-h=kh>h>\hat{T}p^{n}>\bar{T}p^{n},\quad e^{-\lambda
h}=p^{-1}=q=\frac{T-h}{T}=\frac{k}{k+1}.\label{1503_}\end{aligned}$$ For all $i\in\{0,\dots,n\},$ the numbers $hq^i,(T-h)q^i=khq^i$ are natural; now, from $(T-h)q^i>hq^i>\hat{T}$ and $\lambda p^i<1/\bar{T}$ it follows that $\lambda p^i,hq^i,Tq^i$ satisfy the inequalities [$(\ref{150131})$]{},[$(\ref{1501})$]{} for all $i\in\{0,\dots,n\}$. Moreover, $U_{p^{i+1}\lambda}\leq 2$ holds for all $i\in\{0,\dots,n\}$.
[**Forward-tracking.**]{}By $\lambda<1/\hat{T}<1/\bar{T}$, for all $\omega\in\Omega,$ we have $$\begin{aligned}
{U_{\lambda}(\omega)} &{\stackrel{(\ref{150131})}{\leq}}&
{\left[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+e^{-\lambda h}U_{\lambda}(z(h))+{\varepsilon}/2\right]_{\omega}}\\
&\leq&{\left[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+q U_{p\lambda}(z(h))+{\varepsilon}/2+\kappa(\lambda,p)\right]_{\omega}}\\
&\leq&{\left[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+q U_{p\lambda}(z(h))\right]_{\omega}}+{\varepsilon}.\end{aligned}$$
Analogously, by $p\lambda<1/\hat{T}<1/\bar{T}$ and $e^{-p\lambda qh}=e^{-\lambda h}=q$, for all $z\in{{\mathbb{K}}}$, $$\begin{aligned}
qU_{p\lambda}(z(h)) &\leq&
q{\left[\int_{0}^{qh}p\lambda e^{-p\lambda t}g(z_1(t))\,dt+e^{-p\lambda q h}
U_{p^2\lambda}(z_1(qh))\right]_{z(h)}}+q{\varepsilon}/2+q\kappa(p\lambda,p)\\
&=&{\left[\int_{0}^{qh}\lambda e^{-p\lambda t} g(z_1(t))\,dt+q^{2} U_{p^2\lambda}(z_1(qh))\right]_{z(h)}}+q{\varepsilon}.\end{aligned}$$ Substituting it into the relation above, we obtain $$\begin{aligned}
{U_{\lambda}(\omega)} &\leq&{\left[\int_{0}^{h}\!\lambda e^{-\lambda t}g(z(t))\,dt+q U_{p\lambda}(z(h))\right]_{\omega}}\!+{\varepsilon}\\
&\leq&{\bigg[\int_{0}^{h}\!\lambda e^{-\lambda t} g(z(t))\,dt}
+{\Big[\int_{0}^{qh}\!\lambda e^{-p\lambda t} g(z_1(t))\,dt+q^{2}
U_{p^2\lambda}(z_1(qh))\Big]_{z(h)}}{\bigg]_{\omega}}\!\!+{\varepsilon}(1+q).\end{aligned}$$ Repeating, by $\lambda p^2<1/\hat{T}<1/\bar{T}$, we get $$\begin{aligned}
{U_{\lambda}(\omega)}
&\leq&
{\bigg[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+
{\Big[\int_{0}^{qh}\lambda e^{-p\lambda t} g(z_1(t))\,dt}}\\
&\ &
{+{\Big[\int_{0}^{q^2h}\lambda e^{-p^2\lambda t} g(z_2(t))\,dt+q^{3}
U_{p^2\lambda}(z_2(q^2h))\Big]_{z_1(qh)}}\Big]_{z(h)}}{\bigg]_{\omega}}+{\varepsilon}(1+q+q^2).\end{aligned}$$ Proceeding in a similar way, in view of $\lambda p^n<1/\hat{T}<1/\bar{T}$, we obtain $$\begin{aligned}
{U_{\lambda}(\omega)}
&\leq&{\Bigg[\int_{0}^{h}\lambda e^{-\lambda t}g(z(t))\,dt+
{\bigg[\int_{0}^{qh}\lambda e^{-p\lambda t} g(z_1(t))\,dt}}\\
&\ &+ {\Big[\int_{0}^{q^2h}\lambda e^{-p^2\lambda t} g(z_2(t))\,dt+\dots+{\Big[\int_{0}^{q^n h}\lambda e^{-p^n\lambda t}
g(z_n(t))\,dt}}\\
&\ &{{{{+q^{n+1} U_{p^{n+1}\lambda}(z_n(q^n h))\Big]_{z_{n-1}(q^{n-1}
h)}}\dots\Big]_{z_1(qh)}}\bigg]_{z(h)}}\Bigg]_{\omega}}+{\varepsilon}(1+q+q^2+\dots+q^n).\end{aligned}$$
[**Backtracking.**]{} Note that $U_{p^{n+1}\lambda}\leq 2, g\geq 0$. Then, $$\begin{aligned}
{U_{\lambda}(\omega)}
&\leq&{\Bigg[\int_{0}^{h}\lambda g(z(t))\,dt+
{\bigg[\int_{0}^{qh}\lambda g(z_1(t))\,dt+ {\Big[\int_{0}^{q^2h}\lambda g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+{\Big[\int_{0}^{q^n h}\lambda g(z_n(t))\,dt\Big]_{z_{n-1}(q^{n-1}
h)}}\dots\Big]_{z_1(qh)}}\bigg]_{z(h)}}\Bigg]_{\omega}}+2q^{n+1}+\frac{{\varepsilon}}{1-q}.\end{aligned}$$ By the choice of $n$ and $p$, we have $2q^{n+1}<{{\varepsilon}}^{1/2}$ and $\frac{{\varepsilon}}{1-q}=\frac{p{\varepsilon}}{p-1}\leq 2{{\varepsilon}}^{1/2}$. By [$(\ref{1503_})$]{}, $\lambda<p/T$ holds. Thanks to $V[{v}_{q^n (T-h)}]\geq 0$, we obtain $$\begin{aligned}
{U_{\lambda}(\omega)}
&\leq&{\Bigg[\int_{0}^{h}\lambda g(z(t))\,dt+
{\bigg[\int_{0}^{qh}\lambda g(z_1(t))\,dt+ {\Big[\int_{0}^{q^2h}\lambda g(z_2(t))\,dt}}}\\
&\ &{{{+\dots+{\Big[\int_{0}^{q^n h}\lambda g(z_n(t))\,dt\Big]_{z_{n-1}(q^{n-1}
h)}}\dots\Big]_{z_2(q^2h)}}\bigg]_{z_1(qh)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}\\
&{\stackrel{(\ref{conditions2})}{\leq}}&{\Bigg[\frac{p}{T}\int_{0}^{h} g(z(t))\,dt+
{\bigg[\frac{p}{T}\int_{0}^{qh} g(z_1(t))\,dt+\dots+{\Big[\frac{p}{T}\int_{0}^{q^n h} g(z_n(t))\,dt}}}\\
&\ &{{{
+q^{n}V[{v}_{q^n (T-h)}](z_{n}(q^{n} h))\Big]_{z_{n-1}(q^{n-1} h)}}\dots\bigg]_{z(h)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}\\
&{\stackrel{(\ref{conditions1})}{=}}&p{\Bigg[\frac{1}{T}\int_{0}^{h} g(z(t))\,dt+
{\bigg[\frac{1}{T}\int_{0}^{qh} g(z_1(t))\,dt+\dots+{\Big[\frac{1}{T}\int_{0}^{q^n h} g(z_n(t))\,dt}}}\\
&\ &{{{
+q^{n+1}V[{v}_{q^n (T-h)}](z_{n}(q^{n} h))\Big]_{z_{n-1}(q^{n-1} h)}}\dots\bigg]_{z(h)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}.\end{aligned}$$ Recall that $q^nT=q^{n-1}(T-h)>\hat{T}$, $q=\frac{q^nT-q^nh}{q^nT}$ (see [$(\ref{1503_})$]{}); also, [$(\ref{1501})$]{} holds for $q^nh,q^nT$. Thus, $$\begin{aligned}
&\ & {\left[\frac{1}{T}\int_{0}^{q^n h} g(z_n(t))\,dt
+q^{n+1}V[{v}_{q^n (T-h)}](z_{n}(q^{n} h))\right]_{z_{n-1}(q^{n-1} h)}}\\
&=&q^{n}{\left[\frac{1}{q^{n}T}\int_{0}^{q^n h} g(z_n(t))\,dt
+q V[{v}_{q^n (T-h)}](z_{n}(q^{n} h))\right]_{z_{n-1}(q^{n-1} h)}}\\
&{\stackrel{(\ref{1501})}{\leq}}&q^{n}V[{v}_{q^n T}]({z_{n-1}(q^{n-1} h)})+q^{n}{\varepsilon}/3\\
&{\stackrel{(\ref{1503_})}{=}}&
q^{n}V[{v}_{q^{n-1} (T-h)}]
({z_{n-1}(q^{n-1} h)})+q^{n}{\varepsilon}/3.\end{aligned}$$ Then, $$\begin{aligned}
{U_{\lambda}(\omega)}
&\leq&p{\Bigg[\frac{1}{T}\int_{0}^{h} g(z(t))\,dt+
{\bigg[\frac{1}{T}\int_{0}^{qh} g(z_1(t))\,dt
+\dots+{\Big[\frac{1}{T}\int_{0}^{q^{n-1} h} g(z_{n-1}(t))\,dt}}}\\
&\ &\ +{{{
q^{n}V[{v}_{q^{n-1} (T-h)}](z_{n-1}(q^{n-1} h))\Big]_{z_{n-1}(q^{n-1} h)}}\dots\bigg]_{z(h)}}\Bigg]_{\omega}}+3{{\varepsilon}}^{1/2}+pq^{n}{\varepsilon}/3.\end{aligned}$$ Proceeding in a similar way, since $q^{i-1}(T-h){\stackrel{(\ref{1503_})}{=}}q^{i}T$ holds for all $i\in\{0,1,2,\dots,n\}$, we obtain $$\begin{aligned}
{U_{\lambda}(\omega)}
&\leq&p{\Big[\frac{1}{T}\int_{0}^{h} g(z(t))\,dt
+q V[{v}_{T-h}](z(h))\Big]_{\omega}}+3{{\varepsilon}}^{1/2}+p(q^{n}+\dots+q){\varepsilon}/3\\
&\leq&p V[{v}_{T}](\omega)+3{{\varepsilon}}^{1/2}+p(q^{n}+\dots+q+1){\varepsilon}/3\\
&\leq&p-1+V[{v}_{T}](\omega)+5{{\varepsilon}}^{1/2}\\
&\leq& V[{v}_{T}](\omega)+7{{\varepsilon}}^{1/2}\end{aligned}$$
By [$(\ref{1503_})$]{} and $T>\bar{T},$ we get $\kappa(1/T,\frac{p\ln
p}{p-1}){\stackrel{(\ref{1503_})}{=}}\kappa(1/T,T\lambda)\leq\kappa(1/T,p)\leq{\varepsilon}/2$. Hence, $$\begin{aligned}
U_{1/T}\leq U_{\lambda}+{\varepsilon}/2\leq V[{v}_{T}]+7{{\varepsilon}}^{1/2}+{\varepsilon}/2\label{2251}\end{aligned}$$ for all positive $T> k{p^{n+1}}\hat{T}$ if $T/(k+1)^{n+1}$ is also a natural number.
[**The general case.**]{} Consider every positive $T'>k\max\big(p^{n+2}\hat{T},(k+1)^{n+1}\big).$ Now, we can choose a positive $r>1$ such that $\frac{T'}{r(k+1)^{n+1}}$ is natural and $0\leq T'-\frac{T'}{r}\leq (k+1)^{n+1}$ holds. By the choice of $\hat{T}$ and $T',$ we also obtain $$1\leq r\leq \bigg(1-\frac{(k+1)^{n+1}}{T'}\bigg)^{-1}\leq
1+\frac{(k+1)^{n+1}}{T'}< 1+\frac{1}{k}=p.$$ Now, it follows from $T'/r\geq T'/p\geq kp^{n+1}\hat{T}\geq \bar{T}$ that, first, by the definition of $\kappa$, $$U_{1/T'}-U_{r/T'}\leq \kappa(r/T',r)\leq
\kappa(r/T',p)\leq {\varepsilon}/2,$$ secondly, [$(\ref{2251})$]{} holds for $T=T'/r.$ At last, thanks to [$(\ref{2012})$]{}, we also obtain $$V[{v}_{{T'}/{r}}]\leq V[{v}_{T'}]+2(r-1)\leq V[{v}_{T'}]+2(p-1)\leq
V[{v}_{T'}]+{{\varepsilon}}^{1/2}.$$
Thus, $$U_{1/T'}\leq U_{r/T'}+{\varepsilon}/2{\stackrel{(\ref{2251})}{\leq}}
V[{v}_{{T'}/{r}}]+7{{\varepsilon}}^{1/2}+{\varepsilon}\leq
V[{v}_{T'}]+8{{\varepsilon}}^{1/2}+{\varepsilon}\leq V[{v}_{T'}]+9{{\varepsilon}}^{1/2}$$ for all sufficiently large positive $T'.$ By arbitrariness of positive ${\varepsilon}$, the proof is complete.
[^1]: Krasovskii Institute of Mathematics and Mechanics, Russian Academy of Sciences, 16, S.Kovalevskaja St., 620990, Yekaterinburg, Russia; Institute of Mathematics and Computer Science, Ural Federal University, 4, Turgeneva St., 620083, Yekaterinburg, Russia
|
---
author:
- 'Giulio Fabbian[^1]'
- 'Radek Stompor[^2]'
bibliography:
- 'citations.bib'
title: 'High-precision simulations of the weak lensing effect on cosmic microwave background polarization'
---
Introduction
============
The cosmic microwave background (CMB) anisotropies in both temperature and polarization are one of the most studied signals in cosmology and one of the major available sources of constraints of the early-Universe physics. After having decoupled from matter and set free at the time of recombination, CMB photons propagated nearly unperturbed throughout the Universe. The large-scale structures (LSS) emerging in the Universe in the post-recombination period have however left their imprint on them which are referred to as secondary anisotropies. In particular, the gravitational pull of the growing matter inhomogeneities has deviated the paths of primordial CMB photons, modifying somewhat the pattern of the CMB anisotropies observed today. This weak lensing effect on the CMB (see @lewis2006 for an extensive review) therefore offers a unique probe of the matter distribution at intermediate redshift where the forming LSS were still in the nearly-linear regime. Because this depends on the cumulative matter distribution in the Universe, it is expected to be particularly efficient in constraining the properties of all the parameters affecting the growth of LSS, such as neutrino masses and dark energy physics [@deputter2009; @das2012; @hall2012].\
The first observational evidence of the CMB lensing signal had been indirect and obtained through cross-correlation of the CMB maps with high-redshift mass tracers [@smith2007; @hirata2008]. More recently, more direct measurements have become available, thanks to the latest generation of high-precision-and-resolution ground-based CMB temperature experiments, which have collected high-quality data and made possible a direct reconstruction of the power spectra of this deviation using CMB alone [@das2011; @vanengelen2012]. Even more recently, this has been further elaborated on by the Planck results based on the first 15 months of the total intensity data collected by the mission [@PlanckLensing].\
The forthcoming next generation of low-noise CMB polarization experiments such as EBEX [@ebex], POLARBEAR [@kermish2012], SPTpol [@sptpol], and ACTpol [@actpol] and their future upgrades [e.g, POLARBEAR-II, @tomaru2012] will be able to target a CMB observable most affected by weak lensing – the B-mode polarization. Indeed, primordial CMB gradient-like polarization ($E$-modes) is converted into curl-like polarization ($B$-modes) by gravitational lensing [@ZaldarriagaSeljak1998] and is expected to completely dominate the primordial signal at least at small angular scales. The lensing-generated $B$-modes are interesting because of their sensitivity to the large-scale structure distribution, but also because they are the main contaminant of any primordial $B$-modes signal, which is expected in many models of the very early Universe, and which is one of the major goals of the current and future CMB observations. Since sensitivities of the CMB polarization arrays are rapidly improving, the experiments aiming at setting constraints on values of the tensor-to-scalar ratio parameter $r \lesssim 10^{-2}$ are expected to be ultimately limited by the lensing signal [e.g., @Errard2012]. This acts as an extra noise source with a white spectrum shape on large scales and an amplitude of approximately $5\mu K$-arcmin, which could in principle be separated from the primordial signal with the help of an accurate de-lensing procedure [@kesden2002; @seljak2004; @smith2012].
The high quality of forthcoming datasets requires the development, testing and validation through simulations of data-analysis tools capable of fully exploiting the amount of information present there. An important part of this effort involves simulating very accurate, high-resolution maps of the CMB total intensity and polarization, covering a large fraction of the sky and with lensing effects included. The relevant approaches have been studied in the past [e.g. @lewis2005; @basak; @wandelt] and resulted in devising and demonstrating an overall framework for such simulations, as well as in two publicly available numerical codes [@lewis2005; @basak]. Because the computations involved in such a procedure are inherently very time-consuming, the proposed implementations of those ideas unavoidably involve trade-offs between calculation precision and their feasibility, giving rise to a number of problems, practical and more fundamental, which need to be carefully resolved to ensure that these techniques produce high-quality, reliable results. The main objective of this paper is to provide comprehensive answers to some of these problems, with special emphasis on those arising in the context of high-precision and-reliability simulations of the B-mode component of the CMB polarization signal.
Simulating weak lensing of the CMB
==================================
Algebraic background
--------------------
The CMB radiation is completely described by its brightness temperature and polarization fields on the sky, $T(\vartheta,\varphi)$ and $P(\vartheta,\varphi)$. Since both fields are (nearly) Gaussian, they are characterized by their power spectra after their harmonic expansion in a proper basis. Temperature is a scalar field and can be conveniently expanded in terms of scalar spherical harmonics,
$$T(\vartheta,\varphi)=\sum_{l=0}^{l_{max}}\sum_{m=-l}^{l}T_{lm}Y_{lm}(\vartheta,\varphi),
\label{eqn:tempExp}$$
while polarization is described by the Stokes parameters Q and U, which are coordinate-dependent objects, that behave like a spin-2 field on the sphere under rotations [@seljak-zaldarriaga; @kks]. The polarization field must therefore be expanded in terms of spin-2 spherical harmonics, $_{\pm2}Y_{lm}(\vartheta,\varphi)$, $$\begin{aligned}
P(\vartheta,\varphi) &=& (Q+iU)(\vartheta,\varphi)\\ \nonumber
&=& \sum_{l=0}^{l_{max}}\sum_{m=-l}^{l}-(_{2}E_{lm}+i _{2}B_{lm}) _{2}Y_{lm}(\vartheta,\varphi),
\label{eqn:polExp}\end{aligned}$$ where $_{2}E_{lm}$ and $_{2}B_{lm}$ are the gradient and curl harmonic components of a spin-2 field, whose general definitions for and arbitrary spin-s field are $$\begin{aligned}
_{|s|}E_{lm} & \equiv& -\frac{1}{2}\left(_{|s|}a_{lm}+(-1)^{s}_{-|s|}a_{lm}\right) \\
i_{|s|}B_{lm} & \equiv& -\frac{1}{2}\left(_{|s|}a_{lm}-(-1)^{s}_{-|s|}a_{lm}\right). \nonumber\end{aligned}$$
Weak gravitational lensing shifts the light rays coming from an original direction $\vec{\hat{n}}$ on the last scattering surface to the observed direction $\vec{\hat{n}^\prime}$, inducing a mapping between the two directions through the so-called displacement field $\vec{d}$ , i.e., for a CMB observable $X\in\{T,Q,U\}$ $$\tilde{X}(\vec{\hat{n}})=X(\vec{\hat{n}}^\prime )=X(\vec{n}+\vec{d}).
\label{eq:lens}$$ Hereafter, we use a tilde to denote a lensed quantity, we also use a tilde over a multipole number of a lensed quantity, i.e., $\tell{}$, to distinguish it from a multipole number of its unlensed counterpart.
The displacement field is a vector field on the sphere and can be decomposed into a gradient-free and a curl-free component. In most cases we can neglect the gradient-free component and consider the displacement field $\vec{d}$ as the gradient of the so-called lensing potential $\Phi(\vartheta,\varphi)$, the projection of the 3D gravitational potential $\Psi$ on the 2D unit sphere. This quantity can be computed with Boltzmann codes (e.g. CAMB[^3] or CLASS[^4]), from galaxy surveys or N-body simulations [@carbone2008; @das-bode],
$$\Phi(\vec{n})\equiv -2\int_{0}^{\eta_{*}}d_{A}\eta \frac{d_{A}(\eta_{*}-\eta)}{d_{A}(\eta)d_{A}(\eta^{*})}\Psi(\eta,\vec{n}).$$
Here $\eta_{*}$ is the comoving distance to the last scattering surface, $\eta$ is the co-moving distance, $d_{A}$ is the co-moving angular diameter distance. The lensing potential is expected to be correlated on a large scale with temperature anisotropies and $E$-modes of polarization through the integrated Sachs-Wolfe effect; this correlation mainly affects the large angular scales and is of the order of $1\%$ at $\ell\approx100$ and will thus be neglected in the following analysis.\
Since the lensing potential is a scalar function and can be expanded into canonical spherical harmonics, its gradient (a spin-1 curl-free field) can be easily computed in the harmonic domain with a spin-1 spherical harmonic transform (SHT):
$$_{1}E_{lm}=\sqrt{l(l+1)}\Phi_{lm} \qquad _{1}B_{lm}=0.$$
Pixel-domain simulations {#sect:pixdomainsims}
------------------------
### Basics {#s3ect:basics}
Because typical deviations of CMB photons are on the order of few arcminutes (although coherent over the degree scale), we can work in the Born approximation, i.e., considering this deviation as constant between $\vec{\hat{n}}$ and $\vec{\hat{n}^\prime}$, and evaluate the displaced field along the unperturbed direction.\
In practice this means that to compute the lensed CMB at a given point it is sufficient to compute the unlensed CMB at another position on the sky. This observation provides the basis for the pixel-based approaches to simulating lensing effects of the CMB maps. For every direction on the sky corresponding to a pixel center these methods first identify the displaced direction and then compute the corresponding sky signal value, which is used to replace the original value at the pixel center. The implementations of this approach typically involve the following main steps [@lewis2005; @basak; @wandelt]:
1. Generating a random realization of the harmonic coefficients of the unlensed CMB map and its synthesis.
2. Generating a random realization of the harmonic coefficients of the lensing potential and then of the spin-1 displacement field in the harmonic domain. Synthesizing the displacement field.
3. Sampling the displacement field at pixel centers and, for each of them, computing the coordinates of a displaced direction on the sky using the spherical triangle identities on the sphere.
Defining $\alpha$ as the angle between the displacement vector and the $\vec{e_{\vartheta}}$ versor, such that $\vec{d}=d\cos\alpha\,\vec{e_{\vartheta}}+d\sin\alpha\,\vec{e_{\varphi}}$, the value of a lensed field, i.e., $T$, $Q$ and $U$, in a direction $(\vartheta,\varphi)$ is given by the unlensed field at $(\vartheta^\prime,\varphi+\Delta\varphi)$ where, $$\begin{aligned}
\cos\vartheta^\prime &=& \cos d\cos\vartheta-\sin d\sin\vartheta\cos\alpha \\
\sin\Delta\varphi &=& \frac{\sin\alpha\sin d}{\sin\vartheta^\prime}.\end{aligned}$$
4. Computing temperature and polarization fields at displaced positions.\
5. Re-assigning the temperature and polarization from the displaced to new positions to create the simulated lensed map sampled on the original grid. For the polarization, we need also to multiply the lensed field by an extra factor taking into account the different orientation of the basis vector at the two points. Calling $\gamma$ the difference between the angles between $\vec{e_{\vartheta}}$ and the geodesic connecting the two points, and defining $$\begin{aligned}
A&=&\tan\alpha^\prime = \frac{d_{\varphi}}{d\sin d\cot\vartheta + d_{\vartheta}\cos d} \\
e^{2i\gamma}&=&\frac{2(d_{\vartheta}+d_{\varphi}A)^{2}}{d^{2}(1+A^{2})} -1 +\frac{2i(d_{\vartheta}+d_{\varphi}A)(d_{\varphi}-d_{\vartheta}A)}{d^{2}(1+A^{2})},\end{aligned}$$
the lensed polarization field becomes $$\tilde{P}(\vartheta,\varphi)=e^{2\gamma i}P(\vartheta^\prime,\varphi^\prime).$$
6. Smoothing and, potentially, re-pixelizing the lensed map to match a particular experimental resolution, if needed.
### Challenges and goals {#sect:challenges}
There are two main, closely intertwined challenges involved in implementing the approach detailed in the previous section. The first one is related to the bandwidths of fields used in, or produced as a result of, the calculation, and in particular to the need of imposing those on the fields, which are either naturally not band-limited or are band-limited but have too high bandwidths to make them acceptable from the computational efficiency point of view. The other challenge arises from step 4 of the algorithm: the displaced directions do not correspond in general to pixel centers of any iso-latitudinal grid on the sphere, and thus the lensed values of the CMB signal cannot be computed with the aid of a fast SHT algorithm and a more elaborated, and computationally costly approach is needed.
We emphasize that both these problems should be looked at from the perspective of the efficiency of the numerical calculations as well as accuracy of the produced results. We discuss them in some detail below.
{width=".325\textwidth"} {width=".325\textwidth"} {width=".325\textwidth"}
#### Signal bandwidths.
Because the lensing procedure needs to be applied prior to any instrumental response function convolution, the relevant sky signals on all but the last steps above require using a resolution sufficient to support the signal all the way to its intrinsic bandwidth, $\ell_{intr}^X,$ where $X$ is either $T$ for the total intensity, or $P$ for the polarization, or $\Phi$ – for the gravitational potential. However, because mathematically the lensing effects can be seen as a convolution in the harmonic domain [@hu2000; @okamoto2003; @hu2002] of the CMB signal – either the total intensity, $T$, or the polarization, $P$, – and of the potential, $\Phi$, the bandwidth of the resulting lensed field will be broader than that of any unlensed fields and is given roughly by $\ell_{intr}^X+\ell_{intr}^\Phi$. Consequently, the lensed map produced in step 5 should have its resolution appropriately increased to eliminate potential power aliasing effects. The resolution of the unlensed maps produced in steps 1-5 should then coincide with that of the lensed signal but with the number of harmonic modes set by $\ell_{intr}^X$ and $\ell_{intr}^\Phi$ respectively .
One of the problems arising in this context is related to the fact that the unlensed sky signals, $T$, $P$ and $\Phi$, considered here are not truly band-limited even if their power at the small scales decays quite abruptly as a result of Silk damping. Picking an appropriate value for the bandwidth is therefore a matter of a compromise between the precision of the final products and the calculation cost, with both these quantities being quite sensitive to the chosen value, and which will depend in general on a specific application. We emphasize that the presence of the high-$\ell$ power decay plays a dual role in our considerations here. On the one hand, it ensures that the lensing effect at sufficiently large scale can be computed with an arbitrary precision by simply choosing the bandwidth values sufficiently high. On the other hand it does introduce an extra complexity in defining a set of sufficient conditions, which ensure required precision, because these will be typically different in the regime of the high signal power and that of the damping tail. In either case, though, it is clear that whatever the selected bandwidths, the amplitudes of the harmonic modes of the lensed signal close to the highest value of $\tell{X}$ supported by the employed pixelization, i.e., $\tell{X} \sim \ell_{\, intr}^X+\ell_{\, intr}^\Phi$, will generally be unavoidably misestimated, and satisfactory precision can only be achieved for harmonic modes lower than some $\tell{X}_{\, ok} < \ell_{intr}^X$. From the practitioner’s perspective the main problem is therefore, given some precision criterion, $\varepsilon$, which we wish to be fulfilled by the harmonic modes of the lensed signal up to some value of $\tell{X}=\tell{X}_{\,ok}$, how to determine the required bandwidths of the unlensed signals, $\ell_{\,intr}^X = \ell_{\,intr}^X(\tell{Y}_{\,ok}, \varepsilon)$ where $X$ and $Y$ can be the same, e.g., in the case of the $T$ or $E$ signal lensing, or different, e.g., for the potential field or $B$-modes.
One effect of these considerations is that if these are maps of the lensed signals, which are of interest as the final product of the calculation, then the biased high-$\ell$ modes should either be filtered out or suppressed before the map is synthesized from its harmonic coefficients. To ensure that this does not adversely affect the resolution of the final map, the bias should affect only angular scales much smaller than the expected final resolution of the map as produced in step 6 of the algorithm. If the latter is defined by the experimental beam resolution, one therefore needs to ensure that no bias is present for $\tell{X} \simlt \ell_{beam} \sim \sigma_{beam}^{-1}$, where $\sigma_{beam}$ is an experimental beam width.
#### Interpolations. {#sect:interpolation-discussion}
Interpolation is the most popular workaround of the need to directly calculate values of the unlensed fields for every displaced directions, which typically will not correspond to grid points of any iso-latitudinal pixelization. Three interpolation schemes have been considered to date in the context of the polarized signals. @lewis2005 proposed a generic modified bicubic interpolation and demonstrated that it seems to work satisfactorily in a number of cases. This approach together with the direct summation are both implemented in the publicly available code [[LensPix]{}]{}[^5]. Two other methods have been proposed more recently. @basak implemented the general interpolation scheme, which recasts a band-limited function on the sphere as a band-limited function on the 2D torus where a non-equispaced fast Fourier (NFFT) transform algorithm is used to compute the field at the displaced positions. This method would be arbitrarily precise if the sky signals were strictly band-limited. However, the choice of NFFT can become a bottleneck for this algorithm since its numerical workload scales with the number of pixels squared, and its memory requirements are huge. As it is, the NFFT software can be run only on shared-memory architectures, making it more difficult to resolve both these problems. Consequently, the issue of the bandwidth values is becoming of crucial importance for the performance and applicability of the method, and its relevance in particular in the context of simulations of upcoming and future high-resolution experiments needs to be investigated in more detail.\
@wandelt proposed a fast pixel-based method using the spectral characteristics of the field to be lensed to compute the weighting coefficient for the interpolation of this field, without using any spherical harmonic algorithm. Its accuracy is set by the number of neighboring pixels used to interpolate the field at a given point.\
In addition, [@hirata2004] used in their work a polynomial interpolation scheme of arbitrary order and precision, which has been shown to successfully produce temperature maps [@hirata2004; @das-bode] but has not been tested for the polarized case.\
Any interpolation in this context is not without its dangers because interpolations tend to smooth the underlying signals. For a genuinely band-limited function this could in principle be avoided as in, e.g., @basak. However, for the actual CMB signals the bandwidth is only approximate and is a function of the required precision and specific application; the sampling density and interpolation scheme therefore need to be chosen very carefully to render reliable results. Again, the choice of appropriate bandwidth values is therefore central for a successful resolution of this problem.
#### Numerical workload
Numerical cost of the direct calculation per direction is given by ${\cal O}(\ell_{max}^2)$ and corresponds to the cost of calculating an entire set of all $\ell$ and $m$ modes of associated, scalar, or spin-weighted, Legendre functions. For $N_{pix}$ directions the overall cost about ${\cal O}(N_{pix}\,\ell_{max}^2) = {\cal O}(N_{pix})^2$ and is therefore prohibitive for any values of $N_{pix}$ and $\ell_{max}$ of interest. Here, we assumed a relation, $\ell_{max} \propto N_{pix}^{1/2}$, typically fulfilled for the full-sky pixelization with a proportionality coefficient on the order of a few, e.g., for the HEALPix[^6] pixelization [@gorski2005] we have $\ell_{max} = 2\sqrt{3\,N_{pix}}$, while for ECP, $\ell_{max} = 2\,\sqrt{N_{pix}}$. The interpolations can cut on this load, trimming it to the one needed to compute a representation of the signals on an iso-latitudinal grid, with complexity ${\cal O}(N_{pix}^{1/2}\,\ell_{max}^2) = {\cal O}(N_{pix}^{3/2})$ plus the interpolation with the complexity ${\cal O}(N_{pix})$, or ${\cal O}(N_{pix}\,\ln\,N_{pix})$ in the case of NFFT, in both cases with a potentially large pre-factor. Nevertheless, this is clearly a more favorable scaling than the one of the direct method and, as has been shown in the past, makes such calculations feasible in practice. We note, however, that for the sake of the precision of the interpolation one may need to overpixelize the sky, meaning using a higher value of $N_{pix}$ than what would normally be needed to support the harmonic modes all the way to $\ell_{max}$. Hereafter, we denote the overpixelization factor in each of the two directions, $\theta$ and $\phi$, as $\kappa$. Consequently, the number of pixels used is given by $\kappa^2\,N_{pix}$, where $N_{pix}$ is the standard full-sky number of pixels as determined by the selected value of $\ell_{max}$.
#### Goals and methodology.
This paper has two main goals. One is to study internal consistency and convergence of the pixel-domain simulations in the context of the currently viable cosmologies. The other is to study the dependence of the precision of these simulations on some of its most important parameters.
In previous works, analyses of this sort have usually been restricted to comparisons of power spectra of the lensed maps derived by a lensing simulation code and the theoretical predictions computed via an integration of the Boltzmann equation, as implemented in the publicly available codes, CAMB and CLASS. In these works, the effort has been made to find a set of the code parameters for which the resulting spectrum is consistent with the theoretical expectations. Such comparisons are without doubt an important part of a code and method validation. However, they are limited to the cases of the gravitational potentials, $\Phi$, derived in a linear theory, and are not applicable in some other cases where the potential is obtained by some other means such as, N-body simulations. In addition, they may on occasion be misleading because the numerical effects can easily conspire to deliver a spectrum tantalizingly close to the desired one, without any reassurance that the map of the lensed sky characterized by it has correct other statistical properties, such as higher-order statistics. That this is particularly likely and consequential for the $B$-modes spectrum given its low amplitude and the lack of characteristic, fine-scale features. An example of such a conspiracy is shown in Fig. \[bmodesplate\], where the power deficit at the high-$\ell$ end caused by the oversmoothing due to the interpolation nearly perfectly compensates the extra power aliased into the $\ell$-range of interest as a consequence of too crude a resolution of the final map.
We therefore propose to study the robustness of the simulated results by demonstrating their convergence and internal stability with respect to sky sampling and band-limit changes, as expressed by two parameters introduced earlier: the upper value of the signal band, $\ell_{max}$, and the overpixelization factor, $\kappa$. Only once the convergence is reached we compare the results to those computed by other means, if any are available. We note that the convergence tests do not have to, and should not in general, be restricted to the power spectra comparison only and could instead involve other metrics more directly relevant to the simulated maps themselves. In all such tests it is typically required to consider maps with extreme resolutions, which has been traditionally prohibitive for numerical reasons. We overcome this problem with the help of a high-performance lensing code, [[lenS$^{2}$HAT]{}]{}, which we have developed for this purpose.
Our second goal, i.e., to study the dependence of the calculation precision on the two crucial parameters, $\ell_{max}$ and $\kappa$, is complementary and is aimed at providing meaningful and practically useful guidelines of how to select the values of these parameters prior to performing any numerical tests given some predefined precision targets. In this context, we present an in-depth semi-analytical analysis of the impact of these parameters on the lensed signal recovery. Though ultimately they may need to be confirmed numerically case-by-case, e.g., using the convergence tests as discussed earlier, they could be of significant help in providing a reasonable starting point for such tests.
At last we also present a simple, high-performance parallel implementation of the pixel-domain approach, [[lenS$^{2}$HAT]{}]{}, which is capable of reaching extremely high sample density on the sphere thanks to its efficient parallelization and numerical implementation, and which has been instrumental in accomplishing all the other goals of this work.
\
[{width=".5\textwidth"}]{}[{width=".5\textwidth"}]{}\
Exploring the bandlimits {#sect:kers}
========================
CMB lensing in the harmonic domain
----------------------------------
This section addresses the second of goals, as stated above, and describes a semi-analytic study of the impact of the assumed bandwidth values on the precision of the lensed signal. Our discussion is based on the model of @hu2000 and focuses on the lensed $B$-mode signal that is obtained obtained as a result of the lensing acting upon the primordial $E$-mode signal, and is the main target of this paper. Similar considerations can be made, however, for other CMB observable spectra and we present some relevant results calculated for these cases (see Sect. \[sect:accuracy\] for some more details). Using the results of @hu2000, we represent the lensed $B$-mode signal as $$\begin{aligned}
\tilde{C}_{\tell{B}}^{BB}&=&\frac{1}{2}\sum_{\ell^{\,\Phi}\ell^{\,E}}\frac{|_{2}F_{\tell{B}\ell^{\,\Phi}\ell^{\,E}}|^{2}}{2\tell{B}+1} C_{\ell^{\,\Phi}}^{\Phi\Phi}C_{\ell^{\,E}}^{EE}(1-(-1)^{L})
\label{eq:kerb}\end{aligned}$$ where $_{2}F_{\tell{B}\ell^{\,\Phi}\ell^{\,E}}$ is a spin-$2$ coupling kernel (see [@hu2000] for a full expression), $L\equiv\tell{B}+\ell^{\,\Phi}+\ell^{\,E}$ and $C_{{\ell^{\,E}}}^{EE}$ and $C_{{\ell^{\,\Phi}}}^{\Phi\Phi}$ denote the unlensed power spectra of the $E$ mode polarization and of the gravitational potential, respectively. This formula can be obtained by a second-order series expansion around undisplaced direction, which is expected to be accurate to within 1% for multipoles $\tell{B}\simlt 2000$ and then for $\tell{B}\gg2000$, where the CMB amplitude is small and can be modeled by its gradient only, while in the intermediate scales its precision degrades to nearly 5%. The reliability of this analytical model is discussed later in Sec. \[sect:simker\]. We can now introduce 1D kernels, ${\cal H}_{{\ell^{\,E}}}(\tell{B})$, defined as $${\cal H}_{{\ell^{\,E}}} (\tell{B}) \equiv \frac{1}{2}\,C_{{\ell^{\,E}}}^{EE}\,\sum_{{\ell^{\,\Phi}}} \, \frac{|_{2}F_{\tell{B}\,{\ell^{\,\Phi}}\,{\ell^{\,E}}}|^{2}} {2\tell{B}+1} \, C_{{\ell^{\,\Phi}}}^{\Phi\Phi}\,(1-(-1)^{L}).
\label{eq:oneDimKerDef}$$ Summed over ${\ell^{\,E}}$ for a fixed $\tell{B}$, these give the lensed $B$-mode power contained in the mode $\tell{B}$, Eq. \[eq:kerb\], while for a fixed ${\ell^{\,E}}$ they define the power spectrum of the lensed $B$-modes signal, generated via lensing from the $E$ polarization signal that contains non-zero power in a single mode ${\ell^{\,E}}$, and with its amplitude as given by $C_{{\ell^{\,E}}}^{EE}$. The kernels are displayed in Fig. \[fig:1dkers\] together with their analogs for the total intensity and $E$-polarization signals. We find that the kernels computed for different values of $\tell{B}$ are similar, just shifted with respect to each other accordingly. The change in the amplitude simply reflects the change in the assumed power of the $E$ signal, which in turn follows that of the actual $E$ power spectrum. The kernels are flat for values $\tell{B}\ll {\ell^{\,E}}$ and decay as a power law for $\tell{B} \gg {\ell^{\,E}}$, displaying a sharp dip at $\tell{B} = {\ell^{\,E}}$. Similar observations can be made for the $T$ and $E$ kernels, with the exception that unlike their $E$ and $T$ counterparts, the $B$ kernels are not peaked around the dip. This behavior is related to the fact that the lensed $B$-modes signal we discuss here, described by Eq. \[eq:kerb\], is generated by the $E$-polarization, while the main effect of the lensing on $T$ and $E$ is imprinted on these signals themselves. A direct consequence of this is that for any lensed $B$-modes spectrum mode a contribution from local unlensed multipoles will be less dominant, as is the case for the $T$ and $E$ signals, and nonlocal contributions will be relatively more important and therefore required to be accounted for in high-precision calculations.
Indeed, owing to the flat plateau of the kernels at the low-$\ell$ end, in principle all high-$\ell$ unlensed modes contribute to the lensed power at the low-$\ell$ end. The magnitude of their contribution is modulated by the shape of the unlensed $E$ spectrum and therefore eventually becomes negligible only because of the Silk damping, i.e., lack of the power at small angular scales in the unlensed fields. Nevertheless, we can expect that nearly all the modes of the unlensed $E$ spectrum up to the damping scale have to be included in the calculation of the lensed $B$ spectrum to ensure high-precision recovery of the lensed $B$-modes spectrum with $\tell{B} \simlt 1000$. Given some specific target precision, we could and should fine-tune the required $E$-spectrum bandwidth, and whatever is the value selected here, the bandwidth for the potential field will have to be at least the same.
For high-$\ell$ modes of the lensed $B$-modes spectrum, $\tell{B} \gg 1000$, the non-locality of the power transfer due to lensing is even more striking, as due to the low amplitudes of the $E$ spectrum the local contributions are additionally suppressed, and the long power-law tails of the contributions from large and intermediate angular scales, ${\ell^{\,E}}\simlt 1000$ are evidently dominant. Less evident is the fact that also the $E$-power from even smaller angular scales, ${\ell^{\,E}}\simgt \tell{B}$, may be relevant. The contributions from each of these modes may appear small, Fig. \[fig:1dkers\], but are potentially non-negligible due to the large number of those modes. A high-precision recovery of the high-$\ell$ tail of the lensed $B$-modes spectrum will therefore need a careful assessment of the importance of all these contributions, nevertheless, a generic expectation would be that the bandwidth of the unlensed $E$-modes spectrum will have to be higher than the highest value of the lensed $B$-modes signal multipole for which high precision is required, and potentially higher than the scale of Silk damping. Because these very high multipoles of the lensed $B$ spectrum are expected to have a significant contribution from relatively low multipoles of the unlensed $E$ signal, i.e., for which ${\ell^{\,E}}\ll \tell{B}$ given the triangular relations, Eq. \[eq:trRel\], and the definition of the kernels, Eq. \[eq:oneDimKerDef\], we can conclude that the bandwidth of the potential field used in the simulations will have to be at least as large as $\tell{B}$.
There are two main conclusions to be drawn here. First, it is clear that a high-fidelity simulation of the $B$-polarization power spectrum even in a restricted range of angular scales will require broad bandwidths, potentially all the way up to the scale of Silk damping, for both the unlensed $E$-mode polarization signal and the gravitational potential. However, these bandwidth values are not expected to depend very strongly on the maximal $B$-mode multipole that we want to recover, at least as long as it is in the range $\tell{B} \simlt 2000$. Second, because the expected bandwidths are broad, it is important to optimize them to ensure efficiency of the numerical codes without affecting precision of the results.
Thanks to the peaked character of the respective kernels, the lensed modes for the lensed $T$ and $E$ spectra are typically dominated by a local contribution coming from the immediate vicinity of the mode. This in general permits setting the bandwidth for the potential shorter than the mode of the lensed spectrum to be computed. By contrast, the unlensed $T$ and $E$ spectrum have to be known at least up to the multiple of interest of the lensed spectrum, $\tell{X},$ $(X=T$ or $E)$, augmented by the assumed bandwidth of the potential. These observations reflect the usual rule of thumb, [e.g., @lewis2005], indicating that lower bandwidth values can be used in these two cases for the same required accuracy.
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Accuracy {#sect:accuracy}
--------
In this section we aim at turning the consideration presented above into more quantitative prescriptions concerning the bandwidths of the input fields used in the simulations. For this reason we introduce 2D kernels, ${{\cal K}_{\tilde{\ell}^{\,B}}(\ell^{\,E}, \ell^{\,\Phi}) }$, defined as, $${{\cal K}_{\tilde{\ell}^{\,B}}(\ell^{\,E}, \ell^{\,\Phi}) }
\equiv \frac{1}{2} \, \frac{|_{2}F_{\tell{B}\,{\ell^{\,\Phi}}\,{\ell^{\,E}}}|^{2}} {2\tilde{\ell}_B+1} \, C_{{\ell^{\,\Phi}}}^{\Phi\Phi}\,C_{{\ell^{\,E}}}^{EE}\,(1-(-1)^{L}).
\label{eq:twoDimKerDef}$$ These define for a given value of $\ell_B$ a contribution of the $E$ power at $\ell = {\ell^{\,E}}$ and the $\Phi$ power at $\ell = {\ell^{\,\Phi}}$ to the amplitude of the lensed $B$-modes spectrum at that $\ell = \tell{B}$, which can then be computed by summing over ${\ell^{\,E}}$ and ${\ell^{\,\Phi}}$, i.e., $$\tilde{C}_{\tell{B}}^{BB}=\sum_{{\ell^{\,\Phi}},\, {\ell^{\,E}}} \, {{\cal K}_{\tilde{\ell}^{\,B}}(\ell^{\,E}, \ell^{\,\Phi}) }.
\label{eq:ker2dSum}$$ The sum in this equation involves in principle an infinite number of terms and therefore would have to be truncated in any numerical work, either explicitly, e.g., by setting finite limits in the formula above, or implicitly, e.g., by selecting the bandwidths, pixel sizes, etc, in the pixel-domain codes. We therefore used these kernels to study the precision problems involved in this type of calculations. As the expressions for the kernels are approximate, so will be our conclusions. However, as our goal is to provide guidelines on how to select the correct values for the simulations codes, this should not pose any problems. We will return to this point later in this section.
We show a sample of the kernels, ${{\cal K}_{\tilde{\ell}^{\,B}}(\ell^{\,E}, \ell^{\,\Phi}) }$ in Fig. \[fig:kerlog\]. These are computed for selected values of $\ell_{\tilde B}$ for which the approximations involved in their computation are expected to be valid. We note that all elements of the kernel, ${{\cal K}_{\tilde{\ell}^{\,B}}(\ell^{\,E}, \ell^{\,\Phi}) }$, vanish if the quantity $L$, defined in the previous section, is even, as do those for which the triangular relation $$\begin{aligned}
{\displaystyle \left |{\ell^{\,E}}-{\ell^{\,\Phi}}\right | \leq} &{\displaystyle \tell{B} \leq {\ell^{\,E}}+{\ell^{\,\Phi}}}
\label{eq:trRel}\end{aligned}$$ is not satisfied. This last fact is a consequence of the Wigner 3-j symbols in the expressions for $_2F_{\tell{B}\, {\ell^{\,\Phi}}\, {\ell^{\,E}}}$, [@hu2000]. Within these restrictions it is apparent from Fig. \[fig:kerlog\] that each multipole of the lensed $B$-modes spectra $\tilde{\ell}^{B}$ receives contributions from a wide range of harmonic modes of both E and $\Phi$ spectra, extending to values of ${\ell^{\,E}}$ and ${\ell^{\,\Phi}}$ significantly higher than $\tell{B}$ and roughly independent of the latter value at least for $\tell{B} \simlt 2000$. For its higher values a non-negligible fraction of the contribution starts to come from progressively higher multipoles of both $E$ and $\Phi$. Clearly, these trends are consistent with what we have inferred earlier with help of the 1-dim kernels.
As also observed earlier, we find the $B$-modes kernels qualitatively different from those computed for the lensed total intensity and $E$-modes polarization signals, Fig. \[fig:kerlog\], and they are more localized in the harmonic space with the bulk of power coming mainly from scales for which both $\ell^{T, E}$ are relatively close to the considered lensed multipole, $\tell{T,\, E}$.
We note that all the 2D kernels are positive[^7] and therefore including more terms in the sum, Eq. \[eq:ker2dSum\], will always improve the precision of the result. From the efficiency point of view one may want to include in the sum preferably the terms corresponding to the largest 2D kernel amplitudes because they provide the largest contribution to the final lensed result before adding those with progressively smaller kernel amplitudes until the required precision is reached. This approach would in principle ensure that the best accuracy is achieved with the smallest number of included terms. This may therefore look as a potentially attractive option from the perspective of optimizing the calculations. However, in practice, as the recurrence formulae are usually employed in the calculations, e.g., either those needed to compute spherical harmonics in the case of the pixel-domain codes or those needed to calculate the 3-j symbols as in a direct application of Eq. \[eq:ker2dSum\], and therefore all the terms up to a given bandwidth are at our disposal at any time, and it therefore seems efficient and useful to capitalize on those by including all of them in the calculation. Consequently, we estimated what degree of precision can be achieved by such calculations by including all the contributions up to some specific bandwidth values for the $E$ and $\Phi$ multipoles.
For the $B$-modes spectrum we therefore hereafter express the precision of the calculations as $$A_{\tell{B}}^{B}({\ell^{\,\Phi}},{\ell^{\,E}}) \, =\, 1 \, - \, \frac{ \sum_{\ell_{*}^{\Phi}=0}^{{\ell^{\,\Phi}}}\sum_{\ell_{*}^{E}=0}^{{\ell^{\,E}}}{\cal K}_{\tell{B}}(\ell_*^{\,E}, \ell_*^{\,\Phi})}
{\sum_{\ell_*^{\,\Phi}=0}^{{\ell_{max} }} \, \sum_{\ell_*^{\,E}=0}^{{\ell_{max} }}\, {\cal K}_{\tell{B}}(\ell_*^{\,E}, \ell_*^{\,\Phi})},
\label{Baccuracydef}$$ where the sums in the denominator should in principle extend over the infinite range of values of $\ell$, but for practical reasons are truncated to ${\ell_{max} }= 8000$, which for the range of lensed multipoles of interest in this work, $\tell{X} \simlt 5000$, should be sufficient.
This expression can be generalized for all lensed CMB spectra, but in this case our model has to take into account that the main effect due to lensing is to reshuffle the power of the signal and not to convert it into some other component. Therefore the total variance of the signal has to be conserved [e.g. @blanchard1987]. In this case the lensed power spectra of $X=T$ or $=E$ can be written as
$$\begin{aligned}
&\tilde{C}_{\tell{X}}^{X}&=\left(1-(\tell{X\;2}+\tell{X} - \alpha)\,R\right)C_{\tell{X}}^{X} +\sum_{{\ell^{\,X}}, \, {\ell^{\,\Phi}}}{{\cal K}_{\tilde{\ell}^{\,X}}(\ell^{\,X}, \ell^{\,\Phi}) }\\ \label{eq:fullmodelcl}
&R&=\sum_{\ell^{\Phi}=0}^{{\ell_{max} }^{\Phi}}\frac{{\ell^{\,\Phi}}({\ell^{\,\Phi}}+1)(2{\ell^{\,\Phi}}+1)}{8\pi}C_{{\ell^{\,\Phi}}}^{\Phi},\end{aligned}$$
where $\alpha$ is an integer that is different for each CMB spectra
- $\alpha =4$ for X=E
- $\alpha=0$ for X=T
- $\alpha=2$ for X=TE.
We note that the factor $R$ is a smooth function of the cutoff value of the sum over ${\ell^{\,\Phi}}$, which quickly becomes nearly constant for ${\ell_{max} }^{\,\Phi}\gtrsim 1000$, Fig. \[fig:offset\]. Hereafter, we therefore precompute it once assuming ${\ell_{max} }^{\,\Phi}={\ell_{max} }= 8000$ and use it in all subsequent calculations. The generalized expression for the accuracy function in Eq. \[Baccuracydef\] would then be $$A_{\tell{X}}^{X}({\ell^{\,\Phi}},{\ell^{\,X}}) \, = \, 1 \, - \, \frac{ O_{\tell{X}}^{X}+\sum_{\ell_{*}^{\Phi}=0}^{{\ell^{\,\Phi}}}\sum_{\ell_{*}^{X}=0}^{{\ell^{\,X}}}{\cal K}_{\tell{X}}(\ell_*^{\,X}, \ell_*^{\,\Phi})}
{O_{\tell{X}}^{X}+\sum_{\ell_*^{\,\Phi}=0}^{{\ell_{max} }} \, \sum_{\ell_*^{\,X}=0}^{{\ell_{max} }}\, {\cal K}_{\tell{x}}(\ell_*^{\,X}, \ell_*^{\,\Phi})},
\label{Xaccuracydef}$$ where for shortness we have introduced $$\begin{aligned}
O_{\tell{X}}^{X} \equiv \left(1-(\tell{X\;2}+\tell{X} - \alpha)\,R\right)\,C_{\tell{X}}^{X}.
\nonumber\end{aligned}$$ We note that for cosmological models of the current interest, the factor $R$ is typically found to be on the order of $\mathcal{O}(10^{-6})$ and thus the term $O_{\tell{X}}^{X}$ is expected to be negative for most of the values of $\tell{X}$ in the range of interest here, see Fig. \[fig:offset\].\
![Example of the behavior of the offset term, $O_{\tell{T}}^{T}$, as a function of ${\ell_{max} }^{\,\Phi}$ for $\tell{T}=500$ (red), $2000$ (blue), $4000$ (green). The dashed part of the green line represents negative values. $O_{\tell{E}}^{E}$ and $O_{\tell{B}}^{B}$ have a similar shape, but a different amplitude. []{data-label="fig:offset"}](./T_offset_terms){width=".5\textwidth"}
In Fig. \[fig:kerlog\] black solid lines represent the expected error estimates, as expressed by the accuracy function, $A_{\tell{X}}^X({\ell^{\,\Phi}},\ell^{Y})$, for a number of selected values ranging from $25$% to $0.01$%. We note that for the shown range of $\tell{}$ only the sub-percent values of the accuracy are likely to be somewhat biased due to the assumed cutoff in the denominator of Eqs. \[Baccuracydef\] or \[Xaccuracydef\], an effect, which is therefore largely irrelevant for our considerations here. The fact that our accuracy definition is based on an approximate formula is also not a problem because any potential (and small, @challinor2005) discrepancy would affect both the numerator and denominator of Eqs. \[Baccuracydef\] an \[Xaccuracydef\] in the same way. It can therefore be shown that to the first order in the discrepancies amplitude, precision of our accuracy criterion improves progressively when the estimated level of the accuracy, $A_{\tell{X}}({\ell^{\,\Phi}},{\ell^{\,X}})$, tends to $0$ and is degraded to the percent level when $A_{\tell{X}}^{X}({\ell^{\,\Phi}},{\ell^{\,X}}) \approx 90\%$, i.e., when it is well outside of the region of any interest for the high-precision simulations considered here (see Appendix \[appendix:accuracy\_precision\]).\
The differences in the shape of the lensing kernels result in differences in the accuracy contours for different lensed signals and their multipoles as shown in Figs. \[fig:kerlog\] and \[fig:kerlog2\]. In particular, for lensed $B$-modes, the contribution of large-scale power of the CMB to the lensed signal is more significant. In spite of these differences, we, however, find that the overall contours seem to share a similar shape made of two lines nearly aligned with the plot axes which meets at a right angle. Consequently, if one of the two bandwidths is fixed, then the accuracy, which can be reached by such a computation, will be limited and, moreover, starting from some value of the other bandwidth, nearly independent on its value. This has two consequences. First, if the attainable precision is not satisfactory given our goals, it can be improved only by increasing the value of the first bandwidth appropriately. Second, the value of the second bandwidth can be tuned to ensure nearly the best possible accuracy, given the fixed value of the first bandwidth, while keeping it much lower than what the triangular relation, Eq. \[eq:trRel\], would imply. This could lead to a tangible gain in terms of the numerical workload needed to reach some specific accuracy. Turning this reasoning around, we could think of optimizing both bandwidths to minimize the cost of the computation for a desired precision. From this perspective, taking the turnaround point of the contour for a given accuracy may look as the optimal choice. However, this choice would merely minimize the sum of both bandwidths, (or some monotonic function of each of them) for the given accuracy, which may or may not be relevant for a specific case at hand. Instead we may rather select the bandwidths to minimize explicitly actual computational cost of whatever code we plan on using. We present specialized considerations of this sort in the next section.
On a more general level, we find that the standard rule of thumb, interpreting the effects of lensing as a convolution of the unlensed CMB signal with a relatively narrow, $\Delta \ell \sim 500$, convolution kernel due to the lensing potential, applies only for $T$ and $E$ signals and even in these cases only to low and intermediate values of $\tell{T,\, E} \simlt 2000$ and only as long as a computation precision on the order of $\sim 1$% is sufficient. For higher values of the lensed spectrum multipoles or higher levels of the desired accuracy in the case of $T$ and $E$ and for all multipoles of the $B$-polarization signal, the required bandwidths of both the respective, unlensed CMB signal and the gravitational potential are more similar and indeed the latter bandwidth is often found to be broader.
We note that an analysis of this sort is somewhat more prone to problems in the case of the $TE$ power spectrum since the lensing kernels ${{\cal K}_{\tilde{\ell}^{\,TE}}(\ell^{\,TE}, \ell^{\,\Phi}) }$ are not always positive because they contain the products of two different Wigner 3j coefficients and $TE$ power spectra, which may be non-positive, rendering the corresponding accuracy function not strictly monotonic. Hereafter, we excluded this spectrum from our analysis, noting that any band limits prescriptions derived for $T$ and $E$ will also apply directly to $TE$.
Numerical analysis {#lens2sec}
==================
In this section, we present results of simulations of lensed polarized maps of the CMB anisotropies and their spectra. We address two aspects here. First, we numerically study self-consistency of the pixel-domain approach to simulating the lensing effect. Second, we demonstrate how the consideration from the previous section can be used to optimize numerical calculations involved in these simulations.
We start this section by introducing a new implementation of the pixel-domain algorithm, which we refer to as [[lenS$^{2}$HAT]{}]{}.
[[lenS$^{2}$HAT]{}]{} {#sect:lens2hat}
---------------------
[[lenS$^{2}$HAT]{}]{} is a simple implementation of the pixel-domain algorithm for simulating effects of lensing on the CMB anisotropies. The hallmark of the code is its algorithmic simplicity and robustness, with its performance rooted in efficient, memory-distributed parallelization. The code is therefore particularly well-adapted to massively parallel supercomputers. Its implementation follows the blueprint described in @lewis2005 that summarized in Sect. \[s3ect:basics\]. The main features of the code are listed below.
#### Grids.
The code can produce lensed maps in a number of pixelizations used in cosmological applications, but internally it uses grids based on the equidistant cylindrical projection (ECP) pixelization where grid points, or pixel centers, are arranged in a number of equidistant iso-latitudinal rings, with points along each ring assumed to be equidistant. This pixelization supports a perfect quadrature for band-limited functions, which in the context of this work permits minimizing undesirable leakages that typically plague codes of this type. It can be shown, @driscoll_healy_1994, that an ECP grid made of $2\,L$ iso-latitudinal rings, each with $2\,L$ points and a weight, as given by $$w^{j}=\frac{2\pi}{L^{2} }\sin(\theta_{j})\sum_{\ell=0}^{L-1}\frac{\sin\left((2\ell+1)\theta_{j}\right)}{2\ell+1}, \qquad \theta_{j}=\frac{\pi}{2L}j,
\label{weightsdh}$$ is required and sufficient to ensure a perfect quadrature for any function with a band not larger than $L$.
#### Interpolation.
For the interpolation, the code employs the nearest grid point (NGP) assignment, e.g., we assign to every deflected direction a value of the sky signal computed at the nearest center of a pixel of the assumed pixelization scheme, therefore the respective sky signal values are calculable at the fast spherical harmonic speed. The NGP assignment is extremely quick and simple, but it requires the computations to be performed at a very high resolution to ensure that the results are reliable. The sufficient resolution required for this will in general depend on the intrinsic sky signal prior to the lensing procedure, as well as the resolution of the final maps to be produced, as is discussed in Sect. \[sect:codepars\]. As discussed above, in a typical case these are expected to be very high and the computations involved in the problem may quickly become very expensive. Nevertheless, as we show in Sect. \[sec:numPerf\], the overall computational time in this case is only somewhat longer than that involved in some other interpolation schemes, while the memory requirement can be significantly lower. However, the major advantage of this scheme for the purpose of this work is its simplicity and in particular the fact that its precision is driven by a single parameter defining the grid resolution.
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#### Spherical harmonic transforms.
To sidestep the problem of computing spherical harmonic transforms with a huge number of grid points and a very high band limit, [[lenS$^{2}$HAT]{}]{} resorts to parallel computers and massively parallel numerical applications. With these becoming quickly more ubiquitous and affordable this solution is becoming progressively more attractive.
Parallelization of the fast spherical harmonic transforms is difficult due to the character of the input and output objects and the involved computations, where a calculation of each output datum requires knowledge of, and access to, all input data. This is clearly not straightforward to achieve without extensive data redundancy, as done e.g., in [[LensPix]{}]{} or parallel routines of HEALPix, or complex data exchanges between the CPUs involved in the computation. To avoid such problems in our implementation we used the publicly available scalable spherical harmonic transform ([S$^{2}$HAT]{}) library[^8]. This library provides a set of routines designed to perform harmonic analysis of arbitrary spin fields on the sphere on distributed memory architectures (though it has an efficient performance even when working in the serial case). It has a nearly perfect memory scalability obtained via a memory distribution of all main pixel and harmonic domain objects (i.e., maps and harmonic coefficients), and ensures very good load balance from the memory and calculation points of view. It is a very flexible tool that allows a simultaneous, multi-map analysis of any iso-latitude pixelization, symmetric with respect to the equator, with pixels equally distributed in the azimuthal angle, and provides support for a number of pixelization schemes, including the above mentioned ECP; see @szydlarski2011 for more details. The core of the library is written in F90 with a C interface and it uses the message passing interface (MPI) to institute distributed memory communication, which ensures its portability. The latest release of the library also includes routines suitable for general purpose graphic processing units (GPGPUs) coded in CUDA [@hupca2011; @szydlarski2011; @fabbian2012].\
We emphasize that if a sufficient resolution can be indeed attained, the approach implemented here can produce results with essentially arbitrary precision. In the following we demonstrate that thi is indeed the case for the described code.
Code parameters {#sect:codepars}
---------------
### Overview
In this section we describe how we fixed the essential parameters of the code. We first emphasize important relations between them. A detailed description of the procedures used to assign specific values to them, is given in the following sections.
1. We start by defining a target value in terms of the highest value of the harmonic mode, $\tell{Y}_{req}$, that we aim to recover and its desired precision, $\varepsilon$. We then use the reasoning from Sect. \[sect:accuracy\] to translate this requirement into corresponding bandwidths, ${\ell^{\,X}}$ and ${\ell^{\,\Phi}}$, of the relevant unlensed signals, $X$ and $\Phi$. These ensure that the precision of all modes of the lensed signal up to $\tell{Y}_{req}$ will be not lower than $\varepsilon$, barring any unaccounted-for, numerical inaccuracies. The values of ${\ell^{\,X}}$ and ${\ell^{\,\Phi}}$ are then used to estimate the bandwidth of the output, lensed map, $\tell{Y}_{out}$.
2. We then simulate two unlensed maps, $\bm{m}^X$ and $\bm{m}^\Phi$, of the signal $X$ and potential field, $\Phi$, with their band limits set to ${\ell^{\,X}}$ and ${\ell^{\,\Phi}}$, as estimated earlier. The number of pixels of the displacement map, $\bm{m}^\Phi$, is equal to that in the output map of the lensed signal, and for the ECP grid, equal therefore to $N^\Phi_{pix} = 4\,{{\tell{Y}_{out}}\,}^2$. The number of pixels in the $X$-signal map, $\bm{m}^X$ is then given by $N_{pix}^X = 4\,\kappa^2\,{{{\ell^{\,X}}}\,}^2$, where $\kappa$ is an overpixelization factor introduced in Sect. \[s3ect:basics\] and discussed in detail below, Sect. \[sect:pix\]. For simplicity, we assume that the grid for which the unlensed field $X$ is computed is a subgrid of the grid used for $\Phi$.
3. The reassignment procedure (step 5 of the algorithm, Sect. \[s3ect:basics\]) is then straightforwardly performed, leading to the map containing power potentially up to $\tell{Y}_{out}$, which maybe needed to be filtered down to the band limit of $\tell{Y}_{req}$, as initially required.
### Intrinsic bandwidths
We employ the procedure described earlier in this work in Sect. \[sect:accuracy\] to set the intrinsic band limits. Instead of using generic predictions, we aim at optimizing their values to ensure the lowest possible computational overhead. To do so we need to provide a model of the cost of numerical calculations involved in [[lenS$^{2}$HAT]{}]{}. This is dominated by large spherical harmonic transforms, one needed to calculate the map of $\Phi$ and the other to calculate that of signal $X$. Given the parameters introduced above and because the total cost of a spherical harmonic transform is proportional to $N_{pix}\,\ell_{max}$ we therefore obtain $$\begin{aligned}
C\equiv C({\ell^{\,\Phi}},{\ell^{\,X}}) & \propto & 2\,N_{pix}^{\Phi}{\ell^{\,\Phi}}+n_{stokes}\,N_{pix}^{X}{\ell^{\,X}}\nonumber\\
& = & 8\, {\tell{Y}_{out}\,}^2 \, + \, 4\,n_{stokes}\,\kappa^2\,{{\ell^{\,X}}\,}^2\nonumber\\
& = & 8\, \eta^2 \, ({\ell^{\,\Phi}}+{\ell^{\,X}})^2\,{\ell^{\,\Phi}}\, + \, 4\,n_{stokes}\,\kappa^2\,{{\ell^{\,X}}\,}^3.
\label{eq:costfunc}\end{aligned}$$ Here $n_{stokes}$ stands for the number of signal maps, that we aim to produce and is equal $1$ – $T$-only, $2$ – $E$ and $B$, or $3$ – $T$, $E$, and $B$, while for the field $\Phi$ the pre-factor is fixed and equal to $2$, reflecting the number of components of a vector field on the sphere. In deriving the last equation above we have assumed that $\tell{Y}_{out} = \eta\, ({\ell^{\,\Phi}}+{\ell^{\,X}})$. This is justified below, as are the values that should be adopted for $\eta$ and $\kappa$. The expression above includes neither the cost of the interpolation nor reshuffling, but because in both these cases the number of involved operations is proportional to $N_{pix}$, their cost is negligible with respect to that of the transforms.\
Solving for the optimized values of the bandwidths, which simultaneously ensure the desired precision, $\varepsilon$, at a selected multipole, $\tell{Y}_{req}$, involves minimizing the cost function in Eq. \[eq:costfunc\], with a constraint, $A_{\tell{Y}_{req}}^{Y}({\ell^{\,\Phi}},{\ell^{\,X}}) = \varepsilon$, Eqs. \[Baccuracydef\] and \[Xaccuracydef\]. This is implemented as follows. First, we define a grid of levels of the cost function and for each level calculate the best accuracy achievable on its corresponding contour. If this accuracy for some of the levels is close to our target, we find a corresponding pair of bandwidth values, $({\ell^{\,\Phi}},{\ell^{\,X}})$, which then defines our optimized solution. If none of the accuracies is sufficiently close to the required precision, we take two levels for which the assigned accuracies bracket the target value and insert an intermediate level for which we calculate the corresponding best accuracy. We repeat this procedure until the best accuracy found for the newly added contour is sufficiently close to the target one. We then use it to find the pair of the optimized bandwidths as above. As mentioned before, in general, the two optimized bandwidth values will not be equal. This appears to be particularly the case when simulating the CMB spectra at very high multipoles and especially in the cases involving the $B$ modes, which have broader kernels and are more demanding in terms of bandwidth requirements. The procedure allows one to gain a factor of nearly $40$% in terms of runtime inthea range of accuracy of interest for lensed $B$ multipoles close to $4000$, especially if high oversampling is required. For temperature and $E$-mode polarization, where less extra power is required in $\Phi$ to obtain an accurate result, the gain can be quantified to be nearly $20$% - $30$%. We report in Figure \[fig:diag\_bandreq\] the dependence of the optimized bandwidth parameters as a function of the required accuracy imposed at different lensed multipoles of $T$, $E$, and $B$ spectra, in the right column, and contrast them with the bandwidths obtained in the case when both of them are assumed to be equal. In Figure \[fig:costgain\] we show typical runtime gains as a function of the required accuracy.
We note that here that whether we choose to optimize the bandwidths or just assume that they are equal, we find that imposing a certain accuracy level at some multipole, $\tell'$, ensures that the same accuracy requirement will be fulfilled for all $\tell{} \le \tell'$.
### Lensed map band-limit
For the resolution of the final map, we note that in an absence of numerical effects, such as those due to the pixelization and interpolation, the lensing procedure would be described by Eq. \[eq:kerb\] and the bandwidth of the lensed map would be simply given by ${\ell^{\,X}}+ {\ell^{\,\Phi}}$. In the presence of the numerical effects, the output map will have an effective bandwidth typically higher than that, which will lead to some power-aliasing at the high-$\ell$ end if this theoretical band limit is imposed. We find this to be indeed the case in our numerical calculations. However, we also find that once the overpixelization factor is set correctly, the aliasing is localized to at most $25$% of the bandwidth and therefore easy to deal with in post-processing, e.g., step 6 of the algorithm outline in Sect. \[s3ect:basics\]. Consequently, we used $\tell{Y}_{out} = \eta\, ({\ell^{\,X}}+{\ell^{\,\Phi}})$ in our numerical simulations, with $\eta=1.25$ as the band limit.
It is important to emphasize that NGP is one of the sources of the aliasing, because it does not preserve the bandwidth of the interpolated function, like some of the other, ad hoc procedures proposed in this context. Clearly, an interpolation that preserves the function bandwidth would be a significant improvement for this type of algorithms, if it comes without prohibitive numerical cost. We leave such an investigation to future work.
![Comparison between the $E$-modes power spectra of the input displacement field (black) and the displacement field after NGP assignment for several values of the oversampling factor $\kappa$. The input displacement is computed on an ECP grid with a number of pixel $N_{pix}=16384^{2}$ while the discretized one is the result of an NGP assignment on a grid of $\kappa^{2}N_{pix}$. With progressively higher resolution the extra power due to discretization becomes negligible and the two spectra become almost indistinguishable. The discretization-induced error power spectrum is shown as a dotted line for reference; both $E$ and $B$ modes of the discretization error have the same power spectra.[]{data-label="fig:discdisp"}](./discretized-disp.pdf){width=".49\textwidth"}
### Overpixelization factor {#sect:pix}
As we explained already our interpolation procedure consists of two steps: an overpixelization that is followed by an NGP assignment. The overpixelization involves producing maps with the sky signal sampled at significantly higher rate than is necessary given from the signal’s band limit. For the ECP grids used internally by [[lenS$^{2}$HAT]{}]{}, this is implemented by using $\kappa$-times more points in each of the $\phi$ and $\theta$ directions. The remaining problem is then to fix the appropriate value of $\kappa$. To do so, we first observe that for the overpixelized grid, the NGP assignment can be seen in two ways. Either as approximating the true value of the sky signal, which needs to be calculated in one of the displaced directions, which are precisely computed in turn, which is the standard perspective and the only one available if a more sophisticated interpolation scheme is applied. Or it can be seen as approximating each displaced direction by a direction pointing toward the nearest grid point, with a correct sky signal value assigned to it. This second viewpoint provides us with an independent test to check if the density of our overpixelized grid is sufficiently high. The involved procedure involves first calculating the approximate displacement field and its power spectrum, which is then compared with the input power spectrum for the gravitational potential, $\Phi$. We note that the approximation used here can in general generate a non-zero curl and therefore there will be two non-vanishing spectra of the approximate displacement field, corresponding to its $E$ (gradient) and $B$ (curl) components. We then require that the recovered $B$ spectrum is significantly smaller than that of $E$, and that both the recovered $E$ spectrum and the input one agree sufficiently well up to the angular scales, which are of interest given the $\ell$-range of the lensed spectrum we are after and its precision. These latter two are turned into the $\ell$-range requirement using one of the 2D kernels.
Examples of such comparisons are shown in Fig. \[fig:discdisp\] for a number of values of the oversampling factor ranging from $\kappa = 1$ up to $8$. We see that for the latter value the approximate $E$ spectrum is consistent over the entire shown range of $\ell$ values and the recovered $B$ is there significantly smaller. We therefore continue to use this value in the runs discussed later in this work, even if, as noted in the next section, $\kappa = 4$ could be sufficient at least for $\tell{B} \simlt 2000$. We also point out that, as it might have been expected, the departures of the recovered $E$ spectrum for the displacement from the input one are consistent with the presence of the non-zero $B$-type mode in the approximated displacement field with an amplitude similar to that of its $E$-mode spectrum, which renders our two criteria redundant. In addition, if only $T$ and $E$ CMB spectra are of interest, then $\kappa\approx 2$ is usually sufficient to obtain accurate results on the scales of interest because the long-tails of the displacement spectrum are less relevant in these cases.
For completeness, in Fig. \[bmodes\_vs\_sampling\] we show the relevant CMB $B$-spectra computed with the same values $\kappa$ as shown in Fig. \[fig:discdisp\] and aiming at a high-precision reconstruction for $\tell{B} \le 2000$, demonstrating that both overpixelization rates, as inferred above, ensure a satisfactory recovery of this spectrum in the targeted range of $\ell$. We provide more details about this Figure in Sect. \[s3ect:simSpecs\].
![Lensed $B$-modes spectrum computed for different values of over sampling factor compared with the lensed spectrum obtained with the analytical Boltzmann code CAMB (red dashed).[]{data-label="bmodes_vs_sampling"}](./oversampling-b){width=".5\textwidth"}
Validation and tests
--------------------
### Simulated kernels {#sect:simker}
{width=".48\textwidth"} {width=".48\textwidth"}
As a first step of validation of our code, we investigated whether its results agree with the prediction of the semi-analytical approach used to model convolution in the harmonic domain. We focus here on numerically feasible studies of the 1D kernels, as defined in Eq. \[eq:oneDimKerDef\]. For this purpose we assumed that the unlensed CMB signal, i.e., $E$-modes polarization in the case of the lensed $B$-modes spectrum, contains power only in a single harmonic mode, $\ell_0$ i.e., $C_\ell^{EE} \propto \delta^{Kronecker}_{\ell\,\ell_0}$ and computed the resulting lensed $B$-modes spectrum for several values of $\ell_0$ using [[lenS$^{2}$HAT]{}]{}. We compared them with the analytic results obtained for the same multipole and displayed in Fig \[fig:1dkers\]. The results of this calculation are shown in Fig. \[fig:numeric-ker\], where we see that in a range where the analytic model is more reliable the agreement between the two curves is excellent if only a sufficient resolution for the unlensed grid is used. On the other hand, in the region where the analytic approximation we used is not accurate anymore because amplitudes of the CMB signal and its gradient are comparable and therefore the truncation in the series expansion introduces a non-negligible error, the discrepancy between our analytical model and the simulated 1D kernels becomes more evident. Such an approximation tends to overestimate the contribution of each single mode to its neighboring angular scales of a factor of nearly 50% with respect to simulated kernels and to slightly underestimate the contribution of each mode to the kernel tails, i.e., to the multipoles higher than the one in exam. Nevertheless, the analytically-approximated and simulated kernels are found to be qualitatively quite similar, which validates therefore our semi-analytic bandwidth requirements presented earlier.\
### Simulated spectra {#s3ect:simSpecs}
Another batch of performed tests consisted in comparing the spectra obtained from [[lenS$^{2}$HAT]{}]{} and those derived with Boltzmann codes such as CAMB or CLASS. In particular, the black solid line in Fig. \[bmodes\_vs\_sampling\] shows an example of the result obtained for a simulation of lensed $B$-modes designed to reach an accuracy of up to 0.1% at $\ell^{B} \simlt 2000$. Because no band-limit optimization is performed, and it is therefore assumed that ${\ell^{\,\Phi}}={\ell^{\,E}}={\ell_{max} }$, the latter value has to be at least ${\ell_{max} }= 4000$, Fig. \[fig:kerlog2\]. The lensing convolution of signals with such a band limit leads to polarized maps with power up to $2\,\ell_{max}$, which means that to avoid any aliasing, we would need a grid for the lensed sky and the displacement field with at least $N_{\theta}\approx 2{\ell_{max} }$ rings with $N_{\phi}\approx 2{\ell_{max} }$ pixels per ring, i.e., $N_{pix}\approx 16384^{2}$, where we have rounded the number of rings and pixels per rings to a power of $2$. Once the band limit of the signals and the respective grid for the lensed sky is set, we still need to define the overpixelization rate as required by our interpolation. As noted in the previous section, there seem to be a general reasoning based on the discretized displacement spectra, which points toward $\kappa = 8$ as a sufficient value. Because calculating the overpixelized map, albeit with a restricted band-limit, is the most time-consuming part of the code, there may be an interest on occasion to tune $\kappa$ to be as small as possible. In this context we find, as illustrated in Figs. \[fig:discdisp\] and \[bmodes\_vs\_sampling\], that if the extra power introduced by discretization of displacement field does not exceed $10$% of the power in the non-discretized displacement field on scales $\ell\approx 1.5 \,\ell^{B}$, an oversampling factor of $4$ is sufficient to render a power spectrum on scales $\ell\simlt \ell^{B}$ with an accuracy as determined by the assumed bandwidth. However, the factor $4$ should be treated as a lower bound and be used with care because there will typically be a significant amount of extra power in the $B$-mode spectrum for $\ell \simgt \tell{B}$, which may need to be efficiently filtered out before the respective map can be further used. In contrast, if the extra power found in the discretized displacement does not exceed $10$% of the original power for all angular scales up to ${\ell^{\,\Phi}}_{max}$, then no overshooting takes place and the results remain highly accurate also beyond the scale of interest $\ell^{B}$.\
In Fig. \[fig:heroicFig\] we present the spectra for the two polarized components $E$ and $B$ as well as the displacement field, $\Phi$, computed in a run aiming at recovering of these signals in a band up to $\tell{X} \simlt 5500$ with precision better than to $0.1$%. For this run we assumed the value of the required bandlimits to be ${\ell^{\,E}}_{max}={\ell^{\,\Phi}}_{max}=8000$ . These values were extrapolated from Fig. \[fig:diag\_bandreq\], where to obtain a $0.1$% accuracy on $B$-modes on similar angular scales (e.g. $\tell{X}= 4000$) we needed to include power up to ${\ell_{max} }\approx \tell{X}+2500$. Following the same prescription as given for the previously detailed case of Fig. \[bmodes\_vs\_sampling\], we set the resolution of the unlensed sky and displacement field to $N_{pix}=32768^2$ while, to ensure the highest possible reliability of the result, we pushed the oversampling factor to 16. The discretization errors introduced by this setup are found to stay under the $1$% level on all the angular scales involved in the calculation and no significant overshooting is shown (see Fig \[fig:heroicFig\]). Though the band limit and resolution involved may look exaggerated from the practical point of view, they simultaneously demonstrate the capability of the numerical code while illustrating our conclusions concerning the precision of these calculations.\
In general, we find that a simple algorithm as proposed in [[lenS$^{2}$HAT]{}]{} is capable of simulating effects of lensing on CMB over the range of angular scales of interest for current and foreseeable experimental efforts. Moreover, if used properly, it does so with an accuracy that on very small scales is limited rather by the precision of the input power spectrum of unlensed CMB than by the employed numerical algorithm.
{width="\textwidth"}
Convergence tests
-----------------
To investigate the precision and reliability of our approach it is interesting to investigate the numerical convergence of the results without relying on a direct comparison to an external Boltzmann code. Since several experiments in the future will be able to target non-Gaussianities in CMB polarization, i.e., the statistical moments beyond the power spectrum, we decided to study the convergence of the results not only on the power spectrum level, but also in the real domain, i.e., on the map level.\
Power spectrum convergence
--------------------------
{width=".3\textwidth"}{width=".3\textwidth"}{width=".3\textwidth"}\
{width=".3\textwidth"}{width=".3\textwidth"}{width=".3\textwidth"}\
We first investigate the convergence of the power spectrum up to a given scale of interest $\tell{X}$ as a function of the bandwidths. This procedure allows us to simultaneously show the precision of our code and also to indirectly prove the validity of the bandwidth requirements given in Sect. \[sect:lens2hat\]. For this purpose we assumed the bandwidths for CMB and $\Phi$ fields to be equal and then fixed the resolution of the grid following the prescriptions of Sect. \[s3ect:simSpecs\] assuming $\kappa=8$. We simulated CMB maps off all three Stokes parameters $T$, $Q$, and $U$ and then computed the precision of the amplitude of the power in some multipole of interest, $\tell{X}$, recovered from the simulation. The precision is defined as the fractional difference between the amplitudes obtained from two simulations performed for two considered values of ${\ell_{max} }$. For these specific tests we verified that the random realization of the harmonic coefficients used in the simulation is the same when changing the value of the bandwidth from ${\ell_{max} }$ to a value ${\ell_{max} }^{\prime}$ for $\ell\leq{\ell_{max} }$. We report the result of the numerical convergence for $\tell{X}=2000$ in Table \[tab:2000\]. We note that the results agree with the analytic calculation of Sect. \[sect:lens2hat\], where we saw that extending the band limit has no visible effect on the recovered results on the scale of interest if a proper amount of power has already been convolved. As expected, a significant fraction of $E$-modes power is converted into $B$-modes for angular scales $\ell^{E}\in\left[3000,4000\right]$ but no significant improvement is seen if power beyond $\ell^{E}=4000$ is included. We also performed a test case for $\tell{X}=4000$, i.e., in the regime where the gradient approximation is expected to be less accurate, Table \[tab:4000\]. The $B$-modes accuracies are consistent with those derived in Sect. \[s3ect:simSpecs\] except for the last set of bandwidth parameters, where the fractional difference between simulated spectra seems to saturate at a level of $0.1$%. This may be related to a small residual aliasing due to an underestimated oversampling parameter.\
$\tell{X}$ $\Delta_{2000}^{3000}$ $\Delta_{3000}^{4000}$ $\Delta_{4000}^{6000}$ $\Delta_{6000}^{8000}$
------------ ------------------------ ------------------------ ------------------------ ------------------------
TT 43% 0.04% 0.02% 0.003%
EE 31% 0.01% 0.01% 0.005%
BB 35% 3% 0.02% 0.004%
TE 32% 0.04% 0.01% 0.002%
: Numerical convergence of simulated lensed CMB power spectra at multipole $\tell{X}=2000$. Each column shows a fractional change in the lensed spectra amplitude due to an increase of the bandwidths of both the unlensed CMB and potential field, assumed here to be equal, as denoted by super- and sub-scripts of $\Delta_{\ell_{max}}^{\ell_{max}^{\prime}}$.[]{data-label="tab:2000"}
$\tell{X}$ $\Delta_{4000}^{5000}$ $\Delta_{5000}^{6000}$ $\Delta_{6000}^{7000}$ $\Delta_{7000}^{8000}$
------------ ------------------------ ------------------------ ------------------------ ------------------------
TT 31% 0.2% 0.09% 0.1%
EE 32% 0.2% 0.05% 0.03%
BB 7% 4.6% 0.1% 0.09%
TE 21% 0.7% 0.2% 0.2%
: As Table \[tab:2000\], but for $\tell{X}=4000$.[]{data-label="tab:4000"}
Map convergence
---------------
After showing the convergence on the power spectrum level, which provides information on the overall variance of the simulated maps, we investigated if the convergence of our numerical result is also realized in the real domain. For this purpose we first defined an error map obtained as a difference of two maps computed assuming two different bandwidths $\ell_{max,1},\ell_{max,2}$ rescaled by the rms value of one of the two maps, i.e., $$\mathscr{E}^{X}_{\ell_{max,1},\ell_{max,2}}=\frac{\bm{m}^{X}_{{\ell_{max} }2}-\bm{m}^{X}_{{\ell_{max} }1}}{\sqrt{{\rm Var}\,(\bm{m}^{X}_{{\ell_{max} }2})}} \qquad X\in\{T,Q,U\},
\label{eq:relDiffDef}$$ where $\bm{m}^{X}_{\ell_{max,1}}$ is a simulated map of the field $X$ obtained assuming $\ell_{max,1}$ as the bandwidth. After deriving the harmonic coefficients with the procedure outlined in the previous section, we filtered out all modes on angular scales $\ell\geq\tell{X}$ and resampled the signal on a grid that properly samples the signal up to multipole $\tell{X}$. To take advantages of the HEALPix visualization tools, we use for this purpose an HEALPix grid having $nside=1024$ ($2048$) for $\tell{X}=2000$ ($4000$). After resampling the harmonic coefficients we computed the power spectrum of the error field ${\mathscr{E}^{X}_{\ell_{max,1},\ell_{max,2}}}$, which demonstrates the precision obtained on the map level. In Fig. \[fig:errmap2000\] we report the result of this analysis for the test case $\tell{T,E,B}=2000$ . The error-field power spectrum is found to be very similar to a white noise spectra with r.m.s below $1$%. If the power is properly resolved, an accuracy equivalent to $0.1\%$ on the map level can indeed be obtained, while the error slightly increases to $0.3\%$ for polarization (see Fig. \[fig:error-ps2000\]). However, this test case does not include the effect of any realistic experiment setup; in a real-life case the criterion for the convergence is set by the noise level on the pixel, if instrumental noise has to be added to the simulated maps.
![$T$, $E$, and $B$ power spectrum of the error field obtained from maps simulated with different values of bandwidth parameters (colored lines). For reference, the dot-dashed and dashed lines show the spectra of a white-noise process with variance equal to $0.01$ and $0.001$, respectively. []{data-label="fig:error-ps2000"}](convergence_error_2000){width=".48\textwidth"}
The results presented above show that the systematic uncertainties inherent to the pixel-domain simulation method can be controlled with high accuracy, demonstrating that this method can provide a sufficiently precisely framework within which to compare and study different physical assumptions entering such calculations and in particular to investigate the impact of cosmological models on the $B$-mode lensing predictions. We emphasize that the pixel-domain method is sufficiently general to be applicable to a range of diverse physical contexts of this kind. Even more importantly, the applicability of the considerations presented here goes beyond the pixel-domain method and can be straightforwardly extended to, for instance, ray-tracing approaches, which do not involve Born-approximation.
Monte Carlo simulations
-----------------------
To test whether our method produces any significant bias in the power spectrum we produced $N_{r}=100$ independent realizations of $ lensed T$, $Q$, and $U$ maps that were required to reach $0.1\%$ accuracy up to $\tell{B}=3000$ and investigated the statistical properties of the power spectra averaged over these realizations. The latter is expected to be nearly Gaussian-distributed since the non-Gaussian correlations in the lensed power- spectrum covariance induced by lensing itself are negligible for $T$, $E$ and $TE$ spectra. However, for all the power spectra including the $B$ field, we expect the latter statement to be only partially correct since the the covariance of this spectrum is non-Gaussian, especially on small angular scale. Identifying the expected scatter of the averaged spectrum with the theoretical Gaussian sample variance therefore tends to underestimate the scatter itself.\
For each pair of the Stokes parameters, $X$ and $Y$, we define a quantity $$G^{XY}_{\ell}=\sqrt{\frac{(2\ell+1)\cdot N_{r}}{\left[(\bar{C}^{XY}_{\ell})^{2}+{C}^{XX}_{\ell,th}{C}^{YY}_{\ell,th}\right]}}(\bar{C}^{XY}_{\ell}-{C}^{XY}_{\ell,th}),
\label{eq:gxy}$$ where the bar denotes a power spectrum averaged over $N_{r}$ realizations. The ensemble of all values of $G_{\ell}^{XY}$ is expected to be Gaussian-distributed with $0$ mean and variance $1$, which can be tested by means of a Kolmogorov-Smirnov (KS) test. In addition, we define a reduced $\chi^{2}$ statistics, Eq. (\[eq:gxy\]) and following @basak, as $$\tilde{\chi}^{2}_{XY}=\sum_{\ell=2}^{{\ell_{max} }}\frac{\left | G^{XY}_{\ell}\right |^{2}}{{\ell_{max} }-1}.$$
We report in Table \[tab:statistics\] the results of these two tests expressed as the significance level probability of the null hypothesis. We found that the method does not produce any significant biases on $TB$ and $EB$ cross spectra either; these were not shown in the previous analysis but are of potential interest, because they are a sensitive test of any artificially induced correlation. Moreover, the precision and accuracy of the result can be tested quite independently of analytical models by devising a custom convergence procedure as explained in the previous section.
$\bar{C}_{\ell}^{XY}$ Significance $P_{KS}$ Significance $ P_{\tilde{\chi}^{2}_{XY}}$
----------------------- ----------------------- -------------------------------------------
TT 0.19 0.92
EE 0.97 0.65
BB 0.79 0.14
TE 0.58 0.84
TB 0.20 0.34
EB 0.71 0.67
: Results of statistical tests on the recovered lensed power spectra averaged over $100$ realizations. The significance-level probability for the null hypothesis using a Kolmogorov-Smirnov test ($P_{KS}$) and a reduced chi-square $\tilde{\chi}^{2}$ statistics ($P_{\tilde{\chi}_{XY}^{2}}$) show no bias on a statistical level. \[tab:statistics\]
Numerical performance and requirements
--------------------------------------
{width=".325\textwidth"} {width=".325\textwidth"} {width=".325\textwidth"}
\[sec:numPerf\]
In this section we evaluate the strong scaling relations for numerical cost and memory requirements of [[lenS$^{2}$HAT]{}]{}, i.e., we run the code with the same parameters and test its scalability as a function of the number of MPI processes used in the calculation. For this benchmark test we used ${\ell_{max} }^{E/B}={\ell_{max} }^{\Phi}=8000$ and a grid for lensed sky and displacement field of $32768^{2}$ pixels.\
The main data volume involved in the calculations is given by harmonic coefficients and maps that are evenly distributed between processors through the [S$^{2}$HAT]{} library. Their distribution is optimized for all spherical harmonics transform steps involved. The remapping method itself only depends on structures that are also distributed between processors, allowing us to preserve the scalability features inherited from the [S$^{2}$HAT]{} library. The overall memory requirements per processors for a [[lenS$^{2}$HAT]{}]{} run are on the order of $\mathcal{O}(N_{pix}/n)$, where $N_{pix}$ is the total number of pixels of both the lensed map and displacement field and $n$ is the number of MPI tasks (or processors) used for the simulation, which is assumed to be one for the physical core available on our test architecture. The prefactor varies as a function of the oversampling rate and is equal to $(3+\kappa^{2})$ for the temperature and to $(4+(1+\kappa^{2})n_{S\!tokes})$ for polarized cases. We report in Figure \[fig:scaling\] the results of strong- and weak-scaling tests performed on the Cray XE6 Hopper cluster of the NERSC[^9] supercomputing center using the integrated performance monitoring library[^10] (IPM). The discrepancy between our model and the actual memory resource requirements per processors are due to MPI buffer allocations for collective communications and duplications of auxiliary objects describing the properties of the pixelization and observed sky patch used in the simulations as required by the [S$^{2}$HAT]{} library and remapping method. They have a size $\mathcal{O}(11(1+\kappa)\sqrt{N_{pix}})$ and accounts for nearly $25$Mb of data duplicated on each processor. The memory-overhead of the communication part of the [[lenS$^{2}$HAT]{}]{} algorithm depends instead on the number of pixels in the local memory of each processor that is lensed on an area of the map stored on the memory of another processor. This quantity controls the size of MPI buffers, but cannot be precisely determined a priori since it depends on the specific realization of the displacement field used in the simulation.We found for this specific test case that the memory-overhead for the communication has a size of roughly 75Mb per processor.\
The computational cost of our method is driven by the synthesis of the unlensed map, which is the highest-resolution object to be computed and has a number of pixels $\kappa^{2}$ higher than the displacement field. As can also be seen from the right panel of Fig. \[fig:scaling\], the runtime connected to the inverse spherical harmonic transform of the unlensed sky, despite being perfectly load-balanced, tends to flatten due to required internal communication steps and precomputations to initialize the recurrence to compute spherical harmonics. These are per se subdominant steps, but they are expected to play a role for a very fine parallelization [@szydlarski2011]. The overall remapping procedure of pixel values requires a number of operation of about $\mathcal{O}(N_{pix}/n)$ and is subdominant, since it operates on a lower-resolution map, and perfectly scalable because it does not require any communication. The step involving the reconstruction of the lensed map after reshuffling the pixels (denoted as communication part in Fig. \[fig:scaling\]) is subdominant, but the walltime required by this step is expected to slightly grow because it potentially involves the collective communication of small amounts of data between processors and is expected to approach the latency limit for message sending.\
In Fig. \[fig:scaling\] we also mark the performance of the [[LensPix]{}]{} code. However, because these two codes follow different algorithmic approaches and perform different operations to obtain a lensed map, it is not straightforward to set up a proper comparison. The presented results should therefore be viewed as merely indicative. In this case, we have attempted to set the code parameters to obtain an accurate spectrum up to $\ell \simeq 5000$. We assumed the same bandwidth for the [[LensPix]{}]{} runs as for [[lenS$^{2}$HAT]{}]{}, i.e., ${\ell_{max} }=8000$, and used the lowest resolution capable of supporting the corresponding harmonic modes on a HEALPix grid, setting $nside=4096$. This value may be somewhat on the low side given the increase of the bandwidth due to the lensing. [[LensPix]{}]{} also requires as an input an oversampling parameter that defines the unlensed sky resolution in the ECP pixelization. We chose this parameter to be $2$ because this is a value commonly used and has been reported to be sufficient to produce accurate results [e.g. @benoit-levy2012]. With this choice of the input parameters we find that the [[LensPix]{}]{} code displays a better performance in terms of the runtime for an intermediate number of employed MPI-process, but the gain is quickly offset by the superior scaling properties of the [[lenS$^{2}$HAT]{}]{} code and its ability to employ many processors. Moreover, for the sake of comparison, no bandwidth optimization procedure was applied here, which would result in about a factor $\sim 1.4$ improvement in the [[lenS$^{2}$HAT]{}]{} runtimes. We note that the memory and communication bottlenecks prevented us from successfully running [[LensPix]{}]{} on more than $\sim 800$ MPI processors of our computational platform. The performance of [[lenS$^{2}$HAT]{}]{} can be further improved by performing a simultaneous, multi-map analysis (see Appendix \[appendix:code\]), made feasible thanks to its low memory overhead and near perfect scalability of the memory requirements. However, as they are, the two codes seem to be complementary and to address the needs of different computational platforms.
Conclusions
===========
We have investigated and clarified details of modeling and simulations of the gravitational lensing effect on CMB. We particularly aimed at elucidating the role and impact of bandwidths of considered signals on the precision of the pixel-domain approaches [e.g., @lewis2005] to simulating the lensing effect on polarized anisotropies, paying special attention to recovering of the $B$ component. These bandwidths play a crucial role in ensuring a sufficient accuracy of the produced lensed maps and need to be carefully taken into account if numerical effects such as power aliasing are to be kept under control. We developed a semi-analytic approach based on the formalism of [@hu2000] to study these effects, and with its help investigated the necessary requirements for the signal’s bandwidths. In particular, we found out that the simple convolution picture, where the convolution kernel has a limited width of at most few hundred in $\ell$ space due to the gravitational potential, though it works very well for the total intensity, $T$, and $E$ polarization up to $\tell{T/E} \simlt 2000$, is adequate neither for much smaller angular scales in these two cases nor for the $B$-mode signal. Instead, the proposed semi-analytic formalism should be used to guide a selection of the simulation parameters to ensure the final precision of the result, but also to optimize the computational time. We point out that the accuracy considerations we presented are sufficiently generic that they should be applicable to other CMB lensing simulation techniques providing sound guidelines for choices of suitable parameters, that these techniques involve. For the pixel-domain-based methods, our main object of study here, we find that sufficiently high precision can indeed be ensured and permits meaningful simulations of small effects due to different physical assumptions.
Furthermore, we validated our semi-analytic results with the help of extensive numerical computations, for which we developed a simple, massively-parallel lensing simulation code, [[lenS$^{2}$HAT]{}]{}. The code uses an extremely simple but robust approach to the interpolation, involving sky overpixelization and a simple NGP assignment scheme, which, as we showed, leads to easily understandable and controllable numerical effects. These effects are minimized because the code, thanks to its efficient parallelization, permits analyses of extremely large sky maps with very dense sky grids/pixelization. In this way the simulated sky power can be resolved all the way to its actual bandwidths, which are carefully kept track of in the code.
The developed code, [[lenS$^{2}$HAT]{}]{}, is suitable for massively parallel computational platforms, with either shared or distributed memory. It displays nearly perfect scalability in terms of runtime and allocated memory per processor up to the maximal number of CPUs it can employ. This last is determined by the lowest value of the band-limit parameters for either the CMB or the displacement field that is to be used in the runs, $n_{proc}^{max} = \min({\ell^{\,E}}_{max},{\ell^{\,\Phi}}_{max})/2$. It therefore permits extensive simulations involving hundreds of simulated maps in a reasonable time. The major bottleneck of the code performance is due to the need of calculating a single inverse spherical harmonic transform which is required to obtain the overpixelized map of the unlensed signal. This can certainly be alleviated further by using better algorithms and/or numerical implementations, e.g., capitalizing on hardware accelerators such as GPGPU [@hupca2011; @szydlarski2011; @fabbian2012; @reinecke2013]. We leave these code optimizations for future work. The code and its forthcoming version will be publicly available.
This work has been supported in part by French National Research Agency (ANR) through COSINUS program (project MIDAS no. ANR-09-COSI-009). GF acknowledges the support of Università degli Studi di Milano Bicocca and CARIPLO Foundation through the EXTRA program for the preliminary phase of this work. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. We acknowledge use of the publicly available software tools and libraries HEALPix, SPRNG, IPM, S$^2$HAT, CAMB, and LensPix.
Precision of the accuracy formula {#appendix:accuracy_precision}
=================================
The harmonic approximation of [@hu2000], which we used throughout the paper, is known to reproduce the lensed CMB spectra with an accuracy of only a few percent [@challinor2005]. In this appendix we discuss the validity of the definition of the approximate accuracy function, Eq. \[Xaccuracydef\], which is based on this approximation.\
Assuming that we have access to the exact 2D convolution kernels instead of the one derived with the gradient approximation, such that $$C_{\ell}^{X,exact}= \sum_{\ell_{*}^{\Phi}=0}^{\infty}\sum_{\ell_{*}^{X}=0}^{\infty}{\cal K}^{exact}_{\tell{X}}(\ell_*^{\,X}, \ell_*^{\,\Phi}) ,$$ we can express the exact accuracy formula as $$A_{\ell}^{X,true}({\ell^{\,\Phi}},{\ell^{\,X}})=1-\frac{\alpha^{\ell}_{{\ell^{\,X}}, \ell^{\Phi} }+\zeta_{{\ell^{\,X}}, \ell^{\Phi}}}{\alpha^{\ell}_{\ell_{max}^{X}, \ell^{\Phi}_{max} } +\epsilon},$$ where $\alpha^{\ell}_{{\ell^{\,X}}, \ell^{\Phi}}$ and $ \alpha^{\ell}_{{\ell^{\,X}}_{max}, \ell^{\Phi}_{max}}$ correspond to the numerator and dominator in Eq. \[Xaccuracydef\], and the two extra terms that quantify the corresponding, effective errors, which themselves may depend on the cutoff assumed in the computation of both the numerator and denominator, are given as $$\begin{aligned}
\epsilon&=&\sum_{\ell_{*}^{\Phi}=0}^{\infty}\sum_{\ell_{*}^{X}=0}^{\infty}{\cal K}^{exact}_{\tell{X}}(\ell_*^{\,X}, \ell_*^{\,\Phi}) -\alpha^{\ell}_{{\ell^{\,X}}_{max}, \ell^{\Phi}_{max}}\\
\zeta_{{\ell^{\,X}}, \ell^{\Phi}} &=& \sum_{\ell_{*}^{\Phi}=0}^{\ell^{\Phi}}\sum_{\ell_{*}^{X}=0}^{{\ell^{\,X}}}{\cal K}^{exact}_{\tell{X}}(\ell_*^{\,X}, \ell_*^{\,\Phi})-\alpha^{\ell}_{{\ell^{\,X}}, \ell^{\Phi}}.\end{aligned}$$ Hereafter, we assume that the absolute cutoff for the CMB and lensing potential, ${\ell_{max} }$, is chosen sufficiently high so that all the relevant power is included when computing the considered lensed multipole. Because the accuracy of the harmonic expansion $\beta\equiv \epsilon/\alpha^{\ell}_{\ell_{max}^{X}}$ is on the order of percent, we can Taylor-expand the previous expression to the first order in $\beta$, i.e., $$\begin{aligned}
A_{\ell}^{X,true}({\ell^{\,\Phi}},{\ell^{\,X}})&\approx&1- \frac{(\alpha^{\ell}_{{\ell^{\,X}}, \ell^{\Phi} }+\zeta_{{\ell^{\,X}}, \ell^{\Phi}})}{\alpha^{\ell}_{\ell_{max}^{X}, \ell^{\Phi}_{max} }} (1-\beta) +\mathcal{O}(\beta^{2})\\ \nonumber
&=&A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}}) +\beta A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}}) +\beta -\frac{\zeta_{{\ell^{\,X}}, \ell^{\Phi}}}{\alpha^{\ell}_{\ell_{max}^{X}, \ell^{\Phi}_{max} }}.\end{aligned}$$ From now on, we denote the last term on the rhs as $\delta_{{\ell^{\,X}},\ell^{\Phi}}$. We can then rewrite the precision of the accuracy function as $$\begin{aligned}
\left |\frac{\Delta A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}})}{A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}})}\right | &\approx&\left | \frac{1}{A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}})}\left(\beta\left(1- A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}}) \right)- \delta_{{\ell^{\,X}},\ell^{\Phi}}\right) \right | \nonumber\\
&\approx&\left |\frac{\left(+\beta - \delta_{{\ell^{\,X}},\ell^{\Phi}}\right) + \mathcal{O}(\beta^{2})}{A_{\ell}^{X}({\ell^{\,\Phi}},{\ell^{\,X}})}\right |,
\label{eq:accuracy}\end{aligned}$$ where we have assumed that the accuracy function is at most $\mathcal{O}(\beta)$ in the regime of interest here.\
The overall precision of the accuracy function, as expressed by Eq. \[eq:accuracy\], is then driven by the difference between the two terms, which by construction tend to cancel each other because $\zeta_{{\ell^{\,X}}, \ell^{\Phi}}$ goes to $\epsilon$ as ${\ell^{\,X}}, \ell^{\Phi}$ approach ${\ell_{max} }$. The formula therefore becomes more and more accurate as we approach the cutoff limit.
[[lenS$^{2}$HAT]{}]{} code {#appendix:code}
==========================
The code outline follows the general simulation guidelines discussed in Sect. \[s3ect:basics\], but we detail here several features of potential interest of the code structure.
1. When generating a Gaussian random realization of a harmonic coefficients for the unlensed CMB and the displacement field, both the correlation between temperature or $E$-modes and the displacement field generated by the Sachs-Wolfe effect can be taken into account if requested. However, since both are negligible for most multipoles, we neglected them in the runs performed for this paper. We do not expect this correlation to affect the results of our analysis, especially in the high $\ell$ tail of the spectrum because the correlation is confined to large angular scales.\
2. The effects of nonlinear LSS evolution, which consequently affect the lensing potential, are naturally taken into account in the code if they are included in an effective lensing potential power spectrum. Even though nonlinear evolution of matter perturbations induces non-Gaussianities in the matter power spectrum, the contributions of higher-order statistical moments to the lensing potential have been proven to be on the subpercent level [@merkel2010].The assumption of a purely Gaussian lensing potential is thus well applicable and usually sufficient for this kind of simulations. As an alternative, the code can accept pre-computed maps of the potential on the input, which can be therefore arbitrarily non-Gaussian, and which will be used to produce the displacement field.\
3. Since harmonic coefficients are distributed between processors and generated directly in a distributed way, we used the scalable parallel random number generator library[^11] (SPRNG) to avoid correlation between random number streams on each processors.\
4. The computation of the displaced coordinates and the remapping of the pixel locations are managed by two separate routines, one optimized for grids with equidistant rings (e.g. ECP) and the other developed for any pixelization conforming with the requirements imposed by [S$^{2}$HAT]{}. Since maps are distributed between processors, it can happen that the remapping procedure on a given processor identifies the required displaced pixel to be located in a region of the sky map that is not stored in the local memory. For this reason the code has to manage pixel indexing using two different indexing scheme (one for the full-sky map and one for the chunk of the full map stored locally) and has to be able to switch from one to the other. To performed this operation efficiently we have to allocate on each processor an auxiliary bi-dimensional array, that encodes the indices of the pixels required by the processors and that are not present in its memory and the processors on which these are effectively located. The total volume of this structure is therefore equal to that of the part of the lensed sky stored locally and constitutes the only memory overhead required by the remapping procedure.\
5. A collective [MPI\_All2allv]{} communication step is performed to redistribute the information on pixels, that are needed by a processor to build the final lensed map, but is not stored in its local memory. This pattern ensures an even distribution of memory between all cores and a very good scalability up to several thousand MPI processes. On the numerical level the communication time is subdominant, although it can in principle be further optimized with nonblocking MPI local communication calls or by exploiting an hybrid MPI/OpenMP approach.\
6. The code can perform simulations in an arbitrary pixelization scheme that meets the [S$^{2}$HAT]{} requirements. Though we have found that ECP is preferred for the internal computation, the output results can be delivered on a grid selected by the user, e.g. the HEALPix grid.\
7. The code supports simultaneous multi-map analysis on the spherical harmonics step of the algorithm, when memory available for a given processor is sufficient. In this case, the gain in the runtime of the code is roughly equal to the number of maps processed at the same time. This option makes the code particularly appealing for the data analysis steps involving massive use of Montecarlo realizations of lensed CMB maps.
[^1]:
[^2]:
[^3]: <http://camb.info>
[^4]: <http://lesgourg.web.cern.ch/lesgourg/class.php>
[^5]: <http://cosmologist.info/lenspix>
[^6]: <http://healpix.sf.net/>
[^7]: This is not true for the $TE$ kernels, which we comment about later.
[^8]: <http://www.apc.univ-paris7.fr/APC_CS/Recherche/Adamis/MIDAS09/software/s2hat/s2hat.html>
[^9]: <https://www.nersc.gov>
[^10]: <http://ipm-hpc.sourceforge.net/>
[^11]: <http://sprng.cs.fsu.edu>
|
---
abstract: 'A novel approach which combines isogeometric collocation and an equilibrium-based stress recovery technique is applied to analyze laminated composite plates. Isogeometric collocation is an appealing strong form alternative to standard Galerkin approaches, able to achieve high order convergence rates coupled with a significantly reduced computational cost. Laminated composite plates are herein conveniently modeled considering only one element through the thickness with homogenized material properties. This guarantees accurate results in terms of displacements and in-plane stress components. To recover an accurate out-of-plane stress state, equilibrium is imposed in strong form as a post-processing correction step, which requires the shape functions to be highly continuous. This continuity demand is fully granted by isogeometric analysis properties, and excellent results are obtained using a minimal number of collocation points per direction, particularly for increasing values of length-to-thickness plate ratio and number of layers.'
address:
- |
Department of Civil Engineering and Architecture - University of Pavia\
Via Ferrata, 3, 27100 Pavia, Italy
- |
Mechanical and Aerospace Engineering Department\
University of Texas at Arlington\
500 W 1st St, Arlington, TX 76010
- |
Institute of Mathematics - École Polytechnique Fédérale de Lausanne\
CH-1015 Lausanne, Switzerland
- |
Instituto di Matematica Applicata e Tecnologie Informatiche “E. Magenes” (CNR)\
via Ferrata 1, 27100 Pavia, Italy
author:
- Alessia Patton
- 'John-Eric Dufour'
- Pablo Antolin
- Alessandro Reali
title: 'Fast and accurate elastic analysis of laminated composite plates via isogeometric collocation and an equilibrium-based stress recovery approach'
---
Isogeometric Collocation,Splines,Orthotropic materials,Homogenization,Laminated composite plates,Stress recovery procedure
Introduction {#sec:intro}
============
Composite materials consist of two or more materials which combined present enhanced properties that could not be acquired employing any of the constituents alone (see, [[e.g., ]{}]{}[@Gibson1994; @Jones1999; @Reddy2003; @Vinson1986] and references therein). The interest for composite structures in the engineering community has constantly grown in recent years due to their appealing mechanical properties such as increased stiffness and strength, reduced weight, improved corrosion and wear resistance, just to recall some of them. The majority of man-made composite materials consists of reinforced fibers embedded in a base material, called matrix (see, [[e.g., ]{}]{}[@Hashin1972; @Reddy2003]). The matrix material keeps the fibers together, acts as a load-transfer medium between fibers, process which takes place through shear stresses, and protects those elements from being exposed to the environment, while the resistence properties of composites are given by the fibers which are stiffer and stronger than the soft matrix. In this paper we focus on laminated composite materials, which are formed by a collection of building blocks or plies, stacked to achieve the desired stiffness and thickness. For this kind of structures also simple loading conditions, such as traction or bending, cause a complex 3D stress state because of the difference in the material properties between the layers, which may lead to delamination, consequently requiring an accurate stress evaluation through the thickness (see, [[e.g., ]{}]{}[@Reddy2003; @Sridharan2008]). As an alternative to two-dimensional theories, often insufficiently accurate to depict delamination and interlaminar damage, and to layerwise theories, which typically show an high computational cost, a novel method combining an isogeometric analysis (IGA) Galerkin approach with a stress-recovery technique has been recently proposed in [@Dufour2018]. Introduced in 2005 by Hughes et al. [@Hughes2005], IGA aims at integrating design and analysis employing shape functions typically belonging to Computer Aided Design field (such as, B-Splines and NURBS). Using the same shape functions to approximate both geometry and field variables leads to a cost-saving simplification of expensive mesh generation and refinement processes required by standard finite element analyisis. One of the most important features of IGA is the high-regularity of its basis functions leading to superior approximation properties. IGA proved to be successful in a wide variety of problems ranging from solids and structures (see, [[e.g., ]{}]{}[@Auricchio2010b; @Borden2012; @Caseiro2014; @Dhote2014; @Elguedj2014; @Hughes2008; @Hughes2014; @Lipton2010; @Morganti2015]) to fluids (see, [[e.g., ]{}]{}[@Akkerman2008; @Buffa2011; @Gomez2010; @Liu2013]), fluid-structure interaction (see, [[e.g., ]{}]{}[@Bazilevs2011; @Hsu2015]), opening also the door to geometrically flexible discretizations of higher-order partial differential equations in primal form as in [@Auricchio2007; @Gomez2008; @Kiendl2009; @Lorenzo2018]. However, a well-known important issue of IGA is related to the development of efficient integration rules when higher-order approximations are employed (see, [[e.g., ]{}]{}[@Auricchio2012b; @Fahrendof2018; @Hughes2010; @Sangalli2018]). In attempt to address this problem taking full advantage of the special possibilities offered by IGA, isogeometric collocation (IGA-C) schemes have been proposed in [@Auricchio2010a]. The aim was to optimize the computational cost still relying on IGA geometrical flexibility and accuracy. Collocation main idea, in contrast to Galerkin-type formulations, consists in the discretization of the governing partial differential equations in strong form, evaluated at suitable points. Since integration is not required, isogeometric collocation results in a very fast method providing superior performance in terms of accuracy-to-computational effort ratio with respect to Galerkin formulations, in particular when higher-order approximation degrees are adopted (see [@Schillinger2013]). Isogeometric collocation has been particularly successful in the context of structural elements, where isogeometric collocation has proven to be particularly stable in the context of mixed methods. In particular, Bernoulli-Euler beam and Kirchhoff plate elements have been proposed [@Reali2015b], while mixed formulations both for Timoshenko initially-straight planar [@daVeiga2012] and non-prismatic [@Balduzzi2017] beams as well as for curved spatial rods [@Auricchio2013] have been introduced and studied, and then effectively extended to the geometrically nonlinear case [@Kiendl2018; @Marino2016; @Marino2017; @Marino2019; @Weeger2017a; @Weeger2018]. Isogeometric collocation has been moreover successfully applied to the solution of Reissner-Mindlin plate problems in [@Kiendl2015a], and new formulations for shear-deformable beams [@Kiendl2015b; @Kiendl2018], as well as shells [@Kiendl2017; @Maurin2018] have been solved also via IGA collocation. Since its introduction, many promising significant works on isogeometric collocation methods have been published also in other fields, including phase-field modeling [@Gomez2014], contact [@DeLorenzis2015; @Kruse2015; @Weeger2017b] and poromechanics [@Morganti2018]. Moreover, combinations with different spline spaces, like hierarchical splines, generalized B-Splines, and T-Splines, have been successfully tested in [@Casquero2015; @Manni2015; @Schillinger2013], while alternative effective selection strategies for collocation points have been proposed in [@Anitescu2015; @Gomez2014; @Montardini2017]. IGA-Galerkin methods have already been used to solve composite laminate problems, especially relying on high-order theories for enhanced plate and shell theories [@Farzam2018; @Kapoor2013; @Remmers2015; @Shi--Dong2018; @Thai2015]. Recently an isogeometric collocation numerical formulation has been proposed [@Pavan2017] to study Reissner-Mindlin composite plates. Other Galerkin methods [@Guo2014; @Guo2015; @Remmers2015] compute instead a full 3D stress state using isogeometric analysis, applying a layerwise technique. This method can also be applied to isogeometric collocation adopting a multipatch approach, which models each layer as a patch (see Figure \[subfig-1:layerwise\]), enforcing normal stress continuity at the inter-patches boundaries [@Auricchio2012a]. Clearly a layerwise method exploits a number of degrees of freedom directly proportional to the number of layers, inevitably leading to high computational costs. In this paper we apply a single patch 3D isogeometric collocation method to analyze the behavior of composite plates. We adopt a homogenized single-element approach (see Figure \[subfig-2:HSE\]), which conveniently uses one element through the thickness, coupled with a post-processing technique in order to recover a proper out-of-plane stress state. This method is significantly less expensive compared to a layerwise approach since employs a considerably lower number of degrees of freedom.
The post-processing approach, first proposed in [@Dufour2018], takes inspiration from recovery techniques which can be found in [@Daghia2008; @deMiranda2002; @Engblom1985; @Kapoor2013; @Pryor1971; @Ubertini2004] and is based on the direct integration of the equilibrium equations to compute the out-of-plane stress components from the in-plane ones directly derived from a coarse displacement solution. The structure of the paper is organized as follows. In Section \[sec:IGA-Cortho\] the fundamental concepts of multivariate B-Splines and NURBS are presented, followed by an introduction to isogeometric collocation and a description of our IGA-C scheme for orthotropic elasticity. In Section \[sec:IGA-Cstrategies\] we define our isogeometric collocation strategy to study laminated plates, which combines a homogenized single-element approach with an equilibrium-based stress recovery technique. In Section \[sec:results\] we present our reference test case and provide results for the single-element approach. Several numerical benchmarks are displayed, which show a significant improvement between non-treated and post-processed out-of-plane stress components. Finally we provide some mesh sensitivity tests considering an increasing length-to-thickness ratio and numbers of layers to show the effectiveness of the method. We draw our conclusions in Section \[sec:conclusions\].
Isogeometric Collocation: Basics and application to orthotropic elasticity {#sec:IGA-Cortho}
==========================================================================
In this section we introduce the notions of multivariate B-Splines and NURBS, provide some details regarding isogeometric collocation and describe our collocation scheme in the context of linear orthotropic elasticity.
Multivariate B-Splines and NURBS {#subsec:IGAsf}
--------------------------------
In the following, we introduce the basic definitions and notations about multivariate B-Splines and NURBS. For further details, readers may refer to [@Cottrell2007; @Hughes2005; @Piegl1997], and references therein. Multivariate B-Splines are generated through the tensor product of univariate B-Splines. We denote with $d_{p}$ the dimension of the parametric space and therefore $d_{p}$ univariate knot vectors have to be introduced as $$\Theta = \{\theta_{1}^{d},...,\theta_{m_{d}+p_{d}+1}^{d}\}\hspace{1cm}d = 1, ..., d_{p}\,,\label{eq:knotvectors}$$ where $p_{d}$ represents the polynomial degree in the parametric direction $d$, and $m_{d}$ is the associated number of basis functions. Given the univariate basis functions $N^{d}_{i_{d},p_{d}}$ associated to each parametric direction $\xi^{d}$, the multivariate basis functions $B_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})$ are obtained as: $$B_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})=\prod\limits_{d=1}^{d_{p}}N_{i_{d},p_{d}}(\xi^{d})\,,\label{eq:B-Splines}$$ where $\textbf{i} = \{i_{1}, ..., i_{d_{p}}\}$ plays the role of a multi-index which describes the considered position in the tensor product structure, $\textbf{p} = \{{p_{1}, ..., p_{d}}\}$ indicates the polynomial degrees, and $\boldsymbol{\xi} = \{\xi^{1},...,\xi^{d_{p}}\}$ represents the vector of the parametric coordinates in each parametric direction $d$. B-Spline multidimensional geometries are built from a linear combination of multivariate B-Spline basis functions as follows $$\textbf{S}(\boldsymbol{\xi})=\sum\limits_{\textbf{i}}B_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})\textbf{P}_\textbf{i}\,,\label{eq:B-Splinemultigeom}$$ where the coefficients $\textbf{P}_\textbf{i}\in\mathbb{R}^{d_{s}}$ of the linear combination are the so-called control points ($d_{s}$ is the dimension of the physical space) and the summation is extended to all combinations of the multi-index **i**. NURBS geometries in $\mathbb{R}^{d_{s}}$ are instead obtained from a projective transformation of their B-Spline counterparts in $\mathbb{R}^{d_{s}+1}$. Defining $w_{\textbf{i}}$ as the collection of weights according to the multi-index **i**, multivariate NURBS basis functions are obtained as $$R_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})=\frac{B_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})w_{\textbf{i}}}{\sum_{\textbf{j}}B_{\textbf{j},\textbf{p}}(\boldsymbol{\xi})w_{\textbf{j}}}\label{eq:NURBS}$$ and NURBS multidimensional geometries are built as $$\textbf{S}(\boldsymbol{\xi})=\sum\limits_{\textbf{i}}R_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})\textbf{P}_\textbf{i}\,.\label{eq:NURBmultigeom}$$
An Introduction to Isogeometric collocation {#subsec:introIGA-C}
-------------------------------------------
Collocation methods have been introduced within isogeometric analysis as an attempt to address a well-known important issue of early IGA-Galerkin formulations, related to the development of efficient integration rules for higher-order approximations. In fact, element-wise Gauss quadrature, typically used for finite elements and originally adopted for Galerkin-based IGA, does not properly take into account inter-element higher continuity leading to sub-optimal array formation and assembly costs, significantly affecting the performance of IGA methods. Isogeometric collocation aimed at optimizing computational cost, since it may be viewed as a variant of one-point quadrature numerical scheme, still taking advantage of IGA geometrical flexibility and accuracy. Collocation methods are based on the direct discretization in strong form of the differential equations governing the problem evaluated at suitable points. The isoparametric paradigm is adopted and the same basis functions are used to describe both geometry and problem unknowns. Once the approximations are carried out, as in a typical Galerkin-IGA context, by means of a linear combinations of IGA basis functions and control variables, the discrete differential equations are collocated at each collocation point. Consequently a delicate issue is represented by the determination of suitable collocation points. A widespread approach which is proposed in the engineering literature is to collocate at the images of Greville abscissae (see, [[e.g., ]{}]{}[@Johnson2005]), but this represents just the simplest possible option (see, [[e.g., ]{}]{}[@deBoor1973; @Demko1985] for alternative choices). Along each parametric direction $d$, Greville abscissae consist of a set of $m^d$ points, obtained from the knot vector components, $\theta^d_{i}$, as $$\overline{\theta}^d_{i}=\frac{\theta^d_{i+1}+\theta^d_{i+2}+...+\theta^d_{i+p}}{p_d}\hspace{1cm}i = 1,...,m^d\,,\label{eq:greville}$$ $p_d$ being the degree of approximation. Since the approximation is performed through direct collocation of the differential equations, no integrals need to be computed and consequently, evaluation and assembly operations lead to a significantly reduced computational cost.
Numerical formulation for orthotropic elasticity {#subsec:numericalIGA-C}
------------------------------------------------
Once a strategy to select collocation points and compute IGA shape functions is set, a proper description of the equations in strong form for the problem under examination is required, as mentioned in Section \[subsec:introIGA-C\]. We therefore recall the classical elasticity problem in strong form considering a small strain regime and detail equilibrium equations using Einstein’s notation . The following notations are used: $\Omega\subset\mathbb{R}^3$, is an open bounded domain, representing an elastic three-dimensional body, $\Gamma_{N}$ and $\Gamma_{D}$ are defined as boundary portions subjected respectively to Neumann and Dirichlet conditions such that $\Gamma_{N}\cup\Gamma_{D}=\partial\Omega$ and $\Gamma_{N}\cap\Gamma_{D}=\emptyset$. Accordingly, the equilibrium equations and the corresponding boundary conditions are:
$$\begin{aligned}
&\sigma_{ij,j}+b_{i}=0\hspace{2.6cm}\text{in}\;\Omega\label{eq:equilibriumint}\\
&\sigma_{ij}n_{j}=t_{i}\hspace{3.05cm}\text{on}\;\Gamma_{N}\label{eq:equilibriumNeumann}\\
&u_{i}=\overline{u}_{i}\hspace{3.5cm}\text{on}\;\Gamma_{D}\label{eq:equilibriumDirichlet}\end{aligned}$$
\[eq:equilibrium\]
where $\sigma_{ij}$ and $u_{i}$ represent respectively the Cauchy stress and displacement components, while $b_{i}$ and $t_{i}$ the volume and traction forces, $n_{j}$ the outward normal, and $\overline{u}_{i}$ the prescribed displacements. The elasticity problem is finally completed by the kinematic relations in small strain $$\varepsilon_{ij}=\frac{u_{i,j}+u_{j,i}}{2}\,,\\\label{eq:kinematics}$$ as well as by the constitutive equations $$\sigma_{ij}=\mathbb{C}_{ijkm}{\varepsilon}_{km}\,,\label{eq:constlaw}$$ where $\mathbb{C}_{ijkm}$ is the fourth order elasicity tensor.As we described in Section \[sec:intro\], the basic building block of a laminate is a lamina, i.e., a flat arrangement of unidirectional fibers, considering the simplest case, embedded in a matrix. In order to increase the composite resistance properties cross-ply laminates can be employed (i.e., all the plies used to form the composite stacking sequence are piled alternating different fiber layers orientations) in which all unidirectional layers are individually orthotropic. Since the proposed collocation approach uses one element through the thickness to model the composite plate as a homogenized single building block, we focus in this section on the collocation formulation for a plate formed by only one orthotropic elastic lamina. Considering three mutually orthogonal planes of material symmetry for each ply, the number of elastic coefficients of the fourth order elasticity tensor $\mathbb{C}_{ijkm}$ is reduced to 9 in Voigt notation, that can be expressed in terms of engineering constants as $${\scalebox{0.88}{$
\mathbb{C}=\renewcommand\arraystretch{1.75}\begin{bmatrix}
\mathbb{C}_{11} & \mathbb{C}_{12} & \mathbb{C}_{13} & 0 & 0 & 0\\
& \mathbb{C}_{22} & \mathbb{C}_{23} & 0 & 0 & 0\\
& & \mathbb{C}_{33} & 0 & 0 & 0\\
& symm& & \mathbb{C}_{44} & 0 & 0\\
& & & & \mathbb{C}_{55} & 0\\
& & & & & \mathbb{C}_{66}\\
\end{bmatrix}={\bBigg@{17.5}}{[}\begin{matrix}
\cfrac{1}{E_{1}} & -\cfrac{\nu_{12}}{E_{1}} & -\cfrac{\nu_{13}}{E_{1}} & 0 & 0 & 0\\
& \cfrac{1}{E_{2}} & -\cfrac{\nu_{23}}{E_{2}} & 0 & 0 & 0\\
& & \cfrac{1}{E_{3}} & 0 & 0 & 0\\
& symm& & \cfrac{1}{G_{23}} & 0 & 0\\
& & & & \cfrac{1}{G_{13}} & 0\\
& & & & & \cfrac{1}{G_{12}}\\
\end{matrix}{\bBigg@{17.5}}{]}^{-1}\,.$}}\label{eq:CIVeltensor}$$ The displacement field is then approximate as a linear combination of NURBS multivariate shape functions and control points as follows $$\begin{aligned}
&\textbf{u}(\boldsymbol{\xi})=R_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})\hat{\textbf{u}}_{\textbf{i}}\,,\\
&\textbf{v}(\boldsymbol{\xi})=R_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})\hat{\textbf{v}}_{\textbf{i}}\,,\\
&\textbf{w}(\boldsymbol{\xi})=R_{\textbf{i},\textbf{p}}(\boldsymbol{\xi})\hat{\textbf{w}}_{\textbf{i}}\,.
\end{aligned}\label{eq:displapprox}$$ Having defined $\boldsymbol{\tau}$ as the matrix of collocation points, we insert the approximations into kinematics equations and we combine the obtained expressions with the constitutive relations . Finally we substitute into equilibrium equations obtaining
\[eq:IGA-Cinternal\] $$\begin{aligned}
\tag{\ref{eq:IGA-Cinternal}}
&\begin{bmatrix}
\textbf{K}_{11}(\boldsymbol{\tau})&\textbf{K}_{12}(\boldsymbol{\tau})&\textbf{K}_{13}(\boldsymbol{\tau})\\
&\textbf{K}_{22}(\boldsymbol{\tau})&\textbf{K}_{23}(\boldsymbol{\tau})\\
symm&&\textbf{K}_{33}(\boldsymbol{\tau})
\end{bmatrix}\cdot\begin{pmatrix}\hat{\textbf{u}}_{\textbf{i}}\\
\hat{\textbf{v}}_{\textbf{i}}\\
\hat{\textbf{w}}_{\textbf{i}}\end{pmatrix}=-\textbf{b}(\boldsymbol{\tau}),\hspace{1cm}\forall\boldsymbol{\tau}\in\Omega\,,\\
\intertext{where $\textbf{K}_{ij}(\boldsymbol{\tau})$ cofficients can be expressed as}
&\textbf{K}_{11}(\boldsymbol{\tau})=\mathbb{C}_{11}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}^2}+\mathbb{C}_{66}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}^2}+\mathbb{C}_{55}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}^2}\,,\label{eq:IGA-CinternalK11}\\
&\textbf{K}_{22}(\boldsymbol{\tau})=\mathbb{C}_{66}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}^2}+\mathbb{C}_{22}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}^2}+\mathbb{C}_{44}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}^2}\,,\label{eq:IGA-CinternalK22}\\
&\textbf{K}_{33}(\boldsymbol{\tau})=\mathbb{C}_{55}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}^2}+\mathbb{C}_{44}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}^2}+\mathbb{C}_{33}\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}^2}\,,\label{eq:IGA-CinternalK33}\\
&\textbf{K}_{23}(\boldsymbol{\tau})=(\mathbb{C}_{23}+\mathbb{C}_{44})\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}\partial{x_3}}\,,\label{eq:IGA-CinternalK23}\\
&\textbf{K}_{13}(\boldsymbol{\tau})=(\mathbb{C}_{13}+\mathbb{C}_{55})\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}\partial{x_3}}\,,\label{eq:IGA-CinternalK31}\\
&\textbf{K}_{12}(\boldsymbol{\tau})=(\mathbb{C}_{12}+\mathbb{C}_{66})\cfrac{\partial^2{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}\partial{x_2}}\,,\label{eq:IGA-CinternalK12}\end{aligned}$$
and substituting in we obtain:
\[eq:IGA-CNeumann\] $$\begin{aligned}
\tag{\ref{eq:IGA-CNeumann}}
&\begin{bmatrix}
\tilde{\textbf{K}}_{11}(\boldsymbol{\tau})&\tilde{\textbf{K}}_{12}(\boldsymbol{\tau})&\tilde{\textbf{K}}_{13}(\boldsymbol{\tau})\\
&\tilde{\textbf{K}}_{22}(\boldsymbol{\tau})&\tilde{\textbf{K}}_{23}(\boldsymbol{\tau})\\
symm&&\tilde{\textbf{K}}_{33}(\boldsymbol{\tau})
\end{bmatrix}\cdot\begin{pmatrix}\hat{\textbf{u}}_{\textbf{i}}\\
\hat{\textbf{v}}_{\textbf{i}}\\
\hat{\textbf{w}}_{\textbf{i}}\end{pmatrix}=\textbf{t}(\boldsymbol{\tau}),\hspace{1cm}\forall\boldsymbol{\tau}\in\Gamma_{N}\\
\intertext{with $\tilde{\textbf{K}}_{ij}(\boldsymbol{\tau})$ components having the following form}
&\tilde{\textbf{K}}_{11}(\boldsymbol{\tau})=\mathbb{C}_{11}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}}n_{1}+\mathbb{C}_{66}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}}n_{2}+\mathbb{C}_{55}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}}n_{3}\,,\label{eq:IGA-CNeumannK11}\\
&\tilde{\textbf{K}}_{22}(\boldsymbol{\tau})=\mathbb{C}_{66}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}}n_{1}+\mathbb{C}_{22}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}}n_{2}+\mathbb{C}_{44}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}}n_{3}\,,\label{eq:IGA-CNeumannK22}\\
&\tilde{\textbf{K}}_{33}(\boldsymbol{\tau})=\mathbb{C}_{55}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}}n_{1}+\mathbb{C}_{44}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}}n_{2}+\mathbb{C}_{33}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}}n_{3}\,,\label{eq:IGA-CNeumannK33}\\
&\tilde{\textbf{K}}_{23}(\boldsymbol{\tau})=\mathbb{C}_{23}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}}n_{2}+\mathbb{C}_{44}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}}n_{3}\,,\label{eq:IGA-CNeumannK23}\\
&\tilde{\textbf{K}}_{13}(\boldsymbol{\tau})=\mathbb{C}_{13}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_3}}n_{1}+\mathbb{C}_{55}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}}n_{3}\,,\label{eq:IGA-CNeumannK31}\\
&\tilde{\textbf{K}}_{12}(\boldsymbol{\tau})=\mathbb{C}_{12}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_2}}n_{1}+\mathbb{C}_{66}\cfrac{\partial{R_{\textbf{i},\textbf{p}}}(\boldsymbol{\tau})}{\partial{x_1}}n_{2}\,.\label{eq:IGA-CNeumannK12}\end{aligned}$$
As we can see from equations , Neumann boundary conditions are directly imposed as strong equations at the collocation points belonging to the boundary surface (see, [@Auricchio2012a; @DeLorenzis2015]), with the usual physical meaning of prescribed boundary traction.
An IGA collocation approach to model 3D composite plates {#sec:IGA-Cstrategies}
========================================================
In this section we describe our IGA 3D collocation strategy to model composite plates. The proposed method, known as single element approach, relies on a homogenization technique combined with a post-processing approach based on the imposition of equilibrium equations in strong form.
Single-element approach {#subsec:SEA}
-----------------------
The single-element approach considers the plate discretized by a single element through the thickness, which strongly reduces the number of degrees of freedom with respect to layerwise methods. The material matrix is therefore homogenized to account for the presence of the layers as Figure \[subfig-2:HSE\] clearly describes.
Considering a single-element homogenized approach is effective only for through-the-thickness symmetric layer distributions, as for non-symmetric ply stacking sequences the plate middle plane is not balanced. In the case of non-symmetric layer distributions this technique is still applicable when the stacking sequence can be split into two symmetric piles, using one element per homogenized stack with a $C^0$ interface.
This method provides accurate results only in terms of displacements and in-plane stress components and, in order to recover a proper out-of-plane stress state, following [@Dufour2018], we propose to couple it with a post-processing technique. To characterize the variation of the material properties from layer to layer, we homogenize the constitutive behavior to create an equivalent single-layer laminate, referring to [@Sun1988], where explicit expressions for the effective elastic constants of the equivalent laminate are given as
$$\begin{aligned}
&\overline{\mathbb{C}}_{11}=\sum_{k=1}^{N}\overline{t}_{k}\mathbb{C}_{11}^{(k)}+\sum_{k=2}^{N}(\mathbb{C}_{13}^{(k)}-\overline{\mathbb{C}}_{13})\overline{t}_{k}\frac{(\mathbb{C}_{13}^{(1)}-\mathbb{C}_{13}^{(k)})}{\mathbb{C}_{33}^{(k)}}\label{eq:aveC1}\\
&\overline{\mathbb{C}}_{12}=\sum_{k=1}^{N}\overline{t}_{k}\mathbb{C}_{12}^{(k)}+\sum_{k=2}^{N}(\mathbb{C}_{13}^{(k)}-\overline{\mathbb{C}}_{13})\overline{t}_{k}\frac{(\mathbb{C}_{23}^{(1)}-\mathbb{C}_{23}^{(k)})}{\mathbb{C}_{33}^{(k)}}\label{eq:aveC2}\\
&\overline{\mathbb{C}}_{13}=\sum_{k=1}^{N}\overline{t}_{k}\mathbb{C}_{13}^{(k)}+\sum_{k=2}^{N}(\mathbb{C}_{33}^{(k)}-\overline{\mathbb{C}}_{33})\overline{t}_{k}\frac{(\mathbb{C}_{13}^{(1)}-\mathbb{C}_{13}^{(k)})}{\mathbb{C}_{33}^{(k)}}\label{eq:aveC3}\\
&\overline{\mathbb{C}}_{22}=\sum_{k=1}^{N}\overline{t}_{k}\mathbb{C}_{22}^{(k)}+\sum_{k=2}^{N}(\mathbb{C}_{23}^{(k)}-\overline{\mathbb{C}}_{23})\overline{t}_{k}\frac{(\mathbb{C}_{23}^{(1)}-\mathbb{C}_{23}^{(k)})}{\mathbb{C}_{33}^{(k)}}\label{eq:aveC4}\\
&\overline{\mathbb{C}}_{23}=\sum_{k=1}^{N}\overline{t}_{k}\mathbb{C}_{23}^{(k)}+\sum_{k=2}^{N}(\mathbb{C}_{33}^{(k)}-\overline{\mathbb{C}}_{33})\overline{t}_{k}\frac{(\mathbb{C}_{23}^{(1)}-\mathbb{C}_{23}^{(k)})}{\mathbb{C}_{33}^{(k)}}\label{eq:aveC5}\\
&\overline{\mathbb{C}}_{33}=\frac{1}{\bigg(\sum_{k=1}^{N}\cfrac{\overline{t}_{k}}{\mathbb{C}_{33}^{(k)}}\bigg)}\label{eq:aveC6}\\
&\overline{\mathbb{C}}_{44}=\frac{\bigg(\sum_{k=1}^{N}\cfrac{\overline{t}_{k}\mathbb{C}_{44}^{(k)}}{\Delta_{k}}\bigg)}{\Delta},\hspace{0.5cm}\Delta=\bigg(\sum_{k=1}^{N}\frac{\overline{t}_{k}\mathbb{C}_{44}^{(k)}}{\Delta_{k}}\bigg)\bigg(\sum_{k=1}^{N}\frac{\overline{t}_{k}\mathbb{C}_{55}^{(k)}}{\Delta_{k}}\bigg)\label{eq:aveC7}\\
&\overline{\mathbb{C}}_{55}=\frac{\bigg(\sum_{k=1}^{N}\cfrac{\overline{t}_{k}\mathbb{C}_{55}^{(k)}}{\Delta_{k}}\bigg)}{\Delta},\hspace{0.5cm}\Delta_{k}=\mathbb{C}_{44}^{k}\mathbb{C}_{55}^{k}\label{eq:aveC8}\\
&\overline{\mathbb{C}}_{66}=\sum_{k=1}^{N}\overline{t}_{k}\mathbb{C}_{66}^{(k)}\label{eq:aveC9}\end{aligned}$$
where $\mathbb{C}_{ij}^{(k)}$ represents the $ij$-th component of the fourth order elasticity tensor in Voigt notation for the $k$-th layer and $\overline{t}_{k}=\cfrac{t_{k}}{h}$ stands for the volume fraction of the $k$-th lamina, $h$ being the total thickness and $t_{k}$ the $k$-th thickness.
### Post-processing step: Reconstruction from Equilibrium {#subsec:post-processing}
As interlaminar delamination and other fracture processes rely mostly on out-of-plane components, a proper through-the-thickness stress description is required. In order to recover a more accurate stress state, we perform a post-processing step based on the equilibrium equations, following [@Dufour2018], relying on the higher regularity granted by IGA shape functions. This procedure, which takes its roots in [@deMiranda2002; @Engblom1985; @Pryor1971; @Ubertini2004], has already been proven to be successful for IGA-Galerkin. Inside the plate the stresses should satisfy the equilibrium equation that can be expanded as
$$\begin{aligned}
&\sigma_{11,1}+\sigma_{12,2}+\sigma_{13,3}=-b_{1}\,,\label{eq:equilibriumeng1}\\
&\sigma_{12,1}+\sigma_{22,2}+\sigma_{23,3}=-b_{2}\,,\label{eq:equilibriumeng2}\\
&\sigma_{13,1}+\sigma_{23,2}+\sigma_{33,3}=-b_{3}\,.\label{eq:equilibriumeng3}\end{aligned}$$
\[eq:equilibriumeng\]
Assuming the in-plane stress components to well approximate the laminate behaviour, as it will be shown in Section \[sec:results\], we can integrate equation \[eq:equilibriumeng1\] and \[eq:equilibriumeng2\] along the thickness, recovering the out-of-plane shear stresses as
$$\begin{aligned}
\label{eq:ppsigma13}
\sigma_{13}(X_3) &= -\int^{X_3}_{\bar{X_3}}(\sigma_{11,1}(\zeta) + \sigma_{12,2}(\zeta)+b_1(\zeta))\rmd \zeta + \sigma_{13}(\bar{X_3})\,,\\
\label{eq:ppsigma23}
\sigma_{23}(X_3) &= -\int^{X_3}_{\bar{X_3}}(\sigma_{12,1}(\zeta) + \sigma_{22,2}(\zeta)+b_2(\zeta))\rmd \zeta + \sigma_{23}(\bar{X_3})\,,
\end{aligned}$$
where $\zeta$ represents the coordinate along the thickness direction.Finally we can insert equations and into , recovering the $\sigma_{33}$ component as $$\begin{aligned}
\label{eq:ppsigma33}
\sigma_{33}(X_3) &= -\int^{X_3}_{\bar{X_3}}(\sigma_{13,1}(\zeta) + \sigma_{23,2}(\zeta)+b_3(\zeta))\rmd \zeta + \sigma_{33}(\bar{X_3})\,.
\end{aligned}$$ Following [@Dufour2018], the integral constants are chosen to fulfil the boundary conditions at the top or bottom surfaces.Recalling that $$\begin{aligned}
&\sigma_{ij,k}=\overline{\mathbb{C}}_{ijmn}\frac{u_{m,nk}+u_{n,mk}}{2}\,,\label{eq:equilibriumsigmau}
\end{aligned}$$ where the homogenized elasticity tensor $\overline{\mathbb{C}}$ is constant, it is clear the necessity of a highly regular displacement solution in order to recover a proper stress state. Such a condition can be easily achieved using isogeometric collocation, due to the possibility to benefit from the high regularity of B-Splines or NURBS. We also remark that the proposed method strongly relies on the possibility to obtain an accurate description (with a relatively coarse mesh) of the in-plane stress state.
Numerical tests {#sec:results}
===============
In this section, to assess whether the proposed method can effectively reproduce composite plates behaviour, we consider a classical benchmark problem [@Pagano1970] and we address different aspects such as the effectiveness of the proposed post-processing step, the method sensitivity to parameters of interest (i.e., number of layers and length-to-thickness ratio), and its convergence.
Reference solution: the Pagano layered plate {#subsec:Pagano}
--------------------------------------------
A square laminated composite plate of total thickness $t$ made of $N$ orthotropic layers is considered. This structure is simply supported and a normal sinusoidal traction is applied on the upper surface, while the lower surface is traction-free, as shown in Figure \[fig:testproblem\].
![Pagano’s test case [@Pagano1970]. Problem geometry and boundary conditions.[]{data-label="fig:testproblem"}](Images/testproblem.pdf){width=".6\textwidth"}
In the proposed numerical tests we consider different numbers of layers, namely 3, 11 and 33. The thickness of every single layer is set to 1 mm, and the edge length, $L$, is chosen to be $S$ times larger than the total thickness $t$ of the laminate. Different choices of length-to-thickness ratio are considered (i.e., 20, 30, 40, and 50) which allow to draw interesting considerations about the laminate behaviour in the proposed convergence tests. For all examples we consider the same loading conditions proposed by Pagano, i.e., a double sinus with periodicity equal to twice the length of the plate. As depicted in Figure \[fig:testproblem\] the laminated plate is composed of layers organized in an alternated distribution of orthotropic plies (i.e., a 0/90 stacking sequence in our case). Layer material parameters considered in the numerical tests are summarized in Table \[tab:matproperties\] for 0-oriented plies.
\[tab:matproperties\]
The Neumann boundary conditions on the plate surfaces $x_3=\pm\dfrac{t}{2}$ are $$\begin{aligned}
&\sigma_{33}(x_1,x_2,-\dfrac{t}{2})=\sigma_{13}(x_1,x_2,\pm\dfrac{t}{2})=\sigma_{23}(x_1,x_2,\pm\dfrac{t}{2})=0\,,\\
&\sigma_{33}(x_1,x_2,\dfrac{t}{2})=\sigma_0\sin(\dfrac{\pi x_1}{St})\sin(\dfrac{\pi x_2}{St})\,,\label{eq:NBCs}
\end{aligned}$$ where $\sigma_0 =$ 1 MPa.The simple support edge conditions are taken as $$\begin{aligned}
\bullet\;\sigma_{11}=0\;\text{and}\;u_2=u_3=0\;\text{at}\;x_1=0\;\text{and}\;x_1=L\,,\\
\bullet\;\sigma_{22}=0\;\text{and}\;u_1=u_3=0\;\text{at}\;x_2=0\;\text{and}\;x_2=L\,.\label{eq:DBCs}
\end{aligned}$$ All results are then expressed in terms of the following normalized stress components $$\begin{aligned}
&\overline{\sigma}_{ij}=\dfrac{\sigma_{ij}}{\sigma_0S^2},\hspace{1cm}i,j=1,2\,,\\
&\overline{\sigma}_{i3}=\dfrac{\sigma_{i3}}{\sigma_0S},\hspace{1.15cm}i=1,2\,,\\
&\overline{\sigma}_{33}=\dfrac{\sigma_{33}}{\sigma_0}\,.\\\label{eq:normalizedresults}
\end{aligned}$$
Post-processed out-of-plane stresses {#subsec:ppeffects}
------------------------------------
In this section, we comment the results obtained using the proposed IGA-collocation approach, as compared with Pagano’s analytical solution [@Pagano1970]. To give an idea of the improvement granted by the post-processing of out-of-plane stress components, in Figures \[fig:example\_sig3\] and \[fig:example\_sig11\] we compare the reference solution with non-treated and post-processed results for the cases with 3 and 11 layers, considering a length to thickness ratio $S=20$. All numerical simulations are carried out using an in-plane degree of approximation $p=q=6$ and 10 collocation points for each in-plane parametric direction, while we use an approximation degree $r=4$ and one element through the thickness (i.e., $r+1$ collocation points). The sampling point where we show results is located at $x_1=x_2=0.25L$. For both considered cases the in-plane stresses show a good behaviour, as expected, while the out-of-plane stress components, without a post-processing treatment, are erroneusly discontinuous. The proposed results clearly show the improvement granted by the post-process of out-of-plane components.
\
\
\
\
To show the effect of post-processing at different locations of the plate, in Figures \[fig:samplingS13\]-\[fig:samplingS33\] the out-of-plane stress state profile is recovered sampling the laminae every quarter of length in both in-plane directions, for the case of a length-to-thickness ratio equal to 20 and 11 layers.
![Through-the-thickness $\bar{\sigma}_{13}$ profiles for several in plane sampling points. $L$ represents the total length of the plate, that for this case is $L=220\,\text{mm}$ (being $L=S\,t$ with $t=11\,\text{mm}$ and $S=20$), while the number of layers is 11 (post-processed solution, analytical solution [@Pagano1970]).[]{data-label="fig:samplingS13"}](fig_04.pdf)
![Through-the-thickness $\bar{\sigma}_{23}$ profiles for several in plane sampling points. $L$ represents the total length of the plate, that for this case is $L=220\,\text{mm}$ (being $L=S\,t$ with $t=11\,\text{mm}$ and $S=20$), while the number of layers is 11 (post-processed solution, analytical solution [@Pagano1970]).[]{data-label="fig:samplingS23"}](fig_05.pdf)
![Through-the-thickness $\bar{\sigma}_{33}$ profiles for several in plane sampling points. $L$ represents the total length of the plate, that for this case is $L=220\,\text{mm}$ (being $L=S\,t$ with $t=11\,\text{mm}$ and $S=20$), while the number of layers is 11 (post-processed solution, analytical solution [@Pagano1970]).[]{data-label="fig:samplingS33"}](fig_06.pdf)
Convergence behaviour {#subsec:conv}
---------------------
In order to validate the proposed approach in a wider variety of cases, computations with a different ratio between the thickness of the plate and its length are performed respectively for 3, 11, and 33 layers, considering an increasing number of knot spans. Figures \[fig:conv664\] and \[fig:conv666\] assess the convergence behaviour of the method, adopting the following error definition $$\text{e}(\sigma_{ij})=\frac{\max(|\sigma_{ij}^\text{analytic}-\sigma_{ij}^\text{recovered}|)}{\max(|\sigma_{ij}^\text{analytic}|)}\,.\label{eq:error}$$ Note that relation is used only to estimate the error inside the domain to avoid indeterminate forms. Different combinations of degree of approximations have been also considered. A poorer out-of-plane stress approximation is obtained using a degree equal to 4 in every direction, and, in addition, with this choice locking phenomena may occur for increasing values of length-to-thickness ratio. Therefore, we conclude that using a degree of approximation equal to 6 in-plane and equal to 4 through the thickness seems to be a reasonable choice to correctly reproduce the 3D stress state. Using instead uniform approximation degrees $p=q=r=6$ does not seem to significantly improve the results (see Figures \[fig:conv664\], \[fig:conv666\], and Table \[tab:errorl11\]). The post-processing method provides better results for increasing values of length-to-thickness ratio and number of layers and therefore proves to be particularly convenient for very large and thin plates. This is clear since a laminae with a large number of thin layers resembles a plate with average properties. What really stands out looking at the displayed mesh sensitivity results, is the fact that collocation perfectly captures the plates behaviour not only using one element through the thickness but also employing only one knot span in the plane of the plate. A single element of degrees $p=q=6$ and $r=4$, comprising a total of 7x7x5 collocation points, is able to provide for this example maximum percentage errors of 5% or lower (and of 1% or lower in the cases of 11 and 33 layers) for $S=30$ or larger.
\
\
\
\
\
Quantitative results are presented in Table \[tab:errorl11\] for various plate cases, considering a number of layers equal to 11 and 10 collocation points for each in-plane parametric direction. Different number of layers (i.e., 3 and 33) are instead investigated in Appendix \[sec:appendix\]. Increasing length-to-thickness ratios, namely 20, 30, 40, and 50 are considered and the maximum relative error results is reported for a reference point located at $x_1=x_2=0.25L$. Also different degrees of approximation are investigated. Given these results, we conclude that using an out-of-plane degree of approximation equal to 4 leads to a sufficiently accurate stress state. Furthermore the out-of-plane stress profile reconstruction shows a remarkable improvement for increasing values of number of layers and slenderness parameter $S$.
[? l | c ? c | c | c ? c | c | c ?]{}
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& &
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& &$e(\sigma_{13})$ &$e(\sigma_{23})$ &$e(\sigma_{33})$ &$e(\sigma_{13})$ &$e(\sigma_{23})$ &$e(\sigma_{33})$
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\
& &\[%\]&\[%\]&\[%\]&\[%\]&\[%\]&\[%\]
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\
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height 1pt @a xhline
& IGA-C & 97.6 & 56.7 & 6.34 & 96.6 & 56.1 & 6.31
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\
& IGA-C+PP & 0.31 & 2.94 & 0.90 & 1.97 & 1.20 & 0.05
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\
& IGA-C & 98.7 & 55.6 & 6.36 & 98.3 & 55.4 & 6.34
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\
& IGA-C+PP & 0.16 & 1.34 & 0.47 & 0.91 & 0.57 & 0.08
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& IGA-C & 99.2 & 55.3 & 6.37 & 98.9 & 55.2 & 6.36
------------------------------------------------------------------------
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\
& IGA-C+PP & 0.07 & 0.78 & 0.34 & 0.50 & 0.35 & 0.12
------------------------------------------------------------------------
------------------------------------------------------------------------
\
& IGA-C & 99.4 & 55.1 & 6.38 & 99.2 & 55.1 & 6.38
------------------------------------------------------------------------
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\
& IGA-C+PP & 0.03 & 0.52 & 0.29 & 0.30 & 0.25 & 0.15
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\
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height 1pt @a xhline
\[tab:errorl11\]
Conclusions {#sec:conclusions}
===========
In this paper we present a new approach to simulate laminated plates characterized by a symmetric distribution of plies. This technique combines a 3D collocation isogeometric analysis with a post-processing step procedure based on equilibrium equations. Since we adopt a single-element appoach, to take into account variation through the plate thickness of the material properties, we average the constitutive behaviour of each layer considering an homogeneized response. Following this simple approach, we showed that acceptable results can be obtained only in terms of displacements and in-plain stresses. Therefore, we propose to perform a post-processing step which requires the shape functions to be highly continuous. This continuity demand is fully granted by typical IGA shape functions. After the post-processing correction is applied, good results are recovered also in terms of out-of-plane stresses, even for very coarse meshes. The post-processing stress-recovery technique is only based on the integration through the thickness of equilibrium equations, and all the required components can be easily computed differentiating the displacement solution. Several numerical tests are carried out to test the sensitivity of the proposed technique to different length-to-thickness ratios and number of layers. Regardless of the number of layers, the method gives better results the thinner the composites are. Multiple numbers of alternated layers and sequence of stacks (both even and odd) have been studied in our applications. Neverthless only tests which consider an odd number of layers or an odd disposition of an even number of stacks show good results as expected because considering a homogenized response of the material is effective only for symmetric distributions of plies. Further research topics currently under investigation consist in the extension of this approach to more complex problems involving curved geometries and large deformations.
Acknowledgments {#sec:acknowl .unnumbered}
===============
This work was partially supported by Fondazione Cariplo – Regione Lombardia through the project “Verso nuovi strumenti di simulazione super veloci ed accurati basati sull’analisi isogeometrica”, within the program RST – rafforzamento.\
P. Antolin was partially supported by the European Research council through the H2020 ERC Advanced Grant 2015 n.694515 CHANGE.
{#sec:appendix}
Results in terms of maximum relative error considering a plate with a number of layers equal to 3 and 33 are herein presented for a reference point located at $x_1=x_2=0.25L$. Increasing length-to-thickness ratios, namely 20, 30, 40, and 50 are investigated for different degrees of approximations (i.e., $p=q=6$ and $r=4$, and $p=q=r=6$), using 10 collocation points for each in-plane parametric direction and one element through-the-thickness.
[? l | c ? c | c | c ? c | c | c ?]{}
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& &
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& &$e(\sigma_{13})$ &$e(\sigma_{23})$ &$e(\sigma_{33})$ &$e(\sigma_{13})$ &$e(\sigma_{23})$ &$e(\sigma_{33})$
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\
& &\[%\]&\[%\]&\[%\]&\[%\]&\[%\]&\[%\]
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\
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height 1pt @a xhline
& IGA-C & 292 & 57.2 & 5.80& 291 & 57.2 & 5.79
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\
& IGA-C+PP & 10.4 & 3.16 & 0.54 & 11.9 & 1.41 & 0.33
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\
& IGA-C & 311 & 57.5 & 5.77& 311 & 57.5 & 5.77
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\
& IGA-C+PP & 5.05 & 1.40 & 0.28 & 5.75 & 0.63 & 0.11
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\
& IGA-C & 319 & 57.6 & 5.76 & 319 & 57.6 & 5.76
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\
& IGA-C+PP & 2.91 & 0.81 & 0.21 & 3.32 & 0.38 & 0.02
------------------------------------------------------------------------
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\
& IGA-C & 323 & 57.6 & 5.76& 322 & 57.6 & 5.76
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\
& IGA-C+PP & 1.87 & 0.54 & 0.20 & 2.14 & 0.26 & 0.07
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height 1pt @a xhline
\[tab:errorl3\]
[? l | c ? c | c | c ? c | c | c ?]{}
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& &
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& &$e(\sigma_{13})$ &$e(\sigma_{23})$ &$e(\sigma_{33})$ &$e(\sigma_{13})$ &$e(\sigma_{23})$ &$e(\sigma_{33})$
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\
& &\[%\]&\[%\]&\[%\]&\[%\]&\[%\]&\[%\]
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\
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height 1pt @a xhline
& IGA-C & 81.6 & 69.7 & 6.33 & 80.7 & 68.9 & 6.33
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\
& IGA-C+PP & 1.16 & 2.21 & 0.93 & 0.54 & 0.50 & 0.07
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\
& IGA-C & 81.5 & 69.0 & 6.34 & 81.2 & 68.7 & 6.34
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\
& IGA-C+PP & 0.53 & 1.01 & 0.48 & 0.23 & 0.25 & 0.09
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\
& IGA-C & 81.5 & 68.7 & 6.35& 81.3 & 68.6 & 6.34
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\
& IGA-C+PP & 0.32 & 0.59 & 0.34 & 0.11 & 0.16 & 0.12
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\
& IGA-C & 81.6 & 68.6 & 6.35 & 81.4 & 68.5 & 6.35
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\
& IGA-C+PP & 0.23 & 0.40 & 0.30 & 0.05 & 0.13 & 0.16
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\[tab:errorl33\]
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